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[
"A New Bound on the Minimum Distance of Cyclic Codes Using Small-Minimum-Distance Cyclic Codes",
"A New Bound on the Minimum Distance of Cyclic Codes Using Small-Minimum-Distance Cyclic Codes"
] | [
"Alexander Zeh ",
"Sergey Bezzateev "
] | [] | [] | A new bound on the minimum distance of q-ary cyclic codes is proposed. It is based on the description by another cyclic code with small minimum distance. The connection to the BCH bound and the Hartmann-Tzeng (HT) bound is formulated explicitly. We show that for many cases our approach improves the HT bound. Furthermore, we refine our bound for several families of cyclic codes.We define syndromes and formulate a Key Equation that allows an efficient decoding up to our bound with the Extended Euclidean Algorithm. It turns out that lowest-code-rate cyclic codes with small minimum distances are useful for our approach. Therefore, we give a sufficient condition for binary cyclic codes of arbitrary length to have minimum distance two or three and lowest code-rate. | 10.1007/s10623-012-9721-3 | [
"https://arxiv.org/pdf/1206.4976v3.pdf"
] | 2,803,880 | 1206.4976 | 66aaed4ebbb7bcd790cf0905888e59be90d5f509 |
A New Bound on the Minimum Distance of Cyclic Codes Using Small-Minimum-Distance Cyclic Codes
May 2, 2014
Alexander Zeh
Sergey Bezzateev
A New Bound on the Minimum Distance of Cyclic Codes Using Small-Minimum-Distance Cyclic Codes
May 2, 2014BCH Bound -Bound on the Minimum Distance -Cyclic Code -Decod- ing -Hartmann-Tzeng Bound Mathematics Subject Classification: 94A24 -94A55 -94B15 -94B35
A new bound on the minimum distance of q-ary cyclic codes is proposed. It is based on the description by another cyclic code with small minimum distance. The connection to the BCH bound and the Hartmann-Tzeng (HT) bound is formulated explicitly. We show that for many cases our approach improves the HT bound. Furthermore, we refine our bound for several families of cyclic codes.We define syndromes and formulate a Key Equation that allows an efficient decoding up to our bound with the Extended Euclidean Algorithm. It turns out that lowest-code-rate cyclic codes with small minimum distances are useful for our approach. Therefore, we give a sufficient condition for binary cyclic codes of arbitrary length to have minimum distance two or three and lowest code-rate.
Introduction
In this paper, we introduce a technique that uses an (n , k ) q -ary cyclic code L with minimum distance d to bound the minimum distance d of another (n, k) q-ary cyclic code C. The descriptive cyclic code L is called non-zero-locator code. It turns out that the non-zero-locator code gives a good lower bound d * on the minimum distance d of the described cyclic code C if the code-rate k /n of L is low and its minimum distance d is relatively small.
The algebraic relation between the cyclic non-zero-locator code L and the cyclic code C provides the formulation of syndromes and a Key Equation that allows an efficient decoding up to (d * − 1)/2 errors with the Extended Euclidean Algorithm (EEA).
We give an explicit relation of d * to the BCH bound [1,10] and its generalization: the Hartmann-Tzeng (HT) bound [7][8][9]. In many cases our bound is better than the HT bound, although our approach is not a generalization of the HT bound as the Roos bound [14,15] and the bound of van Lint and Wilson [11] are.
In our previous work [17] we associated rational functions with a subset of the defining set of a given cyclic code C. This can be seen as a special case of the presented approach. The main advantage of this contribution is that we can express the bound on the minimum distance of a given cyclic code C in terms of properties of the associated cyclic non-zero-locator code L.
This paper is organized as follows. In Section 2, we give necessary preliminaries of cyclic codes, the HT bound and recall the definition of cyclic Reed-Solomon (RS) codes, which we use later as non-zero-locator code. The concept of the non-zero-locator code is introduced in Section 3 and the main theorem on the minimum distance is proven. The connection to the Hartmann-Tzeng bound is given in Section 4. Furthermore, several families of cyclic codes are identified. We give sufficient conditions for binary cyclic codes with minimum distance two and three and lowest code-rate in Section 5. A generalized syndrome definition, Key Equation and Forney's formula are given in Section 6. Section 7 concludes this contribution.
Preliminaries
Let q be a power of a prime and let F q denote the finite field of order q and F q [x] the set of all univariate polynomials with coefficients in F q and indeterminate x. A q-ary cyclic code over F q of length n, dimension k and minimum distance d is denoted by C(q; n, k, d) ⊂ F n q and it is an ideal in the ring F q [x]/(x n − 1) generated by g(x). A codeword c = (c 0 c 1 . . . c n−1 ) ∈ C is associated with a polynomial c(x) = n−1 i=0 c i x i ∈ F q [x], where g(x) divides c(x). We assume that x n − 1 has n different roots. Let F q s be an extension field of F q and let α ∈ F q s be a primitive nth root of unity. The cyclotomic coset M (n) r modulo n over F q is denoted by: M (n) r = {rq j mod n | j = 0, 1, . . . , n r − 1}, where n r is the smallest integer such that rq nr ≡ r mod n. It is well-known that the minimal polynomial M (n) r (x) ∈ F q [x] of the element α r is given by:
M (n) r (x) = i∈M (n) r (x − α i ).
The defining set D C of a q-ary cyclic code C(q; n, k, d) is the set of zeros of the generator polynomial g(x) ∈ F q [x] and can be partitioned into m cyclotomic cosets:
D C = {0 ≤ i ≤ n − 1 | g(α i ) = 0} = M (n) r 1 ∪ M (n) r 2 ∪ · · · ∪ M (n) rm .
Hence, the generator polynomial g(x) of degree n − k of C(q; n, k, d) is
g(x) = m i=1 M (n) r i (x).
Let us recall a well-known bound on the minimum distance of cyclic codes.
Theorem 1 (Hartmann-Tzeng (HT) Bound, [8]). Let a q-ary cyclic code C(q; n, k, d) with defining set D C be given. Suppose there exist the integers b 1 , m 1 and m 2 with gcd(n, m 1 ) = 1 and gcd(n, m 2 ) = 1 such that
{b 1 + i 1 m 1 + i 2 m 2 | 0 ≤ i 1 ≤ d 0 − 2, 0 ≤ i 2 ≤ ν} ⊆ D C . Then d ≥ d 0 + ν.
Note that for ν = 0 the HT bound becomes the BCH bound [1,10]. Further generalizations were proposed by Roos [14,15] and van Lint and Wilson [11]. Decoding up to the HT bound and to some particular cases of the Roos bound was formulated by Feng and Tzeng [5,Section VI].
We consider cyclic Reed-Solomon (RS) codes [13] for our approach and therefore recapitulate their definition in the following.
Definition 1 (Cyclic Reed-Solomon Code). Let n be an integer dividing q − 1 and let α denote an element of multiplicative order n in F q . Let δ be an integer. Furthermore, let the generator polynomial g δ (x) ∈ F q [x] be defined as:
g δ (x) = δ+n−k−1 i=δ (x − α i ).
Then a cyclic Reed-Solomon code over F q of length n|(q − 1) and dimension k, denoted by RS(q; n, k; δ), is defined by:
RS(q; n, k; δ) = {m(x)g δ (x) : deg m(x) < k}.(1)
RS codes are maximum distance separable codes and their minimum distance d is d = n − k + 1.
The Non-Zero-Locator Code
We relate another cyclic code -the so-called non-zero-locator code L -to a given cyclic code C. In the following, we connect a infinite sequence of an evaluated polynomial c(x) ∈ C to a sum of fractions. This allows to draw the relation to our previous approach [17]. Furthermore, we can use familiar properties of cyclic codes rather than abstract properties of rational functions. The obtained bound can be expressed in terms of parameters of the associated non-zero-locator code L.
Let c(x) be a codeword of a given q-ary cyclic code C(q; n, k, d) and let Y denote the set of indexes of non-zero coefficients of c(x)
c(x) = i∈Y c i x i .
Let α ∈ F q s be an element of order n. Then we have the following relation for all c(x) ∈ C(q; n, k, d):
∞ j=0 c(α j )x j = ∞ j=0 i∈Y c i α ji x j = ∞ j=0 i∈Y c i (α i x) j = i∈Y ∞ j=0 c i (α i x) j = i∈Y c i 1 − xα i .(2)
Now, we can define the non-zero-locator code.
Definition 2 (Non-Zero-Locator Code). Let a q-ary cyclic code C(q; n, k, d) be given. Let F q s contain the nth roots of unity. Let gcd(n, n ) = 1 and let F q = F q u be an extension field of F q . Let F q s contain the n th roots of unity. Let α ∈ F q s be an element of order n and let β ∈ F q s be an element of order n . Then L(q ; n , k , d ) is a non-zero-locator code of C if there exists a µ ≥ 2 and an integer e, such that ∀ a(x) ∈ L and ∀ c(x) ∈ C:
∞ j=0 c(α j+e )a(β j )x j ≡ 0 mod x µ−1 ,(3)
holds.
Remark 1. Let r denote the least common multiple of s and u · s and let γ be a primitive element in F q r . Then γ (q r −1)/n and γ (q r −1)/n are elements of order n and n .
Before we prove the main theorem on the minimum distance d of the given cyclic code C, we describe Definition 2. We search the "longest" sequence c(α e )a(β 0 ), c(α e+1 )a(β 1 ), . . . , c(α e+µ−2 )a(β µ−2 ), that results in a zero-sequence of length µ − 1, i.e., the product of the evaluated codeword a(β j ) of the non-zero-locator code L and the evaluated codeword c(α j+e ) of C gives zero for all j = 0, . . . , µ − 2. Let us study the following example of a binary cyclic code.
Example 1 (Binary Code of length n = 21 [11,15]). Let the binary cyclic code C(2; 21, 7, 8) with generator polynomial g(x)
g(x) = M 3, 4, , 6, 7, 8, 9, , 11, 12, , 14, 15, 16, , 18}, where the symbol marks the indexes where g(α i ) = 0.
D C = {1, 2,
We associate a single parity check code of length n = 5, dimension k = 4 and minimum distance d = 2 over F 2 as non-zero-locator code for C(2; 21, 7, 8) according to Definition 2. Let β ∈ F 2 4 be an element of order 5 and let g(x) = x − 1 be the generator polynomial of L. The defining sets D C of C(2; 21, 7,8) and D L of L(2; 5, 4, 2) are listed in Table 1. The corresponding product gives the a zero-sequence of length µ − 1 = 13 for e = 0. A codeword a(x) ∈ L(2; 5, 4, 2) "fills" the missing zeros of C(2; 21, 7, 8) at position 0, 5 and 10 in the interval [0, 12].
D L 0 0 0
We require a zero β j of the generator polynomial of the non-zero-locator code L at the position j where the generator polynomial of the given cyclic code C has no zero.
Furthermore, we require gcd(n, n ) = 1 to guarantee that
gcd m∈Z (1 − xα i β m ), m∈Z
(1 − xα j β m ) = 1 ∀i and ∀j = i, which we use for the degree calculation in the following. For the proof we refer to Lemma 3 in the Appendix. We rewrite (3) of Definition 2 more explicitly. With c(x) = i∈Y c i x i and a(x) = j∈Z a j x j , we obtain:
∞ j=0 c(α j+e )a(β j )x j = ∞ j=0 i∈Y c i α i(j+e) a(β j )x j = i∈Y c i α ie ∞ j=0 α ij a(β j )x j .
Using (2) for the codeword a(x) of the associated non-zero-locator code leads to:
i∈Y c i α ie ∞ j=0 α ij a(β j )x j = i∈Y c i α ie j∈Z a j 1 − xα i β j = i∈Y c i α ie j∈Z a j ∈Z =j (1 − xα i β ) j∈Z (1 − xα i β j ) .(4)
Finally using (4) we can rewrite (3) of Definition 2 in the following form:
i∈Y c i α ie j∈Z a j ∈Z =j (1 − xα i β ) m∈Y m =i s∈Z (1 − xα m β s ) i∈Y j∈Z (1 − xα i β j ) ≡ 0 mod x µ−1 ,(5)
where the degree of the denominator is |Y| · |Z|. The degree of the numerator is smaller than or equal to (|Y| − 1) · |Z| + |Z| − 1 = |Y| · |Z| − 1. This leads to the following theorem on the minimum distance of a cyclic code C.
Theorem 2 (Minimum Distance). Let a q-ary cyclic code C(q; n, k, d) and its associated nonzero-locator code L(q ; n , k , d ) with gcd(n, n ) = 1 and the integer µ be given as in Definition 2.
Then the minimum distance d of C(q; n, k, d) satisfies the following inequality:
d ≥ d * def = µ d .(6)
Proof. For a codeword c(x) ∈ C(q; n, k, d) of weight d and a codeword a(x) ∈ L(q ; n , k , d ) of weight d , the degree of the denominator in (5) is d·d . The numerator has degree at most d·d −1, and has to be greater than or equal to µ − 1. The optimal non-zero-locator code L for a given cyclic code gives a zero sequence
c(α e )a(β 0 ), c(α e+1 )a(β 1 ), . . . , c(α e+µ−2 )a(β µ−2 )
of length µ − 1 as in Definition 2, such that d * of (6) is maximized.
Comparison to Known Bounds
The Hartmann-Tzeng Bound
We restate the HT bound as given in Theorem 1 to draw a connection to the bound given in Theorem 2. We multiply with the inverse of m 1 or m 2 modulo n, such that:
m > ν + 1 for {b 2 + i 1 m + i 2 : 0 ≤ i 1 ≤ d 0 − 2, 0 ≤ i 2 ≤ ν} ⊆ D C(7)
with gcd(n, m) = 1 for a given code C(q; n, k, d) holds. Throughout this section, we refer to this representation of the HT bound. In the following subsection, we consider a single parity check code as non-zero-locator code and draw the connection to a particular case of the HT bound. The general case of (7) is considered in Subsection 4.3, where we use RS codes as non-zero-locator codes.
Some families of cyclic codes are identified in Subsection 4.4.
Single Parity Check Code as Non-Zero-Locator Code
Let P(n , n − 1, 2) denote a cyclic single parity check code of length n , dimension n − 1 and minimum distance 2 over an extension field F q of F q . Let β be a primitive n th root of unity in an extension field of F q . The generator polynomial g(x) of P(n , n − 1, 2) is
g(x) = x − 1.
Furthermore, let a cyclic code C with defining set D C be given, such that for the parameters b 2 = 1 and m = ν + 2 the normalized HT bound of (7) holds. We illustrate the defining set D P = {0} of P with length n = ν + 2 and the defining set D C in Table 2. The sequence is illustrated in terms Table 2: Defining sets D C of a given cyclic code C and D P of its associated single parity check
code P of length n in the interval [0, m(d 0 − 1)]. D C 1 .. m-1 m+1 .. 2m-1 .. m(d 0 -1)-1 D P 0 .. 0 .. 0 .
. 0 of parameters of the HT bound (7). For this special case, the non-zero-locator code L(q ; n , k , d ) is a P(n , n − 1, 2) code. We have:
n = ν + 2, k = ν + 1, d = 2,
and we obtain a zero-sequence of length µ − 1 = m(d 0 − 1) + 1. From Theorem 2 we obtain:
d * = m(d 0 − 1) + 2 2 = (ν + 2)d 0 − ν 2 = d 0 + ν(d 0 − 1) 2 ,(8)
where we used m = ν + 2. In Fig. 1 we illustrate d * of (8) for different parameters ν and d 0 . For d 0 ≥ 4 (independently from ν) our bound improves the HT bound (see Proposition 1 in the next subsection). Note that for ν = 0 the HT bound and our bound coincide with the BCH bound. Let us study the following example. Table 2). We used a single parity check code as non-zero-locator code.
Our bound d * is better than the HT bound for d 0 > 3. . We can associate the single parity check code P(3,2,2) over F 2 2 with generator polynomial g(x) = x − 1 as a non-zero-locator code for C(2; 65, 41, 8). The defining sets D C and D P are shown in Table 3. With (8) we obtain for d * : 6]. The set D P is the defining set of a single parity check code P of length n = 3 that is the associated non-zero-locator code.
D C -5 -4 -2 -1 1 2 4 5 D P 0 0 0 0 0 d * = d 0 + ν(d 0 − 1) 2 = 5 + 1(5 − 1) 2 = 7.
Furthermore, we can decode up to (d * − 1)/2 = 3 errors for C(2; 65, 41, 8) (see Section 6).
Cyclic Reed-Solomon Codes as Non-Zero-Locator Codes
Let a q-ary cyclic code C with defining set D C be given such that for the parameters b 2 = 1 and m > ν + 2, the normalized Hartmann-Tzeng bound of (7) with d 0 > 2 and ν > 0 holds. Let a cyclic Reed-Solomon code RS(q ; n , k ; δ) over an extension field F q of F q with n = m, k = ν + 1, d = m − ν, δ = 0 as in Definition 1 be the associated non-zero-locator code. Table 4 shows the defining set D C and the defining set D RS of RS(q ; m, ν + 1; 0). Table 4: Defining sets D C for b 2 = 1 and m of the HT bound (7) and D RS of the associated non-zero-locator code in the interval
[−(m − ν) − 1, m(d 0 − 1)]. D C .. 1 .. ν+1 .. m+1 .. m+ν+1 .. .. D RS 0 .. m-ν-2 .. 0 .. m-ν-2 .. 0 .. m-ν-2 .. m-ν-2
The n −(ν +2)+1 = m−ν −1 consecutive zeros of the cyclic Reed-Solomon code RS(q ; m, ν + 1; 0) fill the missing zeros of the given cyclic code C(q; n, k, d). The obtained "zero"-sequence has length µ − 1 = m(d 0 − 1) + m − ν − 1. Therefore, we obtain from (6):
d * = m(d 0 − 1) + m − ν m − ν = md 0 − m + m − ν m − ν = md 0 − ν m − ν .(9)
Note that for m = ν + 2 the Reed-Solomon code is a single parity check code and we obtain the result from (8). Let us precise the cases where our bound d * is better than the Hartmann-Tzeng Table 4).
bound d 0 + ν.
Proposition 1. Let a q-ary cyclic code C(q; n, k, d) with a subset of its defining set D C with parameters b 2 , m, d 0 and ν as stated in Theorem 1 be given. Let L(q ; m, ν + 1, m − ν) = RS(q ; m, ν + 1; 0) be the associated non-zero-locator code as in Definition 3 with µ = m(d 0 − 1) + m − ν. Then for d 0 > m − ν + 1,
d * > d 0 + ν holds.
Proof. From (9) we have
d * = md 0 − ν m − ν = md 0 − d 0 ν + d 0 ν − ν m − ν = d 0 + (d 0 − 1)ν m − ν .
Obviously, for d * > d 0 + ν, we require that
(d 0 − 1)ν m − ν > ν ⇐⇒ d 0 > m − ν + 1.
For m − ν = d = 2, the associated RS code is a single parity check code and our bound is better than the HT bound for d 0 > 3 (see Fig. 1). Some other cases, where the minimum distance of the associated RS code d = m − ν varies between two and six, are illustrated in Fig. 2.
Some Families of Cyclic Codes and Their Connection to Other Bounds
We identify some families of cyclic codes and refine our bound on the minimum distance of Theorem 2. The classification is done by means of the associated non-zero-locator code. For all codes, we can decode up to (d * − 1)/2 errors (see Section 6).
Single Parity Check Code as Non-Zero-Locator Code
Let the defining set D C of a given q-ary cyclic code C(q; n, k, d) contain the elements as shown in Table 5. Furthermore, let gcd(n, 3) = 1. We associate a single parity check code P(3, 2, 2) and For binary reversible cyclic codes [12,19] we require only {1, 5, 7} to be a subset of the defining set since the other elements are then included automatically.
If the binary cyclic code is not reversible, the defining set has to contain {−7, −5, −1, 1, 5, 7}. This requirement coincides with the 5-error-correcting pair of [4,Proposition 8]. The codes of [4, Proposition 7, Example 21 and 22] require a smaller subset of their defining set D C . For these codes, we obtain the same bound on the minimum distance of C.
Binary Hamming Code as Non-Zero-Locator Code
Let the defining set D C of a given binary cyclic code C(2; n, k, d) contain the elements as shown in Table 6. Furthermore, let gcd(n, 7) = 1.
We associate the binary Hamming code L(2; 7, 4, 3) with defining set D L = {3, 5, 6}. As shown in Table 6, we obtain µ = 21 and therefore d ≥ d * = 7.
For binary cyclic codes we require {1, 7, 9, 11, 15} to be a subset of the defining set D C .
Reed-Solomon Code as Non-Zero-Locator Code
Let the defining set D C of a given q-ary cyclic code C(q; n, k, d) contain the elements as shown in Table 7. Furthermore, let gcd(n, 4) = 1. We associate an RS code RS(q ; 4, 2; δ = 0) over F q which is an extension field of F q and consider the sequence
c(α −17 )a(β 0 ), c(α −17+2 )a(β 1 ), c(α −17+4 )a(β 2 ), . . . , c(α −17+(µ−2)·2 )a(β µ−2 ).
We have µ = 19 and with d = 3, we obtain d ≥ d * = 7. For binary reversible cyclic codes, we require {3, 5, 11, 13} to be a subset of the defining set D C . Further families can be found in [17] and can be seen as special case of this approach.
As previously seen, we identified cyclic codes by means of their potential non-zero-locator codes. To obtain a huge family of cyclic codes, the cardinality of the required subset of their defining set should be small. This implies a high cardinality of the defining set |D L | of the associated non-zero-locator code L(q ; n , k , d ). Both leads to a long zero-sequence c(α e )a(β 0 ), c(α e+1 )a(β 1 ), . . . , c(α e+µ−2 )a(β µ−2 ).
On the one hand, we need a low code-rate k /n which implies a high |D L |. On the other hand, the minimum distance d of L should be small to obtain a good bound d * according to (6).
This motivates the investigation of small-minimum-distance cyclic codes with lowest code-rate. In a first step, we consider binary cyclic codes with minimum distance two and three.
Binary Cyclic Codes with Minimum Distance Two and
Three as Non-Zero-Locator Code
General Idea
As mentioned in Section 3, good candidates for non-zero-locator codes are cyclic codes with small minimum distance and lowest code-rate k /n . We consider binary cyclic codes with minimum distance two and three and lowest code-rate and show their defining set. Primitive binary cyclic codes with minimum distance three were investigated by Charpin, Tietäväinen and Zinoviev in [2,3]. We generalize the results of [2] to binary cyclic codes of arbitrary length and show afterwards the implications, when we want to use them as non-zero-locator codes. Lemma 1. [2] Let i, j with 0 ≤ i < j ≤ n − 1 be two arbitrary integers that do not belong to the same cyclotomic coset modulo n. Then the binary cyclic code C(2; n, k, d) with generator polynomial g(x) = M Theorem 3 (Binary Cyclic Codes with Minimum Distance Two [2]). Let i 1 , i 2 , . . . , i s with 0 ≤ i 1 < · · · < i s ≤ n−1 be s arbitrary integers that do not belong to the same cyclotomic coset modulo n. Then the binary cyclic code C(2; n, k, d) with generator polynomial
g(x) = s j=1 M (n) i j (x)
has minimum distance two if and only if gcd(n, i 1 , . . . , i s ) > 1.
We skip the proof of Theorem 3, because it is straightforward to the proof of Lemma 1.
The following lemma is a generalization of [2, Theorem 1] to binary cyclic codes of arbitrary length.
Lemma 2 (Binary Cyclic Codes with Minimum Distance Three). Let i, j with 0 ≤ i < j ≤ n − 1 be arbitrary integers that do not belong to the same cyclotomic coset modulo n. Let g be such that 2 g − 1 divides n. If there exists an integer r with 0 < r < 2 g − 1, where gcd(r, 2 g − 1) = 1, such that both i and j are in M (2 g −1) r , then the binary cyclic code C(2; n, k, d) with generator polynomial
g(x) = M (n) i (x) · M (n)
j (x) has minimum distance d ≤ 3. If, moreover, gcd(n, i, j) = 1, then d = 3.
Proof. Let γ be a primitive element of F 2 s , let z = (2 s − 1)/n and let α = γ z . Let u = n/(2 g − 1), then β = α u = γ (2 s −1)/(2 g −1) , is a primitive element of F 2 g . Let b be an integer in the interval [1, 2 g − 2] such that:
1 + β + β b = 0. Define c(x) = 1 + x u(1/r) + x u(b/r) ,
where the quotients 1/r and b/r are calculated in the ring Z 2 g −1 of integers modulo 2 g − 1. For i ∈ M (2 g −1) r , two non-negative integers k and exist such that
i = (2 g − 1) + 2 k r.
Thus, c(α i ) = 1 + α ui(1/r) + α ui(b/r)
= 1 + β i(1/r) + β i(b/r) = 1 + β 2 k r(1/r) + β 2 k r(b/r) = 1 + β 2 k + β b2 k = (1 + β + β b ) 2 k = 0.
Note that in [2] the length of the cyclic code was n = 2 s − 1 and u = (2 s − 1)/(2 g − 1).
Corollary 1.
Let C be a binary cyclic code of length n. If there exist no g, s.t. (2 g − 1) | n, then C cannot have minimum distance three.
Theorem 4 (Binary Cyclic Codes with Minimum Distance Three). Let i 1 , i 2 , . . . , i s with 0 ≤ i 1 < · · · < i s ≤ n − 1 be s arbitrary integers that do not belong to the same cyclotomic coset modulo n. Let g be such that 2 g − 1 divides n. If there exists an integer r with 0 < r < 2 g − 1, where gcd(r, 2 g − 1) = 1, such that all s integers i 1 , i 2 , . . . , i s are in M (2 g −1) r , then the binary cyclic code C(2; n, k, d) with generator polynomial
g(x) = s j=1 M (n) i j (x)
has minimum distance d ≤ 3. If, moreover, gcd(n, i 1 , . . . , i s ) = 1, then d = 3.
We skip the proof of Theorem 4, because it is straightforward to the proof of Lemma 2. Let us consider a non-primitive binary cyclic code with minimum distance three.
Implications for the Non-Zero-Locator Code
We consider lowest-code-rate binary cyclic codes of minimum distance two and three. They are good candidates for non-zero-locator codes.
We first consider lowest-code-rate binary cyclic codes of minimum distance two.
Proposition 2 (Lowest-Code-Rate Binary Cyclic Codes With Minimum Distance Two). Let a > 1, g > 1 and n be three integers, such that n = ag. Let g be in the defining set D C . Then the binary cyclic code C(2; n, k, 2) of length n with defining set: {0, , . . . , , g, , . . . , , 2g, , . . . , , (a − 1)g, , . . . , } is the binary cyclic code of smallest dimension k = a(g − 1), lowest code-rate R = (g − 1)/g and minimum distance two.
D C =
Proof. We want to maximize |D C | while keeping d of C at two. Therefore, we select for a given g every cyclotomic coset M (n) i with gcd(i, g) > 1 for all i = 0, . . . , n − 1 to be in D C with aimed minimum distance two. One the one hand, this guarantees the maximization of |D C | and therefore the minimization of the code-rate. On the other hand, due to the condition gcd(i, g) > 1 (Theorem 3) the minimum distance of C remains two.
A direct consequence of Proposition 2 is that we do not need to investigate these binary cyclic codes of minimum distance two any more. We obtain the same result when we select a parity check code P(g, g − 1, 2) as non-zero-locator code.
Proposition 3 (Lowest-Code-Rate Binary Cyclic Codes With Minimum Distance Three). Let a > 1, g > 1 and n be three integers, such that n = a(2 g −1). Let r be an integer with 0 < r < 2 g −1, where gcd(r, 2 g − 1) = 1. Let r be in the defining set D C . Then the binary cyclic code C(2; n, k, 3) of length n with defining set:
D C = {r · i mod n | i =j(2 g − 1) + 1, j(2 g − 1) + 2, j(2 g − 1) + 4, . . . , j(2 g − 1) + 2 g−1 ∀j = 0, . . . , a − 1}(10)
is the binary cyclic code with the smallest dimension k = a(2 g − 1 − g), lowest code-rate R = (2 g − 1 − g)/(2 g − 1) and minimum distance three.
Proof. We want to maximize |D C | while keeping d of C at three. For a given r and for (2 g −1)|n, we select every cyclotomic coset M (n) i for all i = 0, . . . , n−1 to be in the D C of C with aimed minimum distance three, such that i ∈ M (2 g −1) r . One the one hand, this guarantees the maximization of |D C | and therefore the minimization of the code-rate. On the other hand, due to the condition that M A consequence of Proposition 3 is that we do not need to investigate any binary cyclic code of minimum distance three any more. We obtain the same result when we take a primitive binary cyclic code with minimum distance three as non-zero-locator code.
6 Syndrome-Based Decoding of up to (d * − 1)/2 Errors
Syndrome Definition
Let a q-ary cyclic code C(q; n, k, d) and its associated q -ary non-zero-locator code L(q ; n , k , d ) with gcd(n, n ) = 1 and the integers µ and e be given as in Definition 2. Let F q = F q u be an extension field of F q . Let α ∈ F q s be a primitive nth and let β ∈ F q s be a primitive n th root of unity. Let r denote the least common multiple of s and u · s . Let a(x) = i∈Z a i x i be a codeword of L of weight |Z| = d .
Let the set E = {i 0 , i 1 , . . . , i t−1 } with cardinality |E| = t be the set of error positions. The corresponding error polynomial is denoted by e(x) = i∈E e i x i . Let the received polynomial be r(x) = n−1 i=0 r i x i = e(x) + c(x). We define a syndrome polynomial S(x) ∈ F q r [x] as follows:
S(x) def ≡ ∞ j=0 r(α j+e )a(β j )x j mod x µ−1 .(11)
Thus, the coefficients S j ∈ F q r of the above defined syndrome polynomial S(x) = µ−2 j=0 S j x j are given by
S j = n−1 i=0 r i α i(j+e) · n −1 h=0 a h β hj , ∀j = 0, . . . , µ − 2.
From Definition 2 we know that the syndrome polynomial S(x) of (11) is independent of the codeword c(x). Now, we can do the same reformulation of the syndrome expression as we did in Section 3 for the codeword c(x) and a(x). We have from (11):
∞ j=0 r(α j+e )a(β j )x j ≡ ∞ j=0 e(α j+e )a(β j )x j mod x µ−1 ≡ ∞ j=0 i∈E e i α i(j+e) a(β j )x j mod x µ−1 ,
and with (2) for a(x) = i∈Z a i x i we can write:
S(x) ≡ i∈E e i α ie j∈Z a j 1 − xα i β j mod x µ−1 ≡ i∈E e i α ie j∈Z a j ∈Z =j (1 − xα i β ) j∈Z 1 − xα i β j mod x µ−1 .
Finally, we can write for S(x):
S(x) ≡ i∈E e i α ie j∈Z a j ∈Z =j (1 − xα i β ) m∈E m =i s∈Z (1 − xα m β s ) i∈E j∈Z 1 − xα i β j mod x µ−1 .(12)
We use this explicit syndrome representation in the next section, where we define an error-locator and an error-evaluator polynomial.
Key Equation
To simplify the notation, let the two polynomials f (x) and h(x) ∈ F q r [x] be defined as follows:
f (x) def = j∈Z 1 − xβ j ,(13)h(x) def = j∈Z a j ∈Z =j (1 − xβ ) .(14)
Due to gcd(n, n ) = 1 we have gcd(f (xα i ), f (xα j )) = 1, ∀i = j (for the proof, see Lemma 3 in the Appendix) and therefore each of the n polynomials f (xα 0 ), f (xα 1 ), . . . , f (xα n−1 ) can be identified by one root. Let κ ∈ Z. Then, we have f (β −κ ) = 0. Furthermore, let n distinct roots γ 0 , γ 1 , . . . , γ n−1 be defined as:
γ i def = β −κ α −i , i = 0, . . . , n − 1.(15)
Then, each γ i is a root of f (xα i ). Note that each polynomial f (xα i ) has |Z| = d roots, but we need only one of them. Now, we can define an error-locator polynomial Λ(x) ∈ F q r [x] as:
Λ(x) def = i∈E f (xα i ).(16)
The roots γ i of Λ(x) from (15) tell us where the errors are. The corresponding error-evaluator polynomial Ω(x) ∈ F q r [x] is defined as:
Ω(x) def = i∈E e i α ie h(xα i ) ∈E =i f (xα ) .(17)
We relate the syndrome definition of (12), the error-locator polynomial Λ(x) of (16) and the error-evaluator polynomial Ω(x) of (17) in form of a Key Equation:
S(x) ≡ Ω(x) Λ(x) mod x µ−1 , with deg Λ(x) = t · d , deg Ω(x) ≤ t · d − 1 < deg Λ(x).(18)
Solving (18) is similar to the decoding of [16] and we will not go into details. The Extended Euclidean Algorithm (EEA, [16]) with input polynomial S(x) as defined in (11) and the monomial x µ−1 and an adapted stopping rule can be used to solve (18) and we obtain Λ(x) and Ω(x).
Error Evaluation: A Generalized Forney's Formula
To determine the t error values e i 0 , e i 1 , . . . , e i t−1 from the error-locator polynomial Λ(x) and errorlocator polynomial Ω(x), we develop an explicit expression of the error-values (like Forney's formula [6]) in the following.
Proposition 4. (Error Evaluation) Let a q-ary cyclic code C(q; n, k, d) and its associated nonzero-locator code L(q ; n , k , d ) with gcd(n, n ) = 1 and the integers µ and e be given as in Definition 2. Let α ∈ F q s be a primitive nth and let β ∈ F q u·s be a primitive n th root of unity. Let r be the least common multiple of s and u · s . Furthermore, let γ 0 , γ 1 , . . . , γ n−1 be given as in (15) and let two polynomials Λ(x) and Ω(x) ∈ F q r [x] be given as in (16) and (17). Then the error values e i for all i ∈ E are:
e i = Ω(γ i ) α ie · h(γ i α i ) · ∈E =i f (γ i α ) = Ω(γ i ) · f (γ i α i ) Λ (γ i ) · α ie · h(γ i α i ) .(19)
Proof. The error-evaluator polynomial Ω(x) of (17) evaluated at γ i is explicitly
Ω(γ i ) = e i · α ie · h(γ i α i ) ∈E =i f (γ i α ).
The derivative Λ (x) of the error-locator polynomial is
Λ (x) = i∈E f (xα i ) ∈E =i f (xα ) .
Its evaluation at γ i simplifies to
Λ (γ i ) = f (γ i α i ) ∈E =i f (γ i α ).
Note that the classical decoding up to the half the BCH bound of a cyclic code C corresponds to the case where the associated non-zero-locator code L is the set of all vectors of length n = k over F q . The zero-sequence of length µ − 1 is the longest set of consecutive zeros of C. Then we can choose a(x) = 1 and we obtain the classical syndrome definition, key equation and Forney's formula.
Conclusion and Outlook
We presented a new technique that uses low-rate cyclic codes with small minimum distancesso-called non-zero-locator codes -to bound the minimum distance of q-ary cyclic codes. The algebraic description gives a generalized Key Equation and allows an efficient decoding. We derived some properties of binary cyclic codes of minimum distance two and three and lowest code-rate.
Future work is to find lowest-code-rate small-minimum-distance non-binary cyclic codes and relate them to our method and bound the minimum distance of other cyclic codes. Combined error-erasure decoding with our proposed method seems to be possible. let γ be a primitive element in F q r . Let N = q r − 1. Then α = γ N/n and β = γ N/n . We consider univariate polynomials in F q r [x]. If gcd(n, n ) = 1 then holds ∀i, j ∈ [n] with i = j.
Proof. We show that the contrary does not hold. If (20) does not hold, then there exist a i and j with i > j and m, m ∈ Z with m = m such that
α i β m = α j β m α i−j = β m −m(21)
holds. Let us express (21) in terms of γ. We obtain: We know that i − j is smaller than n and m − m is smaller than n . This implies that λ is zero. We have:
(i − j)n = (m − m)n ⇒ n |(m − m)n
But (m − m) < n and this implies that gcd(n, n ) = 1.
. Let α ∈ F 2 6 denote an element of order 21. The defining set D C = M
Example 2 (
2Binary Code of length n = 21). Let us again consider the binary code C(2; 21, 7, 8) of Example 1. We have µ − 1 = 13 according to Theorem 2, so d * = 14/2 = 7.The HT bound (Theorem 1) gives also d ≥ 6 (with parameters b 1 = 1, m 1 = 5, m 2 = 1, d 0 = 5 and ν = 1). The Roos bound gives d ≥ 8[11, Example 1], which is the actual minimum distance of C(2; 21,7,8).
Figure 1 :
1Illustration of the fraction d * /(d 0 + ν) of our bound d * of (9) to the Hartmann-Tzeng bound d 0 + ν for ν = 1, . . . , 6 and d 0 = 2, . . . , 20. The parameters of the HT bound are m = ν + 2 (see
Example 3 (.
3Parity Check Code as Non-Zero-Locator Code). Let us consider the binary reversible [12] cyclic code C(2; 65, 41, 8) with the defining set D C = M We know that { , −5, −4, , −2, −1, , 1, 2, , 4, 5, } ⊆ D C . The HT bound gives a lower bound of d ≥ 6 on the minimum distance of C(2; 65, 41, 8) (for b 2 = −5, m = 3, d 0 = 5 and ν = 1)
Figure 2 :
2Illustration of the fraction d * /(d 0 + ν) of our bound d * of (9) to the Hartmann-Tzeng bound d 0 + ν for ν = 6, d 0 = 2, . . . , 20 and m. We used an RS code as non-zero-locator code with minimum distance d = m − ν (see
5 :
5Subset of the defining sets D C of a given cyclic code C in the interval [−10, 10]. The set D L = {0} is the defining set of the single parity check code L(2; = 22 and therefore d ≥ d * = 11.
7 :
7Subset of the defining sets D C of a given cyclic code C in the interval [−17, 17] (only odd indexes are illustrated). The set D L = {0, 1} is the defining set of a Reed-Solomon code RS(q ;
x) has minimum distance two if and only if gcd(n, i, j) > 1.Proof. Let α be an nth root of unity. A binary cyclic code C with generator polynomial g(length n has minimum distance two if there exist a binomial c(x) = x k + x that fulfills c(α i ) = c(α j ) = 0.This holds, if and only if α ki = α i and α kj = α j or, equivalently,(k − )i ≡ (k − )j ≡ 0 mod n.Both congruences are valid if and only if n/ gcd(n, i, j) divides k − . Therefore, such k and exist if and only if gcd(n, i, j) > 1.
Example 4 (.
4Non-primitive Binary Cyclic Code with Minimum Distance Three). Let n = 119 = (2 3 − 1) · 17. In this case g = 3 (see Theorem 4). Then {1, 11, 51} belong to M Therefore the binary cyclic code of length n = 119 with generator polynomial k = 68 and minimum distance d = 3.
=
{1, 2, 4, . . . , 2 g−1 } is the cyclotomic coset of a binary Hamming code of length 2 g − 1. The defining set of the corresponding lowest-code-rate binary cyclic code is a repetition of the defining set of the Hamming code of length 2 g − 1.Example 5 (Non-primitive Binary Cyclic Code with Minimum Distance Three and Lowest Code-Rate). Let us again consider Example 4 with n = 119 = (2 3 − 1) · 17 and k = 68. The binary cyclic code of length n = 119 with generator polynomial g(x) = M with minimum distance three has lowest code-rate R = (2 3 − 1 − 3)/(2 3 − 1) = 68/119. Its defining set D C is:D C = { , 1, 2, , 4, , ,, 8,9, , 11, , , , 15, 16, , 18, , , , 22, . . . , 116, , }.
⇒
(i − j)n − (m − m)n = λ · n · n .
Table 1 :
1Defining sets D C and D L of the binary cyclic code C(2; 21, 7, 8) of Example 1 and its
Table 3 :
3Subset of the defining sets D C of the C(2; 65, 41, 8) code in the interval [−6,
Table
Table 6 :
6Subset of the defining sets D C of a given cyclic code C in the interval [1, 20]. The set
D L = {3, 5, 6} is the defining set of the binary Hamming code L(2; 7, 4, 3).
D C 1 2
4
7 8 9
11
14 15 16
18
D L
3
5 6
3
5 6
3
5 6
Table
AcknowledgmentsWe thank the anonymous referees for valuable comments that improved the presentation of this paper.The authors wish to thank Antonia Wachter-Zeh and Daniel Augot for fruitful discussions. This work has been supported by German Research Council "Deutsche Forschungsgemeinschaft" (DFG) under grant BO 867/22-1.AppendixLemma 3 (Coprimality of n and n ). Let [n] denote the set of integers {0, 1, . . . , n − 1} and let Z be a subset of [n ]. Let α be an element of order n in F q s and let β denote a primitive element of order n in F q s , where F q = F q u . Let r denote the least common multiple of s and u · s and
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| [] |
[
"Logic Against Bias: Textual Entailment Mitigates Stereotypical Sentence Reasoning",
"Logic Against Bias: Textual Entailment Mitigates Stereotypical Sentence Reasoning"
] | [
"Hongyin Luo [email protected] \nMIT Computer Science\nArtificial Intelligence Laboratory Cambridge\n02139MAUSA\n",
"James Glass [email protected] \nMIT Computer Science\nArtificial Intelligence Laboratory Cambridge\n02139MAUSA\n"
] | [
"MIT Computer Science\nArtificial Intelligence Laboratory Cambridge\n02139MAUSA",
"MIT Computer Science\nArtificial Intelligence Laboratory Cambridge\n02139MAUSA"
] | [] | Due to their similarity-based learning objectives, pretrained sentence encoders often internalize stereotypical assumptions that reflect the social biases that exist within their training corpora. In this paper, we describe several kinds of stereotypes concerning different communities that are present in popular sentence representation models, including pretrained next sentence prediction and contrastive sentence representation models. We compare such models to textual entailment models that learn language logic for a variety of downstream language understanding tasks. By comparing strong pretrained models based on text similarity with textual entailment learning, we conclude that the explicit logic learning with textual entailment can significantly reduce bias and improve the recognition of social communities, without an explicit de-biasing process. The code, model, and data associated with this work are publicly available at https: //github.com/luohongyin/ESP.git. | 10.48550/arxiv.2303.05670 | [
"https://export.arxiv.org/pdf/2303.05670v1.pdf"
] | 257,482,682 | 2303.05670 | b3dcd48b68bdbb304fa53299496539c054638e0c |
Logic Against Bias: Textual Entailment Mitigates Stereotypical Sentence Reasoning
Hongyin Luo [email protected]
MIT Computer Science
Artificial Intelligence Laboratory Cambridge
02139MAUSA
James Glass [email protected]
MIT Computer Science
Artificial Intelligence Laboratory Cambridge
02139MAUSA
Logic Against Bias: Textual Entailment Mitigates Stereotypical Sentence Reasoning
Due to their similarity-based learning objectives, pretrained sentence encoders often internalize stereotypical assumptions that reflect the social biases that exist within their training corpora. In this paper, we describe several kinds of stereotypes concerning different communities that are present in popular sentence representation models, including pretrained next sentence prediction and contrastive sentence representation models. We compare such models to textual entailment models that learn language logic for a variety of downstream language understanding tasks. By comparing strong pretrained models based on text similarity with textual entailment learning, we conclude that the explicit logic learning with textual entailment can significantly reduce bias and improve the recognition of social communities, without an explicit de-biasing process. The code, model, and data associated with this work are publicly available at https: //github.com/luohongyin/ESP.git.
Introduction
Recent pretrained language models have achieved significant improvements on natural language understanding tasks (Devlin et al., 2018;Liu et al., 2019;Clark et al., 2020;He et al., 2020;Brown et al., 2020). These models are typically trained based on text similarity of words and sentences. Since the optimization objective maximizes the likelihood of the training corpora, the coherence of words and sentences that often appears together in the training corpora will be increased based on the trained model. However, since the training corpora are generated by humans, they can contain a large amount of social bias and stereotypes, including those concerning gender, race, and religion (Nadeem et al., 2020;Stanczak and Augenstein, 2021;Kiritchenko and Mohammad, 2018).
In contrast, learning by textual entailment (Dagan et al., 2005;Williams et al., 2018)
focuses
The person is a doctor.
The person is feminine.
The person is masculine.
Enc
The person is a nurse.
The person is feminine.
The person is a doctor.
The person is masculine.
Ent CLS E n t a il N e u t r a l C o n t r a d ic t Figure 1: Mitigating stereotypical sentence reasoning bias with textual entailment models. The upper figure stands for calculating text similarities with sentence embeddings generated by a sentence encoder (Enc). The lower figure stands for predicting the sentence relation with a textual entailment classifier (Ent CLS). Both sentence pairs are predicted neutral by the classifier. more on logic than semantic similarity. According to Dagan et al. (2005), textual entailment is not necessarily strict logical entailment. Instead, textual entailment stands for the case where the premise is true so that the hypothesis is likely to be true. Contradiction means that when the premise is true, the hypothesis is likely to be false. A sentence can be entailed, neutral, or contradictory with respect to either semantically similar or unsimilar sentences. As a result, a textual entailment model is less likely to conduct stereotypical reasoning that is caused by text similarity. As illustrated in Figure 1, a sentence encoder model can generate sentence representations that reflect the bias in the pretraining corpora via text similarity calculations. However, a textual entailment model treats both sentence pairs as neutral, indicating that the model should not be biased to either option. The prediction indicates the fact that there is no logical relation between gender and occupation in the example shown.
Besides gender, we also investigate different types of stereotypical sentence reasoning of language models, including race, religion, profession, and emotion using StereoSet (Nadeem et al., 2020), profession and gender terms in (Lu et al., 2020), and emotion terms in (Kiritchenko and Mohammad, 2018). We make the following contributions in this work: 1. Bias in sentence representations. We analyze the different types of stereotypical bias present in pretrained language models and state-of-the-art contrastive sentence representation models. 2. Textual entailment debiases. We demonstrate that textual entailment models perform well on sentence representation tasks, and are significantly less biased than similarity-based sentence encoders, without incorporating any explicit de-biasing. 3. Similarity causes bias, logic leads to fairness. By analyzing the experimental results, we find that the baseline sentence encoders learn human intuitions about text similarity, but contain significantly more stereotypes. In contrast, textual entailment tasks remove the models' perception about text similarity, but produce less biased predictions.
Related Work
Recent advances in language modeling has followed the strategy of learning large-scale models on large-scale unannotated corpora with selfsupervised learning, including masked word and next sentence prediction (Devlin et al., 2018;Liu et al., 2019;He et al., 2020), wrong word detection (Clark et al., 2020), and left-to-right language generation (Brown et al., 2020;Raffel et al., 2020). The training of these models rely on the word and sentence coherence of the pretraining corpora. Word-level language models are the foundation of sentence-level language encoders, including sen-tenceBERT (Reimers and Gurevych, 2019), Sim-CSE (Gao et al., 2021), and DiffCSE (Chuang et al., 2022), that were proposed for generating sentence embeddings with better representation abilities.
Recent studies have revealed that pretrained language models can learn different types of stereotypical and biased reasoning. Recasens et al. (2013) investigated biased languages using Wikipedia texts. Lu et al. (2020) surveyed stereotypical reasoning in word-level language prediction and co-reference resolution. Kiritchenko and Mohammad (2018) probed language models with the sentiment analysis task and measured the different model behav-iors against different social groups. Stereotypical reasoning against race, gender, profession, and religion were also evaluated on recent masked language models and sentence encoders in Nangia et al. (2020) and Nadeem et al. (2020).
The studies about the biases introduced by language models mainly focus on two types of tasks: intra-sentence reasoning and inter-sentence reasoning. Intra-sentence, or word-level, reasoning represents word and co-reference selection in a single sentence, which reveals the bias within word and context representations (Bao and Qiao, 2019;Bartl et al., 2020;Bolukbasi et al., 2016;Cao and Daumé III, 2019;Chaloner and Maldonado, 2019;Manzini et al., 2019;Caliskan et al., 2017). On the other hand, inter-sentence reasoning refers to reasoning biases across sentences. More specifically, a set of given sentences may not have any logical relationship, but a similarity-based language model may be biased towards linking a subset of the sentences, reflecting the coherence bias of the pretraining corpora (May et al., 2019;Kiritchenko and Mohammad, 2018;Nadeem et al., 2020). Recent studies have also investigated the social bias under multilingual settings (Costa-jussà et al., 2019;Elaraby et al., 2018;Font and Costa-Jussa, 2019).
To mitigate the social biases that cause language models to be untrustworthy, recent studies have explored methods to debias the learning and predicting processes of language models. Typical debiasing methods include counterfactual data augmentation (Zmigrod et al., 2019;Dinan et al., 2019;Webster et al., 2020;Barikeri et al., 2021), dropout regularization (Webster et al., 2020), self-debias (Schick et al., 2021), sentence embedding debias (Liang et al., 2020), and iterative nullspace projection (Ravfogel et al., 2020).
Besides the regular similarity-based pretraining method applied by most language models, some sentence encoding models also employ natural language inference (NLI) corpora to learn textual entailment (Bowman et al., 2015;Williams et al., 2018). Superivised SimCSE (Gao et al., 2021) and SentenceBERT (Reimers and Gurevych, 2019) use entailment data as a part of the pretraining corpora, while other studies apply entailment models to handle downstream tasks, including fact-checking (Thorne and Vlachos, 2018), relation extraction (Obamuyide and Vlachos, 2018), and text classification (Yin et al., 2019). The learned textual en-tailment knowledge that encodes logic rather than similarity provides the model a better generalization ability across different tasks and domains.
Method
Measuring Stereotypical Reasoning
In this work, we use data from three different sources to measure the stereotypical biases of sentence encoders. We use the following corpora and corresponding data construction strategies: StereoSet. The StereoSet corpus (Nadeem et al., 2020) contains both intra-and inter-sentence tasks for evaluating stereotypical reasoning, including gender, race, religion, and profession. Each data example contains a context and three options, including a stereotype, an anti-stereotype, and an unrelated sentence. A model is required to score each option and pick one. After selecting an option for each data example, two metrics are evaluated, including (1) the number of stereotypes being selected, and (2) the number of unrelated options being selected.
In this task, an ideal unbiased model selects 50% stereotypes, 50% anti-stereotypes, and 0% unrelated options, while a random model selects 33.3% stereotypical, anti-stereotypical, and unrelated options respectively. We used the idealized Context Association Test (iCAT) score (%) to jointly assess the quality and fairness of the sentence encoders.
iCAT = lms · min(ss, 100 − ss) 50
(1)
where lms (language model score) stands for the percentage that the model selects a related option, and ss (stereotype score) stands for the percentage that the model selects a stereotypical option. The iCAT score highlights the models that tend to select related options with no preference as to stereotypical and anti-stereotypical options. Gender Profession & Emotion Test. We apply the gender and profession vocabulary sets from Lu et al. (2020) and the sentiment vocabulary set from Kiritchenko and Mohammad (2018). With the collected vocabulary, we test if sentence encoders conduct stereotypical reasoning that links some professions and emotions to a specific gender group. We also use an iCAT score to measure the fairness, which is calculated using different metrics iCAT = grs · min(gbs, 100 − gbs) 50
where grs (gender recognition score) stands for the percentage that a model correctly predicts the gender of a gender-indicating noun, and gbs (gender bias score) stands for the percentage that a model links a profession or emotion to the man gender. To calculate the percentages, we use a pool of genderindicating nouns that are associated with different social and family roles.
Data and Prompts
We formulate each context association test as a context sentence and three optional sentences. The task for the models is to score each of the three sentence pairs and select one of the options. We introduce how we construct the test sets and prompts respectively as follows:
StereoSet Inter-sentence. The inter-sentence section of the StereoSet corpus contains 6,374 context association tests, which are straightforward to construct as sentence pair selection tasks, since they are already well-formed for sentence pair scoring.
StereoSet Intra-sentence. The intra-sentence section of the StereoSet corpus contains 6,392 context association tests. Unlike the inter-sentence tests, intra-sentence tests were originally designed for masked language models to select a good word to fill in the BLANK in the context sentence. For example, the context sentence of a test is "Girls tend to be more BLANK than boys", and the option words are "soft", "determined", and "fish" (less related). To conduct sentence-level reasoning, we replace the BLANK in the context sentence with "what" and the candidate words. As a result, a sentence encoder is required to represent the following sentences, "Girls tend to be more what than boys" and "Girls tend to be more soft than boys", etc.
Gender-indicating terms. We collect 71 pairs, or 142 binary gender-indicating terms about social and family roles from Lu et al. (2020), for example, uncle and aunt. 71 of them are masculine and the other 71 are feminine. For each term, for example aunt, we construct a prompt "the person is a(n) aunt". We evaluate if a model successfully reasons "the person is a(n) aunt" → "the person is feminine." The motivation for this gender recognition test is two-fold. First, when people use a genderindicating term, they would like the listener to infer their genders. Second, we want to avoid obtaining a fair but random model that fails to infer genders. Professions and emotions. We collect 65 occupation names from Lu et al. (2020), 20 emotion Table 1: The summary of data, tasks, prompts, metrics, and the scores of an ideal model that will be applied for evaluation in this work. Gender & profession stands for the corresponding vocabulary sets in Lu et al. (2020), and Emotion Vocab stands for the emotion vocabulary set in Kiritchenko and Mohammad (2018).
state terms, and 20 emotional situation terms from Kiritchenko and Mohammad (2018). For an occupation term PRO, we construct a prompt "The person is a PRO"; for an emotion state term ES, we construct a prompt "The person feels ES"; and for an emotion situation term ESIT, we construct a prompt "The person told us about the ESIT event."
We evaluate whether a model tends to link the construct profession and emotion prompts to one of the genders or not. A summary of the data, tasks, prompts, metrics, and scores of an ideal model is shown in Table 1. We define an "ideal model" as a fair and perfectly understanding model.
Textual Entailment
Training. We train the textual entailment models with the MultiNLI corpus (Williams et al., 2018). In MultiNLI, each data example contains a premise and a hypothesis, and the task is to predict if the hypothesis is likely to be true or false given the premise. Each sentence pair is classified into three classes: entailed, neutral, and contradictory. For a premise p and a hypothesis h, we construct the following supposition for the entailment model, h is entailed by p.
The classifier model is trained to output true, false, and neutral for each input supposition, and the entailment relations of each sentence pair can be directly inferred from the truth value of the corresponding prompt. In this work, we train entailment classifiers based on BERT (Devlin et al., 2018), RoBERTa (Liu et al., 2019), and DeBERTa (He et al., 2020). Evaluation. Standard sentence reasoning methods are based on the inner product of the embeddings of two sentences. With the textual entailment models, we can calculate three scores for each sentence pair, including entail, neutral, and contradictory scores. With these scores, we can calculate a prediction about the logical relation between two sentences. In summary, we have two strategies to score sen-tence pairs: 1. continuous sentence pair scoring with entail, neutral, or contradiction scores, and 2. discrete scoring using entailment predictions (entail = 0, neutral = 1, and contradictory = 2). Given a context, we prefer an option with a higher entailment score, lower contradictory score, and smaller entailment labels.
For the continuous scoring strategy, we calculate the language model score with the number of tests where the stereotype or anti-stereotype option score is higher than the unrelated option, and calculate the stereotype score with the number tests where stereotype option score is higher than antistereotype option. For the discrete scoring strategy where we assign each option an entailment label, the language score is calculated with the number of tests where the unrelated option is predicted to be less entailed than the stereotype or anti-stereotype. The stereotype score is calculated with the number of tests where the label {0, 1, 2} of the stereotype option is lower then the anti-stereotype.
Experiments
Language Understanding
To ensure that the fairness of the entailment-based language model does not come from a lack of language understanding ability, we first show the zero-shot adaptation performance of the entailmentbased language models. On the MNLI-mismatch task, The RoBERTa model achieves 89.0% accuracy, and the DeBERTa model achieves 83.4%. We compare different language models on other tasks in the GLUE benchmark (Wang et al., 2018), including QNLI, QQP, RTE, and SST2 tasks. For each task, we construct suppositions for classification according to the corresponding task description as shown in Table 2.
We compare the zero-shot adaptation performance of our entailment-based supposition (ESP) language models with weakly supervised baseline models of different scales as follows: Few-shot 350M models.
We compare our entailment-based models with LM-BFF and UPT (Wang et al., 2022) models. Both baseline models are based on RoBERTa-large that contains 350M parameters with 32 humanannotated training samples. Few-shot 137B models. We also compare the entailment-based models with large-scale language models (LLMs), LaMDA (Thoppilan et al., 2022) and FLAN (Wei et al., 2021) containing 137B parameters, which are about 400 times larger than the entailment-based models. The LLMs are adapted to the tasks with 4 to 8 training samples. The results are shown in Table 3. We found that overall, both RoBERTa and DeBERTa-based entailment models outperform all baselines, without using any task-specific training data. This proves the computation and data efficiency of entailmentbased language models.
Fairness
We evaluate pretrained language models, supervised/unsupervised SimCSE (Gao et al., 2021), and entailment models based on BERT, RoBERTa, and DeBERTa. The overall experiment results are shown in Table 4. StereoSet-Intrasentence. In Table 4, we use the fairness score (FS) to assess the bias of the models. We have F S = min(ss,1−ss) 0.5 , where ss stands for the stereotype score defined in (Nadeem et al., 2020). All baselines are sentence reasoning models pretrained with the next sentence prediction (NSP) task. We noticed that stronger sentence encoders can lead to more biased reasoning results. For BERT-based models, the unsupervised SimCSE model achieves a much higher language model score than the BERT-NSP model, outperforming by over 10%. The supervised SimCSE also marginally outperforms the baseline model. However, both SimCSE models are more biased. The fair score of the supervised SimCSE is 15% lower than the baseline BERT model. Because of the high sentence retrieval performance, the unsupervised SimCSE model achieves the best iCAT score, outperforming the pretrained BERT model by 4%.
The result remains the same for RoBERTa-based models. Both supervised and unsupervised Sim-CSE models significantly outperform the pretrained model, by 27% and 32%, respectively. As with the BERT-based models, RoBERTa SimCSE models are also more biased. According to the low language modeling score, the baseline RoBERTa pertrained model is almost random. As a result, the fairness score is as high as 96%. The SimCSE models achieve higher iCAT scores mainly because of the improvement on the language model score. We found that the DeBERTa model achieves the highest iCAT score among all NSP models. It achieves a very high fairness score (99.68%), but a relatively low language model score of 76.24%. As a result, the iCAT score of DeBERTa is only marginally higher than the BERT-based unsupervised SimCSE model, which achieves a 89.46% language model score.
The entailment models achieve the best iCAT score, and both entailment scoring strategies outperform baseline sentence embedding models. Comparing with the best BERT, RoBERTa, and De-BERTa based baselines, the corresponding discrete entailment model achieved a 12.5%, 39%, and 25% improvement in iCAT score. We observed that the discrete scoring models are generally better than the continuous scoring method. Although the continuous scoring method has certain biases, a discrete model can prevent biased prediction. For example, although the entailment score of option a is higher than option b, both options can be both classified into the neutral category. Table 4: Performance of pretrained language models and textual entailment models on StereoSet, gender recognition (rec.), profession, and emotion tests. LMS stands for language model score, FS stands for fairness score, and iCAT stands for ideal context association test score. NSP stands for next sentence prediction. The profession and emotion iCAT scores are calculated by multiplying the gender recognition score and the corresponding fairness scores. All scores are in percentage (%).
StereoSet-Intersentence. In general, the Intersentence task had similar trends as the Intrasentence task. The performance of the pretrained baseline models perform much better than the intrasentence tasks since the options are more diverse, making it easier for the models to identify the more related options. The difference within the baseline models are that the supervised SimCSE models perform better than the unsupervised sentence embedding models.
The entailment models are also significantly better than all the baseline models. All discrete scoring models achieve higher than 99% language modeling scores, and the fairness scores are all higher than 94%. The iCAT scores of the discrete entailment models are at least 93.4%, outperforming the best baseline model, supervised SimCSE with RoBERTa by 18%. On the other hand, the continuous entailment models also outperform the best SimCSE model by at least 9% in iCAT score. We also note that the discrete entailment models outperform the continuous models by a significant margin because the labels prevent a large amount of stereotypical reasoning. Gender recognition. We evaluated the models' ability to recognize the gender of binary genderindicating nouns, for example, (uncle, aunt) and (brother, sister). We use the set of 71 pairs, 142 gender-indicating nouns from Lu et al. (2020). The RoBERTa-based, supervised SimCSE model achieves high gender recognition accuracy (as high as 99%), while the performance of the pretrained DeBERTa model is close to random at around 50%. We found that the supervised SimCSE models are significantly better than other baseline models on this task.
On the other hand, we found that the continuous entailment scoring strategy achieves very high gender recognition performance. All three models achieve an accuracy higher than 99% with very low standard deviations. In contrast to the previous tasks, the discrete scoring models have decreased performance. We hypothesize that this is because the continuous models are good enough, but the discrete model score blurs the selective bias, which is needed in this task since we need diverse predictions. Despite this fact, the DeBERTa based discrete model still achieves high gender recognition accuracy (97%). Profession bias test. We use a vocabulary set from Lu et al. (2020) consisting of 65 profession nouns which are expected be gender-neutral, but possibly being affected by stereotypes. For the baseline models, we found that the stronger sentence representation models, supervised and unsupervised SimCSE, are significantly more biased than pretrained language models. Since the SimCSE models learns better sentence embeddings based on text similarity, they perform better at gender recognition, but retain more stereotypes in the pretraining corpora. Combined with the high gender recognition performance, the unsupervised BERT SimCSE model achieves the best iCAT score among all baseline models.
For this task, all entailment models outperform all baseline models. The DeBERTa and RoBERTa models are significantly better than BERT-based models. For the continuous scoring models, the RoBERTa-based entailment model achieves the highest iCAT score (86.93%), outperforming the best baseline model by 27%. As for previous tasks, the discrete entailment scoring strategy is more fair. The best discrete entailment model, DeBERTa, achieves a high iCAT score (95.1%), outperforming the best baseline model by 36%. The exception is the RoBERTa-based entailment model. The continuous RoBERTa model outperforms the discrete model by almost 2% iCAT score.
Emotion bias test. We use the emotion vocabulary sets, including 40 emotion state and situation words. We conduct context association tests on the gender-indicating nouns with the emotion words. On this task, the BERT and RoBERTa models have different behaviors. The RoBERTa-based SimCSE models outperform the pretrained RoBERTa model on both fairness and iCAT scores. However, the BERT SimCSE models are outperformed by the pretrained BERT model. The supervised RoBERTa model performs best among all baseline models, achieving 76% iCAT score.
The entailment models outperform most base-line models. The only exception is that the BERTbased entailment model is outperformed by the supervised RoBERTa SimCSE model. However, the discrete entailment RoBERTa and DeBERTa entailment models outperform all baseline models by a large margin. The discrete RoBERTa entailment model outperforms the best baseline model by more than 11%, and the DeBERTa entailment model outperforms the best baseline by 17%.
Summary. We make the following observations:
• SimCSE models achieve higher language model and gender recognition scores than pretrained models, but they are more biased.
• The entailment models achieve significantly better performance than all baseline models in both language modeling and fairness metrics. The discrete scoring strategy is more fair than the continuous strategy, in general.
RoBERTa-SimCSE
Analysis
Performance Breakdown
In the previous section, we reported the overall performance of each task. In this section, we analyze the performance of all sub-tasks. The StereoSet corpus has four sub-tasks, including gender, religion, profession, and race. The profession bias task has 65 different profession nouns as sub-tasks, and similarly, the emotion bias task has 40 sub-tasks. We break down and analyze the performance of the sub-tasks to investigate if the models conduct biased reasoning on sub-tasks, but achieve high average fairness scores.
StereoSet. The breakdown iCAT scores of Stere-oSet sub-tasks is shown in Figure 2.a, including the four sub-tasks under the intra-and inter-sentence settings. We do not find the entailment models to be biased on some of the sub-tasks. Instead, the entail-ment models consistently outperform the baseline pretrained models. We also note that the pretrained models based on different architectures achieve varying results on different tasks. In contrast, the entailment model based on different architectures achieve stable iCAT scores. We also notice that the entailment models perform better on race and religion tasks. As shown in Table 4, the performance of the discrete scoring models achieve better and more stable iCAT scores.
Profession bias test. We compare the breakdown performance of RoBERTa-based entailment and SimCSE models. As shown in Figure 2.b, the iCAT scores on most profession terms of the entailment model outperforms the SimCSE model by more than 20%. The only exception where the pretrained model outperforms the entailment models is the word "Bartender." The most significant improvement we achieved is almost 50% iCAT score on the term "dental hygienist." Emotion bias test. We also test the RoBERTabased models on different emotion state and situation terms. In all 40 emotion words, the entailment model outperforms the SimCSE model in 35 subtasks. The most biased emotion word of SimCSE is "disappointed," which is improved using the entailment model. On the other hand, the most biased emotion word of the entailment model is "devastated." Both models are relatively biased on the word "sad," achieving lower than 40% iCAT scores. The most significant improvement is on the word "relieved." The sub-tasks that the entailment model does not outperform the pretrained models are "scared," "terrified," "depressed," "devastated," and "miserable."
Prompt Embedding Analysis
We have found that the language modeling and fairness performance of entailment models are significantly higher than pretrained language models. In this section, we attempt to explain this phenomenon. To understand the difference between the entailment and pretrained models, we analyze the embedding of the gender terms and profession and emotion nouns. The results of the RoBERTabased SimCSE and entailment models are shown in Figure 3 with t-SNE (Van der Maaten and Hinton, 2008). The profession bias test results on RoBERTa-SimCSE is shown in Figure 3.a. We find that because of the strong representation ability of Sim-CSE, the embeddings of the profession and gender terms reflect the word similarities that aligns with human intuition. The boundary of the gender terms is detected by a linear SVM model (Hearst et al., 1998;Pedregosa et al., 2011). We find that the learned boundary separates terms of different genders with high accuracy. In addition, we notice that related profession terms group closely, as shown in the circles in Figure 3.a. In contrast, the word embedding distribution produced by the entailment model shown Figure 3.b appears to be more random. A similar phenomenon is observed on the emotion bias test. In Figure 3.c, nouns representing different genders are well-separated, and related words cluster closely. However in Figure 3.d, similar words are less correlated based on the entailment prompt embeddings.
The experimental results of both tasks and models indicate that the prompt embeddings learned by the entailment models contribute to logical reasoning rather than word coherence representation. Considering the fact that the entailment models perform significantly better than the pretrained models, we conclude that the biases are caused by the similarity-based learning objectives because such algorithms learn and reflect the biases in the training corpora. However, the textual entailment models learn logic without preserving textual similarities, leading to fairer performance.
Conclusion
In this work, we found that textual entailment learning reduces the bias of pretrained language models for sentence representation. We evaluated BERT, RoBERTa, and DeBERTa-based pretrained, Sim-CSE, and entailment models on stereotype, profession, and emotion bias tests. The textual entailment models outperform other models with significantly lower bias without other explicit debiasing processes, while preserving the language modeling ability, which results in significantly better idealized context association test scores. By analyzing the sentence embeddings, we found that the models relying on textual entailment produce less biased results by learning logic and reducing the amount of text coherence knowledge retained from the pretraining corpora containing existing social biases.
Ethics Statement
We investigate the stereotypes and biases of pretrained language models and introduce the less biased textual entailment models that reduce bias on gender, profession, religion, and race. We noticed that the existing gender-related bias studies and corpora mainly focus on the binary gender setting, and we also follow this line of research because of data limitations. While such data limitation might disappoint a number of communities, we will extend this work to non-binary settings in future work.
Limitations
As we described in the previous section, we studied the stereotypes including gender biases. However, we investigated under the binary gender setting, because of the limitation of the existing benchmarks. Furthermore, we evaluated medium-sized language models with around 350M parameters, but have not tested the largest language models yet. We only analyze the predictive bias on a set of genderindicating vocabulary, but do not look into every example and explain the source of the learned bias in the pretraining corpora or social traditions.
On the other hand, there are further limitations in the benchmarks we study in this work, as pointed out by Blodgett et al. (2021) that StereoSet is not perfect. On the other hand, some words in the vocabulary collected by (Lu et al., 2020) are rarely used, for example, "poetess" and "manageress". In future work, we will explore building more inclusive and comprehensive benchmarks to mitigate the limitations.
Figure 2 :
2Breakdown performance of pretrained and entailment language models on StereoSet, profession, and emotion bias tests. In StereoSet, we present the performance of all models. In the profession and emotion bias tests, we compare the performance of RoBERTa-SimCSE and the continuous RoBERTa entailment model.
Figure 3 :
3The prompt analysis with RoBERTa-based SimCSE and entailment models on profession and emotion bias tests. In figures a. and c, different gender terms are separated by a boundary learned by a linear SVM, and the gray circles highlight correlated words. InFigure a., the circled clusters are [singer, designer, writer, filmmaker, artist, musician], [carpenter, plumber], [barber, butcher], [lawyer, judge], and [economist, scientist, professor]. In Figure c., the circled clusters are [devastated, depressed, anxious], [relieved, ecstatic, glad, happy, excited], and [angry, annoyed].
Table 2 :
2The suppositions constructed based on the definitions of different GLUE tasks(Wang et al., 2018).Method
QNLI QQP RTE SST2 Avg.
Few-shot 350M models
LM-BFF
69.2
69.8 83.9 90.3 78.3
UPT
70.1
72.1 68.9 92.9 76.0
Few-shot Large-scale 137B models
LaMDA
55.7
58.9 70.8 92.3 69.4
FLAN
63.3
75.9 84.5 94.6 79.6
Zero-shot entailment-based 350M model
RoBERTa 71.5
78.6 81.2 87.7 79.8
DeBERTa 77.3
79.9 84.5 90.1 82.9
Table 3 :
3The performance of zero-shot entailmentbased models and strong few-shot supervised baselines.
AcknowledgementsWe are grateful for the insightful comments and suggestions from the reviewers.
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| [] |
[
"Stochastic formulation of incompressible fluid flows in wall-bounded regions",
"Stochastic formulation of incompressible fluid flows in wall-bounded regions"
] | [
"Zhongmin Qian "
] | [] | [] | In this paper we establish a mathematical framework which may be used to design Monte-Carlo simulations for a class of time irreversible dynamic systems, such as incompressible fluid flows, including turbulent flows in wall-bounded regions, and some other (non-linear) dynamic systems. Path integral representations for solutions of forward parabolic equations are obtained, and, in combining with the vorticity transport equations, probabilistic formulations for solutions of the Navier-Stokes equations are therefore derived in terms of (forward) McKean-Vlasov stochastic differential equations (SDEs), which provides us with the mathematical framework for Monte-Carlo simulations of wall-bounded turbulent flows. | null | [
"https://export.arxiv.org/pdf/2206.05198v2.pdf"
] | 249,605,691 | 2206.05198 | ae65d9766a371f8c02f7ca695fcf9efd7d19fb25 |
Stochastic formulation of incompressible fluid flows in wall-bounded regions
Zhongmin Qian
Stochastic formulation of incompressible fluid flows in wall-bounded regions
arXiv:2206.05198v2 [physics.flu-dyn] 8 Oct 2022Diffusion processFeynman-Kac formulasMcKean-Vlasov SDEsMonte Carlo methodNavier-Stokes equationparabolic equationspinned diffusion measuresturbulent flowsvelocity fieldvorticityvorticity transport equationwall-bounded fluid flows MSC classifications: 60H3035Q3035Q3576D0376D0576D17
In this paper we establish a mathematical framework which may be used to design Monte-Carlo simulations for a class of time irreversible dynamic systems, such as incompressible fluid flows, including turbulent flows in wall-bounded regions, and some other (non-linear) dynamic systems. Path integral representations for solutions of forward parabolic equations are obtained, and, in combining with the vorticity transport equations, probabilistic formulations for solutions of the Navier-Stokes equations are therefore derived in terms of (forward) McKean-Vlasov stochastic differential equations (SDEs), which provides us with the mathematical framework for Monte-Carlo simulations of wall-bounded turbulent flows.
Introduction
It was pointed out that, in a report [32] by von Neumann about 70 years ago, turbulence may be understood with the assistance of powerful electronic computer simulations. With the increasing computational power it is now possible to obtain information about turbulent flows via direct numerical simulations (DNS), large-eddy simulation (LES) and other simulation technologies (cf. [13,33,21,39] for example). One of the common difficulties for implementing turbulent simulations (such as the finite element method, the finite difference method and the finite volume method) lies in the non-linear and non-local nature of the governing fluid dynamic equations, the Navier-Stokes equations. According to Kolmogorov's small scale theory of isotropic turbulence, in order to obtain meaningful simulations of turbulent flows, the space separation has to be smaller than Re −3/4 L, where Re is the Reynolds number and L the typical length. This requirement for a small mesh in numerical schemes leads to huge computation hours for simulating turbulence, see [12,Chapter 7] for a very useful analysis about this aspect. It is thus desirable to develop Monte-Carlo schemes for simulations of turbulent flows which may avoid the use of small grids in order to numerically solve non-linear fluid dynamic equations.
Monte-Carlo simulations are based on the law of large numbers, which says the expectation of a random variable can be approximated by the average of independent samplings. Although convergence rates of Monte-Carlo methods are in general slow, but the advantage of Monte-Carlo schemes in most cases lies in their capability of dealing with multivariate dynamic variables. To implement Monte-Carlo schemes for numerically calculating solutions of some linear and non-linear partial differential equations, explicit representations of solutions in terms of some distributions, or in terms of stochastic differential equations, have to be established. An archetypal example is the numerical scheme for computing solutions of Schrödinger type partial differential equations, where the Feynman-Kac formula provides us with a useful representation to the solutions of the Schrdinger equations.
In this paper we establish a mathematical framework for designing Monte-Carlo schemes for numerically computing solutions of incompressible fluid flows in wall-bounded regions. The basic idea in our approach is to derive an equivalent formulation of the fluid dynamic equations in terms of McKean-Vlasov type (ordinary) stochastic differential equations which can be solved numerically. Let us describe the basic idea and the main technical issues which will be solved in the present paper.
The basic dynamic variable for a fluid flow in a region D is its velocity u(x,t) for x ∈ D and t ≥ 0, a time dependent vector field. The main scientific question in turbulence is to give a good description of u(x,t). Mathematically determining the velocity u(x,t) is equivalent to describing its integral curve X (ξ ,t) which are solutions to the following differential equation
d dt X (ξ ,t) = u(X (ξ ,t),t), X (ξ , 0) = ξ . (1.1)
For a viscous fluid flow with viscosity constant ν > 0, according to Taylor [42], one may consider the Brownian trajectories X (ξ ,t) instead which are solutions to the random dynamic system dX (ξ ,t) = u(X (ξ ,t),t)dt + √ 2νdB(t), X (ξ , 0) = ξ , (1.2) where B(t) is a Brownian motion on some probability space. The main idea is to rewrite the velocity u(x,t) in terms of the distribution of the Brownian particles X (ξ ,t), substituting it into (1.2), so that (1.2) becomes a stochastic differential equation involving the distribution of its solution. We may call this technique as solving the closure problem of Taylor's diffusion. The closure problem for incompressible fluid flows freely moving in the whole space (without boundary) has been solved in a recent paper [37] (see also for a similar probabilistic representation in [9]) by using an idea from the vortex methods [4,5,11,29]. The most important case of fluid flows in wall-bounded regions will be solved in the present paper. Let us recall some basic ideas in the study of the vortex dynamics and the vorticityvelocity formulation, for details one may refer to [6,7,11,28,29]. Consider an incompressible fluid flow with viscosity constant ν > 0, whose velocity u = (u 1 , u 2 , u 3 ) and pressure P solve the Navier-Stokes equations
∂ ∂t u i + (u · ∇)u i − ν∆u i + ∂ ∂ x i P = 0 (1.3) and 3 ∑ j=1 ∂ ∂ x j u j = 0 (1.4)
where i = 1, 2, 3. The sum in (1.4) is the divergence of u, ∇ · u at any instance, so the second equation means that the velocity u(x,t) is divergence free (which can be formulated in the distribution sense on R 3 ). u 0 (x) = u(x, 0) is called the initial data. The boundary condition at infinity has to be supplied such as the velocity is bounded at infinity, but let us ignore this kind of technical issues. It is known that the vorticity ω = ∇ ∧ u, whose components ω i = ε i jk ∂ ∂ x j u k , plays a dominate rôle in the description of turbulent flows. The vorticity ω evolves according to the vorticity transport equations
∂ ∂t ω i + (u · ∇)ω i − ν∆ω i − 3 ∑ j=1 S i j ω j = 0 (1.5)
for i = 1, 2, 3, which are obtained by differentiating the Navier-Stokes equations. Here
S j i = 1 2 ∂ u j ∂ x i + ∂ u i ∂ x j (1.6)
is the symmetric tensor of rate-of-strain. Note that the dynamic variables u, ω and S depend on (x,t) and are time irreversible. Our goal is to express the velocity u(x,t) in terms of the distribution of Taylor's diffusion X (ξ ,t) defined by (1.2). To achieve this goal, we observe that X (ξ ,t) is a diffusion process with its infinitesimal generator L u = ν∆ + u · ∇, whose transition probability density p u (τ, ξ ,t, x) is the fundamental solution to L u + ∂ ∂t , cf. [41], [23] for example, According to a general fact from the parabolic theory (cf. [18]), p u (τ, ξ ,t, x) coincides with the fundamental solution to the forward heat operator L ⋆ u − ∂ ∂t . Since u is divergence-free, so that L ⋆ u = L −u , hence the vorticity transport equations (1.5) may be written as Schrödinger type equations
L −u − ∂ ∂t ω i + 3 ∑ j=1 S i j ω j = 0, (1.7)
and therefore it is possible to express ω i in terms of the distribution of the Taylor diffusion, the data S i j and the initial vorticity ω 0 . For fluid flows for which the "potential" term ∑ 3 j=1 S i j ω j vanish identically (for i = 1, 2, 3); in fact it is the case for 2D fluid flows; the vorticity transport equations become L −u − ∂ ∂t ω i = 0, thus ω i can be written as an integral of the initial vorticity ω 0 = ∇ ∧ u 0 against the fundamental solution p u (τ, ξ ,t, x). Therefore the closure problem for this case can be solved by using the Biot-Savart law, cf. [28,29]. For turbulent flows, it is typical that the non-linear vorticity stretching term ∑ 3 j=1 S i j ω j do not vanish. For this case one may apply the Feynman-Kac formula to (1.7) in terms of the time reversed diffusion with generator L −u (but not L u ). To resolve the closure problem to (1.2), we need to rewrite the Feynman-Kac formula in terms of the law of the Taylor diffusion (1.2), which will be achieved by using the duality between the conditional law of the time reversed L −u -diffusion and the conditional law of L u -diffusion, cf. [37] for details.
In this paper we aim to solve the closure problem for incompressible fluid flows constrained in a domain D with a boundary ∂ D, so that schemes may be devised for numerically calculating the solutions, including the flows within their boundary layers. For a wall-bounded flow, the velocity u(x,t) satisfies the Navier-Stokes equations (1.3, 1.4) only for x ∈ D and t > 0, and has to satisfy the no-slip condition, i.e. u(x,t) vanishes for x ∈ ∂ D. We extend the definition of u(x,t) to all x ∈ R 3 such that ∇ · u = 0 on R 3 in the distribution sense. This latter requirement is needed in order to ensure again the duality for the conditional laws of some diffusions involved. Hence we can define the Taylor diffusion again by (1.2) on the whole space R 3 .
To solve the closure problem for the boundary problem of the fluid dynamic equations, we develop several technical tools, mainly the duality of conditional laws, and the forward type Feynman-Kac formula for a general class of diffusion processes. We believe that these results have independent interest by their own and may be useful for the study of other problems associated with diffusion-reaction equations (cf. [16,17]). More precisely we will establish these results for the laws of diffusion processes with infinitesimal generator being an elliptic operator of second order
L = d ∑ i, j=1 ∂ ∂ x j a i j (x,t) ∂ ∂ x i + d ∑ i=1 b i (x,t) ∂ ∂ x i ,
although some results are established for more general elliptic operators, i.e. for elliptic operators not in divergence form. This paper is organised as the following. In Section 2, we introduce a class of time dependent elliptic operators of second order which set up the basic data used throughout the paper. We recall in this section a few analytic and probabilistic structures, and recall several well known relations among forward, backward fundamental solutions and transition probability density functions, scattered in literature, stated as Lemma 2.2, Lemma 2.3 and Lemma 2.5. In Section 3, by using the classical Feynman-Kac formula, we establish a general duality relation between the fundamental solution and the transition probability density function (Lemma 3.2 and Lemma 3.3), and under the divergence free condition on the drift vector field b(x,t) establish the duality (Lemma 3.4). In Section 4, we prove the main technical tool, a time reverse theorem (Theorem 4.1) for the conditional laws of the L -diffusion when the vector field b(x,t) is divergence free. Thus far we have established the basic tools for proving our main results. In Section 5, we prove a forward Feynman-Kac formula for solutions of Schrödinger type parabolic systems on the whole space, which generalises the results in [37]. Section 6 and Section 7 contain the main contributions of this paper. In Section 6, we establish a forward Feynman-Kac formula for solutions of the (non homogeneous) boundary value problems, Theorem 5.3, and in Section 7 we apply the theory to solve the closure problem for solutions of the Navier-Stokes equations satisfying the no-slip boundary condition.
Numerical experiments are not included in the present paper for the reason that numerical simulations based on the formulation of the Navier-Stokes equations in Section 7 must be done case by case, and some of numerical experiments will be published in separate papers.
In the past decades, many excellent works addressing some probabilistic aspects of fluid dynamic equations have been published by various authors, although not from a view-point of numerically calculating solutions, cf. [2], [3], [7,8], [9], [44,45] for example and the literature therein. There are still many papers the author may be not aware of, to the best knowledge of the present author however, only the Navier-Stokes equations on R d or T d (d = 2, 3) were considered in a view of stochastic analysis in particular via the stochastic flow method, and the boundary value problems are not treated yet in the existing literature. Also in this paper we adopt an approach of weak solutions both for PDEs and SDEs, and therefore stochastic flows do not play a rôle in our study. This makes distinct difference between our approach and the existing methods.
Assumptions and notations
In this section, we introduce several notions and notations which will be used throughout the paper.
Let Let ν > 0 be a positive constant representing "the viscosity constant". Let a(x,t) = (a i j (x,t)) i, j≤d be a Borel measurable, d × d symmetric matrix-valued function defined for x ∈ R d and t ∈ R. It is assumed that a(x,t) is uniformly elliptic in the sense that there is a constant λ ≥ 1 such that
λ −1 |ξ | 2 ≤ d ∑ i, j=1 ξ i ξ j a i j (x,t) ≤ λ |ξ | 2 (2.1)
for every ξ = (ξ 1 , · · · , ξ d ) ∈ R d and (x,t) ∈ R d × R. It is also assumed that there is a symmetric matrixvalued function σ (x,t) = (σ i j (x,t)) such that σ i j (x,t) = σ j i (x,t) and
a i j (x,t) = d ∑ l=1 σ i l (x,t)σ j l (x,t) (2.2) for all i, j ≤ d. Let b(x,t) = (b 1 (x,t), . . . , b d (x,t)
) be a Borel measurable, time-dependent vector field on R d , and c(x,t) be a Borel measurable scalar function on R d × R. a(x,t), b(x,t) and c(x,t) may be defined originally only for (x,t) ∈ D × J (where J ⊂ R is an interval), but without further specification, these functions are automatically extended to all (x,t) ∈ R d × R according to the following rules: b(x,t) = 0, c(x,t) = 0 and a i j (x,t) = δ i j for (x,t) / ∈ D × J unless otherwise specified. Define a time-dependent differential operator of second order
L a;b,c = ν d ∑ i, j=1 a i j ∂ 2 ∂ x j ∂ x i + d ∑ i=1 b i ∂ ∂ x i + c. (2.
3)
The first term involving a i j (x,t) on the right-hand side is called the diffusion part, b(x,t) is called the drift vector field, and the scalar multiplication part c(x,t) is called the zero-th order term. If c = 0, then L a;b,0 is a diffusion operator, which will be denoted by L a;b for simplicity. If a i j (x,t) = δ i j for all (x,t) ∈ R d × R, then L a;b,c is denoted by L b,c , and similarly, if a i j (x,t) = δ i j and c(x,t) = 0 for all (x,t) ∈ R d × R, then L a;b,c is denoted by L b . L a;b,c may be written in a form whose diffusion part is written in divergence form
L a;b,c = ν d ∑ i, j=1 ∂ ∂ x j a i j ∂ ∂ x i + d ∑ i=1 b i − ν d ∑ j=1 ∂ a i j ∂ x j ∂ ∂ x i + c, (2.4)
as long as ∂ a i j ∂ x j exist. According to integration by parts, the formal adjoint L ⋆ a;b,c of L a;b,c is again a second order differentiable operator and
L ⋆ a;b,c = L a;b ⋆ ,c ⋆ , where b i ⋆ (x,t) = −b i (x,t) + 2ν d ∑ j=1 ∂ a i j (x,t) ∂ x j (2.5) and c ⋆ (x,t) = c(x,t) − ∇ · b(x,t) + ν d ∑ i, j=1 ∂ 2 a i j (x,t) ∂ x j ∂ x i (2.6)
respectively. The formal adjoint of the (forward) heat operator L a;b,c − ∂ ∂t is the (backward) heat operator
L a;b ⋆ ,c ⋆ + ∂ ∂t . Example 2.1. If a i j = δ i j , then L b,c = ν∆ + b · ∇ + c and L ⋆ b,c = L −b,c−∇·b ,
which is a very important relation in the study of random vortex methods, cf. [4,11,29] for details.
Similarly
L a;b,c = ν d ∑ i, j=1 ∂ ∂ x j a i j ∂ ∂ x i + d ∑ i=1 b i ∂ ∂ x i + c (2.7)
is an elliptic operator of second order, and L a;b,c = L a;b,c , wherê
b i = b i + ν d ∑ j=1 ∂ a i j ∂ x j (2.8)
as long as ∂ a i j ∂ x j exist. Note thatb is independent of c. In general Einstein's convention that repeated indices are summed up in their ranges is applied, unless said otherwise. The following convention will also be applied to quantities which rely on a, b and c. If a quantity depends on a, b and c then it may be labelled with a lower subscript a; b, c. If a i j (x,t) = δ i j for all x and t, then the part a; will be omitted, so that
L b,c = ν∆ + d ∑ i=1 b i ∂ ∂ x i + c. (2.9)
If c = 0, then the part , c will be omitted, hence
L a;b = ν d ∑ i, j=1 a i j ∂ 2 ∂ x i ∂ x j + d ∑ i=1 b i ∂ ∂ x i (2.10)
and therefore
L b = ν∆ + d ∑ i=1 b i ∂ ∂ x i . (2.11)
This convention will be applied to other quantities such as the fundamental solutions, transition probability density functions, and so on. In particular L a;b = L a;b . We finally make some comments on further regularity assumptions on the data a, b, c and q (to be introduced later on) in additional to the assumptions we have already made. For simplicity, we assume that all data a, b, c and q are smooth and bounded, although these regularity conditions may be too demanding for applications. In fact, in additional to the assumptions made, the following assumptions are sufficient for our arguments to be true.
1) The results and arguments about L a;b are valid if a i j (x,t) are uniformly continuous in (x,t), b i (x,t), c(x,t) and q(x,t) are bounded, Borel measurable and continuous in t, although the boundedness conditions may be weaken.
2) The results and arguments about L a;b are valid if a i j (x,t) are Borel measurable and continuous in t, b i (x,t), c(x,t) and q(x,t) are bounded, Borel measurable and continuous in t, although the boundedness conditions may be weaken.
Analytic structures
In this section we recall several analytic structures associated with the basic data a(x,t), b(x,t) and c(x,t).
The fundamental solutions are the basic analytic quantities associated with the elliptic operators of second order, and the reader may refer to [1] and [18,25] for the most fundamental results in this aspect.
Γ a;b,c (x,t; ξ , τ), defined for x, ξ ∈ R d and τ < t,
is the fundamental solution of the forward parabolic equation
L a;b,c − ∂ ∂t f (x,t) = 0 (2.12)
in the sense that for every fixed
τ ≥ 0 and ξ ∈ R d , f (x,t) = Γ a;b,c (x,t; ξ , τ) solves (2.12), and for every bounded continuous function ϕ on R d lim t↓τ R d Γ a;b,c (x,t; ξ , τ)ϕ(ξ )dξ = ϕ(x) (2.13) for every x ∈ R d . Similarly Γ ⋆ a;b,c (x,t; ξ , τ) defined for x, ξ ∈ R d and for t < τ is a fundamental solution of the back- ward heat operator L a;b,c + ∂ ∂t , if for any ξ ∈ R d and τ, f (x,t) = Γ ⋆ a;b,c (x,t; ξ , τ) satisfies the backward parabolic equation L a;b,c + ∂ ∂t f (x,t) = 0 (2.14)
and lim
t↑τ R d Γ ⋆ a;b,c (x,t; ξ , τ)ϕ(ξ )dξ = ϕ(x)
for every continuous function ϕ on R d and for every x ∈ R d .
Lemma 2.2. The following relation holds:
Γ a;b,c (x,t; ξ , τ) = Γ ⋆ a;b ⋆ ,c ⋆ (ξ , τ; x,t) (2.15)
for any τ < t and x, ξ ∈ R d .
For a proof of this basic fact, see [18, Theorem 15, page 28].
Probabilistic structures
In this section we recall several probabilistic structures associated with the elliptic operator of second order L a;b . If a i j (x,t) are uniformly continuous and b i (x,t) are bounded and Borel measurable, then according to [40] there is a unique family of probability measures P ξ ,τ a;b (where τ ≥ 0 and ξ ∈ R d ) on the path space Ω = C([0, ∞), R d ) of all continuous paths in R d , equipped with its Borel σ -algebra, such that 1) the diffusion starts from ξ at time τ in the following sense that
P ξ ,τ a;b [ψ ∈ Ω : ψ(t) = ξ for all 0 ≤ t ≤ τ] = 1. (2.16) 2) (local martingale property) for every f ∈ C 2,1 (R d × [τ, ∞)), M [ f ] t = f (ψ(t),t) − f (ψ(τ), τ) − t τ L a;b + ∂ ∂ s f (ψ(s), s)ds for t ≥ τ ≥ 0 and M [ f ] t = 0 for 0 ≤ t ≤ τ, is a local martingale under the probability measure P ξ ,τ a;b . The family P ξ ,τ a;b (where ξ ∈ R d , τ ≥ 0) of probability measures is simply called the L a;b -diffusion. If a i j (x,t) = σ i k (x,t)σ k j (x,t) and σ i j (x,t) and b i (x,t)
are Lipschitz continuous, then the L a;b -diffusion may be constructed by solving Itô's stochastic differential equations:
dX i = b i (X ,t)dt + √ 2ν d ∑ k=1 σ i k (X ,t)dB k , X s = ξ for s ≤ τ (2.17) for i = 1, . . . , d, where B = (B 1 , · · · , B d ) is the standard Brownian motion in R d . The distribution of the strong solution X is the probability measure P ξ ,τ a;b . Let P a;b (τ, ξ ,t, dx) = P ξ ,τ a;b [ψ ∈ Ω : ψ(t) ∈ dx] (where t > τ ≥ 0)
, called the transition probability function of the L a;b -diffusion. Since a(x,t) is uniformly elliptic, P a;b (τ, ξ ,t, dx) has a probability density function p a;b (τ, ξ ,t, x) with respect to the Lebesgue measure, so that
P a;b (τ, ξ ,t, dx) = p a;b (τ, ξ ,t, x)dx for t > τ ≥ 0 and ξ ∈ R d .
The transition density function p a;b (τ, ξ ,t, x) (for t > τ ≥ 0) associated with the L a;b -diffusion is positive and Hölder continuous (cf. [1]).
Lemma 2.3. Let p a;b (τ, ξ ,t,
x) be the transition probability density function of the L a;b -diffusion. Then
p a;b (τ, ξ ,t, x) = Γ ⋆ a;b (ξ , τ; x,t) = Γ a;b,c (x,t; ξ , τ) for all 0 ≤ τ < t and x, ξ ∈ R d , wherẽ b i (x,t) = −b i (x,t) + 2ν d ∑ j=1 ∂ a i j (x,t) ∂ x j (2.18) andc (x,t) = −∇ · b(x,t) + ν d ∑ i, j=1 ∂ 2 a i j (x,t) ∂ x j ∂ x i . (2.19)
This follows immediately from Lemma 2.2 and the well known fact that
p a;b (τ, ξ ,t, x) = Γ ⋆ a;b (ξ , τ; x,t) for all 0 ≤ τ < t and x, ξ ∈ R d (cf. [41]). Remark 2.4. If a i j (x,t) = δ i j and b(x,t) is divergence free for every t, that is, ∇ · b = 0 in the distribution sense, then p b (τ, ξ ,t, x) = Γ ⋆ b (ξ , τ; x,t) = Γ −b (x,t; ξ , τ) (2.20) for all 0 ≤ t < τ. In particular, if ∇ · b = 0, then Γ b (x,t; ξ , τ) = p −b (τ, ξ ,t, x) (2.21)
for all t ≥ τ ≥ 0. The relationship (2.21) plays the crucial rôle in the random vortex method (cf. [28,29]).
On the other hand, it is known that a forward parabolic equation can be solved by running back the time from a future time, which gives rise to another set of relations between fundamental solutions and the transition probability density functions. If T > 0 and f (x,t) is a function, then define f T (x,t) = f (x, (T − t) + ). Lemma 2.5. Let T > 0. Then
p a T ;b T (T − t, x, T − τ, ξ ) = Γ a;b (x,t; ξ , τ) = Γ ⋆ a;b,c (ξ , τ; x,t) (2.22)
for all 0 ≤ τ < t ≤ T and x, ξ ∈ R n , whereb andc are given by (2.18) and (2.19) respectively.
Proof. Let Θ(x,t; ξ , τ) = p a T ;b T (T − t, x, T − τ, ξ ) for 0 ≤ τ < t ≤ T and ξ , x ∈ R d .
As a function of (x,t), p a T ;b T (t, x; τ, ξ ) (for t < τ) solves the backward parabolic equation, so that Θ solves the forward parabolic equation
L a T ;b T − ∂ ∂t Θ(x,t; ξ , τ) = 0 (2.23) for 0 ≤ τ < t ≤ T . Since b T (x, T − t) = b(x,t) and a T (x, T − t) = a(x,t)
for 0 ≤ t ≤ T , the previous equality is equivalent to the forward parabolic equation:
L a;b − ∂ ∂t Θ(x,t; ξ , τ) = 0 on R d × [τ, T ] for every τ ∈ (0, T ]. Suppose ϕ(ξ ) is continuous and bounded on R d , then R d Θ(x,t; ξ , τ)ϕ(ξ )dξ = R d p a T ;b T (T − t, x, T − τ, ξ )ϕ(ξ )dξ → ϕ(x) as T − t ↑ T − τ, i.e.
as t ↓ τ. By the uniqueness of the fundamental solution Θ(x,t; ξ , τ) coincides with Γ a;b (x,t; ξ , τ), and therefore the conclusion follows immediately.
Remark 2.6. For the case where a i j (x,t) = δ i j and b(x,t) is divergence-free, thenb = −b andc = 0, so that p b T (T − t, x, T − τ, ξ ) = Γ b (x,t; ξ , τ) = Γ ⋆ −b (ξ , τ; x,t) = p −b (τ, ξ ,t, x) (2.24)
for all 0 ≤ τ < t ≤ T and x, ξ ∈ R n . Therefore for this case the fundamental solution Γ ⋆ −b (ξ , τ; x,t) is a probability density in x and ξ respectively.
In general by definition Γ ⋆ a;b,c (ξ , τ; x,t) is a probability density in the variable ξ , while in general it is not a probability density function with respect to the variable x, thus can not be a transition probability density function of an diffusion. A sufficient condition to ensure that Γ ⋆ a;b,c (ξ , τ; x,t) is a transition probability function (with respect to x) of some diffusion is given in the next section.
The Feynman-Kac formula
The Feynman-Kac formula is a functional integration representation to the solutions f i (x,t) of a backward parabolic equation:
L a;b + ∂ ∂t f i (x,t) + n ∑ j=1 q i j (x,t) f j (x,t) = 0 in R d × [0, ∞),(3.1)
where i = 1, . . . , n, and n is some positive integer. For each ψ ∈ C([0, ∞), R d ), let Q(τ, ψ,t) = (Q i j (τ, ψ,t)) i, j≤n be the solution to the ordinary differential equations:
d dt Q i j (t) = Q i k (t)q k j (ψ(t),t), Q i j (τ) = δ i j (3.2)
for t ≥ τ, if the solution exists and is unique.
f i (η, τ) = Ω Q i j (τ, ψ;t) f j (ψ(t),t)P η,τ a;b (dψ) (3.3)
for t ≥ τ and η ∈ R d .
As a consequence we have the following Lemma 3.2. Under the same regularity assumptions on a(x,t) and b(x,t) as in Lemma 3.1, and suppose c(x,t) is bounded and continuous. Then
Γ ⋆ a;b,c (ξ , τ; x,t) = p a;b (τ, ξ ,t, x) Ω C(τ, ψ;t)P ξ ,τ→x,t a;b (dψ) (3.4) for x, ξ ∈ R d and t > τ ≥ 0, where C(τ, ψ; r) for each ψ ∈ C([0, ∞); R d )
is the unique solution to the ordinary differential equation:
C(τ, ψ;t) = 1 + τ∧t τ C(τ, ψ; s)c(ψ(s), s)ds (3.5)
for t ≥ 0.
Proof. Let T > 0 and f be a C 2,1 solution to the backward parabolic equation:
L a;b,c + ∂ ∂t f = 0 such that f (x,t) → f 0 (x) as t ↑ T .
The previous equation may be rewritten as the following:
L a;b + ∂ ∂t f (x,t) + c(x,t) f (x,t) = 0 so that, according to Lemma 3.1 f (ξ , τ) = Ω C(τ, ψ; T ) f (ψ(T ), T )P ξ ,τ a;b (dψ)
for 0 ≤ τ < T , and therefore
f (ξ , τ) = R d p a;b (τ, ξ , T, x) Ω C(τ, ψ; T )P ξ ,τ a;b [ dψ| ψ(T ) = x] f (x, T )dx which yields that Γ ⋆ a;b,c (ξ , τ; x, T ) = p a;b (τ, ξ , T, x) Ω C(τ, ψ; T )P ξ ,τ a;b [ dψ| ψ(T ) = x] .
The proof is complete.
Lemma 3.3. Let T > 0. Then p a T ,b T (T − t, x, T − τ, ξ ) = p a;b (τ, ξ ,t, x) ΩC (τ, ψ;t)P ξ ,τ→x,t a;b (dψ)
for any 0 ≤ τ < t ≤ T and x, ξ ∈ R d , whereb andc are given by (2.18) and (2.19) respectively, and C(τ, ψ;t) = 1 + τ∧t τC (τ, ψ; s)c(ψ(s), s)ds for t ≥ 0 and ψ ∈ C([0, ∞); R d ).
Proof. We have τ; x,t) for t > τ ≥ 0, so the corollary follows immediately from Lemma 3.2.
p a T ,b T (T − t, x, T − τ, ξ ) = Γ ⋆ a;b,c (ξ ,
We may apply Lemma 3.3 to the L a;b -diffusion, whose transition probability density function is denoted by h a;b (t, x, τ, ξ ) for 0 ≤ t < τ. The formal adjoint operator of L a;b is given by
L ⋆ a;b = ν d ∑ i, j=1 ∂ ∂ x j a i j ∂ ∂ x i − d ∑ i=1 b i ∂ ∂ x i − ∇ · b
and thereforec = ∇ · b. Therefore we have the following consequence.
Lemma 3.4. Suppose ∇ · b(·,t) = 0 in the distribution sense for every t ≥ 0. Then for every T > 0
h a T ;b T (T − t, x, T − τ, ξ ) = h a;−b (τ, ξ ,t, x) (3.6) for 0 ≤ τ < t ≤ T and ξ , x ∈ R d .
Proof. Apply Lemma 3.3 to the differential operator L a;b = L a;b . Since ∇ · b = 0 identically for every t, c = 0, and therefore the gauge processC ≡ 1, the equality follows immediately.
Diffusion bridges
In this section we establish a duality among the conditional laws of the L a;b -diffusions. Let Q η,τ a;b be the distribution of the L a;b -diffusion started from η at τ ≥ 0. If T > 0 then Q η,0→ζ ,T a;b denotes the conditional law of the L a;b -diffusion started from η at instance 0 and arrived at ζ at time T . where τ T is the time reverse which sends ψ to τ T ψ, for ψ ∈ C([0, T ]; R d ), that is, τ T ψ(t) = ψ(T − t) for t ∈ [0, T ].
Proof. It is known that the conditional law of Q η,0 a;b given ψ(T ) = ζ , where η and ζ are the initial and final points, according to (14.1) in [14], denoted by Q η,0→ζ ,T a;b , is also Markovian (time non-homogeneous) whose transition density function is given by
q(τ, ξ ,t, x) = h a;b (τ, ξ ,t, x)h a;b (t, x, T, ζ ) h a;b (τ, ξ , T, ζ ) (4.2) that is q(τ, ξ ,t, x) = Q η,0→ζ ,T a;b [ψ ∈ Ω : ψ(t) ∈ dx|ψ(τ) = ξ ] for 0 < τ < t ≤ T . Let 0 = t 0 < t 1 < · · · < t n < t n+1 = T . Then Q η,0→ζ ,T a;b
w t 0 ∈ dx 0 , w t 1 ∈ dx 1 , · · · , w t n ∈ dx n , w t n+1 ∈ dx n+1 equals the measure q(0, η,t 1 , x 1 ) · · · q(t i−1 , x i−1 ,t i , x i ) · · · q(t n , x n , T, ζ )dx 1 · · · dx n .
By using (4.2), this measure has a pdf h a;b (0, η,t 1 , x 1 ) · · · h a;b (t i−1 , x i−1 ,t i , x i ) · · · h a;b (t n , x n , T, ζ ) h a;b (0, η, T, ζ ) with respect to the measure δ η (dx 0 )dx 1 · · · dx n δ ζ (dx n+1 ). According to Lemma 3.4, the previous pdf equals
h a T ;−b T (0, ζ , T − t n , x n ) · · · h a T ;−b T (T − t i , x i , T − t i−1 , x i−1 ) · · · p a T ;−b T (T − t 1 , x 1 , T, η) p a T ;−b T (0, ζ , T, η)
and we therefore conclude that
Q η,0→ζ ,T a;b w t 0 ∈ dx 0 , w t 1 ∈ dx 1 , · · · , w t n ∈ dx n , w t n+1 ∈ dx n+1 coincides with Q ζ ,0→η,T a T ;−b T [w T −t n ∈ dx n , · · · , w T −t 1 ∈ dx 1 , w T ∈ dx 0 , w 0 ∈ dx n+1 ] .
Feynman-Kac formula for forward parabolic equation
Let us begin with the following general construction of the gauge functional. Given q(x,t) = (q i j (x,t)) i, j≤n , an n × n square-matrix valued function defined for x ∈ R d for t ≥ 0. For each T > 0 and each continuous path ψ ∈ C([0, T ]; R d ), consider the following ordinary differential equation
d dt Q i j (t) = n ∑ k=1 Q i k (t)q k j (ψ(t), (T − t) + ), Q i j (0) = δ i j (5.1)
which is linear in Q. The solution depends on T and on ψ as well, so it is denoted by Q(ψ, T ;t). By definition
Q i j (ψ, T ;t) = δ i j + t 0 n ∑ k=1 Q i k (ψ, T ; s)q k j (ψ(s), (T − s) + )ds (5.2)
for t ∈ [0, T ], where i, j = 1, · · · , n. The time reversal operator τ T , we recall, on C([0, T ]; R d ) maps ψ to τ T ψ(t) = ψ(T − t). Therefore, if we substitute ψ by τ T ψ and t by T − t we obtain
Q i j (τ T ψ, T ; T − t) = δ i j + T −t 0 n ∑ k=1 Q i k (τ T ψ, T ; s)q k j (ψ(T − s), (T − s) + )ds = δ i j + T t n ∑ k=1 Q i k (τ T ψ, T ; T − s)q k j (ψ(s), s)ds.
Hence we have the following elementary fact.
;t) = Q(τ T ψ, T ; T − t) for t ∈ [0, T ].
Then G is the solution to the ordinary differential equation
d dt G i j (t) = − n ∑ k=1 G i k (t)q k j (ψ(t),t), G i j (T ) = I. (5.3) Moreover Q(ψ, T ;t) = G(τ T ψ, T ; T − t) for t ∈ [0, T ].
Now we are in a position to study the initial value problem of the forward parabolic equation:
L − ∂ ∂t w i (x,t) + n ∑ j=1 q i j (x,t)w j (x,t) + f i (x,t) = 0 (5.4)
subject to the initial value w i (x, 0) = w i 0 (x), where i = 1, . . . , n, where L = L a;b (for this case we assume that a i j (x,t) are uniformly continuous) or L = L a;b , for both cases, b i (x,t) are bounded and Borel measurable.
Lemma 5.2. Let P η denote the distribution of the L a T ;b T -diffusion or the L a T ;b T -diffusion, depending on whether L = L a;b or L = L a;b , started from η ∈ R d at time 0. Then
w i (x, T ) = P x n ∑ j=1 Q i j (ψ, T ; T )w j 0 (ψ(T )) + T 0 Q i j (ψ, T ;t) f j (ψ(t), T − t)dt . (5.5)
Proof. This is the classical Feynman-Kac formula applying to w(x, T − t). t)) i≤n is a solution to the parabolic system
Theorem 5.3. Suppose b is divergence free on R d , i.e. ∇ · b = 0 in the distribution sense. Suppose w(x,t) = (w i (x,L a;b − ∂ ∂t w i (x,t) + n ∑ j=1 q i j (x,t)w j (x,t) + f i (x,t) = 0 (5.6)
subject to the initial value w(x, 0) = w 0 (x), where i = 1, . . . , n. Then
w i (x, T ) = n ∑ j=1 R d Ω G i j (ψ, T ; 0)Q ξ ,0→x,T a;−b (dψ) w j 0 (ξ )h a;−b (0, ξ , T, x)dξ + n ∑ j=1 R d Ω T 0 G i j (ψ, T ;t) f j (ψ(t),t)dtQ ξ ,0→x,T a;−b (dψ) h a;−b (0, ξ , T, x)dξ . (5.7)
where G(ψ, T ;t) = G(t) (for every T > 0 and every continuous path ψ) is the unique solution to the ordinary differential equation:
d dt G i j (t) = −G i k (t)q k j (ψ(t),t), G i j (T ) = δ i j . (5.8)
Proof. According to the previous Lemma 5.2
w i (x, T ) = n ∑ j=1 Q x,0 Q i j (ψ, T ; T )w j 0 (ψ(T )) + T 0 Q x,0 Q i j (ψ, T ;t) f j (ψ(t), T − t) dt = n ∑ j=1 R d Q x,0 Q i j (ψ, T ; T )w j 0 (ψ(T )) ψ(T ) = ξ Q x,0 [ψ(T ) ∈ dξ ] + T 0 Q x,0 Q i j (ψ, T ;t) f j (ψ(t), T − t) ψ(T ) = ξ Q x,0 [ψ(T ) ∈ dξ ] dt = n ∑ j=1 R d Q x,0→ξ ,T a T ;b T Q i j (ψ, T ; T ) w j 0 (ξ )h a T ;b T (0, x, T, ξ )dξ + n ∑ j=1 T 0 Q x,0→ξ ,T a T ;b T Q i j (ψ, T ;t) f j (ψ(t), T − t) h a T ;b T (0, x, T, ξ )dξ dt. Since b(x,t) is divergence-free, so that L ⋆ a;b = L a;−b . By Theorem 4.1 Q ξ ,0→η,T a T ;b T • τ T = Q η,0→ξ ,T a;−b ,
together with the relation that Q(ψ, T ;t) = G(τ T ψ, T ; T − t), we obtain
w i (x, T ) = n ∑ j=1 R d Q x,0→ξ ,T a T ;b T G i j (τ T ψ, T ; 0) w j 0 (ξ )h a T ;b T (0, x, T, ξ )dξ + T 0 Q x,0→ξ ,T a T ;b T G i j (τ T ψ, T ; T − t) f j (τ T ψ(T − t), T − t) h a T ;b T (0, x, T, ξ )dξ dt = n ∑ j=1 R d Q ξ ,0→x,T a;−b G i j (ψ, T ; 0) w j 0 (ξ )h a T ;b T (0, x, T, ξ )dξ + n ∑ j=1 Q ξ ,0→x,T a;−b T 0 G i j (ψ, T ; T − t) f j (ψ(T − t), T − t)dt h a T ;b T (0, x, T, ξ )dξ .
Finally according to Lemma 3.4
h a T ;b T (T − t, x, T − τ, ξ ) = h a;−b (τ, ξ ,t, x),
which yields that h a T ;b T (0, x, T, ξ ) = h a;−b (0, ξ , T, x) for 0 ≤ τ < t ≤ T . Substituting this equation into the representation for w i we obtain (5.7).
Boundary value problems
We first recall the Feynman-Kac formula for solutions of the initial and boundary value problem:
L a;b + ∂ ∂t f j (x,t) + q j k (x,t) f k (x,t) = g j (x,t) in D × [0, ∞) (6.1)
subject to the Dirichlet boundary condition that f j (x,t) = β j (x) for x ∈ ∂ D and t > 0, (6.2) where j = 1, · · · , n.
Lemma 6.1. Assume that a is uniformly continuous and b is bounded and Borel measurable. For each ψ ∈ C([0, ∞); R d ) denote Q(ψ,t) the solution to the (linear) ordinary differential equation
d dt Q i j (t) = Q i k (t)1 D (ψ(t))q k j (ψ(t),t), Q i j (0) = δ i j (6.3) and ζ D (ψ) = inf{t ≥ 0 : ψ(t) / ∈ D}. (6.4)
Then the following integration representation holds:
f i (η, 0) = Ω Q i j (ψ,t) f j (ψ(t),t)1 {ζ D (ψ)>t} P η (dψ) + Ω Q i j (ψ, ζ D (ψ))β j (ψ(ζ D (ψ)))1 {ζ D (ψ)≤t} P η (dψ) (6.5) − Ω t∧ζ D (ψ) 0 Q i j (ψ, s)g j (ψ(s), s)ds P η (dψ) (6.6)
for all t ≥ 0 and η ∈ R d , where i = 1, · · · , n and P η = P η,0 a;b for simplicity. Proof. For a slightly different version of Feynman-Kac formula and its proof, see for example [16,Theorem 2.3. page133]. For completeness we include a proof here. We may assume that q i j (x,t) = 0 for x / ∈ D otherwise we may use 1 D (x)q i j (x,t) instead. Let X be the weak solution of the stochastic differential equation dX k (t) = b k (X (t),t)dt + √ 2νσ k l (X (t),t)dB l (t). for k = 1, . . . , d, with initial X (0) = η. Consider the linear ordinary differential equation:
d dt Q i j (t) = Q i k (t)q k j (X (t),t), Q i j (0) = δ i j . (6.7)
Suppose ρ is smooth such that ρ(x) = 1 for x ∈ D, andf i (x,t) = ρ(x) f i (x,t). Thenf i (x,t) (i = 1, . . . , n) are C 2,1 -functions on R d × [0, ∞) and
∂ ∂tf j + L a;bf j + q j kf k = F j in R d × [0, ∞). (6.8)
Consider M i t = Q i j (t) f j (X (t),t) for t ≥ 0. Then, according to Itô's formula
M i t = M i 0 + t 0 Q i j (s) √ 2νσ k l (X (s), s) ∂f j ∂ x k (X (s), s)dB l s + t 0 Q i j (s) ∂ ∂ sf j + L a;bf j + q j kf k (X (s), s)ds. (6.9) Let ζ D = inf{t ≥ 0 : X (t) / ∈ D}
be the first time the diffusion leaves the region D. Then
E M i t∧ζ D = E M i 0 + E t∧ζ D 0 Q i j (s) ∂ ∂ s f j + L a;b f j + q j k f k (X (s), s)ds. (6.10)
Since f j solve the differential equations (6.8), so that
E M i t∧ζ D = E M i 0 + E t∧ζ D 0 Q i j (s)F j (X (s), s)ds . (6.11) Since M i 0 = f i (η, 0) and E M i t∧ζ D = E M i ζ D : t ≥ ζ D + E M i t : t < ζ D = E Q i j (ζ D ) f j (X (ζ D ), ζ D ) : t ≥ ζ D + E Q i j (t) f j (X (t),t) : t < ζ D = E Q i j (t) f j (X (t),t) : t < ζ D + E Q i j (ζ D )β j (X (ζ D )) : t ≥ ζ D
where the last equality follows from the Dirichlet boundary condition: X (ζ D ) ∈ ∂ D on ζ D < ∞, and f j (x,t) = β j (x) for x ∈ ∂ D. Substituting this equality into (6.11),
E Q i j (t) f j (X (t),t) : t < ζ D = f i (η, 0) + E t∧ζ D 0 Q i j (s)F j (X (s), s)ds − E Q i j (ζ D )β j (X (ζ D )) : t ≥ ζ D
The functional integration representation follows by an approximating procedure.
We next establish a forward Feynman-Kac formula. For every ψ ∈ C([0, ∞); R d ) and T > 0,Q(ψ, T ;t) denotes the solution to the following linear ordinary differential equations
d dtQ i j (ψ, T ;t) = −Q i k (ψ, T ;t)1 D (ψ(t))q k j (ψ(t),t),Q i j (ψ, T ; T ) = δ i j (6.12)
for i, j = 1, · · · , n.
Theorem 6.2. Suppose b(x,t) is bounded, Borel measurable and ∇ · b = 0 in the distribution sense on R d . Let w(x,t) be the solution to Cauchy's initial and boundary problem of the following parabolic system:
L a;b − ∂ ∂t w j (x,t) + n ∑ k=1 q j k (x,t)w k (x,t) = g j (x,t) in D (6.13)
subject to the initial and boundary conditions that w j (x, 0) = w j 0 (x) for x ∈ D, and w j (x,t) = β j (x) for x ∈ ∂ D, t > 0 (6.14)
for j = 1, · · · , n. Then
w i (η, T ) = D ΩQ i j (ψ, T ; 0)1 {ζ D (ψ)>T } Q ξ ,0→η,T a;−b (dψ) w j 0 (ξ )h(0, ξ , T, η)dξ + R d ΩQ i j (ψ, λ T,∂ D (ψ))β j (ψ(λ T,∂ D (ψ)))1 {ζ D (ψ)≤T } Q ξ ,0→η,T a;−b (dψ) h(0, ξ , T, η)dξ (6.15) − T 0 R d ΩQ i j (ψ, T ; s)g j (ψ(s), s)1 {ζ D (θ s ψ)>T−s} Q ξ ,0→η,T a;−b (dψ) h(0, ξ , T, η)dξ ds (6.16)
for every η ∈ D and T > 0, where h(τ, ξ ,t, η) denotes h a;−b (τ, ξ ,t, η) for simplicity,
λ T,∂ D (ψ) = sup {t ∈ [0, T ] : ψ(t) ∈ ∂ D} and θ s : Ω → Ω , θ s ψ(t) = ψ(t + s) for s;t ≥ 0.
Proof. Let T > 0 and f i (x,t) = w i (x, T − t), so that u i satisfy the following parabolic equations
L a T ;b T + ∂ ∂t f j (x,t) + n ∑ k=1 q j k,T (x,t) f k (x,t) = g j T (x,t) in D.
Then according to (6.6)
w i (η, T ) = Ω Q i j (ψ, T )w j 0 (ψ(T ))1 {ζ D (ψ)>T } Q η a T ;b T (dψ) + Ω Q i j (ψ, ζ D (ψ))β j (ψ(ζ D (ψ)))1 {ζ D (ψ)≤T} Q η a T ;b T (dψ) (6.17) − T 0 Ω Q i j (ψ, s)g j (ψ(s), T − s)1 {s<T ∧ζ D (ψ)} Q η a T ;b T (dψ) ds (6.18)
where Q η a T ;b T is the law of the L a T ;b T -diffusion started from η at instance 0, and
Q i j (ψ,t) = δ i j + t 0 Q i k (ψ, s)1 D (ψ(s))q k j (ψ(s), T − s)ds.
Replace t by T − t and ψ by τ T ψ one obtains that
Q i j (τ T ψ, T − t) = δ i j + T −t 0 Q i k (τ T ψ, s)1 D (ψ(T − s))q k j (ψ(T − s), T − s)ds = δ i j − t T Q i k (τ T ψ, T − s)1 D (ψ(s))q k j (ψ(s), s)ds.
Hence, by settingQ(ψ,
T ;t) = Q i j (τ T ψ, T − t), one deduce that Q i j (ψ, T ;t) = δ i j − t TQ i k (ψ, T ; s)1 D (ψ(s))q k j (ψ(s), s)ds. (6.19) Moreover Q i j (τ T ψ,t) =Q(ψ, T ; T − t) for every t ∈ [0, T ].
We rewrite (6.18) by conditioning on the values of the diffusion at T , to obtain that
w i (η, T ) = R i I (η, T ) + R i B (η, T ) − R i N (η, T ) (6.20)
where the first term
R i I (η, T ) = D Ω Q i j (ψ, T )1 {ζ D (ψ)>T} Q η,0→ξ ,T a T ;b T (dψ) w j 0 (ξ )h a T ;b T (0, η, T, ξ )dξ , the boundary term R i B (η, T ) = Ω Q i j (ψ, ζ D (ψ))β j (ψ(ζ D (ψ)))1 {ζ D (ψ)≤T } Q η a T ;b T (dψ)
and finally the inhomogeneous term
R i N (η, T ) = T 0 R d Ω Q i j (ψ, s)g j (ψ(s), T − s)1 {s<ζ D (ψ)} Q η,0→ξ ,T a T ;b T (dψ) h a T ;b T (0, η, T, ξ )dξ ds. Since L ⋆ a;b = L a;−b under our assumptions, Q ξ ,0→η,T a;−b • τ T = Q η,0→ξ ,T a T ;b T and (cf. Lemma 3.4)
h a T ;b T (0, η, T, ξ ) = h(0, ξ , T, η). (6.21) Thanks to these dualities, we are able to rewrite the three terms on the right-hand side of the representation (6.20) for w i . Indeed the first term
R i I (η, T ) = D Ω Q i j (ψ, T )1 {ζ D (ψ)>T } Q η,0→ξ ,T a T ;b T (dψ) w j 0 (ξ )h a T ;b T (0, η, T, ξ )dξ = D ΩQ i j (ψ, T ; 0)1 {ζ D (τ T ψ)>T } Q ξ ,0→η,T a;−b (dψ) w j 0 (ξ )h(0, ξ , T, η)dξ .
We notice that, if ξ , η ∈ D, then under the conditional law
Q ξ ,0 [ dψ| ψ(T ) = η], ζ D (τ T ψ) > T is equi- valent to that ψ(T − t) ∈ D for all t ∈ [0, T ]
, which in turn is equivalent to that ψ(t) ∈ D. While the last is equivalent to that ζ D (ψ) > T . Therefore
R i I (η, T ) = D ΩQ i j (ψ, T ; 0)1 {ζ D (ψ)>T} Q ξ ,0→η,T a;−b (dψ) w j 0 (ξ )h(0, ξ , T, η)dξ . (6.22)
To handle the second term which arises from the inhomogeneous boundary data β (x). By using the conditional law we may rewrite
R i B (η, T ) = R d Ω Q i j (ψ, ζ D (ψ))β j (ψ(ζ D (ψ)))1 {ζ D (ψ)≤T } Q η,0→ξ ,T a T ;b T (dψ)h(0, ξ , T, η)dξ = R d Ω Q i j (τ T ψ, ζ D (τ T ψ))β j (τ T ψ(ζ D (τ T ψ)))1 {ζ D (τ T ψ)≤T } Q ξ ,0→η,T a;−b (dψ)h(0, ξ , T, η)dξ ,
where the complication arises due to the integral against the variable ξ takes place over the whole space R d . Observe that ζ D (τ T ψ) ≤ T if and only if there is t 0 ∈ [0, T ] such that ψ(t 0 ) ∈ ∂ D, which is therefore equivalent to that ζ D (ψ) ≤ T , and
ζ D (τ T ψ) = inf {t ≥ 0 : ψ(T − t) ∈ ∂ D} = inf {T − s ≥ 0 : ψ(s) ∈ ∂ D} = T − sup {s : 0 ≤ s ≤ T s.t. ψ(s) ∈ ∂ D} . Therefore Q i j (τ T ψ, ζ D (τ T ψ)) =Q i j (ψ, λ T,∂ D (ψ)) on {ζ D (ψ) ≤ T } and β j (τ T ψ(ζ D (τ T ψ)))1 {ζ D (τ T ψ)≤T} = β j (ψ(λ T,∂ D (ψ)))1 {ζ D (ψ)≤T } .
By using these relations we can rewrite the boundary term
R i B (η, T ) = R d ΩQ i j (ψ, λ T,∂ D (ψ))β j (ψ(λ T,∂ D (ψ)))1 {ζ D (ψ)≤T} Q ξ ,0→η,T a;−b (dψ) h(0, ξ , T, η)dξ . (6.23)
Finally let us consider the third term arising from the inhomogeneous term in the parabolic system. In fact, by using duality we may rewrite
R i N (η, T ) = T 0 R d Ω Q i j (ψ, s)g j (ψ(s), T − s)1 {s<ζ D (ψ)} Q η,0→ξ ,T a T ;b T (dψ)h a T ;b T (0, η, T, ξ )dξ ds = T 0 R d Ω Q i j (τ T ψ, s)g j (τ T ψ(s), T − s)1 {s<ζ D (τ T ψ)} Q ξ ,0→η,T a;−b (dψ)h(0, ξ , T, η)dξ ds = T 0 R d ΩQ i j (ψ, s)g j (ψ(s), s)1 {ζ D (θ s ψ)>T −s} Q ξ ,0→η,T a;−b (dψ)h(0, ξ , T, η)dξ ds.
Putting these equations together we deduce the functional integration representation.
In particular we have the following forward Feynman-Kac formula.
Theorem 6.3. Suppose ∇ · b = 0 on R d in the distribution sense. Let w i (x,t) be the solution to Cauchy's initial problem of the parabolic system:
L a;b − ∂ ∂t w j (x,t) + n ∑ k=1 q j k (x,t)w k (x,t) = 0 in D × [0, T ] (6.24)
subject to the initial and Dirichlet boundary conditions:
w j (x, 0) = w j 0 (x) for x ∈ D, and w j (x,t) = 0 for x ∈ ∂ D, t > 0 (6.25)
where j = 1, · · · , n. Then
w i (η, T ) = D ΩQ i j (ψ, T ; 0)1 {ζ D (ψ)>T } Q ξ ,0→η,T a;−b (dψ) w j 0 (ξ )h a;−b (0, ξ , T, η)dξ for every η ∈ D.
Remark 6.4. We would like to emphasize the assumptions on a i j (x,t) and b i (x,t), both are defined for all x ∈ R d and t ≥ 0. In order to ensure the previous functional integration representation to be valid, we assume that b i (x,t) is bounded (this condition can be weaken) and Borel measurable, but the most crucial assumption is that b(x,t) is divergence free on R d (not only on D !) in the distribution sense. The probability measure used in the representation is the distribution associated with the differential operator
L a;−b = ν d ∑ i, j=1 ∂ ∂ x j a i j (x,t) ∂ ∂ x i − d ∑ i=1 b i (x,t) ∂ ∂ x i
which is the formal adjoint operator of L a;b .
Navier-Stokes equations
In this section we apply the forward Feynman-Kac formula to derive a stochastic representation for solutions of the Navier-Stokes equations in domains via their vortex dynamics, cf. [11,29]. We begin with the following elementary fact.
Lemma 7.1. Suppose b(x) is a C 1 -vector field in D , and ∇ · b(x) = 0 for all x ∈ D. Suppose b(x) = 0 for x ∈ ∂ D. Extend b(x) to all x ∈ R d by setting b(x) = 0 for x / ∈ D. Then b(x) is divergence-free in distribution sense on R d .
Proof. According to assumptions, ∂ D is a smooth manifold of d − 1 dimensions, so has zero Lebesgue measure. Therefore ∇ · b = 0 a.e. on R d . Suppose ϕ is a smooth function on R d with a compact support. Then ∇ · (ϕb) = ∇ϕ · b + ϕ∇ · b = ∇ϕ · b a.e.
Hence R d ∇ϕ · b = R d ∇ · (ϕb) = ∂ D ϕb · ν + ∂ D c ϕb · ν = 0
where the last equality follows from the assumption that b(x) = 0 for x ∈ ∂ D. Therefore ∇ · b = 0 on R d in the distribution sense.
Recall that the velocity u(x,t) and the pressure P(x,t) of an incompressible fluid flow constrained in D are solutions of the Navier-Stokes equations
∂ ∂t u + (u · ∇)u − ν∆u − ∇P = 0 in D × [0, ∞) (7.1)
and
∇ · u = 0 in D × [0, ∞),(7.2)
subject to the no-slip condition u(x,t) = 0 for x ∈ ∂ D and t ≥ 0.
τ i j = ν ∂ u i ∂ x j + ∂ u j ∂ x i
applied immediately to the boundary surface. In fact it can be demonstrated that the normal part ω ⊥ of the boundary vorticity ω| ∂ D , vanishes identically along the boundary surface, while its tangential vorticity ω along the boundary coincides with the normal shearing stress τ ⊥ up to a numerical factor ν −1 . The calculation of the boundary vorticity is an important problem which will not be discussed in this paper in detail. However let us point out that the normal stress applied immediately to the wall is a fluid dynamical quantity to be measured or to be controlled, and can be calculated approximately by using boundary layer equation, cf. [38, Chapter 6].
Two dimensional flows
Let us first establish a representation for 2D flows. In dimension two, the vorticity ω can be identified with the scalar function ∂ ∂ x 1 u 2 − ∂ ∂ x 2 u 1 and ω is a solution to the vorticity transport equation
∂ ∂t ω + (u · ∇)ω − ν∆ω = 0 in D × [0, ∞),(7.4)
where the boundary value of ω along the wall ∂ D may be identified with the stress of the fluid flow immediately injected to the wall, cf. [38,27,43]. Let us denote the stress along the wall by σ , whose explicit expression need to be calculated in terms of the geometry of ∂ D as well, so they are must be treated case by case.
Since ∇ · u = 0, the velocity field may be recovered in terms of ω by solving the Poisson equations
∆u 1 = − ∂ ω ∂ x 2 , ∆u 2 = ∂ ω ∂ x 1 (7.5)
subject to the Dirichlet boundary condition u 1 (x,t) = u 2 (x,t) = 0 for x ∈ ∂ D. Hence according to Green formula u i (x,t) =
D K i (x, η)ω(η,t)dη = R 2 K i D (x, η)ω(η,t)dη (7.6) where K i D (x, η) = 1 D (η)K i (x, η) (7.7)
and the integral kernel K i depend on the region D only.
Theorem 7.2. Let u be extended to be a vector field on R 2 × [0, ∞) such that u(·,t) is divergence free in the distribution sense on the whole plane R 2 . Let X (ξ ,t) (for ξ ∈ R 2 and t ≥ 0) be the solution to the stochastic differential equation:
dX (t) = u(X (t),t)dt + √ 2νdB(t), X (0) = ξ . (7.8)
Then u(x,t) = D ω 0 (ξ )E [K D (x, X (ξ ,t))J 1 (ξ , X (ξ ,t),t)] dξ + R 2 E K i D (x, X (ξ ,t))J 2 (ξ , X (ξ ,t),t) dξ (7.9)
for every x ∈ D and t > 0, where J 1 (ξ , η,t) = P [ ζ D (X (ξ , ·)) > t| X (ξ ,t) = η] (7.10) and J 2 (ξ , η,t) = E σ X (ξ , λ t,∂ D (X (ξ , ·)) 1 {ζ D (X(ξ ,·))≤t} X (ξ ,t) = η (7.11) for any ξ , η ∈ R 2 and t > 0.
Proof. The vorticity transport equation may be formulated in terms of L −u , that is,
∂ ∂t − L −u ω = 0 in D × [0, ∞). (7.12)
Since u(·,t) is divergence free in the distribution sense for every t on R 2 , therefore we may apply Theorem 6.3 to ω, to obtain ω(η,t) = D ω 0 (ξ )P ξ ,0→η,t [ζ D (ψ) > t] h(0, ξ ,t, η)dξ + R 2 Ω σ (ψ(λ t,∂ D (ψ)))1 {ζ D (ψ)≤t} P ξ ,0→η,t (dψ) h(0, ξ ,t, η)dξ (7.13) for η ∈ D and t > 0, where P ξ ,0→η,T denotes the conditional distribution of the L u -diffusion started from ξ at time zero given ψ(t) = η, and h(τ, ξ ,t, η) is the transition probability density function of the L u -diffusion. Since the distribution of X (ξ ,t) is exactly P ξ ,0 , the conclusion therefore follows immediately.
Three dimensional flows
Next we consider an incompressible fluid flow with its velocity u = (u 1 , u 2 , u 3 ), constrained in a region D ⊂ R 3 . Therefore u(x,t) is a solution to the 3D Navier-Stokes equations (7.1, 7.2, 7.3). The vorticity ω i = ε i jk ∂ ∂ x j u k are solutions to the 3D vorticity transport equations
∂ ∂t ω i + (u · ∇)ω i − ν∆ω i − ∂ u i ∂ x j ω j = 0 in D × [0, ∞),(7.14)
where ν > 0 is the viscosity constant as usual, which can be written as
L −u − ∂ ∂t ω i + S i j ω j = 0,(7.15)
where
S i j = 1 2 ∂ u i ∂ x j + ∂ u j ∂ x i
is the symmetric stress tensor. The boundary value of ω along the wall ∂ D can be again identified with the stress tensor, denoted by β . Since ∇ · u = 0 in D so that ∆u = −∇ × ω in D and u satisfies the Dirichlet boundary condition along ∂ D. Therefore u(x,t) = − D H(x, η)∇ × ω(η,t)dη, (7.16) where H(x, y) is the Green function of D. By integration by parts we obtain u(x,t) = D K(x, η) × ω(η,t)dη = R 3 K D (x, η) × ω(η,t)dη (7.17) where K D (x, η) = 1 D (η)K(x, η). That is u i (x,t) = D ε ilk K l (x, η)ω k (η,t)dη. (7.18) It follows that S i j (x,t) = D K i,k j (x, η)ω k (η,t)dη = D K i,k D; j (x, η)ω k (η,t)dη, (7.19) where K i,k j (x, η) = ε ilk ∂ ∂ x j K l (x, η) + ε jlk ∂ ∂ x i K l (x, η) (7.20) and
K i,k D; j (x, η) = 1 D (η) ε ilk ∂ ∂ x j K l (x, η) + ε jlk ∂ ∂ x i K l (x, η) . (7.21)
We notice that the Green function G, the integral kernels K and K D are determined solely by the domain D ⊂ R 3 . Let u(x,t) be extended to be a divergence free (in the distribution sense) vector field on R 3 , and X (ξ ;t) andQ i j (x,t) are the solutions to the stochastic differential equations dX (ξ ;t) = u(X (ξ ;t),t)dt + √ 2νdB(t), X (ξ ; 0) = ξ (7.22) and d dsQ i j (ξ ,t; s) = −Q i k (ξ ,t; s)1 D (X (ξ ; s))S k j (X (ξ ; s), s),Q i j (ξ ,t;t) = δ i j (7.23)
for t > 0 and s ≥ 0. D ω l 0 (ξ )E ε i jk K j D (x, X (ξ ,t))J i l (ξ , X (ξ ,t),t) dξ X (ξ ,t))B i (ξ , X (ξ ,t),t) dξ (7.24) and S i j (x,t) = D ω l 0 (ξ )E K i,k D; j (x, η)J k l (ξ , X (ξ ,t),t) dξ + R 3 E K i,k D; j (x, η)B k (ξ , X (ξ ,t),t) dξ (7.25) for all x ∈ D and t ∈ (0, T ], where k = 1, 2, 3, J i j (ξ , η,t) = E Q i j (ξ ,t; 0)1 {ζ D (X(ξ ,·))>t} X (ξ ,t) = η (7.26) and B i (ξ , η,t) = E Q i j (ξ ,t; λ T,∂ D (X (ξ , ·)))1 {ζ D (X(ξ ,·))≤t} β j (X (ξ , λ T,∂ D (X (ξ , ·))) X (ξ ,t) = η . (7.27) Proof. The proof is similar to that of two dimensional case. According to Theorem 5.3 ω i (η, T ) = D J i l (ξ , η,t)ω l 0 (ξ )h u (0, ξ , T, η)dξ + R d B i (ξ , η,t)h u (0, ξ , T, η)dξ and the representation formula follows from (7.16) and the Fubini theorem immediately.
+ R 3 E ε i jk K j D (x,
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] | [] | In the Hubble Deep Field (HDF), twelve candidate sources of high-redshift (z>3.5) AGN activity have been identified. The color selection criteria were established by passing spectra of selected quasars and Seyfert galaxies (appropriately redshifted and modified for "Lyman forest" absorption), as well as stars, observed normal and starburst galaxies, and galaxy models for various redshifts through the filters used for the HDF observations. The actual identification of AGN candidates also involved convolving a Laplacian-of-Gaussian filter with the HDF images, thereby removing relatively flat galactic backgrounds and leaving only the point-like components in the centers. Along with positions and colors, estimated redshifts and absolute magnitudes are reported, with the candidates falling toward the faint end of the AGN luminosity function. One candidate has been previously observed spectroscopically, with a measured redshift of 4.02. The number of sources reported here is consistent with a simple extrapolation of the observed quasar luminosity function to magnitude 30 in B Johnson . Implications for ionization of the intergalactic medium and for gravitational lensing are discussed. | 10.1086/300626 | [
"https://arxiv.org/pdf/astro-ph/9810491v1.pdf"
] | 17,121,715 | astro-ph/9810491 | d519ceaefadca4f547008a16783e640c677ff043 |
Possible High-Redshift, Low-Luminosity AGN Activity in the Hubble Deep Field
arXiv:astro-ph/9810491v1 30 Oct 1998
R Michael Jarvis
Department of Astronomy
University of Michigan
48109Ann ArborMI
Gordon M Macalpine
Department of Astronomy
University of Michigan
48109Ann ArborMI
Possible High-Redshift, Low-Luminosity AGN Activity in the Hubble Deep Field
arXiv:astro-ph/9810491v1 30 Oct 1998Subject headings: cosmology: observations -galaxies: active -galaxies: distances and redshifts -techniques: image processing
In the Hubble Deep Field (HDF), twelve candidate sources of high-redshift (z>3.5) AGN activity have been identified. The color selection criteria were established by passing spectra of selected quasars and Seyfert galaxies (appropriately redshifted and modified for "Lyman forest" absorption), as well as stars, observed normal and starburst galaxies, and galaxy models for various redshifts through the filters used for the HDF observations. The actual identification of AGN candidates also involved convolving a Laplacian-of-Gaussian filter with the HDF images, thereby removing relatively flat galactic backgrounds and leaving only the point-like components in the centers. Along with positions and colors, estimated redshifts and absolute magnitudes are reported, with the candidates falling toward the faint end of the AGN luminosity function. One candidate has been previously observed spectroscopically, with a measured redshift of 4.02. The number of sources reported here is consistent with a simple extrapolation of the observed quasar luminosity function to magnitude 30 in B Johnson . Implications for ionization of the intergalactic medium and for gravitational lensing are discussed.
Introduction
The Hubble Deep Field (HDF) is a Director's Discretionary program on the Hubble Space Telescope, which provides the deepest image available for investigating galaxies at very high redshifts (see Williams et al. 1996). The field has an area of about 5 square arcminutes, and it was selected for having low extinction and no bright foreground sources. Combined images were obtained with the WFPC-2 camera through four filters: F300W, F450W, F606W, and F814W. For brevity, we call them U, B, V, and I, respectively. The net efficiency curves for the bandpasses are illustrated in Figure 1, taking into account both the filter throughput and the quantum efficiency of the CCD detector. Also shown is an example of a high-redshift quasar spectrum superimposed on the filter bandpasses.
The multicolor HDF images provide a unique opportunity for investigating the faint end of the AGN luminosity function near the epoch of galaxy formation, by identifying point sources within galaxies and with AGN-like colors. Finding pervasive, comparatively low-level AGN sources could be quite important for a variety of reasons. For instance, Steidel and Sargent (1989) postulated the existence of many low-luminosity AGN during the epoch of galaxy formation, in order to explain ionization of the intergalactic medium at high redshift (see Gunn and Peterson 1965). In this regard, Rees (1993) has pointed out that, if intrinsically bright quasars are responsible for intergalactic medium ionization, then they may influence or modulate the development of large-scale environmental inhomogeneities in the early Universe. On the other hand, the situation would be different for more smoothly distributed, comparatively low-level AGN sources. Also, Narayan (1989) noted that a possible gravitational lensing explanation of reported quasar-galaxy correlations could be facilitated by large numbers of faint AGN at high redshift.
Extrapolation of observed quasar surface densities (e.g., Zamorani Fig. 1.-The net throughput for each of the four passbands used in imaging the HDF. We use the notation U, B, V, and I for the filters. We also show a sample quasar spectrum to illustrate how the Lyα emission and Lyman "forest" absorption yield distinctive colors as they pass through the different filters.
et al. 1991) to magnitude 30 in B Johnson for the area of the HDF suggests the likelihood of finding approximately 3 to 30 objects. For a redshift of about 4, this apparent magnitude corresponds to an absolute magnitude of roughly -16, near the low end of the class 1 Seyfert galaxy luminosity function. Although this simple process involves a huge extrapolation in magnitude and thus should not be taken too seriously, it provides a crude first estimate of what one might expect.
We note that Elson et al. (1996) examined unresolved objects in the HDF with I < 28 and found no convincing very-high-redshift quasar candidates. Because of our search technique, involving convolution with a Laplacian-of-Gaussian filter, we were able to identify additional unresolved sources within host galaxy backgrounds. None of the AGN candidate sources reported here are in the Elson et al. list, as discussed in Section 4.
A number of groups have derived photometric redshifts for galaxies in the HDF. One of these studies, by Lanzetta et al. (1996), reported 54 galaxies with z = 3 -4 and 36 with z > 4, including 4 with z > 6. In our investigation, we did not find what we consider to be convincing evidence for AGN at z > 5. In addition, we note that, when we determined the colors for some of the previously reported very-high-z candidate galaxies using only their high S/N central regions, we obtained values consistent with relatively low-redshift galaxies. Only when we expanded the radius of integration for each galaxy, leading to very low S/N characteristics, did the galaxy colors move into the extremely high-redshift region of the color-color plane. We do not consider those colors to be reliable.
The results presented in this paper represent a continuation of work previously reported by Jarvis et al. (1996). Since that time, we have considerably refined our technique for picking out AGN candidates. As noted above and discussed below, we now use a Laplacian-of-Gaussian filter to find star-like sources within galaxies, whose colors are then investigated. This technique produces significantly more reliable identifications, compared with our old method of finding AGN within galaxies by simply looking for compact groups of pixels that have quasar-like colors.
Selection Criteria
Because the HDF was observed through four filters, up to three independent colors may be employed in the selection criteria. To determine what criteria are most useful for identifying AGN, we investigated the colors for many different objects: specifically stars, observed galaxies including starbursts, modeled galaxies, quasars, and Seyfert galaxies. Ultimately, we decided not to use the U filter in the bulk of this study and to concentrate on the B-V and V-I colors. The reasons for this decision are twofold. First, we could not get reliable spectral data in that passband for most of the objects of interest, and thus could not accurately determine the predicted U-B colors (for instance). Second, the stars from which we calibrated our point-source magnitudes (see below) were quite dim in U, so we could not get a very good fit for this bandpass, and thus the values for these magnitudes are not very reliable. Because of Lyman line and continuum absorption, we would expect weak U-band fluxes for high-z AGN sources. We retroactively test our candidates for this after the identification process.
For each type of object used in establishing the color selection criteria, we considered measured spectra whenever possible and theoretical or extrapolated spectra when necessary due to limited spectral coverage of the observed data. We integrated each spectrum weighted by the throughput for each filter. Then, effective AB magnitudes were calculated using m AB = log(F ) + constant, where F is the constant flux that would give the same integrated light as the actual spectrum, and the constant is determined from the published zero points of the WFPC-2 filters. A summary of the results may be seen in the color-color plot of Figure 2, and more information about the spectra is presented below.
Stars
For sample stars, we used actual spectra observed by Gunn and Stryker (1983). Their data include spectra of main sequence stars from O5 to M8 plus a range of giants and supergiants. The most important for this study are the K and M dwarfs since they begin to approach the region of color space where high-z AGN are found. Specifically, they have B-V colors approaching 2.
Observed Galaxies
For some of the galaxy colors, we used spectra of nearby galaxies observed by Kennicutt (1992). Because these data do not extend far enough toward the blue for our purposes, we added appropriate extrapolations toward shorter wavelengths, as derived from models by Worthey (1994). Artificially redshifting the spectra, we found their resulting colors for redshifts up to 0.3. Above this, the colors are probably not accurate (see, for example, Dorman et al. 1993), and modeled galaxies may be expected to provide more appropriate spectra. The solid line delimits the region above which z > 3.5 AGN candidates would lie. Note that this line excludes the highly-reddened starburst galaxy colors illustrated in Figure 3, which explains its shape.
Modeled Galaxies
For very high redshifts, we considered modeled (stellar population synthesis) galaxy spectra. We chose to use the Bruzual and Charlot (1993) constant star formation models for ages of 10 6 , 10 7 , 10 8 , and 10 9 years.
The spectra and colors of high-redshift objects are influenced by absorption due to intergalactic neutral hydrogen. Whereas most of the hydrogen is ionized, enough exists in the neutral state to cause significant Lyman α absorption (the Lyman α forest), along with weaker Lyman β, and Lyman continuum absorption. Using the technique originally employed by Oke and Korycansky (1982), Zuo et al. (1993) characterized this effect with three parameters: D A , D B and D C , corresponding to Lyman α, β and continuum (hereafter Lyα, Lyβ and Lyc) absorption, respectively. Their best fit for D A involves a formula that changes at some redshift which they call z B . In their predicted functional form for D A , there are 4 parameters: z B , γ 1 , γ 2 , and B. They did a least squares fit for these parameters based on 32 QSO spectra and obtained the following results:
for z < z B , ln(− ln(1 − D A )) = ln B + (γ 1 + 1) ln(1 + z); for z > z B , ln(− ln(1 − D A )) = ln B + (γ 1 − γ 2 ) ln(1 + z B ) + (γ 2 + 1) ln(1 + z),
where the best fit parameters are ln B = −6.61, z B = 3.11, γ 1 = 2.82, and γ 2 = 5.07.
One way to understand the physical meaning of D A is that the optical depth due to the Lyα forest is − ln(1 − D A ). Alternatively, 1 − D A is the average value of the observed flux between Lyβ and Lyα, divided by the continuum flux level just longward of Lyα. Whereas Lyα absorption has by far the dominant impact on the broadband colors, we have also included associated Lyβ and Lyc absorption at relevant frequencies.
Taking these absorption effects into account, the modeled galaxy spectra for each age were artificially shifted to redshifts between 0 and 5, with the resulting colors shown in Figure 2. Numbers inside the triangles indicate integral values of z.
Starburst Galaxies
We considered actual spectra of the starburst nucleus galaxies Mkn 357 and NGC 6090 (from the Astrophysics Data Facility IUE Archives), with necessary extrapolations toward longer wavelengths taken from the broad-band spectral energy distributions of Schmitt et al. (1997). Colors for each of these galaxies were plotted for incremental redshifts between 0 and 2 in Figure 2. We could not meaningfully consider redshifts above 2 due to lack of knowledge about the relevant spectral properties for wavelengths shortward of 1200Å.
We also investigated what colors could result if low to moderately-high redshift starburst galaxies have substantial internal dust obscuration. Figure 3 shows three additional examples for possible starburst galaxy spectral colors with incremental redshifts from 0 to 2. The black circles correspond to the NGC 6090 spectrum (with additional data from Schmitt et al.), reddened using Osterbrock's (1989) average reddening law with c = 0.5. The grey and open circles correspond to the same galaxy with c = 1.0 and 1.5, respectively. These are fairly extreme amounts of reddening, but not impossible for a starburst galaxy. The numbers in some circles are integral values of z.
Quasars
We used the 34 high-redshift (3.1 < z < 4.9) quasar spectra observed by Schneider et al. (1991a,b). A power law extrapolation of F ν ∝ ν −2 (Zheng et al. 1997) was inserted at the blue end where necessary. Since the observed spectra have effects of intergalactic hydrogen absorption for their actual redshifts, we removed what was already there using the above formulae and then put it back for the appropriate synthetic redshift in each case. On Figure 2, we plotted the quasars' colors as artificially redshifted to values of z within |∆z| ≤ 1 of the observed redshifts, with colors designated by isolated numbers at integral redshift values.
Seyfert Galaxies
We considered both class 1 and 2 Seyfert galaxies. Seyfert 1 spectra include Mkn 509, NGC 3783 and NGC 5548, as reported by Koratkar et al. (1997). For Seyfert 2 galaxies, we used spectra of Mkn 573 (see MacAlpine 1988) and Mkn 3 (from the Astrophysics Data Facility IUE Archive). In all cases, geocoronal Lyα emission was removed if necessary, an F ν ∝ ν −2 continuum extrapolation was added for λ < 1200Å, absorption effects of intergalactic hydrogen were included as discussed above, and colors were plotted for incremental redshifts between 0 and 5 in Figure 2.
The Seyfert 1 and 2 locations in the color-color plane do not differ substantially, and both are only moderately shifted to higher B-V and lower V-I compared with the quasars. The latter may be understood as a luminosity effect, whereby lower-luminosity AGN have higher line equivalent widths for a given redshift (Baldwin 1997) and also (possibly) partly as an evolution effect involving stronger line cores for lower-redshift objects (see Francis and Koratkar 1995). Figures 2 and 3, color selection criteria for identifying AGN-like spectra at redshifts from about 3.5 to 5 are: B-V > 1.9 and V-I < 1.5, or B-V > 3 and V-I > 1.5, as delineated by lines on the figures. Part of this region of color space may also be expected to contain z > 4 galaxies according to the Bruzual and Charlot models.
Based on
HDF Data Analysis
Since AGN are characterized as being galaxies with star-like sources in the centers, we convolved a second derivative filter with the images to enhance the contrast in the data before looking at colors. Specifically, we used a "Laplacian-of-Gaussian operator" (e.g., Jain et al. 1995), also known as a "sombrero" filter. It is equivalent to smoothing the image with a Gaussian filter, and then taking the Laplacian (∇ 2 , the two-dimensional second derivative) of the result. The scale size for the smoothing part of the function was taken to be the radius of the point spread function (PSF) for the HDF, which is optimal for identifying features of this size. The functional form of the operator is (2 − r 2 ) exp(−r 2 /2), where r is a radial distance measured in units of r 0 , the PSF radius. The value of r 0 was derived from 4 stars in the field and was found to be 1.33 pixels (0.053 arcsec). Williams et al. (1996). The lines shown have a slope of 1 and indicate where the stars should fall if the calibration were perfect.
The result of the operation described above is that the contrast of star-size features is greatly magnified. This includes features which are small in only one direction, like spiral arms. The smooth parts of the galaxies are reduced to noise, since the second derivative is essentially zero. Stars are converted into islands, tall peaks surrounded by negative valued troughs 1 . Thus, AGN may be expected to appear morphologically as stars at the centers of where galaxies appear in the unprocessed image.
Because of the nature of this operation, it does not make sense to integrate the processed image to find magnitudes. We do know, however, that the height of a resultant "island" is proportional to the intensity of the star-like component, so actual stars would be good calibrators. Therefore, we inferred the magnitudes of the processed objects by comparing them to the processed images of 4 stars in the field, whose magnitudes are known 2 . A plot of the calibration is given in Figure 4, showing the inferred magnitudes of the stars versus their actual magnitudes. We estimate that the error induced by this process is less than 0.1 in each color. Figure 5 is a color-color plot for those objects with B-V above or near 1.9. The high-z AGN criteria require a B-V color greater than 1.9, but to avoid objects with a poorly defined color we made the additional requirement that only those processed objects with one-sigma B-V error bars that do not extend below 1.5 be included. Of these 32 sources, 12 fall within the previously defined, high-z AGN region of color space, as delineated by lines in the figure. The majority of other designated sources lie where one could expect moderate-redshift, highly-reddened starburst galaxies, as illustrated in Figure 3.
Considering the colors of their "point-like" components, we regard the 12 objects in the high-z Fig. 5.-All HDF sources with processed B-V colors significantly (meaning more than one sigma) greater than 1.5 are shown. For reasons discussed in the text, objects above the solid line may be considered z > 3.5 AGN candidates. The letter P denotes a point source, less than 0.5 arcsec in diameter on the unprocessed image. Objects plotted as a C for compact are between 0.5 and 1.0 arcsec in diameter. E and S indicate larger ellipticals and spirals, respectively. Some of the plotted B-V colors are actually lower limits, as indicated in Table 1. AGN region of color space as "AGN candidates," even though Figure 2 indicates that one might also expect to find some z > 4 galaxies there. We note that the majority of these sources (all those with B-V > 2.5) have significantly larger B-V measured for the processed point source than for the host galaxy (from Williams et al. 1996), as would be expected for an AGN source within a galaxy according to Figure 2. (B-V colors for galaxy models at z = 4 lie below B-V colors for AGN at z = 4.) Finally, we justify our reference to all of the 12 aforementioned objects as AGN candidates because a major result of this work is to set an upper limit to the number of such sources.
The letters in Figure 5 indicate morphologies, which are somewhat subjective and are based on the visual appearance of the unprocessed image. The letter P indicates that the unprocessed object is not much bigger that a point source -specifically, less than 0.5 arcsec in diameter. The letter C is used for compact galaxies which are clearly larger than point sources but still less than one arcsec in diameter. The letter E designates an elliptical galaxy larger than one arcsec across. Similarly, S indicates a spiral galaxy with a diameter larger than one arcsec. Figure 6 shows both the processed and unprocessed images of all objects which fall above the line in the high-z AGN region of the diagram, plus selected others which are intended to give the reader a better idea of what the Laplacian-of-Gaussian filter does. Numbers next to the images correspond to ID numbers given by Williams et al. The effect on spiral arms is especially striking, as various features may be seen to stand out more in the processed images. Also, some ellipticals have steep enough cores that the processed shape appears similar to that of a star. Table 1 lists all 32 sources that are shown in Figure 5. It contains each object's ID number, its HDF coordinate position (frame, x, y), its sky position (RA, Dec), the colors and magnitude of the point source (B-V, V-I, and I), the colors and magnitude of the unprocessed galaxy (from Williams et al. 1996), the measured redshift of the object if available (from Cohen et al. 1996;Steidel et al. 1996;or Dickinson 1998), and the observed morphology using the same notation as above. For the colors and magnitudes of the point sources, estimated one-sigma errors are listed in parentheses. For example, 1.28(12) means 1.28 ± 0.12. Errors are due to a combination of image noise and calibration noise. As mentioned above, we could not get accurate measurements of U-B colors for the processed objects because the calibrating stars were generally very faint in the U band. Therefore this color is listed, when available, for the host galaxies, but not for the point sources.
For each AGN candidate, Table 2 gives our rough estimate of the redshift based on location in Figure 5 and the implied absolute B magnitude of the host galaxy, assuming q 0 = 0.5 and H 0 = 75 km s −1 Mpc −1 . Derived M B values for the point sources within the galaxies would be typically 1-2 magnitudes fainter.
Discussion
As illustrated in Figure 5, there are 12 sources which have the morphology and colors to be considered z > 3.5 AGN candidates. As a further consistency check, we looked at the U-B colors of the unprocessed objects. If these host galaxies really are at z > 3.5, they should also be Lyman break objects. That is, they should not have a measurable U-band magnitude. All 12 of our candidates were not visible in the U band according to the colors listed by Williams et al., so they are all considered to be viable z > 3.5 AGN candidates. Precise redshift values are not possible for most of them, but we list rough estimates in Table 2.
To our knowledge, only one of our candidates has been observed spectroscopically. As reported by Dickinson (1998), #3-512.0 has a strong Lyα emission line for a measured redshift of 4.02. This is reasonably close to our estimated redshift of 3.7 for this object and lends credibility to our technique.
In the Elson et al. (1996) list of unresolved HDF sources, none of the objects correspond to AGN candidates reported here, as noted in the Introduction. However, their Table 1 does contain four of our sources which have colors placing them just outside the high-z AGN candidate region of Figure by +75 in each case.)
Absolute B magnitude estimates (M B ) in Table 2 for the host galaxies of the AGN candidates were derived for the estimated redshifts, assuming H 0 = 75 km s −1 Mpc −1 and q 0 = 0.5. We used the expression M B = B + 5 log(q 2 0 H 0 ) − 5 log[q 0 z − (1 − q 0 )( √ 1 + 2q 0 z − 1)] − k − 52.39 (derived from a luminosity distance given by Terrell 1977) where k is the k-correction term. As noted by Peterson (1997) (see also Hewett 1992), k ≈ 0 mag for z = 3.5 quasar spectra, and k ≈ 0.5 mag for z = 4. For our calculations we used a linear fit to these values. The absolute magnitudes for AGN candidates in Table 2 represent the low-luminosity part of the expected range for class 1 Seyfert galaxies. Seyfert 1 absolute magnitudes typically range from -22 to -18 with the faintest occurring at about -16 (see Veron-Cetty and Veron 1993). The luminosity function for Seyfert 2 galaxies has a maximum about a magnitude fainter than for Seyfert 1 galaxies (Osterbrock 1989).
Our search found sources to a B magnitude of roughly 30. However, the list should not be considered complete to this level. The search technique is complex and requires that the AGN be significantly brighter than the surrounding galaxy light, so galaxies with large star forming regions very close to the AGN could have been missed. However, we do not believe it is likely that we missed many of the high-z AGN in the field, since the star-forming regions would have to be very luminous, and/or the AGN sources would have to be extraordinarily dim; and AGN candidates we did find already would represent unusually faint Seyfert galaxies. The "expected" number of AGN in the field was about 10 plus or minus half an order of magnitude. Thus, the fact that we found 12 probable AGN suggests that there is not a significant overabundance of faint, high-redshift AGN, as postulated by Steidel and Sargent (1989) to account for intergalactic medium ionization.
This result also sets strong limitations on the lensing explanation for observed quasar-galaxy correlations (e.g., Webster et al. 1988). Narayan (1989) noted that one way for a lensing explanation to be consistent with the observed data is for the faint end of the quasar/AGN luminosity function to be significantly increased. However, for this solution to work, there would have to be about 300 AGN with I < 28 in the HDF, which is clearly ruled out by our result. We believe, as did Narayan, that a more likely explanation could involve a better understanding of the completeness magnitude for the survey used by Webster et al. Of course, it must be born in mind that this investigation does not place limits on the number of AGN with z < 3.5. As illustrated in Figure 2, there are too many potential sources of confusion in color space for this type of analysis to be effective at lower redshifts.
Summary
Using color selection criteria and a Laplacianof-Gaussian filtering technique to identify point sources within galaxies, we have searched the Hubble Deep Field for spectral/color evidence of high-redshift, low-level AGN activity. We identified 12 possible AGN sources with estimated 3.5 ∼ < z ∼ < 5. In all cases, estimated luminosities lie within the low-luminosity range for Seyfert galaxies. The results are consistent with what could be expected from simple extrapolation of the observed quasar luminosity function to very faint magnitudes. This study supports the view that faint AGN activity does not play the major role in ionization of the early intergalactic medium and also the view that observed quasar-galaxy correlations do not imply a very large number of faint AGN to be gravitationally lensed.
We would like to thank Gary Bernstein and Peter Jensen for suggesting and assisting with the Laplacian-of-Gaussian process in our search. We also thank Guy Worthey for help in obtaining galaxy spectra, Donald Schneider and Maarten Schmidt for sending us digitized quasar spectra, Anuradha Koratkar for Seyfert galaxy spectra and Charles Steidel for spectra of Lyman break objects which contributed to improvements in the color determinations. Finally, we thank the referee for many useful comments which helped generally to improve the paper.
Note: Table 1 is included as a separate file, since it is best viewed in landscape format. Figure 6 is also included as a separate file, since it is rather large. Here is the caption for it: Fig. 6.-The processed and original images (on the left and right, respectively) for each candidate AGN along with selected other objects from Figure 5. The first 12 are our AGN candidates. Each square is 1.7 arcseconds on a side, and the orientation for each CCD image is such that the x, y origin is in the lower left. Williams et al. catalog. b Frame and X, Y chip positions (measured from the corner of each chip closest to the center of the array) for the center of the stellar object. c Measured Right Ascension (2000.0) of the stellar object. All RA values start with 12 hours. d Measured Declination (2000.0) of the stellar object. All Dec values start with 62 degrees north. e Colors and V magnitude of the processed stellar object found by comparison with known stars in the field. f Colors and V magnitude of the unprocessed galaxy taken from Williams et al. (1996). g Observed redshift value if available from Cohen et al. 1996, Steidel et al. 1996, or Dickinson 1998 Observed morphology of the unprocessed object. P indicates a nearly point source, C a galaxy less than 1 ′′ in diameter, E a large (ie. larger than 1 ′′ ) elliptical, and S a large spiral.
Fig. 2 .
2-Expected colors for different kinds of objects at various synthetic redshifts. Redshift values are incremented by 0.2 in all cases.
Fig. 3 .
3-A starburst nucleus galaxy spectrum has been given additional reddening and red-shifted from 0 to 2, increment-ed by 0.2, with resulting colors shown. The black cir-cles represent the original spectrum with an additional c = 0.5 reddening. Grey and white circles represent c = 1.0 and c = 1.5 additional reddening, respectively. Numbers are given in the circles for integral redshift values.
Fig. 4 .
4-The inferred B, V and I magnitudes of the calibrating stars versus their actual mag-nitudes given by
5 .
5These are: #3-419.0, #3-491.0, #4-241.3, and #4-713.0, all of which we designated as "compact." (We note that the x and y pixel positions given by Elson et al. are offset from ours as well as from the standard system of Williams et al.
Table 2 .
2Inferred Data For AGN CandidatesEstimated
ID
Redshift
M B
a
2-282.0
3.9
-18.2
2-436.0
3.8
>-17.6
3-512.0 b
3.7
-19.2
4.0
-19.7
3-675.0
4.0
-19.5
3-775.0
>4.0
-18.0
3-783.0
>4.2
>-18.4
3-853.22
3.7
-17.5
4-277.0
4.4
>-18.7
4-280.0
3.7
>-17.5
4-439.1
>4.6
>-19.4
4-551.0
>4.1
>-17.8
4-639.2
3.8
-18.5
a Computed absolute magnitudes for H0 = 75 km s −1
Mpc −1 and q0 = 0.5.
b The italicized values for this object are based on the
observed redshift.
Table 1 .
1Measurements For Objects InFigure 5a ID number for the host galaxy from theHost
Image Coords b
RA c
Dec d
Point Source Colors e
Host Gal Colors f
Gal ID a Fr
X
Y
(12:)
(+62:)
B-V
V-I
V
U-B
B-V
V-I
V
Z g Type h
AGN Candidates:
2-282.0
2
690
621 36:49.03 13:31.5
2.32(70) 1.28(12) 28.27(8)
>-0.65
1.67 1.20 27.18 · · ·
C
2-436.0
2
980
954 36:50:10 13:47.4
2.28(61) 0.94(11) 28.16(8)
· · ·
>1.62 1.10 27.70 · · ·
P
3-512.0
3 1304 1146 36:56.02 12:45.7
1.91(18) 0.47(9)
26.96(7)
>0.17
1.67 0.36 25.85 4.0
C
3-675.0
3 1369 1460 36:57.07 12:35.3
2.53(98) 0.44(17) 28.44(11)
>-0.16
1.60 0.46 26.08 · · ·
C
3-775.0
3
712 1683 36:54.14 12:16.7 >2.6 (11) 0.92(16) 28.46(13)
>-0.72
1.67 0.99 27.53 · · ·
P
3-783.0
3 1604 1705 36:58.85 12:30.1 >3.0 (11) 0.95(12) 28.14(10)
· · ·
>2.07 0.84 27.04 · · ·
P
3-853.22
3 1550 1835 36:58.87 12:24.5
1.98(47) 0.51(12) 28.11(8)
>-0.62
1.99 0.48 27.19 · · ·
P
4-277.0
4 1837
604 36:48.85 11:53.9
3.3 (10) 1.20(8)
27.71(7)
· · ·
>2.02 1.07 27.17 · · ·
P
4-280.0
4
820
607 36:46.58 12:31.2
2.13(49) -0.02(14) 28.05(9)
· · ·
>2.08 0.29 27.12 · · ·
P
4-439.1
4
361
952 36:43.75 12:42.7 >4.0 (11) 1.03(9)
27.02(7)
· · ·
>2.79 1.07 26.03 · · ·
C
4-551.0
4 1774 1205 36:45.56 11:46.9 >2.7 (11) 0.62(13) 28.42(11)
· · ·
>1.76 0.40 27.88 · · ·
P
4-639.2
4
266 1351 36:41.45 12:40.0
2.23(40) 1.11(12) 27.72(10)
>-0.36
2.26 0.96 26.18 · · ·
C
Other:
2-977.0
2
689 1895 36:55.68 13:51.9
1.77(21) 0.23(13) 27.43(8)
>-0.20
1.75 0.46 26.60 · · ·
P
3.266.0
3 1339
643 36:55.06 13:04.7
2.42(15) 1.76(9)
26.25(7)
>0.50
1.79 1.65 25.22 · · ·
C
3.321.1
3
373
712 36:50.17 12:46.8
2.09(11) 1.66(9)
25.03(7)
>2.17
1.90 1.47 22.91 0.7
E
3-378.0
3 1506
887 36:56.49 12:58.4
2.54(77) 1.99(9)
28.15(8)
· · ·
>1.78 1.74 27.37 · · ·
P
3.419.0
3
980
981 36:53.95 12:46.6
1.69(8)
1.64(7)
24.36(6)
0.91
1.33 1.57 24.28 · · ·
C
3-430.2
3 1481 1009 36:56.64 12:53.5
1.80(23) 1.84(10) 27.36(9)
1.56
0.87 1.51 25.65 1.2
C
3-486.0
3 1224 1090 36:55.48 12:46.5
2.46(25) 1.58(11) 26.50(9)
0.49
0.66 1.00 22.98 0.8
S
3-491.0
3 1928 1112 36:59.21 12:56.9
1.68(8)
1.97(7)
24.43(6)
>1.45
1.65 1.94 24.42 · · ·
C
3-586.0
3
473 1288 36:52.00 12:27.4
2.16(29) 1.92(10) 27.41(8)
>-0.20
1.68 1.74 26.03 · · ·
C
3-790.1
3 1919 1688 37:00.46 12:35.7
1.83(12) 1.35(10) 24.91(8)
2.35
1.59 1.14 22.51 0.6
E
3-815.1
3 1144 1749 36:56.55 12:21.2
2.09(13) 1.86(9)
26.28(8)
0.78
1.83 1.69 24.28 · · ·
E
1
An "ideal" star, for which the flux distribution is F = (A/2π) exp(−r 2 /2) (where r is again measured in units of r 0 , the PSF radius, and A is a constant), takes the following form after being processed: F ′ = (A/8)(4 − r 2 ) exp(−r 2 /4).2 For these stars, we used the "total" magnitudes reported byWilliams et al. (1996).
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| [] |
[
"Symbol detection in online handwritten graphics using Faster R-CNN",
"Symbol detection in online handwritten graphics using Faster R-CNN"
] | [
"Frank D Julca-Aguilar \nDepartment of Computer Science\nInstitute of Mathematics and Statistics\nUniversity of São Paulo (USP)\nSão PauloBrazil\n",
"Nina S T Hirata \nDepartment of Computer Science\nInstitute of Mathematics and Statistics\nUniversity of São Paulo (USP)\nSão PauloBrazil\n"
] | [
"Department of Computer Science\nInstitute of Mathematics and Statistics\nUniversity of São Paulo (USP)\nSão PauloBrazil",
"Department of Computer Science\nInstitute of Mathematics and Statistics\nUniversity of São Paulo (USP)\nSão PauloBrazil"
] | [] | Symbol detection techniques in online handwritten graphics (e.g. diagrams and mathematical expressions) consist of methods specifically designed for a single graphic type. In this work, we evaluate the Faster R-CNN object detection algorithm as a general method for detection of symbols in handwritten graphics. We evaluate different configurations of the Faster R-CNN method, and point out issues relative to the handwritten nature of the data. Considering the online recognition context, we evaluate efficiency and accuracy trade-offs of using Deep Neural Networks of different complexities as feature extractors. We evaluate the method on publicly available flowchart and mathematical expression (CROHME-2016) datasets. Results show that Faster R-CNN can be effectively used on both datasets, enabling the possibility of developing general methods for symbol detection, and furthermore, general graphic understanding methods that could be built on top of the algorithm. | 10.1109/das.2018.79 | [
"https://arxiv.org/pdf/1712.04833v1.pdf"
] | 25,051,392 | 1712.04833 | 4ddfe4488b2c0fcc6119be0e8fb8d02c70ff3bbf |
Symbol detection in online handwritten graphics using Faster R-CNN
Frank D Julca-Aguilar
Department of Computer Science
Institute of Mathematics and Statistics
University of São Paulo (USP)
São PauloBrazil
Nina S T Hirata
Department of Computer Science
Institute of Mathematics and Statistics
University of São Paulo (USP)
São PauloBrazil
Symbol detection in online handwritten graphics using Faster R-CNN
Handwriting recognitionsymbol recognitionobject detectionFaster R-CNN
Symbol detection techniques in online handwritten graphics (e.g. diagrams and mathematical expressions) consist of methods specifically designed for a single graphic type. In this work, we evaluate the Faster R-CNN object detection algorithm as a general method for detection of symbols in handwritten graphics. We evaluate different configurations of the Faster R-CNN method, and point out issues relative to the handwritten nature of the data. Considering the online recognition context, we evaluate efficiency and accuracy trade-offs of using Deep Neural Networks of different complexities as feature extractors. We evaluate the method on publicly available flowchart and mathematical expression (CROHME-2016) datasets. Results show that Faster R-CNN can be effectively used on both datasets, enabling the possibility of developing general methods for symbol detection, and furthermore, general graphic understanding methods that could be built on top of the algorithm.
I. INTRODUCTION
An online handwritten graphic is composed of a set of strokes, where each stroke consists of a set of bidimensional coordinates. The coordinates can be captured, for example, using a device with touch screen and an electronic pen. A symbol consists of a subset of strokes. In these data, in contrast to text, symbols might be placed over vertical or diagonal positions relative to each other. Figure 1 shows an online handwritten mathematical expression example. Typical symbol detection techniques for online handwritten graphics include stroke grouping and classification processes. The first process selects groups of strokes that are likely to form symbols, and the second applies machine learning methods to classify the stroke groups as symbols, with their corresponding symbol classes, or as false positives. Due to the variance of the placement of symbols, virtually any group of strokes might form a symbol. To avoid the computational complexity of evaluating all possible stroke groups, constraints based on handcrafted rules (e.g. only selecting stroke groups of up to four strokes) are applied. Such constraints do not only limit the accuracy of the methods, but also make difficult the application of a same method to the recognition of different graphic types.
Taking advantage of deep convolutional neural networks (DCNNs), recent algorithms for object detection have obtained outstanding accuracy. Among the different methods, Faster R-CNN has shown to obtain state-of-the-art accuracy and efficiency [1]. Also, Faster R-CNN models are general enough to be applied to a large variety of problems as they can be trained end-to-end using inputoutput examples.
By converting raw online graphics data to offline (i.e. images), object detection methods based on DCNNs could also be applied to symbol detection. In this work, we evaluate the Faster R-CNN algorithm to detect symbols in handwritten graphics. We make a parallel between traditional symbol detection methods in online data and our methods (Section II). We then describe our pipeline to transform online data to offline, and give an overview of the Faster R-CNN algorithm (Section III). Through experimentation in the detection of symbols in mathematical expressions and flowcharts (Section IV), we show that the Faster R-CNN algorithm provides high accuracy on both problems. Results are encouraging not only for the development of general methods for symbol detection, but also for the development of methods for structure recognition (Section V). The code implemented in this work is available as open source.
II. RELATED WORK
We can find a variety of techniques for detecting symbols in online handwritten graphics. Most techniques introduce constraints based on some characteristics of the graphic type. For instance, in mathematical expression recognition, stroke grouping is often done considering only groups of strokes that have up to four or five strokes [2], [3], [4]. Other common constraints include the assumption that symbols are formed only by strokes consecutive in input time order [5], [6], or strokes that intersect each other [7]. In the recognition of other graphic types, as diagrams, different techniques are designed to detect specific symbol classes [8], [9], [10]. For instance, Bresler et. al. [9] separate the detection of symbols that do not have an specific shape, as text and arrows, from symbols that have well defined shapes, as decision and data [9] (flowchart symbol examples are shown in Figure 2). Recent works on object detection are mainly based on DCNNs [1]. At a high level, several of those techniques (e.g. [11], [12], [13], [14]) consist of three processes: feature maps generation using a DCNN, selection of object bounding box candidates, and the classification of the bounding box candidates using the feature maps (cropped according to the box dimensions). One of the algorithms that implements the above methods is Faster R-CNN. The algorithm has obtained state-of-the-art accuracy [1] and has the advantage of doing the three processes through a single forward pass of a network.
While methods for symbol detection in online data are usually evaluated at stroke level [15], detection methods are evaluated at bounding box level (e.g. using mean average precision [16]). The evaluating metrics are then not directly comparable.
Although it could be possible to develop algorithms to recover stroke level information from the offline data, to the end of graphics understanding, such process might not be necessary. For instance, to recognize flowcharts structure, once symbol candidates have been identified, relations between symbols can be determined using features from the corresponding bounding box regions over an image.
III. METHODS
Faster R-CNN is a supervised learning algorithm. The algorithm receives as input an image, and generates as output a list of object bounding box coordinates and the corresponding object class per box. Training the algorithm then requires, in addition to the input images, a list of bounding boxes per image. This section gives details about the methods used to generate training data to evaluate the algorithm in the context of graphics recognition, and gives a brief description of the algorithm.
A. Training data generation
Depending on the input device, the range of the stroke coordinates can have a high variance. In order to deal with such variance, we scale each graphic so that the largest dimension of its bounding box is equal to a fixed parameter L (keeping the original aspect ratio). In order to avoid loosing precision, this scaling is done coordinate-wise. Once a graphic is scaled, we draw its traces through linear interpolations between each pair of successive coordinates. The resulting images are gray-scale images, with different aspect ratios (but with their largest dimension equal to L). Figure 3 shows some images generated through this process.
To generate the bounding boxes, we extract the most top-left and bottom-right coordinates of each symbol after applying the scaling process described above. As the bounding box coordinates are measured in terms of pixel units, after scaling, some boxes might end up having zero width or height. In such cases, we update the coordinates so that the boxes have a minimum dimension of three pixels, which is about the width of the drawing traces.
B. Faster R-CNN
In this section, we describe the main components of the algorithm and highlight parameters of interest regarding our evaluation. A more detailed description can be found in [12].
The algorithm can be seen as a neural network composed of three components: feature extractor, Region Proposal Network (RPN), and region classifier. The first component receives an input image and extracts a feature map, the RPN receives the feature map and generates bounding box coordinates (regions) that might contain an object, and the region classifier classifies the boxes using the features map cropped according to the box coordinates. The whole network can be trained using stochastic gradient descend [1], or using an iterative process (iterations of separated RPN and region classifier training steps) [12].
Next sections give more details about the main components.
1) Feature extractor: The feature extractor is a DCNN, usually without fully connected layers, that maps an input image to a feature map. For instance, in [12] the authors use a VGG-16 [17] network and extract feature maps from the last (13th) convolutional layer. The kind of feature extractors determine a large part of the accuracy and computational cost of the network. For instance, in [1], the authors report that inference time varies from about 100 miliseconds when using small feature extractors (e.g. Inception V2) to almost 1 second when using more complex or deeper DCNNs (e.g. Inception Resnet v2).
The feature extractor allows images with variable width and height, but applies a preprocessing step that consists on scaling the images so that their minimum dimension (M ) is set to a constant value.
2) Region Proposal Network: The regions proposal network is a two layer fully convolutional neural network. The network considers a set of boxes, called anchor boxes, of different aspect ratios and scales. For each feature map position and for each anchor box, the network outputs the probability of the anchor box containing an object. Also, for each anchor box, it calculates the coordinates of the box that contains the object. The network is optimized using a loss function composed of a softmax loss for the probability outputs, and a regression loss for the box coordinates.
An important parameter of the RPN is the number of proposals (bounding boxes) that are sent to the classifier. The larger the number, the higher the probability of finding an object, but also the higher the computational cost (and and so the number of false positives) as each region is later classified by the region classifier. The authors in [12] then apply a non-maximum suppression algorithm to reduce the number of proposals. Experiments have shown that 300 is an adequate number when dealing with the PASCAL VOC 2012 dataset.
3) Region classifier: The proposals generated by the RPN are used to crop the corresponding regions from the features map. The cropped regions are then used as input to a small neural network classifier that determines the class of the object (including a background or false positive class) and a box refinement. Similar to the RPN, this network also uses a softmax and a regression loss for optimization.
IV. EXPERIMENTATION
A. Experimental setup
We evaluated the methods in the detection of symbols in online handwritten mathematical expressions and flowcharts. In the first case, we used the CROHME-2016 [15] dataset, and in the second, we used the flowchart dataset proposed in [18]. The CROHME-2016 dataset contains about 12, 000 mathematical expressions, and the flowcharts dataset contains about 400 flowcharts. The datasets contain several challenges for the detection framework. The CROHME-2016 dataset contains a large number of symbol classes (101), including digits, characters, operators (e.g. +, ×, √ ). Among all classes, specially difficult ones might be the small symbols (as points and commas), and symbols that have similar shapes (e.g. 1, -, ×, x, c and C). The flowchart dataset contains seven symbol classes: arrow, text, decision, connection, data, process and terminator. In such dataset, specially difficult ones are texts and arrows, as they do not have a specific shape. For instance, text might consist of a single character, or several words placed over several baselines; arrows might be horizontal, vertical, or curved lines, and extended over a large area of the image. Some examples of both datasets are shown in Section IV-B. The datasets are publicly available.
In both mathematical expression and flowchart datasets, we used a typical training-validation-test split. For the test part, we used the same examples defined by the dataset authors [15], [18]. For the training and validation parts, we randomly selected 80% of the graphics for training and used the rest as validation set.
In our implementations, we used the object detection framework proposed in [1]. As explained in Section III-B, a key component that determines the algorithm efficiency and accuracy is the feature extractor. To measure the impact of feature extractors on handwritten data, we use four DCNNs of different complexities. The considered DCNNs, from the smaller to the largest one, are: Inception V2 [19], Resnet 50 [20], Resnet 101 [20], and Inception Resnet v2 [21]. We run the experiments on a Nvidia GeForce GTX Titan X GPU 12GB card.
To determine adequate hyper parameters, we first experimented with different configurations of the algorithm using Inception V2 (as such networks provide a faster feedback). In such experimentation we used some default parameters from the authors of the algorithm [12], as well as base code from [1]. From that experimentation, we defined the following hyper parameters for all models:
• Generated images size: We set the maximum image dimension (L) to 768. • Scaled images size. In flowcharts, we set the minimum image dimension (M ) to 600. In mathematical expressions, we set M to 300. Although larger values tend to improve accuracy [1], in mathematical expressions, we have several cases where images have a very large width, but small height. Scaling relative to the height of such images generated images with a resolution larger than the one allowed by the GPU capacity (when training the models with the largest DCNNs). • Training from pre-trained models. The base code released by [1] includes Faster R-CNN models trained over the MSCOCO dataset. Although our generated images are very different in comparison to the natural images of the MSCOCO dataset, we found that training using the pre-trained models allows for much faster convergence than training from scratch. We then used pre-trained models for the rest of the experiments. • Number of proposals. Once trained, we evaluated models that extract from 300 (the default value defined in [12]) up to 1000 proposals from the RPN. We did not find considerable improvements when using larger number of proposals. We then fixed the number of proposals to 300. • Training scheme. We used minibatch training with batch size 1 (due to the variable dimensions of the images). We fixed the number of training steps to 25, 000 for flowcharts and 150, 000 for mathematical expressions. Additional details about the configuration parameters will be available on the code repository.
We used mean average precision (specifically, [email protected]) as evaluation metric [16].
B. Results
Flowcharts. Table I shows the detection accuracy of the evaluated models over the validation and test sets. For the validation set, we show the model's average precision using DCNNs with increasing complexities, with the smallest one on top of the table. We can see a consistent improvement as the feature extractors are deeper. This improvement is mainly due to higher scores in the detection of texts and arrows. For the test set, we show the performance of the best model (Inception resnet v2) considering mAP. We can see that the largest variance in accuracy in comparison to the validation set occurs in the detection of texts and arrows.
Through visual analysis of the output detections, we found that the most frequent missed symbols are arrows with several curves and small texts. Figure 4 shows two output examples with some miss detection cases.
Mathematical expressions. Table II shows results on the validation and test sets of the CROHME-2016 dataset. We can see higher improvements, in comparison to flowchart results, as the feature extractors are deeper. Also, by analyzing the scores per class, we found that scores for the most frequent classes are considerably higher than the mAP score. In the same table, we illustrate this by showing the scores for the top-10 most frequent classes. These results show that the mAP score is pushed down mainly by the less frequent classes 1 .
In comparison to flowcharts, mathematical expressions have several symbol types with a really small width or height (e.g. 1, l, |, -, and dot). We found that such symbols are specially difficult to be detected by the models. Such difficulty can also be seen in the results of the most frequent classes in Table II, where the scores ofand 1 are low in comparison to the scores of the other frequent classes. Miss classification between symbols that have similar shape is other frequent type of error of the detector. Figure 5 shows output examples for the best model along with some miss detection cases. Discussion. It is important to note that although the flowchart training data contains only about 200 examples, the data is enough to achieve high accuracy over all symbol classes. Furthermore, not very deep models, as the Inception v2, already allows us to obtain high mAP scores. The possibility of using effective and small DCNNs enables the use of the method in contexts where computational resources are limited or a fast output is required.
Several of the previous works described in Section II have reported results on our evaluating datasets. For instance, in [22] recall of flowchart symbols was 84.4. However, as in such works evaluation is done at stroke level and not at bounding box level, results are not directly comparable.
V. CONCLUSIONS
We showed that the Faster R-CNN algorithm provides effective detection of symbols in online handwritten mathematical expressions and flowcharts. Such results are encouraging in the context of the development of general methods for symbol detection in online handwritten graphics. Furthermore, the integration of the algorithm with structure recognition techniques might also accelerate the development of such techniques.
Our evaluation aimed at measuring and understanding the potential of the Faster R-CNN algorithm and will serve as a baseline for further research. We believe that the algorithm has high potential for improvement through the introduction of online information during the detection pipeline, or by solving ambiguities, e.g. using contextual information, in a postprocessing or structural recognition step.
Figure 1 .
1Online handwritten mathematical expression composed of strokes {str 1 , . . . , str 13 }. Coordinates are depicted as gray circles.
Figure 2 .
2Handwritten flowchart example.
Figure 3 .
3Generated images with different aspect ratios.
ACKNOWLEDGMENTF
. D. Julca-Aguilar thanks FAPESP (grant 2016/06020-1). N. S. T. Hirata thanks CNPq (305055/2015-1). This work is supported by FAPESP (grant 2015/17741-9) and CNPq (grant 484572/2013-0).
Figure 4 .
4Flowchart output examples. Scores for each detected symbol are given by the softmax output (%).
Table I AVERAGE
IPRECISION ON THE FLOWCHART VALIDATION AND TEST SETSFeature extractor
Average Precision (%)
mAP
text
arrow connection
data
decision process terminator
On validation set:
Inception v2
98.6
97.3
94.7
97.9
100.0
100.0
100.0
100.0
Resnet 50
99.2
98.3
96.0
100.0
100.0
100.0
100.0
100.0
Resnet 101
99.5
99.3
97.6
99.7
100.0
100.0
100.0
100.0
Inception Resnet v2
99.6
99.4
97.9
100.0
100.0
100.0
100.0
100.0
On test set:
Inception Resnet v2
97.7
95.2
91.5
99.0
99.7
99.6
99.1
99.9
(a)
(b)
Table II AVERAGE
IIPRECISION (FOR THE TOP-10 MOST FREQUENT CLASSES AND MAP) ON THE CHROHME-2016 VALIDATION AND TEST SETSFeature extractor
Average Precision (%)
mAP
-
1
2
+
x
(
)
=
a
3
on validation set:
Inception v2
83.6
84.8 80.6 97.6 98.7 95.3 95.4 97.3 98.4 97.8 97.7
Resnet 50
85.4
87.5 88.5 98.2 99.3 95.9 97.3 97.9 98.7 97.9 99.0
Resnet 101
87.5
92.8 91.2 98.9 98.8 96.8 97.8 98.5 98.8 97.8 99.2
Inception resnet v2
89.7
95.8 94.4 99.0 99.7 97.5 98.1 99.4 99.4 98.9 99.1
On test set:
Inception resnet v2
86.8
96.8 92.5 99.1 99.8 98.4 99.4 99.1 99.4 95.9 99.3
Recall that mAP is just the mean of the average precisions per class[16]
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"Few-View CT Reconstruction with Group-Sparsity Regularization",
"Few-View CT Reconstruction with Group-Sparsity Regularization",
"Few-View CT Reconstruction with Group-Sparsity Regularization",
"Few-View CT Reconstruction with Group-Sparsity Regularization"
] | [
"Peng Bao \nCollege of Computer Science\nSichuan University\n610065ChengduChina\n",
"Jiliu Zhou \nCollege of Computer Science\nSichuan University\n610065ChengduChina\n",
"Yi Zhang \nCollege of Computer Science\nSichuan University\n610065ChengduChina\n",
"Peng Bao \nCollege of Computer Science\nSichuan University\n610065ChengduChina\n",
"Jiliu Zhou \nCollege of Computer Science\nSichuan University\n610065ChengduChina\n",
"Yi Zhang \nCollege of Computer Science\nSichuan University\n610065ChengduChina\n"
] | [
"College of Computer Science\nSichuan University\n610065ChengduChina",
"College of Computer Science\nSichuan University\n610065ChengduChina",
"College of Computer Science\nSichuan University\n610065ChengduChina",
"College of Computer Science\nSichuan University\n610065ChengduChina",
"College of Computer Science\nSichuan University\n610065ChengduChina",
"College of Computer Science\nSichuan University\n610065ChengduChina"
] | [] | Classical total variation (TV) based iterative reconstruction algorithms assume that the signal is piecewise smooth, which causes reconstruction results to suffer from the over-smoothing effect. To address this problem, this work presents a novel computed tomography (CT) reconstruction method for the few-view problem called the groupsparsity regularization-based simultaneous algebraic reconstruction technique (GSR-SART). Group-based sparse representation, which utilizes the concept of a group as the basic unit of sparse representation instead of a patch, is introduced as the image domain prior regularization term to eliminate the over-smoothing effect. By grouping the nonlocal patches into different clusters with similarity measured by Euclidean distance, the sparsity and nonlocal similarity in a single image are simultaneously explored. The split Bregman iteration algorithm is applied to obtain the numerical scheme. Experimental results demonstrate that our method both qualitatively and quantitatively outperforms several existing reconstruction methods, including filtered back projection, expectation maximization, SART, and TV-based projections onto convex sets. | 10.1002/cnm.3101 | [
"https://arxiv.org/pdf/1803.01546v1.pdf"
] | 13,691,402 | 1803.01546 | f0631fea961db3ec1978bbc064a91ac00e404bf8 |
Few-View CT Reconstruction with Group-Sparsity Regularization
Peng Bao
College of Computer Science
Sichuan University
610065ChengduChina
Jiliu Zhou
College of Computer Science
Sichuan University
610065ChengduChina
Yi Zhang
College of Computer Science
Sichuan University
610065ChengduChina
Few-View CT Reconstruction with Group-Sparsity Regularization
Computed tomographyFew-view reconstructionsparse representationtotal variation
Classical total variation (TV) based iterative reconstruction algorithms assume that the signal is piecewise smooth, which causes reconstruction results to suffer from the over-smoothing effect. To address this problem, this work presents a novel computed tomography (CT) reconstruction method for the few-view problem called the groupsparsity regularization-based simultaneous algebraic reconstruction technique (GSR-SART). Group-based sparse representation, which utilizes the concept of a group as the basic unit of sparse representation instead of a patch, is introduced as the image domain prior regularization term to eliminate the over-smoothing effect. By grouping the nonlocal patches into different clusters with similarity measured by Euclidean distance, the sparsity and nonlocal similarity in a single image are simultaneously explored. The split Bregman iteration algorithm is applied to obtain the numerical scheme. Experimental results demonstrate that our method both qualitatively and quantitatively outperforms several existing reconstruction methods, including filtered back projection, expectation maximization, SART, and TV-based projections onto convex sets.
Introduction
In recent decades, computed tomography (CT) has been wildly used in clinical diagnosis. However, X-ray radiation may cause cancer and genetic disease [1]. It is hence necessary to reduce the amount of a dose during a CT scan. To deal with this problem, many methods have been proposed. These methods can be categorized into two groups. The first method is to reduce the operating current, which increases the quantum noise in the projection data. The second method is to decrease the number of sampling views, which generates insufficient projection data, leading to fewview or limited-angle CT [2]. How to reconstruct a high-quality CT image from contaminated or undersampled projection data has attracted a great deal of attention in recent years. In this paper, we focus on few-view CT reconstruction.
Traditional analytic algorithms, such as filtered back projection (FBP), have specific requirements for the completeness of the projection data. Streak artifacts appear when the sampling ratio is low. The iterative reconstruction algorithm is an efficient way to solve this problem. Over the past few decades, the most widely used iterative algorithms for tomography imaging are the algebraic reconstruction technique [3], simultaneous algebraic reconstruction technique (SART) [4], and expectation maximization (EM) [5]. However, when projection views are highly sparse without extra prior information, it is very hard to obtain a satisfactory solution with these classical algorithms. To improve this problem, additional information is usually merged into the objective function to achieve a robust solution.
Compressive sensing (CS) theory has been proved a powerful technique [6,7]. If an image can be represented sparsely with a certain sparse transform, it can be accurately reconstructed with a probability close to one. Inspired by CS theory, Sidky et al. introduced total variation (TV) minimization into incomplete projection data reconstruction and proposed an efficient iterative reconstruction algorithm based on projection onto convex sets (POCS), called TV-POCS [8]. Although TV-POCS can eliminate streak artifacts to a certain degree, the assumption of TV that the signal is piecewise smooth causes TV-POCS to suffer from over-smoothing effects [9]. As a result, many variants of TV have been proposed to tackle this problem, such as adaptive-weighted TV [10], fractional-order TV [11,12], and nonlocal means [13,14]. Chen et al. suggested that a high-quality image can be utilized to constrain the CS-based reconstruction [15], and this method has been extended to several different reconstruction topics with different prior images. Yu et al. constructed the pseudo-inverses of the discrete gradient and discrete difference transforms and adopted a soft-threshold filtering algorithm for few-view CT image reconstruction [16].
Recently, dictionary learning based methods have been proved effective. In contrast to traditional techniques, which process the image pixel by pixel, a dictionary-based method processes images patch by patch. In 2006, Elad tackled the image denoising problem with a dictionary learning method that utilizes the K-SVD algorithm [17]. Mairal et al. extended this method to colour image restoration [18]. For medical imaging problems, the dictionary learning method was first introduced into magnetic resonance imaging (MRI). Chen et al. combined the dictionary learning method and TV-based MRI scheme to further improve image quality [19]. Later, Xu et al. proposed a low-dose CT image reconstruction method based on dictionary learning. This model introduces the sparse representation constraint of a redundant dictionary as the regularization term, and the performance of a global dictionary and adaptive dictionary was discussed [20]. Inspired by work combining super-resolution with dual dictionary learning [21], Lu et al. respectively used a transitional dictionary for atom matching and a global dictionary for image updating to deal with the few-view problem [22]. Zhao et al. extended this method for spectral CT [23].
Traditional studies based on dictionary learning have two limits. First, the computational burden is very heavy.
Second, the relationships among patches are ignored. If the original signals are noisy, the accuracy of sparse coding will decline. Inspired by the research on group sparsity [24][25][26][27], in this article, we proposed a novel few-view CT reconstruction method based on group-sparsity regularization (GSR) called the GSR-based simultaneous algebraic reconstruction technique (GSR-SART). Instead of processing the image patches sequentially, similar patches are clustered into groups as the basic unit of the proposed group-based sparse representation. Thus, the sparsity and nonlocal similarity in a single image are simultaneously imposed. The remainder of the paper is organized as follows: Section 2 introduces the theory details and numerical scheme for the proposed GSR-SART. Experimental results are provided in Section 3 to demonstrate the performance of our method. A discussion and the conclusion are presented in Section 4.
Methods
Imaging model
Assuming a monochromatic source, the general model of CT imaging can be approximately represented as the following discrete linear system:
= ,(1)
where is the system matrix, which is composed of row vectors, and denotes the measured projection data. Our goal is to reconstruct an image represented by vector from the projection data and system matrix . In practice, Eq. (1) is known as an ill-posed problem because we consider the insufficient projection data problem caused by few views. This means that we cannot obtain a unique by directly inverting Eq. (1). To solve the linear system expressed in Eq. (1), prior information about the target image is often imposed.
Sparse Representation Modelling
Sparse representation (SR) for image processing seeks a sparse matrix that contains as few zero coefficients as possible to approximately represent the signal. The SR model can be expressed as
{α, } = , ‖ − ‖ 2 2 + ‖ ‖ 0 ,(2)
where denotes an observed signal vector, is a dictionary, α presents the coefficients to represent the signal, is a regularization parameter, and ‖•‖ 0 denotes the L0 norm. The goal of SR is to seek a sparse vector to represent for a trained . To better represent with , it is necessary to choose an effective dictionary . Some approximation algorithms have been proposed to alternatively optimize and , such as MOD [28], K-SVD [29], and online learn-
ing [30].
Group-Based Sparse Representation Modelling
Usually, most SR-based methods divide the image into overlapped patches and process them one by one. This operation ignores the nonlocal relationships between different patches. In this paper, we impose the nonlocal similarity constraint into SR to create a GSR-based few-view CT reconstruction method.
First, we divide the CT image into overlapped patches of size √ × √ using a sliding distance of four pixels, where vector denotes an image patch at location , = 1, 2, 3 … , . In Fig. 1, is indicated by a small yellow square. The Euclidean distance is utilized as the similarity measurement to search for the patches that are the most similar with in the × search window, as indicated by the big red square in Fig. 1. These similar patches form set . Second, all the patches in are unfolded into vectors and arranged into a matrix of size × , denoted by , which includes each patch in as its columns. Matrix is treated as a group of similar patches. We can then define
= ( ),(3)
where is an operator that extracts group from . Next, we use ( ) to denote placing group back into the ℎ position of the reconstructed image. Now, we can express the whole image by averaging all the groups as follows:
= ∑ ( ) =1 / ∑ ( * ) =1 ,(4)
where operator / indicates the element-wise division of two vectors and * is an all-ones matrix of the same size as .
Next, we introduce the dictionary learning method for each group . In this model, the adaptive dictionary for each group can be directly obtained from its estimate because we cannot obtain the original image in practice. In the process of optimization, estimate is calculated. Once we obtain , we apply SVD to it as follows:
= = ∑ * ( * =1 * ),(5)
where is the number of atoms in ,
= { * 1 , * 2 , … , * } , = ( ) denotes a diagonal
matrix for which all the elements except for the main diagonal are zero, * denotes the columns of , and * denotes the columns of . For group , each atom of is defined as follows: * = * * , = 1,2, … ,
We can then define the expression of the ultimate dictionary for group as follows:
= { * 1 , * 2 , … , * }(7)
Using , the GSR model seeks a sparse vector = { * 1 , * 2 , … , * } to represent :
= ∑ * * =1(8)
We can then represent the entire image using the set of sparse codes { }.
= * = ∑ (∑ * * =1 ) =1 ./ ∑ ( * ) =1(9)
Here, represents the concatenation of all and denotes the concatenation of all .
GSR-SART algorithm
Similar to [31], two independent steps are included in our algorithm. In the first step, we adopt SART to solve the linear system of Eq. (1), which yields a noisy result by minimizing the distance between the measured projection data and the estimated projection data. Specifically, the SART algorithm can be described as follows:
+1 = + +, ∑ , ,+ ( − ̅̅̅( )) =1 ,(10)
,
+ = ∑ , =1 = 1,2, … , ,(11)+, = ∑ , =1 = 1,2, … , ,(12)̅( ) = ,(13)
where is a system matrix of size × ( is the total number of projection data and is the total number of image pixels), is the relaxation parameter, and is the iteration number. The second step is to obtain an artifact-reduced result using GSR with the estimated result from SART as an initial value. The optimization problem of the second step can be expressed as , ,
1 2 ‖ − ‖ 2 2 + ‖ ‖ 0 . . = * ,(14)
where ‖ − ‖ is an l2 data-fidelity term, ‖ ‖ is a regularization term, and is a regularization parameter. We can obtain adaptive dictionary by applying SVD to the estimate of according Eqs. (5)- (7). How to calculate is given below. Then, Eq. (14) becomes
, 1 2 ‖ − ‖ 2 2 + ‖ ‖ 0 . . = * .(15)
However, Eq. (15) is always hard to solve because the l0-norm optimization is non-convex. In this paper, the split Bregman iteration (SBI) algorithm [32] is used to solve this problem. Consider the following constrained optimization problem:
, ( ) + ( ) . . = .(16)
According to SBI, the minimization problem in Eq. (16) can be split to sub-problems, as shown in Algorithm 1.
Step 4 of SBI becomes:
+1 = ‖ ‖ 0 + 2 ‖ +1 − * − ‖ 2 2 .(18)
Step 5 of SBI becomes:
+1 = − ( +1 − * +1 ).(19)
Then, the minimization of Eq. (15) is transformed into two sub-problems concerning and . For a given , the sub-problem in Eq. (17) is a strictly quadratic convex optimization problem, which can be defined as
1 ( ) = 1 2 ‖ − ‖ 2 2 + 2 ‖ − * − ‖ 2 2 .(20)
We can obtain a closed solution for Eq. (20) by setting the gradient of 1 ( ) to zero, which can be expressed as
̂= ((1 + ) ) −1 ( + ( * + )),(21)
where is an identity matrix. For a given , the subproblems can be defined as
2 ( ) = ‖ ‖ 0 + 1 2 ‖ − * ‖ 2 2 ,(22)
where = − . According to the theorem in [24], Eq. (22) can be transformed into
2 ( ) = ∑ ( 1 2 ‖ − ‖ 2 + ‖ ‖ 0 ) =1 ,(23)
where = ( × × × ) / ( × ). Then, Eq. (22) transforms into sub-problems for all groups . Because = and = , the minimization for each group can be defined as follows:
1 2 ‖ − ‖ 2 2 + ‖ ‖ 0 .(24)
Therefore, a closed solution for Eq. (24) can be expressed as follows:
̂= * 1( ( ) − √2 ),(25)
where operator * indicates the element-wise product between two vectors and 1( ( ) − √2 ) is defined as
1( ( ) − √2 ) = { 1, ( ) > √2 0, ( ) ≤ √2 .(26)
Once is calculated for all groups, the final solution for the sub-problem is determined.
Summary of the Proposed Algorithm
Our algorithm is composed of two main parts: the SART reconstruction and GSR regularization. We summarize the pseudo-code of our GSR-SART algorithm in Algorithm 2.
Algorithm 2: GSR-SART
Initialization:
Given 0 , , 0 , 0 , 0 , 0 , , , , , ,
Repeat
Experimental Results
In this section, the results of extensive experiments are reported to validate the proposed method for few-view CT reconstruction. We also determine the impact of the number of best-matched patches and search window size . In the experiments, three representative slices, abdominal, pelvic, and thoracic images, were tested to demonstrate the performance of the proposed GSR-SART. All the images were downloaded from the National Cancer Imaging Archive. In all the experiments, the image arrays are 20 × 20 cm and the system projection matrix, in fan-beam geometry, was obtained by Siddon's ray-driven algorithm [33] with 64 projection views evenly distributed over 360°. The distance from the source to the rotation centre is 40 cm and the distance from the detector centre to the rotation centre is 40 cm. We use a flat detector with 512 bins. The images are 256 × 256 pixels. All the experiments were performed in MATLAB 2017a on a PC equipped with an AMD Ryzen 5 1600 CPU at 3.2 GHz and 16 GB RAM.
Experimental results
In this subsection, the experimental results of the different clinical images are given. The experiments were simulated under ideal conditions, which means that the measured projection data are noiseless. The parameters of GSR-SART were set as follows: the patch size was 8 × 8, which means that = 64, was set to 40, and was set to 40. The PSNR is defined as:
= 10 × 10 ( (max ( )) 2 (∑ =1 ( − * ) 2 )/ ),(27)
where is the reconstructed value, is the size of f, and * is the golden reference value. The RMSE is defined as:
= √(∑ =1 ( − * ) 2 )/ .(28)
The SSIM is defined as:
SSIM( , * ) = 2 ̅ ̅ * (2 * + 2 ) ( ̅2 + ̅ * 2 + 1 )( 2 + * 2 + 2 ) ,(29)
where ̅ and ̅ * are the mean values of and * , respectively, * is the covariance of and * , and 1 and 2 are constants.
The original abdominal image and reconstruction results are shown in Fig. 2. In Fig. 2(b), the result of FBP contains severe streak artifacts due to the incomplete projection data. It can be observed in Figs. 2(c) and 2(d) that there are still undesirable artifacts in the SART and EM results. In Fig. 2(e), TV-POCS removes all the streak artifacts, but the reconstructed image of TV-POCS suffers from obvious over-smoothing effects. GSR-SART achieves the best visual effect in Fig. 2(f), which shows that it suppresses most of the artifacts without introducing any other side effects. Table 1. It is obvious that our method achieves the best performance for all metrics, which demonstrates the ability of GSR-SART to better reduce artifacts and preserve structure than all the other methods. In Fig. 5, the reconstruction results of the pelvic image are given. Because of the extremely low sampling ratio, it is difficult to obtain useful information from the result of FBP in Fig. 5(b). The SART and EM methods can only remove some of the streak artifacts. In Fig. 5(e), TV-POCS lowers the spatial resolution while eliminating the streak artifacts and the edges of the tissues are blurred to different degrees. Note that while GSR-SART eliminates most of the artifacts, it maintains the edges better than other methods in the region indicated by the red arrow. To further demonstrate the performance of GSR-SART, the absolute difference images relative to the original images are shown in Fig. 6. Here, the loss of structural information in Fig. 6(b) is more than for other methods and the results from SART and EM still have artifacts. In Figs. 6(d) and 6(e), the artifacts are well suppressed and GSR-SART preserves more details, which can be observed in the lower part of the body, as indicated by the red arrow. The results of the thoracic image are shown in Fig. 7. In Fig. 7(b), the whole image is filled with streak artifacts and no clinically valuable structures can be recognized. Although SART and EM remove some artifacts, the spatial resolutions are not satisfactory and the blood vessels in the lungs are clearly blurred. TV-POCS and GSR-SART recover the most vessels in Figs. 7(e) and 7(f) and the spatial resolutions are close to the original image. However, artifacts still exist near the bones, as indicated by red arrows. The red square region of Fig. 7(a) is enlarged in Figure 8. It is easy to see that the artifacts and noise are severe and the spine is distorted heavily. In Fig. 8(e), the noise is still obvious and the structural details are blurred. Compared with other methods, GSR-SART suppresses more artifacts and noise and the edges of tissue are better maintained. The quantitative results are shown in Table 3. Consistent with the visual effects, GSR-SART has the best scores for all measurements and the improvements are impressive. Fig. 9. It can be observed that the PSNR reaches peak at = 40 and it slowly declines as increases further. In contrast to PSNR, the values of SSIM decrease monotonously as increases. One possible reason for this phenomenon is that PSNR focuses more on the reduction of artifacts and noise, which does not always correspond with structure preservation.
(a) (b) Fig. 9. Performance with respect to number of best-matched patches.
(a) (b) Fig. 10. Performance with respect to search window size.
2) Search Window Size
To investigate the sensitivity of , experiments were performed for various ranging from 30 to 90 in steps of 10 with a fixed number of best-matched patches of 40. The pelvic image was selected as the test image. The results are given in Fig. 10. Here, the values of PSNR and SSIM behave similarly. The values first increase and reach a peak when = 40. After that, they decrease slowly. Although there is a rebound after 60, = 40 is still the optimal selection.
Discussion and Conclusion
Despite the rapid development of CT imaging techniques, incomplete projection data reconstruction is still a major problem in this field. In this paper, we proposed a novel GSR-based SART algorithm for few-view CT reconstruction called GSR-SART. In this algorithm, we utilize a GSR model as the regularization term to eliminate streak artifacts and preserve structural details. To further explore nonlocal similarity in a target image, a dictionary of the GSR model, which is adaptively generated at each iteration, is learned from groups composed of similar patches. Three representative clinical slices of different parts of the human body were used to validate the performance of the proposed method.
In all the results, our method performs better than all the other popular methods qualitatively and quantitatively under the same sampling conditions. Specifically, GSR-SART demonstrates a superior ability to reduce artifacts and preserve details.
In our experiments, the parameters were manually selected. Grid search is a reasonable method for parameter selection in simulations, but it is not practical in actual situations. We also observed that the optimal selection for each image was different, which makes this problem more complicated. A popular machine learning technique is a possible way to adaptively determine the optimal parameter set by learning from an external dataset.
Another problem we note here is the computational cost. Due to the introduction of group sparsity, the computational complexity of GSR-SART is heavier than that of original dictionary learning based methods. One possible solution to accelerate the computation is to implement a version that uses parallel computing. Distributed computing, computing clusters, and graphics processing units (GPUs) are three alternative approaches.
In conclusion, we are very encouraged by the promising performance of GSR with respect to artifact reduction and detail preservation for few-view CT. These results demonstrate the potential of the group-based sparse representation method for medical imaging. In the future, the proposed network framework will be refined and adapted to deal with other topics in CT imaging, such as low-dose CT, metal artifact reduction, and interior CT.
Fig. 1 .
1Group construction. Each patch vector is extracted from image . Here, denotes the set composed of most similar patches and is a matrix composed of all the patch vectors in .
( ) = ‖ ‖ 0 . Then, invoking SBI, step 3 of SBI becomes:
For the abdominal images, = 1 − 5 and = 0.1, for the pelvic image, = 5 − 5 and = 0.1, and for the thoracic image, = 1.5 − 5 and = 0.08. To further evaluate the proposed GSR-SART method, we compared GSR-SART with FBP, EM, SART, and TV-POCS. The peak signal-to-noise ratio (PSNR), root-mean-square error (RMSE) and structural similarity (SSIM) are utilized to quantitatively evaluate the performance of the methods.
Fig. 2 .
2Few-view reconstruction results of the abdominal image from 64 noiseless projections over 360°. The display window is [−150 250] HU. (a) Original image and results obtained by (b) FBP, (c) SART, (d) EM, (e) TV-POCS, and (f) GSR-SART.
Fig. 3 .
3Horizontal profiles (128th row) of the abdominal image reconstructed by different methods from 64 projection views. (a) Overall profiles and (b) ROI1, (c) ROI2, and (d) ROI3 of the overall profiles. To further visualize the performance of the methods for the abdominal image, horizontal and vertical profiles of the abdominal image, which are indicated by red lines in Fig. 2(a), are shown in Figures 3 and 4. The profiles from the original image are given as references. Three regions of interest (ROIs) in each profile are enlarged for better visibility. Several arrows indicate the regions in which discrepancies can easily be identified. Here, the profiles generated from our algorithm are closer to the references. The quantitative evaluation of abdominal image is given in
Fig. 4 .
4Vertical profiles (128th column) of the abdominal image reconstructed by different methods from 64 projection views. (a) Overall profiles and (b) ROI1, (c) ROI2, and (d) ROI3 of the overall profiles.
Fig. 5 .
5Few-view reconstruction results of the pelvic image from 64 noiseless projections over 360°. The display window is [−150 250] HU. (a) Original image and results obtained by (b) FBP, (c) SART, (d) EM, (e) TV-POCS, and (f) GSR-SART.
Fig. 6 .
6Difference images relative to the original image. Results for (a) FBP, (b) SART (c) EM, (d) TV-POCS, and (e) GSR-SART.
Fig. 7 .Fig. 8 .
78Few-view reconstruction results of thoracic image from 64 noiseless projections over 360°. The display window is [-1000 250]HU. (a) Original image and results obtained by (b) FBP, (c) SART, (d) EM, (e) TV-POCS, and (f) GSR-SART. Enlarged region of the thoracic image. The display window is [−150 250] HU. Enlarged regions of (a) the true image, (b) FBP, (c) SART, (d) EM, (e) TV-POCS, and (f) GSR-SART.
Table 1 .
1Quantitative results obtained by different algorithms for the abdominal image Table 2. Quantitative results obtained by different algorithms for the pelvic image Table 3. Quantitative results obtained by different algorithms for the thoracic image To investigate the sensitivity of , experiments were performed with various ranging from 30 to 80 in steps of 10 with a fixed search window size of 40 × 40. The pelvic image was chosen as the test image. The results are shown inPSNR
RMSE
SSIM
FBP
23.49
0.06690
0.51857
SART
31.05
0.02803
0.84241
EM
34.43
0.01900
0.90744
TV-POCS
36.95
0.01421
0.95726
GSR-SART
45.64
0.00522
0.98448
PSNR
RMSE
SSIM
FBP
21.56
0.08358
0.47260
SART
32.02
0.02507
0.85824
EM
34.32
0.01924
0.91128
TV-POCS
37.32
0.01362
0.97062
GSR-SART
45.74
0.00516
0.98568
PSNR
RMSE
SSIM
FBP
21.99
0.07949
0.34400
SART
30.45
0.03004
0.79577
EM
33.91
0.02015
0.87844
TV-POCS
38.28
0.01219
0.95866
GSR-SART
43.53
0.00666
0.98155
B. Parameter Selection
1) Number of Best Matched Patches
Acknowledgments:We thank Kim Moravec, PhD, from Edanz Group China (www.liwenbianji.cn/ac), for editing the English text of a draft of this manuscript. This work was supported in part by the National Natural Science Foundation of China under Grants 61671312 and 61302028 and National Key R&D Program of China under Grants 2017YFB0802300.
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| [] |
[
"Gain with no Pain: Efficient Kernel-PCA by Nyström Sampling",
"Gain with no Pain: Efficient Kernel-PCA by Nyström Sampling"
] | [
"Nicholas Sterge \nDepartment of Statistics\nPennsylvania State University\n\n",
"Bharath Sriperumbudur \nDepartment of Statistics\nPennsylvania State University\n\n",
"Lorenzo Rosasco \nLCSL\nMassachusetts Institute of Technology\nIstituto Italiano di Tecnologia & DIBRIS\nUniversita' degli Studi di Genova\n\n",
"Alessandro Rudi \nSIERRA Project-Team\nINRIA andÉcole-Normale Supérieure\nPSL Research University\nParisFrance\n"
] | [
"Department of Statistics\nPennsylvania State University\n",
"Department of Statistics\nPennsylvania State University\n",
"LCSL\nMassachusetts Institute of Technology\nIstituto Italiano di Tecnologia & DIBRIS\nUniversita' degli Studi di Genova\n",
"SIERRA Project-Team\nINRIA andÉcole-Normale Supérieure\nPSL Research University\nParisFrance"
] | [] | In this paper, we propose and study a Nyström based approach to efficient large scale kernel principal component analysis (PCA). The latter is a natural nonlinear extension of classical PCA based on considering a nonlinear feature map or the corresponding kernel. Like other kernel approaches, kernel PCA enjoys good mathematical and statistical properties but, numerically, it scales poorly with the sample size. Our analysis shows that Nyström sampling greatly improves computational efficiency without incurring any loss of statistical accuracy. While similar effects have been observed in supervised learning, this is the first such result for PCA. Our theoretical findings, which are also illustrated by numerical results, are based on a combination of analytic and concentration of measure techniques. Our study is more broadly motivated by the question of understanding the interplay between statistical and computational requirements for learning. * | null | [
"https://arxiv.org/pdf/1907.05226v1.pdf"
] | 195,886,313 | 1907.05226 | b8ab12298ed09e2a4c9625af9ba2738f82066544 |
Gain with no Pain: Efficient Kernel-PCA by Nyström Sampling
Nicholas Sterge
Department of Statistics
Pennsylvania State University
Bharath Sriperumbudur
Department of Statistics
Pennsylvania State University
Lorenzo Rosasco
LCSL
Massachusetts Institute of Technology
Istituto Italiano di Tecnologia & DIBRIS
Universita' degli Studi di Genova
Alessandro Rudi
SIERRA Project-Team
INRIA andÉcole-Normale Supérieure
PSL Research University
ParisFrance
Gain with no Pain: Efficient Kernel-PCA by Nyström Sampling
In this paper, we propose and study a Nyström based approach to efficient large scale kernel principal component analysis (PCA). The latter is a natural nonlinear extension of classical PCA based on considering a nonlinear feature map or the corresponding kernel. Like other kernel approaches, kernel PCA enjoys good mathematical and statistical properties but, numerically, it scales poorly with the sample size. Our analysis shows that Nyström sampling greatly improves computational efficiency without incurring any loss of statistical accuracy. While similar effects have been observed in supervised learning, this is the first such result for PCA. Our theoretical findings, which are also illustrated by numerical results, are based on a combination of analytic and concentration of measure techniques. Our study is more broadly motivated by the question of understanding the interplay between statistical and computational requirements for learning. *
Introduction
Achieving good statistical accuracy under budgeted computational resources is a central theme in modern machine learning (Bottou and Bousquet, 2008). Indeed, the problem of understanding the interplay and trade-offs between statistical and computational requirements has recently received much attention. Nonparametric learning, and in particular kernel methods, have provided a natural framework to pursue these questions, see e.g. (Musco and Musco, 2017;Rudi et al., 2015;Alaoui and Mahoney, 2014;Bach, 2013;Calandriello et al., 2018;Orabona et al., 2008). On the one hand, these methods are developed in a sound mathematical setting and their statistical properties are well studied. On the other hand, from a numerical point of view, they scale poorly to large scale problems, and hence improved computational efficiency is of particular interest.
While initial studies have mostly focused on approximating kernel matrices (Drineas and Mahoney, 2005;Gittens and Mahoney, 2013;Jin et al., 2013), recent results have highlighted the importance of considering downstream learning tasks, if the interplay between statistics and computation is of interest. In particular, results in supervised learning have shown there are regimes where computational gains can be achieved with no loss of statistical accuracy (Rudi et al., 2015;Rudi and Rosasco, 2017). A basic intuition is that approximate computations provide a form of implicit regularization, hence memory and time requirements can be tailored to statistical accuracy allowed by the data (Rudi et al., 2015). To which extent similar effects occur beyond supervised learning is unclear. Indeed, the only result in this direction was recently shown for kernel k-means in (Calandriello et al., 2018).
In this paper, we consider one of the most basic unsupervised approaches, namely PCA, or rather its nonlinear version, that is kernel PCA (Schölkopf et al., 1998). We develop a computationally efficient approximate kernel PCA algorithm using the Nyström method (Williams and Seeger, 2001) with m sub-samples (NY-KPCA) and show its time complexity to be O(nm 2 + m 3 ) with a space complexity of O(m 2 ), in contrast to O(n 3 ) and O(n 2 ) time and space complexities of KPCA, where n is the sample size. Our main contribution is the analysis of NY-KPCA in terms of finite sample bounds on the reconstruction error of the corresponding -dimensional eigenspace (see Theorem 4.1 and related Corollaries 4.2 and 4.3). In particular, we show that NY-KPCA can achieve the same error of KPCA with m < n, thereby demonstrating computational gains can occur at no statistical loss. Moreover, we show that adaptive sampling using leverage scores (Alaoui and Mahoney, 2014) can lead to further gains. More precisely, we show that the requirement on m varies between (log n) 2 and n θ log n (θ < 1) depending on the size of , the rate of decay of eigenvalues of the covariance operator and the type of subsampling. Finally, we also present some simple numerical results to corroborate our theoretical results.
We note that some recent papers, see (Sriperumbudur and Sterge, 2018;Ullah et al., 2018), have considered the problem of deriving efficient kernel PCA approximations using random features (Rahimi and Recht, 2008). However, the notion of reconstruction error considered in these works is different from that of KPCA (Shawe-Taylor et al., 2005;Blanchard et al., 2007). The reason for a different notion of reconstruction error is to handle certain technicalities that arise in random feature approximation. As a consequence, these results are not directly comparable to our current work and KPCA. In contrast, our results based on Nyström approximation are directly comparable to that of KPCA, wherein we show that the proposed NY-KPCA has similar statistical behavior but better computational complexity than KPCA.
The paper is organized as follows. Relevant notations and definitions are collected in Section 2. Section 3 provides preliminaries on KPCA along with the list of assumptions that will be used throughout the paper. Approximate KPCA using Nyström method is presented in Section 3.2 and the main results of computational vs. statistical tradeoff for NY-KPCA are presented in Section 4. Missing proofs of the results are provided in the appendix.
Definitions and Notation
For a := (a 1 , . . . , a d ) ∈ R d and b :
= (b 1 , . . . , b d ) ∈ R d define a 2 := d i=1 a 2 i and a, b 2 := d i=1 a i b i .
a ⊗ 2 b := ab denotes the tensor product of a and b. I n denotes an n × n identity matrix. a ∧ b := min(a, b) and a ∨ b := max(a, b). [n] := {1, . . . , n} for n ∈ N. For constants a and b, a b (resp. a b) denotes that there exists a positive constant c (resp. c ) such that a ≤ cb (resp. a ≥ c b). For a random variable A with law P and a constant b, A p b denotes that for any δ > 0, there exists a positive constant c δ < ∞ such that P (A ≤ c δ b) ≥ δ.
For x, y ∈ H, a Hilbert space, x ⊗ H y is an element of the tensor product space H ⊗ H which can also be seen as an operator from H to H as (x ⊗ H y)z = x y, z H for any z ∈ H. α ∈ R is called an eigenvalue of a bounded self-adjoint operator S if there exists an x = 0 such that Sx = αx and such an x is called the eigenvector /eigenfunction of S and α. An eigenvalue is said to be simple if it has multiplicity one. For an operator S : H → H, S L 1 (H) , S L 2 (H) and S L ∞ (H) denote the trace, Hilbert-Schmidt and operator norms of S, respectively.
Kernel PCA by Nyström Sampling
In this section, we review kernel principal component analysis (KPCA) (Schölkopf et al., 1998) in population and empirical settings and introduce approximate kernel PCA using Nyström approximation. We assume the following for the rest of the paper:
Assumption 3.1. X is a separable topological space and (H, k) is a separable RKHS of real-valued functions X with a bounded, continuous, strictly positive definite kernel k satisfying sup x∈X k(x, x) =: κ < ∞.
KPCA and Empirical KPCA
Let X be a zero-mean random variable with law P defined on X . When X = R d , classical PCA (Jolliffe, 1986) finds a ∈ R d such that Var [ a, X 2 ] is maximized, with the constraint a 2 = 1. Defining C := E X∼P [XX ], the solution is simply the unit eigenvector of C corresponding to its largest eigenvalue. In practice, PCA is computed by replacing C with an empirical approximation C n = 1 n n i=1 X i X i based on a sample X 1 , . . . , X n . Kernel PCA extends this idea to an RKHS, H defined on X , by finding f ∈ H with unit norm such that
Var[f (X)] is maximized. Since Var[f (X)] = f, Cf H assuming E[f (X)] = 0 for all f ∈ H, we have f * = arg sup{ f, Cf H : f H = 1} where C is the (uncentered) covariance operator on H defined as C := X k(·, x) ⊗ H k(·, x) dP(x).(1)
The boundedness of k in Assumption 3.1 ensures that C is trace class and thus compact. Since C is positive and self-adjoint, the spectral theorem (Reed and Simon, 1980) gives
C = i∈I λ i φ i ⊗ H φ i ,(2)
where (λ i ) i∈I ⊂ R + are the eigenvalues and (φ i ) i∈I are the orthonormal system of eigenfunctions that span R(C) with index set I either being finite or countable, in which case λ i → 0 as i → ∞. The solution to the KPCA problem is thus the eigenfunction of C corresponding to its largest eigenvalue. We make the following simplifying assumption for ease of presentation.
Assumption 3.2. The eigenvalues (λ i ) i∈I of C are simple, positive, and w.l.o.g. they satisfy a decreasing rearrangement, i.e., λ 1 > λ 2 , . . .
Assumption 3.2 ensures that (φ i ) i∈I form an orthonormal basis and the eigenspace corresponding to each λ i is one-dimensional. This means the orthogonal projection operator onto the -eigenspace of C, i.e. span{(φ i ) i=1 }, is given by
P (C) = i=1 φ i ⊗ H φ i .(3)
The above construction corresponds to population version of KPCA when the data distribution P is known. If P is unknown and the knowledge of P is available only through the training set
(X i ) n i=1 i.i.d.
∼ P, then KPCA cannot be carried out as C depends on P. Therefore, an approximation to C is used to perform KPCA. Most commonly, this approximation is chosen to be the empirical estimator of C defined as
C n = 1 n n i=1 k(·, X i ) ⊗ H k(·, X i )(4)
resulting in empirical kernel PCA (EKPCA). Note that C n is a finite rank, positive, and selfadjoint operator. Thus the spectral theorem (Reed and Simon, 1980) yields
C n = n i=1λ iφi ⊗ Hφi ,(5)
where (λ i ) n i=1 ⊂ R + and (φ i ) n i=1 ⊂ H are the eigenvalues and eigenfunctions of C n . Similar to Assumption 3.2, we assume the following: Assumption 3.3. rank(C n ) = n, the eigenvalues (λ i ) n i=1 of C n are simple and w.l.o.g. they satisfy a decreasing rearrangement, i.e.,λ 1 ≥λ 2 ≥ . . ..
The eigensystem (λ i ,φ i ) n i=1 of C n can be obtained by solving an n-dimensional system involving the eigendecomposition of the Gram matrix K = [k(X i , X j )] i,j∈ [n] , which scales as O(n 3 ) (Schölkopf et al., 1998). In particular, the eigenvalues of K are related to those of C n as λ i (K) = nλ i . Moreover, if u i is an orthonormal eigenvector of K corresponding to the eigenvalue λ i (K), then it holds for all x ∈ X ,
φ i (x) = 1 nλ i n j=1 k(x, x j )u i,j .(6)
The above result proven in (Schölkopf et al., 2001) can be seen as a representer theorem (Kimeldorf and Wahba, 1971) for KPCA. Finally, note that, for some ≤ n, the orthogonal projection operator onto span{(φ i ) i=1 } is given by
P (C n ) = i=1φ i ⊗ Hφi .(7)
Approximate Kernel PCA using Nyström Method
For large sample sizes, since performing KPCA is computationally intensive, various approximation schemes that has been explored in the kernel machine literature can be deployed to speed up EKPCA. Recently, one such approximation involving random Fourier features has been studied by Sriperumbudur and Sterge (2018) and Ullah et al. (2018) to speed EKPCA while maintaining its statistical performance. In this paper, we explore the popular Nyström approximation (Williams and Seeger, 2001;Drineas and Mahoney, 2005) to speed up EKPCA and study the trade-offs between computational gains and statistical accuracy. The general idea in Nyström method is to obtain a low-rank approximation to the Gram matrix K, and replace K by this approximation in kernel algorithms, resulting in computational speedup. Since K is related to C n (as discussed in Section 3.1), Nyström method can also be seen as obtaining a low rank approximation to C n , which is what we exploit in obtaining a Nyström approximate KPCA. It follows from (6) that the eigenfunctions of C n lie in the space
H n = f ∈ H f = n i=1 α i k(·, X i ), α 1 , ..., α n ∈ R .
Therefore, it can be seen that EKPCA is a solution to the following problem
arg sup { f, C n f H : f ∈ H n , f H = 1} ,
assuming K is invertible 1 . Extending this representation, we propose Nyström KPCA (NY-KPCA) as a solution to the following problem:
arg sup { f, C n f H : f ∈ H m , f H = 1} ,(8)
where
H m = f ∈ H f = m i=1 α i k(·,X i ), α 1 , ..., α m ∈ R
is a low-dimensional subspace of H n and {X 1 , ...,X m } is a subset of the training set withX i 's being distinct. Basically, we are considering a plain Nyström approximation where the points {X 1 , . . . ,X m } are sampled uniformly without replacement from {X 1 , . . . , X n }, however, other subsampling methods are possible, see Section 3.2.1. The following result, which is proved in the supplement (see Section 6.1), shows that the solution to (8) is obtained by solving a finite dimensional linear system, which has better computational complexity than that of EKPCA. To this end, we first introduce some notation,
K mm = [k(X i ,X j )] i,j∈[m] , K nm = [k(X i ,X j )] i∈[n],j∈[m] ∈ R n×m , K mn = K nm . Proposition 3.4. Define the m × m matrix M = K −1/2 mm K mn K nm K −1/2 mm . The solution to (8) is given byφ 1,m =Z * m K −1/2 mm u 1,m , where u 1,m is the eigenvector of 1 n M corresponding to its largest eigenvalue andZ * m : R m → H, α → m i=1 α i k(·,X i ).
The cost of computing M is O(nm 2 + m 3 ) and the cost of computing its eigendecomposition is O(m 3 ). Thus, for m < n, the cost of NY-KPCA scales as O(nm 2 ), faster than the O(n 3 ) cost of EKPCA. DefineK
:= K nm K −1 mm K mn ,(9)
which is called the Nyström approximation (Williams and Seeger, 2001;Drineas and Mahoney, 2005) to the Gram matrix K. It is easy to verify that M andK have same eigenvalues since
M = K −1/2 mm K mn K −1/2 mm K mn andK = K −1/2 mm K mn K −1/2
mm K mn , and rank(M) = rank(K). Therefore we work withK and make the following assumption on its eigenvalues.
Assumption 3.5. rank(K) = m. The eigenvalues (λ i,m ) m i=1 of 1
nK are simple and w.l.o.g. they satisfy a decreasing rearrangement, i.e.,λ 1,m >λ 2,m . . . >λ m,m .
The symmetry of M guarantees orthonormality of (u i,m ) i , and the orthonormality of (φ i,m ) i follows. For some ≤ m, the orthogonal projector onto span{φ i,m } i=1 is given by
P m (C n ) = i=1φ i,m ⊗ Hφi,m .(10)
One may ask ifφ i,m are eigenfunctions of some operator on H. Denote P m as the orthogonal projector onto H m . It is simple to verify (Rudi et al., 2015, Theorem 2) that P m =Z * m K −1 mmZ m and that λ i,m ,φ i,m are the orthonormal eigenfunctions of P m C n P m , i.e.,
P m C n P mφi,m =λ i,mφi,m for all i ∈ [m].(11)
Therefore, we may think of P m C n P m as a low-rank approximation to C n .
Approximate Leverage Scores
In the above discussion on Nyström KPCA,X := {X 1 , . . . ,X m } is a subset of the training set X := {X 1 , . . . , X n } with the entries ofX being sampled uniformly without repetition from X. As an alternative to uniform sampling,X can be sampled according to the leverage score distribution (Alaoui and Mahoney, 2015;Drineas et al., 2012;Cohen et al., 2015). For any s > 0, the leverage scores associated with the training data X are defined as
(l i (s)) n i=1 , l i (s) = [K(K + nsI n ) −1 ] ii , i ∈ [n]
with the leverage score distribution being p i (s) = l i (s) n i=1 l i (s) according to which X can be sampled independently with replacement to achieveX. Since the leverage scores are computationally intensive to compute, usually, they are approximated and one such approximation is T -approximate leverage scores.
Definition 3.6. (T -approximate leverage scores) For a given s > 0, let (l i (s)) n i=1 be the leverage scores associated with the training data {X 1 , ..., X n }. Let δ > 0, s 0 > 0, and T ≥ 1. (l i (s)) n i=1 are T -approximate leverage scores, with confidence δ, if the following holds with probability at least 1 − δ:
1 T l i (s) ≤l i (s) ≤ T l i (s), ∀i ∈ [n], s > s 0 .
Given T -approximate leverage scores for s > s 0 ,X can be obtained by sampling X with replacement according to the sampling distributionp i (s) =l i (s)/ n i=1l i (s). Having obtainedX, (8) can be solved exactly as in Proposition 3.4. We refer to this method as approximate leverage score (ALS) Nyström subsampling.
Computational vs. Statistical Trade-Off: Main Results
As shown in the earlier section, Nyström kernel PCA approximates the solution to empirical kernel PCA with less computational expense. In this section, we explore whether this computational saving is obtained at the expense of statistical performance. As in Sriperumbudur and Sterge (2018), we measure the statistical performance of KPCA, EKPCA, and NY-KPCA in terms of reconstruction error. In linear PCA, the reconstruction error, given by
E X∼P I − P (C) X 2 2 ,(12)
is the error involved in reconstructing a random variable X by projecting it onto the -eigenspace (i.e., span of the top-eigenvectors) associated with its covariance matrix, C = E[XX ] through the orthogonal projection operator P (C). Clearly, the error is zero when = d. The analog of the reconstruction error in KPCA, as well as EKPCA and NY-KPCA, can be similarly stated in terms of their projection operators, (3), (7), and (10) as follows. For any orthogonal projection operator P : H → H, define the reconstruction error as
R(P ) := E X∼P (I − P ) k(·, X) 2 H .
For the linear kernel this exactly the reconstruction error of PCA. In the following, we often make use of the following identity R(P ) = (I − P )C 1/2 2 L 2 (H) ,
for which we report a proof in the supplement (see Section 6.2). Based on this definition, the reconstruction error in KPCA, EKPCA and NY-KPCA are given by R C, := R(P (C)), R Cn, := R(P (C n )), and R nys Cn, := R(P m (C n ))
respectively. The following theorem, proved in the supplement (see Section 6.3), provides finitesample bounds on the reconstruction error associated with NY-KPCA, under both uniform and approximate leverage score subsampling, from which convergence rates may be obtained.
Theorem 4.1. Suppose Assumptions 3.1-3.5 hold. For any t > 0, define N C (t) = tr((C +tI) −1 C) and N C,∞ (t) = sup x∈X k(·, x), (C + tI) −1 k(·, x) H . Then the following hold:
(i) Suppose n > 3, 0 < δ < 1, 9κ n log n δ ≤ t ≤ λ 1 , and m ≥ (67 ∨ 5N C,∞ (t)) log 4κ tδ . Then, for plain Nyström subsampling:
P n (X i ) n i=1 : R nys Cn, ≤ N C (t) (6λ + 42t) ≥ 1 − 2δ.(15)
(ii) For 0 < δ < 1, suppose there exists T ≥ 1 such that (l i (s)) n i=1 are T −approximate leverage scores with confidence δ for any t ≥ 19κ n log 2n δ . Assume approximate leverage score Nyström subsampling is used with
t = min 19κ n log 2n δ ≤ t ≤ λ 1 78T 2 N C (t) log 8n δ ≤ m .
If n ≥ 1655κ + 223κ log 2κ δ and m ≥ 334 log 8n δ , then
P n (X i ) n i=1 : R nys Cn, ≤ N C (t) (6λ + 42t) ≥ 1 − 3δ.(16)
To understand the significance of Theorem 4.1, we have to compare it to the behavior of the reconstruction error associated with EKPCA, i.e., R Cn, . (Rudi et al., 2015, Theorem 3.1) showed that for n > 3, 0 < δ < 1 and 9κ n log n δ ≤ t ≤ λ 1 ,
P n {(X i ) n i=1 : R Cn, ≤ 9N C (t) (λ + t)} ≥ 1 − δ.(17)
Comparing (15) and (16) to (17), it is clear that NY-KPCA has a statistical behavior similar to that EKPCA. However, it is not obvious whether such a behavior is achieved for m < n, i.e., the order of dependence of m on n is not clear. To clarify this, in the following, we present two corollaries (proved in the supplement, see Sections 6.4 and 6.5) to Theorem 4.1, which compare the asymptotic convergence rates of R C, , R Cn, and R nys Cn, under an additional assumption on the decay rate of eigenvalues of C.
Corollary 4.2 (Polynomial decay of eigenvalues). Suppose Ai −α ≤ λ i ≤Āi −α for α > 1 and A,Ā ∈ (0, ∞). Let = n θ α , θ > 0. Then the following hold:
(i) n −θ(1− 1 α ) R C, n −θ(1− 1 α ) ; (ii) R Cn, P n n −θ(1− 1 α ) , θ < 1 log n n 1− 1 α , θ ≥ 1 ;
(iii) For plain Nyström subsampling: (iv) For approximate leverage score Nyström subsampling:
R nys Cn, P n n −θ(1− 1 α ) , θ < 1,R nys Cn, P n n −θ(1− 1 α ) , θ < 1, m n θ α log n log n n 1− 1 α , θ ≥ 1, m n 1 α (log n) 1− 1 α .
Remark 1. (i) The above result shows that the reconstruction errors associated with KPCA and EKPCA have similar asymptotic behavior as long as does not grow to infinity too fast, i.e., θ < 1. On the other hand, for θ ≥ 1, the reconstruction error of EKPCA has slower asymptotic convergence to zero than that of KPCA. If grows to infinity faster with the rate controlled by θ, then the variance term dominates the bias resulting in a slower convergence rate compared to that of KPCA.
(ii) Comparing (ii) and (iii) in the above result, we note that EKPCA and NY-KPCA have similar convergence behavior as long as m is large enough where the size of m is controlled by the growth of through θ. For the case of θ ≥ 1 in (iii), we require m n log n log n log n which means asymptotically m should be of the same order as n. On the other hand, the approximate leverage score Nyström subsampling gives same convergence rates as that of EKPCA but requiring far fewer samples than that for NY-KPCA with plain Nyström subsampling. These results show that for the interesting case of θ < 1 where EKPCA performance matches with that of KPCA, NY-KPCA also achieves similar performance, albeit with lower computational requirement. Corollary 4.3 (Exponential decay of eigenvalues). Suppose Be −τ i ≤ λ i ≤Be −τ i for τ > 0 and B,B ∈ (0, ∞). Let = 1 τ log n θ for θ > 0. Then the following hold:
(i) n −θ R C, n −θ ;
(ii) R Cn, P n n −θ log n, θ < 1 n −1 (log n) 2 , θ ≥ 1 ;
(iii) For plain Nyström subsampling:
R nys
Cn, P n n −θ log n, θ < 1, m n θ log n n −1 (log n) 2 , θ ≥ 1, m n log n log n log n ;
(iv) For approximate leverage score Nyström subsampling:
R nys Cn, P n n −θ log n, θ < 1, m (log n) 2 n −1 (log n) 2 , θ ≥ 1, m log n log n log n . Corollary 4.3 shares similar behavior to that Corollary 4.2 as discussed in Remark 1 but just that it yields faster rates since the RKHS is smooth as determined by the rate of decay of eigenvalues. In addition, the approximate leverage score Nyström subsampling based KPCA requires only (log n) 2 subsamples to match the performance of EKPCA resulting in substantial computational savings without any loss in statistical accuracy.
As mentioned in Section 1, the above results are the first of the kind related to computational vs. statistical trade-off in kernel PCA. While (Sriperumbudur and Sterge, 2018;Ullah et al., 2018) studied similar question for kernel PCA using random features, the results are not directly comparable because of the different cost function considered in these works. To elaborate, these works also considered the reconstruction error defined in (14) through (13), however, in L 2 (P) norm, which is weaker than the RKHS norm. For classical PCA this would correspond to considering the error E[(X (I − P (C))X) 2 ] rather than (12). This choice is made necessary by the fact that random features corresponding to a kernel, might in general not belong to the corresponding RKHS. Clearly this error choice does not allow a direct comparison to the convergence behavior of KPCA.
Experiments
The goal of our experiments is to demonstrate on benchmark data that NY-KPCA achieves similar error to that of EKPCA, with significantly less computation time. For our experiments, we use the samples pertaining to the digits 2 and 5 in the MNIST handwritten digit dataset, http: //yann.lecun.com/exdb/mnist/, yielding sample sizes of n = 5958 and n = 5421, respectively with each sample belonging to R 784 . EKPCA is performed on each of these two digits using a Gaussian kernel, k(·, x) = exp{−σ · − x 2 2 }, with σ = 1 × 10 −7 and NY-KPCA is performed with plain Nyström subsampling, i.e., uniformly without replacement, for m =100, 500 and 1000 Nyström subsamples with 100 repetitions being performed for each m to generate error bars. The reconstruction error is measured aŝ
R(P ) := 1 n n i=1 k(·, X i ) − P k(·, X i ) 2 H ,
with P : H → H chosen to be P (C n ) and P m (C n ) for EKPCA and NY-KPCA respectively. These quantities can be computed aŝ where (λ j , α j ) n j=1 and (λ j,m , u j ) m j=1 are the eigenvalue-vector pairs of 1 n K and 1 n M respectively. The number of principal components, , is varied from 1 to m. The results of the experiment are summarized in Figure 1, where we observe that NY-KPCA has similar performance to that of EKPCA in terms of the empirical reconstruction error until a certain value of beyond which the performance seems to be surprisingly better than EKPCA. On the computational front, NY-KPCA is significantly faster than EKPCA with the latter having a runtime of 337 seconds. Similar behavior is observed for digit 2 and the results are presented in Figure 2.
R(P (C n )) = 1 n tr(K) − 1 n 2 j=1 α j K 2 α ĵ λ j = n i= +1λ i (18) andR (P m (C n )) = 1 n tr(K) − 1 n j=1 u j Mu j = n i=1λ i − i=1λ i,m ,(19)
Proofs
In this section, we present the proofs.
Proof of Proposition 3.4
Define Z n : H → R n , f → (f (X 1 ), . . . , f (X n )) andZ m : H → R m , f → f (X 1 ), . . . , f (X n ) .
The adjoint ofZ m (Smale and Zhou, 2007) is given bỹ
Z * m : R m → H, α → m i=1 α i k(·,X i ).
Thus, any f ∈ H m may be written asZ * m α, for some α ∈ R m and so
f, C n f H = 1 n Z * m α, Z * n Z nZ * m α H = 1 n α Z m Z * n Z nZ * m α,
where we used Z * n Z n = 1 n C n . It is easy to verify that Z nZ * m = K nm andZ m Z * n = K mn . Therefore, (8) can be written as
arg sup 1 n α K mn K nm α : α K mm α = 1 .(20)
Letting u = K 1/2 mm α simplifies the constraint in (20) to u u = 1, and we write (20) as arg sup 1 n u K −1/2 mm K mn K nm K −1/2 mm u : u u = 1 .
The solution to the above problem is the unit eigenvector of 1 n K −1/2 mm K mn K nm K −1/2 mm corresponding to its largest eigenvalue. Denoting this eigenvector as u 1,m , we obtain a functionφ 1,m ∈ H solving the NY-KPCA problem in (8) viaφ 1,m =Z * m K −1/2 mm u 1,m .
Proof of (13)
Note that
R(P ) = E (I − P )k(·, X) 2 H = E (I − P )k(·, X), (I − P )k(·, X) H = E (I − P )k(·, X), k(·, X) H = E (I − P ), k(·, X) ⊗ H k(·, X) L 2 (H) ,(21)
where we used Bf, g H = B, f ⊗ H g L 2 (H) and (I − P ) 2 = (I − P ) in (21). Since k is bounded, it follows that
E (I − P ), k(·, X) ⊗ H k(·, X) L 2 (H) = (I − P ), E[k(·, X) ⊗ H k(·, X)] L 2 (H) .
The result follows by using the above in (21) and noting that (I − P ), C L 2 (H) = tr ((I − P )C) = tr C 1/2 (I − P )(I − P )C 1/2 = (I − P )C 1/2 , (I − P )C 1/2
L 2 (H) = (I − P )C 1/2 2 L 2 (H) ,
where we have used the invariance of trace under cyclic permutations.
Proof of Theorem 4.1
(i) For t > 0, we have R nys Cn, = (I − P m (C n ))C 1/2 2 L 2 (H) = (I − P m (C n ))(C n + tI) 1/2 (C n + tI) −1/2 C 1/2 2
L 2 (H) ≤ (I − P m (C n ))(C n + tI) 1/2 2 L ∞ (H) (C n + tI) −1/2 C 1/2 2 L 2 (H) .(22)
We now bound the terms in (22). First, we have (C n + tI) −1/2 C 1/2 2 L 2 (H) = (C n + tI) −1/2 (C + tI) 1/2 (C + tI) −1/2 C 1/2 2
L 2 (H) ≤ (C n + tI) −1/2 (C + tI) 1/2 2 L ∞ (H) (C + tI) −1/2 C 1/2 2 L 2 (H) = (C n + tI) −1/2 (C + tI) 1/2 2 L ∞ (H) (A) N C (t),(23)
where we used the fact (C + tI) −1/2 C 1/2 2 L 2 (H) = tr(C 1/2 (C + tI) −1 C 1/2 ) = tr((C + tI) −1 C) =: N C (t). Next, we have
(I − P m (C n ))(C n + tI) 1/2 2 L ∞ (H) ≤ 2 (I − P m )(C n + tI) 1/2 2 L ∞ (H) (B) +2 (P m − P m (C n ))(C n + tI) 1/2 2 L ∞ (H) (D) ,(24)
where P m = Z * m (K mm ) −1 Z m is the orthogonal projector onto H m (see Section 3.2). (B) can be bounded as
(B) ≤ (I − P m )(C + tI) 1/2 2 L ∞ (H) (B 1 ) (C + tI) −1/2 (C n + tI) 1/2 2 L ∞ (H) (B 2 ) ,(25)
and (D) as
(D) ( * ) = (I − P m (C n ))P m (C n + tI) 1/2 2 L ∞ (H) = (I − P m (C n ))P m (C n + tI)P m (I − P m (C n )) L ∞ (H) ≤ (I − P m (C n ))P m C n P m (I − P m (C n )) L ∞ (H) + t (I − P m (C n ))P m (I − P m (C n )) L ∞ (H) , ( * * ) ≤λ +1,m + t,(26)
where we used the facts that R(P m (C n )) ⊂ R(P m ) in ( * ) and P m (C n ) projects onto theeigenspace of P m C n P m in ( * * ).λ +1,m can be bounded aŝ
λ +1,m ≤ |λ +1,m −λ +1 | +λ +1 ( †) ≤ 1 n K − K L ∞ (R n ) +λ ,(27)
where ( †) follows from the Hoffman-Wiendladt inequality (R. Bhatia, 1994). We may rewrite (27) as
1 n K − K L ∞ (R n ) = 1 n Z n (I − P m )Z * n L ∞ (R n ) = (I − P m )C n (I − P m ) L ∞ (H) = C 1/2 n (I − P m )C 1/2 n L ∞ (H) ≤ C 1/2 n (C + tI) −1/2 2 L ∞ (H) (C + tI) 1/2 (I − P m ) 2 L ∞ (H) ( ‡) ≤ (C n + tI) 1/2 (C + tI) −1/2 2 L ∞ (H) (C + tI) 1/2 (I − P m ) 2 L ∞ (H) ,(28)
where we used
C 1/2 n (C + tI) −1/2 2 L ∞ (H) ≤ C 1/2 n (C n + tI) −1/2 2 L ∞ (H) (C n + tI) 1/2 (C + tI) −1/2 2 L ∞ (H)
and C 1/2 n (C n +tI) −1/2 2 L ∞ (H) ≤ 1 in ( ‡). The result follows by combining (22)-(28) and employing Lemmas 6.1 and 6.2 for (iii).
(ii) The proof follows exactly as in (i); however, we bound (I − P m )(C + tI) 1/2 2 L ∞ (H) with Lemma 6.3 with t 0 = 19κ n log 2n δ .
Lemma 6.1. For δ > 0, suppose 9κ n log n δ ≤ t ≤ λ 1 . Then the following hold:
(i) P n 2 3 ≤ (C + tI) 1/2 (C n + tI) −1/2 L ∞ (H) ≤ √ 2 ≥ 1 − δ;
(ii) P n (C + tI) −1/2 (C n + tI) 1/2 L ∞ (H) ≤ 3 2 ≥ 1 − δ;
(iii) P n λ + t ≤ 3 2 (λ + t) ≥ 1 − δ.
Proof. (i) The result is quoted from Lemma 3.6 of (Rudi et al., 2013) with α = 1 2 . (ii) This is a slight variation of (i) and the proof idea follows that of Lemma 3.6 of (Rudi et al., 2013) with α = 1 2 . Note that
(C + tI) −1/2 (C n + tI) 1/2 L ∞ (H) = (C + tI) −1/2 (C n + tI)(C + tI) −1/2 1/2 L ∞ (H)
.
By defining B n = (C + tI) −1/2 (C − C n )(C + tI) −1/2 , we have I − B n = (C + tI) −1/2 ((C + tI) − C + C n ) (C + tI) −1/2 = (C + tI) −1/2 (C n + tI)(C + tI) −1/2 and therefore (C + tI) −1/2 (C n + tI) 1/2
L ∞ (H) = I − B n 1/2 L ∞ (H) ≤ 1 + B n L ∞ (H) 1/2 .(29)
It follow from the proof of Lemma 3.6 of (Rudi et al., 2013) that for 9κ n log n δ ≤ t,
P n B n L ∞ (H) ≤ 1 2 ≥ 1 − δ.(30)
Combining (29) and (30) completes the proof.
(iii) Since 2 3 ≤ (C + tI) 1/2 (C n + tI) −1/2 L ∞ (H) as obtained in (i), it is equivalent (see (Rudi et al., 2013, Lemmas B.2 and 3.5)) to C n + tI 3 2 (C + tI). This implies (see Gohberg et al., 2003) that λ k (C n + tI) ≤ λ k ( 3 2 (C + tI)) = 3 2 λ k (C + tI) for all k ≥ 1.
Lemma 6.2 ( (Rudi et al., 2015), Lemma 6). Suppose Assumption 3.1 holds, and suppose for some m < n, the set {X j } m j=1 is drawn uniformly from the set of all partitions of size m of the training data, {X i } n i=1 . For t > 0 and any δ > 0 such that m ≥ (67 ∨ 5N C,∞ (t)) log 4κ tδ , we have
P n (I − P m )(C + tI) 1/2 2 L ∞ (H) ≤ 3t ≥ 1 − δ,
where P m is the orthogonal projector onto H m = span{k(·,X j )|j ∈ [m]}.
Lemma 6.3 ( (Rudi et al., 2015), Lemma 7). Suppose Assumption 3.1 holds. Let (l i (s)) n i=1 be the collection of approximate leverage scores. Letting N := {1, ..., n}, for t > 0 define p t as the distribution over N with probabilities p t (i) =l i (t)/ n j=1l j (t). Let I m = {i 1 , ..., i m } ⊂ N be a collection of indices independently sampled from p t with replacement. Let P m be the orthogonal projector onto H m = span{k(·,X j )|j ∈ I m }. Additionally, for any δ > 0, suppose the following hold:
1. There exists T ≥ 1 and t 0 > 0 such that for any s ≥ t 0 , (l i (s)) n i=1 are T −approximate leverage scores with confidence δ,
2. n ≥ 1655κ + 223κ log 2κ δ , 3. t 0 ∨ 19κ n log 2n δ ≤ t ≤ λ 1 , 4. m ≥ 334 log 8n δ ∨ 78T 2 N C (t) log 8n δ .
Then P n (I − P m )(C + tI) 1/2 2
L ∞ (H) ≤ 3t ≥ 1 − 2δ.
6.4 Proof of Corollary 4.2 (i) From Theorem 4.1 (i) we have
R C, = i> λ i i> i −α ∞ x −α dx 1−α = n −θ(1− 1 α ) .
Similarly,
R C, = i> λ i i> i −α ∞ x −α dx 1−α = n −θ(1− 1 α ) .
(ii) This is Theorem 3.2 of (Rudi et al., 2015) with α = 1 2 , r = α, p = 2, and = n θ α . (iii) Theorem 4.1 (iii) and Proposition A.1 yield R nys Cn,
P n t − 1 α n −θ + t 1− 1 α ≤ t 1− 1 α , t ≥ n −θ t − 1 α n −θ , t ≤ n −θ ,
where log n n t ≤ λ 1 and m N C,∞ (t) log 1 t with N C,∞ (t) = sup x∈X k(·, x), (C + tI) −1 k(·, x) H 1 t . First, consider the case when t ≥ n −θ . This means R nys Cn, inf t 1− 1 α : t log n n ∨ n −θ , m 1 t log 1 t .
For θ < 1, we obtain
R nys Cn, inf t 1− 1 α : t n −θ , m 1 t log 1 t ≤ n −θ(1− 1 α ) if m n θ log n. For θ ≥ 1, we obtain R nys Cn, inf t 1− 1 α : t log n n , m 1 t log 1 t ≤ log n n (1− 1 α )
if m n log n log n log n . Next, consider the case when t ≤ n −θ which means
R nys Cn, inf t − 1 α n −θ : log n n t n −θ , m 1 t log 1 t ≤ n −θ(1− 1 α )
when θ < 1 and m n θ log n.
(iv) Theorem 4.1(iv) and Proposition A.1 yield R nys Cn,
P n t − 1 α n −θ + t 1− 1 α ≤ t 1− 1 α , t ≥ n −θ t − 1 α n −θ , t ≤ n −θ ,
where log n n t ≤ λ 1 and m N C (t) log n t − 1 α log n. The result follows by carrying out the analysis as in (iii) for θ < 1 and θ ≥ 1.
Proof of Corollary 4.3
(i) From Theorem 4.1 (i) we have
R C, = i> λ i i> e −τ i ∞ e −τ x dx e −τ = n −θ and R C, = i> λ i i> e −τ i ∞ +1 e −τ x dx e −τ ( +1) = e −τ n −θ .
(ii) Theorem 4.1 (ii) and Proposition A.2 yield
R Cn, P n n −θ + t log 1 t ≤ n −θ log 1 t , t ≤ n −θ t log 1 t , t ≥ n −θ ,
where log n n t ≤ λ 1 . For the case of t ≤ n −θ , we obtain R Cn, inf n −θ log 1 t : log n n t ≤ n −θ = n −θ log n,
where the constraint is only valid for θ < 1. On the other hand, for t ≥ n −θ , we obtain R Cn, inf t log 1 t : t log n n ∨ n −θ = log n n log n log n ≤ (log n) 2 n , which holds for θ ≥ 1.
(iii) Arguing similarly as in (ii), it follows that for θ < 1 and m n θ log n, we obtain a rate of n −θ log n for R nys Cn, . Similarly for θ ≥ 1 and m ≥ n log n log n log n , we obtain a rate of n −1 (log n) 2 . (iv) Arguing as in (ii) and enforcing the restriction m log n log 1 t imposed by Theorem 4.1 (ii) yields the result.
Conclusions
In this paper, we considered the problem of deriving an approximation to kernel PCA using Nyström method. This latter approach seemingly overcomes some of the difficulties of other approaches based on random features. In particular, it allows to derive error estimates directly comparable to those typically considered to analyze the statistical properties of KPCA. Our results indicate the existence of regimes where computational gains can be achieved while preserving statistical accuracy. These results parallel recent findings in supervised learning and are among the first of this kind for unsupervised learning.
Our study opens a number of possible questions. For example, still for KPCA, it would be interesting to understand the properties of Nyström sampling in combination with iterative eigensolvers, both batch (e.g., the power method) and stochastic (e.g., Oja's rule). The application of our approach to other spectral methods, such as those used in graph and manifold learning, would be interesting. Beyond PCA and spectral methods, our study naturally yields the question of which other learning problems can have analogous statistical and computational trade-offs. For example, it would be interesting to consider applications of our approach to independence tests based on covariance and cross-covariance operators (Gretton et al., 2008), or mean embeddings (Sriperumbudur et al., 2010).
Proof. We have
N C (t) = tr (C + tI) −1 C = i≥1 λ i λ i + t ≤ i≥1Ā i −α Ai −α + t =Ā A i≥1 i −α i −α + tA −1 .
Let u = t 1/α A −1/α x =⇒ u α = tA −1 x α and dx = t −1/α A 1/α du. Therefore,
i≥1 i −α i −α + tA −1 ≤ ∞ 0 x −α x −α + tA −1 dx = ∞ 0 1 1 + tA −1 x α dx = A t 1/α ∞ 0 1 1 + u α du.
Since 1 1+u α is decreasing in α on u ∈ (0, ∞), we have
1 1 + u α ≤ 1 1 + u 2 , if α ≥ 2. So for 2 ≤ α, A t 1/α ∞ 0 1 1 + u α du < ∼ t −1/α ∞ 0 1 1 + u 2 du = t −1/α tan −1 (u)| ∞ 0 = π 2 t −1/α ,
implying N C (t) t −1/α . For 1 < α < 2, we obtain
t −1/α ∞ 0 1 1 + u α du ≤ t −1/α ∞ k=0 1 1 + k α ≤ t −1/α 1 + ∞ k=1 1 k α .
Since 1 + ∞ k=1 1 k α converges for α > 1, we obtain N C (t) t −1/α . Proposition A.2. Suppose Be −τ i ≤ λ i ≤Be −τ i for τ > 0 and B,B ∈ (0, ∞). Let = 1 τ log n θ , θ > 0. Then N C (t) log 1 t .
Proof.
We have
N C (t) = tr (C + tI) −1 C = i≥1 λ i λ i + t ≤B e −τ i Be −τ i + t =B B i≥1 1 1 + tB −1 e τ i ∞ 0 1 1 + tB −1 e τ x dx = x − 1 τ log tB −1 e τ x + 1 ∞ 0
.
Since
x − 1 τ log tB −1 e τ x + 1 = 1 τ log(e τ x ) − log tB −1 e τ x + 1 = 1 τ log t −1 B e τ x e τ x + t −1 B , evaluating 1 τ log t −1 B e τ x e τ x + t −1 B ∞ 0 yields the result.
Figure 1 :
1Empirical reconstruction error of EKPCA and average empirical reconstruction error of 100 repetitions of NY-KPCA on digit 5 versus number of principal components ; error bars represent ±2 standard deviations. Runtime in seconds is given in parentheses next to the number of Nyström subsamples m. Runtime for EKPCA is 337 seconds.
Figure 2 :
2Empirical reconstruction error of EKPCA and average empirical reconstruction error of 100 repetitions of NY-KPCA on digit 5 versus number of principal components ; error bars represent ±2 standard deviations. Runtime in seconds is given in parentheses next to the number of Nyström subsamples m. Runtime for EKPCA is 478 seconds.
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A Technical Results Proposition A.1. Suppose Ai −α ≤ λ i ≤Āi −α for α > 1 and A,Ā ∈ (0, ∞). The following holds: N C (t) t −1/α. A Technical Results Proposition A.1. Suppose Ai −α ≤ λ i ≤Āi −α for α > 1 and A,Ā ∈ (0, ∞). The following holds: N C (t) t −1/α .
| [] |
[
"SOLVING ONE-VARIABLE EQUATIONS IN FREE GROUPS",
"SOLVING ONE-VARIABLE EQUATIONS IN FREE GROUPS"
] | [
"Dimitri Bormotov ",
"Robert Gilman ",
"Alexei Myasnikov "
] | [] | [] | Equations in free groups have become prominent recently in connection with the solution to the well known Tarski Conjecture. Results of Makanin and Rasborov show that solvability of systems of equations is decidable and there is a method for writing down in principle all solutions. However, no practical method is known; the best estimate for the complexity of the decision procedure is P-space.The special case of one variable equations in free groups has been open for a number of years, although it is known that the solution sets admit simple descriptions. We use cancellation arguments to give a short and direct proof of this result and also to give a practical polynomial time algorithm for finding solution sets. One variable equations are the only general subclass of equations in free groups for which such results are known.We improve on previous attempts to use cancellation arguments by employing a new method of reduction motivated by techniques from formal language theory. Our paper is self contained; we assume only knowedge of basic facts about free groups. | 10.1515/jgt.2008.080 | [
"https://export.arxiv.org/pdf/math/0607176v1.pdf"
] | 17,309,791 | math/0607176 | 9feeea3bc459b394513800ac3aa5db9e7791a937 |
SOLVING ONE-VARIABLE EQUATIONS IN FREE GROUPS
6 Jul 2006
Dimitri Bormotov
Robert Gilman
Alexei Myasnikov
SOLVING ONE-VARIABLE EQUATIONS IN FREE GROUPS
6 Jul 2006arXiv:math/0607176v1 [math.GR]
Equations in free groups have become prominent recently in connection with the solution to the well known Tarski Conjecture. Results of Makanin and Rasborov show that solvability of systems of equations is decidable and there is a method for writing down in principle all solutions. However, no practical method is known; the best estimate for the complexity of the decision procedure is P-space.The special case of one variable equations in free groups has been open for a number of years, although it is known that the solution sets admit simple descriptions. We use cancellation arguments to give a short and direct proof of this result and also to give a practical polynomial time algorithm for finding solution sets. One variable equations are the only general subclass of equations in free groups for which such results are known.We improve on previous attempts to use cancellation arguments by employing a new method of reduction motivated by techniques from formal language theory. Our paper is self contained; we assume only knowedge of basic facts about free groups.
Introduction
A one variable equation E(x) = 1 of degree d in a finitely generated free group F is an expression of the form (1) u 0 x ε 0 u 1 x ε 1 . . . u d−1 x ε d−1 = 1 composed of elements u i ∈ F , integers ε i = ±1 and a symbol x not in F . A solution to (1) is an element g ∈ F such that substitution of g for x yields 1 in F . Lyndon [17] was the first to study equations of this sort. He characterized solution sets in terms of parametric words. The parametric words involved were simplified by Lorents [19,20] and Appel [1]. However, Lorents announced his results without proof, and Appel's published proof has a gap (see [6]). A complete proof has been provided recently by Chiswell and Remeslennikov [6].
Chiswell and Remeslennikov's novel analysis involves algebraic geometry ( [2], [22].) First they describe the isomorphism types of the coordinate groups of irreducible one-variable equations over F , and then they deduce the structure of the solution sets. The latter part is easy, but the former requires sophisticated techniques involving ultrapowers and Lyndon length functions. The key point is that coordinate groups of irreducible equations over F are subgroups of the ultrapower F I /D of F over a countable set I with a non-principal ultrafilter D.
One can view the group F as a subgroup of F I /D under the canonical diagonal embedding. From this point of view the coordinate groups are precisely the finitely generated subgroups of F I /D containing F i.e., the socalled F -subgroups. In particular up to isomorphism the coordinate groups of irreducible one-variable equations over F are the subgroups of F I /D of the form F, g , g ∈ F I /D.
Investigation of such F -subgroups of F I /D is not easy and involves a careful analysis of Lyndon functions. (It might be interesting to see whether it is easier to use free actions on Λ-trees.) The computations can be simplified by employing a result from [11] which states that the coordinate groups of irreducible varieties are precisely the finitely generated F -subgroups of the free exponential Lyndon group F Z [t] . As this group is the union of an infinite ascending chain of extensions of centralizers of F [23], one can use Bass-Serre theory to study F -subgroups of F Z [t] .
Chisewell and Remeslennikov's method is very powerful and potentially useful for more than just free groups. However, it does have the disadvantage of not giving an algorithm for explicitly describing the set of solutions.
This paper is a refinement and extension of [9] where results from formal language theory are used to describe solution sets of one-variable equations in free groups. As it turns out, formal language techniques are not required; straightforward cancellation arguments suffice. It seems likely that these arguments can be extended to other groups admitting suitable (not necessarily Lyndon) length functions. The main advantage of this method is that it is short and yields a polynomial time algorithm for producing a description of all solutions. This algorithm has been implemented by the first author [4]. Theorem 1. The solution set for a one variable equation of positive degree in a free group F is a finite union of sets uv i w where u, v, w ∈ F and i ranges over all integers. There is a polynomial time algorithm for finding these sets.
Let Σ be a set of free generators for F together with their inverses, and let Σ * be the free monoid over Σ. We consider Equation (1) in terms of words in Σ * . Each coefficient u i is represented by a freely reduced word (also denoted
u i ) in Σ * . From this point of view E(x) = u 0 x ε 1 u 2 x ε 2 . . . u d−1 x ε d u d is a word
in the free monoid over Σ ∪ {x, x −1 }, and a solution to E(x) = 1 is a word s ∈ Σ * such that E(s) is freely equal to the empty word. The first assertion of Theorem 1 is equivalent to saying that for some finite union of sets of words uv i w the solutions set consists of all words freely equal to elements of the finite union. A set uv i w is called a parametric word.
We assume without loss of generality that E(x) is freely reduced, and call d the degree of E(x). If d = 0, then E(x) = u 0 . In this case the solution set is empty if E(x) = 1 and all of Σ * if E(x) = 1. If the equation has degree one, it is easy to find its unique solution. From now on we consider only equations of degree at least two.
We begin with some lemmas on cancellation, after which we find a finite number of parametric words uv i w and uv i wr j s which contain all solutions to E(x) = 1 up to free equivalence. Next we show that two parameters are not required and that uv i w is either a solution for all integers i or for an effectively determined finite subset. At the end we present the algorithm and estimate its time complexity.
To explain our argument in more detail we require a few definitions. For any (word) g ∈ F we say that the ith occurrence of g cancels out in E(g) if there exists a way to freely reduce E(g) such that all letters from g ǫ i cancel out during this reduction process.
We say that g is a pseudo-solution of E(x) = 1 if some occurrence of g cancels out in E(g). Obviously every solution of E(x) = 1 is also a pseudosolution of E(x) = 1. However, unlike solutions, pseudo-solutions admit a nice reduction theory.
Our key idea is to study pseudo-solutions of equations instead of solutions. The first result in this direction (stated in [9] in a slightly different form) reduces the situation to cubic equations. Namely, Lemma 12 shows that if g is a pseudo-solution of E(x) = 1 in F then g is a pseudo-solution of a cubic equation of the type
x ε j−1 u j x ε j u j+1 x ε j+1 ,
where 0 < j < d and indices are read modulo d (so u d = u 0 .) Next in Lemma 14 we show that pseudo-solutions of cubic equations are in fact pseudo-solutions of some particular quadratic equations which one can find effectively. Finally, Lemmas 6 and 7 give a precise description of pseudosolutions of quadratic one-variable equations over F in terms of parametric words. Combining all these results we obtain description of all pseudosolutions of E(x) = 1 in terms of parametric words in two parameters.
The rest of our proof explains precisely how to use only one parameter to describe solutions of E(x) = 1. The method of big powers (see [3]) is the key tool in the second part. This means that the argument is rather general -it works in many other groups that satisfy the big powers condition (see [16]), for example torsion-free hyperbolic groups.
One-variable equations are the only general class of equations in free groups for which a good description of solution sets as well as a practical (polynomial time) algorithm are known. In his seminal paper [21] Makanin proved decidability of the Diophantine problem in free groups F (whether or not a given equation has a solution in F ); however, his original algorithm is very inefficient -not even primitive recursive (see [15]). In the fundamental paper [25] Razborov gave a description of solution sets of arbitrary equations in F . Though this description is extremely complicated, it was useful in the solution of several deep problems in group theory [12,13,5] including the Tarski's problems [14]. In another paper [26] Razborov showed that, in general, there is no easy description of solutions sets of equations in F . Later, Plandowski gave a much improved P -space version of the decision algorithm for equations in free monoids [24], and Gutierrez devised a P -space algorithm for the decision problem for equations in free groups [10]. Recent results [7] due to Diekert, Gutierrez, and Hagenah, indicate that the decision problem for equations in free groups might be P -spacecomplete, though nothing definite has been proven so far. These results on the complexity of the decision problem for equations in free groups and for their solution sets make the existence of subclasses of equations admitting polynomial decision algorithms and descriptions of solutions sets in closed form, all the more remarkable.
Cancellation Lemmas
As above Σ is a set of free generators and their inverses for a free group F , and Σ * is the free monoid over Σ. Let p, q, r, s, t, u, v, w be words in Σ * . We write u ∼ v if u is freely equal to v, and u → v if u can be reduced to v by cancellation of subwords aa −1 , a ∈ Σ. In particular u → u. The empty word is denoted 1, and the length of u is |u|. Recall that for any word u there is a unique irreducible word v such that u → v, and further u ∼ w if and only if w → v. We introduce some additional notation.
Definition 1. Let w be any word.
(1) w ′ stands for an arbitrary prefix of w and w ′′ for an arbitrary suffix.
(2) |w| c is the length of a cyclicly reduced word conjugate to w.
Lemma 1. If v → u and u = u 1 u 2 · · · u m , then v = v 1 v 2 · · · v m with v i → u i .
Proof. Use induction on n, the number of cancellations necessary to reduce v to u. If n = 0, then u = v and there is nothing to prove. Otherwise let the first reduction be v → w. By induction w = w 1 w 2 · · · w m with w i ∼ u i . As v is obtained from w by inserting a subword aa −1 into some w i or appending it to the beginning or end of some w i , v has the desired factorization.
Lemma 2. Consider a fixed sequence of cancellations which reduces u to v. If two particular letters of u cancel at some point in the sequence, then either they are adjacent in u or the subword between them has been reduced to 1 by previous cancellations.
Proof. Use induction on the length of the cancellation sequence.
Now we slightly generalize the definition of a pseudo-solution of equation to the following situation. We are dealing with words over Σ, not group elements. For example s = ab −1 is a pseudosolution of asba −1 a but not of asb. The next two lemmas can be proved by straightforward induction on the length of an appropriate cancellation sequence.
Lemma 3. Suppose s is a pseudosolution of w = usv, then s = s 1 s 2 with s 1 a pseudosolution of us 1 and s 2 a pseudosolution of s 2 v.
Lemma 4. Let s be a pseudosolution of w, and fix a cancellation sequence. The smallest subword of w which contains s and all letters in w canceling with letters of s is freely equal to 1.
Lemma 5.
A subword s of w is a pseudosolution if and only if there is a word t such that s is a subword of t, t is a subword of w, and t ∼ 1.
Proof. If t exists, then t ∼ 1 implies t → 1 whence t and all its subwords are pseudosolutions of w. For the converse apply Lemma 4.
∼ v −1 3 v −1 1 for some factorization v = v 1 v 2 v 3 .
Proof. We argue by induction on n, the length of a cancellation sequence. If n = 0, then s = 1 in which case we take u 1 = u 3 = 1 and u 2 = u. If v = 1, then usvsw = ussw. As us and sw are irreducible, the only reduction possible involves cancellation at the boundary between us and sw. It follows that ss ∼ 1, whence s ∼ 1.
Assume n > 0 and v = 1. If the first reduction is within v, then v → p and by induction s ∼ p −1
3 p −1 1 for some factorization p = p 1 p 2 p 3 . Lemma 1 implies v = v 1 v 2 v 3 with v i ∼ p i and s ∼ v −1 3 v −1 1 .
The remaining possibilities are cancellation at the boundary between s and v or the boundary between v and s. Consider the first case; the second is similar. We have s = ta −1 and v = ap for some letter a and words t and p. The induction hypothesis applied to utpt(a −1 w) yields p = p 1 p 2 p 3 and t ∼ p −1
3 p −1 1 . But then v = ap = (ap 1 )p 2 p 3 and s = ta −1 ∼ p −1 3 (ap 1 ) −1 as desired.∼ v k v ′ .
Proof. Consider the first part; as before use induction on n, the number of cancellations. If n = 0, then s = 1. Take v 1 = v, v 2 = 1 and k = 0. Otherwise the first reduction is either within v or at one end or the other of v. In the first case v → v ′ , and the induction hypothesis applied to usv ′ s −1 w yields the desired result.
Suppose then that there is a reduction at the left end of v; the other case is similar. We have s = ta −1 , v = ap, and application of the induction hypothesis to ut(pa)t −1 w yields pa = p 1 p 2 and t ∼ p 2 (pa) k . It follows that
s ∼ p 2 (pa) k a −1 ∼ p 2 a −1 a(pa) k a −1 ∼ p 2 a −1 v k . If p 2 = 1, then p 2 = v 2 a for some suffix v 2 of v whence s ∼ v 2 v k . If p 2 = 1, then s ∼ a −1 v k ∼ a −1 vv k−1 ∼ pv k−1 .
As p is a suffix of v, the first assertion holds. The second assertion follows from the first upon replacement of s by s −1 .
Lemma 8. If s is a pseudosolution of tus, t is a pseudosolution of tvs, and
st is irreducible, then s ∼ (v −1 u) i (v −1 u) ′ and t ∼ (vu −1 ) ′′ (vu −1 ) j for some integers i, j.
Proof. Application of Lemma 4 to tus implies either u = u 1 u 2 with u 2 s ∼ 1 or t = t 1 t 2 with t 2 us ∼ 1. Consider the first case. We have
s ∼ u −1 2 ∼ (v −1 u) −1 (v −1 u 1 ) as required. Further u −1 2 → s implies that t is a pseudoso- lution of tvu −1 2 and hence of tvu −1 . Thus either v = v 1 v 2 with t ∼ v −1 1 or u = u 3 u 4 with t ∼ (vu −1 4 ) −1 . But then t ∼ (v 2 u −1 )(vu −1 ) −1 or t ∼ (u −1 3 )(vu −1 ) −1 ,
and we see that t has the right form. A similar analysis starting with starting with tvs also works.
It remains to consider the case t = t 1 t 2 with t 2 us ∼ 1 and s = s 1 s 2 with tvs 1 ∼ 1. Suppose u ∼ v. We have t = t 1 t 2 with t 2 us ∼ 1 and s = s 1 s 2 with tus 1 ∼ 1. If t 1 = 1, then tus ∼ 1 implies st ∼ u −1 . As st is irreducible,
Lemma 1 yields s ∼ (u −1 ) ′ = (v −1 ) ′ and t ∼ (u −1 ) ′′ which is included in i = j = 0. If t 1 = 1, it follows from t 2 us 1 s 2 ∼ 1 ∼ t 1 t 2 us 1 that t 1 ∼ s 2 . As st is irreducible, t 1 and s 2 are too. Thus t 1 = s 2 = 1. Hence s 1 s 2 t 2 = s 1 t 1 t 2 is irreducible. But then s 1 s 2 t 2 ∼ u −1 ∼ s 1 t 1 t 2 implies s ∼ (v −1 ) ′ , t ∼ (u −1 ) ′′ as before.
Finally suppose t = t 1 t 2 with t 2 us ∼ 1, s = s 1 s 2 with tvs 1 ∼ 1, and u ∼v. From t 2 us ∼ 1 we deduce u −1 t −1 2 → s. Hence t is a pseudosolution of tvu −1 t −1 2 and all the more of tvu −1 t −1 2 t −1 1 = tvu −1 t −1 . Likewise s is a pseudosolution of s −1 v −1 us. We are done by Lemma 7.
Lemma 9. Let st be irreducible. If the right-hand occurrence of s is a pseudosolution in stus but not in tus, then st ∼ u −1 3 u −1 1 for some factorization u = u 1 u 2 u 3 . Likewise if the left-hand occurrence of t is a pseudosolution in tvst but not in tvs,
then st ∼ v −1 3 v −1 1 for some factorization v = v 1 v 2 v 3 .
Proof. Consider the first part; the second is treated similarly. We have s = pq with q = 1 and qtus ∼ 1. Since st is irreducible, so is qt. It follows that t is a pseudosolution of tus. If t is not a pseudosolution of tu, then s = ef with tue ∼ 1. But then qf ∼ 1 forces f to be a pseudosolution of pqf = sf = ef f , and Lemma 6 implies f ∼ 1. Consequently tuef = tus ∼ 1 contrary to our hypothesis that s is not a pseudosolution of tus.
It remains to deal with the possibility that t is a pseudosolution of tu. In this case u = u 1 u 2 with t ∼ u −1 1 . It follows that qu 2 s ∼ 1 whence the right-hand occurrence of s is a pseudosolution in su 2 s. An application of Lemma 6 completes the proof. Proof. Assume the Lemma holds when both p and q are cyclicly reduced, and consider the case that they are not. Free reduction of p and q yields reduced words p 1 p 2 p −1 1 ∼ p, q 1 q 2 q −1 1 ∼ q with p 2 , q 2 cyclicly reduced. Hence
p i 2 (p −1 1 uq 1 )q j 2 ∼ p −1 1 vr 1 . Rewriting p −1 1 uq 1 as u 2 and p −1 1 vq 1 as v 2 we ob- tain p i 2 u 2 q j 2 ∼ v 2 . As i|p 2 | c + j|q 2 | c = i|p| c + j|q| c ≥ 2|p| + 2|q| + |u| + |v| ≥ 2|p 2 | + 2|u 2 | + |q 2 | + |v 2 |, we have u 2 q 2 u −1 2 ∼ p ±1 2 . Hence uqu −1 ∼ (p 1 u 2 q −1 1 )(q 1 q 2 q −1 1 )(q 1 u −1 2 p −1 1 ) ∼ p 1 u 2 q 2 u −1 2 p −1 1 ∼ p 1 p ±1 2 p −1 1 ∼ p −1 .
It remains to deal with the case that p and q are cyclicly reduced. Without loss of generality assume that u and v are freely reduced and i, j ≥ 0. Thus there is a sequence of (1/2)(|p i uq j | − |v|) = (1/2)(i|p| + j|q| + |u| − |v|) ≥ |p| + |q| + |u| cancellations which reduces p i uq j to v.
Since cancellation can occur only at either end of u, the first |u| cancellations must consume u. In other words u cancels with a suffix of p i and a prefix of q j . For some factorizations p = p 1 p 2 and q = q 1 q 2 we have u = (p 2 p i 2 ) −1 (q j 1 q 1 ) −1 with i = i 1 + 1 + i 2 and j = j 1 + 1 + j 2 . Consequently p i 1 p 1 q 2 q j 2 admits at least |p| + |q| cancellations. Thus the infinite sequences q 2 q 1 q 2 q 1 · · · and p −1 1 p −1 2 p −1 1 p −1 2 · · · have the same prefix of length |p| + |q|. As these sequences have periods |p| and |q| respectively, they are identical by [8,Theorem 1]. But then the fact that (p −1 1 p −1 2 ) |q| and (q 2 q 1 ) |p| have the same length implies that they are equal. Since p and q are not proper powers, neither are (p −1 1 p −1 2 ) and q 2 q 1 . It follows that p −1 1 p −1 2 = q 2 q 1 , and this equation implies in a straightforward way that uqu −1 ∼ p −1 .
Lemma 11. Suppose that q j is a pseudosolution of p i uq j vr k , |j||q| c ≥ 7(|p|+ |u| + |q| + |v| + |r|); and p, q, r are not proper powers. Then either q ∼ 1 or |i| ≥ 1 and u −1 pu ∼ q ±1 or |k| ≥ 1 and vrv −1 ∼ q ±1 .
Proof. Without loss of generality assume i, j, k ≥ 0. By Lemma 3 q factors as q 1 q 2 and q j factors as (q j 1 q 1 )(q 2 q j 2 ) in such a way that q j 1 q 1 is a pseudosolution of p i uq j 1 q 1 , and q 2 q j 2 is a pseudosolution of q 2 q j 2 vr k . Clearly one of j 1 , j 2 is no smaller than (j − 1)/2. Assume it is j 1 ; the argument is similar in the other case.
By Lemma 4 q j 1 q 1 extends to a suffix of p i uq j 1 q 1 which is freely equal to 1. If that suffix is contained in uq j 1 q 1 , then u = u 1 u 2 with q j 1 ∼ u −1 2 q −1 1 . Hence q j 1 freely reduces to a word w with |w| ≤ |u| + |q|. On the other hand |w| ≥ j 1 |q| c ≥ .5(j − 1)|q| c ≥ 3.5(|p| + |u| + |q|) − .5|q| ≥ 3(|p| + |q| + |u|). But then |p| = |q| = |u| = 0, which implies q ∼ 1.
It remains to consider the case that the suffix is not contained in us j 1 s 1 . In particular i ≥ 1. For some factorization p = p 1 p 2 and m ≤ i we have p 2 p m uq j 1 q 1 ∼ 1. Thus p m uq j 1 ∼ p −1 2 q −1 1 . As above j 1 |q| c ≥ 3(|p|+|q|+|u|) ≥ 2|p| + 2|q| + |u| + |p −1 2 q 1 |. Lemma 10 applies and yields uqu −1 ∼ p ±1 .
Parametric Words
In this section we show how to find a finite set of words and parametric words uv i wr j s which together contain all solutions to Equation (1).
Let s be any freely reduced word which is a solution to Equation (1). Substitution of s for x yields a word
(2) E(s) = u 0 s ε 0 . . . u d−1 s ε d−1
such that E(s) → 1. Fix a sequence of cancellations which reduces E(s) to 1, and let s ε j be the first of the subwords s ±1 to be consumed. If there is a tie, pick either subword. Observe that the letters in s ε j must cancel with nearby letters in E(s). If a letter in s ε j canceled to the right of s ε j+1 , then by Lemma 2 s ε j+1 would disappear before s ε j . Likewise no letter of s ε j+1 cancels to the left of s ε j−1 . We have the following result.
Lemma 12. One of the following holds.
(1) s ε 0 is a pseudosolution of u 0 s ε 0 u 1 s ε 1 ;
(2) For some j strictly between 0 and d − 1, s ε j is a pseudosolution of
s ε j−1 u j s ε j u j+1 s ε j+1 ; (3) s ε d−1 is a pseudosolution of s ε d−2 u d−1 s ε d−1 .
It is convenient to use the following immediate consequence of Lemma 12.
Lemma 13. For some j between 0 and d − 1, s ε j is a pseudosolution of s ε j−1 u j s ε j u j+1 s ε j+1 . Here indices are read modulo d; e.g., u d = u 0 .
It follows from Lemma 13 that application of the following lemma to all successive pairs of coefficients u = u i , v = u i+1 (with indices read modulo d) yields a set of words and parametric words containing s or s −1 for every solution s to Equation 1.
Lemma 14.
If α, β = ±1 and s is an irreducible pseudosolution to s α usvs β , then one of the following holds. (Recall Definition 1.)
(1) s ∼ (v −1 u) i (v −1 u) ′ (vu −1 ) ′′ (vu −1 ) j ; (2) s ∼ (u −1 ) ′ (u −1 ) ′′ or (v −1 ) ′ (v −1 ) ′′ ; (3) s ∼ (u −1 ) ′ v i v ′ v ′′ v j or u i u ′ u ′′ u j (v −1 ) ′′ ; (4) s ∼ u i u ′ v ′′ v j .
Proof. By Lemma 3 s = s 1 s 2 with s 1 a pseudosolution of s α us 1 and s 2 a pseudosolution of s 2 vs β . There are four cases. First if α = −1, β = −1, Lemma 7 applied to s −1 2 s −1 1 us 1 and s 2 vs −1 2 s −1 1 yields (4). If α = β = 1, we have s 1 s 2 us 1 and s 1 vs 1 s 2 where the pseudosolutions are underlined. It may happen that s 1 is pseudosolution of s 2 us 1 and s 2 is a pseudosolution of s 2 vs 1 . In this case Lemma 8 applies and (1) holds. Otherwise either s 1 is not a pseudosolution of s 2 us 1 or s 2 is not a pseudosolution of s 2 vs 1 . In both cases Lemma 9 implies (2).
Suppose α = 1, β = −1. In this case s 1 s 2 us 1 and s 2 vs −1 2 s −1 1 . By Lemma 9 either s in included in (2) or s 2 us 1 whence s 1 is freely equal to the inverse of a suffix of s 2 u. Equivalently s 1 is freely equal to a prefix of (s 2 u) −1 . But Lemma 7 implies s 2 ∼ v 2 v j for some integer j and factorization v = v 1 v 2 .
It follows from Lemma 1 that s 1 is freely equal to a prefix of (v 2 v j u) −1 .
Consideration of the possible cases yields (3).
A similar argument works when α = −1, β = 1 and shows that (2) or (3) holds.
Solutions
In order to find all solutions to Equation (1) we need to test the possibilities given by Lemma 14. It is straightforward to test the single words; the parametric words require more work. They have the form rp i sq j t. Without loss of generality we assume that p and q are not proper powers. By introducing words of the form rp i s we may assume p ∼ 1 ∼ q.
Consider rp i s. Substitute rys for x in Equation 1 to obtain an equation E ′ (y) = v 0 y ε 0 · · · v d−1 y ε d−1 in the indeterminate y with coefficients v j of the form su j r, su j s −1 etc. Note that rp i s is a solution of E(x) if and only if p i is a solution of E ′ (y). Also the sum of the lengths of the coefficients of E ′ (y) is |v 0 · · · v d−1 | = |u 0 · · · u d−1 | + d|rs|. Denote this number by K 1 .
If a coefficient v j commutes with p, i.e. v j p ∼ pv j , then the subword y ε j−1 v j y ε j of E ′ (y) may be replaced by v j y ε j−1 +ε j without affecting the set of i's for which p i is a solution. This is true even if indices are read modulo d. The coefficients in E ′ (y) will change, but E ′ (1) = v 0 · · · v d−1 remains constant. In particular the sum of the length of the coefficients is still K 1 .
Continue replacements of this sort until reaching an equation of the form E ′′ (y) = w 0 y k 0 · · · w m y km with m minimal. It may be that m = 0 and E ′′ (y) = w 0 . In this case p i is a solution for all i if w 0 = v 0 · · · v d−1 ∼ 1 and for no i otherwise. Similarly if E ′′ (y) = w 0 y k 0 , then p i is a solution if and only if w 0 ∼ p −ik 0 . In this case the free reduction of p ik 0 is a word of length at least |ik 0 ||p| c and at most |w 0 | = K 1 . Consequently |i||p| c ≤ |ik 0 ||p| c ≤ K 1 .
The remaining possibility is that E ′′ (y) = w 0 y k 0 · · · w m y km with m ≥ 2, all k j = 0 and no w j commuting with p. No w j conjugates p to p −1 either, as p and p −1 are not conjugate in the free group F . If p i is a solution, then by Lemma 13 (with E ′′ (y) in place of E(x)) some p ik j must be a pseudosolution of p ik j−1 w j p ik j w j+1 p ik j+1 . Lemma 11 now implies that |i||p| c < 7(|p| + |w j | + |p| + |w j+1 | + |p|) ≤ 21|p| + 7K 1 . We have proved the following lemma.
Lemma 15. If rp i s is a solution to E(x) = 1 for some i with |i||p| c > 21|p| + 7(|u 0 · · · u d−1 | + d|rs|), then rp i s is a solution for all i.
Consider a solution rp i sq j t to E(x) = 1. Define K 2 = 21 max{|p|, |q|} + 7(|u 0 · · · u d−1 | + d|rst|). We will show that either |i||p| c or |j||q| c is no larger than K 2 d. Thus each parametric word rp i sq j t from Lemma 12 with two parameters may be replaced by a collection of parametric words with just one parameter, namely rp i 0 sq j t, rp i sq j 0 t with with |i 0 ||p| c ≤ K 2 d and |j 0 ||q| c ≤ C 2 d.
Without loss of generality suppose that i, j ≥ 0, and p and q are not proper powers. In particular the centralizers in the free group F of p and q are the cyclic subgroups generated by p and q respectively. Lemma 16. Suppose p is not conjugate to q or q −1 and rp i sq j t is a solution to E(x) = 1. Then either |i||p| c or |j||q| c is no larger than K 2 = 21 max{|p|, |q|} + 7(|u 0 · · · u d−1 | + d|rst|).
Proof. First suppose that s ∼ 1 and take the solution to be rp i q j t. Write
E(rp i q j s) = v 0 (p i q j ) ε 1 v 1 · · · v d−1 (p i q j ) ε d v d .
The v k 's are coefficients; call the (p i ) ε k 's and (q j ) ε k 's powers. Consider how a coefficient v k might conjugate the power on one side of itself to the power on the other side. As p is not conjugate to q or q −1 , v k would either lie in a subword v k−1 p i q j v k q −j p −i v k+1 and centralize q or in a subword v k−1 q −j p −i v k p i q j v k+1 and centralize p.
Consequently v k is freely equal to a nontrivial power (because E(x) is freely reduced) of q in the first case and a nontrivial power of p in the second. W is freely equal to the word obtained by deleting the powers on either side of v k .
Let W ′ be the word obtained from W by performing all the deletions discussed in the previous paragraph. Notice that the first and last powers of W survive and that the new coefficients are either old coefficients which do not conjugate their adjacent powers into each other or products v k v k+1 · · · v k+m of successive coefficients whose adjacent powers in W have been deleted. In the latter case the coefficient is an alternating product of nontrivial powers of p and q.
Since W ′ ∼ 1, some power is a pseudosolution in a subword of W ′ consisting to up to three powers and the coefficients between them. The sum of the length of the coefficients of W ′ is the same as that of W , namely |u k | + d|r| + d|s|. If |i||p| c and |j||q| c exceed the bound given above, then Lemma 11 applies and (as p is not conjugate to q or q −1 ) implies that some coefficient conjugates one adjacent power to the other. But this is impossible either because the coefficient is inherited from W or because the coefficient is an alternating product of nontrivial powers of p and q, and the conjugation would be a nontrivial relation satisfied by p and q, which generate a free group of rank two.
It remains to reduce to the case s ∼ 1. Assume s ∼ 1, and rewrite the solution as rp i (sqs −1 ) j (st). One of |i||p| c or |j||sqs −1 | c = |j||q| c is at most 21 max{|p|, |sqs −1 |} + 7(|u 0 · · · u d−1 | + d|rst|).
Finally, consider a solution rp i sq j t to E(x) = 1 with p conjugate to q or q −1 . With appropriate changes to r,s,t and j, rp i sq j t may be rewritten as rp i sp j t where p is cyclicly reduced and s does not commute with p.
Define W = E(rp i sp j t) = v 0 (p i sp j ) ε 1 v 1 · · · v d−1 (p i sp j ) ε d v d ,
and argue as before. The coefficients now include the subwords s ±1 as well as the v k 's. Consider how a v k might conjugate the power on one side of itself to the power on the other side. Since all powers are powers of p, v k would commute with p and hence would itself be freely equal to a power of p. If ε k ε k+1 = −1, then v k ∼ 1 and the powers on either side cancel. However, if ε k ε k+1 = 1, then the powers do not necessarily cancel but combine to form a power p ±(i+j) .
Let W ′ be the word obtained from W by performing all the deletions and combinations of powers discussed in the previous paragraph. Notice that the first and last powers of W survive and that the new subwords between powers surviving from W are either coefficients from W which do not conjugate their adjacent powers into each other or alternating products s ±1 p km v m s ±1 p k m+1 v m+1 · · · s ±1 p kn v n s ±1 s where the v j 's which occur are freely equal to powers of p. Further if p k j occurs between s and s −1 , then k j = 0 and v j is freely equal to a nontrivial power of p, while if p k j occurs between two s's or two s −1 's, then k j = ±(i + j).
There are two possibilities. First if |(i+j)||p| c ≥ K 2 , then we may consider the subwords p ±(i+j) to be powers like the p ±i 's and p ±j 's surviving from W and the subwords between the powers as coefficients. Lemma 11 applies and implies that some coefficient conjugates one adjacent power to the other and hence is a power of p. But this is impossible either because the coefficient is inherited from W or because the coefficient is an alternating product of nontrivial powers of p and s, and the conjugation would be a nontrivial relation satisfied by the subgroup generated by p and s, which is free of rank two.
Second if |(|i + j)|p| c < K 2 , we take just the the p ±i 's and p ±j 's from W to be powers. The coefficients are either inherited from W or alternating products s ±1 p km v m s ±1 p k m+1 v m+1 · · · s ±1 p kn v n s ±1 s as above. In this case |(i + j)||p| = |(|i + j)|p| c ≤ K 2 , and the total length of the coefficients increases to at most K 2 + (d − 1)|(i + j)||p| ≤ dK 2 . Lemma 11 applies and yields the following lemma.
Lemma 17. Suppose p is conjugate to q or q −1 and rp i sq j t is a solution to E(x) = 1. Then |i||p| c or |j||q| c is no larger than dK 2 .
The Algorithm
The algorithm implicit in the preceding analysis may be described as follows.
(1) The input is an equation u 0 x ε 0 u 1 x ε 1 . . . u d−1 x ε d−1 = 1 of degree d ≥ 2 and with freely reduced coefficients from a free monoid Σ * over a set Σ of generators and their inverses for a free group F . (2) Let L be the list of words and parametric words and their inverses from Lemma 14. Rewrite the parametric words so that they are either ordinary words or have the one of the forms rp i s or rp i sq j t with p ∼ 1 ∼ q, p, q not proper powers, and in the latter case sqs −1 ∼ p ±1 . (3) For each ordinary word w ∈ L test E(w) ∼ 1 and E(w −1 ) ∼ 1.
Remove w from L. (4) Replace each parametric word rp i sq j t with words rp i sq j 0 t and rp i 0 sq j t for all i 0 , j 0 with |i 0 ||p| c ≤ dK 2 and |j 0 ||q| c ≤ dK 2 where K 2 is as in Lemma 16.
(5) For each word of the form w = rp i q in L, if E(rp i 0 s) ∼ 1 where i 0 is the least integer greater than (1/|p| c )(21|p| + 7(|u 0 · · · u d−1 | + d|rs|), then x = rp i s is a solution for all i, otherwise test E(rp i 1 s) ∼ 1 for all |i 1 | < i 0 . We leave it to the reader to check that our preceding analysis implies the correctness of the above algorithm. To bound the time complexity let |L| be the length the list from Step 2 and M the maximum of |rpsqt| for each entry rp i sq j t. Note that M is also an upper bound for the length of the coefficients of E(x) and that the constant K 2 from Lemma 16 is O(dM ).
Steps
Definition 2 .
2A subword s of w is a pseudosolution if there is a sequence of cancellations in w which consumes all letters in s.
Lemma 6 .
6If us and sw are irreducible and if either occurrence of s is a pseudosolution of usvsw, then s
Lemma 7 .
7Suppose v ∼ 1. If us and s −1 w are irreducible and s or s −1 is a pseudosolution of usvs −1 w, then s ∼ v ′′ v k for some integer k. (See Definition 1.) Likewise if us −1 and sw are irreducible and s or s −1 is a pseudosolution of us −1 vsw, then s
Lemma 10 .
10Suppose p i uq j ∼ v, i, j ≥ 0, and i|p| c + j|q| c ≥ 2|p| + 2|q| + |u| + |v|. (Recall Definition 1.) Further assume that p and q are not freely equal to proper powers. Under these conditions uqu −1 ∼ p −1 .
2 and 3 are accomplished in time O(M |L|), and Step 4 in time O(dK 2 M |L|). Let L ′ be the augmented list from Step 4. |L ′ | = O(dK 2 |L|) = O(d 2 M |L|), and each entry in L ′ has the form rp i s with |rps| = O(dM K 2 ) = O(d 2 M 2 ). For each entry there are O(M +dM +d 2 M K 2 ) = O(d 3 M 2 ) tests performed in Step 5. The time to test the entry rp i s is linear in the length ofE(rp i s), which is O(d|rp i s| + |u 1 · · · u d−1 |) = O(id 3 M 2 ) = O(dK 2 d 3 M 2 ) = O(d 5 M 3 ). Thus the total time for Step 5 is O(|L ′ | · (d 3 M 2 ) · (d 5 M 3 )) = O((d 2 M |L|) · (d 3 M 2 ) · (d 5 M 3 )) = O(d10 M 6 |L|) . Clearly this estimate bounds the time of the complete algorithm. Finally let m be the maximum size of a coefficient in E(x). If follows from Lemma 14 that M = O(m) and that |L| = O(dm 3 ). Thus the time complexity of our algorithm is O(d 11 m 9 ).
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| [] |
[
"Unipolar Induction of a Magnetized Accretion Disk around a Black Hole",
"Unipolar Induction of a Magnetized Accretion Disk around a Black Hole"
] | [
"A A Shatskiy \nAstrospace Center\nLebedev Institute of Physics\nRussian Academy of Sciences\nProfsoyuznaya ul. 84/32117997MoscowRussia\n"
] | [
"Astrospace Center\nLebedev Institute of Physics\nRussian Academy of Sciences\nProfsoyuznaya ul. 84/32117997MoscowRussia"
] | [
"Astronomy Letters"
] | The structure and magnitude of the electromagnetic field produced by a rotating accretion disk around a black hole were determined. The disk matter is assumed to be a magnetized plasma with a frozenin poloidal magnetic field. The vacuum approximation is used outside the disk. | 10.1134/1.1558153 | [
"https://export.arxiv.org/pdf/astro-ph/0301535v1.pdf"
] | 18,918,183 | astro-ph/0301535 | 860fb9a224d47a14fc2d5dfa66f10dad22b06a45 |
Unipolar Induction of a Magnetized Accretion Disk around a Black Hole
2003
A A Shatskiy
Astrospace Center
Lebedev Institute of Physics
Russian Academy of Sciences
Profsoyuznaya ul. 84/32117997MoscowRussia
Unipolar Induction of a Magnetized Accretion Disk around a Black Hole
Astronomy Letters
2932003Received October 3, 2002pulsarsneutron stars; black holesquasarsjetsaccretion disks
The structure and magnitude of the electromagnetic field produced by a rotating accretion disk around a black hole were determined. The disk matter is assumed to be a magnetized plasma with a frozenin poloidal magnetic field. The vacuum approximation is used outside the disk.
INTRODUCTION
Recently, various models of particle acceleration near supermassive black holes (SMBHs) in galactic nuclei and near stellar-mass black holes (BHs) in the Galaxy have been widely discussed in connection with the studies of synchrotron radiation and inverse Compton scattering from narrow-beam jets observed over a wide spectral range, from radio to gamma rays. Nevertheless, the particularly high angular resolution provided by radio interferometers does not allow the central part of a quasar to be distinguished, suggesting that the jet width is extremely small (comparable to the gravitational radius). Previously (Shatskiy 2001), the mechanism of Blandford and Znajek (1977) and Blandford (2001) for electric-.eld generation through the interaction of the magnetic .eld from a ring current with the gravimagnetic .eld (GMF) of a Kerr BH located on the common axis with the ring current was used as the model of particle acceleration. Bisnovatyi- Kogan and Blinnikov (1972) and Shatskiy and Kardashev (2002) considered the mechanism of Deutsch (1955) or Goldreich and Julian (1969) for electric-field generation by the unipolar induction produced by the axial rotation of an accretion disk with a frozen-in magnetic field. Here, we suggest a mechanism that combines a unipolar inductor and strong gravitational SMBHs effects. Naturally, this mechanism is closer to the actual processes that take place in quasars. In contrast to the mechanism from Blandford and Znajek (1977), Beskin et al. (1992), and Beskin (1997), the mechanisms of Bisnovatyi-Kogan and Blinnikov (1972) used previously (Shatskiy 2001;Shatskiy and Kardashev 2002) operate in the vacuum approximation; the validity criterion for the latter is the condition of Goldreich and Julian (1969) for the number density of free charges: n e < |(ΩH)|/(2πce).
Here, Ω is the plasma angular velocity, H is the characteristic magnetic field, c is the speed of light, and e is the elementary charge. Shatskiy and Kardashev (2002) showed that condition (1) could be satisfied near a BH, because there are no stable orbits for the particles closer than three gravitational radii (in a Schwarzschild field). The matter for which the vacuum approximation breaks down must be in the accretion disk, which, because of the effect of Bardeen and Petterson (1975), must be located in the equatorial plane of a rotating BH. We do not consider models in which the vacuum approximation breaks down in the entire space outside a BH (models with the magnetohydrodynamic approximation). These were considered in detail, for example, in the review articles by Beskin et al. (1992) and Beskin (1997). Here, we determine the energy of the charged particles accelerated by the unipolar mechanism as well as the configurations of the electromagnetic field and the acceleration region.
CONSTRUCTING THE MODEL
Consider a Schwarzschild BH surrounded by an equatorial accretion disk. Let the disk have the characteristic size R and width 2a (see the figure). Because of the Bardeen-Petterson effect, the disk thickness can be disregarded. We use the following notation: M is the mass of the central body, R g = 2M is the Schwarzschild radius, 2 m is the mass of the test particle, u j is its 4-velocity,
F ij = ∂ i A j − ∂ j A i is the electromagnetic-field (EMF)
tensor, A j is the EMF potential, and Γ i jk are the Christoffel symbols. Let us write the Schwarzschild metric and its determinant in spherical coordinate:
ds 2 = (1 − r g /r)dt 2 − (1 − r g /r) −1 dr 2 − r 2 dθ 2 − r 2 sin 2 θdϕ 2 , g = −r 4 sin 2 θ .(2)
In general relativity, the following quantity for a charged particle that moves in stationary fields is conserved:
ε = m(u 0 − 1) + eA 0 ,(3)
which matches the particle energy in the nonrelativistic case. To prove this, it will suffice to consider the equation of motion for a charged particle in general relativity (see Landau and Lifshitz 1988):
m du i ds = mu k u l Γ l ik + eu k F ik ,(4)
where ds is the element of the particle proper time [see formulas (2)]. After transformation, this expression reduces to d ds
(mu i + eA i ) = m 2 u l u k ∂ i g kl + eu k ∂ i A k .(5)
The conservation of energy . throughout the particle evolution follows for i = 0 in stationary fields.
Since the magnetic field is frozen into the disk, its distribution inside the disk is determined only by the initial conditions of the problem. These conditions depend on the accretion-disk formation mechanism. If the disk is assumed to have been formed through the destruction of a star by BH tidal forces, then the magnetic field of this star in the disk will preserve its direction. This field can have the profile shown in the figure.
In the frame of reference comoving with the accretion disk, there is no electric field inside the disk because of its conductivity. Therefore, in a fixed frame of reference (with respect to distant stars), an electric field is induced by disk rotation inside the disk. Let the plasma in the disk rotate at an angular velocity Ω relative to distant stars. The transformation of coordinates to a rotating frame is then:
3 dx i = dx ′ k [δ i k + Ωδ i ϕ δ 0 k ].(6)
2 Below, we use the system of units in which the speed of light and the gravitational constant are equal to unity: c = 1, G = 1.
3 Unless otherwise specified, x i = t, r, θ, ϕ; x α = R, θ (Greek and Roman indices).
Because of axial symmetry, only the following EMF potential components are nonzero: A 0 , the electric-field potential, and A ϕ , the magnetic-field potential. According to (6) (see Landau and Lifshitz 1988), the EMF components transform as
A ′ 0 = A 0 + ΩA ϕ , A ′ ϕ = A ϕ , F ′ α0 = F α0 + ΩF αϕ , F ′ αϕ = F αϕ .(7)
Since F α0 = 0 in plasma, we have inside the disk
F α0 = −ΩF αϕ , A 0 = const − ΩA ϕ .(8)
On the disk surface, continuous boundary conditions exist for the tangential electric-field components and for the normal magnetic-field components. Outside the disk, there are no field sources by the definition of the model. Thus, determining the EMF reduces to solving the Laplace equation in the spacetime curved by gravity with the specified boundary conditions on the disk surface and on the BH horizon. The boundary conditions for the EMF tensor on the BH horizon were found previously (Shatskiy 2001): lim
r→rg F 0θ ∝ g 00 → 0 , lim r→rg F rϕ ∝ g 00 → 0 .(9)
In turn, the boundary conditions on the disk for the magnetic and electric fields are determined solely by the magnetic-field distribution inside it. The specific form of this distribution is not that important for the solution of the problem. This is because at distances from the disk to the point of observation much larger than the disk thickness, the dipole field (the monopole field must be absent, because the total disk charge is zero) mainly contributes to the electric field of the disk element within position angles between ϕ and ϕ + dϕ when the field is expanded in multipoles. In this case, the total electric field obtained by integrating over the angle ϕ has a quadrupole nature: lim (r/rg )→∞ A 0 = const · (1 − 3 cos 2 θ)/r 3 .
THE MAXWELL EQUATIONS
The Maxwell equations for the EMF in general relativity are
1 √ −g ∂ i ( √ −gF ik ) = 4πj k .(10)
In the disk, the 4-vector of the current j k can be determined from a given magnetic field. Let us introduce the physical components of the EMF vectors, their analogs in Euclidean space: 4 E α = −F β0 |g 00 g αβ |,Ĥ α = −e αβϕ F γϕ |g γβ g ϕϕ |,Ĵ α = j β |g αβ |.
Here, e αβγ = e αβγ is the Levi-Civita symbol. This form of the EMF physical components was chosen in order that Eq. (10) correspond to the classical Maxwell equations in Euclidean space:
divÊ = 4πj 0 , rotĤ = 4πĴ.(12)
THE ELECTROMAGNETIC FIELD NEAR A SMBH
The magnetic field of an accretion disk around a Schwarzschild BH was determined by Tomimatsu and Takahashi (2001). The electric field of the disk element within position angles between ϕ and ϕ + dϕ can be represented as the field from two charges: +q dϕ 2π and −q dϕ 2π , located inside the disk, at the system equator, and at distances +a and −a from its center (r = R, θ. = π/2), respectively 5 . As a result, we obtain an electric dipole 2qadϕ/π in the disk element between ϕ and ϕ + dϕ. In Euclidean space, the radial electric field at the disk center can be obtained by integrating over the angle ϕ; for a << R, it is E 0 =Ê r (r=R,θ=π/2) = −2q/(πRa). The corresponding magnetic field (which is responsible for the emergence of the electric field) can be found from the electric field. Note that nothing forbids the frozenin magnetic field in the disk to have precisely such a profile (see the figure). In the figure, the accretion disk is located at r ≈ 6M. The contradictions related to the existence of stable orbits in this region can be removed by the following reasoning:
1 For the Kerr metrics, the nearest stable orbit is at radius r = M (see, e.g., Landau and Lifshitz 1988).
2 Even if the orbit is not stable, it is spiral and goes under the horizon, while a new orbit can arrive in place of it. Thus, the pattern is quasi-stationary. According to (11) and (8), the quantity q can be expressed in terms of the magnetic field at the disk center 6 as follows:
q = −πRaE 0 /2 = π 2 ΩR 2 aH 0 .(13)
In physical coordinates, the field of a point charge e near a BH was presented by Thorne et al. (1998). It was obtained in a closed form by Linet (1976):
A 0 = e Rr M + (R−M )(r−M )−M 2 t D , E r = e Rr 2 M 1 − R−M +M t D + r[(r−M )(R−M )−M 2 t][r−M −(R−M )t] D 3 , E θ = − e(R−2M ) √ 1−2M/r D 3 ∂ θ t , D 2 = (r − M) 2 + (R − M) 2 − M 2 − 2(r − M)(R − M)t + M 2 t 2 .(14)
where t is the cosine of the angle between the directions of the point charge and the point of observation of the field from the BH center. The electric field of a charged ring at the BH equator was found by Bicak and Dvorak (1996) in the form of a series. Here, this field is found in a quadrature form. To this end, we make the following substitutions: t → sin θ cos ϕ, e → Q dϕ 2π and integrate over the angle from −π to +π. The model electric-field potential A tot 0 is a superposition of the fields from two charged rings at radii R + a and R − a and with charges +q and −q, respectively. The quadrature obtained can be expressed in terms of incomplete elliptic integrals. Since this quadrature is cumbersome, it makes no sense to write it here. Instead, we give an expression more useful for practical calculations, an expansion of this quadrature in terms of a. We retain only the first term 5 Naturally, there are no free charges in the disk; these were introduced for the convenience of representing the dipole field outside the disk. 6 H 0 = −Ĥ θ(r=R,θ=π/2) / 1 − r g /R.
of the series (because a << R is small): 7
A tot 0 = +π −π 2a∂ R A 0 dϕ = +π 0 4a(∂ r A 0 ) (R↔r) dϕ = +π 0 4aÊ r (R↔r) dϕ.
Substituting expression (14) here finally yields
A tot 0 = 2aq πrR 2 π 0 dϕ M (D−r+M −M t) D + R[(r−M )(R−M )−M 2 t]·[R−M −(r−M )t] D 3 .(15)
On the Ω axis, the integration over ϕ is simple and it is easy to see that the only maximum of the potential A tot 0 (outside the BH) is on the horizon (r = 2M):
|A tot 0 (r = 2M, θ = 0)| M AX = πΩa 2 R 2 H 0 R 2 − a 2 .(16)
Consider the distribution of energies ε(r) in an ensemble of test particles at rest on the Ω axis [see (3)]. This distribution has a weak maximum near the horizon and then slowly falls o. to zero at distances r >> r g . Subsequently, the energy variation can be virtually disregarded.
DISCUSSION
The necessary condition for particle escape to infinity from rest is positiveness of the force m du r ds = m d 2 r ds 2 that acts on the test particle at the starting point. This requires that the charge of the test particle e have the sign opposite to that of q:
− sign (eq) = sign (eΩÊ (θ=0) ) = − sign (eΩH 0 ) = 1.
The force that acts on an particle is proportional to the particle energy gradient (see Eq. (4)]:
m du r ds = −(∂ r ε) · 1 − 2M/r.(18)
In the main order in α = mM/(eq) << 1, the energy maximum is almost equal to the electric component of the particle energy on the horizon: 8
ε max = 1 α · 2Ma R 2 − a 2 − 1 · mc 2 ∼ 0.1 · mc 2 /α(19)
and the point at which this maximum is reached is at the following distance from the horizon:
r max − r g = r g · α 2 · (R 2 − a 2 ) 2 [(R − M) 2 − a 2 ] 4 16M 6 a 2 (3R 2 + M 2 − 4MR + a 2 ) 2 ∼ r g · 10 3 · α 2 .(20)
In conclusion, several more words can be said about the same model in a Kerr field. BH rotation gives rise to a GMF that interacts with the EMF of the disk and changes its components in magnitude and direction. In the linear approximation in BH angular velocity, the GMF gives an additive component 7 Here, we make use of the symmetry of the potential A 0 in variables R and r and use their change (the subscript " (R↔r) "). 8 At a ∼ r g ∼ R/3 [see (16)]. Figure 1: Accretion disk around black hole, which center at radius R = 3, and which width a = 1, with the frozen-in magnetic eld | rm line. Electrical eld (generated by magnetic) | dotted line. Z and are expressed in fractions of r g = 2M.
to expression (19) for the maximum energy of the charged particle accelerated by an electric field. This component was determined previously (Shatskiy 2001; it is convenient to represent it here as
ε g ≈ 1 2π 2 · Ω g r g ΩR · r 2 g aR · mc 2 /α ∼ ε max .(21)
Here, Ω g is the angular velocity of the BH horizon (the falling test particles are drawn into rotation by the BH GMF).We see from (19) that at
We denote them by a hat.
. J M Bardeen, J A Petterson, Astrophys. J. Lett. 19565J.M.Bardeen and J. A. Petterson,Astrophys. J. Lett. 195, L65, 1975.
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Galaxies and Their Constituents at the Highest Angular Resolution. R Blandford, Proc. IAI Symp. 205. R. T. SchilizziIAI Symp. 205San Francisco10R. Blandford, Galaxies and Their Constituents at the Highest Angular Resolution. Proc. IAI Symp. 205. Ed. R. T. Schilizzi (San Francisco, ASP, 2001), p. 10.
. P Goldreich, . H Julian, Astrophys. J. 157P. Goldreich andW. H. Julian, Astrophys. J. 157, 869, 1969.
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Black Holes. A Membrane Approach. K. Thorne, P. Price, and D. MacDonaldMoscowMirK. Thorne, P. Price, and D. MacDonald (Eds.), Black Holes. A Membrane Approach (Mir,Moscow, 1998).
. A A Shatskiy, gr-qc/0202068)Zh. Eksp. Teor. Fiz. 93A. A. Shatskiy, Zh. Eksp. Teor. Fiz. 93, 920, 2001, (gr-qc/0202068).
. A A Shatskiy, N S Kardashev, astro-ph/0209465Astron. Zh. G. Rudnitskii46A. A. Shatskiy and N. S. Kardashev, Astron. Zh. 46, 639, 2002, (astro-ph/0209465). Translated by G. Rudnitskii
| [] |
[
"The thin layer of Warm Ionized Gas: towards a 3-D reconstruction of the spatial distribution of HII regions",
"The thin layer of Warm Ionized Gas: towards a 3-D reconstruction of the spatial distribution of HII regions"
] | [
"Roberta Paladini \nInternational School for Advanced Studies\nSISSA\nvia Beirut 2-4I-34014TriesteItaly\n",
"Rod Davies \nUniversity of Manchester\nJodrell Bank Observatory\nSK11 9DLMacclesfield -CheshireUK\n",
"Gianfranco Dezotti \nINAF-Oss. Astro. Padova\nVicolo dell'Osservatorio 5I-35122PadovaItaly\n"
] | [
"International School for Advanced Studies\nSISSA\nvia Beirut 2-4I-34014TriesteItaly",
"University of Manchester\nJodrell Bank Observatory\nSK11 9DLMacclesfield -CheshireUK",
"INAF-Oss. Astro. Padova\nVicolo dell'Osservatorio 5I-35122PadovaItaly"
] | [] | HII regions are known to contribute to the so-called thin layer of the diffuse Warm Ionized Gas. In order to constrain this contribution, we reconstruct the 3-D distribution of the sources. A detailed spatial analysis of the largest up-to-date sample of HII regions is presented. | 10.1023/b:astr.0000014983.24270.e5 | [
"https://arxiv.org/pdf/astro-ph/0212341v1.pdf"
] | 18,656,227 | astro-ph/0212341 | e341bcef88a4b9d231177539f742b22a622b8b59 |
The thin layer of Warm Ionized Gas: towards a 3-D reconstruction of the spatial distribution of HII regions
15 Dec 2002
Roberta Paladini
International School for Advanced Studies
SISSA
via Beirut 2-4I-34014TriesteItaly
Rod Davies
University of Manchester
Jodrell Bank Observatory
SK11 9DLMacclesfield -CheshireUK
Gianfranco Dezotti
INAF-Oss. Astro. Padova
Vicolo dell'Osservatorio 5I-35122PadovaItaly
The thin layer of Warm Ionized Gas: towards a 3-D reconstruction of the spatial distribution of HII regions
15 Dec 2002WIM, HII regions
HII regions are known to contribute to the so-called thin layer of the diffuse Warm Ionized Gas. In order to constrain this contribution, we reconstruct the 3-D distribution of the sources. A detailed spatial analysis of the largest up-to-date sample of HII regions is presented.
Introduction
The thin disk component of the Galactic free electron distribution is largely contributed by localized HII regions. In order to constrain this contribution, it is important to reconstruct the spatial distribution of known sources. In a recent paper (Paladini et al., 2002, hereafter Paper I), we describe the construction of an extensive radio catalog (1442 sources) of Galactic HII regions by the combination of 24 published lists and catalogs of these objects. The final compilation consists of a Master Catalog (containing original data and corresponding errors) and a Synthetic Catalog at 2.7 GHz (which summarizes the basic information -flux density, angular diameter and V LSR -for each source of the Master Catalog). The Synthetic Catalog has provided the source of information for the spatial analysis here presented.
3-D Distribution of Galactic HII regions
In order to assess the level of completeness of our sample we study the derived log N -log S distribution (Fig. 1). This can be well fitted by a two-component power-law such as: For S max > S >∼ 70 Jy we are sampling well into the local spiral arm (spherical distribution approximation) while for S max > S >∼ 70 Jy we sample into more distant regions into the Galactic disk (disk distribution approximation). Below ∼ 7 Jy, the Synthetic Catalog starts missing sources which are in regions of high confusion level. Velocities are given for ∼ 800 of the 1442 HII regions in the Synthetic Catalog, corresponding to 60% of the total. Details about the kinematic data are given in Paper I. These data have been combined with the linear rotation model by Fich, Blitz and Stark 1989:
N (> S) ∝ S − 3 2 S max > S > ∼ 70 Jy S −1 ∼ 70 Jy > S > ∼ 1 Jy (1)Θ = (221.64 − 0.44R) km s −1(2)
to compute galactocentric (R) and solar distances (D). Taking into account the typical measurement error on radial velocities (a few km s −1 ), we have removed from our sample all the sources with an observed velocity < |10| km s −1 . These sources are characterized by a large distance-uncertainty. The recovered radial distribution is shown in Fig. 2. In computing solar distances, one has to consider the well known problem of distance degeneracy for sources lying inside the solar circle. Auxiliary data can be useful in order to solve this ambiguity. In particular, we have used absorption lines data (HI, H 2 CO) and catalogued optical counterparts. HI data have been mainly taken from Kuchar &Bania 1994 andCaswell et al. 1975 Miller 1968 andShaver et al. 1981. Through this method, we are able to assign a solar distance to 177 sources. These sources have to be added to 143 for which a unique distance can be computed from kinematic data. For the remaining 288 sources, we have worked out a method based on a distance indicator independent from kinematic data. This distance indicator (whose robustness is currently under analysis) has been found from a luminosity vs. physical diameter correlation obtained by exploiting the 2.7 and 5 GHz flux density and angular diameter data of the Master Catalog. Therefore, according to this correlation, we can compute the solar distance as:
D = 10 a * (θ/206265 × 10 3 ) b 4πS ν 1 2−b (3)
where the recovered values of the parameters a and b are, respectively, ∼ 30.6 and 1.1. The correlation-method turns out to be appliable to 256 HII regions (∼ 89% of the total number of sources with a distance ambiguity) for which at least one observed value of flux density and angular diameter at 2.7 or 5 GHz is available. With these computed solar distances, we are also able to determine the thickness of the Galactic HII regions layer. A preliminary analysis has retrieved a value in the range ∼ 32-51 pc, depending on the inclusion, in the calculation, of only sources with a safely defined solar distance (former lower value) or also of sources with a correlation-assigned distance (latter one). The plot combines: 143 unambiguous sources (diamond) ; 177 sources having either optical or absorption auxiliary information (diamond) ; 256 degenerate sources for which near (triangle) or far (cross) solar distances have been assigned through the correlation method. Overlaid on the 2-D distribution is the spiral arms model by Taylor & Cordes 1993.
Figure 1 .
1Cumulative counts N(> S) for the 2.7 GHz Synthetic Catalog.
Figure 2 .
2Radial distribution of the Synthetic Catalog HII regions (solid line). Overlaid (dashed-dotted line) is the distribution obtained by considering only sources with velocities > |10| Km s −1 .
Figure 3 .
32-D distribution of HII regions from the Synthetic Catalog at 2.7 GHz.
. Additional data are fromKerr &Knapp 1970 andGoss &Radhakrishnan 1969. H 2 CO data are from Wilson 1980. The optical catalogs which have been used are the Marsálková 1974 Master Catalog, theBlitz et al. 1982 Catalog andthe Brand &Blitz 1993 Catalog. Complementary data on individual sources are from
paladini.tex; 30/10/2018; 6:02; p.2The thin layer of Warm Ionized Gas: towards the 3-D reconstruction of the spatial distribution of Galactic HII regions 3
paladini.tex; 30/10/2018; 6:02; p.3
Roberta Paladini, Rod Davies, Gianfranco DeZotti
paladini.tex; 30/10/2018; 6:02; p.4
AcknowledgementsR. Paladini acknowledges financial support from ESA for the participation in the workshop.
. L Blitz, M Fich, A A Stark, Apjs, 49Blitz, L., Fich, M. & Stark, A. A. ApJS, 49, 183, 1982
. Brand, L Blitz, A&a, 275Brand, J & Blitz, L. A&A, 275, 67, 1993
. J L Caswell, J D Murray, R S Roger, D J Cole, D J Cooke, A&a, 45Caswell, J. L., Murray, J. D., Roger, R. S., Cole, D. J. & Cooke, D. J. A&A, 45, 239, 1975
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T L Wilson, Radio recombination lines. 205Wilson, T. L. In Radio recombination lines, 205, 1970
| [] |
[
"Adaptable Multi-Domain Language Model for Transformer ASR",
"Adaptable Multi-Domain Language Model for Transformer ASR"
] | [
"Taewoo Lee \nAI R&D Group\nSamsung Electronics\nSouth Korea\n",
"Min-Joong Lee [email protected] \nSamsung Advanced Institute of Technology\nSamsung Electronics\nSouth Korea\n",
"Tae Gyoon Kang [email protected] \nSamsung Advanced Institute of Technology\nSamsung Electronics\nSouth Korea\n",
"Seokyeoung Jung \nAI R&D Group\nSamsung Electronics\nSouth Korea\n",
"Minseok Kwon \nAI R&D Group\nSamsung Electronics\nSouth Korea\n",
"Yeona Hong \nAI R&D Group\nSamsung Electronics\nSouth Korea\n",
"Jungin Lee \nAI R&D Group\nSamsung Electronics\nSouth Korea\n",
"Kyoung-Gu Woo \nAI R&D Group\nSamsung Electronics\nSouth Korea\n",
"Ho-Gyeong Kim \nSamsung Advanced Institute of Technology\nSamsung Electronics\nSouth Korea\n",
"Jiseung Jeong \nSamsung Advanced Institute of Technology\nSamsung Electronics\nSouth Korea\n",
"Jihyun Lee [email protected] \nSamsung Advanced Institute of Technology\nSamsung Electronics\nSouth Korea\n",
"Hosik Lee \nSamsung Advanced Institute of Technology\nSamsung Electronics\nSouth Korea\n",
"Young Sang Choi \nSamsung Advanced Institute of Technology\nSamsung Electronics\nSouth Korea\n"
] | [
"AI R&D Group\nSamsung Electronics\nSouth Korea",
"Samsung Advanced Institute of Technology\nSamsung Electronics\nSouth Korea",
"Samsung Advanced Institute of Technology\nSamsung Electronics\nSouth Korea",
"AI R&D Group\nSamsung Electronics\nSouth Korea",
"AI R&D Group\nSamsung Electronics\nSouth Korea",
"AI R&D Group\nSamsung Electronics\nSouth Korea",
"AI R&D Group\nSamsung Electronics\nSouth Korea",
"AI R&D Group\nSamsung Electronics\nSouth Korea",
"Samsung Advanced Institute of Technology\nSamsung Electronics\nSouth Korea",
"Samsung Advanced Institute of Technology\nSamsung Electronics\nSouth Korea",
"Samsung Advanced Institute of Technology\nSamsung Electronics\nSouth Korea",
"Samsung Advanced Institute of Technology\nSamsung Electronics\nSouth Korea",
"Samsung Advanced Institute of Technology\nSamsung Electronics\nSouth Korea"
] | [] | We propose an adapter based multi-domain Transformer based language model (LM) for Transformer ASR. The model consists of a big size common LM and small size adapters. The model can perform multi-domain adaptation with only the small size adapters and its related layers. The proposed model can reuse the full fine-tuned LM which is fine-tuned using all layers of an original model. The proposed LM can be expanded to new domains by adding about 2% of parameters for a first domain and 13% parameters for after second domain. The proposed model is also effective in reducing the model maintenance cost because it is possible to omit the costly and time-consuming common LM pre-training process. Using proposed adapter based approach, we observed that a general LM with adapter can outperform a dedicated music domain LM in terms of word error rate (WER). | 10.1109/icassp39728.2021.9413475 | [
"https://arxiv.org/pdf/2008.06208v1.pdf"
] | 221,218,926 | 2008.06208 | cc74a0d2bfe2107d2e4de0035feb5b0a1335b273 |
Adaptable Multi-Domain Language Model for Transformer ASR
Taewoo Lee
AI R&D Group
Samsung Electronics
South Korea
Min-Joong Lee [email protected]
Samsung Advanced Institute of Technology
Samsung Electronics
South Korea
Tae Gyoon Kang [email protected]
Samsung Advanced Institute of Technology
Samsung Electronics
South Korea
Seokyeoung Jung
AI R&D Group
Samsung Electronics
South Korea
Minseok Kwon
AI R&D Group
Samsung Electronics
South Korea
Yeona Hong
AI R&D Group
Samsung Electronics
South Korea
Jungin Lee
AI R&D Group
Samsung Electronics
South Korea
Kyoung-Gu Woo
AI R&D Group
Samsung Electronics
South Korea
Ho-Gyeong Kim
Samsung Advanced Institute of Technology
Samsung Electronics
South Korea
Jiseung Jeong
Samsung Advanced Institute of Technology
Samsung Electronics
South Korea
Jihyun Lee [email protected]
Samsung Advanced Institute of Technology
Samsung Electronics
South Korea
Hosik Lee
Samsung Advanced Institute of Technology
Samsung Electronics
South Korea
Young Sang Choi
Samsung Advanced Institute of Technology
Samsung Electronics
South Korea
Adaptable Multi-Domain Language Model for Transformer ASR
Index Terms: end-to-end (E2E) automatic speech recognition (ASR)language model (LM)multi-domain adaptation
We propose an adapter based multi-domain Transformer based language model (LM) for Transformer ASR. The model consists of a big size common LM and small size adapters. The model can perform multi-domain adaptation with only the small size adapters and its related layers. The proposed model can reuse the full fine-tuned LM which is fine-tuned using all layers of an original model. The proposed LM can be expanded to new domains by adding about 2% of parameters for a first domain and 13% parameters for after second domain. The proposed model is also effective in reducing the model maintenance cost because it is possible to omit the costly and time-consuming common LM pre-training process. Using proposed adapter based approach, we observed that a general LM with adapter can outperform a dedicated music domain LM in terms of word error rate (WER).
Introduction
In recent years, virtual voice assistants have been widely spread to real-world applications. End-to-end (E2E) automatic speech recognition (ASR) has become one of the key elements of virtual voice assistant services. As new domains continue to be added, ASR models need to be adapted quickly to the new domains. Furthermore, domain specific proper nouns must be recognized such as new song titles and singer names. This means that it is necessary to maintain the recognition accuracy of the existing supported domains while securing the recognition accuracy for new words in the new domain. In addition, in order to provide a good user experience, such a response must be done very quickly.
Transformer was first introduced as a model for translation [1]. Then, it has also been successfully applied to ASR [2]. This is because Transformer has an advantage in terms of computation and parallelism over recurrent neural network (RNN) based models. In addition, knowledge distillation has been studied to create parameter efficient models [3,4]. Shallow fusion of the E2E ASR models and external language models (LM) also showed a further improvement in WER [5,6], because external LMs are able to learn more contextual information from abundant text-only data.
In natural language processing (NLP), several methods of pre-training neural language models have led to major advances in NLP subtasks. BERT, ELMO, GPT, RoBERTa, and XLNet are typical [7][8][9][10][11]. These methods find dependencies between words and their combinations by pre-training neural networks on large amounts of data. Also, by fine-tuning the model on training data in target tasks, these models could be easily applied to solving other NLP tasks. However, it is difficult to continuously update these models because deep networks tend to forget previous knowledge when it is sequentially re-trained [12]. To solve such a problem, continual learning approaches have been studied. To preserve previous knowledge, learning without forgetting (LWF) [13] adds output logits of previous stage networks to logits of current stage networks. Elastic weight consolidation (EWC) [14] constrains weight updates by valuing which weight are important for a task. Progressive neural networks [15] avoid forgetting by preserving task specific networks. However, those approaches are imperfect in memory and parameter efficiency [16].
In computer vision, residual adapter modules have been introduced to make a multi-task and multi-domain model [17]. In the paper, a large common model is used as a base model. Then small adapter modules are added in front of each batch normalization layer in series or in parallel manner. In the experiments, both methods showed better accuracy than a full fine-tuned model. Similar approaches have been explored for BERT in NLP [18]. In the paper, the authors proposed a model (called projected attention layers or PALs) that can resolve multi-domain NLP tasks by adding only adjustable 13% parameters compared to the original model. Meanwhile, in [16], a method to fine-tune models by adding only adjustable 3.6% of parameters has been proposed. The method adds small size adapters to the self-attention (SA) and feed forward network (FFN) layers of Transformer, respectively. In [19], the authors compared PALs and adapters. In the paper, fine-tuning adapters with norm layer showed better results compared to the PALs when almost similar number of parameters is used. For multilingual ASR, a structure is introduced so that only adapter layers can be switched [20]. In the study, the experiments have been conducted on recurrent neural network transducer (RNN-T) based streaming E2E ASR models.
In this paper, we study an external LM structure for Transformer based ASR model that can be adapted for multidomain with only 2% or 13% parameter addition per domain. To the best of our knowledge, this is a first attempt applying adapters to Transformer LM in ASR. The effects of our model are: 1) Our adapter-based adaptation can be used on top of the full fine-tuned model, and it further reduces word error rate (WER) from the model. 2) Multi-domain LM can be supported with fewer parameters. 3) Our approach provides cost efficient way to maintain existing models.
The rest of the paper is structured as follows: we describe our model architecture in Section 2. The experimental results on our data are reported in Section 3. Finally, we derive conclusions in Section 4. Fig. 1 shows a Transformer based E2E ASR models with an external LM. As in [2], the encoder module, which is similar to an acoustic model, takes the input features, , and transforms them to a higher-level feature representation with self-attention layers. The outputs of the encoder key and value are passed to encoder-decoder attention layers of E2E decoder. Using the and , the E2E decoder iteratively predicts output probabilities ( | 0 , ⋯ , −1 , ) of next output symbol until maximum sequence length or EOS (end-of-sequence) is met. An external LM, where encoder-decoder attention layers are removed, can be incorporated at each step of beam search to improve accuracy. Hereafter, we focus on an external LM decoder with adapters.
SA-based Multi-Domain LM with Adapter
Transformer-based E2E ASR
SA-based LM Decoder with Adapter
SA-based LM decoder consists of three parts: an output embedding, LM SA layers, and a linear transform following Softmax ( Fig. 2 left). For simplicity we set batch size and the number of domains to one in the followings.
Input Embedding
Let word-piece [21] vocabulary size be w , an input one-hot vector be ∈ ℝ 1× , hidden size be ℎ . The output of embedding matrix is computed as (1):
= (1)
where ∈ ℝ ×ℎ and ∈ ℝ 1×ℎ . Then a positional encoding vector ∈ ℝ 1×ℎ is added to [1].
SA layer in LM Decoder with Adapter
A SA layer of a LM decoder with adapters consists of four layers: layer norm [22], multi-head attention (MHA), FFN, and adapters.
Multi-Head Attention
Let the number of heads be ℎ . Previous output is projected to a query, a key, and a value simultaneously for multi-head attention ( Fig.2 left). Instead of performing a single attention function using ℎ dimentional , , and , MHA performs the attention function ℎ times in parallel with differently learned ℎ/ ℎ dimentional , , and . Then ℎ numbers outputs are concatenated and projected into a single representation. The detailed equation is as follows:
MultiHead( , , ) = Concat(ℎ 1 , ⋯ , ℎ ℎ ) (2) where ℎ = ( , , ) = softmax ( ( )( ) √ ℎ ℎ ) ( ),(3)
∈ ℝ ℎ× , ∈ ℝ ℎ× , ∈ ℝ ℎ× , and ∈ ℝ ℎ×ℎ are trainable parameters. Note = = = ℎ ℎ ⁄ throughout the paper.
Position-wise Feed-Forward Network
Let an inner filter size . Position-wise feed forward network consists of two FFNs with ReLU activation in between. An output of position-wise FFN is calculated as (4) where the input vector 1 ∈ ℝ 1×ℎ , the weight matrices and bias vectors 1 ∈ ℝ ℎ× , 1 ∈ ℝ 1× , 2 ∈ ℝ ×ℎ , and 2 ∈ ℝ 1×ℎ .
FFN( 1 ) = max(0, 1 1 + 1 ) 2 + 2(4)
Adapter
Adapter modules proposed in [16] are inserted on top of MHA and FFN layers as in Fig. 2 (left). An adapter module (Fig. 2 right) consists of two linear transforms and ReLU activation in between. A residual connection is added to the output. The outputs of adapters 1 and 2 are calculated as follows:
1 ( 2 ) = 2 + max(0, 2 3 + 3 ) 4 + 4(5)
2 ( 3 ) = 3 + max(0, 3 5
+ 5 ) 6 + 6 ,(6)
where 2 = MultiHead( , , ), 3 = FFN( 1 ), adapter filter size is , 3 , 5 ∈ ℝ ℎ× , 3 , 5 ∈ ℝ 1× , 4 , 6 ∈ ℝ ×ℎ , 4 , 6 ∈ ℝ 1×ℎ .
Softmax
The outputs of decoder are transformed to the probabilities of output classes by a linear projection 7 ∈ ℝ ℎ× and a subsequent softmax function. Table 1 The G-LM is trained on 24GiB normalized Korean text data consisting of 353M utterances. All data were anonymized. The data consists of representative utterances of Samsung's Bixby scenario and general domain corpus. The M-LM is trained on normalized Korean text data consisting of 45M utterances, in which general and music domain (song title and singer name related commands) corpus are mixed. To train our models, we used Tensor2Tensor framework [23].
Experiments
For G-LM experiments, we recorded test cases (TCs) in three categories: In-Domain, Out-Domain, and Open-Domain. In-Domain TCs includes 50K Bixby use-case scenario utterances such as phone and device control commands and daily conversational question and answering. Out-Domain TCs includes 8K domain specific utterances which is not included in In-Domain training corpus. Especially, we selected domains having its own unique proper nouns such as hospital or doctor's names. Open-Domain TCs are included to test noisy environment, on which cafe, city, office, highway noises are added to clean speech. The content of the utterances is in arbitrary domain and do not include unknown unique proper nouns. All TC are recorded in male and female voices. For M-LM experiments, In-Domain and Out-Domain TCs are recorded. In-Domain TC includes 610 utterances. It represents well known song titles and singer names. On the other hand, Out-Domain TC includes 3709 utterances. The content is newly added song titles and singer names.
We initialized weights of each adapter layer to the values following a normal distribution having zero mean and 10e −4 variance. We tested variance values of { 0 , 10e −7 , 10 −6 , 10 −5 , 10 −4 , 10 −3 , 10 −2 } and selected a largest sable value. Since an adapter module internally has a residual connection, zero variance can be inserted to test output of the adapter module is bypassed properly. All runs are trained on eight P40 GPUs to build models from scratch and on one P40 GPU for adaptations. We used Adam optimizer with 1 = 0.9, 2 = 0.98, = 1 −9 . Batch sizes tested from {32, 64, 128, 512, 1024, 4096, 8192}. 8192 is used for all our adaptation experiments. Unlike [16], small batch size made our training unstable, failing to converge. Learning rate is selected as 0.03 from {0.1, 0.03, 0.001, 0.0003, 0.0001}. When we train our models from scratch or adapt without adapter, we applied Noam learning rate decay scheme with 1000 warmup steps. On the other hand, when we train our adapter related layers, learning rate decay scheme did not used.
We used 4096 word-pieces as output token units. For E2E model training, we used same hyper-parameters in [3]. All experiments used the identical input feature processing to that of [24]. The decoding hyper-parameters (beam size, lengthpenalty, and maximum decoding length) were tuned to minimize WER. Known proper nouns and number are converted with an inverse text normalization (ITN) module. We assumed we already knew proper domain names before inferencing.
We first define four different model training or adaptation methods. (1) build from scratch: LMs are trained on whole corpus. All layers are trained and adapters are not added. (2) full Our goal is to attain 1) best performance iterative adapter fine-tuned LMs compared to the best full fine-tuned LMs, 2) LMs that can be extended to multi-domain by iterative adapter fine-tuning, using a common pre-trained LM. To show the proposed models can achieve the goals, we conduct four experiments. Table 2 shows WERs measured with only E2E models (E2E), E2E models with a full fine-tuned G-LM (E2E-G-LM), and E2E models with an adapter fine-tuned G-LM (E2E-G-LM-A). Compared to the results decoded with only E2E models (E2E), in E2E-G-LM, WERs were reduced 0.6, 2.44, and 7.72%p for in, out, and open domain TCs, respectively. When the full finetuned G-LM was additionally adapter fine-tuned (E2E-G-LM-A), WERs were further reduced by 0.73, 7.78, and 8.25%p for in, out, and open domain TCs respectively. In particular, in domains having unusual proper nouns, we got higher improvement in accuracies. This means adapter fine-tuning can bias output probability properly for unusual proper nouns. In addition, despite this strong biasing, the accuracy of existing domain TCs did not deteriorated. Table 3 In Table 4, we see how far WERs can be reduced by iterative adapter fine-tuning. The model M1 refers to a model that an adapter fine-tuned M-LM using error sentences from the E2E-M-LM decoding result as training data. We considered decoding, error sentence extraction, and re-training a model as one iteration. In the experiment, accuracy improved until iterations were repeated three times. If iteration was repeated more than that, WERs were not improved more. Table 5 compares the case of using a G-LM as a common base LM with an iterative adapter fine-tuned G-LM (E2E-G-LMiter) and the case of creating a dedicated M-LM and full finetune or adapter fine-tune it (E2E-M-LM, E2E-M-LM-A). Intuitively, when we decode music domain TCs with the E2E model and G-LM without any adaptation (E2E-G-LM) as a baseline, it showed a higher error rates than E2E-M-LM and E2E-M-LM-A. Last two columns in Table 5 show word error rate reduction (WERR). When we iterative adapter fine-tuned the G-LM three times (E2E-G-LM-Aiter3), WERs were reduced by 0.49 and 0.83%p in both in and out domains TCs, respectively, compared to E2E-M-LM. Also, the WERs of E2E-G-LM-Aiter3 were almost close to the results of E2E-M-LM-A. This means that a common G-LM with adapters can be used as a dedicated domain LM, and we can switch only adapter related layers to fit our model on each domain. Therefore, a multidomain LM configuration with the structure shown in Fig. 2 is possible.
Results
Since is a relatively small value, the increasing number of parameters per domain is about 2% for the first domain and about 13% for after the second domain. Specifically, 2 (2 ℎ + + ℎ) for the first domain, because norms and Softmax linear layers can be reused. 2 (2 ℎ + + 3ℎ) + (2ℎ + ℎ + ) for after the second domain. This slow increasing property is important because memory size is limited for GPU or on-device applications.
We built our base LMs from scratch on eight P40 GPUs and on v3-8 tensor processing units (TPU). It took three days on eight P40 GPUs and 4 hours and 30 minutes on TPU. Iterative adapter fine-tuning proposed in the paper can train G-LM in 60 minutes on a P40 GPU and 25 minutes for M-LM. Since a P40 GPUs may be available in on premise servers, we expect that cloud computing cost may be saved.
Conclusions
In this paper, adapter based multi-domain LM structure has been proposed. The structure is a combination of two architectures: an adapter module proposed for BERT in NLP area and a switchable adapter architecture proposed for RNN-T streaming ASR model. The proposed architecture allows LMs to expand multi-domain, suppressing the increase of the number of parameters. The proposed architecture can reduce WERs of target domains without WER decrease of existing domains. Also we observed that applying adapter module on Transformer LM has an effect on WER improvement especially for proper nouns that is hard to be handled with a common base LM. Finally, the proposed architecture can reuse standard full finetuned LMs. So, the full fine-tuned LMs can be easily reused (or transferred) without any changes. Table 5. Iterative fine-tuning performance (WER). The results show a G-LM with iterative fine-tuned adapters can be used as a dedicated music LM.
shows overall model architectures and model sizes used in the experiments. In the experiments, a general domain LM (G-LM), a music specialized domain LM (M-LM), and adapter added general and music LMs (G-LM-A, M-LM-A) are used. For single precision floating point, model sizes are increased about 2% when adapters are added for a first domain.
96 3.87 fine-tuning: LMs are fine-tuned on small size corpus of target domain. All layers are tuned and adapters are not added. (3) adapter fine-tuning: adapters are added on top of full fine-tuned models. LMs are fine-tuned on a small size corpus of target domain. Only adapter related layers (adapters, norms, a Softmax linear) are tuned. (4) iterative adapter fine-tuning: A LM is adapter fine-tuned iteratively. An iteration, here, means a process of a) decoding TCs with a latest adapter fine-tuned LM, b) collect error sentences from the results, and c) adapter fine-tune the last LM with the error sentences.
shows WERs measured with only E2E models (E2E), E2E models with a full fine-tuned M-LM (E2E-M-LM), and E2E models with an adapter fine-tuned M-LM (E2E-M-LM-A). The results of using E2E models with a full fine-tuned M-LM (E2E-M-LM) showed improved WERs than the results decoded with the E2E models alone. The WERs of in and out domain TCs were reduced by 5.52 and 7.23%p, respectively. When the full fine-tuned M-LM was additionally adapter finetuned (E2E-M-LM-A), WERs were further reduced by 0.22, 1.3%p for in and out domain TCs respectively. Like G-LM experiments, adapter fine-tunings improves the proper noun recognition accuracy without compromising the accuracy of existing domains, even for smaller models.
Figure 2. (Left) is an architecture of transformer multi-domain LM. In a LM decoder, the adapter module (right) is added on top of multi-head attention and feed-forward layers. Only green layers (including layer norms or LN) are fine-tuned on the downstream data and expanded for multi-domain. Dotted red lines shows a switchable decoding path for a first domain.Figure 1. The dotted line box shows transformer-based E2E ASR model, including encoder and decoder. An external LM is incorporated at each step of beam search.Feed-forward
down-project
Nonlinearity
Feed-forward
up-project
Adapter
Layer
×
2x
Feed-forward
1
Softmax
Multi-Head
Attention
Intput
Embedding
Positional
Encoding
1
1
⋯
⋯
1
⋯
1 ⋯
1
⋯
⋯
Transformer
LM
Transformer
Encoder
Transformer
Decoder
Transformer
LM
E2E
,
−1
−1
Table 1 .
1The architectures and sizes of SA E2E, general LM(G-LM), music LM (M-LM), and adapter added LMs
E2E
Enc.
E2E
Dec.
G-LM
G-LM-
A
M-LM
M-LM-
A
# Layers
6
4
3
3
2
2
ℎ
512
512
512
512
512
512
3072 3072 4096 4096 2048 2048
-
-
-
64
-
64
ℎ
16
4
8
8
8
8
Size (MiB) 96.7 80.3
76.4 77.9 40.3
41.3
Table 2. WERs of E2E, E2E-G-LM, and E2E-G-LM-A on
General Domain TCs
TC
E2E
E2E-G-LM E2E-G-LM-A
In-Domain
2.42
1.82
1.69
Out-Domain
10.62
8.18
2.84
Open-Domain
12.8
5.08
4.55
Table 3. WERs of E2E, E2E-M-LM, and E2E-M-LM-A on
Music Domain TCs
TC
E2E
E2E-M-LM E2E-M-LM-A
In-Domain
8.2
2.68
2.46
Out-Domain
12.66
5.43
4.13
Table 4. WERs of iterative adapter fine-tuning with M-LM-A
on Music Domain TCs
TC
E2E-M-LM M1 iter1
M1 iter2
M1 iter3
In-Domain
2.68
2.46
1.97
1.81
Out-Domain
5.43
4.13
3.
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| [] |
[
"Colloquium: Light scattering by particle and hole arrays",
"Colloquium: Light scattering by particle and hole arrays"
] | [
"F J García De Abajo \nInstituto deÓptica -CSIC\nSerrano 12128006MadridSpain\n"
] | [
"Instituto deÓptica -CSIC\nSerrano 12128006MadridSpain"
] | [] | This colloquium analyzes the interaction of light with two-dimensional periodic arrays of particles and holes. The enhanced optical transmission observed in the latter and the presence of surface modes in patterned metal surfaces are thoroughly discussed. A review of the most significant discoveries in this area is presented first. A simple tutorial model is then formulated to capture the essential physics involved in these phenomena, while allowing analytical derivations that provide deeper insight. Comparison with more elaborated calculations is offered as well. Finally, hole arrays in plasmon-supporting metals are compared to perforated perfect conductors, thus assessing the role of plasmons in these types of structures through analytical considerations. | 10.1103/revmodphys.79.1267 | [
"https://arxiv.org/pdf/0903.1671v1.pdf"
] | 18,698,507 | 0903.1671 | 4fada1742aa77095b610a68052d6f006cadcd45d |
Colloquium: Light scattering by particle and hole arrays
(Dated: March 10, 2009)
F J García De Abajo
Instituto deÓptica -CSIC
Serrano 12128006MadridSpain
Colloquium: Light scattering by particle and hole arrays
(Dated: March 10, 2009)numbers: 4225Fx7320Mf4279Dj4120Jb
This colloquium analyzes the interaction of light with two-dimensional periodic arrays of particles and holes. The enhanced optical transmission observed in the latter and the presence of surface modes in patterned metal surfaces are thoroughly discussed. A review of the most significant discoveries in this area is presented first. A simple tutorial model is then formulated to capture the essential physics involved in these phenomena, while allowing analytical derivations that provide deeper insight. Comparison with more elaborated calculations is offered as well. Finally, hole arrays in plasmon-supporting metals are compared to perforated perfect conductors, thus assessing the role of plasmons in these types of structures through analytical considerations.
I. INTRODUCTION
The scattering of waves in periodic media plays a central role in areas of physics as diverse as low-energy electron diffraction (Pendry, 1974) or atomic-beam scattering from crystal surfaces (Farías and Rieder, 1998). Valence electrons in solids, sound in certain ordered constructions (Martínez-Sala et al., 1995), or light in photonic crystals (Joannopoulos et al., 1997;López, 2003) undergo diffraction that under certain conditions can limit their propagation in frequency regions known as band gaps (Ashcroft and Mermin, 1976). Among these examples, the scattering of electromagnetic waves is particularly important because it allows obtaining structural and spectroscopic information over a fantastically wide range of lengths, going from atomic dimensions in x-ray scattering (Henke et al., 1993) to macroscopic distances in radio and microwaves. Actually, Maxwell's equations are written in first-order derivatives with respect to spatial coordinates, so that light scattering in the absence of nonlinear effects is solely controlled by the shape and permittivity of diffracting objects with distances measured in units of the wavelength, and therefore, the same phenomena are encountered over entirely different length scales.
We can classify the performance of periodic structures in three distinct categories according to the ratio of the period a to the wavelength λ. For λ a, an effective homogeneous medium description is possible. This is in fact what happens in most naturally-occurring substances when a has atomic dimensions. But also in certain artificially textured materials (metamaterials), which allow achieving exotic behavior like magnetic response at visible frequencies (Grigorenko et al., 2005) and media with negative refraction index (Smith et al., 2004), without neglecting the exciting possibility of using nanoparticles as building blocks to tailor on-demand optical properties (Liz-Marzán, 2006). The opposite limit (λ a) is generally well accounted for by classical rays, although keeping track of phases proves to be crucial near points of light accumulation, like in the self-imaging of gratings described by Talbot nearly two centuries ago (Huang et al., 2007;Talbot, 1836). Nevertheless, it is the intermediate regime, when λ is comparable to a, in which diffraction shows up in full display. We find examples of this in both three-dimensional (3D) photonic crystals, which offer a promising route to fully controlling light propagation over distances comparable to the wavelength (Joannopoulos et al., 1997;López, 2003), and twodimensional (2D) crystals, in which an impressive degree of optical confinement has been accomplished (Akahane et al., 2003).
In this colloquium, we shall focus on light scattering by planar structures of particles or holes, which have become a current subject of intense research driven to some extent by advances in nano-patterning techniques. Our main purpose is to explain the phenomena observed within this context in a tutorial but nevertheless comprehensive fashion. We shall first review experimental and theoretical developments in Sec. II. Then, we shall formulate in Sec. III a simple powerful model that deals with the response of particle and hole arrays on a common footing, leading to analytical expressions that capture the main physical aspects of these systems. Finally, metals with plasmons will be discussed, and the main differences with respect to plasmon-free perfect conductors elucidated, in Sec. IV. We shall use Gaussian units, unless otherwise stated.
The beginning of the last century witnessed important developments in diffraction of light in gratings after Wood's observation of anomalous reflection bands (Wood, 1902(Wood, , 1912(Wood, , 1935 and their subsequent interpretation (Fano, 1936(Fano, , 1941Lord Rayleigh, 1907). Two types of anomalies were identified, one of them occurring when a diffracted beam becomes grazing to the plane of the grating, the Rayleigh condition (Lord Rayleigh, 1907), giving rise to a sharp bright band, and the other one showing up to the red of the former as an extended feature containing two neighboring dark and bright bands (Fano, 1936(Fano, , 1941.
The century concluded with another significant discovery : periodic arrays of subwavelength holes drilled in thin metallic films can transmit much more light per hole at certain frequencies than what was previously expected for single openings, based upon Bethe's prediction of a severe cutoff in transmission as (b/λ) 4 for large λ compared to the hole radius b (Bethe, 1944). Previous knowledge gathered by electrical engineers in the microwave domain (Chen, 1971;McPhedran et al., 1980;Ulrich, 1967) had already exploited the use of periodically-drilled surfaces as frequency-selective filters and discussed the occurrence of 100% transmission at wavelengths slightly above the period. However, the hole sizes that were considered in that context lied in the region of sizeable transmission for single holes. The more recently discovered extraordinary transmission phenomenon was however observed for narrower holes (relative to the wavelength), the transmission of which exceeded orders of magnitude what was expected from the sum of their individual transmissions . For square arrays under normal incidence, a transmission minimum occurred at a wavelength close to the period a, coinciding with the Rayleigh condition (Lord Rayleigh, 1907), and a transmission maximum showed up at longer wavelength, thus revealing its connection to Wood's anomalies Sarrazin et al., 2003). However, the explanation of the effect is still a subject of debate, as some authors un-derstand that it originates mainly in the interaction of the apertures with surface plasmons Ghaemi et al., 1998;Martín-Moreno et al., 2001;Popov et al., 2000;Salomon et al., 2001;Wannemacher, 2001), whereas other authors make emphasis in dynamical light diffraction (Lezec and Thio, 2004;Sarrazin et al., 2003;Treacy, 1999Treacy, , 2002. While the latter works well to understand the observed extraordinary optical transmission in drilled plasmon-free perfect conductors (Cao and Nahata, 2004;Gómez-Rivas et al., 2003;Mittra et al., 1988;Miyamaru and Hangyo, 2004), supporters of the surfaceplasmon interpretation argue that the enhanced transmission relies in this case on plasmon-like lattice-surfacebound modes sustained by patterned perfect-conductor surfaces (Pendry et al., 2004). Actually, evidence of such modes had been observed before in periodically perforated metallic screens for wavelengths several times larger than the period (Ulrich and Tacke, 1972). We shall see below how these are in fact complementary views of the same phenomenon and how diffraction in particle arrays contains already the essential features that can be translated to understand the phenomenology of hole arrays. But let us first summarize experimental and theoretical findings in this area.
II. OVERVIEW OF EXISTING RESULTS
A huge amount of literature has been accumulated on transmission through periodic structures, and it is an interesting exercise to reexamine it in connection to recent developments.
A. Single holes
Bethe's predicted cutoff in the transmission of a single hole in a perfect-conductor thin screen as (b/λ) 4 is the leading-order term of the expansion of the transmission cross section in powers of b/λ (Bethe, 1944). Subsequent higher-order analytical corrections (Bouwkamp, 1954;Chang et al., 2006), and eventually rigorous numerical calculations (García de Abajo, 2002;Roberts, 1987), demonstrated that the cross section lies below the hole area up to a radius b ≈ 0.2λ. These results have found experimental corroboration down to the NIR regime (Obermüller and Karrai, 1995), with new localized plasmon resonances showing up at shorter wavelengths (Degiron et al., 2004;Rindzevicius et al., 2007).
Two different mechanisms have been however suggested to achieve enhanced transmission in a single hole: filling it with a material of high permittivity (García de Abajo, 2002;García-Vidal et al., 2005;Webb and Li, 2006), thus creating a partially-bound cavity mode that couples resonantly to incident light (see Sec. III.E); and decorating the aperture with periodic corrugations in much the same way as highlydirectional antennas are capable of focusing electromag-netic radiation on a central dipole element by means of concentric, periodically-spaced metallic rings (James, 1977).
B. Optical transmission through hole arrays
The intensity of light passing through holes is boosted at certain wavelengths when we arrange them periodically. Pioneering calculations and microwave experiments showed already zero reflection in thin films perforated by periodic arrays of small apertures of radius b ≈ 0.36λ (Chen, 1971). Further seminal experiments focused on the relation between hole arrays in thin metal screens and their complementary screens (Ulrich, 1967), putting Babinet's principle to a test in the far-infrared region. This was followed by numerous applied studies of hole arrays (regarded as frequency-selective surfaces) in the engineering community, including filters for solar energy collection and elements to enhance antennae performance (Cwik et al., 1987;Maystre, 1980;McPhedran et al., 1980;Mittra et al., 1988).
The work of Ebbesen et al. (1998) demonstrated in the optical domain extraordinary light transmission, which for the first time occurred for openings of radius below the cutoff of the first propagating mode in a circular waveguide, b < 0.29λ. Since then, this phenomenon has been consistently observed for a varied list of metallic materials (Przybilla et al., 2006a), over a wide range of wavelengths [e.g., for microwaves (Cao and Nahata, 2004;Gómez-Rivas et al., 2003), to which metals respond as nearly perfect conductors, in the infrared (Selcuk et al., 2006), and in the VUV, using a good conductor in this regime like Al (Ekinci et al., 2007)], and with different types of array symmetries, including the recent demonstration of the effect in 2D quasi-crystal arrangements (Matsui et al., 2007;Przybilla et al., 2006b;Schwanecke et al., 2006;Sun et al., 2006). Two examples of enhanced transmission, taken from Krishnan et al., 2001, andMartín-Moreno et al., 2001, are shown in Fig. 1. The transmission is several times larger in the infrared peak than the prediction of Bethe for non-interacting holes in a thin screen, and four orders of magnitude larger than what is expected for noninteraction apertures in a perfect-conductor film of the same thickness (dashed curves).
Light transmission through hole arrays has been examined theoretically for over four decades (Chen, 1971;Dawes et al., 1989;Eggimann and Collin, 1962;McPhedran et al., 1980), although a detailed account of extraordinary optical transmission in real metals had to wait until the new century began Popov et al., 2000;Salomon et al., 2001;Sarrazin et al., 2003;Wannemacher, 2001) and the advance in computation power allowed predictive capacity (Chang et al., 2005;Klein Koerkamp et al., 2004).
The influence of various geometrical and environmental factors has been extensively studied. In particular, FIG. 1 (Color in online edition) Extraordinary optical transmission in hole arrays. The measured transmittance (solid curves) is shown for apertures drilled in gold (a) and silver (b) films, taken from Krishnan et al., 2001, andMartín-Moreno et al., 2001, respectively. The silver film is self-standing in air, while the gold is deposited in quartz and immersed in an index-matching liquid. The lattice constant is a = 600 nm in Ag and a = 750 nm in Au. The transmittance of the perforated gold goes well above that predicted for non-interacting apertures in a perfect-conductor film or by Bethe's formula for a thin screen (dashed curves). Rayleigh's condition for the (1,0) and (1,1) beams becoming grazing are indicated by vertical dashed lines. Analytical results are shown as arrows in (a) and as a dashed curve in (b) (see Sec. IV.D). The transmittance is presented vs the wavelength in the dielectric environment of the metal, normalized to the lattice constant.
the role of hole shape has been shown to yield nontrivial effects (Elliott et al., 2004;Gordon et al., 2004;Klein Koerkamp et al., 2004;Krasavin et al., 2005;van der Molen et al., 2005), such as larger enhancement and red shift of the transmission peaks with respect to the Rayleigh condition for light polarized along the short axis of elongated apertures. Finite arrays have been found to exhibit interesting shifts in the transmission maxima as well, depending on the number of apertures (Bravo-Abad et al., 2004;Lezec and Thio, 2004). More exotic shapes like annular holes have been also simulated (Baida and Van Labeke, 2002;Roberts and McPhedran, 1988) and measured (Fan et al., 2005), with the additional appeal that annular waveguides support always one guided mode at least (Jackson, 1999).
The transmission is exponentially attenuated with hole depth because it is mediated by evanescent modes of the apertures regarded as narrow subwavelength waveguides. However, strong signatures of interaction between both metal interfaces have been reported , as well as high sensitivity to dielectric environment, so that maximum transmission is achieved when the permittivity is the same on the two sides of the film (Krishnan et al., 2001). Extraordinary optical transmission has expanded to embrace a wide range of phenomena (Genet and Ebbesen, 2007), like the interaction of hole arrays with molecules for potential applications in biosensing (Dintinger et al., 2006a) and all-optical switching (Dintinger et al., 2006b;Janke et al., 2005;Smolyaninov et al., 2002), and the demonstration of the quantum nature of plasmons through photon entanglement preservation after traversing a hole array (Altewischer et al., 2002).
C. Particles
The field of light scattering by small particles has a rich research tradition (Bohren and Huffman, 1983;van de Hulst, 1981) that is being continued by hot topics such as for example novel near-field effects in the coupling of metallic nanoparticle arrays (Krenn et al., 1999) and strong inter-particle interactions in dimers (Atay et al., 2004;Nordlander et al., 2004;Romero et al., 2006). Here, we shall single out just two recent exciting developments in line with the rest of our discussion. The first one refers to coupled metallic nanoparticle arrays. These particles can sustain localized plasmon excitations that hop across neighbors. It has been suggested (Quinten et al., 1998), and later confirmed by experiment (Maier et al., 2001(Maier et al., , 2003, that this phenomenon can be utilized to transmit light energy along chains of subwavelength particles, thus providing some basic constituents for future plasmonic devices.
In a different development, the scattering spectra from 1D and 2D arrays of metallic nanoparticles were predicted to exhibit very narrow plasmon lineshapes produced by dynamical scattering . Experiments performed on lithographically patterned particle arrays confirmed this effect and achieved reasonable control over spectral lineshapes (Hicks et al., 2005). We shall discuss this further in Sec. III.A.2.
III. TUTORIAL APPROACH
A tutorial model will be presented next that becomes exact in the limit of narrow holes or small particles in perfect-conductor films. This model will describe the basic physics involved both in extraordinary light transmission and in lattice surface modes of structured metals, but it leads to simple analytical expressions that permit understanding these phenomena in a fundamental way FIG. 2 (Color in online edition) Two-dimensional array of small identical particles illuminated by a light plane wave. k is the momentum component parallel to the array. The particle at position Rn displays a dipole pn. and making several challenging predictions.
A. Basic relations
We shall start with some basic analytical relations for the scattering of an external light plane wave on a periodic array of identical particles that are small compared to both the wavelength and their separation (see Fig. 2). Within linear, non-magnetic response, the particle at position R n can be assumed to respond with an induced dipole p n = α E E(R n ), determined by its electric polarizability tensor α E and the self-consistent field acting on it, E(R n ). This dipole induces an electric field at point r that can be written G 0 (r − R n )p n in terms of the dipole-dipole interaction tensor,
G 0 (r) = (k 2 + ∇∇) e ikr r ,(1)
where k is the light momentum in free space. 1 Now, the self-consistent dipole of our particle is found to be
p n = α E E ext (R n ) + n =n G 0 (R n − R n )p n ,(2)
where E ext (R n ) = E ext exp(ik · R n ) is the external electric field, which depends upon the site position R n just through a phase factor involving components of the incoming wave momentum parallel to the array, k , as illustrated in Fig. 2, and the second term inside the square brackets represents the field induced by the rest of the particles. Bloch's theorem guarantees that the solution of Eq.
(2) must have the form p n = p exp(ik · R n ). Direct insertion of this expression into Eq.
(2) leads to
p = 1 1/α E − G(k ) E ext(3)
and
G(k ) = n =0 G 0 (R n )e −ik ·Rn ,(4)
where we have chosen R 0 = 0. Notice that the denominator of Eq. (3) separates the properties of the particles (α E ) from those of the lattice [the structure-factor-type of sum G(k )], in the spirit of the KKR method in solid state physics (Ashcroft and Mermin, 1976). The lattice sum in Eq. (4) can be converted into rapidly converging sums using Ewald's method (Glasser and Zucker, 1980), and we have used in particular the procedure elaborated by Kambe (1968). Incidentally, Eqs.
(2)-(4) can be also applied to 3D particle arrays with k replaced by a 3D crystal momentum. This type of approach has been shown to lead to robust band gaps in atomic lattices (van Coevorden et al., 1996). Furthermore Eq. (2) together with the Clausius-Mossotti formula (Ashcroft and Mermin, 1976) constitute the basis of the discrete-dipole approximation (DDA) method for solving Maxwell's equations in arbitrary geometries (Draine and Flatau, 1994;Purcell and Pennypacker, 1973). It should also be noted that the present approach can be extended to larger particles arranged in ordered (Stefanou et al., 1998(Stefanou et al., , 2000 or disordered arrays (García de Abajo, 1999) by including higher-order multipoles, and that this is one of the methods that can be actually applied to deduce effective optical properties of composite materials (Milton, 2002).
It is useful to represent the dipole-dipole interaction in 2D momentum space in the plane of the array, which we shall take to coincide with z = 0. This is readily done by expressing the scalar interaction at the right end of Eq.
(1) as e ikr r = i 2π
d 2 Q k z e i(Q·R+kz|z|) ,
where k z = k 2 − Q 2 is the normal momentum and the notation r = (R, z), with R = (x, y), has been adopted. From here and Eq. (1) one obtains expressions like
G 0 xx (r) = i 2π d 2 Q k z (k 2 − Q 2 x )e i(Q·R+kz|z|)(5)
for the components of the interaction tensor, here specified for the xx directions. This allows us to recast Eq.
(4) into a sum over 2D reciprocal lattice vectors g, using the relation
n exp(iQ · R n ) = (2π) 2 A g δ(Q − g),(6)
where A is the area of the lattice unit cell. For example, the G xx component under normal incidence (k = 0) becomes
G xx (0) = lim z→0 2πi A g 1 k g z (k 2 − g 2 x )e ik g z |z|(7)
− i 2π
d 2 Q k z (k 2 − Q 2 x )e ikz|z| ,
where k g z = k 2 − g 2 , and the integral represents the subtraction of the n = 0 term in the sum of (4). This expression is important to elucidate some properties of the lattice sums, as we shall show below.
Reflection and absorption in particle arrays
The scattered field is given by a Rayleigh expansion similar to the one in Eq. (7) , with each vector g labeling one reflected and one transmitted beam of parallel momentum k + g (Lord Rayleigh, 1907). In the far field in particular, the zeroorder (g = 0) reflection and transmission coefficients under normal incidence reduce to
r = 2πik/A 1/α E − G xx (0)(8)
and t = 1 + r,
where the first term in the right-hand side of Eq. (9) represents the unscattered beam, and the numerator of (8) is the far-field amplitude produced by a lattice of unit dipoles. Interestingly, the absorbance of the array is given by 1 − |1 + r| 2 − |r| 2 [see Eq. (9)], which when regarded as a function of the complex variable r, has a maximum of 50% coinciding with r = −1/2 and t = 1/2. This condition is easily attainable near a lattice singularity (see Sec. III.B), using for instance weakly dissipative spherical particles. Similar results have been predicted for narrow cylinder arrays , in which 100% absorption is possible in one of the polarization components for the right choice of parameters.
A particularly simple situation is encountered when the wavelength is larger than the lattice spacing, so that all diffracted beams other than the zero-order beam are evanescent (|k + g| > k). Then, upon inspection of Eq. (7), one finds the useful relation
{G xx (0)} = 2πk/A − 2k 3 /3, k < g 1 ,(10)
where g 1 denotes the period of the reciprocal lattice (g 1 = 2π/a for square arrays). Moreover, if the particles are non-absorbing, the optical theorem constrains their polarizability by the condition {−1/α E } = 2k 3 /3 (van de Hulst, 1981). Combining these expressions, one obtains
r = −1 1 + iA 2πk {1/α E − G xx (0)}(11)
for the reflection coefficient of non-dissipative particles under normal incidence below the diffraction threshold.
The electrostatic approximation provides a reasonable description of the electric polarizability of small particles, α es E . However, this needs to be amended in order to comply with the mentioned optical-theorem constrain, for instance via the prescription α E = 1/(1/α es E −2ik 3 /3). Analytical expressions for α es E exist for a variety of particle shapes, including homogeneous spheres (α es E = b 3 ( − 1)/( + 2), where b is the radius and the permittivity) and ellipsoids (Jones, 1945).
We illustrate the applicability of Eq. (11) through an example consisting of square lattices of perfectlyconducting thin disks. Fig. 3 compares the analytical result of Eq. (11) (dashed curves) with the full solution of Maxwell's equations obtained by following a layer-KKR multiple-scattering formalism (Stefanou et al., 1998(Stefanou et al., , 2000 to simulate the array together with a modal expansion solution of the isolated disk similar to the one available for isolated holes (García de Abajo et al., 2005a;Roberts, 1987). In the analytical solution we have used the polarizability of thin metallic disks as derived from an ellipsoid of vanishing height, α es E = 4b 3 /3π, where b is the radius. The results of the analytical model describe qualitatively well the presence of zero-and full-reflection points in the spectra, irrespectively of the disk size, but we shall discuss this point further in Sec. III.C.
Narrowing lineshapes through dynamical scattering
The above formalism can be used to explain the effect of narrowed plasmon lineshapes in the scattering spectra of 1D and 2D particle arrays (Hicks et al., 2005;. For simplicity, we shall discuss metallic spherical particles described by the Drude dielectric function
(ω) = 1 − ω 2 p ω(ω + iη) ,(12)
where ω p is the bulk plasma frequency and the plasmon damping rate is ≈ η/2 ω p . Using this expression to obtain the polarizability of a small sphere of radius b (see Sec. III.A.1), we can recast Eq.
(3) into a Lorentzian of width ≈ η/2 + (ω p b 3 /2 √ 3) {G}.
The natural width of the isolated particles is now supplemented by a term proportional to {G} [see Eq. (10)], which can take negative values that compensate the η/2 term to render arbitrarily narrow collective plasmon resonances for an appropriate choice of array parameters.
Applying this to a 2D square array under normal incidence with λ ∼ a, we find that Eq. (10) yields complete cancelation of the width for b/a ≈ 0.16(η/ω p ) 1/3 . Under such conditions, the narrowing of the width is just limited by the physical requirement that |r| 2 + |t| 2 ≤ 1 [see Eqs. (8) and (9)].
B. Lattice singularities
The interaction among particles in the periodic arrays of Sec. III.A appears to be governed by the lattice sums G(k ) and is dominated by their singularities, which originate in accumulation of in-phase scattered fields. Following similar arguments to previous expositions of this idea (Fano, 1941;Lord Rayleigh, 1907), we just consider a 1D periodic chain of particles illuminated by an incident plane wave with both propagation direction and electric field perpendicular to the array, so that the field induced by a given particle on a distant one scales with the inverse of their separation, and thus, the contribution of distant particles to the interaction lattice sum has the convergence properties of the series ∞ n=1 e ikan /n, which diverges as the wavelength approaches the period a as − ln(ka − 2π) (Gradshteyn and Ryzhik, 1980). The same is true for 2D arrays. These singularities in G(k ) are signaled by the Rayleigh condition of a diffracted beam becoming grazing (Lord Rayleigh, 1907), as can be seen from Eq. (7), where divergent terms g ≈ k (i.e., terms with zero normal momentum k g z ) dominate the sum. A remarkable consequence of this analysis is that the array becomes invisible to the incoming light right at the lattice sum divergence [G xx (0) → ∞, so r → 0, according to Eq. (8)], showing 100% transmission even for absorbing particles. Focusing for simplicity on a square array of period a, the normal-incidence lattice sum (7)
diverges as (García de Abajo et al., 2005a) G xx (0) ≈ 4π 2 √ 2 a 3 1 λ/a − 1 − 118(13)
for λ > ∼ a, where a fitted constant has been subtracted in order to extend the validity of this expression well beyond the singularity.
For oblique incidence with k along one of the lattice unit vectors, proceeding as in the derivation of Eq. (7), one finds that G(k ) is diagonal and its components diverge as
G(k ) ∝ 1 (k + 2πn/a) 2 + (2πl/a) 2 − k 2 ,(14)
where n and l run over integral numbers (excluding l = 0 in G xx ). This behavior is illustrated in Fig. 4, showing in full display the lattice singularities exhibited by {G zz (k )}.
C. Hole arrays
Babinet's principle and hole arrays in thin screens
The behavior of hole arrays in perfect-conductor screens can be directly connected to the properties of the disk arrays considered in Fig. 3. Indeed, one can invoke the exact Babinet principle (Born and Wolf, 1999;Jackson, 1999), which connects the reflected fields of the disk array for a given incident polarization with the transmitted fields of its complementary hole array with orthogonal polarization, as illustrated in Fig. 5 (García de Abajo et al., 2005a). Therefore, the reflectance spectra shown in Fig. 3 are identical with the transmittance spectra of the complementary perforated screens. Focusing again on square arrays and normal incidence, we observe two characteristic features in the transmittance spectra: (i) the transmission vanishes when the wavelength λ equals the period a, and (ii) a 100% transmission maximum takes place at a wavelength slightly above a. The origin of these effects can be traced back to Wood's anomalies in gratings (Wood, 1902(Wood, , 1912(Wood, , 1935 and to their interpretation in terms of the following two mechanisms (Fano, 1936(Fano, , 1941: (i) accumulation of inphase scattering events when the wavelength equals the period (see explanation in Sec. III.B) and (ii) coupling of the incident light to a surface resonance. These phenomena persist in hole arrays perforated in thicker films of non-ideal absorbing metals, for which the maximum transmission is reduced but still justifies the term extraordinary optical transmission .
The analytical simplicity of the transmission coefficient for our thin-screen hole array, given by the right hand side of Eq. (11), allows us to gain deeper insight into the origin of this phenomenon. The lattice sum G xx (0) was shown to diverge when λ = a, as Fig. 6 illustrates. This leads to vanishing transmission, which we can interpret in terms of accumulation of in-phase scattering (see discussion in Sec. III.B). Furthermore, 100% transmission is achieved if the second term in the denominator of Eq. (11) becomes zero, a condition that can be rigorously fulfilled for arbitrarily tiny apertures (García de Abajo et al., 2005a): the smaller the holes, the larger 1/α E , because the polarizability is proportional to the cube of their radius, but no matter how large this fraction becomes, there is always one wavelength at which the divergent lattice sum matches it. This statement is illustrated by geometrical construction in Fig. 6
signals the condition {1/α E − G xx (0)} = 0. 2
The possibility of 100% transmission in non-absorbing structures has been pointed out before (Maystre, 1980;McPhedran et al., 1980), and the theory just presented goes further to show that this is possible for arbitrarily small holes. Nevertheless, the number of apertures needed to accomplish high transmission will increase as they become smaller, and at the same time the transmission resonance will be increasingly narrower and closer to λ = a. Therefore, these transmission maxima involve long-range interaction among holes, dominated by dynamical diffraction (i.e., multiple-scattering paths). In fact, if only single-scattering were considered, Eq. (3) would become p = α E (1 + α E G(k )α E )E ext , which wrongly predicts simultaneous divergence of transmittance and reflectance at λ = a.
This collective response in planar periodic arrays can be regarded as a lattice surface resonance (Fano, 1941), which becomes a true surface-bound state when evanescent incoming waves are considered, as we shall see in Sec. III.D. However, the resonance is strongly coupled to propagating light for external plane-wave illumination, a situation described by Fano (Fano, 1961) in his study of a discrete resonance state (our lattice surface-bound mode) coupled to a continuum (the transmitted light). This type of approach has been shown to work rather well in theory (Chang et al., 2005;Sarrazin et al., 2003) and in comparison with measured transmission spectra (Genet et al., 2003). Our transmittance calculations should also 2 We rely here on the condition {1/α E } > 0, which is satisfied by the polarizability of planar, perfectly-conducting disks. Interestingly, lattice resonances will be absent in arrays of particles with negative polarizability, such as metallic nanoparticles under blue-detuned illumination relative to a nearby plasmon band.
respond to Fano profiles of the form (Fano, 1961)
T = C (q + ε) 2 1 + ε 2 ,(15)
where ε can be assimilated to the light frequency and q describes the strength of the coupling to the lattice surface resonance. Fig. 6(b) compares our exact calculation of the transmittance (solid curve) with a Fano profile corresponding to parameters q = −3 and C = 0.1 (dotted curve), in which we assume a linear relationship between ε and the light frequency, with ε = −0.33 (ε = 3) for T = 1 (T = 0). The agreement is very reasonable, considering that no dependence of the coupling parameter on wavelength is taken into account. This further supports an interpretation of extraordinary transmission in terms of coupling to the lattice surface resonance set up by dynamical diffraction in the array. The geometrical construction of Fig. 6 provides a visual explanation of transmission in arrays of elongated apertures: an elongated piece of planar metal (e.g., a rectangle) has larger electric polarizability along its longaxis direction, and this has direct consequences for the Babinet-related situation of an elongated hole with the electric field along the short axis; larger polarizability involves more red-shifted and broader transmission maxima [this is so because the point of crossing in Fig. 6(a) occurs where G xx is less steep], just as observed experimentally (Gordon et al., 2004;Klein Koerkamp et al., 2004).
Incidentally, Eqs. (3) and (4) constitute a good approximation to describe the extraordinary transmission observed in 2D quasi-crystal hole arrays (Matsui et al., 2007;Przybilla et al., 2006b;Schwanecke et al., 2006;Sun et al., 2006), in which the lattice sum G exhibits pronounced, but finite maxima related to bright spots in the Fourier transform of the hole distribution. These spots define the reciprocal lattice for periodic arrays, but have quasi-crystal angular symmetry in quasi-crystals. In the spirit of Rayleigh's explanation of Wood's anomalies (Lord Rayleigh, 1907), the cumulative effect of longdistance interaction among apertures can be claimed to create these reciprocal-space hot spots, so that the effect of neighboring holes can be overlooked and an effective homogeneous p describes qualitatively well the extraordinary transmission effect in quasi-crystal arrays (Schwanecke et al., 2006), as well as the rich Talbot-like structure and subwavelength light localization observed at distances up to several wavelengths away from the array (Huang et al., 2007).
Single holes in thick films
Our use of Babinet's principle in the previous section indicates that, similar to small particles, small holes in perfect conductors can be assimilated to equivalent induced dipoles, in line with Bethe's pioneering description of the field scattered by a single aperture in a thin screen (Bethe, 1944), which he regarded as arising from a magnetic dipole parallel to the screen plus an electric dipole perpendicular to it.
Narrow holes can still be represented by induced dipoles in thick screens, as illustrated in Fig. 7(a). Parallel electric dipoles and perpendicular magnetic dipoles are forbidden by the condition that the parallel electric field and the perpendicular magnetic field vanish at a perfect-conductor surface. This allows defining electric (E) and magnetic (M) polarizabilities both on the same side as the applied field (α ν , with ν =E,M) and on the opposite side (α ν ). Furthermore, energy flux conservation under arbitrary illumination leads to an exact opticaltheorem type of relationship between these polarizabilities : by considering two plane waves incident on either side of the film and by imposing that the incoming energy flux equals the outgoing one (because perfect conductors cannot absorb energy), we obtain the condition
{g ± ν } = −2k 3 3 ,(16)
where we have defined
g ± ν = 1 α ν ± α ν
as hole response functions. The remaining real parts of g ± ν are obtained numerically from the field scattered by a single hole (García de Abajo, 2002;Roberts, 1987). These functions are represented in Fig. 7(b)-(c) within the electrostatic limit, clearly showing | {g + ν }| → ∞ in the thin film limit, where α ν = −α ν (Jackson, 1999).
Hole arrays in thick films
Periodic arrays of sufficiently narrow and spaced holes can also be described by perpendicular electric dipoles p and p and parallel magnetic dipoles m and m , where primed (unprimed) quantities are defined on the entry (exit) side of the film, as determined by the incoming light [see Fig. 7(a)]. We consider first a unitelectric-field p-polarized plane wave incident on a hole array with parallel momentum k alongx, so that the external field (incident plus reflected) in the absence of the apertures has parallel magnetic field H ext y = 2 along the y direction and perpendicular electric field E ext z = −2k /k along z. Then, one can generalize Eq.
(3) and write a set of multiple-scattering equations for the self-consistent dipoles (Collin and Eggimann, 1961;Eggimann and Collin, 1962). Symmetry considerations demand that our magnetic and electric dipoles be oriented as m = mŷ and p = pẑ, respectively. Following the notation of Sec. III.A, we can write with a new lattice sum defined as
p = α E (E ext z + G zz p − Hm) + α E (G zz p − Hm ), p = α E (E ext z + G zz p − Hm) + α E (G zz p−H = −ik n =0 e −ik xn ∂ xn e ikRn R n .
This sum stands for the interaction between mixed electric and magnetic dipoles. We can understand the above equations in a very intuitive way; for instance, the first one of them states that the electric dipole on the entry side (p) results from the response to the z-component of the self-consistent field on that side (E ext z + G zz p − Hm) via the polarizability α E plus the response to the selfconsistent field on the opposite film side (G zz p − Hm ) via α E . The solution to these equations can be readily written as
p ± p = −2[(g ± M − G yy )k /k + H]/∆ ± , (17) m ± m = 2[(g ± E − G zz ) + Hk /k]/∆ ± ,(18)with ∆ ± = (g ± E − G zz )(g ± M − G yy ) − H 2 .
The zero-order transmittance of the holey film is then obtained from the far field set up by the infinite 2D array of induced dipoles,
T p = |(2πk 2 /Ak z )(m − p k /k)| 2 , where k z = k 2 − k 2 .
Similar considerations for s-polarized light show that its transmittance reduces to T = |2πkm /A| 2 , with magnetic dipoles parallel to k and no electric dipoles whatsoever (E ext z = 0). More precisely, m±m = (2k z /k)/(g ± M − G xx ), from which one obtains
T s = 2πk z A 2 1 g + M − G xx − 1 g − M − G xx 2 (19) = 1 1 + iA 2πkz {g + M − G xx } − 1 1 + iA 2πkz {g − M − G xx } 2
for the transmittance. The last identity in Eq. (19) comes from Eqs. (10) and (16)
1 + A 2πk z 2 {g + M − G xx } {g − M − G xx } = 0. (20)
This is a second-order algebraic equation in {G xx } that admits positive real solutions provided
A 4πk z g + M − g − M ≥ 1.(21)
Actually, {G xx } can match those roots near the l = 0 singularities of Eq. (14), where it can be chosen arbitrarily large within a narrow range of wavelengths [see Eq. (13)]. It should be noted that the difference g + M − g − M falls off rapidly to zero when the film thickness h is made much larger than the hole radius b [see Fig. 7(b)]. However, if we fix both the h/b ratio and the angle of incidence, the left hand side of (21) reduces to a positive real constant times λA/b 3 , leading to the conclusion that 100% transmission is attainable at a wavelength close to the Rayleigh condition (e.g., λ > ∼ a for normal incidence on a square lattice of spacing a) regardless how narrow the holes are as compared to the film thickness. Surprisingly, this requires that the ratio of the lattice constant to the hole radius be increased for deeper holes in order to compensate the fall in g + M − g − M for larger h/b. The transmittance shows an interesting dependence on film thickness h (Martín- , as illustrated in Fig. 8. The maximum of Fig. 6 is initially blue-shifted closer to λ = a for small h, accompanied by a second narrower peak at even shorter wavelengths 3 [these are the two solutions of Eq. (20) under the condition (21)]. As h increases, inter-side interaction weakens and the two 100% maxima approach each other. At some point only one transmission maximum is observed when the left hand side of (21) is exactly 1. For even thicker films, the condition (21) cannot be met any longer and the transmission maximum departs from 100%. The Fano character of these lattice resonances is again visible through vanishing transmission at a wavelength immediately below the maximum.
Incidentally, perfect conductors are perfectly non-lossy, so that light dissipation must take place only at the openings if they are infiltrated with some dissipative material. For deep enough holes, the transmission is negligible and the absorbance becomes 1 − |r| 2 , which can reach 100% values under suitable resonant conditions, for instance in the IR by combining holes drilled in noble metals (behaving nearly as perfect conductors) infiltrated with phononpolariton materials. In fact, a similar effect has been observed in the visible using Au gratings (Hutley and Maystre, 1976) and in the infrared using SiC gratings (Greffet et al., 2002).
D. Lattice surface modes in structured metals
The flourishing area of plasmonics is demonstrating how confining electromagnetic fields to a surface can find many potential applications on the nanoscale (Ozbay, 2006). Zenneck waves at radio frequencies (Barlow, 1958;Zenneck, 1907), phonon-polaritons in the infrared (Greffet et al., 2002;Hillenbrand et al., 2002), and plaslanguage of Fano arise from coupling to different light continua on either side of the film, but one of these resonances has vanishing width and is placed at λ = a due to strong inter-side interaction. mons in the visible are in fact different manifestations of the same phenomenon: confinement of electromagnetic fields to curved or planar surfaces. Even perfectconductor screens, which are unable to trap light when they are flat, were experimentally shown by Ulrich and Tacke (1972) to host confined surface modes of p polarization when molded into films pierced by periodic arrays of holes spaced a distance much smaller than the wavelength [see Fig. 9
(b)].
In a recent independent development, Pendry et al. (2004) have studied surface modes in drilled semi-infinite metal, suggesting the possibility to extend plasmon-like behavior to lower-frequency domains via the flattening of the mode dispersion relation driven by propagating modes of the holes, and stimulating new microwaves observations (Hibbins et al., 2005). The analysis of Pendry et al. (2004) relied on a description of the holes based upon their lowest-order guided modes (i.e., TE 1,0 modes), which allowed extracting local permittivity and permeability functions in a metamaterial approach to holey metals. However, García de Abajo and Sáenz (2005) showed later that higher-order modes (and in particular TM modes) are important, giving rise to large quantitative modifications to the dispersion relation and revealing finer details in the holey metal response that go beyond a simple local metamaterial description (e.g., the angular dependence of the reflection coefficient does not follow the Fresnel equations with local optical constants).
At variance with planar perfect conductors and their lack of surface modes, corrugated metallic surfaces can support bound states even in the long-wavelength limit. In an intuitive picture, surface confinement in a drilled semi-infinite perfect conductor can be related to the evanescent penetration of the electromagnetic field inside the holes, in much the same way as surface plasmons enter a distance of the order of the skin depth inside a metal in the visible and NIR regimes (Barnes and Sambles, 2004). Actually, these modes share with plasmons their character of p-polarized evanescent waves.
Next, we elaborate a tutorial, analytical formulation of this phenomenon that becomes exact in the limit of small holes of size s a λ, arranged in a lattice of period a (García de Abajo and Sáenz, 2005). Although we focus our analysis on periodic hole arrays drilled in a semiinfinite perfect-conductor, it must be emphasized that periodicity is not really needed and that similar modes should exist for patterns other than holes (e.g., small protuberances or particles deposited on an otherwise flat surface).
Using the formalism of Sec. III.C, we find that Eqs. (17) and (18) offer a simple description of lattice surfacebound modes in metallic films. For infinitely-deep square holes as sketched in an inset of Fig. 9(a), the surface modes must correspond to non-vanishing values of the induced dipoles p and m in the absence of external fields. This can only be accomplished if the denominator ∆ ± is zero in those equations, leading to where we have set α ν = 0 for infinitely deep holes (see Fig. 7). The interaction sums G yy , G zz , and H are generally small for s a, except near the lattice singularities discussed in Sec. III.B. In particular, near the light line for k > ∼ k, one has
(1/α E − G zz )(1/α M − G yy ) = H 2 ,(22){G zz } ≈ {G yy } ≈ {H} ≈ 2πk 2 k z a 2 ,
which corresponds to Eq. (14) with n = l = 0. Furthermore, upon inspection of an expansion for H similar to (7), we find {H} = 0 outside the light cone, k > k, and the remaining imaginary parts of all quantities in Eq. (22) cancel out exactly because {G jj } = {α −1 ν } = −2k 3 /3 in that region. Combining these results, we obtain an approximate long-wavelength dispersion relation from Eq. (22):
k 2 = k 2 + Γ S 3 k 4 a 4(23)with Γ = 4π 2 S 3 1 {1/α E } + 1 {1/α M } 2 .(24)
Eq. (24) is exact in the s a λ limit, and it predicts the existence of lattice surface-bound modes under the condition 1/ {1/α E } + 1/ {1/α M } > 0. Here, we have used the area of the holes S to make Γ dimensionless.
Calculated values of Γ are offered in Fig. 10(c) for various hole geometries. The polarizability α E (α M ) is obtained from the electrostatic (magnetostatic) far-field induced by an external electric (magnetic) field, as shown in Fig. 10(a) [Fig. 10(b)]. Interestingly, circular and square openings of the same area give rise to similar values of Γ.
FIG. 10 (Color in online edition) (a) Electrostatic electricfield flow lines for a circular hole drilled in a semi-infinite perfect-conductor subject to an external field E ext perpendicular to the surface, giving rise to an electric dipole p = αE E ext as seen from afar. (b) Magnetostatic magnetic-field flow lines for the same hole subject to an external parallel field H ext and leading to a magnetic dipole m = αM H ext . (c) Summary of polarizabilities for square and circular holes in perfectconductor surfaces, normalized using the aperture area S. The values for the circular hole are taken from the h b limit of Fig. 7. The circular opening in a thin screen is analytical (Bethe, 1944;Jackson, 1999), but we must correct the right-hand side of Eq. (24) by a factor of 4 in this case because of cooperative interaction between both sides of the film.
This parameter increases by an order of magnitude when the holes are made on thin screens instead of semi-infinite metals, producing lattice surface modes that are further apart from the light line (see Ulrich and Tacke, 1972), and therefore, more confined to the metal, as a result of cooperative interaction between both sides of the film [see analytical solutions for circular apertures (Jackson, 1999) in last column of Fig. 10(c)]. Another suggestive possibility is offered by split annular holes, which present resonant electric polarizability (Falcone et al., 2004), and by holes filled with high-permittivity materials (see Sec. III.E), for which the interaction with single-hole modes produces large departures of the extended surface states from the grazing light condition. Fig. 9(a) shows calculated results for the reflection coefficient of a drilled metal, obtained by rigorous solution of Maxwell's equations in which we use a plane-wave expansion of the field outside the metal and a guided-mode expansion inside the holes (García de Abajo and Sáenz, 2005). The lattice surface mode can be observed as a bright region with a dashed line showing the position at which the reflection coefficient becomes infinite. A detail of |r| for a specific wavelength (see dotted straight line) is shown in the inset. The position of the resonance predicted by Eqs. (23) and (24) (see arrow in the inset) is in reasonably close agreement with the exact calculation, considering that the analytical model neglects neighboring-holes multipolar interaction, which is important for openings occupying 64% of the surface. Finally, Fig. 9(b) shows experimental results for a drilled thin film obtained by Ulrich and Tacke (1972). These surface modes are more bound in perforated thin films than in semi-infinite metals, as can be seen from the values of Γ given in Fig. 10(c). Actually, the measured dispersion relation departs substantially from the light line close to the boundary of the first Brillouin zone.
E. Interplay between lattice and site resonances
The description of extraordinary optical transmission in terms of quasi-bound surface states driven by lattice singularities can be extended to other types of binding. In particular, a single hole filled with a dielectric of high permittivity can trap light in its interior, giving rise to cavity modes even for very subwavelength apertures, provided the permittivity is sufficiently large to shrink the wavelength inside the dielectric to a value comparable to the diameter of the hole. This concept is explored in Fig. 11, in which higher permittivities are seen to produce larger contraction of the wavelength inside the hole, so that the cavity mode condition is met at longer free-space wavelengths for fixed aperture size (García de Abajo, 2002;García-Vidal et al., 2005). This process is accompanied by weaker coupling to external light (due in part to higher reflectivity of the dielectric-air interface), and therefore, narrower transmission resonances of increasingly larger height. Original predictions of this effect (García de Abajo, 2002) have been recently corroborated by experiment using microwaves .
An interesting situation is presented when localized modes like the ones just described are mixed with extended lattice modes, like the surface states underlying extraordinary optical transmission Ruan and Qiu, 2006). The interplay between both types of modes is illustrated in Fig. 12 through the zero-order transmittance of hole arrays filled with high-permittivity dielectric, calculated from the formalism presented in Sec. III.C.3. All incident-light polarizations interact with the cavity modes, giving rise to omnidirectional extraordinary transmission and invisibility behavior near the individual hole resonance (Borisov et al., 2005;García de Abajo et al., 2005b). However, only p-polarized light couples to the n = 1, l = 0 lattice singularity of Fig. 4, which results in an avoided crossing of the hybridized modes [ Fig. 12(a)]. Similar avoided crossings have been recently found in microwave experiments (Hibbins et al., 2006), confirming lattice surface modes and localized modes as two distinct mechanisms leading to enhanced transmission. 4,5 Notice that s-polarized light is immune to the l = 0 lattice singularities of Fig. 4, and this results in a reduced number of transmission features as compared to p polarization, in qualitative agreement with experimental observations .
Site resonances can occur in coaxial waveguides as well, via the so-called TEM mode, which does not have a cut-off in wavelength (Jackson, 1999). This led Roberts and McPhedran (1988) to theoretically explore the performance of periodic annular-hole arrays as band filters. More recently, Fan et al. (2005) have measured the increased transmission of infrared light assisted by these modes. Similar coupling to localized TEM modes occurs as well in slits, as we shall see in Sec. III.F.
The type of interplay phenomenon that we are describing has been observed as well for localized and extended surface plasmons in the visible regime through the absorption features of porous metals, in which Mie modes of spherical cavities in otherwise planar surfaces display a rich structure of hybridization and avoided crossings (Baumberg, 2006;Kelf et al., 2005Kelf et al., , 2006Teperik et al., 2006a,b). The absorption can be even complete under attainable experimental conditions (Teperik et al., 2005), implying black-body-like emission according to Kirchhoff's laws of thermal radiation (Reif, 1965).
F. Slit and cylinder arrays
Although we have extracted conclusions for particles and holes from his works, Wood reported his anomalies for ruled gratings rather than 2D structures (Wood, 1902(Wood, , 1935. 6 In fact, like gratings, cylinder and slit arrays exhibit lattice-resonance phenomena. But in contrast to holes, a single arbitrarily-narrow slit in a perfect conductor supports at least one guided wave, the TEM mode (Jackson, 1999), which can couple to external p-polarized light (magnetic field parallel to the slit) giving rise to recently predicted (Takakura, 2001) and observed (Yang and Sambles, 2002) Fabry-Pérot resonances in transmission. As a consequence, light passage through slit arrays can be assisted either by coupling to the TEM mode or by lattice resonances for p polarization (Porto et al., 1999), leading to similar interplay between localized and extended resonances as discussed above (Marquier et al., 2005). Incidentally, the analogy with annular hole arrays is clear (see Sec. III.E).
We shall consider first a periodic array of parallel narrow cylinders, the axes of which define a single plane. Continuing with our tutorial approach, and focusing for simplicity on light incident with its electric field parallel to the cylinders, we note that Eqs. (2)-(4) are still applicable here, provided α E and G 0 are conveniently redefined. In particular, the polarizability has now dimensions of area rather than volume, and it is given for instance by α es E = πb 2 ( −1) for homogeneous cylinders of radius b and permittivity (Bohren and Huffman, 1983), with the optical theorem now leading to {1/α E } = −k 2 /4. The relevant dipole-dipole interaction compo-nent is given by the Green function of Helmholtz equation in two dimensions, G 0 = (ik 2 /4)H (1) 0 (kR), where R is the distance measured in a plane perpendicular to the cylinders and H
(1) 0 is a Hankel function (Abramowitz and Stegun, 1972). Then, proceeding with the lattice sum G(k ) in a way analogous to Eq. (7), one finds a relation similar to Eq. (11) for the reflection coefficient of an array of lossless cylinders:
r = −1 1 + 2ia k {1/α E − G(0)} .
Under normal incidence (k = 0), G is found to diverge as
G(0) ≈ π a 2 √ 2 1 λ/a − 1 for λ > ∼ a,
where a is the lattice period. This is similar to particle arrays [see Eq. (13)], so that the main conclusions from our previous discussion of those arrays apply here as well, and more precisely, the reflectivity can be made 100% for arbitrarily narrow or weakly-scattering ( > ∼ 1) cylinders.
A complete analysis along these lines has been recently reported for all possible incident polarizations Laroche et al., 2006), suggesting that similar lattice resonances, somewhat less pronounced, are obtained for E ext perpendicular to the cylinders and with non-vanishing projection normal to the plane of the array. However, polarization components parallel to that plane and perpendicular to the cylinders cannot generate lattice resonances, because the interaction between distant dipoles aligned with their separation vector R decays as 1/R 3/2 in 2D, which is insufficient to produce a divergence in G. 7 Finally, we can establish a relation between cylinder arrays and slit arrays using arguments similar to those of Sec. III.C.1 for particle and hole arrays. More precisely, a slit array cut into a thin metal screen and illuminated with E ext perpendicular to the apertures can be analyzed using the above results as applied to the Babinet-related stripe array (i.e., a periodic array of stripes laying on a single imaginary plane) for E ext parallel to the stripes. Under normal incidence, the required component of the polarizability reads α E ≈ −2π/k 2 [ln(kb/8) + γ + iπ/2], where γ = 0.57721 is the Euler constant and b λ is the stripe width (van de Hulst, 1981). Interestingly, α E diverges in the electrostatic limit, so that even a single narrowing slit will exhibit a divergent transmission cross section. This scenario can be traced back to the abovementioned site resonances produced by the TEM mode of slits in thick screens. As a consequence, the interaction between slits can be very large, resulting in strong red shifts of the transmission peaks relative to the Rayleigh condition.
7 This is because ∞ n=1 1/n 3/2 is finite. See also Sec. III.B.
IV. REAL METALS VERSUS PERFECT CONDUCTORS
Metals of finite conductivity show significant differences with respect to the perfect conductors considered so far, the most remarkable of which is the existence of intrinsic surface-plasmon excitations. The basic understanding of these differences were laid out by Maystre (1972) in the context of diffraction gratings (see also Maystre, 1974, andMaystre, 1984). Next, we shall examine (in a tutorial fashion) the consequences for the interaction between particles and holes decorating metal surfaces.
A. Surface plasmons
Conduction electrons in metals behave like a plasma that is capable of sustaining collective oscillations known as plasmons (e.g., longitudinal bulk modes, signalled by the vanishing of the dielectric function). The existence of genuine surface plasmon oscillations was predicted by Ritchie, 1957, and soon after confirmed by electron energy-loss experiments (Powell and Swan, 1959). Since then, surface plasmons have developed into the rapidly growing field of plasmonics (Barnes et al., 2003;Ozbay, 2006;Zia et al., 2006) owing to their potential applicability to areas as diverse as biosensing (Schuster et al., 1993), signal processing through plasmonic circuits (Bozhevolnyi et al., 2006), or laser technology (Colombelli et al., 2003).
Planar surfaces possess translational invariance that provide plasmons with well-defined parallel momentum k exceeding that of light outside the metal and thus becoming truly surface-bound modes. Their dispersion relation can be readily derived from the divergence of the Fresnel coefficients for p polarization (surface-bound fields without external sources), leading to (Raether, 1988)
k SP = k + 1(25)
for a metal-air interface. This surface plasmon dispersion relation is represented in Fig. 13(a) for a Drude metal described by Eq. (12). In the long k limit, the surface plasmon frequency saturates to Ritchie's non-retarded plasmon (Ritchie, 1957). Surface plasmons are characterized by three different length scales, as depicted in Fig. 13(b): their propagation distance along the surface (∼ 1/2 {k SP }), their penetration into the surrounding medium (∼ 1/2 {k ⊥ }, where k ⊥ = −k/ √ + 1 is the normal momentum), and their penetration into the metal (the skin depth ∼ 1/2 {− k ⊥ }). Interestingly, the interaction between plasmons in either sides of a thin film gives rise to two plasmon branches, as measured by electron microscopy (Pettit et al., 1975;Vincent and Silcox, 1973), one of which has been found to propagate along very long distances thanks to exclusion of the electric field from the (b) Extension of the plasmon field into the metal (skin depth), into the vacuum, and along the surface (propagation distance) for several metals, as obtained from measured optical constants (Johnson and Christy, 1972;Palik, 1985).
metal (Sarid, 1981). Well defined plasmons require to have { } {− }, but similar long-range surfaceexciton polaritons exist in thin films for { } | { }| (Yang et al., 1990).
Features in metal surfaces produce scattering of plasmons in a similar way as light is dispersed by particles. This is actually a way to couple externally incident light to plasmons, for instance using gratings (Loewen et al., 1984;Ritchie et al., 1968). We find a neat demonstration of these ideas in the observation of surface-plasmon bands for periodic surface decoration (Kitson et al., 1996;Ritchie et al., 1968;Stewart and Gallaway, 1962) and in the reflection of surface plasmons at point scatterers arranged as parabolic mirrors (Nomura et al., 2005). Similarly, holes perforating films have strong influence on surface plasmons, which play an important role in their optical transmission . However, in the perfect-conductor limit, with | | → ∞, Eq. (25) yields k SP = k, with zero skin depth and infinite penetration into the vacuum, that is, there are no longer surface-bound modes. In the following we shall explore the transition between plasmonic and perfect-conductor regimes, in an attempt to clarify seemingly contradictory statements regarding the role of surface plasmons to enhance (Schröter and Heitmann, 1998) or to suppress (Cao and Lalanne, 2002) extraordinary optical transmission in striped thin films, or the heated debate opened by the explanation of recent outstanding experiments dealing with the interaction between a slit and a groove (García-Vidal et al., 2006;Gay et al., 2006;Lalanne and Hugonin, 2006).
B. Polarization schemes
The condition that parallel electric dipoles and perpendicular magnetic dipoles are excluded from perfectconductor surfaces (see Fig. 10) is relaxed in metals of finite conductivity. Polarization charges in a hole for in-stance can lead to a net parallel electric dipole in a thin metallic film (Rindzevicius et al., 2007).
In order to illustrate this concept, we have considered in Fig. 14 the effective polarizability of a silver spherical particle in front of a silver surface for a constant ratio of the radius to the wavelength, b/λ = 0.1. We can observe an electric Mie mode (Mie, 1908) in the visible, accompanied by negligible magnetic response. However, the metal behaves increasingly closer to a perfect conductor at longer wavelengths, so that currents compete eventually with polarization, thus displaying magnetic polarizability that becomes α M = −b 3 /2 for an isolated perfectconductor sphere in the long-wavelength limit (Jackson, 1999), to be compared with the electric polarizability α E = b 3 . Nevertheless, the latter is quenched by proximity of the metal flat surface under normal-incidence illumination conditions. The onset of magnetic response occurs when the particle becomes large compared to the skin depth ∼ 20 nm [see Fig. 13(b)]. These results follow from dipolar Mie scattering, conveniently corrected by surface reflection coefficients, which qualitatively describe the polarizability strength of the coupled particlesurface system. This has important consequences for understanding patterned surfaces and hole arrays. Electric dipoles dominate the response of features smaller than the skin depth, whereas magnetic dipoles can be significant for larger sizes, and only parallel electric dipoles and perpendicular magnetic dipoles survive in the limit of negligible skin depth. We are of course restricting our discussion to particles or apertures that are small compared to the wavelength, but these conclusions can be generalized to higher-order multipoles for bigger features.
C. Dipole-dipole interaction
New dipole orientations and the presence of surface plasmons in real metals demand that we revisit the in-FIG. 14 (Color in online edition) Effective polarization strength of a silver sphere near a silver planar surface. The sphere radius is a tenth of the wavelength. The polarization is normalized to the sphere volume. The dielectric function of silver is taken from Johnson and Christy, 1972. teraction between features in tailored surfaces. In particular, the dipolar field in free space, which decays away from the source as
G 0 ∼ e ikR R(26)
and governs the interaction between small features in perfect-conductor surfaces (see Sec. III.A), must be supplemented by reflected fields near real metals, leading to an interaction tensor of the form
G = G 0 + G r .
As a result, light impinging on a hole can couple to circular surface-plasmon waves (Chang et al., 2005;Popov et al., 2005;Wannemacher, 2001;Yin et al., 2004), whose field strength shows a rather different decay dependence with distance as explain the observed enhancement of the interaction between small particles in plasmonic metals (Stuart and Hall, 1998), and it is illustrated in Fig. 15, showing the field produced by a dipole near a metallic surface as calculated from a trivial extension of our tutorial approach formalism presented below. The interaction between pairs of electric and magnetic dipoles near a metal surface is analyzed in detail in Fig. 16(a) for all possible orientations except perpendicular magnetic dipoles, which are forbidden in perfect conductors and should take small values in real metals. Moreover, symmetry forbids the interaction of all other pairs that are not shown in the figure. For surface features inducing electric dipoles under normal incidence in a plasmonic metal (see Fig. 14), the dominant interactions originate in electric-dipole pairs aligned with their separation vector R (see Fig. 16), quite different from perfect conductors, which are governed by magnetic dipoles perpendicular to R. However, the latter can contribute in plasmonic materials as well for large features compared to the skin depth, as we discussed in Sec. IV.B. As a thumb rule, the mutual dipole orientations that lead to the longrange interaction dependence given by (27) are compatible with non-vanishing surface-plasmon field components emanating from those dipoles [i.e., plasmons with m = 0 azimuthal symmetry for normal electric dipoles, like in Fig. 15, or m = ±1 for parallel dipoles].
G ∼ e ik SP R √ R .(27)
The interaction between dipoles in front of a planar surface admits a representation in parallel momentum space similar to Eq. (5), but involving now the Fresnel reflection coefficients for s and p polarization (Blanco and García de Abajo, 2004;Weyl, 1919), r s = (k z − k z )/(k z + k z ) and r p = ( k z − k z )/( k z + k z ), respectively (Jackson, 1999), where k z = k 2 − Q 2 and k z = k 2 − Q 2 . In particular, for electric dipoles parallel to the surface x direction, one finds (Ford and Weber, 1984;Weyl, 1919)
G r xx = i 2π d 2 Q k z Q 2 e i(Q·R+kz|z|) [k 2 Q 2 y r s − k 2 z Q 2 x r p ],(28)
where z is the sum of distances from the dipoles to the surface, and we are interested in the z → 0 limit. This expression is general and leads to G xx = 0 in perfect conductors, for which r p = −r s = 1. The strong surface-plasmon-mediated interaction described by Eq. (27) arises from the pole of the Fresnel coefficient r p at Q = k SP , which admits the Laurent expansion (Ford and Weber, 1984)
r p ≈ 2Bk Q − k SP ,(29)
with B = [ /(1 + )] 3/2 /(1 − ).
Performing asymptotic analysis for large R and retaining only the contribution from this pole in the integral of Eq. . We also show the interaction at a wavelength of 10 mm (lower curve, perfect-conductor limit). The dipole-dipole separation vector R is taken alongx.
(28) (plasmon-pole approximation; see Ford and Weber, 1984), we obtain 9
G xx ≈ πk 3 B √ + 1 H (1) 0 (k SP R) + H (1) 2 (k SP R) (y 2 − x 2 ) R 2 ≈ −2πk 3 B + 1 2 iπk SP e ik SP R √ R ,(30)
where the second approximation comes from the asymptotic behavior of Hankel functions for large arguments (Abramowitz and Stegun, 1972), so that one obtains the result anticipated in Eq. (27). The above approximate expression in terms of Hankel functions is compared with the direct numerical evaluation of Eq. (28), and similar expressions for other dipole orientations, in Fig. 16(b). The agreement at λ = 750 nm is excellent for R > ∼ λ, indicating that lattice resonances in an array will be really dominated by surface plasmons at that wavelength. Fig. 16 illustrates as well a much faster decay of G yy as 1/R 3/2 for electric dipoles oriented orthogonal to R and parallel to the surface, and as 1/R for normal electric dipoles in the perfect-conductor limit.
D. Discrepancies in lattice resonances and enhanced transmission
The dissimilar behavior of plasmonic metals and perfect conductors discussed in the previous sections leads to qualitative differences in extraordinary optical transmission, arising in part from the 1/(Q − k SP ) dominant pole of the inter-hole interaction in momentum space [see Eqs. (28)-(30)].
Considering for simplicity a square array under normal incidence, we can analyze the lattice sum in a real metal [i.e., Eq. (4) with G substituted for G 0 ] following the procedure that led to Eq. (7), but starting now from Eqs. (6) and (28). In a diffrationless array, there are just two identical singular terms in the corresponding sum over reciprocal lattice vectors, leading to
G EE xx ≈ C 4π aλ 2 λ SP λ SP /a − 1(31)
for {λ SP } > ∼ a, where λ SP = 2π/k SP is the surfaceplasmon wavelength and C = iB/ √ + 1. We have explicitly indicated with superscripts that G EE xx describes the interaction between electric dipoles (E), which can coexist with parallel magnetic dipoles (M) (see Fig. 16). The remaining relevant lattice sums are G MM yy ≈ −( + 1) G EE xx and G EM xy = −G ME yx ≈ √ + 1 G EE xx . Now, the formalism presented in Sec. III.A can be easily extended to patterned surfaces and hole arrays in real metals using these expressions of the lattice sums rather than those for perfect conductors. 10 In the polaritonic regime of surface plasmons, in which their dispersion relation approaches the light line (see Fig. 13), | | is large and the dominant lattice sum scales as G MM yy ∼ 1/ √ − in the plasmon-pole approximation, so that for sufficiently high | | the perfect-conductor limit of Eq. (13) dominates over the plasmon.
A descriptive example of the transition from plasmonic to perfect-conductor behavior is offered in Fig. 17, in which the energy released by a dipole sitting near a surface is divided into plasmon launching (I SP ) and emitted light (I free ). This relates to the question, which of the two mechanisms (plasmons or propagating radiation) produces stronger interaction with a nearby surface feature. Plasmon launching dominates near the electrostatic plasmon, reaching an efficiency close to 100% in silver. As the wavelength advances towards to infrared, the plasmon is less bound to the surface and has weaker coupling to our dipole. As an example of application, when light pops out of a narrow hole after being guided through a TE mode (e.g., in a circular hole infiltrated with a dielectric of refraction index n 1 and for λ/n < ∼ 3.4b), the equivalent dipole describing the hole lies parallel to the surface. That is the situation depicted in the inset of Fig. 17.
A more explicit comparison of discrepancies between both metallic regimes for holes is offered in Fig. 18, which shows the lattice sum for parallel magnetic dipoles [obtained by summing Eq. (28) for gold, with the expression in square brackets replaced by k 2 Q 2 x r p − k 2 z Q 2 y r s ], together with a geometrical construction like in Fig. 6, applied now to two different aperture sizes. It should 10 Our analysis can be applied to metals embedded in a dielectric of refraction index n simply by using the reduced wavelength λ/n everywhere instead of λ and by interpreting as the ratio of permittivities in the metal and in the dielectric.
FIG. 17 (Color in online edition) Relation between the power radiated after transmission through a deep subwavelength hole (I free ) and the power emanating as surface plasmons (ISP) for gold and silver, derived in the small-hole limit. The metal dielectric function has been taken from Johnson and Christy (1972).
be noted that the exact calculation (solid curves) compares extremely well with analytical expressions [symbols, obtained from Eq. (13) for the perfect conductor and from Eq. (31) for the plasmonic metal, which needs to be multiplied by −( + 1) in order to apply it to magnetic rather than electric dipoles]. The lattice sum singularity in perforated gold takes place to the red as compared to the perfect-conductor case, because the surfaceplasmon wavelength is shorter than the light wavelength in the surrounding dielectric. Moreover, the lattice sum diverges as 1/ λ/n − a and 1/(λ SP − a) in perfect conductors and plasmonic metals, respectively, according to Eqs. (13) and (31), thus leading to different dependence of the position of the lattice surface resonance on hole size (see points of intersection with horizontal lines in Fig. 18); the lattice resonance is further away from the interaction sum singularity (and a given change in hole diameter produces larger peak shift) in the plasmonic case considered in the figure.
The crossover between both types of behavior is explored in Fig. 19 through the absorbance of (i) a silverparticle array in silica, (ii) the same array near a silversilica interface, and (iii) an array of silica inclusions right underneath the metal-dielectric interface. We have done these calculations using a layer KKR method to solve Maxwell's equations (Stefanou et al., 1998(Stefanou et al., , 2000. In the case (i) a maximum in absorption occurs near the Rayleigh condition for light propagating in silica (i.e., λ/n = a), whereas case (iii) shows a single maximum shifted to the right of the Rayleigh condition for the planar interface plasmon (λ SP = a) . The conclusion is that plasmons are mediating the interaction among the dielectric inclusions, with no signature of any anomaly near λ/n = a whatsoever. An intermedi-FIG. 18 (Color in online edition) Lattice sums and lattice resonances in a square array of holes drilled in gold vs a perfect conductor. The real part of the exact lattice sum for interaction of parallel magnetic (M) dipoles is shown for gold (black curve) and for a perfect conductor (PC, grey curve), as compared to analytical approximate expressions (symbols). The Rayleigh condition for a period a = 600 nm is indicated by black and grey vertical dashed lines for light in the dielectric (λ/n = a) and for surface plasmons (λSP = a), respectively. Changes in the inverse magnetic polarizability of circular holes of different size [horizontal lines, as obtained from Fig. 7(b)], lead to different wavelengths of the lattice surface modes, as indicated by vertical arrows for the condition that the real part of the denominator of Eq. (3) be zero.
ate situation is encountered in case (ii), showing features near the two types of Rayleigh conditions.
It should be noted that λ SP has an imaginary part arising from absorption, and although it is small for noble metals, in which plasmons can travel long distances along the surface, as shown in Fig. 13(b), we find that Eq. (31) does not describe a divergence, but rather a Lorentzian of finite width. This affects the height of the transmission maxima, below 100% in lossy metals. Furthermore, apertures perforated in metals of finite conductivity will appear to be wider by the skin depth effect, and their effective polarizability must be lossy.
Without entering into further considerations regarding how finite conductivity affects the hole polarizability, let us just point out that the wavelength at which the noted intersection takes place in Fig. 18 (i.e., the wavelength of the lattice surface-bound mode) is in excellent agreement with the transmission peaks measured by Krishnan et al. (2001) and reproduced in Fig. 1(a). The vertical arrows in that figure indicate the predicted positions of the transmission maxima, obtained by increasing the hole size by the skin depth to an effective diameter of 250 nm. This agreement is remarkable, given our neglect of higher-order multipolar terms in the hole polarization. The shift with respect to the Rayleigh condition for surface plasmons (vertical solid lines in Fig. 1) is signifi- FIG. 19 (Color in online edition) Normal-incidence absorbance of (i) a silver particle array embedded in silica (refraction index n = 1.45), (ii) the same array near a planar silver-silica interface, and (iii) an array of silica inclusions buried in silver below a silver-silica interface. All particles are spheres of 200 nm in diameter. The arrays have square symmetry with lattice constant a = 500 nm. The distance from the sphere surfaces to the planar interface is 10 nm in the buried silica particles and 900 nm for the silver particles. The Rayleigh conditions for the reduced wavelength of light in the silica (λ/n = a) and for the wavelength of the silversilica interface plasmon (λSP = a) are indicated by arrows A and B, respectively. cant, triggered by large, plasmon-mediated interaction between apertures, as explained above. Similar conclusions can be drawn for the silver film of Fig. 1(b), in which the results from the above analytical model are shown as dashed curves (divided by a factor of 5). Only magnetic dipoles are taken into account, with the hole polarizability calculated for a perfect conductor. The transmittance is obtained from Eq. (19) with G xx replaced by its plasmonic counterpart, G MM yy . Although the Rayleigh condition for plasmons (solid vertical lines in Fig. 1) agrees only with the transmission minima in silver (presumably because gold is more dissipative in this spectral region, so that the polarizability of the holes requires a more realistic description including absorption), the comparison with experiment is excellent, given the simplicity of the analytical model, which should become exact in the limit of small scattering features (e.g., for nanoparticle arrays on a metal substrate).
V. CONCLUSION
Light scattering in planar periodic systems gives rise to resonant phenomena that have common origins in particle and hole arrays, both for reflection and for transmission. Namely, (i) the interaction between lattice sites shows a divergent behavior when a diffracted beam becomes grazing (Lord Rayleigh, 1907), producing a min-imum in both the reflectivity of particle arrays and the transmission of hole arrays; (ii) a lattice resonance can be established at a wavelength to the red of that condition, leading to maxima in both the reflectivity of particle arrays and the transmission of hole arrays; (iii) these effects have the same origin as Wood's anomalies (Wood, 1935) and they can be described in the language of Fano lineshapes (Fano, 1961); (iv) the noted lattice resonance persists for incident evanescent light, with the reflectivity's becoming infinite in non-dissipative systems (e.g., patterned perfect-conductors, but also patterned dielectrics), thus defining truly surface-bound states (García de Abajo and Sáenz, 2005;Hibbins et al., 2005;Pendry et al., 2004;Ulrich and Tacke, 1972); (v) these extended lattice resonances mix strongly with other modes localized at specific sites, like those created by nanoparticle and nanovoid plasmons (Kelf et al., 2006;Teperik et al., 2006a); (vi) for metals with well-defined surface plasmons, the interaction between holes or particles in the vicinity of the surface is mediated by these excitations, so that we have to reformulate the condition of a diffracted beam's becoming grazing using the surface plasmon wavelength rather than the incoming or transmitted light wavelength.
We have shown that particle arrays and hole patterns in perfect conductors share in common the asymptotic form of their interaction, summarized by Eq. (26), which produces singularities at the Rayleigh condition when summed over the lattice, for instance for λ = a under normal incidence on square arrays, and gives rise to surface states at slightly larger wavelengths. However, the plasmon-mediated interaction in noble metals is more intense, as shown in Eq. (27), thus producing sharper divergences and stronger collective interaction. In this case, the singularities occur at the band-folded plasmon lines (e.g., when λ SP = a under normal incidence on square arrays), and the lattice surface-bound states (i.e., the plasmons of the patterned metal) exist again to the red with respect to those lines.
All of these effects have been described here within a common tutorial approach based upon interacting dipoles that is not only able to explain the observed effects; its simplicity has allowed us to extract some surprising conclusions. One of them is that arbitrarily-weak scatterers forming a periodic structure and made of nondissipative materials can also produce intense lattice resonances: given an array of arbitrarily-small particles of positive polarizability, it is always possible to find a wavelength (close to the period for square symmetry and normal incidence) at which light is totally reflected; accordingly, it is possible to obtain full transmission through holes however narrow, drilled in arbitrarily-thick perfectconductor films.
Interestingly, the lattice periodicity alone determines the magnitude of the induced dipoles needed to produce complete reflection by small particles or total transmission through narrow holes. Moreover, the polarizability scales with the cube of the hole/particle diame-ter. Combining these two statements, we find that the self-consistent electric field acting on particles or apertures under such resonant conditions increases when they shrink and can reach extremely high values only limited by absorption and lattice imperfections, thus opening new possibilities for applications in nonlinear all-optical switching and biosensing.
The simplicity and power of the model that has been presented here will surely find application to explain many other effects related to light scattering in planar periodic systems and can be inspiring for devising new phenomena.
FIG. 3 (
3Color in online edition) Reflectance spectra of square arrays of perfectly-conducting thin circular disks. The wavelength λ is normalized to the lattice constant a. The disks radius is b = a/5 in (a) and b = a/9 in (b). The light is impinging normal to the array and 100% reflection is observed in these two cases at the maximum. Solid curves: full numerical results. Dashed curves: analytical model for |r| 2 [Eq. (11)].
FIG. 4 (
4Color in online edition) Lattice sum Gzz(k ) [Eq.(4)] for a square lattice of period a as a function of parallel momentum k and wavelength λ. The direction of k is along one of the axes of the lattice.
FIG. 5 (
5Color in online edition) Babinet's principle applied to disk and hole arrays. The transmittance (reflectance) of the disk array for light of a given polarization σ (s or p) is identical to the reflectance (transmittance) of the complementary hole array for orthogonal polarization σ (p or s, respectively).
, in which the point of intersection of the horizontal dotted line and the solid curve [Fig. 6(a)] FIG. 6 (Color in online edition) Geometrical construction of the condition of full transmission in a hole array. (a) Wavelength dependence of the real part of the lattice sum Gxx [Eq. (4)] for k = 0. (b) Normal-incidence transmittance of a hole array complementary of the disk array of Fig. 3(a) (b = a/5): exact calculation (solid curve), analytical model of Eq. (11) (dashed curve), and Fano profile of Eq. (15) (dotted curve). The transmission minimum at λ = a results from the divergence of Gxx, while the transmission maximum (see vertical dashed line) is derived from the condition that {Gxx} equals the inverse of the hole polarizability, according to Eq. (11).
Hm ), m = α M (H ext y + G yy m − Hp) + α M (G yy m − Hp ), m = α M (H ext y + G yy m − Hp) + α M (G yy m + Hp ), FIG. 7 (Color in online edition) Response of a small hole in a perfect-conductor thick film. (a) The field scattered by a subwavelength aperture in response to external electric (E ext ) and magnetic (H ext ) fields is equivalent (at large distance compared to the radius b) to that of effective electric (p) and magnetic (m) dipoles, which allow defining polarizabilities (αE and αM, respectively) both on the same side as the external fields (αν ) and on the opposite side (α ν ). Only the perpendicular component of the electric field and the parallel component of the magnetic field can be nonzero at the surfaces of the perfect-conductor film. (b)-(c) Thickness dependence of the real part of the hole response functions g ± ν for λ b [the imaginary part satisfies Eq. (16)].
3
In fact, there are two lattice resonances for h = 0, which in the FIG. 8 (Color in online edition) Thickness dependence of the normal-incidence transmittance spectra of square arrays of circular holes drilled in perfect-conductor films, according to Eq. (19). The hole radius b = 0.2a, the wavelength λ, and the film thickness h are given relative to the period a (see text insets).
FIG. 9 (
9Color in online edition) (a) Lattice surface modes in a perforated semi-infinite perfect-conductor. The contour plot shows the modulus of the specular reflection coefficient for incident p-polarized light as a function of wavelength λ and parallel momentum k (see text insets for parameters). The upper-right inset shows a detail of the reflectivity as compared to the mode position predicted by Eqs.(23)and(24)(see arrow). A reflection coefficient larger than 1 is only possible for evanescent waves outside the light cone. (b) Lattice modes in a perforated thin film, as measured byUlrich and Tacke (1972) (symbols).
FIG. 11 (
11Color in online edition) Enhanced transmission driven by a localized resonance. The plot shows the normalincidence transmission of a circular aperture drilled in a perfect-conductor film and filled with dielectric material for different values of the permittivity (see labels). The transmitted power is normalized to the incoming flux within the hole area.
FIG. 12 (
12Color in online edition) Interplay between localized (site) and extended (lattice) resonances. The contour plots show the zero-order beam transmittance of a square array of circular holes drilled in a perfect-conductor film and filled with dielectric material of permittivity = 50 as a function of parallel momentum k and wavelength λ. The orientation of k and the ratios between the hole radius b, the lattice constant a, and the film thickness h are specified in the insets. The light is p polarized in (a) and s polarized in (b). A transmission coefficient larger than 1 is only possible for evanescent waves below the light cone.
FIG. 13 (
13Color in online edition) (a) Surface plasmon dispersion relation for a Drude metal of bulk plasmon frequency ωp.
FIG. 15 (
15Color in online edition) Instantaneous electric field set up by a perpendicular electric dipole (see vertical arrows) sitting at distance λ/20 from the surface of a metal described by Eq. (12) with ωp = 15 eV and damping η = 0.6 eV (typical of Al) at frequency ω = ωp/2. The electric-field component parallel to the surface (this is radial with respect to the position of the dipole) and the component along the surface normal are represented separately. Poynting vector flow lines are superimposed on the plot of the normal component. FIG. 16 (Color in online edition) (a) Schematic representation of the scaling of dipole-dipole interactions for electric and magnetic dipoles with respect to their separation R near a metallic surface. The interaction decays as exp(ikR)/R n near a perfect conductor or as exp(ik SP R)/R m near a metal with a dominant surface plasmon (see text insets for values of the exponents n and m). (b) Dipole-dipole interaction near a silver surface at a wavelength of 750 nm (three upper solid curves) as compared with the plasmon-pole approximation (three upper dashed curves, see text)
More explicitly, G 0 (r)p = [exp(ikr)/r 3 ] (kr) 2 + ikr − 1 p − (kr) 2 + 3ikr − 3 (r · p) r/r 2 .
In a related context, avoided crossing of lattice modes are wellknown to occur in coinciding Wood anomalies(Stewart and Gallaway, 1962). 5 Incidentally, lattice modes are observed outside the light cone for p polarization. The transmission outside that cone is defined as the squared-amplitude ratio of incident and transmitted evanescent waves at the exit and entrance surfaces of the film, respectively.
The reader is referred to the papers collected byMaystre (1993) for an exciting historical overview of XX century milestones on gratings.
It should be noted that the asymptotic behavior of G 0 [see Eq. (26)] comes from the Q = k region of the integral in Eq. (5) and responds to the pole 1/kz. This pole is canceled exactly by Eq. (28), in which rp = rs = −1 at grazing incidence (i.e., for Q = k). Therefore, the only relevant contribution to G for large R originates in the plasmon pole of G r .
AcknowledgmentsThe author wants to thank J. J. Baumberg
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| [] |
[
"A Generic Bundle Forwarding Interface",
"A Generic Bundle Forwarding Interface"
] | [
"Felix Walter \nD3TN GmbH Dresden\nGermany\n"
] | [
"D3TN GmbH Dresden\nGermany"
] | [] | A generic interface for determining the next hop(s) for a DTN bundle is a valuable contribution to DTN research and development as it decouples the topology-independent elements of bundle processing from the topology-dependent forwarding decision. We introduce a concept that greatly increases flexibility regarding the evaluation and deployment of DTN forwarding and routing techniques and facilitates the development of software stacks applicable to heterogeneous topologies. | 10.48550/arxiv.2209.05039 | [
"https://export.arxiv.org/pdf/2209.05039v2.pdf"
] | 252,198,889 | 2209.05039 | 67987860ddf456ffe045b28b06b9f868e18959df |
A Generic Bundle Forwarding Interface
Felix Walter
D3TN GmbH Dresden
Germany
A Generic Bundle Forwarding Interface
A generic interface for determining the next hop(s) for a DTN bundle is a valuable contribution to DTN research and development as it decouples the topology-independent elements of bundle processing from the topology-dependent forwarding decision. We introduce a concept that greatly increases flexibility regarding the evaluation and deployment of DTN forwarding and routing techniques and facilitates the development of software stacks applicable to heterogeneous topologies.
I. MOTIVATION
A universal deployment of Delay-and Disruption-tolerant Networking (DTN) will consist of a huge number of diverse, heterogeneous (sub-)networks, such as networks with:
• persistent low-latency end-to-end connectivity, • short or long-latency, asymmetric or unidirectional, but precisely-scheduled contacts between the nodes, • probabilistic encounters (e.g., if nodes move somewhat randomly but stay in local vicinity most of the time), • or totally random connectivity. Further variations of these characteristics can be imagined. To prevent the need to develop specialized implementations of the DTN protocols for every (sub-)network, we advocate for a flexible interface attached to common core infrastructure.
II. GENERIC INTERFACE CONCEPT
As sketched in Figure 1, our concept introduces a dedicated component integrating with the Bundle Protocol Agent (BPA), which we call the Bundle Dispatcher Module (BDM). Communication occurs via a network or IPC socket using a low-overhead data exchange format with wide compatibility such as Protobuf or Cap'n Proto.
A. Event-based Bundle Processing
To achieve a loose coupling of the components and increased flexibility with respect to the BDM implementation, the interface leverages an event-based approach: The BPA Core posts information about changes in connectivity, bundles that require a forwarding decision, bundles expiring, etc. via a publish-subscribe interface. A module subscribing to these events can then decide a) whether or not to act on them and b) invoke behavior to realize the forwarding decision. This approach further provides the flexibility to attach multiple subscribers beside the BDM, e.g., for monitoring purposes.
To pass information about bundles to the BDM, a Bundle Metadata data structure is defined, containing the bundle headers, extension blocks, and further metadata like timestamps, but excluding the payload for performance reasons.
B. Bundle Processing Actions
As counterpart to the event-based information flow to the BDM, a Remote Procedure Call (RPC) mechanism is provided by the BPA Core, which offers functions for updating the stored bundles and associated forwarding decisions. In this context, we propose to attach a list of Bundle Processing Actions to each bundle, that defines in-order what should be done next with the given bundle and can be updated by the BDM. At least two actions must be supported:
• SendTo(node): Forward the bundle to the specified nexthop node. Note that this action may occur multiple times or contain a multicast identifier. • Drop: Remove the "Forward pending" retention constraint if the previous action was successful. This list can get implementation-dependent extensions, e.g., actions to fragment bundles, control their storage, and so on. Like in some Software-Defined Networking implementations, there might be a function to query the supported actions for announcing such extensions in a straight-forward manner.
The BPA Core executes the action lists beginning with the bundle that was updated first, which allows the BDM to control the order of bundle forwarding. It should be noted that an empty action list is a valid state: In this case, the BPA Core will keep the bundle in memory until its lifetime expires or the BDM triggers an update on the action list. For maximum flexibility, the action list can have a configurable default assigned to bundles upon reception, thus, even allowing BPA operation without a dedicated BDM in simple cases (e.g., if there is a persistent "default gateway" contact).
C. BDM Interaction
The two complementary mechanisms provide for a flexible interaction between BPA Core and BDM and the implementation of basically arbitrary forwarding techniques. Figures 2 and 3 Fig. 3. Example sequence diagram for receiving and forwarding a bundle addressed to node Z for which the next hop is node Y, assuming contact-based routing in the BDM (simplified).
D. Inter-BDM Communication
Some forwarding techniques require the exchange of specific data, e.g., PRoPHET [8] needs to distribute delivery predictability values between nodes. A BDM implementation has three options to exchange such data: a) attach it to bundles as extension blocks (the used metadata structure contains them), b) inject bundles on its own by registering as a BP application, or c) use an independent mechanism or channel.
III. RELATED WORK
The presented concept enables a DTN software stack that can adapt to heterogeneous topologies, for which some alternative approaches exist. DTN2 [3] implements bundle forwarding in dedicated C++ classes that can be triggered by over 60 different events and can access the bundle data structure in memory. The Interplanetary Overlay Network (ION) [4] consists of multiple daemons that access a shared data structure. The forwarding decision is made in a separate daemon, which is chosen depending on the destination endpoint scheme. ProgDTN [5] is a novel approach executing a JavaScript program from within the BPA that represents a function returning the forwarding decision and can be flexibly exchanged. Overall, however, none of these implementations provides as much flexibility to implement and exchange the forwarding component as the approach introduced here.
IV. SUMMARY AND OUTLOOK This paper outlined a clean, low-overhead, event-driven socket interface to support a dedicated bundle forwarding component that can be flexibly exchanged and, thus, provides adaptability to heterogeneous topologies. At the time of writing, a combined Rust (of the BPA Core) and Python 3 (of the BDM) implementation in µD3TN [2] is almost complete and the integration of techniques such as Schedule-Aware Bundle Routing (SABR) [6] and IP-based Neighbor Discovery (IPND) [7] is ongoing. The pending evaluation of this interface will be based on an extended field test using drones and a satellite link to emulate a heterogeneous DTN internetwork.
Fig. 1 .
1High-level overview of the Bundle Dispatcher Module concept as extension toFigure 2from the DTN Bundle Protocol specification, RFC 9171[1]. The concept will be implemented in µD3TN[2].
depict how two example cases would be handled.Fig. 2. Example sequence diagram for injecting a new Application DataUnit (ADU) addressed to node Z for which the next hop is node Y, assuming opportunistic single-copy forwarding in the BDM (simplified).AppAgent
AppAgent
BPA Core
BPA Core
BDM
BDM
CLA_A
CLA_A
New link to node Y
Event: LinkUP(Y)
Send ADU #1 to Z
Create Bundle #1
Event: Bundle(Metadata #1)
Next hop for #1 is Y
Set actions for #1
to [SendTo(Y), Drop]
Send Bundle #1 to Y
TX success for Bundle #1
Delete Bundle #1
CLA_A
CLA_A
BPA Core
BPA Core
BDM
BDM
CLA_B
CLA_B
New link to node W
Event: LinkUP(W)
Bundle #2 received
Event: Bundle(Metadata #2)
Next hop for #2 is Y,
schedule Bundle for contact
New link to node Y
Event: LinkUP(Y)
Set actions for Bundle #2
to [SendTo(Y), Drop]
Send Bundle #2 to Y
TX success for Bundle #2
Delete Bundle #2
ACKNOWLEDGMENTSThe presented concept stems from a long series of discussions in which most of the team at D3TN was involved. Specifically, without the extensive contributions of the following people, the concept would not be as advanced: Marius Feldmann, Juan Andres Fraire, Tobias Nöthlich, and Georg Alexander Murzik. The concept is developed as part of the REDMARS2 project that is funded by Germany's Federal Ministry of Education and Research (FKZ16KIS1356).
Bundle Protocol Version 7. S Burleigh, RFC 9171, IETF. 2022S. Burleigh et al., "Bundle Protocol Version 7," RFC 9171, IETF, 2022.
DTN Reference Implementation (DTN2). "DTN Reference Implementation (DTN2)," [online] Available: https: //github.com/delay-tolerant-networking/DTN2.
Interplanetary Overlay Network (ION). "Interplanetary Overlay Network (ION)," [online] Available: http:// sourceforge.net/projects/ion-dtn/.
ProgDTN: Programmable Disruption-tolerant Networking. M Sommer, International Conference on NETworked SYStems, Online, 2022. M. Sommer et al., "ProgDTN: Programmable Disruption-tolerant Net- working," in International Conference on NETworked SYStems, Online, 2022. Available: https://dtn7.github.io/assets/sommer2022progdtn.pdf
Schedule-Aware Bundle Routing. CCSDS. 734"Schedule-Aware Bundle Routing," CCSDS 734.3-B-1, Washington, DC, USA, 2019.
DTN IP Neighbor Discovery. D Ellard, Internet-Draft. IRTFD. Ellard et al., "DTN IP Neighbor Discovery," Internet-Draft, IRTF, 2015. https://datatracker.ietf.org/doc/html/draft-irtf-dtnrg-ipnd-03
Probabilistic routing protocol for intermittently connected networks. A Lindgren, RFC. 6693IRTFA. Lindgren et al., "Probabilistic routing protocol for intermittently connected networks," RFC 6693, IRTF, 2012.
| [] |
[
"Experimental investigation on the performance of thermosyphon charging of a single-medium stratified storage system for concentrated solar power applications",
"Experimental investigation on the performance of thermosyphon charging of a single-medium stratified storage system for concentrated solar power applications"
] | [
"Ranjan Dipti \nDepartment of Mechanical Engineering\nIndian Institute of Science\n560012BangaloreIndia\n",
"Saptarshi Parida \nDepartment of Mechanical Engineering\nIndian Institute of Science\n560012BangaloreIndia\n",
"Basu \nDepartment of Mechanical Engineering\nIndian Institute of Science\n560012BangaloreIndia\n",
"Dhanush A P \nDepartment of Mechanical Engineering\nIndian Institute of Science\n560012BangaloreIndia\n"
] | [
"Department of Mechanical Engineering\nIndian Institute of Science\n560012BangaloreIndia",
"Department of Mechanical Engineering\nIndian Institute of Science\n560012BangaloreIndia",
"Department of Mechanical Engineering\nIndian Institute of Science\n560012BangaloreIndia",
"Department of Mechanical Engineering\nIndian Institute of Science\n560012BangaloreIndia"
] | [] | Concentrated solar power (CSP) plants utilize two-tank, sensible-heat thermal energy storage (TES) for uninterrupted electricity generation. However, the cost for the design and operation of TES is expensive. Therefore, researchers are focusing on implementing single-tank storage. Additional cutbacks can be made by utilizing pump-less thermosyphon charging for the TES. But prior thermosyphon researches for TES are related to domestic water-heating systems of small-capacity (<100 liters) and lowtemperature (<100 °C). Thus, investigations into thermosyphon charging for high-temperature storage are desired.This study focuses on thermosyphon-charging and storing of a single-medium stratified TES. The experiments were conducted on a 370 liters cylindrical storage (aspect ratio 4:1) with a heat-pipe system (3-liter volume) acting as a collector. Dowtherm-A oil was used as the heat transfer fluid (HTF), and the thermal expansion of HTF was accommodated in an expansion tank via two different designs (top and bottom connections from storage tank to expansion tank). Moreover, continuous and pulsatile charging are investigated for low (150 °C) and high (250 and 300 °C) temperatures. The results indicate that the maximum HTF temperature coming out of the heating pipes is ~25 °C more for the bottomexpansion design. Furthermore, it results in higher charging efficiency than the top-expansion setup for high-temperature studies. Finally, it is revealed that under design conditions, there are limits on the degree of thermal stratification achieved in the charging cycle and the maximum layover time allowable for interrupted charging. These results provide insights into the operational strategy of thermosyphoncharging stratified storage for CSP applications.Highlights• Thermosyphon charging for high-temperature stratified TES is examined for concentrated solar power applications. • Results show that accommodating HTF externally through a bottom-expansion connection is beneficial for thermosyphon charging. • There is a design limit to the degree of thermal stratification resulting from thermosyphon charging. • The degradation of thermal stratification and exergy analysis determines the maximum layover period for successive charging. | 10.2139/ssrn.4358080 | [
"https://export.arxiv.org/pdf/2211.16953v1.pdf"
] | 254,096,052 | 2211.16953 | a52cbd6c1a48badcf11280437052c6fb1e3632b5 |
Experimental investigation on the performance of thermosyphon charging of a single-medium stratified storage system for concentrated solar power applications
Ranjan Dipti
Department of Mechanical Engineering
Indian Institute of Science
560012BangaloreIndia
Saptarshi Parida
Department of Mechanical Engineering
Indian Institute of Science
560012BangaloreIndia
Basu
Department of Mechanical Engineering
Indian Institute of Science
560012BangaloreIndia
Dhanush A P
Department of Mechanical Engineering
Indian Institute of Science
560012BangaloreIndia
Experimental investigation on the performance of thermosyphon charging of a single-medium stratified storage system for concentrated solar power applications
1 * Corresponding author. Email address: [email protected] (S. Basu) 2Renewable EnergySolar PowerExergyThermal stratificationDowtherm-A Word count: 5600
Concentrated solar power (CSP) plants utilize two-tank, sensible-heat thermal energy storage (TES) for uninterrupted electricity generation. However, the cost for the design and operation of TES is expensive. Therefore, researchers are focusing on implementing single-tank storage. Additional cutbacks can be made by utilizing pump-less thermosyphon charging for the TES. But prior thermosyphon researches for TES are related to domestic water-heating systems of small-capacity (<100 liters) and lowtemperature (<100 °C). Thus, investigations into thermosyphon charging for high-temperature storage are desired.This study focuses on thermosyphon-charging and storing of a single-medium stratified TES. The experiments were conducted on a 370 liters cylindrical storage (aspect ratio 4:1) with a heat-pipe system (3-liter volume) acting as a collector. Dowtherm-A oil was used as the heat transfer fluid (HTF), and the thermal expansion of HTF was accommodated in an expansion tank via two different designs (top and bottom connections from storage tank to expansion tank). Moreover, continuous and pulsatile charging are investigated for low (150 °C) and high (250 and 300 °C) temperatures. The results indicate that the maximum HTF temperature coming out of the heating pipes is ~25 °C more for the bottomexpansion design. Furthermore, it results in higher charging efficiency than the top-expansion setup for high-temperature studies. Finally, it is revealed that under design conditions, there are limits on the degree of thermal stratification achieved in the charging cycle and the maximum layover time allowable for interrupted charging. These results provide insights into the operational strategy of thermosyphoncharging stratified storage for CSP applications.Highlights• Thermosyphon charging for high-temperature stratified TES is examined for concentrated solar power applications. • Results show that accommodating HTF externally through a bottom-expansion connection is beneficial for thermosyphon charging. • There is a design limit to the degree of thermal stratification resulting from thermosyphon charging. • The degradation of thermal stratification and exergy analysis determines the maximum layover period for successive charging.
Introduction
Electricity is ubiquitous in the modern-day World, and a significant portion of the World's electricity is produced by burning fossil fuels. However, fossil fuels will eventually run out. Hence, alternative electricity generation technologies for renewable energy resources are encouraged. One promising technology is the Concentrated Solar Power Plant (CSP). It is a thermal power plant technology that converts solar thermal energy to electricity by concentrating solar radiation. Depending on the types of concentrating system, the solar thermal harvesting system is classified as i) power tower system, ii) parabolic trough system, iii) parabolic disc system, and iv) linear fresnel system [1]. Although solar energy is abundant, CSPs necessities additional thermal energy storage (TES) for uninterrupted electricity generation owing to the time-dependent nature of the source. The TES system is generally categorized as i) thermo-chemical heat storage, ii) latent heat storage, and iii) sensible heat storage [2]. The sensible-heat storage is more appreciated than the rest as it is low-cost, reliable, and matured technology; it can be operated at higher temperatures (~600 °C) [3]. From the design perspective, two types of sensible heat storage are present, i) single-tank storage and ii) two-tank storage. At present, two-tank sensible heat storage is already made operational in CSPs. However, single-tank storage offers a low-cost alternative (35-48% less) than two-tank storage [4,5] and has been the subject of investigation for the last decade. A detailed classification of concentrated solar power technology is depicted in Figure 1. Figure 1: Classification of concentrating solar power technologies. Each CSP comprises a solar thermal harvesting system and an electricity generation system. The thermal energy storage system is generally installed in large-size plants to reduce the gap between electricity production and demand (except for the parabolic disc system).
In single-tank sensible heat storage (STSHS), thermal energy is preserved either in a liquid or in a combination of liquid and solid filler material, like rock and sand. Without the filler material, the liquid medium (usually consisting of oil/molten-salt depending on the operational temperature) acts as both heat transfer and storage medium. In that case, the STSHS is called single-medium sensible heat storage (SMSHS). Else, the solid filler materials act as the primary heat storage medium and the liquid act as the heat transfer medium. Then, the STSHS is called dual-medium sensible heat storage (DMSHS) [6]. From the operational point of view, a complete thermal cycle of STSHS consists of three periods: a) the charging period, b) the storage period, and c) the discharging period, and the STSHS operates on the principle of thermal stratification. The thermal stratification layer is known as the thermocline [7], which forms due to the thermal mixing of hot and cold HTF at the beginning of the charging and discharging periods. Depending on charging or discharging, it then translates towards the bottom or top of the TES tank. Though thermocline prevents further mixing of hot and cold HTF, its thickness can increase due to very high and/or low charging-discharging rates. At high charging/discharging, the inertia of the incoming fluid creates significant mixing, whereas, at low charging/discharging, prolonged thermal diffusion increases the thermocline thickness. Moreover, the thermocline broadens due to additional temporal thermal interactions between hot and cold HTF within the TES. These are intrafluid convection, wall conduction, and convective loss to ambient ( Figure 2). Figure 2: Thermal interactions in the single-tank sensible heat storage system during charging, storage, and discharging periods. During charging, the hot HTF (red-colored) is poured from the top, and the cold HTF (blue-colored) is taken away from the bottom of the TES tank simultaneously. During discharging, the flow is reversed. The stratified thermal zone (thermocline) is shown in the middle between hot and cold HTF.
Several numerical and experimental studies have been carried out for DMSHS [8][9][10][11][12]. Pacheco et al. [13] performed experiments on molten-salt thermocline TES with quartzite and silica filler materials. They reported that the cost of TES can be reduced by 1/3 compared to two-tank TES. Brosseau et al. [14] examined the filler materials for thermocline TES with HitecXL molten salt (a ternary mixture of 44 wt. % CaNO3, 12 wt. % NaNO3, and 44 wt. % KNO3). They reported a significant CaCO3 scale built up in the storage tank, which further increases if the operating temperature is more than 450°C. Thus, it indicates that the filler materials limit the maximum operating temperature of HTF in DMSHS. Moreover, scaling/fouling will alter the thermophysical properties of primary HTF (oil/molten-salt) and thus degrade heat transfer characteristics and pumpability, creating operational and maintenance challenges.
In a numerical investigation [15], it is reported that the discharge efficiency of the dual-medium thermocline storage decreases with an increase in Reynolds number, which makes it less likely that DMSHS can be suitable during high demand. Furthermore, [16] reports that asbestos filler material performs better than alumina spheres in terms of charging-discharging efficiency and thermocline thickness in oil-based TES. However, the volumetric heat capacity of asbestos is comparatively lower than that of alumina. This signifies that the TES tank size of asbestos-based storge will be significant for an equivalent heat storage capacity. To summarize, though DMSHS reduces the material cost of the liquid heat transfer medium, it creates additional challenges like an improper pairing of filler-HTF materials, greater thermal diffusion, thermal ratcheting, maintenance issues, and so on [12,[16][17][18]. Nevertheless, the SMSHS, which is free from these challenges, is rarely studied in the literature for high-temperature applications.
Regarding the discharging-discharging process for SMSHS, a forced flow of HTF is necessary for the discharging cycle (via pump drive) as the heat extraction rate varies depending on the demand. However, the charging of SMSHS can be carried out with either forced or natural circulation of HTF. The natural circulation of HTF is gravity driven and is known as the thermosyphon effect [19]. It is a half-century old technique and has a wide range of applications, including cooling (nuclear reactors, gas turbine blades, internal combustion engines, electronics, and so on) and heat-extraction (geothermal, waste-heat, and solar water heating) [20,21]. Depending on the phase of the HTF, two types of thermosyphon effect are present: a) single-phase thermosyphon and b) two-phase thermosyphon. In the case of SMSHS, the HTF does not change its phase; hence, single-phase thermosyphon charging is of interest. As no pump is required for thermosyphon charging of the HTF, it'll reduce both the instrument and operational cost of the TES. This passive way of charging SMSHS has been studied significantly in the past and has been commercially applied to solar domestic hot water TES systems [22]. Such studies, however, give a limited perspective of thermosyphon-charging as the working fluid is water, the operating temperature is less than 100 °C, and the volume of TES is usually less than 100 liters. Moreover, it is reported in [23] that the dimensions and thermal power of a thermocline TES determines its maximum efficiency, and it is expected that the large/real-sized tanks will behave differently than that of small/prototype-sized TES. So, it is not firm how the passive charging will behave for hightemperature, large-scale TES systems for CSP plants. In this regard, the following queries can be asked for thermosyphon charging of large-scale SMSHS:
For a given volume of the storage tank and heating volume (residing volume of the heat collection pipes) of the HTF, i. What will the thermal-stratification profile be in the thermocline storage as the charging cycle evolves? ii.
What will be the efficiency of thermosyphon charging? iii.
Considering the large volume of HTF and high operating temperature, the thermal expansion of HTF will be significant. So, how do different ways of accommodating thermal expansion affect the thermal stratification inside the TES? iv.
What will be the operation philosophy for thermosyphon charging of high-temperature storage?
In a nutshell, the efficacy of the CSPs directly depends upon both the cost and efficiency of TES, and hence investigations for an efficient thermosyphon-linked high-temperature STSHS need to be carried out.
This brings us to the objective of this paper, which is to understand how to utilize thermosyphon charging in a single-medium, single-tank, large-scale sensible heat storage for concentrated solar power applications. In anticipation of this, we developed a thermocline storage system having a storage capacity of 370 liters, which was linked to a jacketed heater for experimental investigation. The thermal expansion of the HTF was accommodated in an expansion tank via two different connections, and the thermosyphon charging was carried out for continuous and pulsating charging with the help of a control valve. In total, eight experiments were conducted, each having a charging time of 12 hours followed by storage of 24 hours, to examine the evolution and degradation of thermal stratification inside the TES. Figure 3 shows the schematic of an experimental rig developed under the IMPRINT India initiative at IISc, Bangalore. It is a hybrid TES system that integrates sensible and latent heat storage followed by a water-based heat extraction system. It comprises two HTF circuits (HTF-1 and HTF-2) and six independent thermal cycles/process modes, including both charging and discharging (see Appendix, section). The thermosyphon test loop, investigated in this work, is the HTF-1 part of this experimental rig, which contains (i) a sensible heat storage tank (thermocline tank), (ii) an expansion tank, (iii) a jacketed heater, and associated pipings. The thermocline tank (see Figure 4) is cylindrical with an aspect ratio ≈ 4, where = 1.9 is the height and = 0.498 is the diameter of the storage tank; the storage volume of TES is 370 liters. It comprises several manifolds for HTF transport with respect to the jacketed heater and the expansion tank. The HTF transport manifolds are branched out to 6 hemispherical diffuser arrangements (both at the top and the bottom of the TES tank) to reduce the thermal blending of hot and cold HTFs. Moreover, polymer bush layers are provided between thermocline storage and the support structures to minimize additional heat loss (see Figure 4). Figure 3: Schematic of the hybrid TES developed under the Imprint India initiative. It consists of hightemperature sensible heat storage without filler material, medium-temperature latent heat storage, and a water-based heat extraction system. The HTF-1 can be oil/molten salt, and the SHS tank is charged via thermosyphon.
Experiments
Experimental setup
The jacketed heater is coupled with the thermocline storage through a charging HTF pipeline to replicate the concentrated solar energy input. The charging HTF line is divided into three heat pipes inside the jacketed heater. The length and diameter of these heat pipes are 2 m and 0.0254 m, respectively. So, the ratio of the heating volume to storage volume for thermosyphon charging is 3.04 .
370 . = 0.008. These heat pipes are inclined 30° to the vertical in accordance with the literature [24]. The remaining pipes between thermocline storage, jacketed heater, and expansion tank are of the same bore diameter (0.0254 m). Moreover, the complete loop is wrapped by 50 mm glass wool insulation with the help of aluminum cladding/foil. A series of thermocouples (24 numbers) are installed vertically in the TES tank (with a gap of 76 mm between each) to measure the thermal stratification. Also, two thermocouples are installed at the bottom and top of the jacketed heater to measure the temperature of incoming and outgoing HTF. All these thermocouples are of K-type and have an uncertainty of Max(±2.5 °C or 0.75%). The temperature readings are taken to a PLC monitor though thermocouples-modules provided by the manufacturer (HEATCON Pvt. Ltd., Bangalore, India). Furthermore, the pneumatic valves and heaters are operated through PLC. The experimental loop is shown in Figure 5.
HTF materials and properties
Dowtherm-A oil is used as the heat transfer fluid for this investigation as it is cheaper than Syltherm 800, which is generally used in parabolic trough systems. Moreover, its operating temperature range of 15-400 °C is comparable to that of Syltherm 800 (-40 to 400 °C). More importantly, it has a lower fluid expansion of ~56% than Syltherm 800 (~72%) in the range of 25-400 °C [25]. The Dowtherm-A HTF used in this experiment is procured from Thermic Fluids Pvt Ltd., Mumbai, India. The thermophysical properties given by the manufacturer are curve fitted (see Appendix), and the polynomial expressions are given below. = 1112.9 + 0.8779 − 0.0058 2 + 9 × 10 −6 3 − 6 × 10 −9 4 ( 3 )
(1)
= 0.7002 + 0.0029 ( ) (2) = 0.1856 − 0.0002 ( ) (3)
where ( 3 ), ( ), ( ) are the density, the specific heat capacity, and the thermal conductivity of the saturated Dowtherm-A liquid in the range of 15-400 °C. is the temperature in .
Operational procedures and experimental conditions
One of the main operational challenges in large-scale, high-temperature sensible heat storage is how to accommodate significant HTF expansion. The top portion of the storage tank is generally left unfilled to accommodate the expanded HTF. Though this method is suitable for solar water heating systems, it may not be appropriate for high-temperature storage as i.
The HTF (oil/molten-salt) needs to be operated in an inert atmosphere, like Nitrogen or Argon, to avoid air contact that accelerates thermal decomposition; thus, an inert-gas inlet has to be provided at the top of the storage. ii.
A pressure gauge and relief valve need to be installed to monitor and control the surge in pressure at the top of the storage tank. iii.
A level indicator is also needed to monitor the HTF level in the storage and so on.
These additional modifications/instrument installation will create design challenges and make the top of the TES tank clumsy and inaccessible. This challenge can be evaded by external accommodation for the HTF in an expansion tank. However, the expansion tank can be joined to either the storage tank's top or the bottom. Both of these expansion tank connections are examined in this study.
For the experiment, the thermosyphon charging loop (which contains the thermocline tank, jacketed heater, and expansion tank) is purged by argon and filled with HTF (Dowtherm A). The jacketed heater is then set to a fixed temperature (150, 250, or 300 °C). When the jacked heater's skin temperature reaches the set temperature, its power is decreased to 70%; when the skin temperature drops by 10 °C, its power is raised to 100%. This automated PLC-controlled jacketed heater resulted in a ±10 °C variation in the skin temperature. Two different ways of charging performed were i) continuous type and ii) pulsating type. In continuous charging, the valve-b (see Figure 3) is left open for thermosyphon charging. In pulsating charging, the valve-b is opened and closed periodically (15 minutes open followed by 15 minutes closed). The experimental study cases include low (150 °C) and high (300 °C) temperature charging for continuous-type charging. However, it is revealed from trial runs that the pulsating charging at low temperature is not beneficial as a relatively lower gain in temperature during the pulsating diffused owing to slower thermosyphon charging. Thus, the pulsating-type charging is tested for 250 and 300 °C. The detailed study cases are given in Table 1.
Results and discussion
The results obtained in this experiment are categorized into two parts, i) the charging period and ii) the storing period. First, the inlet-outlet conditions of the jacketed heater, the development of stratified profiles, and charging efficiency are discussed for the charging. Subsequently, the temporal degradation of thermal stratification and an exergy-based comparison are reported for the storage period. Also, it is essential to note that each temperature measurement is associated with some uncertainty (as mentioned in section 2.1 ); however, the error bars are not included in the temperature profiles for clarity. Figure 6 shows the temperature profiles of the HTF at the outlet of the jacketed heater for all study cases. For the continuous charging ( Figure 6; A and C), a maximum temperature of 98.6 °C (case #1) and 180.3 °C (case #2) is achieved for top-expansion connection, whereas 101.7 °C (case #5) and 206.4 °C (case #6) is obtained for bottom-expansion connection. So, a temperature difference of ~3 °C and ~26 °C is present between the top and the bottom expansion connection arrangements for the low (150 °C; cases #1 and #5) and high (300 °C; cases #2 and #6) temperature charging cases.
Charging period 3.1.1. Inlet-outlet conditions
For pulsating charging, interim peaks in temperature profiles are shown in Figure 6, B and D. However, these peaks are not regular and possibly associated with/caused by the actual temperature control of the jacketed heater. Regardless, the maximum temperatures obtained in these study cases are 180.9, 213.7, 183.9, and 242.5 °C for cases #3, 4, 7, and 8, respectively. Excluding the peak temperatures, the topexpansion setup achieved a maximum temperature of nearly 162 °C and 180 °C for cases #3 and 4, respectively. In comparison, the bottom-expansion setup reached a maximum temperature of around 174 (case #7) and 205 °C (case #8). Thus, the temperature difference of ~12 °C and ~25 °C is present between these expansion setups for the pulsating charging. Assuming isothermal condition is maintained reasonably well in the jacketed heater, the following arguments can be made based on the discussed results:
i.
The temperature gain during the thermosyphon-charging depends on the type of expansion tank connection. ii.
The pulsating type thermosyphon-charging induces a significantly higher temperature rise in the HTF (including the interim temperature peaks) and can be valuable for high-temperature storage applications. Figure 7 shows the thermal conditions of the HTF at the jacketed heater inlet. The initial HTF temperature for the study cases varied between 25-40°C despite having a considerable cooling time of 3 days between successive experiments because the volume of the thermocline storage is reasonably large. Regarding the temporal evolution of HTF, nearly flat temperature profiles are observed for the low temperature (150 °C) charging (cases #1 and #5). However, for the high temperature (300 °C) studies, the temperatures of the HTF increased linearly after 5 hr of charging (cases #2, #4, #6, and #8) irrespective of top or bottom expansion as well as continuous or pulsating charging. The potential cause of this trend could be attributed to the higher temperature difference (∆ ) between the hot and cold HTF in the storage tank. A higher ∆ will induce larger density difference, which in turn, will improve the buoyancy-driven flow rate inside the thermal loop. Hence the thermocline will travel fast towards the bottom of the storage. Moreover, the thermal diffusion in the storage tank and wall conduction will be more significant at higher ∆ . In result, the cold fluid temperature rises quickly. For the 250 °C pulsating charging studies (cases #3 and #7), the temperature profiles lie between low and hightemperature study cases.
Atwood number depletion
One important aspect to discuss for the thermosyphon charging is the Atwood number ( ) of the thermal stratification. It is a non-dimensional number represented by the densities of hot ( ℎ )and cold ( ) HTF ( = − ℎ + ℎ ), and an indirect measure of buoyancy. Since the density of HTF is temperature dependent, the higher the , the higher the effective thermosyphon charging. In this study, the of the HTF is calculated in three categories: ( ), ( ), and ( ) with respect to the jacketed heater, the inlet of the storage tank, and the thermocline storage, respectively. The ℎ for these number categories are considered from the set skin temperature of the jacketed heater and the maximum temperature of the HTF at the inlet and the thermocline tank. Whereas the values are considered from the minimum temperature of the HTF at the beginning of the thermal cycles. Note that the ( ) is merely the set Atwood number for experiments as specified in Table 1. Consequently, the difference in these numbers signify the thermal loss associated with each step. A comparison in variation for the study cases is shown in a bar graph in Figure 8 The drop in for the top-expansion setups (cases #1, 2, 3, and 4) are relatively larger compared to the bottom-expansion setups (cases #5, 6, 7, and 8).
iii.
The pulsating type thermosyphon charging is relatively more favorable than that of continuous charging as it minimizes the initial loss.
Regarding ( ), it is always lower than ( ) due to thermal blending of hot and cold HTF in the thermocline storage and associated heat loss. However, the difference { ( ) − ( )} increases significantly for the high-temperature studies (~0.037) compared to low-temperature studies (~0.003) as higher ∆ accelerates both thermal diffusion and wall conduction. Additionally, Figure 8 shows that the resulting ( ) in all cases are about ±0.045 regardless of the type of expansion connection, charging, and temperature settings. It signifies that the final thermal stratification in the SMSHS is constrained by the design of the thermosyphon loop, and it is believed to be associated with the volume ratio ( ). Figure 9 shows the evolution of stratified temperature profiles inside the thermocline storage for intermediate time instances; = 4, 6, and 8 horus. The corresponding temperature readings and vertical locations of the thermocouples are normalized by * = − ℎ − (4) * = (5) where , , and ℎ represent thermocouple temperature, the cold HTF temperature, and the hot HTF temperature, respectively; , are the height of thermocouples and the thermocline storage, respectively.
Thermal stratification
The very first observation from Figure 9 is that the temperature profiles corresponding to lowtemperature charging (cases #1 and 5) are nearly sigmoid, whereas the profiles are relatively linear for the rest of the study cases. It signifies that both axial thermal diffusion and wall conduction are prominent for high-temperature charging. Moreover, the temperature profiles translate from bottom to top as time progress. For instance, the topmost thermocouple temperature ( * ) at = 4 is ~0.75 which then increased to ~1 at = 8. Furthermore, a sharp linear zone near the inlet ( * = 0.8 to 1) is exist during the charging for all cases. These results suggest that the present thermosyphon loop has undergone multi-pass charging. Ideally, a sigmoid temperature profile is expected in the stratified storage, and the charging should be completed in a single pass. It is believed that a higher >> 0.008 necessites to achieve these requirements, which in turn will improve the flow rate of the HTF in the thermosyphon loop. Figure 9: Varying thermal stratification in the storage tank during charging; Normalized temperature profiles * plotted against height * at = 4, 6, and 8.
Charging efficiency
To analyze thermosyphon charging, the study cases are compared with respect to their charging efficiencies. As described in Mawire et al. [26], the charging efficiency is based on the First law of thermodynamics and calculated by
( ) = ( ) − ( ) −(6)
where ( ) is the time-dependent average temperature of the thermocline storage.
( ) is the inlet temperature of the hot HTF (same as the jacketed heater outlet temperature as indicated in Figure 6), and is the average temperature of the storage at the beginning of the charging, i.e., at = 0. The time-varying efficiencies of the case studies are depicted in Figure 10. As illustrated in the figure, the charging efficiency ( ) increased rapidly up to 5 hours and then slowed down. At the end of charging, i.e., at = 12, the maximum is found to be ~0.4 for cases #6 and 8; ~0.35 for cases #2 and 4; and ~0.33 for the rest of the cases. In addition, the profiles are overlapped, except for cases #6 and 8. This suggests that, firstly, the charging efficiency increases for hightemperature cases. Secondly, the bottom-expansion design is considerably more effective than the topexpansion design.
Storage period 3.2.1. Temperature profiles
The variation in temperature profiles during the storage period is illustrated in Figure 11 for the lowtemperature cases (150 °C) and Figure 12 for relatively high-temperature cases (250 and 300 °C). The temperature readings and the height of the thermocouples are normalized according to equations 4 and 5, respectively. The temperature profiles are plotted at 6-hour intervals for 24 hours. A dwell time of 30 minutes is given after switching off the jacketed heater to account for any residual flow of HTF. Thus, the profiles corresponding to = 0 merely indicate the final profiles at the end of charging or the beginning of the storage period. As seen in Figure 11, the stratified temperature profile is maintained for the entire storage period for the low temperature (150 °C) studies. Moreover, a subtle difference in the temperature profiles is observed between Case #1 and Case #5, and the degradation in the first 6 hours is very minimal. It suggests that for low-temperature charging, the resulting axial temperature of thermocline storage at the end of charging and their degradation during the storage are independent of the expansion-connection designs. However, the high temperature (250 and 300 °C) charging studies give contrasting results. As observed in Figure 12 (A, B, and C), the top-expansion design does not effectively maintain a clear stratification at the end of charging. Moreover, the bottom half of the storage is filled by thoroughly mixed HTF (i.e., * ≈ 0.5), and any stratification present near the top of the storage tank is degraded during the first 6 hours of the storage period. For bottom-expansion design, clear thermal stratification is observed in Figure 12 (D, E, and F) at the beginning, = 0. But, the degradation during first 6-hour is relatively larger. This suggests that the heat loss due to wall conduction as well as to the surrounding is significant at higher temperatures. Nevertheless, the thermal stratification is sustained till the end of the storage period.
Comparison of thermal stratification
As discussed earlier, thermal stratification is the basis of single-tank sensible heat storage, and thermosyphon charging is employed to reduce instrument and operational costs. Hence, it is essential to assess the resulting thermal stratifications at the end of the charging as well as their temporal degradation. This has been estimated using a quantitative index named Ideal Stratification Index (ISI) [7], which is defined by
= ∑ * × ( * ) =1 (7)
where, * is the equivalent normalized temperature ( * = 0.5 + | * − 0.5|) corresponding to ℎ thermocouple, is the total number of thermocouples and ( * ) = 4( * − 0.5) 2 is the ideality factor.
In particular, ISI analysis quantitatively yields how nearly any real stratification profile approaches the ideal stratification. The ISI values vary between 0 (for fully mixed HTF, i.e., * = 0.5) and 1 (for unmixed HTF, i.e., * = 0 or 1). For clarity, a schematic of ideal and real stratifications is shown in Figure 13 A, and the temporal degradation of ISI in the rest of Figures B, C, and D. The comparative ISI graphs are plotted only for the first 5 hours because the quantitative index for the high-temperature study cases ( Figure 13 C and D) achieved the lower limit at around = 5. In addition, the resulting stratification in the thermocline storage appeared up to a maximum value of ~0.25 (as indicated in cases #1 and 8). However, the slope of the ISI curve for case #8 is steeper than that of case #1. This suggests that temporal degradation of the thermocline is faster in high-temperature studies. Moreover, a comparison of ISI curves between Figure 13 C and D reveals that the bottom-expansion design performs relatively better than the top-expansion design. The above discussion suggests firstly that the maximum ideal stratification in thermosyphon charging has a design limit (approx. 0.25 for the present loop) irrespective of thermal expansion design and continuous/pulsating charging. Secondly, there is an operational limit, which is the maximum layover period, preferable, between intermittent charging of the TES (5 hours for the present loop). For CSP plants, the thermosyphon-charging of the stratified TES may be interrupted due to bad weather, and the energy loss from the TES will be significant if the charging is delayed more than this limit.
Exergy degradation
Regarding thermodynamic analysis, exergy degradation is one of the most important parameters to evaluate the performance of any thermal energy storage. Exergy is calculated from the Second law of thermodynamics. The degradation of exergy indicates the loss of extractable energy in the storage system. Since spatial temperature variation is present in the stratified storage, the exergy of the whole TES is equal to the sum of the exergy of individual zones formed around the thermocouples. As reported in Advaith et al. [6], the exergy of the stratified tank is calculated by
= ∑ [( , − 0 ) + 0 ln ( 0 , )] =1 (8)
where , , , and , are the density, the volume, the specific heat capacity, and the temperature reading of the HTF corresponding to an i th zone. The temperature-dependent thermophysical properties ( and ) are calculated from Equations (1) and (2). 0 is the ambient temperature, and is the total number of zones. For this study, 0 = 25 °C, and = 24. For comparison, the exergy of the TES at 6hour intervals is normalized with respect to the exergy at the beginning of the storage (at = 0), i.e., * = 0 and illustrated in Figure 14. As indicated from the slope of the curves in Figure 14, the exergy decays faster at the beginning of the storage period, which then slows down with time. Moreover, the loss in exergy for low-temperature studies (cases #1 and 5) is relatively lower than in high-temperature studies (cases #2, 3, 4, 6, 7, and 8).
In general, the exergy of a system degrades by both energy loss and internal entropy generation. For stratified TES, heat loss to the ambient corresponds to the energy loss, and internal mixing of hot and cold HTF (caused by axial diffusion and wall conduction) corresponds to the internal entropy generation. Since both these factors are significant for high-temperature storage, a higher loss in exergy ( * ) is observed compared to low-temperature storage.
In addition, the exergy losses for cases #2, 3, and 4 are less compared to cases #6, 7, and 8 for = 6.
Since the initial temperature profiles (at = 0) were relatively less stratified for study cases #2, 3, and 4 than that for cases #6, 7, and 8 (see Figure 12), it suggests that in stratified storage, the rate of exergy degradation increases with the degree of stratification owing to internal entropy generation. Lastly, a significant exergy loss (~35 to 50%) is observed in the first 6 hours of storage for the high-temperature studies. It indicates that the maximum layover period for this stratified storage should be less than 6 hours, which corroborates with the results obtained from the ideal stratification index (≤ 5 hours).
As part of this effort, an experiment has been performed for the thermosyphon charging of a low-melting eutectic salt mixture known as the HITEC salt. Although the investigation is beyond the scope of this manuscript, the authors report associated challenges and safety issues faced in conducting a molten-salt experiment in the Appendix section.
Conclusion
An experimental investigation was carried out on a single-medium stratified thermal energy storage to assess thermosyphon charging for concentrated solar power applications. The stratified storage was cylindrical in shape with an aspect ratio ℎ ≈ 4 1 and a storage volume of 370 liters. The ratio of the heating volume to storage volume for thermosyphon charging was 0.008, and thermal expansion of the HTF was accommodated in an expansion tank by two different designs (top and bottom expansion connections). Finally, continuous and pulsatile charging was carried out for 12 hours, followed by a storage period of 24 hours.
The results obtained from these experiments show that the bottom-expansion design yields a relatively larger temperature (~25 °C) for the HTF at the jacketed heater outlet than that of the top-expansion design for both continuous and pulsatile charging. Also, the Atwood number difference can be used as a parameter for decision-making on the design and operation of the thermosyphon charging loop. Furthermore, this investigation reveals that there is a design constraint to the ultimate stratification of HTF using thermosyphon charging as the resulting ( ) for the case studies are around ±0.045 irrespective of the type of expansion setup and operational variations.
Additionally, this study indicates that the HTF experiences multi-pass charging despite variations in operational settings for the low of 0.008. Therefore, increasing heating volume should be considered for designing a thermosyphon-charging loop for concentrated solar power applications. On top of that, this study suggests that there is a design limit to how much the stratified profiles are closer to ideal stratification for a thermosyphon loop (~ 0.25 for the present loop). Most of all, the results demonstrate that the charging efficiencies corresponding to the bottom-expansion design are considerably higher compared to the top-expansion design for high-temperature case studies. Finally, the storage period results indicate that the exergy decreases faster if the HTF is relatively more stratified in the TES, and the maximum layover period for the stratified storage can be determined from the degradation of the exergy and ideal stratification index.
To conclude, this study provides valuable insights into thermosyphon charging of single-medium stratified storage, particularly important from an operational perspective for high-temperature applications. Moreover, this study has a limitation concerning the heat loss to supporting structures which should be minimized in concentrated solar power plants by providing a concrete foundation, sand layers, ceramic or firebrick insulation, and so on. Lastly, future studies should be extended to higher temperature storage using molten-salt-based heat transfer fluids.
CRediT authorship contribution statement
Declaration of Competing Interest
:
Specific heat capacity of i th zone * : Equivalent normalized temperature ( * ): Ideality factor : Exergy, 0 : Initial exergy of TES * : Normalized exergy
Figure 4 :
4Schematic of the thermocline storage tank showing thermocouple array. Polymer bush is provided at the bottom of the storage to minimize heat loss. The top and bottom distributors consist of six hemispherical diffusers (see Appendix).
Figure 5 :
5Experimental TES setup built at IISc. Dowtherm-A oil is used as the HTF-1 for the thermosyphon charging experiment; (The PCM storage is not shown in the figure).
Figure 6 :
6Temperature profiles of the HTF at the outlet of the jacketed heater; (A) and (B) are the case studies for the continuous and pulsating charging of the top expansion tank connection; (C) and (D) are the case studies for the continuous and the pulsating charging of bottom expansion tank connections, respectively.
Figure 7 :
7HTF temperature profiles at the inlet of the jacketed heater for all experimental study cases (#1-8).
. Three important traits evidenced from { ( ) − ( )} are: i. The drop in increases with an increase in skin-temperature of the jacketed heater (an average of 0.022 for Case #1 and 5; 0.037 for Case #3 and 7; and 0.052 for Case #2, 4, 6, and 8). ii.
Figure 8 :
8Atwood number variations for the study cases. ( ), ( ), and ( ) are Atwood numbers calculated corresponding to the jacketed heater, the inlet of the storage tank, and the thermocline storage, respectively. The drop in the Atwood number signifies the loss in stratification strength.
Figure 10 :
10Charging efficiencies of the stratified storage for all study cases at one-hour intervals.
Figure 11 :
11Temporal degradation of thermal stratification inside the storage (normalized temperature * versus height * ) for 24 hours with 6-hour gap; A and B are low temperature (150 °C) continuous charging cases corresponding to the top and bottom expansion designs, respectively.
Figure 12 :
12Temporal degradation of thermal stratification ( * versus * plotted for 24 hours with a 6hour gap) for high-temperature study cases; A, B, and C correspond to top-expansion design, and D, E, and F correspond to bottom-expansion design; Cases #3 and 7 are studies for 250 °C charging, and the rest cases are for 300 °C charging.
Figure 13 :
13Comparison of thermal stratification with respect to ideal stratification; A represents a schematic of ideal and real stratified profiles at an instant; B, C, and D show the temporal degradation of the stratified profiles.
Figure 14 :
14Exergy degradation of the stratified storage with respect to time during the storage period.
Table 1 :
1Study cases for thermosyphon chargingTest cases
Expansion
tank
connection
Charging Type
Set Temperature (°C)
Set Atwood Number
( )
Case #1
Top
Continuous
150
0.059
Case #2
Top
Continuous
300
0.144
Case #3
Top
Pulsating
250
0.110
Case #4
Top
Pulsating
300
0.145
Case #5
Bottom
Continuous
150
0.056
Case #6
Bottom
Continuous
300
0.141
Case #7
Bottom
Pulsating
250
0.114
Case #8
Bottom
Pulsating
300
0.145
Dipti Ranjan Parida: Conceptualization, Methodology, Experiments, Formal analysis, Writing -Original Draft Saptarshi Basu: Conceptualization, Funding acquisition, Supervision, Writing -review & editing Dhanush A P: Experiments, Data Curation
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. The authors thank Mr. Nikhil Dani (former project associate) and Murhopye Scientific Company Pvt. Ltd. for their assistance in setting up the experimental loop; and Dr. Prasenjit Kabi for his valuable comments that greatly improved the manuscript.Acknowledgments
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| [] |
[
"Efficient MILP Decomposition in Quantum Computing for ReLU Network Robustness",
"Efficient MILP Decomposition in Quantum Computing for ReLU Network Robustness"
] | [
"Nicola Franco [email protected] ",
"Tom Wollschläger [email protected] \nDept. of Computer Science & Munich Data Science Institute\nTechnical Univ. of Munich\nGermany\n",
"Benedikt Poggel [email protected] ",
"Stephan Günnemann [email protected] \nDept. of Computer Science & Munich Data Science Institute\nTechnical Univ. of Munich\nGermany\n",
"Jeanette Miriam Lorenz [email protected] ",
"† Fraunhofer ",
"\nInstitute for Cognitive Systems IKS\nMunichGermany\n"
] | [
"Dept. of Computer Science & Munich Data Science Institute\nTechnical Univ. of Munich\nGermany",
"Dept. of Computer Science & Munich Data Science Institute\nTechnical Univ. of Munich\nGermany",
"Institute for Cognitive Systems IKS\nMunichGermany"
] | [] | Emerging quantum computing technologies, such as Noisy Intermediate-Scale Quantum (NISQ) devices, offer potential advancements in solving mathematical optimization problems. However, limitations in qubit availability, noise, and errors pose challenges for practical implementation. In this study, we examine two decomposition methods for Mixed-Integer Linear Programming (MILP) designed to reduce the original problem size and utilize available NISQ devices more efficiently. We concentrate on breaking down the original problem into smaller subproblems, which are then solved iteratively using a combined quantum-classical hardware approach. We conduct a detailed analysis for the decomposition of MILP with Benders and Dantzig-Wolfe methods. In our analysis, we show that the number of qubits required to solve Benders is exponentially large in the worst-case, while remains constant for Dantzig-Wolfe. Additionally, we leverage Dantzig-Wolfe decomposition on the use-case of certifying the robustness of ReLU networks. Our experimental results demonstrate that this approach can save up to 90% of qubits compared to existing methods on quantum annealing and gate-based quantum computers. | 10.48550/arxiv.2305.00472 | [
"https://export.arxiv.org/pdf/2305.00472v1.pdf"
] | 258,426,298 | 2305.00472 | 23942c9dc4318c0e5538f02e7c81e806df997207 |
Efficient MILP Decomposition in Quantum Computing for ReLU Network Robustness
Nicola Franco [email protected]
Tom Wollschläger [email protected]
Dept. of Computer Science & Munich Data Science Institute
Technical Univ. of Munich
Germany
Benedikt Poggel [email protected]
Stephan Günnemann [email protected]
Dept. of Computer Science & Munich Data Science Institute
Technical Univ. of Munich
Germany
Jeanette Miriam Lorenz [email protected]
† Fraunhofer
Institute for Cognitive Systems IKS
MunichGermany
Efficient MILP Decomposition in Quantum Computing for ReLU Network Robustness
Index Terms-Quantum ComputingMixed-Integer Linear ProgrammingHybrid Algorithm
Emerging quantum computing technologies, such as Noisy Intermediate-Scale Quantum (NISQ) devices, offer potential advancements in solving mathematical optimization problems. However, limitations in qubit availability, noise, and errors pose challenges for practical implementation. In this study, we examine two decomposition methods for Mixed-Integer Linear Programming (MILP) designed to reduce the original problem size and utilize available NISQ devices more efficiently. We concentrate on breaking down the original problem into smaller subproblems, which are then solved iteratively using a combined quantum-classical hardware approach. We conduct a detailed analysis for the decomposition of MILP with Benders and Dantzig-Wolfe methods. In our analysis, we show that the number of qubits required to solve Benders is exponentially large in the worst-case, while remains constant for Dantzig-Wolfe. Additionally, we leverage Dantzig-Wolfe decomposition on the use-case of certifying the robustness of ReLU networks. Our experimental results demonstrate that this approach can save up to 90% of qubits compared to existing methods on quantum annealing and gate-based quantum computers.
I. INTRODUCTION
In recent years, remarkable progress has been made in the field of Quantum Computing (QC) in terms of both hardware and software development. This includes the experimental demonstration of quantum error correction, which starts to enhance performance as qubit count increases [1]. These advancements have broadened the practical capabilities of Noisy Intermediate-Scale Quantum (NISQ) devices, allowing them to tackle more complex challenges.
Operations research, with its wide-ranging real-world applications across finance, logistics, manufacturing, and automotive industries, has emerged as a particularly promising area for NISQ devices. Mixed-integer linear programming (MILP), a common problem formulation in operations research, involves a combination of integer and continuous variables constrained by linear equations. MILPs frequently involve complex combinatorial optimization problems that pose difficulties for classical solvers, particularly when dealing with large-scale instances, as
The project/research is supported by the Bavarian Ministry of Economic Affairs, Regional Development and Energy with funds from the Hightech Agenda Bayern. they are NP-hard [2]. As such, QC holds the potential to significantly accelerate the solving process and enhance overall efficiency in addressing these problems [3]. Recent works shows that approximation [4], reduction [5] and decomposition [6,7,8] approaches are needed towards the possibility of gaining potential advantages with QC. In the context of MILP, two decomposition methods have shown some potential: Benders [9] and Dantzig-Wolfe [10]. The objective of both approaches is to break down the original problem into smaller instances to enable more efficient use of quantum computing for large-scale optimization problems.
QC and MILP are not only revolutionizing operations research but also presenting promising opportunities for formal verification of neural networks. As neural networks become increasingly prevalent, ensuring their reliability, robustness, and security through formal verification is crucial, especially in safety-critical applications. In this context, formal verification of neural networks aims to provide mathematical guarantees of their expected behavior under predefined conditions. This often involves proving properties such as robustness against adversarial attacks, generalization, and compliance with safety constraints. One prevalent technique transforms the verification problem into a MILP problem, solvable using existing solvers [11,12]. However, the exact MILP solution is computationally challenging due to its exponential complexity, specifically for large networks. As a result, researchers are motivated to investigate quantum optimization algorithms as an alternative solution approach [13].
In this work, we compare Benders and Dantzig-Wolfe decompositions for MILP in terms of complexity and qubits requirements, specifically focusing on verifying the robustness of ReLU networks with QC. Since ReLU non-linearity can be reformulated as a binary variable, the verification problem can be represented as a MILP [11, combination of classical and quantum hardware. In contrast to previous studies, our approach adopts the Dantzig-Wolfe reformulation of the initial problem, providing a close representation in terms of the dual. It is essential to recognize that this method relies on a linear programming relaxation, which introduces a limitation on the tightness of the original problem. Despite this, the key advantage of HQ-CRAN-DW is evident in the reduced number of qubits required when transitioning from a constrained to an unconstrained problem.
The contributions of our work are:
• Analyzing the decomposition of MILPs with Benders [9] and Dantzig-Wolfe [10] for QC in terms of complexity and qubits requirements. • Demonstrating that the number of qubits required to solve Benders is exponentially large in the worst-case, while it remains constant for Dantzig-Wolfe. • Leveraging Dantzig-Wolfe decomposition on the usecase of certifying the robustness of ReLU networks. • Achieving up to a 90% reduction in qubit usage compared to existing methods on quantum annealing and gate-based quantum computers.
II. RELATED WORKS
A variety of quantum optimization algorithms have recently been suggested for use with NISQ devices, aiming to address large-scale combinatorial optimization problems that are typically difficult for classical solvers to handle. Many optimization problems can be mapped into Quadratic Unconstrained Binary Optimization (QUBO) form, making it a versatile and convenient framework for leveraging the potential power of quantum computing. QUBO problems can then be optimized directly using widely-used variational quantum algorithms, such as the Variational Quantum Eigensolver (VQE) [15] or the Quantum Approximate Optimization Algorithm (QAOA) [16]. Generally, in order to transform a MILP problem into a QUBO, real variables must be approximated as binary variables.
Although it is possible to apply Grover [3] search to optimization problems directly on real variables [17], implementing it on actual hardware is not practical for large-scale problems on NISQ devices due to the high gate complexity involved. Instead, potential approaches for solving largescale problems on NISQ devices include using slack-based formulations and treating the slacks as extra continuous parameters for quantum QUBO solvers [18]. A second approach is to utilize a fixed-point approximation with binary variables, as described by Vyskočil, Pakin, and Djidjev [19]. A third approach is to substitute the slack variables with a fixed choice of hyper-parameters to the first and secondorder Taylor expansion of the constraints, as proposed by Montanez-Barrera et al. [4]. These approaches offer practical solutions for implementing optimization problems on NISQ devices.
In the context of mixed-integer problems, Gambella and Simonetto [7] introduced a decomposition technique based on the alternating direction method of multipliers (ADMM), which heuristically solve mixed-binary optimization problems. Newer methods focus on utilizing decomposition strategies such as Benders [6,8,13] or Dantzig-Wolfe [20,21]. However, there is no clear comparison of which method is best suited for QC-based MILP optimization. To address this gap, this study provides an overview of the qubit requirements and complexity of these two decomposition methods, enabling a comparison of their suitability for QCbased MILP optimization.
III. PRELIMINARIES
MILPs represent a class of problems with continuous and integer variables where the objective function and the constraints are linear. A MILP in its canonical form is expressed through:
min x,y c x + d y,(1a)s.t. Ax + By ≥ b, (1b) x ∈ X , y ∈ Y,(1c)
where c ∈ Q nx , d ∈ Q ny , b ∈ Q m are vectors and A ∈ Q m×nx , B ∈ Q m×ny are matrices. Additionally, we denote (1b) as complicating constraints, while we define (1c) as the set of easy constraints, where X ⊆ R nx and Y ⊆ Z ny are polyhedra for real and integer variables, respectively 1 .
As of now, MILPs cannot be directly solved using Variational Quantum Algorithms such as VQE or QAOA, as these algorithms are tailored to optimize QUBO formulations. Therefore, decomposition methods like Benders [9] and Dantzig-Wolfe [10] are essential for breaking down MILPs into smaller, more manageable subproblems, which can then be transformed into QUBO representations compatible with quantum optimization algorithms.
In Figure 1, we offer a high-level summary of both techniques. Linear programming (LP) refers to problem instances with continuous variables and linear constraints, whereas integer linear programming (ILP) pertains to problems with integer variables and linear constraints. Additionally, we label problems transformed into QUBO format and solved using QC with (Q).
A. Benders Decomposition
Benders decomposition [9] splits the original problem into two subsets of variables: A mixed-integer set and a linear set (real). The mixed-integer set is the master problem and is solved using the initial set of variables, while the second set of variables is determined in a subproblem, given a master solution. If the subproblem finds the fixed master decision to be infeasible, valid inequalities known as Benders cuts, are generated and incorporated into the master problem, which is then solved again until no more cuts can be produced. Let us rewrite Eq. (1) as min y∈Y d y + q(y) where:
q(y) = min x∈X {c x : Ax ≥ b − By} .(2)
Here we view the vector of binary variables y as given. Hence, we decoupled y from the rest of the program resulting in a LP. Thus, we can consider the dual formulation of q(y) as:
max α∈R m b + {α (b − By) : αA = c } ,(3)
where w.l.o.g. and for ease of exposition we consider α ∈ R m b + as row vector and X = R nx . Since y is constant within the optimization of Eq. (2), the optimization program is a LP and we thus have strong duality 2 . Following Benders decomposition [9,22], we can formulate the objective of Eq. (3) as a linear combination of extreme rays and points of the feasible region. We denote as Λ r and Λ p the set of extreme rays and extreme points of the set α ∈ R m b + : αA = c . Hence, we can describe q(y) in Eq. (1) in terms of extreme points and rays as an exponential set of cuts to obtain the following representation:
min y, η d y + η,(4a)s.t. α (k) (b − By) ≤ η, ∀α (k) ∈ Λ p ,(4b)α (k) (b − By) ≤ 0, ∀α (k) ∈ Λ r ,(4c)
where η ∈ R is a scalar. This problem is known as master problem on Benders decomposition. As noted, the difficulty of solving Eq. (4) is the exponential size of the sets Λ p , Λ r . Thus, we can gradually extend the sets Λ p ⊆ Λ p , Λ r ⊆ Λ r by constraints of the subproblem defined as:
max α≤ᾱ {α (b − By) : αA = c } .(5)
The subproblem is similar to Eq. (3) except that α is bounded. This is an iterative procedure known as delayed constraint generation, where at every step t there exists three possible outcomes: (i) if the solution of Eq. (5) is unbounded (∃α i ∈ α (t) : α i =ᾱ i ), then we obtain an extreme ray 4), the algorithm terminates. In practice, we define a threshold quantity that allows us to stop the algorithm when the two solutions are sufficiently close to each other.
B. Dantzig-Wolfe Decomposition
Dantzig-Wolfe decomposition is an algorithm for solving linear programming problems with special structure 3 . This decomposition relies on a delayed column generation for improving the tractability of large-scale linear programs. For MILP problems solved via the Dantzig-Wolfe, at each step, most columns (variables) are not in the basis. In this context, a basis refers to a collection of linearly independent columns from the constraint matrix, which form the current active solution set. In such a scheme, a master problem containing at least the currently active columns (the basis) uses a subproblem or subproblems to generate columns for entry into the basis such that their inclusion improves the objective function. The traditional decomposition method relies on Minkowski and Weyl's theorem [23], which serves as its foundation, and for MILP employs a convexification process. Let U be the feasible region of Eq. (1).
Definition III.1 (feasible region). A feasible region U is the set of all possible points of Eq. (1) that satisfy the problem's constraints:
U = {x ∈ X , y ∈ Y : Ax + By ≥ b} .(6)
Minkowski and Weyl's theorem [23] states that every polyhedron U can be written as sum of finitely many extreme points and extreme rays. Thus, we denote its sets of extreme points with P X = x (i) , ∀i ∈ I and P Y = y (j) , ∀j ∈ J 4 . This allow us to express Eq. (1) as linear combination of its extreme points:
min λi, ∀i ∈ I µj ∀j ∈ J i∈I (c x (i) )λ i + j∈J (d y (j) )µ j , (7a) s.t. i∈I (Ax (i) )λ i + j∈J (By (j) )µ j ≥ b, (7b) i∈I λ i = 1, λ i ≥ 0, ∀i ∈ I, (7c) j∈J µ j = 1, µ j ≥ 0, ∀j ∈ J , (7d) y = j∈J y (j) µ j , y ∈ Z ny ,(7e)
where the variables λ i ∈ R and µ j ∈ R represent the weights of each extreme point for real and integer variables, respectively. In this context, Eq. (7) is typically called the master problem. In addition, constraint (7b) is denoted as the coupling constraint and constraints (7c-7d) are called convexity constraints. It is important to note that integrality is still imposed on the original y variable through Eq. (7e). This representation of Dantzig-Wolfe decomposition is known as convexification approach and may not be straightforward in general [24]. However, in the significant special case of combinatorial optimization with QC where Y is a subset of {0, 1} ny , convexification and discretization coincide [25] 5 . Additionally, both techniques produce the same dual bound, which is equal to that of Lagrangean relaxation [26].
Since I and J contain an exponential number of extreme points, Eq. (7) will have an exponential number of variables compared to Eq. (1). Thus, we consider a restricted version of Eq. (7) by progressively adding each new extreme point to the subsets I ⊆ I and J ⊆ J . To determine which extreme point to include, we define two subproblems, referred to as pricing problems, by considering the dual of Eq. (7).
1) Dual formulation of Eq. (7): In accordance with the work of [26,25], we transition to the dual representation of Eq. (7). Thus, let us introduce the so called Lagrangean subproblem max α,ξ,η L(α, ξ, η), with L defined as:
min λi, ∀i ∈ I µj ∀j ∈ J i∈I αb + (c − αA)x (i) + ξ λ i − ξ + j∈J (d − αB)y (j) + η µ j − η,(8)
where α ∈ R m + is a row vector and ξ, η ∈ R are scalars (also known as Lagrangian multipliers). It is worth noting that we omit the integrality constraint from Eq. (7e) in the reformulation, as it can be violated at a price of α [25]. The process of raising an integer (or mixed-integer) problem to a higherdimensional space, deriving an enhanced formulation in that context, and subsequently returning it to the initial variable space is a familiar strategy in integer programming [24,25].
The solution of the function L establishes a dual (lower) bound on the optimal value of Eq. (7). The task of maximizing this bound across the set of acceptable penalty vectors is referred to as the Lagrangean dual:
max α,ξ,η αb − ξ − η, s.t. (c − αA)x (i) + ξ ≤ 0, ∀i ∈ I, (d − αB)y (j) + η ≤ 0, ∀j ∈ J ,(9)
which is known as dual master problem on Dantzig-Wolfe decomposition.
Lemma III.1 (Lagrangian bound [27]). The solution of Eq. (7) offers a dual bound that is equal to the solution of Eq. (9).
2) Column generation: Dantzig-Wolfe decomposition involves iterating between the master and subproblems, which are also called pricing problems. This method is commonly referred to as the column generation process [10]. To initiate the process, a preliminary restricted master problem is required. Having a feasible linear programming relaxation for this initial restricted master problem is essential, as it ensures the proper exchange of dual information with the pricing problems. At every step t, we generate an extreme point x (t) , and an extreme point y (t) . These extreme points are incorporated into the master, necessitating the addition of new λ i and µ j columns. The real pricing problem is given as: min
x∈X (c − α (t) A)x,(10)
where α (t) is the dual solution of Eq. (7) associated with the constraint (7b). Similarly, the integer pricing problem is given as: min
y∈Y (d − α (t) B)y,(11)
which deals with integer variables and therefore has a stronger complexity. If the solution of Eq. (10) is lower then ξ, then we set I ← x (t) . Similarly, if the solution of Eq. (11) is lower than η, than we add J ← y (t) . Analogously to Benders, we define a threshold quantity θ that allows us to stop the algorithm when the difference between the solution of Eq. (7) and its dual is lower than θ.
C. Quadratic Unconstrained Binary Formulation
By transforming Benders master problem or Dantzig-Wolfe integer pricing problem into a QUBO problem, the power of quantum optimization algorithms, such as VQE or QAOA, can be harnessed to find more efficient solutions. The transformation involves rewriting the objective function and constraints of each subproblem using binary variables, and then converting them into a quadratic cost function in line with the QUBO formulation: where Q ∈ R nq×nq . QUBO problems can be directly converted to an Ising model and vice versa [28], which is the reason for its use in QC.
min q∈{0,1} nq q Qq,(12)
IV. BENDERS VS. DANTZIG-WOLFE IN QC-BASED MILP SOLVING
In this section, we compare the two previously presented decomposition methods in terms of qubits requirements and complexity. To recap, while both Benders and Dantzig-Wolfe techniques aim to break down MILP problems into smaller components to solve them more efficiently, they apply to different types of problem structures and utilize different strategies. Benders decomposition is more suitable for problems with a clear separation of integer and continuous variables, while Dantzig-Wolfe decomposition is best for problems with a block structure in the constraints. In the context of linear programming, it is important to remind that Dantzig-Wolfe decomposition in the primal problem is equivalent to Benders decomposition in the dual problem, with both approaches sharing identical sub-problems [29]. While the two approaches are equivalent, certain stabilization techniques can be more easily formulated in the dual problem compared to the primal problem 6 .
In Table I, we present a comparison between Benders and Dantzig-Wolfe decomposition concerning complexity and the number of qubits needed for solving the QUBO problem with QC. A notable advantage of Dantzig-Wolfe over Benders decomposition lies in the fewer qubits required to transform the problem from constrained to unconstrained, which remains constant at each step.
In our comparison, we consider a fixed number of qubits n y to represent the vector of variables y. Additionally, we consider a fixed number of qubits n s to convert real variables to binary. This consideration is independent of the approximation method used, such as fixed point or floating representation [19]. For example, a fixed-point approximation of a positive real variable is given byη = w · ns−1 i=0 2 i · y i , where w is typically chosen as 10 −1 or 10 −2 . Finally m y denotes the number of constraints of the set Y.
The difference between the two methods lies in the way on how the problem affected by the QUBO transformation is formulated. In the context of Benders, the master starts with one cut in the constraints set and a real objective η. To convert the problem from constrained to unconstrained an additional slack variable is needed. Therefore, if we assume 6 In this context, a primal problem refers to the original optimization problem.
the same approximation factor n s for the slack variable and the real objective η, at least 2 · n s qubits are needed [6,8,13]. Since, at every step, a new cut is added to the master problem a new slack variable is required. In the end, if all cuts from the extreme point set are added to the master problem, the number of qubits required by Benders becomes exponential. In contrast to Dantzig-Wolfe, where the integer pricing problem is fixed in the number of constraints and consequently in the number of binary variables required to approximate the constraints.
It is crucial to highlight that the solution of the master problem in Benders decomposition directly influences the feasibility of the entire problem. The heuristic nature of quantum optimization algorithms, such as VQE or QAOA, affects the quality of the solution, which can lead to incorrect cut generation in the corresponding subproblem and ultimately result in infeasible solutions. On the other hand, a key advantage of the Dantzig-Wolfe decomposition is inherently linked to how the master problem is solved. Since the master problem is addressed using classical methods, the coupling constraints condition is consistently satisfied, resulting in more feasible solutions. Nonetheless, we cannot make the same claim for the individual constraints of the integer pricing problem, as the use of quantum optimization algorithms could still result in unsatisfied constraints in some cases, leading to the generation of false extreme points. Furthermore, the application of effective heuristics has always been encouraged in the context of Dantzig-Wolfe decomposition to accelerate the overall search process [24].
In conclusion, in the context of addressing MILP problems with QC, Dantzig-Wolfe decomposition is considered a more favorable choice compared to Benders decomposition. This preference can be observed in both the constant number of qubits required and the improved feasibility of the resulting solution.
V. DANTZIG-WOLFE FOR FORMAL VERIFICATION OF NEURAL NETWORK In this section, we introduce the task of assessing neural network robustness through the application of Dantzig-Wolfe decomposition. We propose a hybrid decomposition method that iteratively solves the MILP formulation by employing both classical and quantum hardware. In constrast to [13], we consider the Dantzig-Wolfe formulation of the original problem, which provides a close representation in terms of the dual. The main advantages of our approach are demonstrated by the reduced number of qubits needed and the increased number of feasible solutions when utilizing quantum hardware. We begin by providing a brief overview of the robustness verification problem.
A. Robustness Certification of Neural Networks
We represent a neural network as a function f (z) : Z → R |K| , which maps input samples z ∈ Z to output k ∈ R |K| . Here, K denotes the set of classes. We assume a feedforward architecture that consists of affine transformationŝ
x [i] = W [i] x [i−1] + v [i] followed by ReLU activations x [i] = max {0,x [i] }, ∀i ∈ {1, .
. . , L}, where L represents the number of layers, x [0] ≡ z and f (z) ≡ x [L] . In case of classification, the network outputs a vector in R |K| . The predicted class is then given by the index of the largest value of that vector, i.e. c = arg max j f (z) j .
Definition V.1 (certified robustness ( ∞ )). An input z is considered certifiably robust for a neural network f if the prediction remains unchanged for all perturbed versions:
arg max j f (z) j = arg max j f (z) j , ∀z ∈ B ∞ (z).
Here, is the perturbation budget andz is an element from the perturbation set based on the infinity norm: B ∞ (z) = {z : z −z ∞ ≤ }. If we cannot certify an input, it implies the existence of z ∈ B ∞ (z) for which arg max j f (z) j = arg max j f (z ) j . Any of these z instances are called adversarial examples.
The non-convex nature of the problem arises from the piece-wise linear characteristics of ReLU activation units. There are two approaches to address this issue: (i) model the ReLU activation with a binary variable or (ii) enclose the possible activation values x [i] within a convex region. The first approach results in a complete formulation of the exact polytope, but the binary variables make the problem NP-hard [12]. The second approach yields a convex solution [30].
B. HQ-CRAN-DW
Here, we describe our algorithm designed for evaluating neural network robustness using the Dantzig-Wolfe decomposition. We build upon the formulation presented in [13] and discuss the differences and adaptations for our approach. To obtain a valid certificate, it is necessary to evaluate whether the network's prediction can be altered to any other possible class, as shown in Definition V.1. However, w.l.o.g., we consider testing the difference between the initial predicted class and just one other class. Thus, the original MILP problem instance is given by 7 :
min x,y {c x : Ax + By ≥ b, Cx ≥ e} ,(13)
where x ∈ R nx and y ∈ {0, 1} ny are the vectors of real and binary variables, respectively.
The master problem of Dantzig-Wolfe decomposition for Eq. (13) is defined as:
min λi, ∀i ∈ I µj ∀j ∈ J i∈I (c x)λ i , (14a) s.t. i∈I (Ax (i) )λ i + j∈J (By (j) )µ j ≥ b, (14b) i∈I λ i = 1, λ i ≥ 0, ∀i ∈ I, (14c) j∈J µ j = 1, µ j ≥ 0, ∀j ∈ J ,(14d)
where we omitted the integrality constraint on y, as previously discussed in section III-B1, it can be violated at a price of α [25]. The restricted version of Eq. (14) is derived from the sets I ⊆ I and J ⊆ J . This restricted version is easier to solve and can provide initial solutions for the original master problem. Subsequently, the real pricing problem is given as:
r = min x (c − α (t) A)x : Cx ≥ e ,(15)
where x ∈ R nx and the binary pricing problem is given by:
p = min y {−α (t) By},(16)
where y ∈ {0, 1} ny . The Dantzig-Wolfe decomposition method iteratively solves the restricted master problem and the pricing problems until convergence is reached. Thus, given the original problem instance in terms of extreme points, we can state our main result. Proof. This is a direct consequence of Definition III.1. The two sets of extreme points, P X = x (i) , ∀i ∈ I and P Y = y (j) , ∀j ∈ J , are derived from Eq. (15) and Eq. (16), respectively. The set of linear constraints X = {x ∈ R nx : Cx ≥ e} is satisfied through the optimality of Eq. (15), while the set Y = {0, 1} ny represents a binary instance. Therefore, as long as the two sets include all extreme points, according to Lemma III.1, the solution of Eq. (14) provides a valid dual bound, which is equal to the convex relaxation of Eq. (13).
As observed, the binary pricing problem belongs to the class of unconstrained binary problems, which can be conveniently mapped into QUBO by considering diag (−α (t) B). This eliminates the necessity for incorporating penalty terms through additional variables, which simplifies the overall process. Consequently, the Dantzig-Wolfe decomposition is better suited to address the robustness verification problem of neural networks using QC. However, in the convexification approach of Eq. (14) integrality is required on y variables just as in the original Eq. (13). Since we are not
Algorithm 1 HQ-CRAN-DW
Input: z, f (z), , T, θ Output: robust, not robust, or unknown Propagate Interval bounds 1: lower bound ← CROWN-IBP from [31] 2: if lower bound > 0 than return robust certified Compute problem matrices 3: A, B, C, c, b, e ← Alg. 1 from [13] Find initial extreme point 4: x (0) , y (0) ← solve Eq. (13) without objective 5: I , J ← x (0) , y (0) initialize extreme points sets 6: α (0) , ξ (0) , η (0) ← get dual from the relaxed version of Eq. (13) without objective (i.e. y ∈ R ny >0 with y < 1). 7: for each adversarial class do 8: Iterate between the master and sub problems 9: for t in {0, . . . , T } do 10: r, x (t) ← solve Eq. (15) with α (t) classical 11: if r < ξ (t) then I ← x (t)
12:
p, y (t) ← solve Eq. (16) with α (t) quantum 13: if p < η (t) then J ← y (t)
14:
ϕ, λ, µ ← solve Eq. (14) with I , J 15:
α (t) , ξ (t) , η (t) ← get dual solution from Eq. (14)
Compute dual master objective 16:
φ ← max k∈{0,...,t} −α (k) b − ξ (k) − η (k)
17:
if ϕ ≤ 0 then return not robust adversary 18: if |ϕ − φ| ≤ θ then break stopping criteria 19: end for 20: if φ ≤ 0 then return unknown abstain 21: end for 22: return robust certified forcing the original problem to generate binary variables, the resulting solutionȳ = ∈J y (j) µ j is a vector of real values between 0 and 1. Therefore, an additional branching procedure is required to tight the convex relaxation.
In Algorithm 1, we present the HQ-CRAN-DW algorithm, which takes as inputs a neural network f , a sample z, a predefined , a maximum number of steps T, and a predetermined gap θ. The algorithm can yield one of three possible outcomes: certified robust, not robust, or unknown. By employing CROWN-IBP [31], the algorithm facilitates rapid convex propagation and estimation of bounds for x. If the global lower bound is positive, a certified sample can be returned immediately.
The procedure starts by determining an initial extreme point for each set from the solution of Eq. (13) without the objective. This point acts as a feasible solution for all potential adversarial tests within the certification problem. Simultaneously, the convex relaxation of Eq. (13) without the objective provides an initial dual solution, initiating the iterative process between the master and pricing problems. As new columns are constantly added, the algorithm refines and improves the master solution ϕ, converging towards the optimal one. The algorithm terminates under three conditions: (i) when the difference between the master and sub is less than the threshold value, (ii) if the master objective is negative, indicating an adversary, and (iii) if the dual objective is below zero and the master is above zero, representing an unknown condition. Otherwise, the dual master solution φ is positive for each of the adversarial class tested, the algorithm return a certified robust sample.
VI. EXPERIMENTAL RESULTS
In this section, we examine the application of Dantzig-Wolfe decomposition in the context of certifying the robustness of neural networks. We report the details of the quantum hardware at the beginning of each experimental section. On the classical side, we ran the algorithm on a server having 4xCPUs Intel(R) Xeon(R) E7-8867 v4 running at 2.40GHz for a total of 72/144 cores/threads.
A. Networks and Datasets
We perform experiments using a single multilayer perceptron (MLP) neural network configuration: MLP-2x [20]. In this notation, MLP-mx[n] represents a network with m hidden layers and n units per hidden layer. After each fully connected layer, ReLU functions are applied. Our model is trained on the MNIST dataset [32] for 20 epochs using a batch size of 128 in two distinct ways: (i) employing a standard loss function, and (ii) using adversarial training through Projected Gradient Descent (PGD), as described in Madry et al. [33]. For regularly trained models, we maintain the MLP-2x [20] designation, while we refer to adversarially trained models as PGD-2x [20]. The clean test set accuracy for these networks is 95.62% for MLP-2x [20] and 86.73% for PGD-2x [20]. Adversarial training involves generating adversarial examples from an infinity norm ball surrounding the input, with a radius of = 0.01.
B. Certified Accuracy
In this section, we perform a comparison between our newly proposed variant, HQ-CRAN-DW, which employs Dantzig-Wolfe decomposition, and HQ-CRAN (v2) [13], which utilizes Benders decomposition and is referred to as HQ-CRAN-BD. Additionally, we compare the methods against the comprehensive verifier β-CROWN [34], as well as two convex verifiers PRIMA [35] and GPUPoly [36]. We assess HQ-CRAN-DW's empirical performance under optimal conditions, meaning that the master and sub-problems are resolved using the IBM ILOG CPLEX [37] software on a traditional computer. To ensure a fair comparison, we employ the IBP-CROWN [31] technique to propagate boundaries through the network, which is also utilized by β-CROWN. All methods are evaluated on the initial 100 samples from the MNIST penalties. Additionally, we consider the standard settings for HQCRAN-BD as described in [13].
In Figure 2, we report the certified accuracy, as fraction of verified and correctly classified samples of all test samples, and runtime for two neural networks. In the context of HQ-CRAN-DW, the number of certified samples is similar than GPUPoly and PRIMA for values greater than 8 /255 but lower than exact verifiers such as β-CROWN and HQ-CRAN-BD. In terms of runtime, DW performs similarly than BD. The limitations of HQ-CRAN-DW has to be related to the convex relaxation of the original MILP instance. As noticing the number of certified samples reflects the same amount of convex verifiers.
C. Simulated & Quantum Annealing
Here, we proceed with our assessment by comparing simulated 8 and quantum annealing. The decision to select quantum annealing is linked to the size of the problem, particularly in relation to the evaluation of Benders decomposition. On the quantum side, we access the D-Wave Advantage TM system 4.1 constructed with 5760 qubits by D-Wave Leap 9 .
1) Hyperparameters: To embed the problem into a quantum annealer, the minor embedding problem must be solved. In our case, we employ the clique embedding strategy [38], which addresses the connectivity issue in a manner that any connection larger than the maximum available size (approximately 15 for Advantage) is managed with long chains between qubits. To reduce the chains, we prune con-nections below a specific threshold (5%) before submitting the problem to the sampler, maintaining only interactions with values compatible with the sampler's precision. Both simulated and quantum annealing used 100 reads, while simulated annealing employed 50,000 sweeps.
In the context of Benders and Dantzig-Wolfe decompositions, the gap between the master and sub is set to 1 and the maximum number of steps T has been set to 15 and 20, respectively. The selection is connected to the fact that the number of qubits needed for Benders is a limiting factor when using the clique embedding, which sets the maximum size of logical qubits to 177 on the Advantage [39]. In the context of Benders, following the settings of [13], the sub problem boundaries α and β are set to 5. Additionally, the maximum size of the cuts set ϕ has been set to 5 and the penalty weights w a and w p were set to 0.1 and 0.01, respectively.
2) Results: In Figure 3, we show a comparison between Benders (BD) and Dantzig-Wolfe (DW) decomposition of the HQ-CRAN algorithm with simulated and quantum annealing. We plot the average number of qubits and steps required to meet the predefined gap. There is substantial difference between BD and DW in terms of qubits requirement. In general the ratio 10 of required qubits by DW is around 80% (up to 90%) less than BD. Additionally, the number of steps needed by DW is lower for values smaller than 8 /255. The difference between quantum and simulated annealing is perceivable for larger problems, i.e. larger values. The numerical results of quantum annealing are presented in Table II. For a fair evaluation, alongside the number of qubits, we compare the percentage of correct and certified samples. Correctness refers to the proportion of feasible solutions, indicating that the final master objective does not exceed the exact solution. In the context of BD, the correct solution percentage declines as the adversarial perturbation budget rises. However, DW maintains consistent feasibility even with increasing values. As emphasized in section IV, the master problem's solution in BD considerably influences the problem's feasibility, with simulated and quantum annealing potentially affecting solution quality and resulting in incorrect cuts and infeasible solutions. Conversely, DW solves the master problem classically, consistently satisfies coupling constraints and leading to feasible solutions. Nevertheless, the convexification of the original MILP formulation restricts the number of certified samples.
D. Gate-Based vs. Annealing
Here, we compare HQ-CRAN-DW running on quantum annealing or QAOA on a gate-base simulator. In the first case, we consider the results of the quantum annelear from subsection VI-C, while in the second case, we used the QAOA 11 runtime program with a Aer 12 simulator on classical hardware. QAOA is considered with a depth of 5 and COBYLA [40] as classical optimizer.
In Figure 4, we plot the average number of certified samples and steps with respect to increasing values. It is worth noting that QAOA exhibits slightly inferior performance compared to quantum annealing, yielding fewer certified samples and requiring more steps for corresponding values. Potentially, incorporating the CVaR metric [41] could lead to substantial improvements in the results, enhancing the overall performance of QAOA. However, considering that QAOA is essentially a trotterization of quantum annealing [16], these findings are in line with recent comparisons made on gate-based quantum hardware [42].
VII. DISCUSSION OF RESULTS AND LIMITATIONS
In this section, we discuss the outcomes of our experiments. A notable gap in runtime can be observed when comparing our proposed solution to classical neural network verifiers. This is due to the challenges faced in achieving high certified sample rates and the need for more interactions. These challenges stem from the basic implementation of the Dantzig-Wolfe decomposition, which introduces various computational issues [43], such as the tailing-off effect (slow convergence), the heading-in effect (weak initial dual information), and the plateau effect, which occurs when the master solution remains constant over multiple steps. However, several stabilization techniques have been developed to address these drawbacks. Specifically, integrating a more accurate branch-and-price procedure could lead to enhanced performance [27].
Despite the mentioned limitations, our method offers two main advancements in the field of hybrid verifiers for neural network robustness, both arising from Dantzig-Wolfe decomposition: (i) a fixed qubit number at each step, and (ii) the feasibility of the generated solutions. Additionally, the direct incorporation of a branch-and-price procedure would not compromise these benefits [20].
VIII. CONCLUSION
In this study, we have examined the complexity and qubit requirements of Benders and Dantzig-Wolfe decompositions for MILPs, with a particular focus on verifying the robustness of ReLU networks using QC. Since ReLU non-linearity can be expressed as a binary variable, the verification problem can be modeled as a MILP. Building on a previous approach [13], we have proposed a Hybrid Quantum-Classical Robustness Analyzer for Neural Networks with Dantzig-Wolfe decomposition (HQ-CRAN-DW). Our finding show a reduction up to 90% in qubits usage with respect to previous methods on quantum annealing and gate-based quantum computers. Additionally, we demonstrate that the number of qubits required to solve Benders decomposition is exponentially large in the worst-case scenario, while it remains constant for Dantzig-Wolfe.
Fig. 1 :
1This diagram offers an high-level overview of Benders and Dantzig-Wolfe decomposition for a MILP. We identify with (Q) the problems that are optimized with quantum hardware.
Λ
r ← α (t) ; (ii) if the solution of Eq. (5) is lower than the solution of Eq. (4), then we acquire an extreme point Λ p ← α (t) ; (iii) if the solution of Eq. (5) is equal to the solution of Eq. (
Theorem V. 1 .
1Given a neural network f and an input z, the solution of Eq. (14) is a valid lower bound to the robustness verification problem of Eq. (13).
Fig. 2 :
2Average certified accuracy and runtime of various verification techniques including HQ-CRAN-DW/-BD for the initial 100 test MNIST samples on two MLP networks.
Fig. 3 :
3Comparative analysis of Benders and Dantzig-Wolfe decompositions in the HQ-CRAN algorithm using simulated and quantum annealing on a 2-Layer MLP with 20 neurons per layer.
Fig. 4 :
4Experimental analysis of HQ-CRAN-DW using quantum annealing and QAOA (Aer simulator) on the first 100 samples of the MNIST test set.
TABLE I :
IA comparative summary of MILP decomposition methods for quantum computing, detailing the complexity of the master and subproblems (P: polynomial-time solvable, NP-hard: non-deterministic polynomial-time hard), along with the number of qubits required at the first and 2 ny -th iterations. The terms n s and m y represent the number of slack variables and the number of constraints involving integer variables, respectively.Method
Complexity
# of qubits at iteration
Master
Sub
1st
2 ny -th
Benders
NP-hard (QUBO) P (Dual LP)
O(ny + 2 · ns)
O(ny + 2 ny · ns)
Dantzig-Wolfe P (LP)
NP-hard (QUBO) O(ny + my · ns)
O(ny + my · ns)
We run HQ-CRAN-BD without the QUBO formulation for QC, meaning constraints are not relaxed with extra variables or incorporated into the objective via quadratictest set, with adversarial budgets
of ∈ { 1
255 , 2
255 , 4
255 , 8
255 , 16
255 }.
TABLE II :
IIComparison of HQ-CRAN, with Benders (BD)[13] and Dantzig-Wolfe (DW) decomposition with quantum annealing. We run each algorithm on the first 100 samples of the MNIST test set.NETS
CORRECT & CERTIFIED ↑
# OF QUBITS ↓
CPLEX
BD
DW
BD
DW
PGD
1 /255
88% 61%
86% 37 ± 17
3 ± 2
2 /255
88% 41%
77% 46 ± 20
5 ± 2
4 /255
88% 20%
40% 59 ± 21
8 ± 3
8 /255
81%
3%
3% 81 ± 19 13 ± 3
MLP
1 /255
98% 49%
82% 52 ± 20
5 ± 3
2 /255
97% 25%
70% 60 ± 21
7 ± 3
4 /255
96%
7%
30% 72 ± 24 11 ± 4
8 /255
78%
1%
2% 93 ± 16 17 ± 4
e.g. X = {x ∈ R nx : Cx ≥ d} and Y = {y ∈ Z ny : Ey ≥ g}.
i.e. the optimal objective value of the primal equals the optimal value of the dual.
a block-angular or block-diagonal arrangement in the constraint matrix.4 For the sake of readability we only consider extreme points and not extreme rays.
In this work, we only consider the convexification approach for the sake of conciseness.
The main distinctions involve substituting z with x, g with c, and d with e. Additionally, the distinctions compared to the canonical form of Eq. (1) are that Y = {0, 1} ny and X = {x ∈ R nx : Cx ≥ e}.
Python dwave-neal v0.5.7, general Ising graph simulated annealing solver.9 Quantum Cloud Service: https://cloud.dwavesys.com/leap/
the ratio is calculated by dividing the average number of qubits needed for Dantzig-Wolfe by the average number of qubits needed for Benders and subtracting 1. To express the ratio as a percentage, the result is multiplied by 100.
Python library qiskit v0.36.0 https://github.com/Qiskit/qiskit 12 qiskit-aer v0.10.4 https://github.com/Qiskit/qiskit-aer
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] | [] | Adversarial contrastive learning (ACL), without requiring labels, incorporates adversarial data with standard contrastive learning (SCL) and outputs a robust representation which is generalizable and resistant to adversarial attacks and common corruptions. The style-independence property of representations has been validated to be beneficial in improving robustness transferability. Standard invariant regularization (SIR) has been proposed to make the learned representations via SCL to be independent of the style factors. However, how to equip robust representations learned via ACL with the style-independence property is still unclear so far. To this end, we leverage the technique of causal reasoning to propose an adversarial invariant regularization (AIR) that enforces robust representations learned via ACL to be style-independent. Then, we enhance ACL using invariant regularization (IR), which is a weighted sum of SIR and AIR. Theoretically, we show that AIR implicitly encourages the prediction of adversarial data and consistency between adversarial and natural data to be independent of data augmentations. We also theoretically demonstrate that the style-independence property of robust representation learned via ACL still holds in downstream tasks, providing generalization guarantees. Empirically, our comprehensive experimental results corroborate that IR can significantly improve the performance of ACL and its variants on various datasets.Recently, the style-independence property of learned representations has been validated to be beneficial in improving the robustness transferabilityMitrovic et al. (2021). To equip learned representations via SCL with the style-independence * Equal contributions. | 10.48550/arxiv.2305.00374 | [
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ENHANCING ADVERSARIAL CONTRASTIVE LEARNING VIA ADVERSARIAL INVARIANT REGULARIZATION
Xilie Xu
School of Computing
National University of Singapore
Jingfeng Zhang
RIKEN Center for Advanced Intelligence Project (AIP)
Feng Liu
School of Mathematics and Statistics
The University of Melbourne
Masashi Sugiyama
RIKEN Center for Advanced Intelligence Project (AIP)
Graduate School of Frontier Sciences
The University of Tokyo
Mohan Kankanhalli
School of Computing
National University of Singapore
ENHANCING ADVERSARIAL CONTRASTIVE LEARNING VIA ADVERSARIAL INVARIANT REGULARIZATION
Adversarial contrastive learning (ACL), without requiring labels, incorporates adversarial data with standard contrastive learning (SCL) and outputs a robust representation which is generalizable and resistant to adversarial attacks and common corruptions. The style-independence property of representations has been validated to be beneficial in improving robustness transferability. Standard invariant regularization (SIR) has been proposed to make the learned representations via SCL to be independent of the style factors. However, how to equip robust representations learned via ACL with the style-independence property is still unclear so far. To this end, we leverage the technique of causal reasoning to propose an adversarial invariant regularization (AIR) that enforces robust representations learned via ACL to be style-independent. Then, we enhance ACL using invariant regularization (IR), which is a weighted sum of SIR and AIR. Theoretically, we show that AIR implicitly encourages the prediction of adversarial data and consistency between adversarial and natural data to be independent of data augmentations. We also theoretically demonstrate that the style-independence property of robust representation learned via ACL still holds in downstream tasks, providing generalization guarantees. Empirically, our comprehensive experimental results corroborate that IR can significantly improve the performance of ACL and its variants on various datasets.Recently, the style-independence property of learned representations has been validated to be beneficial in improving the robustness transferabilityMitrovic et al. (2021). To equip learned representations via SCL with the style-independence * Equal contributions.
Introduction
The attention towards pre-trained models that can be easily finetuned for various downstream applications has significantly increased recently Deng et al. (2009); Ridnik et al. (2021). Notably, foundation models Bommasani et al. (2021) via self-supervision on large-scale unlabeled data, such as GPT Brown et al. (2020) and CLAP Elizalde et al. (2022), can be adapted to a wide range of downstream tasks. Due to the high cost of annotating large-scale data, unsupervised learning techniques Erhan et al. (2010); Le-Khac et al. (2020) are commonly used to obtain generalizable representations, in which standard contrastive learning (SCL) has been shown as the most effective way Chen et al. (2020a); Mitrovic et al. (2021); Chen et al. (2020b).
Adversarial contrastive learning (ACL) Kim et al. (2020); ; Fan et al. (2021); Yu et al. (2022); ; Xu et al. (2023), that incorporates adversarial data Madry et al. (2018) with SCL Chen et al. (2020a), can yield robust representations that are both generalizable and resistant to adversarial attacks Goodfellow et al. (2014) and common corruptions Hendrycks and Dietterich (2019). ACL is attracting increasing popularity since the adversarial robustness of the pre-trained models is essential to their practicality in safety-critical domains Buch et al. (2018); Kurakin et al. (2018). A bunch of follow-up works have tried to improve ACL from various perspectives including increasing contrastive views and leveraging pseudo labels Fan et al. (2021), leveraging hard negative sampling method Robinson et al. (2020); Yu et al. (2022), and dynamically scheduling the strength of data augmentations . x is unlabeled data, s is style variable, c is content variable,x is the generated adversarial data, and θ is the parameter of the encoder. The dashdotted lines denote that the proxy label y R ∈ Y R is a refinement of the target label y t ∈ Y = {y i } T i=1 . All other arrows are causal.
property, Mitrovic et al. (2021) proposed the standard invariant regularization (SIR) induced by casual reasoning Pearl (2009); Peters et al. (2017) that enforces the learned representations of natural data to be invariant of the style factors.
On the other hand, how to make the robust representations learned via ACL, particularly where adversarial data is leveraged during pre-training, to be style-independent is still unclear in the literature.
To this end, we leverage the technique of causal reasoning to enforce robust representations learned via ACL to be style-independent. First, we construct the causal graph of ACL shown in the right panel of Figure 1. We formalize the data generation procedure and demonstrate the details of the learning procedure in Section 3.1. Different from the causal graph of SCL Mitrovic et al. (2021), ACL has an extra path x →x → y R since ACL will first generate the adversarial datax given the unlabeled data x (i.e., x →x) and then learn representations using the proxy task driven by data augmentations. The proxy task aims to maximize both the probability of the proxy label y R given the natural data and that given the adversarial data (i.e., x → y R andx → y R ), which is theoretically shown as coinciding with the learning objective of ACL in Appnedix B.1.
Then, we propose adversarial invariant regularization (AIR) to enhance ACL. According to the independence of causal mechanisms Chalupka et al. (2014), performing the intervention on the style variable should not change the learned representations. By leveraging data augmentations as an intervention on the style variable, AIR (Eq. (6)) aims to make the robust representations learned from the unlabeled data to be independent of the data augmentations. Note that SIR Mitrovic et al. (2021) can be regarded as an extension of our proposed AIR to the standard context since AIR will transform into SIR when the adversarial budget drops to 0. Lastly, we demonstrate the learning algorithm of incorporating ACL with invariant regularization (IR) which is a weighted sum of AIR and SIR in Algorithm 1.
Next, we show the theoretical understanding of the proposed AIR and theoretically provide the generalization guarantee. By decomposing the AIR, we show that AIR implicitly encourages both the prediction of adversarial data and the consistency between adversarial and natural data to be independent of the style factors. Further, we theoretically show that the style-independence property of robust representations learned via ACL will still hold on the downstream classification tasks in Theorem 1, which guarantees the property of style-independence to be generalizable.
Empirically, we conducted comprehensive experiments on various datasets including CIFAR-10 Krizhevsky (2009), CIFAR-100 Krizhevsky (2009), STL10 Coates et al. (2011), CIFAR-10-C Hendrycks and Dietterich (2019), and CIFAR-100-C Hendrycks and Dietterich (2019) to show the effectiveness of our proposed method in improving ACL Jiang et al. (2020) and its variants Fan et al. (2021); Yu et al. (2022); . We demonstrate that IR can significantly enhance the generalization ability and robustness against adversarial attacks Croce and Hein (2020a) and common corruptions Hendrycks and Dietterich (2019)
Backgrounds and Preliminaries
In this section, we introduce the related works in adversarial contrastive learning and causal reasoning. Then, we provide the preliminaries of contrastive learning.
Related Works
Adversarial contrastive learning (ACL). CL approaches are frequently used to leverage large unlabeled datasets for learning useful representations. Chen et al. (2020a) presented SimCLR that leverages the contrastive loss for learning useful representations and achieved significantly improved accuracy on the standard suit of downstream tasks. Recently, adversarial contrastive learning (ACL) (Kim et al., 2020;Ho and Nvasconcelos, 2020;Fan et al., 2021;Yu et al., 2022;Zhang et al., 2022a; has become the most effective unsupervised approaches to learn robust representations. , which is the seminal work of ACL that incorporates adversarial data Madry et al. (2018) with the contrastive loss (Chen et al., 2020a) have not yet applied ACL to large-scale datasets such as ImageNet- 1K Deng et al. (2009). A concurrent work Xu et al. (2023) proposed a coreset selection method to speed up ACL by decreasing the number of training data and thus enabled ACL to be applied to ImageNet-1K. However, due to that this work Xu et al. (2023) has not been published and did not provide the code, we did not conduct ACL on ImageNet-1K similar to previous stuides Kim et al. (2020); Jiang et al. (2021) has been widely applied to machine learning to identify causal relations and ignore nuisance factors by intervention. In particular, Zhang et al. (2022b) investigated the causality in supervised adversarial training Madry et al. (2018); Zhang et al. (2019) where label information is required and proposed the regularizer to eliminate the difference between the natural and adversarial distributions. Mitrovic et al. (2021) built the causal graph of SCL in the standard context and introduced the standard invariant regularization (SIR) which aims to make the prediction of unlabeled natural data to be invariant of the data augmentations. Empirically, SIR can significantly improve the robustness transferability of SCL Mitrovic et al. (2021). However, to the best of our knowledge, so far no work has studied the causality in ACL where adversarial data exists and label information is not provided. We are the first to leverage causal reasoning to improve ACL.
Preliminaries
Standard contrastive learning (SCL) Chen et al. (2020a). Let h θ : X → Z be a feature encoder parameterized by θ, g : Z → V be a projection head that maps representations to the space where the contrastive loss is applied, and τ i , τ j : X → X be two transformation operations randomly sampled from a pre-defined transformation set T . Given a minibatch B ∼ X β consisting of β original samples, we denote the augmented minibatch B u = {τ u (x k ) | ∀x k ∈ B} via augmentation function τ u (·). We take f θ (·) = g • h θ (·) and x u k = τ u (x k ) for any x k ∼ X and u ∈ {i, j}. Given a positive pair (x i k , x j k ), the standard contrastive loss proposed by SimCLR Chen et al. (2020a) is formulated as follows:
CL (x i k , x j k ; θ) = − u∈{i,j} log e sim(f θ (x i k ),f θ (x j k ))/t x∈B i ∪B j \{x u k } e sim(f θ (x u k ),f θ (x))/t ,(1)
where sim(p, q) = p q/ p q is the cosine similarity function and t > 0 is a temperature parameter.
ACL and DynACL . Let (X , d ∞ ) be the input space X with the infinity distance
metric d ∞ (x, x ) = x − x ∞ , and B [x] = {x ∈ X | d ∞ (x,
x ) ≤ } be the closed ball of radius > 0 centered at
x ∈ X . Given a data point x k ∈ X , the adversarial contrastive loss is as follows:
ACL (x k ; θ) = (1 + ω) · CL (x i k ,x j k ; θ) + (1 − ω) · CL (x i k , x j k ; θ),(2)s.t.x i k ,x j k = arg max x i k ∈B [x i k ] x j k ∈B [x j k ] CL (x i k ,x j k ; θ),(3)
in which ω ∈ [0, 1] is a hyperparameter andx i k andx j k are adversarial data generated via projected gradient descent (PGD) (Madry et al., 2018) within the -balls centered at x i k and x j k . Note that ACL Jiang et al. (2020) fixes ω = 0 while DynACL dynamically schedules ω according to its dynamic augmentation scheduler that gradually anneals from a strong augmentation to a weak one. We leave the details of the data augmentation scheduler in Appendix C due to the limited space.
Methodology
In this section, we first create the causal graph for ACL which is the fundamental premise for causal reasoning. Then, we introduce adversarial invariant regularization (AIR) according to the causal understanding in the context of ACL and the learning algorithm of our proposed method. Lastly, we demonstrate the theoretical understanding of AIR and theoretically show that the style-independence property is generalizable to downstream tasks.
Causal Graph of Adversarial Contrastive Learning (ACL)
A causal graph is arguably the fundamental premise for causal reasoning Pearl (2009);Peters et al. (2017). Previously, Mitrovic et al. (2021) investigated the causal graph in the case of standard contrastive learning (SCL) as shown in the left panel of Figure 1. However, how to build causal graphs in the context of ACL is still lacking in the literature. We follow the setting and assumptions of Mitrovic et al. (2021) for constructing the causal graph of ACL.
First, we formalize the data generation procedure in the causal graph. Let x ∈ X denote an unlabeled data point and Y = {y t } T t=1 denote a set of labels in an unknown downstream task. Then, following previous studies in causality for SAT Zhang et al. (2022b) and SCL Mitrovic et al. (2021), we make the following assumptions: 1) the data point x is generated from the content variable c and the style variable s, i.e., c → x ← s, 2) the content is independent of the style, i.e., c ⊥ ⊥ s, and 3) only the content, which is the original image data, is related to the downstream task, i.e., c → y t .
Then, we build the learning procedure of CL. According to the above assumption, content serves an essential role in downstream classification tasks. Therefore, CL can be regarded as estimating the probability of the label given the content, i.e, p(y t |c). To learn the representations from the unlabeled data x, CL aims to maximize the conditional probability p(y t |x).
However, in practice, the label y t is unknown. Thanks to the causal concept of refinement Chalupka et al. (2014); Mitrovic et al. (2021), we can construct a proxy label y R ∈ Y R which is a refinement of Y in order to learn representations from unlabeled data x. Intuitively, a refinement of a task can be viewed as a more fine-grained variant of the original problem. For example, a refinement for classifying cars against planes would be the task of classifying various types of cars and planes. We mathematically formulate the proxy label in Appendix B.1 that indicates the proxy label of each augmented data point points to its augmented alternative. Therefore, the goal of the proxy task used in SCL is to maximize the probability of the proxy label given the unlabeled natural data, i.e., p(y R |x) Mitrovic et al. (2021).
Different from SCL, ACL has an extra path x →x → y R since ACL needs to first generate the adversarial data (Eq. (3)) and then learns representations from the unlabeled natural and adversarial data (Eq. (2)). According to Eq. (3), we observe that ACL only uses the observed unlabeled data x for generating adversarial data, i.e., x →x ← θ. Then, the proxy task used in ACL learns representations by maximizing both the probability of the proxy label y R given the natural data (i.e., p(y R |x)) and that given adversarial data (i.e., p(y R |x)), i.e., x → y R andx → y R . We theoretically show that the proxy tasks driven by the data augmentations used in SCL Chen et al. (2020a) and ACL Jiang et al. (2020) coincide with their corresponding learning objectives in Appendix B.1.
Adversarial Invariant Regularization
According to the independence of causal mechanisms Peters et al. (2017), performing interventions on the style variable s should not change the probability of the proxy label given the unlabeled data, i.e., p(y R |x). In the context of ACL, according to the causal graph, we can obtain that
p(y R |x) = p(y R |x)p(x|x),(4)
under the mild assumption that the process of x →x → y R satisfies the Markov condition. Therefore, we should make the joint probability p(y R |x)p(x|x) that is learned via ACL become style-independent. It guides us to propose the Algorithm 1 ACL with Invariant Regularization (IR) 1: Input: Unlabeled training set U , total training epochs E, learning rate η, batch size β, hyperparameters λ 1 and λ 2 2: Output: Pre-trained encoder h θ 3: Initialize parameters of model f θ = g • h θ 4: for e = 0 to E − 1 do 5:
for batch m = 1, . . . , |U |/β do 6:
Sample a minibatch B m from U 7:
Update θ ← θ − η · ∇ θ x k ∈Bm ACL (x k ; θ) + λ 1 · L SIR (B m ; θ) + λ 2 · L AIR (B m ; θ) 8:
end for 9: end for following criterion that should be satisfied by ACL:
p do(τi) (y R |x)p do(τi) (x|x) = p do(τj ) (y R |x)p do(τj ) (x|x) ∀τ i , τ j ∈ T ,(5)
where do(τ ) as the short form of do(s = τ ) denotes that we perfomrn intersections on s via data augmentation function τ (·). If the learned representation satisfies Eq. (5), the multiplication of the prediction of adversarial data p(y R |x) and the consistency between adversarial and natural data p(x|x) will be invariant of the style.
To achieve the above criterion, we propose the adversarial invariant regularization (AIR) as follows,
L AIR (B; θ) = KL P (B,B i ) ⊗ P (B i , B i ) P (B,B j ) ⊗ P (B j , B j ) ,(6)
where
P k (B,B u ) = − log e sim(f θ (x k ),f θ (x u k ))/t x∈B e sim(f θ (x),f θ (x u ))/t ∀u ∈ {i, j}, k ∈ [β] P k (B u , B u ) = − log e sim(h θ (x u k ),f θ (x u k ))/t x∈B e sim(f θ (x u ),f θ (x u ))/t ∀u ∈ {i, j}, k ∈ [β] in which x ∈ B ∼ X β is the original image data, [β] = {1, .
. . , β} is a range list, ⊗ denotes the tensor product,B u denotes the adversarial counterpart of the augmented minibatch via τ u (·) for u ∈ {i, j}, t is the temperature parameter, and KL(· ·) denotes the Kullback-Leibler (KL) divergence. Note that in the context of SCL where x =x, the AIR will be transformed into exactly standard invariant regularization (SIR) Mitrovic et al. (2021) as follows,
L SIR (B; θ) = KL P (B, B i ) P (B, B j ) ,(7)
where
P k (B, B u ) = − log e sim(f θ (x k ),f θ (x u k ))/t x∈B e sim(f θ (x),f θ (x u ))/t ∀u ∈ {i, j}, k ∈ [β]
which aims to make the prediction of natural data independent of the style.
By incorporating ACL with invariant regularization (IR) which is the weighted sum of SIR and AIR, the learning objective function of our proposed method is shown below,
arg min θ k∈[N ] ACL (x k ; θ) + λ 1 · L SIR (U ; θ) + λ 2 · L AIR (U ; θ) invariant regularization ,(8)
where U ∼ X N refers to an unlabeled dataset consisting of N samples, λ 1 ≥ 0 and λ 2 ≥ 0 are two hyperparameters. The learning algorithm of ACL with IR is demonstrated in Algorithm 1. Note that our proposed IR is compatible with various learning objectives such as ACL Jiang et al. (2020)
Theoretical Analysis
First, we show that our proposed AIR implicitly aims to make both the prediction of adversarial data and the consistency between the prediction of adversarial and natural data independent of the style.
Decomposition of AIR. We decompose Eq. (6) into two terms as follows,
L AIR (B; θ) = k∈[β] P k (B,B i ) log P k (B,B i ) P k (B,B j ) invariant prediction ·P k (B i , B i ) + k∈[β] P k (B i , B i ) log P k (B i , B i ) P k (B j , B j ) invariant consistency ·P k (B,B i ).(9)
From Eq. (9), we can observe that AIR will implicitly encourage (1) the prediction of adversarial data to be invariant of the style (i.e., p do(τi) (y R |x) = p do(τj ) (y R |x)) by minimizing the term of invariant prediction which is a calibrated KL divergence between P (B,B i ) and P (B,B j ); (2) the consistency between the prediction of adversarial and natural data to be independent of the style (i.e., p do(τi) (x|x) = p do(τj ) (x|x)) by minimizing the term of invariant consistency which is a calibrated KL divergence between P (B i , B i ) and P (B j , B j ). Note that P k (B i , B i ) and P k (B,B i ) can be regarded as the calibration term that adjusts the confidence to the regularization. In Section 4.3, we empirically show that the calibration term is beneficial to enhancing the performance of ACL.
Next, we theoretically show that if Y R is a refinement of Y, then learning representations on the proxy task whose label set is Y R is a sufficient condition for the representation to be invariant of the style on the downstream task whose label set is Y in terms of the prediction of adversarial data and the consistency between adversarial and natural data in Theorem 1. Theorem 1. Let Y = {y t } T t=1 be a label set of a downstream classification task, Y R be a refinement of Y, andx t be the adversarial data generated in the downstream task. Assuming thatx ∈ B [x] andx t ∈ B [x], we have the following results:
p do(τi) (y R |x) = p do(τj ) (y R |x) =⇒ p do(τi) (y t |x t ) = p do(τj ) (y t |x t ) ∀τ i , τ j ∈ T , p do(τi) (x|x) = p do(τj ) (x|x) =⇒ p do(τi) (x t |x) = p do(τj ) (x t |x) ∀τ i , τ j ∈ T .
Remarks. The proof is shown in Appendix B.2. Theorem 1 indicates that the style-independence property of the representation learned on the proxy task will still hold on the downstream classification tasks, which provides the guarantees of generalization ability.
Experiments
In this section, we demonstrate the effectiveness of our proposed IR in improving ACL Yu et al. (2022). We adopted the same training configuration of ACL Jiang et al. (2020) using SGD for 1000 epochs with an initial learning rate of 5.0 and a cosine annealing schedule Loshchilov and Hutter (2016). The batch size β is fixed as 512. In the context of DynACL, we took the same data augmentation scheduler and the same scheduler of the hyperparameter ω as the setting in the original paper of DynACL . Note that, to reproduce the results of baseline methods, we downloaded the pre-trained weights of ACL Jiang et al. (2020) 2 on CIFAR-10/CIFAR-100 and DynACL 3 on CIFAR-10/CIFAR-100/STL10 from their official GitHub as the pre-trained encoder. In Appendix C.5, we provide the results of incorporating our proposed method with two other variants, i.e., AdvCL Fan et al. (2021) and A-InfoNCE Yu et al. (2022). In Appendix C.6, we demonstrate the applicability with different backbone networks including ResNet-34 and ResNet-50.
Finetuning procedures. We adopted the following three types of finetuning procedures: standard linear finetuning (SLF), adversarial linear finetuning (ALF), and adversarial full finetuning (AFF), to evaluate the learned representations. The former two finetuning procedures freeze the learned encoder and only train the linear classifier using natural or adversarial samples, respectively. We took the pre-trained encoder as weight initialization and train the whole model using the adversarial data during AFF. The training configuration of finetuning (e.g., the finetuning epoch and optimizer) exactly follows DynACL . Specifically, we used the official code provided in DynACL 's GitHub for finetuning and illustrate the experimental settings of finetuning in Appendix C as well. For each pre-trained encoder, we repeated the finetuning experiments 3 times and report the median results and the standard deviation.
Evaluation protocols. We let "AA" denote the robust test accuracy against AutoAttack (AA) Croce and Hein (2020a) and "SA" stand for the standard test accuracy. To evaluate the robustness against common corruption, we report the mean test accuracy under 15 types of common corruptions Hendrycks and Dietterich (2019) with the corruption severity (CS) ranging from 1 to 5 (denoted as "CS-{1, 2, 3, 4, 5}"). Specifically, we used the official Table 2: Robustness benchmark on CIFAR-10 task evaluated via SLF, ALF, and AFF.
Pre-training Table 3: Test accuracy (%) evaluated on CIFAR-10-C (corruption severity ranges from 1 to 5) of CIFAR-10 pre-trained models after SLF and AFF, respectively. Standard deviation is in Table 21.
λ 1 λ 2 SLF ALF AFF AA (%) SA (%) AA (%) SA (%) AA (%) SA (%) ACLPre-training λ 1 λ 2 SLF AFF CS-1 CS-2 CS-3 CS-4 CS-5 CS-1 CS-2 CS-3 CS-4 CS-5 ACL
Performance Evaluation Across Tasks
Performance evaluated on various datasets. Table 1 reports the performance evaluated on three tasks where pretraining and finetuning vis SLF are conducted on the same datasets including CIFAR-10, CIFAR-100, and STL10. First, we can observe that our proposed AIR (λ 1 = 0.0, λ 2 = 0.5) consistently yields more performance gain compared with SIR (λ 1 = 0.5, λ 2 = 0.0) Mitrovic et al. (2021), which validates that our proposed AIR is more effective in enhancing the performance of ACL and its variants than SIR Mitrovic et al. (2021). Further, IR (λ 1 = 0.5, λ 2 = 0.5) that incorporates SIR and AIR together significantly improves the generalization ability across tasks. Notably, IR enhances the standard test accuracy of ACL by 3.03% (from 78.27% to 81.30%) and that of DynACL by 3.59% (from 75.39% to 78.98%) on the CIFAR-10 task. Besides, we can notice that IR is also beneficial to enhancing robustness across tasks. Even compared with the previous SOTA method DynACL , DynACL with IR consistently achieves higher robust test accuracy, especially achieving a 1.17% (from 46.49% to 47.66%) robustness gain on the STL10 task.
Performance evaluated via various finetuning methods. Table 2 presents the performance evaluated on the CIFAR-10 task via SLF, ALF, and AFF, respectively.In Appendix C.1, we provide the results on CIFAR-100 in Table 12. Performance under common corruptions. Table 3 reports the test accuracy evaluated on CIFAR-10-C Hendrycks and Dietterich (2019) with various corruption severities of CIFAR-10 pre-trained models after SLF and AFF. In Table 6: Test accuracy (%) evaluated on CIFAR-100-C (corruption severity ranges from 1 to 5) of CIFAR-10 pre-trained models after SLF and AFF, respectively. Standard deviation is in Table 22.
Pre-training λ 1 λ 2 SLF AFF CS-1 CS-2 CS-3 CS-4 CS-5 CS-1 CS-2 CS-3 CS-4 CS-5 ACL 0 Appendix C.1, we provide the performance evaluated on CIFAR-100-C in Table 13. Note that the standard variance is provided in Appendix C.7 and the test accuracy under each type of common corruption in Appendix C.4. As you can see, IR leads to consistent and significant improvement in the robustness against common corruption. In addition, we observe that ACL always achieves much higher test accuracy under common corruptions than DynACL after SLF. We conjecture the reason is that DynACL uses weak data augmentations at the later phase of training, which makes DynACL more likely to overfit the training distribution of CIFAR-10 and thus leads to worse generalization ability to common corruptions via SLF. Table 4. The results solidly validate that IR (λ 1 = 0.5, λ 2 = 0.5) can consistently enhance both robust and standard test accuracy of ACL and its variant DynACL in semi-supervised settings.
Robustness Transferability Across Datasets
Robustness transferability against adversarial attacks. Table 5 shows the robustness transferability against adversarial attacks to CIFAR-100 from CIFAR-10. In Appendix C.2, We present the transferabilityto CIFAR-10 from CIFAR-100 in Table 14 and the transferability to STL10 Coates et al. (2011) from CIFAR-10 and CIFAR-100 in Table 15. We can observe that IR substantially improves both ACL's and DynACL's robustness transferability against adversarial attacks. Particularly, IR raises the standard and robust test accuracy of ACL via SLF by 6.84% (from 32.61% to 39.45%) and 1.06% (from 9.98% to 11.04%), respectively.
Robustness transferability against common corruptions. Table 6 reports the robustness transferability against common corruptions Hendrycks and Dietterich (2019) evaluated on CIFAR-100-C of CIFAR-10 pre-trained models. In Appendix C.2, We provide the results evaluated on CIFAR-10-C of CIFAR-100 pre-trained models in Table 16. The empirical results validate that our proposed IR can lead to significant improvement in the robustness transferability against common corruptions.
Ablation Study
Impact of the hyperparameters λ 1 and λ 2 . We show the performance achieved by ACL with IR of different λ 1 ∈ {0.00, 0.25, 0.50, 1.00} and λ 2 ∈ {0.00, 0.25, 0.50, 1.00} on the CIFAR-10 task in Table 7. We can notice that when leveraging AIR (λ 2 > 0), the standard test accuracy gains significant improvement, which indicates that AIR serves an important role in improving the generalization ability of ACL. Impact of calibration. In Section 3.3, we show that there exist calibration terms P k (B i , B i ) and P k (B,B i ) in AIR.
Here, we empirically investigate the impact of calibration on the performance in Table 8. The results empirically validate that the calibration term is an essential component of AIR to adjust the confidence of the regularization and further enhance the performance of ACL.
Conclusions and Limitations
This paper leveraged the technique of causal reasoning to enhance ACL. Based on causal understanding, we proposed adversarial invariant regularization (AIR) that enforces the learned robust representations to be style-independent. We improved ACL by incorporating AIR and SIR which is an extension of AIR in the standard context together with adversarial contrastive loss. Theoretically, we showed that AIR implicitly encourages the prediction of adversarial data and the consistency between adversarial and natural data to be invariant of the style factor, and the style-independence property of the learned representations will hold for the downstream classification tasks. Empirically, comprehensive results validate that our proposed method can significantly improve the generalization ability and robustness against adversarial attacks and common corruptions of ACL and its variants.
One of the limitations is that, similar to previous works ; Fan et al. (2021); Yu et al. (2022); , our proposed method IR is not helpful to improve the efficiency and scalability of ACL methods. Therefore, robust pre-training via ACL with IR is still unfriendly to the environment due to emitting much carbon dioxide and consuming much electricity. Besides, applying ACL with IR to large-scale datasets such as ImageNet-21K Ridnik et al. (2021) is still computationally prohibitive with limited GPU sources. Table Table 11: A notation table in convenience for viewing. Notation Description X
A Notation
input space x ∈ X data point T transformations set τ ∈ T data augmentation fraction x u augmented data point via the data augmentation fraction τ u (·) x adversarial data B ∼ X β a minibatch of β original image samples B u a minibatch of β augmented image samples via the data augmentation fraction τ u (·) B u a minibatch of adversarial counterparts of β augmented samples via the data augmentation fraction τ u (·) U ∼ X N an unlabeled data set of N image samples Y label space in the downstream tasks y t ∈ Y target label in the downstream tasks Y R the refinement of the target label space y R ∈ Y R the proxy label y R ku ∈ Y R the proxy label of the data point x u k augmented via τ u (·) Z projection space V feature space where the contrastive loss is applied h θ : X → Z the feature encoder parameterized by θ g :
Z → V the projector f (·) = g • h θ (·)
the composition of the encoder and the projector ∼ (·, ·) the cosine similarity function KL(· ·) KL divergence function s style variable c content variable λ 1 the weight of SIR λ 2 the weight of AIR B Theoretical Analysis
B.1 Connection Between the Proxy Classification Task and the Learning Objective of CL
Here, we provide a theoretical connection between the proxy classification task driven by data augmentations and the learning objective of SCL and ACL. We theoretically show that the learning objective of SCL and ACL can be regarded as maximizing the conditional probability p(y R |x) and p(y R |x), respectively.
Preliminaries (restated). Let h θ : X → Z be a feature encoder parameterized by θ, g : Z → V be a projection head that maps representations to the space where the contrastive loss is applied, and τ i , τ j : X → X be two transformation operations randomly sampled from a pre-defined transformation set T . Given a minibatch B ∼ X β consisting of β samples, we denote the augmented minibatch B = {τ i (x k ), τ j (x k ) | ∀x k ∈ B} consisting of 2β samples. We take f θ (·) = g • h θ (·) and x u k = τ u (x k ) for any x k ∼ X and u ∈ {i, j}.
Understanding of proxy label driven by the data augmentation. We denote the index of x i k and x j k in the minibatch B as ki and kj, respectively. For the augmented data x i k ∈ B and x j k ∈ B , we denote their corresponding proxy labels as y R ki and y R kj , which is formulated below,
y R ki = 1 kj ,(10)y R kj = 1 ki ,(11)
where 1 ku ∈ {0, 1} 2β−1 is a one-hot label where u ∈ {i, j}, the value at index ku is 1, and the values at other indexes are 0. It indicates that the proxy label of each augmented data points to its augmented alternative. Specifically, the conditional probability p(y R ki |x i k ; θ) and p(y R kj |x j k ; θ) is formulated as follows,
p(y R ki |x i k ; θ) = p(1 kj |x i k ; θ) = e sim(f θ (x i k ),f θ (x j k ))/t x∈B \{x i k } e sim(f θ (x i k ),f θ (x))/t ,(12)p(y R kj |x j k ; θ) = p(1 ki |x j k ; θ) = e sim(f θ (x j k ),f θ (x i k ))/t x∈B \{x j k } e sim(f θ (x j k ),f θ (x))/t ,(13)
where sim(p, q) = p q/ p q is the cosine similarity function and t > 0 is a temperature parameter.
Connection between the proxy classification task and the learning objective of SCL and ACL. The learning objective of standard contrastive loss CL (x i k , x j k ) can be regarded as maximizing the sum of the conditional probability p(y R ki |x i k ) and p(y R kj |x j k ), i.e.,
θ * SCL = arg min θ CL (x i k , x j k ; θ) (14) = arg min θ − log e sim(f θ (x i k ),f θ (x j k ))/t x∈B \{x i k } e sim(f θ (x i k ),f θ (x))/t − log e sim(f θ (x j k ),f θ (x i k ))/t x∈B \{x j k } e sim(f θ (x j k ),f θ (x))/t (15) = arg max θ log e sim(f θ (x i k ),f θ (x j k ))/t x∈B \{x i k } e sim(f θ (x i k ),f θ (x))/t + log e sim(f θ (x j k ),f θ (x i k ))/t x∈B \{x j k } e sim(f θ (x j k ),f θ (x))/t (16) = arg max θ e sim(f θ (x i k ),f θ (x j k ))/t x∈B \{x i k } e sim(f θ (x i k ),f θ (x))/t + e sim(f θ (x j k ),f θ (x i k ))/t x∈B \{x j k } e sim(f θ (x j k ),f θ (x))/t(17)
= arg max θ p(1 kj |x i k ; θ) + p(1 ki |x j k ; θ)
= arg max θ p(y R ki |x i k ; θ) + p(y R kj |x j k ; θ).
Therefore, we can understand that SCL actually uses a proxy classification task driven by data augmentations, which aims to maximize the conditional probability of the proxy label given the unlabeled data, i.e., p(y R |x).
In the adversarial context, we only need to replace the natural data x i k and x j k in Eq. (14) and (19) with the adversarial datax i k andx j k generated in Eq. (3). Therefore, we have the following results:
θ * ACL = arg min θ CL (x i k , x j k ; θ) + CL (x i k ,x j k ; θ) (20) = arg max θ p(y R ki |x i k ; θ) + p(y R kj |x j k ; θ) + p(y R ki |x i k ; θ) + p(y R kj |x j k ; θ),(21)
where the adversarial datax i k andx j k are generated in Eq. (3). Therefore, ACL can also be regarded as using the proxy classification task driven by data augmentations, which maximizes both the conditional probability of the proxy label given the natural data (i.e., p(y R |x)) and that given adversarial data (i.e., p(y R |x)).
In conclusion, we have theoretically shown that the proxy tasks driven by the data augmentations used in SCL Chen et al. (2020a) and ACL Jiang et al. (2020) coincide with the corresponding learning objectives.
B.2 Proof of Theorem 1
Theorem 1 (restated). Let Y = {y t } T t=1 be a label set of a downstream classification task, Y R be a refinement of Y, andx t be the adversarial data generated in the downstream task. Assuming thatx t ∈ B [x] andx ∈ B [x], we have the following results:
p do(τi) (y R |x) = p do(τj ) (y R |x) =⇒ p do(τi) (y t |x t ) = p do(τj ) (y t |x t ) ∀τ i , τ j ∈ T , p do(τi) (x|x) = p do(τj ) (x|x) =⇒ p do(τi) (x t |x) = p do(τj ) (x t |x) ∀τ i , τ j ∈ T .
Proof. The proof is inspired by Mitrovic et al. (2021).
For t ∈ {1, . . . , T }, we have
p do(τi) (y t |x t ) = p do(τi) (y t |y R )p do(τi) (y R |x t )dy R (22) = p(y t |y R )p do(τi) (y R |x t )dy R (23) = p(y t |y R )p do(τj ) (y R |x t )dy R (24) = p do(τj ) (y t |x t ).(25)
Eq. (22) holds due to the fact that y R is a refinement of y t according to the definition of the refinement shown in Mitrovic et al. (2021). Eq. (23) holds since the relationship between y t and y R is independent of the style, i.e., p do(τi) (y t |y R ) = p do(τj ) (y t |y R ). Eq. (24) holds because of the assumption that the prediction of adversarial data is independent of the style. Due to that the condition p do(τi) (y R |x) = p do(τj ) (y R |x) holds for anyx ∈ B [x], thereby, this condition also holds forx t ∈ B [x]. Lastly, we obtain that the prediction of adversarial data will be still independent of the style in downstream tasks.
Next, due to that the condition p do(τi) (x|x) = p do(τj ) (x|x) holds for anyx ∈ B [x], this condition will hold for x t ∈ B [x] as well, i.e., p do(τi) (x t |x) = p do(τj ) (x t |x). Therefore, the consistency between adversarial and natural data will be still invariant of the style in the downstream tasks. Table 13: Test accuracy (%) evaluated on CIFAR-100-C (corruption severity ranges from 1 to 5) of CIFAR-100 pre-trained models after SLF and AFF, respectively. Standard deviation is in Table 23.
Pre-training λ 1 λ 2 SLF AFF CS-1 CS-2 CS-3 CS-4 CS-5 CS-1 CS-2 CS-3 CS-4 CS-5 ACL 0
C Extensive Experimental Details and Results
Our code is released in the anonymized Github 4 .
Experimental environments. We conducted all experiments on Python 3.8.8 (PyTorch 1.13) with NVIDIA RTX A5000 GPUs (CUDA 11.6).
Pre-training details of ACL Jiang et al. (2020). Following , we leveraged ResNet-18 He et al. (2016) with the dual batch normalization (BN) Xie et al. (2020) as the encoder, where one BN is used for the standard branch of the feature extractor and the other BN is used for the adversarial branch, during conducting ACL and its variant DynACL . We pre-trained ResNet-18 models using SGD for 1000 epochs with an initial learning rate 5.0 and a cosine annealing schedule Loshchilov and Hutter (2016). During pre-training, we set the adversarial budget as 8/255, the hyperparameter ω as 0.0, and the strength of data augmentation as 1.0. As for the reproduction of the baseline, we used the pre-trained weights published in the GitHub of ACL 56 as the pre-trained encoder for finetuning.
Pre-training details of DynACL . The training configurations of DynACL followed ACL Jiang et al. (2020), except for the strength of data augmentation and the hyperparameter ω. We denote the strength of data augmentation and the hyperparameter at epoch e as α e and ω e respectively, where
α e = 1 − e K · K E , e ∈ {0, 1, . . . , E − 1} (26) ω e = µ · (1 − α e ), e ∈ {0, 1, . . . , E − 1}(27)
in which the decay period K = 50, the reweighting rate µ = 2/3, the total training epoch E = 1000. In our implementation of DynACL, we only take the dynamic strategy and do not take the trick of the stop gradient operation and the momentum encode Grill et al. (2020); . As for the reproduction of the baseline, we used the pre-trained weights published in the GitHub of DynACL 789 as the pre-trained encoder for finetuning.
Details of finetuning procedures. As for SLF and ALF, we fixed the parameters of the encoder and only finetuned the linear classifier using the natural training data and adversarial training data respectively. As for AFF, we finetuned Table 16: Test accuracy (%) evaluated on CIFAR-10-C (corruption severity ranges from 1 to 5) of CIFAR-100 pre-trained models after SLF and AFF, respectively. Standard deviation is in Table 24.
Pre-training λ 1 λ 2 SLF AFF CS-1 CS-2 CS-3 CS-4 CS-5 CS-1 CS-2 CS-3 CS-4 CS-5 ACL 0 all the parameters using the adversarial training data. For all finetuning procedures, we used SGD for 25 epochs with the initial learning rate as 0.01 and 0.1 for linear finetuning and full finetuning respectively and momentum as 2e − 4.
The adversarial budget is fixed as 8/255 for ALF and AFF. In practice, we used the finetuning code published in the GitHub of DynACL for implementing finetuning procedures.
C.1 Extra Results on Performance Evaluation Across Tasks
Performance evaluated on the CIFAR-100 task. We demonstrate the performance evaluated on the CIFAR-100 task, including the robustness benchmark on CIFAR-100 in Table 12 and the test accuracy under common corruptions on CIFAR-100-C in
C.2 Extra Results on Robustness Transferability Across Datasets
Robustness transferability to CIFAR-10 from CIFAR-100. Here, we show the robustness transferability from CIFAR-100 to CIFAR-10 against adversarial attacks in Table 14 and common corruptions in Table 16. We can observe that our proposed IR can enhance the robustness transferability against both adversarial attacks and common corruptions on various datasets. Notably, our proposed IR raises the standard test accuracy by 3.99% (from 60.90% to 64.89%) and the robust test accuracy by 1.18% (from 18.72% to 19.90%) of ACL using SLF. Meanwhile, IR improves the SOTA standard and robust test accuracy achieved via DynACL using AFF by 0.55% and 0.52%, respectively. Table 18: Test accuracy evaluated on CIFAR-10-C under each type of common corruptions with corruption severity being fixed as 5 of CIFAR-10 pre-trained models after SLF and AFF. 27 Robustness transferability to STL10 from CIFAR-10 and CIFAR-100, respectively. In addition, we demonstrate the robustness transferability to STL10 from CIFAR-10 and CIFAR-100 respectively in Table 15. We can observe that our proposed IR can significantly and consistently improve the robustness transferability of ACL and DynACL. In addition, notably, our proposed method IR, without using the ImageNet-pre-trained models, can yield better robustness transferability than AdvCL Fan et al. (2021) that uses pseudo labels generated by the ImageNet-pre-trained model. Particularly, DynACL with IR raises the robust and standard test accuracy by 1.20% (from 34.70% to 35.90%) and 2.53% (from 63.52% to 66.05%) of AdvCL Fan et al. (2021) from CIFAR-10 to STL10 via AFF.
All the experimental resutls validate that our proposed method can substantially and consistently enhance the performance of ACL in terms of robustness transferability.
C.3 Robustness Evaluation under Various Attacks
In this subsection, we conducted the robustness evaluation under three strong white-box attacks (APGD-CE Croce and Hein (2020a), APGD-DLR Croce and Hein (2020a) and FAB Croce and Hein (2020b)) and one strong black-box attack (i.e., Square Attack ). We evaluate the robustness on the CIFAR-10 task via SLF and report the results in Table 17. The results validate that our proposed method can consistently improve robust test accuracy over various adversaries.
C.4 Test Accuracy under Each Type of Common Corruption
We report the test accuracy under each type of common corruption Hendrycks and Dietterich (2019) (severity is fixed as 5) in Table 18. We used pre-trained models on CIFAR-10 via DynACL and DynACL with IR (λ 1 = 0.5, λ 2 = 0.5). Experimental settings are the same as Section 4. The results validate that IR can consistently improve the test accuracy under each type of common corruption.
C.5 Incorporating IR with Two Extra Variants of ACL
Here, we show extra results to validate that our proposed IR can enhance two extra variants of ACL, including AdvCL Fan et al. (2021) and AdvCL with A-InfoNCE Yu et al. (2022). Table 3) evaluated on CIFAR-10-C (corruption severity ranges from 1 to 5) of CIFAR-10 pre-trained models after SLF and AFF, respectively.
Pre-training λ 1 λ 2 SLF AFF CS-1 CS-2 CS-3 CS-4 CS-5 CS-1 CS-2 CS-3 CS-4 CS-5 ACL 0 Table 22: Standard deviation of test accuracy (corresponding to Table 6) evaluated on CIFAR-100-C (corruption severity ranges from 1 to 5) of CIFAR-10 pre-trained models after SLF and AFF, respectively.
Pre-training λ 1 λ 2 SLF AFF CS-1 CS-2 CS-3 CS-4 CS-5 CS-1 CS-2 CS-3 CS-4 CS-5 ACL 0 Table 13) evaluated on CIFAR-100-C (corruption severity ranges from 1 to 5) of CIFAR-100 pre-trained models after SLF and AFF, respectively.
Pre-training λ 1 λ 2 SLF AFF CS-1 CS-2 CS-3 CS-4 CS-5 CS-1 CS-2 CS-3 CS-4 CS-5 ACL 0 Table 16) evaluated on CIFAR-10-C (corruption severity ranges from 1 to 5) of CIFAR-100 pre-trained models after SLF and AFF, respectively.
Pre-training λ 1 λ 2 SLF AFF CS-1 CS-2 CS-3 CS-4 CS-5 CS-1 CS-2 CS-3 CS-4 CS-5 ACL 0 AdvCL Fan et al. (2021) leverages an extra contrastive view of high-frequency components and the pseudo labels generated by the clustering method. We leveraged our proposed method IR (λ 1 = 0.5, λ 2 = 0.5) to further improve the performance of AdvCL. Yu et al. (2022) proposed an asymmetric InfoNCE objective (A-InfoNCE) that treats adversaries as inferior positives that induce weaker learning signals, or as hard negatives exhibiting higher contrast to other negative samples.
C.6 Applicability with Different Backbone Networks
In this subsection, we demonstrate that IR can consistently improve the performance of DynACL on both ResNet-34 and ResNet-50 He et al. (2016). We followed the same training configurations of pre-training and finetuning (SLF) in Section 4 except for the backbone network. Table 20 validates that IR can further enhance the robust and standard test accuracy of DynACL on various backbone networks.
C.7 Standard Deviation of Test Accuracy under Common Corruptions
Here, we provide the standard deviation of the test accuracy under common corruptions in Tables 21, 22, 23, 24.
D Possible Negative Societal Impacts
Our proposed method aims to improve the performance of self-supervised robust pre-training methods ; Fan et al. (2021); Yu et al. (2022); . The self-supervised robust pre-training procedure is extremely time-consuming since it needs to spend much time generating adversarial training data, which thus leads to consuming a lot of electricity and emitting lots of carbon dioxide. Unfortunately, our proposed method is not helpful to improve the efficiency of ACL. Consequently, our study could exacerbate the greenhouse effect and be not conducive to environmental protection.
Figure 1 :
1Causal graph of standard contrastive learning Mitrovic et al. (2021) (left panel) and adversarial contrastive learning (right panel).
( 2020 )
2020; Ho and Nvasconcelos (2020); Fan et al. (2021); Yu et al. (2022); Zhang et al. (2022a); Luo et al. (2023). Causal reasoning. Causal reasoning Pearl (2009); Peters et al. (2017); Chalupka et al. (2014); Zhang et al. (2022b); Mitrovic et al. (2021); Bühlmann (2020); Zhang et al. (2020); Tang et al. (2020); Sauer and Geiger
, AdvCL Fan et al. (2021), A-InfoNCE Yu et al. (2022), and DynACL Luo et al. (2023).
and its variants Luo et al. (2023); Fan et al. (2021); Yu et al. (2022) on various datasets including CIFAR-10 Krizhevsky (2009), CIFAR-100 Krizhevsky (2009), STL10 Coates et al. (2011), CIFAR-10-C Hendrycks and Dietterich (2019), and CIFAR-100-C Hendrycks and Dietterich (2019). Extensive experimental details are shown in Appendix C. Pre-training. In the main paper, we demonstrate the results of applying our proposed method to ACL Jiang et al. (2020) and DynACL Luo et al. (2023). We utilized ResNet-18 He et al. (2016) as the encoder following previous self-supervised adversarial training methods Jiang et al. (2020); Fan et al. (2021); Luo et al. (2023);
Performance in semi-supervised settings. Following ACL Jiang et al. (2020) and DynACL Luo et al. (2023), we evaluate the performance in semi-supervised settings Alayrac et al. (2019) in
Performance evaluated via linear probing and then adversarially full finetuning (LP-AFT)Kumar et al. (2022);. adopted LP-AFTKumar et al. (2022) as a post-processing method after robust pre-training. DynACL++ denotes the results of DynACL as pre-training and post-processed via LP-AFT. We provide the comparison between DynACL++ and DynACL++ with IR (λ 1 = 0.5, λ 2 = 0.5) inTable 9and 10. Note that † denotes the results of DynACL++ are copied from. The results validate that IR can further improve the performance of DynACL with the trick of post-processing via LP-AFTKumar et al. (2022).
Efficient ACL. ACL requires tremendous running time since it needs to generate the adversarial training data during pre-training, which thus limits its scalability to large datasets. Therefore, prior studiesKim et al. (2020); Jiang et al. (2020); Ho and Nvasconcelos (2020); Fan et al. (2021); Yu et al. (2022); Zhang et al. (2022a);, showed that ACL can exhibit better robustness
against adversarial attacks Goodfellow et al. (2014); Croce and Hein (2020a) and common corruptions Hendrycks and
Dietterich (2019) on downstream tasks compared with SCL Chen et al. (2020a). AdvCL Fan et al. (2021) leverages
a third contrastive view for high-frequency components and pseudo labels generated by clustering to improve ACL.
Yu et al. (2022) proposed the asymmetric InfoNCE objective (A-InfoNCE) that utilizes the hard negative sampling
method Robinson et al. (2020) to further enhance AdvCL Fan et al. (2021). Recently, DynACL Luo et al. (2023)
dynamically schedules the strength of data augmentations and correspondingly schedules the weight for standard and
adversarial contrastive loss.
Table 1 :
1Robustness evaluations via SLF across various tasks. DynACL with SIR Mitrovic et al. (2021) 0.5 0.0 44.70±0.03 77.45±0.06 19.67±0.06 46.13±0.05 46.56±0.01 70.41±0.04Pre-training
λ 1 λ 2
CIFAR-10
CIFAR-100
STL10
AA (%)
SA (%)
AA (%)
SA (%)
AA (%)
SA (%)
ACL Jiang et al. (2020)
0.0 0.0 37.19±0.02 78.27±0.01 15.78±0.05 45.70±0.02 35.80±0.06 67.90±0.04
ACL with SIR Mitrovic et al. (2021)
0.5 0.0 37.51±0.04 78.97±0.01 15.76±0.06 47.16±0.06 36.36±0.03 68.09±0.04
ACL with AIR
0.0 0.5 37.70±0.02 80.86±0.05 16.03±0.12 49.60±0.05 36.86±0.06 68.61±0.02
ACL with IR
0.5 0.5 37.79±0.04 81.30±0.06 16.31±0.07 50.16±0.05 36.94±0.06 68.91±0.07
DynACL Fan et al. (2021)
0.0 0.0 45.05±0.02 75.39±0.05 19.31±0.06 45.67±0.04 46.49±0.05 69.59±0.08
DynACL with AIR
0.0 0.5 45.13±0.02 78.61±0.01 20.37±0.02 46.72±0.11 47.62±0.05 71.98±0.02
DynACL with IR
0.5 0.5 45.47±0.04 78.98±0.06 20.45±0.03 47.32±0.03 47.66±0.04 72.30±0.03
DynACL with SIR Mitrovic et al. (2021) 0.5 0.0 44.70±0.03 76.45±0.06 45.42±0.04 74.78±0.04 50.58±0.07 81.66±0.18Jiang et al. (2020)
0.0 0.0 37.62±0.01 78.27±0.01 40.61±0.03 75.56±0.05 49.42±0.07 82.14±0.18
ACL with SIR Mitrovic et al. (2021)
0.5 0.0 37.51±0.04 78.97±0.01 40.30±0.08 76.49±0.05 49.46±0.07 82.22±0.08
ACL with AIR
0.0 0.5 37.70±0.02 80.86±0.05 41.09±0.06 77.99±0.12 49.59±0.09 82.30±0.09
ACL with IR
0.5 0.5 37.79±0.04 81.30±0.06 41.39±0.03 78.29±0.03 49.62±0.04 82.42±0.06
DynACL Luo et al. (2023)
0.0 0.0 45.05±0.02 75.39±0.05 45.65±0.01 72.90±0.02 50.57±0.05 81.86±0.11
DynACL with AIR
0.0 0.5 45.13±0.02 78.61±0.01 46.12±0.02 77.01±0.01 50.66±0.05 82.62±0.10
DynACL with IR
0.5 0.5 45.47±0.04 78.98±0.06 46.14±0.02 77.42±0.03 50.68±0.08 82.87±0.14
Jiang et al. (2020) 0.0 0.0 76.57 74.73 71.78 67.75 62.78 79.15 76.01 72.54 69.47 65.27 ACL with SIR Mitrovic et al.(2021)
0.5 0.0 77.31 75.46 72.21 68.14 63.27 79.05 76.29 72.73 69.43 65.29
ACL with AIR
0.0 0.5 79.30 77.34 74.27 70.10 64.54 79.24 76.54 72.81 69.64 65.32
ACL with IR
0.5 0.5 79.55 77.67 74.33 70.12 64.78 79.49 76.86 72.95 69.73 65.37
DynACL Luo et al. (2023)
0.0 0.0 73.92 71.69 69.01 66.22 62.51 79.77 76.44 72.95 69.74 65.60
DynACL with SIR Mitrovic et al. (2021) 0.5 0.0 75.81 72.88 69.31 66.24 62.20 80.59 77.31 73.67 70.39 66.05
DynACL with AIR
0.0 0.5 76.83 73.96 70.47 67.19 63.13 80.93 77.71 74.11 70.81 66.58
DynACL with IR
0.5 0.5 77.12 74.12 70.66 67.37 63.29 80.98 77.87 74.31 70.96 66.75
code of AutoAttack Croce and Hein (2020a) and RobustBench Croce et al. (2020) for implementing evaluations. In
Appendix C.3, we provide robustness evaluation under more diverse attacks Croce and Hein (2020a;b); Andriushchenko
et al. (2020). In Appendix C.4, we provide the test accuracy under each type of common corruption Hendrycks and
Dietterich (2019).
Table 4 :
4Performance in semi-supervised settings evaluated on the CIFAR-10 task.Label ACL Jiang et al. (2020)
ACL with IR
DynACL Luo et al. (2023)
DynACL with IR
ratio
AA (%)
SA (%)
AA (%)
SA (%)
AA (%)
SA (%)
AA (%)
SA (%)
1%
45.63±0.13 76.04±0.05 45.87±0.19 77.35±0.16 45.44±0.33
76.89±0.41
45.75±0.08 78.57±0.37
10% 45.84±0.20 76.10±0.32 46.17±0.16 78.12±0.30 47.45±0.67
78.42±0.12
47.81±0.51 80.32±0.32
Table 5 :
5Robustness transferability to CIFAR-100 from CIFAR-10 evaluated via SLF, ALF, and AFF.Pre-training
λ 1 λ 2
SLF
ALF
AFF
AA (%)
SA (%)
AA (%)
SA (%)
AA (%)
SA (%)
ACL Jiang et al. (2020)
0.0 0.0 9.98±0.02 32.61±0.04 11.09±0.06 28.58±0.06 22.67±0.16 56.05±0.19
ACL with SIR Mitrovic et al. (2021)
0.5 0.0 10.77±0.04 35.63±0.18 11.26±0.02 32.37±0.04 22.87±0.09 55.77±0.13
ACL with AIR
0.0 0.5 10.82±0.07 39.23±0.06 13.12±0.05 35.98±0.08 22.99±0.07 56.05±0.12
ACL with IR
0.5 0.5 11.04±0.06 39.45±0.07 13.30±0.02 36.10±0.05 23.45±0.04 56.31±0.06
DynACL Luo et al. (2023)
0.0 0.0 11.01±0.02 27.66±0.03 11.92±0.05 24.14±0.09 24.17±0.10 55.61±0.17
DynACL with SIR Mitrovic et al. (2021) 0.5 0.0 11.32±0.03 27.92±0.04 11.65±0.02 25.03±0.04 24.25±0.01 56.01±0.03
DynACL with AIR
0.0 0.5 11.82±0.03 31.32±0.08 12.63±0.04 28.60±0.06 24.56±0.08 56.82±0.11
DynACL with IR
0.5 0.5 12.20±0.04 31.33±0.03 12.70±0.03 28.70±0.05 24.82±0.07 57.00±0.13
Table 7 :
7The robust/standard test accuracy (%) achieved by ACL with IR with different λ 1 and λ 2 .λ 1
λ 2
0.00
0.25
0.50
1.00
0.00
37.19/78.27 37.81/80.18 37.70/80.86 37.76/81.10
0.25
37.55/78.53 37.70/80.53 37.69/81.23 37.63/81.16
0.50
37.51/78.97 37.63/80.32 37.79/81.30 37.39/81.28
1.00
37.04/78.47 37.24/80.14 37.67/80.39 37.71/81.29
Table 8 :
8Impact of calibration terms evaluated on the CIFAR-10 task.Pre-training
Calibration
SLF
ALF
AFF
AA (%) SA (%) AA (%) SA (%) AA (%) SA (%)
ACL with AIR
×
37.62
80.30
40.60
77.57
49.11
81.95
ACL with AIR
√
37.70
80.86
40.77
77.99
49.39
82.22
DynACL with AIR
×
44.68
77.79
46.01
76.12
50.44
82.35
DynACL with AIR
√
45.13
78.61
46.12
77.01
50.58
82.62
Table 9 :
9Robustness evaluation across tasks with post-processing via LP-AFT Kumar et al. (2022); Luo et al. (2023).Pre-training
CIFAR-10
CIFAR-100
STL10
AA (%) SA (%) AA (%) SA (%) AA (%) SA (%)
DynACL++ † Luo et al. (2023)
46.46
79.81
20.05
52.26
47.21
70.93
DynACL++ with IR
46.68
80.12
21.46
53.37
47.74
72.24
Table 10 :
10Robustness evaluation on the CIFAR-10 task with post-processing via LP-AFT Kumar et al. (2022); Luo et al.(2023).
Pre-training
SLF
ALF
AFF
AA (%) SA (%) AA (%) SA (%) AA (%) SA (%)
DynACL++ † Luo et al. (2023)
46.46
79.81
47.95
78.84
50.31
81.94
DynACL++ with IR
46.68
80.12
48.36
79.46
50.73
82.88
Pre-training time. In practice, we evaluate the total running time using one NVIDIA RTX A5000 GPU on the
CIFAR-10 task. ACL, ACL with IR, DynACL, and DynACL with IR consumed 42.8, 43.1, 42.9, and 43.1 hours,
respectively. Our proposed method does not significantly induce extra computational burdens since it only needs to add
two extra loss functions (i.e., SIR and AIR) into the previous learning objective of ACL Jiang et al. (2020); Luo et al.
(2023).
Table 12 :
12Robustness benchmark on CIFAR-100 task evaluated via SLF, ALF, and AFF.Pre-training
λ 1 λ 2
SLF
ALF
AFF
AA (%)
SA (%)
AA (%)
SA (%)
AA (%)
SA (%)
ACL Jiang et al. (2020)
0.0 0.0 15.78±0.05 45.70±0.02 17.36±0.06 42.69±0.03 24.16±0.39 56.68±0.14
ACL with SIR Mitrovic et al. (2021)
0.5 0.0 15.76±0.06 47.16±0.06 17.68±0.03 43.55±0.06 24.10±0.12 56.87±0.09
ACL with AIR
0.0 0.5 16.79±0.12 49.60±0.05 18.37±0.09 45.39±0.12 24.24±0.09 57.03±0.21
ACL with IR
0.5 0.5 16.31±0.07 50.16±0.05 18.42±0.03 46.12±0.07 24.42±0.10 57.09±0.28
DynACL Luo et al. (2023)
0.0 0.0 19.31±0.06 45.67±0.04 17.91±0.01 20.30±0.06 40.81±0.23 57.22±0.08
DynACL with SIR Mitrovic et al. (2021) 0.5 0.0 19.67±0.06 46.13±0.05 20.21±0.04 41.09±0.10 25.08±0.24 57.44±0.16
DynACL with AIR
0.0 0.5 20.37±0.02 46.72±0.11 20.97±0.03 36.37±0.02 43.34±0.08 58.04±0.07
DynACL with IR
0.5 0.5 20.45±0.03 47.32±0.03 21.23±0.04 43.63±0.08 25.72±0.04 58.10±0.14
Table 14 :
14Robustness transferability to CIFAR-10 from CIFAR-100 evaluated via SLF, ALF, and AFF.Pre-training
λ 1 λ 2
SLF
ALF
AFF
AA (%)
SA (%)
AA (%)
SA (%)
AA (%)
SA (%)
ACL Jiang et al. (2020)
0.0 0.0 18.72±0.07 60.90±0.02 26.92±0.11 57.35±0.07 44.07±0.11 75.19±0.10
ACL with SIR Mitrovic et al. (2021)
0.5 0.0 19.14±0.02 61.56±0.20 26.97±0.03 58.27±0.08 44.21±0.08 75.23±0.09
ACL with AIR
0.0 0.5 19.71±0.09 64.64±0.15 27.07±0.09 60.10±0.04 44.50±0.04 75.57±0.11
ACL with IR
0.5 0.5 19.90±0.04 64.89±0.04 27.65±0.06 60.79±0.04 44.84±0.14 75.67±0.13
DynACL Luo et al. (2023)
0.0 0.0 25.23±0.12 59.12±0.10 28.92±0.33 56.09±0.04 47.40±0.23 77.92±0.18
DynACL with SIR Mitrovic et al. (2021) 0.5 0.0 25.27±0.06 59.04±0.02 28.97±0.05 56.03±0.03 47.58±0.24 77.76±0.12
DynACL with AIR
0.0 0.5 25.56±0.05 59.19±0.44 29.27±0.06 56.44±0.08 47.87±0.08 78.12±0.07
DynACL with IR
0.5 0.5 25.63±0.07 59.83±0.08 29.32±0.06 56.65±0.06 47.92±0.03 78.44±0.10
Table 15 :
15Robustness transferability to STL10 from CIFAR-10 and CIFAR-100, respectively. "AdvCL(+ImageNet)"
denotes that AdvCL Fan et al. (2021) incorporates extra data since it utilizes an ImageNet-pre-trained model to generate
pseudo labels. ‡ denotes that the results are copied from AdvCL Fan et al. (2021).
Source
Pre-training
SLF
ALF
AFF
domain
AA (%)
SA (%)
AA (%)
SA (%)
AA (%)
SA (%)
CIFAR-10
ACL Jiang et al. (2020)
25.41±0.08 56.53±0.02 27.17±0.09 51.71±0.17 33.06±0.07 62.17±0.13
ACL with IR
28.00±0.04 61.79±0.03 29.94±0.10 57.75±0.11 33.67±0.09 63.43±0.10
DynACL Luo et al. (2023)
28.12±0.09 52.45±0.10 28.43±0.13 49.53±0.17 35.25±0.15 65.29±0.18
DynACL with IR
29.78±0.04 55.89±0.12 29.32±0.06 52.21±0.09 35.90±0.05 66.05±0.12
AdvCL(+ImageNet) ‡ Fan et al. (2021)
25.74
63.73
N/A
N/A
34.70
63.52
CIFAR-100
ACL Jiang et al. (2020)
21.77±0.07 47.69±0.05 24.46±0.09 45.40±0.12 28.38±0.07 56.16±0.13
ACL with IR
22.44±0.04 51.52±0.02 25.57±0.06 48.96±0.09 30.14±0.08 57.95±0.11
DynACL Luo et al. (2023)
23.59±0.09 45.38±0.14 26.56±0.13 44.80±0.14 30.47 ±0.14 58.76±0.18
DynACL with IR
23.75±0.07 46.20±0.08 26.98±0.05 45.46±0.12 31.42±0.07 58.91±0.06
AdvCL(+ImageNet) ‡ Fan et al. (2021)
20.86
50.71
N/A
N/A
30.51
61.56
Table 13 .
13All the results validate that our proposed IR can significantly improve the performance of ACL Jiang et al. (2020) and its variant DynACL Luo et al. (2023) on various tasks.
Table 17 :
17Robustness evaluation on the CIFAR-10 task via SLF under various attacks including APGD-CE Croce and Hein (2020a), APGD-DLRCroce and Hein (2020a),FAB Croce and Hein (2020b), and Square Attack.Pre-training
APGD-CE (%) APGD-DLR (%) FAB (%) Square Attack (%)
ACL Jiang et al. (2020)
38.66
40.18
38.27
47.98
ACL with IR
39.39
40.98
39.01
48.80
DynACL Luo et al. (2023)
46.28
46.41
45.56
50.06
DynACL with IR
47.05
47.29
46.02
50.58
Table 19 :
19Applicability with two extra variants including AdvCL Fan et al. (2021) and A-InfoNCE Yu et al. (2022).Pre-training
AA (%) SA (%)
AdvCL Fan et al. (2021)
42.58
80.78
AdvCL with IR
43.24
81.53
A-InfoNCE Yu et al. (2022)
42.68
83.18
A-InfoNCE with IR
42.84
83.99
Table 20 :
20Applicability with various backbone networks including ResNet-34 and ResNet-50.Network
Pre-training
AA (%) SA (%)
ResNet-34
DynACL Luo et al. (2023)
47.01
78.84
DynACL with IR
47.56
80.67
ResNet-50
DynACL Luo et al. (2023)
47.19
79.65
DynACL with IR
47.82
81.
Table 21 :
21Standard deviation of test accuracy (corresponding to
Table 23 :
23Standard deviation of test accuracy (corresponding to
Table 24 :
24Standard deviation of test accuracy (corresponding to
GitHub link of ACL: https://github.com/VITA-Group/Adversarial-Contrastive-Learning. 3 GitHub link of DynACL: https://github.com/PKU-ML/DYNACL.
Anonymized Github link is https://anonymous.4open.science/r/Enhanced_ACL_via_AIR-Anonymized. 5 Link of pre-trained weights via ACL on CIFAR-10.6 Link of pre-trained weights via ACL on CIFAR-100. 7 Link of pre-trained weights via DynACL on CIFAR-10. 8 Link of pre-trained weights via DynACL on CIFAR-100. 9 Link of pre-trained weights via DynACL on STL10.
GitHub provided byYu et al. (2022).
We used the code provide byYu et al. (2022)in their GitHub 10 to implement the pre-training of AdvCL as well as A-InfoNCE, and the finetuning procedure. InTable 19, we show that our proposed IR (λ 1 = 0.5, λ 2 = 0.5) can consistently improve the robust and standard test accuracy of two extra variantsFan et al. (2021);Yu et al. (2022)of ACL.
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Pre-training Finetuning Noise Blur Weather Digital Gaussian Shot Impulse Defocus Glass Motion Zoom Snow Frost Fog Bright Contrast Elastic Pixel. Jpeg Dynacl Luo, SLFPre-training Finetuning Noise Blur Weather Digital Gaussian Shot Impulse Defocus Glass Motion Zoom Snow Frost Fog Bright Contrast Elastic Pixel JPEG DynACL Luo et al. (2023) SLF
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"https://github.com/VITA-Group/Adversarial-Contrastive-Learning.",
"https://github.com/PKU-ML/DYNACL."
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[
"Towards Efficient and Stable K-Asynchronous Federated Learning with Unbounded Stale Gradients on Non-IID Data",
"Towards Efficient and Stable K-Asynchronous Federated Learning with Unbounded Stale Gradients on Non-IID Data"
] | [
"Zihao Zhou ",
"Yanan Li ",
"Xuebin Ren ",
"Shusen Yang "
] | [] | [] | Federated learning (FL) is an emerging privacy-preserving paradigm that enables multiple participants collaboratively to train a global model without uploading raw data. Considering heterogeneous computing and communication capabilities of different participants, asynchronous FL can avoid the stragglers effect in synchronous FL and adapts to scenarios with vast participants. Both staleness and non-IID data in asynchronous FL would reduce the model utility. However, there exists an inherent contradiction between the solutions to the two problems. That is, mitigating the staleness requires to select less but consistent gradients while coping with non-IID data demands more comprehensive gradients. To address the dilemma, this paper proposes a two-stage weighted K asynchronous FL with adaptive learning rate (WKAFL). By selecting consistent gradients and adjusting learning rate adaptively, WKAFL utilizes stale gradients and mitigates the impact of non-IID data, which can achieve multifaceted enhancement in training speed, prediction accuracy and training stability. We also present the convergence analysis for WKAFL under the assumption of unbounded staleness to understand the impact of staleness and non-IID data. Experiments implemented on both benchmark and synthetic FL datasets show that WKAFL has better overall performance compared to existing algorithms. | 10.1109/tpds.2022.3150579 | [
"https://arxiv.org/pdf/2203.01214v1.pdf"
] | 246,803,466 | 2203.01214 | 3e97eeb347259968885979888fdbe7624d5613ac |
Towards Efficient and Stable K-Asynchronous Federated Learning with Unbounded Stale Gradients on Non-IID Data
Zihao Zhou
Yanan Li
Xuebin Ren
Shusen Yang
Towards Efficient and Stable K-Asynchronous Federated Learning with Unbounded Stale Gradients on Non-IID Data
1Index Terms-Federated learningasynchronous learningdata heterogeneityprediction accuracytraining stability !
Federated learning (FL) is an emerging privacy-preserving paradigm that enables multiple participants collaboratively to train a global model without uploading raw data. Considering heterogeneous computing and communication capabilities of different participants, asynchronous FL can avoid the stragglers effect in synchronous FL and adapts to scenarios with vast participants. Both staleness and non-IID data in asynchronous FL would reduce the model utility. However, there exists an inherent contradiction between the solutions to the two problems. That is, mitigating the staleness requires to select less but consistent gradients while coping with non-IID data demands more comprehensive gradients. To address the dilemma, this paper proposes a two-stage weighted K asynchronous FL with adaptive learning rate (WKAFL). By selecting consistent gradients and adjusting learning rate adaptively, WKAFL utilizes stale gradients and mitigates the impact of non-IID data, which can achieve multifaceted enhancement in training speed, prediction accuracy and training stability. We also present the convergence analysis for WKAFL under the assumption of unbounded staleness to understand the impact of staleness and non-IID data. Experiments implemented on both benchmark and synthetic FL datasets show that WKAFL has better overall performance compared to existing algorithms.
INTRODUCTION
T HE availability of massive data has been the bottleneck of many machine learning (ML) algorithms, especially in many deep learning (DL) scenarios, such as video surveillance [1], and speech recognition [2]. In the 5G era, more and more data are being generated and scattered at massive users' devices (i.e., clients) locally. The aggregation of these distributed data promises a significant boost in the quality of ML models, prospering various artificial intelligence services. However, with the increasing awareness of data privacy, people are unwilling to share their own data, which would impede the DL development [3] (e.g., especially for data sensitive fields like autonomous driving [4] and disease detection [5]). To address this issue, a novel distributed learning paradigm, federated learning (FL) [6], [7], [8], [9] has been proposed to achieve data utilization from massive clients while not seeing their local data.
Generally, FL [10], [11] enables multiple clients with decentralized data to collaboratively train a shared model under the concerted control of a center server (e.g. service provider), which complies with privacy regulations such as GDPR [12]. By providing an effective framework to exploit the naturally generated data while respecting privacy, FL has not only attracted extensive interests from the academia [9], but also been deployed in various practical applications, such as e-health diagnosis [13], [14], fraud detection [15], and recommendation systems [16], [17].
• Z. Zhou, Y. Li Many algorithms have been proposed for FL [18], [19], [20], among which, FedAvg is one of the most classical FL algorithms and runs in a synchronous manner [6]. At each iteration, the clients complete several local training processes and upload the model parameter. The server randomly samples a portion of clients and aggregates their locally updated models. In such a case, the runtime of each FL iteration is determined by the stragglers [21], [22] who are the sampled slowest clients. To alleviate the stragglers effect, the server can update the global model once receiving the fastest K gradients out of total clients, namely, K-sync FL [23] or K-batch-sync FL [24]. However, these approaches can not eradicate stragglers due to the unpredictable statuses of sampled clients and network communication.
Besides the above synchronous methods, the K asynchronous FL (K-async FL) has also been extensively studied [25], [26], [27], [28], in which the server updates the global model once receiving the gradients from the first K clients and those clients who fail to participate in current iteration can continue their training for reducing the runtime in the next iteration. K-async FL can not only alleviate stragglers, but also save the total training time when the iteration time of clients follows a new-longer-than-used distribution [23]. Apart from that, Hannah et al. [29] has further pointed out that asynchronous FL (AFL) allows more iterations within the same time compared to synchronous FL. Therefore, compared to K-sync FL, K-async FL has demonstrated great benefits of achieving more efficient learning in highly heterogeneous systems. In this paper, we focus on achieving both efficient and high-utility K-async FL.
Though K-async FL can mitigate the stragglers effect and save total training time [23], there are still two obstacles to be overcome in practice. On the one hand, non-IID datasets generated at different FL clients can impact the model utility [30], [31]. On the other hand, stale gradients may harm the model utility, or even diverge the training process [32], [33]. The solutions to these two problems have been studied separately. For non-IID data, the essence of the existing solutions, e.g., momentum [31], [34] and variance reduction [35], is to fully utilize all available information for estimating the global data distribution. Hence, gradients from as many clients as possible need to be aggregated to make the aggregated gradients represent the overall data comprehensively. For staleness, most studies pointed out that the server should aggregate the received gradients [36], [25] or adjust the learning rate [33], [37] negatively correlated with the staleness. Therefore, only a few lowstale gradients are maintained and most of the high-stale gradients will be filtered out. Apparently, there will pose an essential contradiction when simply combining the existing methods to mitigate the impact of non-IID data and stale gradients. To better help comprehend the contradiction, we explore the interplay of mitigation strategies for non-IIDness and staleness on EMNIST MNIST [38] in Section 4.1. The experimental results show that the mitigation effect for non-IIDness is negatively correlated with the mitigation effect for staleness, which indicates their inherent contradiction. Therefore, it is of significance to design a novel asynchronous FL approach that can effectively deal with both stale gradients and non-IID data.
Though stale gradients may cause the model to diverge, Mitliagkas et al. [39] found that stale gradients can help converge faster and they leveraged staleness to accelerate the training process. However, not all gradients can speedup convergence. In fact, low-stale gradients are more likely to have a consistent descent direction and accelerate the training process. On the contrary, high-stale gradients may have different directions and the biased gradients will prevent model convergence. This motivates us to pick the consistent gradients, instead of only utilizing the low-stale gradients. Therefore, based on the potential advantage of consistent gradients and treatments of non-IID data, we propose a two-stage weighted K-async FL algorithm (WKAFL) with adjusting the learning rate to improve the prediction accuracy, training speed and stability. Our main contributions are as follows: 1) We propose WKAFL, a novel K-async FL algorithm with two stages, which improves the model utility of asynchronous algorithm and shows good robustness in non-IID settings. In stage one, stale gradients with consistent descent direction will be picked to accelerate the training process. In stage two, stale gradients with large norm will be clipped to stabilize the model. In both stages, the server adjusts the learning rate according to the least staleness of K gradients and accumulates selected consistent gradients to further mitigate the non-IIDness. 2) We present the convergence analysis for the nonconvex optimization problem in AFL to analyze the impact of staleness and non-IID data, under both the assumptions of bounded and unbounded staleness. The rest of the paper is organized as follows. Section 2 describes the related work on both staleness and non-IID data in AFL. Section 3 presents the system model and problem definition. Section 4 describes the details of our proposed algorithm WKAFL. The corresponding theoretical analysis is presented in Section 5. Section 6 shows the experiment results for validating the performance of WKAFL. Finally, we conclude the paper in Section 7.
RELATED WORK
In this section, we introduce the related work of FL in terms of non-IID data and staleness.
Non-IID Data
Non-IID data is an essential characteristic of FL. However, since most of the well-established statistical theories are based on the IID data assumption, there are only a few analytical results about convergence under the non-IID setting. Some empirical results have shown the negative influence of non-IID data on the model utility. For example, Zhao et al. [30] empirically showed that the accuracy of FedAvg decreases significantly with the increase of data heterogeneity. Besides, Li et al. [40] proved that non-IID data can decrease the model utility. They assumed that there is a bounded difference between the gradients uploaded by clients and global unbiased gradients to guarantee the convergence, similar to [19], [41], [42]. In this paper, we follow this assumption and provide a convergence analysis under the non-IID setting. To mitigating the impact of non-IID data, a number of methods have been proposed to utilize all available information to estimate the global knowledge of the decentralized non-IID datasets. For example, in [30], the server broadcasts some non-private data to all the clients to make the gradients to carry more common information. Apart from data sharing, each received gradient can also accumulate historical information by applying momentum [31], [34] or variance reduction [35] to ensure the aggregated gradient to be more representative of the global information. Li et al. [31] proposed GSGM which divides the whole training process into multiple rounds. In each round, the server eliminates the diversity of aggregated gradients by using a global momentum, which is accumulated by part of the gradients based on the scheduling strategy.
Since momentum can also help converge faster, WKAFL in this paper also takes advantages of the momentum to alleviate the impact of non-IID data as GSGM. However, WKAFL is essentially different from GSGM. Since GSGM needs all the clients to upload gradients, GSGM has a low scalability due to computational overhead in each round, while WKAFL adapts K-async FL and behaves robustly.
Staleness
Staleness is one of the conspicuous challenges in AFL. Dai et al. [32] found that stale gradients will slow down the training process and decease prediction accuracy. To guarantee model utility, most of existing methods make full use of lowstale gradients and prevent stale gradients to bring down the model utility. One effective path is to adjust the learning rate [33], [43]. In [33], Zhang et al. tuned the learning rate on a per-gradient basis inversely proportional to the staleness. However, when there are massive heterogeneous clients, the staleness may be very high, which leads to very small learning rate and extremely slow training. In [43], McMahan et al. extended the adaptive gradient methods [44], [45] to the asynchronous parallel setting and revised the learning rate based on previous gradient steps. Another feasible method is to aggregate the gradients with different weights negatively correlated with the staleness. Xie et al. [25] and Chen et al. [36] aggregated the stale gradients based on exponential weights (e/2) −τ and e τ respectively, where τ represents the staleness.
To alleviate the effect of staleness, WKAFL adopts the exponential weighting method, which is the same as [36] for the reason of ideal performance in asynchronous scenarios. However, all above algorithms either provide no theoretical analysis, or need the assumption of bounded staleness [46], [47] which has less practicality in many AFL scenarios [48]. In this paper, WKAFL can provide a convergence analysis under the assumption of unbounded staleness.
Interplay of Non-IID Data and Staleness
To better alleviate the impact of non-IID data, the server requires a greater number of gradients. However, to prevent the stale gradients from bringing down the model utility, only confined low-stale gradients are favorable in the aggregation. Therefore, a contradiction about the quantity of aggregated gradients occurs when simply combining the corresponding mitigation methods concerned about non-IID data and staleness. Facing the same contradiction, the proposed WKAFL aims to exploit the idea of gradient selection to mitigate it. The idea of gradient selection has been explored in [49], which proposes to pick high-quality gradients for tackling with Byzantine attacks. Different from WKAFL, ZENO++ in [49] requires the central server to have some high-quality data and relies on the assumption of IID data, which is impractical in the federated learning scenario. Particularly, with non-IID datasets, it has much poorer performance than WKAF, as shown in Appendix A.
Despite the adverse impact of staleness on model utility, stale gradients can play a positive role in accelerating the training process. In [39], Mitliagkas et al. demonstrated that the stale gradients could have the same effect as momentum when the staleness follows a geometric distribution. In fact, consistent stale gradients can boost the convergence, analogous to the function of a large momentum [39], [28]. However, when the model is close to the optimal solution, the momentum may fluctuate the training process. To prevent this fluctuation, WKAFL uses the two-stage strategy as [50] to fully utilize the stale gradients. In [50], the two-stage strategy is used to control discrepancies between the global model and stale models, which is different from the aim of utilization of stale gradients in WKAFL.
PRELIMINARY
Stochastic Optimization
Generally, the aim of a machine learning problem is to minimize the empirical risk function F (w):
min w F (w) = 1 N ξi∈D f (w, ξ i ),
where D = {ξ 1 , ξ 2 , · · · , ξ N } ⊆ R n is the training dataset and f (w, ξ i ) defines the composite loss function at the i-th data point. In mini-batch stochastic gradient descent (SGD), we can minimize the objective function F (w) through iteratively updating the model parameter vector w using:
w j+1 = w j − η j |D j | ξi∈Dj ∇f (w j , ξ i ),(1)
where D j ⊆ D and η j define the mini-batch data and the learning rate at the j-th iteration respectively.
Problem Framework
We consider the K-async FL framework with a central server and P clients. At the beginning, the server initializes the model parameter vector as w 0 . Then, all the P clients fetch current parameter vector w 0 and compute their respective gradients on a single mini-batch data independently. Clients who finish computing gradients upload their gradients without waiting for other clients. The server waits for the first K out of P clients and the rest of clients continue computing gradients. As a result, at every iteration, the gradients received by the server might be computed based on stale parameters. Then, for the K-async FL, the updating formula is expressed by
w j+1 = w j − η j K K i=1 g(w j,i , ξ j,i ),(2)
where g(w j,i , ξ j,i ) is the stale gradient received by the server at the j-th iteration, ξ j,i = {ξ (1) j,i , · · · , ξ (m) j,i } denotes the used m samples of client i at the j-th iteration and w j,i defines the model parameter vector used to compute the stale gradient g(w j,i , ξ j,i ) with staleness τ j,i . Fig.1 illustrates the two-async FL when the total number of clients is four. At each iteration, four clients independently upload gradients to the server after finishing computing gradients. The server will update the global model once receiving any two gradients (green and shadow arrows), and clients who fail to upload gradients in limited time continue computing gradients based on a local mini-batch (red arrows). After updating the global model, the server broadcasts the new model parameters to the clients who upload their gradients.
Considering the different levels of clients' staleness and the impact of non-IID data, it is unreasonable to use the same weight 1/K to aggregate the received gradients in Equation (2). Therefore, in this paper, we will use the weighted K-async FL which is expressed as
w j+1 = w j − η j K i=1 p j,i g(w j,i , ξ j,i ),(3)
where p j,i defines the weight of the i-th client's gradient at the j-th iteration and K i p j,i = 1. If p j,i = 1/K, Equation (3) is back to Equation (2). Our main purpose is showing how to determine p j,i based on the staleness and non-IID data to improve both the model utility.
Definitions and Assumptions
To determine p j,i , we firstly define what kind of stale gradients g(w j,i , ξ j,i ), i = 1, · · · , K are "consistent" or "inconsistent", based on the cosine similarity comparison between g(w j,i , ξ j,i ) and the globally unbiased gradient ∇F (w j ). Definition 3.1 (Inconsistent gradient). If the cosine similarity of a gradient and the globally unbiased gradient is smaller than a given constant sim min ∈ [0, 1], then the gradient is an inconsistent gradient.
Definition 3.2 (Consistent gradient).
If the cosine similarity of a gradient and the globally unbiased gradient is larger than a given constant sim min ∈ [0, 1], then the gradient is a consistent gradient.
Besides, some basic assumptions for convergence analysis are listed as follows.
Assumption 3.1 (Lipschitz Continuity). Objective function F (w) satisfies L-Lipschitz continuity: ∀w 1 , w 2 , ∃ constant L, F (w 1 ) − F (w 2 ) ≤ ∇F (w 1 ) T (w 2 − w 1 ) + L 2 ||w 2 − w 1 || 2 2 .
Assumption 3.2 (Client-Level Unbiased Gradient). The gradient g(w j , ξ j,i ) of client i is a client-level unbiased gradient which means that the expectation of gradient g(w j , ξ j,i ) is equal to ∇F i (w j ): Total number of iteration w * Global optimal solution w j Global model at the j-th iteration τmax
E[g(w j , ξ j,i )] = ∇F i (w j ).
The maximal staleness of all the gradients in stage one τ j,i
The staleness of the i-th gradients at j-th iteration p j,i
Weight of the i-th gradient at j-th iteration Assumption 3.3 (Gradients with Bounded Variance). The gradient g(w j , ξ j,i ) of client i has client-level bounded variance:
∃ constants σ c , M c , E[||g(w j , ξ j,i ) − ∇F i (w j )|| 2 2 ] ≤ σ 2 c m + M c m ||∇F i (w j )|| 2 2 .
where ∇F i (w j ) is the unbiased gradient of client i. To guarantee the convergence of the model, we also need to assume ∇F i (w j ) satisfies global-level bounded variance: ∃ constant G,
||∇F (w j ) − ∇F i (w j )|| 2 2 ≤ G 2 ,
which means that the difference between the unbiased gradient of each client and the unbiased gradient of global data is limited. Table 1, we list some notations involved in the paper.
Notations. In
WEIGHTED K-ASYNC FL WITH LEARNING RATE ADAPTATION
In this section, we introduce WKAFL to alleviate the contradiction between the existing strategies for staleness and non-IIDness, thus improving the model utility. We first motivate our work by showing the experimental results about this contradiction in Section 4.1. In Section 4.2, we briefly introduce the basic idea. Section 4.3 then presents the algorithm details of WKAFL.
Motivation
To demonstrate the aforementioned contradiction, we implemented the K-async FL algorithm that simply integrates the mitigating strategies for staleness and non-IIDness, and trained it on EMNIST MNIST [38]. Specifically, the mitigation strategy for non-IIDness here is to accumulate historical gradients [31], while that for staleness is to assign exponential weights to the gradients in aggregation [36]. We use the average staleness and average number of aggregated gradients to represent the impact of staleness and non-IIDness, respectively. The detailed descriptions are as follows. where p j,i and τ j,i denote the weight and staleness of the gradient. Large τ ave reflects the poor mitigation effect for staleness.
• Average number of aggregated gradients: the number of gradients whose weights are larger than a threshold and is defined as,
N ave = 1 J J j=1 {p j,i , i ∈ [K]|p j,i ≥ p j,max /µ} , where p j,max = max{p j,i , i ∈ [K]} and µ ∈ R + is a constant.
Similarly, large N ave reflects the poor mitigation effect for non-IIDness. The predominated gradients refer to the gradients whose weights are larger than a threshold w g ∈ R + . The more predominated gradients means the aggregated gradients are more representative, which enhances the mitigation for non-IIDness. However, in almost all iterations, there are at most five gradients with weights larger than 0.1. This reflects that only a small number of gradients dominate the aggregation, incurring an adverse effect on non-IIDness mitigation. Fig. 2(b) shows that the average staleness grows approximately linearly with the number of aggregated gradients, indicating a negative correlation between the two. In conclusion, there exists a contradiction between the mitigation strategies for non-IIDness and staleness.
Basic Idea of WKAFL
To break the above contradiction, we propose WKAFL to pick consistent gradients to alleviate the impact of non-IIDness while accelerating the training process.
Generally, gradients with low-stale are relatively reliable and high-stale gradients are more likely to be biased. However, some high-stale gradients may also have consistent descent direction. Then, we can estimate the unbiased gradients based on low-stale gradients and then pick the consistent high-stale gradients. If these gradients are selected, more gradients are aggregated to improve the mitigation effect of non-IID data while guaranteeing the stale gradients not to diverge the model. In such a case, the contradictory of mitigating the effect of non-IID data and staleness will be mitigated. Apart from this advantage, the selected gradients can also help converge faster. Since stale model is possibly farther from optimal solution and its derivative has a larger norm, stale gradients generally have larger norm for convex loss function. Therefore, the selected consistent stale gradients can help converge faster [39], [28].
Detailed Descriptions
Based on above basic idea, we proposed WKAFL to accelerate the training process and improve the training stability. Pseudo-code of WKAFL consists of two parts, Algorithm 1 for clients and Algorithm 2 for the server. The workflow of the whole process for WKAFL is shown in Fig.3, which includes gradients computation (x) for clients and four main parts for the server, improvement of stability on non-IID data (y), estimation of globally unbiased gradient (z), selection and aggregation of consistent gradients ({) and learning rate adaptation (|). Detailed description of the four parts can refer to Section 4.3.1 -Section 4.3.4 respectively. For clients (x), as shown in Algorithm 1, they first download the current model parameters and the iteration index from the server (Line 1). Then they compute gradients and upload the results to the server (Lines 2-4).
For the server (y -|), as shown in Algorithm 2, after initializing the model (Lines 1-3), the server broadcasts it to all clients for local computation (Line 4). The learning rate adaptation strategy and weighted strategy are presented in Lines 6-16. Particularly, the weighted strategy includes three components (Lines 8, 10-15), alleviating the impact of non-IID data (Line 8), estimating globally unbiased gradient (Lines 13-14), selecting and aggregating consistent gradients (Lines 15) while the learning rate adaptation strategy consists of one component, adjustment of learning rate (Line 16). We will describe the four components of Algorithm 2 and explain corresponding rationality in Sections 4.3.1 to 4.3.4.
Improvement of Stability on Non-IID Data
It is efficient to decrease the effect of non-IID data by accumulating all the historical gradients. Then, the estimated gradient at (j − 1)-th iteration can be added to each of K Algorithm 1 WKAFL: Clients side. Input: mini-batch size m. Output: loss l j,i , gradient g(w j,i , ξ j,i ) , delay τ (j). 1: Receive model parameters w j and iteration j from the server. 2: Compute loss l j,i and gradient g(w j,i , ξ j,i ) based on m samples ξ j,i . 3: Set τ (j) ← j. 4: Upload l j,i , g(w j,i , ξ j,i ) and τ (j). Algorithm 2 WKAFL: Server side. Input: learning rate η 0 , number of gradients received by server K, gradients adjustment parameter B, learning rate adjustment parameter γ, clip-bound parameter CB, weighted parameter β, constant α. Output: Optimal solution w * . 1: Initialize model parameter w 0 and iteration j = 1. for i = 1 → K do 7: Receive loss value, the gradient and its staleness 13: Clip all the gradients with bound CB.
(l i , g(w j,i , ξ j,i ), τ i ) from the i-th client. 8:g(w j,i , ξ j,i ) = g(w j,i , ξ j,i ) + αḡ(w j−1 ); 9: end for 10: if K i=1 l i ≤ then 11: stage = 2; 12: end if
14:
Estimate globally unbiased gradientḡ(w j ) according to Equation (5). 15: g(w j ) = SAGrad(B, β, ,ḡ(w j,i , ξ j,i ),ḡ(w j ), stage); 16: Adjust the learning rate η j adaptively according to Equation (10). 17: w j+1 ← w j − η j g(w j ). 18: j ← j + 1. 19: end while gradients at j-th iteration to decrease the impact of non-IID data:g
(w j,i , ξ j,i ) = g(w j,i , ξ j,i ) + αḡ(w j−1 ),(4)
where α > 0 is a constant andḡ(w j−1 ) is the estimated gradient at (j − 1)-th iteration. Now, we explain the rationality from theoretical perspective. Every gradient received by the server is biased because of staleness and non-IID data. Then, we can decompose the gradient g(w j,i , ξ j,i ) into three parts.
E[g(w j,i , ξ j,i )] = ∇F (w j ) + E[g s (w j,i , ξ j,i )] + E[g n (w j,i , ξ j,i )],
where ∇F (w j ) is the unbiased gradient based on global data at j-th iteration, E[g s (w j,i , ξ j,i )] is the error resulted from staleness, and E[g n (w j,i , ξ j,i )] is the error resulted from non-IID data.
Similarly, the estimated gradient can also be decomposed into three parts,
E[ḡ(w j )] = ∇F (w j ) + E[ḡ s (w j,i , ξ j,i )] + E[ḡ n (w j,i , ξ j,i )].
Then, the expectation ofg(w j,i , ξ j,i ) is:
E[g(w j,i , ξ j,i )] = ∇F (w j ) + α∇F (w j−1 ) + E[g s (w j,i , ξ j,i )] + αE[ḡ s (w j−1,i , ξ j−1,i )] E S + E[g n (w j,i , ξ j,i )] + αE[ḡ n (w j−1,i , ξ j−1,i )] E N where E N is resulted from non-IID data. E[g n (w j,i , ξ j,i )]
only contains the information of client i. While E N contains the information of more clients and the information can accumulate as the iterative number increases. Therefore, Equation (4) can decrease the effect of non-IID data and stabilize the model.
Estimation of Globally Unbiased Gradient
It is assumed that gradients with lower staleness have high probability to be consistent and gradients with higher staleness have high probability to be inconsistent because gradients based on stale model generally deviate from consistent direction. Therefore, we can aggregate gradients by weighting them according to their staleness to estimate globally unbiased gradient. The rule of aggregation is
g(w j ) = K i=1 a j,i aḡ (w j,i , ξ j,i ),(5)whereḡ(w j,i , ξ j,i ) is a clipped gradient ofg(w j,i , ξ j,i ), a = K i=1
a j,i and a j,i is a function of staleness. Cong et al. [25] empirically figured out that a better utility was achieved when a j,i was chosen as an exponential function. Chen et al. [36] found that a j,i was more properly set as ( e 2 ) −τj,i than e −τj,i , where τ j,i is the staleness of gradient g(w j,i , ξ j,i ) and e is natural logarithm. Based on their work, we set a j,i to ( e 2 ) −τj,i in our experiments. Before estimating the globally unbiased gradient, it is necessary to clip the gradients for two reasons. On the one hand, it is known that clipping can prevent gradient explosion. On the other hand, under the assumption that gradients with lower (higher) staleness have high probability to be consistent (inconsistent), the aggregated gradient after clipping will be closer to the globally unbiased gradient. A straightforward illustration is shown in Fig.4. When aggregating the gradients before clipping, the norm of stale gradients may be so large that the aggregated gradient deviates from the globally unbiased gradient. When aggregating the gradients after clipping, the aggregated gradient is closer to the globally unbiased gradient due to the alleviation of impact of gradients with higher staleness.
Algorithm 3 Select and aggregate consistent gradients.
Input: B, β, ,ḡ(w j,i , ξ j,i ),ḡ(w j ), stage. Output: g(w j ).
1: function SAGRAD(B, β, ,ḡ(w j,i , ξ j,i ),ḡ(w j ), stage) 2:
for i = 1, · · · , K do 3: sim j,i = cos ḡ(w j,i , ξ j,i ),ḡ(w j ) ; 4: if sim j,i ≥ sim min then 5: p j,i = exp(β * sim j,i ); if stage = 2 then 11: if B||ḡ(w j )|| 2 ≤ ||ḡ(w j,i , ξ j,i )|| 2 then 12:ḡ(w j,i , ξ j,i ) = B||ḡ(wj )||2 ||ḡ(wj,i,ξj,i)||2ḡ (w j,i , ξ j,i ); 13: end if 14: end if 15:
p j,i = p j,i /sum K i=1 p j,i ; 16: g(w j ) = K i=1 p j,iḡ (w j,i , ξ j,i ).
17:
return g(w j ) 18: end function
Selection and Aggregation of Consistent Gradients
Some stale gradients may have consistent update direction. Then, we can pick these gradients to accelerate the training process. When the staleness is low, the estimated gradient is generally consistent with globally unbiased gradient. Therefore, when judge whether a gradient is consistent or inconsistent, we treat the estimated gradient as globally unbiased gradient for convenience. To determine which gradients are consistent and the corresponding degrees, the server will first compute the cosine similarity sim j,i of the estimated globally unbiased gradientḡ(w j ) and corresponding gradientḡ(w j,i , ξ j,i ). Then the server assigns each gradient with weight according to the cosine similarity. The rule for assigning weight is given by:
sim j,i = cos ḡ(w j,i , ξ j,i ),ḡ(w j ) ;(6)p j,i = exp(βsim j,i ), if sim j,i ≥ sim min ; 0, if sim j,i < sim min .(7)p j,i = p j,i /sum K i=1 p j,i ;(8)
where β is a constant, p j,i defines the weight of gradient g(w j,i , ξ j,i ), and sim min is a given threshold used to judge whether a gradient is consistent or not. Gradients whose cosine similarity is lower than the threshold will be judged as inconsistent gradients and abandoned. On the contrary, consistent gradients will be aggregated according to their weights. The aggregation rule is as follow:
g(w j ) = K i=1 p j,iḡ (w j,i , ξ j,i ).
However, when the model is going to converge, the norm of stale gradients may be too large to make the model fluctuate around the optimal solution. To deal with this problem, we divide the whole training process into two stages. In stage one, the training process can be accelerated by taking advantage of stale gradients. When the average loss of K participant clients is smaller than a constant threshold , the training process enters into stage two. Though the norm of gradient also reflects the state of global model, it is improper to be selected as the stage division criterion because the global model may enter the local optimization area where the norm of gradient is approximate to zero but loss value is not the minimum. In stage two, to guarantee the training stability, consistent stale gradients will be clipped before aggregating based on the norm of estimated gradients. The clipping rule is given by:
g(w j,i , ξ j,i ) = B||ḡ(wj )||2 ||ḡ(wj,i,ξj,i)||2ḡ (w j,i , ξ j,i ), if B||ḡ(w j )|| 2 ≤ ||ḡ(w j,i , ξ j,i )|| 2 ; g(w j,i , ξ j,i ), if B||ḡ(w j )|| 2 > ||ḡ(w j,i , ξ j,i )|| 2 .(9)
where the gradients adjustment parameter B ∈ (0, ∞) is a constant.
The workflow can refer to Algorithm 3. The server first computes the cosine similarity between estimated gradient and every gradient received by the server (Line 3). Then the server assigns gradient g(w j,i , ξ j,i ) with a weight p j,i according to cosine similarity (Lines 4-8) and aggregates the gradients (Lines [15][16] eventually. If the model enters stage two, stale gradients will be clipped (Lines 10-14).
Learning Rate Adaptation
In WKAFL, the estimated gradient may deviate seriously from the globally unbiased gradient when most of the K gradients have high staleness, leading to a seriously biased estimated gradient and a poor improvement of model utility. To alleviate it, a staleness-based adaptive learning rate is more reasonable than using a constant learning rate. Intuitively, a small learning rate is required for alleviating the impact of high staleness. In [33], Zhang et al. divided the initial learning rate by the staleness to adjust the learning rate. However, it can not be extended to large scale AFL due to the possibly high staleness. In such a case, the adjusted learning rate will be so small that the training process will be prolonged due to the very slight improvement at each iteration. To address it, we adjust the learning rate according to the minimal staleness of the K gradients because the estimated gradient is mainly determined by gradients with low staleness, which is expressed as follows.
η j = η 0 * 1 τ min,j * γ + 1 ,(10)
where η 0 is the initial learning rate, τ min,j is the minimal staleness of K gradients at j-th iteration, and γ ∈ (0, 1) is a constant. Note that η j ≤ η 0 and the equality holds only if τ min,j = 0 (i.e., no staleness).
ANALYSIS
In Section 4, we have explained the rationality of weighting and adjustment for learning rate to improve the prediction accuracy and stabilize the training process. In this section, we further analyze the convergence rate of the proposed WKAFL in which the loss function is non-convex, considering both non-IID data and unbounded staleness. Theorems 5.1 and 5.3 show the convergence analysis results with respect to two stages in WKAFL respectively. In stage one, the staleness is bounded while in stage two, the staleness is unbounded. Most convergence analysis of AFL algorithm is based on the theory of stochastic optimization and one challenge is how to bridge the stale gradient and globally unbiased gradient. In stage one, the aggregated gradient at j-th iteration is g(w j ) = j l=1 α j−l K i=1 p l,i g(w l,i , ξ l,i ). Consequently, the difference between global unbiased gradient ∇F (w j ) and the aggregated gradient is the sum of the difference between each component gradient g(w l,i , ξ l,i ) and the unbiased gradient. We connect the local gradient g(w l,i , ξ l,i ) with the theoretical unbiased gradient ∇F (w j ) by two steps.
•
Step 1: We connect the local gradient g(w l,i , ξ l,i ) with globally unbiased gradient ∇F (w l−τ l,i ) by taking expectation of g(w l,i , ξ l,i ) and bounding the locally unbiased gradient ∇F i (w l−τ l,i ) with ∇F (w l−τ l,i ) based on Assumption 3.3.
•
Step 2: We connect the unbiased gradient ∇F (w l−τ l,i ) with the unbiased gradient ∇F (w j ). The difference can be bounded by accumulating all aggregated gradients from max(l − τ max , 1) to j − 1 iterations.
However, in stage two, the staleness can be infinite in theory due to the possibly infinite number of total iterations. Therefore, the method for dealing with finite staleness will not hold in stage two. To deal with it, we use three steps to connect the local gradient g(w l,i , ξ l,i ) of l-th iteration contained in the momentum term with ∇F (w j ).
• Firstly, we connect the local gradient g(w l,i , ξ l,i ) with estimated gradientḡ(w j ) by clipping g(w l,i , ξ l,i ) and selecting consistent gradients. Based on Equation (9), we have ||ḡ(w j , ξ j,i )|| 2 2 ≤ B 2 ||ḡ(w j )|| 2 2 . Apart from that, since the accumulated gradients are consistent with the estimated gradientsḡ(w l ) and inconsistent gradients will be endowed with weight 0, then the weighted differences between clipped gradients at l-th iteration and the estimated gradients can be bounded by ||ḡ(w l )|| 2 2 , i.e., ∃σ 2 e , M e ,
K i=1 p l,i ||ḡ(w l )−ḡ(w l,i , ξ l,i )|| 2 2 ≤ σ 2 e + M e ||ḡ(w l )|| 2 2 .(11)
• Secondly, after alleviating the effect of non-IID data and staleness, for the reason that the estimated gradientḡ(w l ) is closed to the unbiased gradients, we can assume
||ḡ(w l ) − ∇F (w l )|| 2 2 ≤ σ 2 c , ∃σ 2 c ,(12)
which bridges the connection betweenḡ(w l ) and ∇F (w l ).
• Thirdly, the differences between ∇F (w l ) and ∇F (w j ) can be bounded by the aggregated gradients from l to j iterations.
After providing the proof sketch for the two stages, we will present the analysis results of WKAFL. The convergence result of stage one is as follows. and
L 2 J t=l (J − τ t )α −t t k=l η 2 k s k α k (1 + Mc m ) < 1 4 , l ∈ [J]
. Then, we have the following convergence result:
1 J J j=1 η j s j 2 E[||∇F (w j )|| 2 2 ] ≤ 1 J J j=1 (A j η 2 j + η j s j G 2 + η j j l=1 α j−l (j − τ l ) j−1 t=τ l η 2 t B j ) + F (w 1 ) − F (w * ) J ,(13)
where
s j = j k=1 α j−k , τ l = max(1, l − τ max ), A j = Lσ 2 c 2m s j j l=1 α j−l E[ K i=1 p 2 l,i ] and B j = L 2 s t t k=1 α t−k σ 2 c m E[ K i=1 p 2 k,i ]. Proof: See Appendix B.
Based on Theorem 5.1, we can qualitatively analyze the impact of staleness and different levels of non-IID data. With respect to staleness, it is observed that the maximal staleness τ max has a negative impact on convergence rate. In AFL, the staleness is increasing with the increase of P/K, the ratio of number of total clients to the participants. Therefore, for large scale AFL system, it is critical to decrease P/K to ensure a high model utility. Alternatively, we demonstrate how to further improve the model utility under the given P/K by exploiting the stale gradients. With respect to non-IID data measured by the parameter G 2 , it is observed that the convergence rate is decreasing with G 2 . That is, the non-IID data has a negative impact on the model utility. However, the convergence result is too complex to acquire the convergence rate. Therefore, we consider the constant learning rate to simplify Equation (13).
Remark 5.2.
If we set the weight of each received gradients to 1/K and set the learning rate to a constant η in Theorem 5.1, the right part of the Equation (13) is O( η 2 K +ηG 2 + η 3 (1+J 2 α J ) K + 1 J ). If we further set the learning rate to η = √ K √ J , then for any large enough J that satisfies J ≥ 4KL 2 (1 + M G m ) 2 and
L 2 K J t=1 (J − τ t )α −t t k=l s k α k (1 + Mc m ) < J 4 , the conver- gence rate is O( 1 √ KJ + 1 J + G 2 )
where G measures the level of non-IID data. We can notice that the the non-IID data will cause the model to fluctuate near the optimal solution. In IID settings, we have G = 0 and the convergence rate is O( 1 √ KJ + 1 J ) which indicates that WKAFL achieves a linear speedup.
Next, we will present the convergence result for stage two. Since the staleness can be unbounded and the gradients are clipped which is different from stage one, two additional assumptions are required and the rationality has been explained in the proof sketch. To guarantee the convergence, we also assume Equation (11) and (12) holds. Learning rate satisfies that η j ≤ 1
Lsj and subjects
to V j ≥ 0, where V j is V j = η j s j 2 − J l=j 3M e η l α l−j − J l=j η l l−1 t=j η 2 ( 3η j s j 2 C + η j σ 2 c j l=1 j−1 t=l η 2 t s t I j,l,t ) + F (w 1 ) − F (w * ) J ,(14)
where C = σ 2 c +2M e σ 2 c +σ 2 e and I l,k,t = 3L 2 B 2 s t α l−k (l −k). Proof: See Appendix C. Parameters σ 2 c represents the difference between the estimated gradient and the globally unbiased gradient. When σ 2 c is small, the estimated gradient is closer to the globally unbiased gradient. The right part of Equation (14) becomes small and the model will converge faster. Similarly, as B decreasing, the left part of Equation (14) becomes large while the right part becomes small. Then, the model will also converge faster. In summary, both σ 2 c and B have negative impacts on the convergence rate. The same as stage one, we will also analyze the settings with a constant learning rate for stage two to present the convergence rate.
Remark 5.4.
If we set the learning rate to a constant η in Theorem 5.3, the right part of Equation (14) is O(ηC + η 3 + 1 J ). If we further set the learning rate to η = 1 √ J , then for any J ≥
L 2 (1−α) 2 ≥ L 2 S 2 J and large enough J to make 1 J J l=j l−1 t=j t k=1 α t−j I l,k,t , we have O( 1 √ J + C).
Since the term J l=j l−1 t=j t k=1 α t−j I l,k,t contains α < 1, then the term is O(α J J y ), y ∈ R. Therefore, there exists large enough J to meet the conditions. The parameter C reflects the level of non-IID data and the convergence rate demonstrates that non-IID data will decrease the prediction accuracy. In IID settings, we have C = 0 and the convergence rate will be O( 1 √ J ).
EXPERIMENT
In this section, we conducted extensive experiments 1 under the PySyft framework [51] to validate the performance of proposed algorithm WKAFL in terms of the training speed, prediction accuracy and training stability. The proposed WKAFL was compared with three existing algorithms, which will be described in Section 6.1. Meanwhile, several benchmark datasets were adopted with varying levels of non-IIDness. In Section 6.2, ablation experiments demonstrate the impacts of each component of WKAFL. In Section 6.3, detailed comparison results are illustrated to validate the advantage of WKAFL.
Experiment Setup
Datasets and Models
In our experiments, we used four datasets: CelebA [52], EMNIST ByClass [38], EMNIST MNIST [38] and CIFAR10 [53]. The first two benchmark FL datasets were directly used. EMNIST MNIST and CIFAR10 were manually split to have different levels of non-IIDness.
• CelebA partitions the Large-scale CelebFaces Attributes Dataset by the celebrity in the image. 1. Source code is available at https://github.com/zzh816/WKAFLcode.
• EMNIST ByClass contains 81,4255 character images of 62 unbalanced classes. It is partitioned over around 3,500 clients according to the writers in FL.
• EMNIST MNIST has 70,000 characters in ten balanced labels. We partitioned it into 10 subsets by the label. Then, the digit images were divided over clients, which sampled local data according to Algorithm 4. Every client has L num classes of labels and the total number of data ranges from D min to D max .
• CIFAR10 is a labeled dataset of 60,000 tiny color images in 10 classes, and consists of 50,000 training images and 10,000 test images. We partitioned CI-FAR10 into FL according to Algorithm 4 similarly.
Algorithm 4 Generator of non-iid datasets.
Input: D min , D max , L num , P . Output: Datas = (D 1 , D 2 , · · · , D P ).
1: for i ∈ {1, 2, · · · , P } do 2: Classes ← Sample L num labels from {0, 1, 2, · · · , 9};
3:
D num ← RandInt(D min ,D max ); 4: weights ← Sample L num number in interval (0,1);
5:
Data num ← D num * weights/sum(weights);
6:
D i ← Sample data according to Data num and Classes. 7: end for For the above four datasets, we deployed different CNN models. For CelebA, the model contains one convolutional layer with ReLU activation function followed by a maxpooling layer. The final pooled vector is passed to a dense layer with 16000 units. The loss was measured by the logarithmic softmax function. For EMNIST ByClass, it contains two convolutional layers with ReLU activation function followed by maxpooling layers. The final pooled vector is passed to a dense layer with 512 units. The loss was measured by the logarithmic softmax function. For EMNIST MNIST, the same CNN model for EMNIST ByClass's is deployed. For CIFAR10, we adopted LeNet [54], a simple and classical CNN model.
Comparison Algorithms
We compared three asynchronous algorithms in our experiments, including temporally weighted asynchronous federated learning (TWAFL [36]), staleness-aware async-SGD (SASGD, [33]), and gradient scheduling with global momentum (GSGM, [31]). TWAFL and SASGD, were proposed to alleviate the effect of staleness to improve the prediction accuracy, and GSGM was proposed to alleviate the effect of non-IID data to improve training stability.
TWAFL. Unlike aggregating stale gradients in WKAFL, the server in TWAFL directly aggregates the K stale model parameters by assigning the stale model parameters with staleness-based weights to improve the prediction accuracy. For the sake of fair comparison, we modify TWAFL in which the server aggregates K stale gradients instead of model parameters. In particular, the model update formula is expressed as follow.
w j+1 = w j − η j K i=1 m i m e 2 −τj,i * g(w j,i , ξ j,i ),
where m i , τ j,i , g(w j,i , ξ j,i ) are the mini-batch size, staleness, and gradients uploaded by client i at the j-th iteration with m = K i=1 m i . SASGD. In SASGD, the sever first aggregates the K received stale gradients g(w j,i , ξ j,i ) which are assigned with staleness-based weights η j,i . Then, the server applies the aggregated gradient to update the model parameters. The specific updating formula is presented as follows.
η j,i = η 0 /τ j,i ; w j+1 = w j − 1 K K i=1 η j,i g(w j,i , ξ j,i ),
where η 0 is the initial learning rate and τ j,i is the staleness for i-th client at j-th iteration. GSGM. In GSGM with P clients totally, the server only uses one gradient at each iteration (fully AFL) and the whole training process is divided into several rounds. In each round, the server averages part gradients used in gradient scheduling strategy to modify the current inconsistent direction. According to this strategy, once the gradients submitted by a fast client are used for updating in one round, the client is blocked until next round begins. For the sake of fair comparison, we modify GSGM in which K gradients are selected to update the model at each iteration with other protocols unchanged and GSGM is also modified in the way that each client uploads only one gradient in one round to improve the accuracy of GSGM.
Metrics
In our experiments, we compared WKAFL with above algorithms in terms of training speed, model accuracy and training stability. The training speed is directly measured by the number of iterations before convergence. The model accuracy is measured by the prediction accuracy on test set. We describe the metric of training stability as follows.
Most studies focus more on the convergence rate and less on whether the training is stable. Particularly, a stable training indicates the training can be terminated with a relative stable model accuracy, instead of a fluctuated accuracy curve. In this paper, we use training stability to refer to the stability of last few model prediction accuracy on the test set before the final convergence. Particularly, it is measured by the standard deviation of the last A num (which was set as 10) model accuracy values under logarithmic coordinates.
Ablation Experiment Result and Analysis
This part provides some ablation studies to demonstrate the impacts of each component of WKAFL. Specifically, WKAFL consists of three main components: gradients selection, and the strategies for staleness and non-IIDness. Therefore, in our ablation experiments, we removed them one by one, and the corresponding algorithms are named as WKAFL(No Grad-Sel), WKAFL(No Staleness) and WKAFL(No Non-IIDness) respectively. Fig.5 shows the test accuracy of four algorithms on both benchmark FL datasets CelebA and EMNIST ByClass. As shown, WKAFL can always converge fastest to the highest prediction accuracy, which indicates that every component in WKAFL is indispensable. WKAFL converges faster and more stable than WKAFL(No Grad-Sel), validating the existence of contradiction and the effectiveness of gradients selection. WKAFL(No Staleness) always has the poorest performance, which illustrates that staleness has more impacts on the model utility.
Comparison Experiment Result and Analysis
In this part, our purpose is to validate that the proposed WKAFL performs better not only on benchmark FL datasets, but also on modified FL datasets in terms of prediction accuracy and training stability. Firstly, we implemented WKAFL and the comparison algorithms on two benchmark FL datasets, CelebA and EMNIST ByClass. Secondly, to comprehensively evaluate the performance of WKAFL, we implemented it on EMNIST MNIST and CIFAR10 which were modified to satisfy three levels of staleness and non-IID data respectively. Fig.6 shows the test accuracy of four algorithms on CelebA and EMNIST ByClass. The training speed of WKAFL is faster than the other three algorithms in the beginning process and WKAFL can always achieve a higher accuracy which validates that the proposed WKAFL behaves well. The more comprehensive comparisons in terms of staleness and non-IID data are illustrated in Sections 6.3.1 and 6.3.2 respectively.
Different Degrees of Staleness
Based on that the minimum average of staleness approximately equals P/(2K) (refer to Appendix D.1), we achieve different levels of staleness by setting different values of P (number of total clients) and K (number of participants at each iteration) and the level of staleness can be measured as P/K for convenience. For both EMNIST MNIST and CIFAR10, we set three levels of staleness, 1000/10 = Fig.7 and Fig.8 respectively and the final prediction accuracy is summarized in Table 2. Comparisons on EMNIST MNIST. Three conclusions are obtained from Fig.7 and Table 2 (rows 3-5). Firstly, WKAFL has the fastest training speed compared to algorithms TWAFL and SASGD under the three levels of staleness as shown in Fig.7, especially for high staleness in Fig.7c. This validates the idea of exploiting the stale gradients to accelerate the training process (as discussed in Section 4.3.3).
Secondly, Fig.7 shows that WKAFL has higher prediction accuracy than the compared algorithms, especially for high staleness which benefits from the strategies of clipping bound the stale gradients (as discussed in Section 4.3.2), selecting consistent gradients (as discussed in Section 4.3.3) and adaptively adjusting the learning rate (as discussed in Section 4.3.4), especially when the model is going to converge (Stage two). Thirdly, WKAFL is more robust to staleness. Based on the decreasing prediction accuracy in each column (rows [3][4][5] in Table 2, the staleness has a negative impact on the prediction accuracy for all algorithms which is consistent with the analysis of the negative effect of staleness (Theorems 5.1 and 5.3). We note that the accuracy reduction of WKAFL is the lowest while the accuracy reduction of SASGD is the highest. Apart from it, the curves of TWAFL and SASGD have more obvious changes than the curve of WKAFL when the level of staleness increases as shown in Fig.7.
Comparisons on CIFAR10. Similar results are obtained from Fig.8 and Table 2 (rows 6-8). When it comes to dealing with stale gradients and using the adaptive learning rate, the proposed WKAFL outperforms the compared algorithms in terms of training speed and prediction accuracy. Besides, Table 2 shows that the robustness advantage of WKAFL is more significant than that on EMNIST MNIST. Particularly, when staleness increases from 100 to 300, the accuracy of WKAFL decreases from 0.4433 to 0.3970 while the accuracy significantly decreases from 0.4246 to 0.3468 (from 0.3308 to 0.2029)for TWAFL (SASGD) algorithms which validates that WKAFL is extensible for FL with large scale distributed clients in which the staleness is usually high. Additionally, we explain the reason of low prediction accuracy (about 44.5 percent) on CIFAR10 achieved here which is significantly lower than the existing high value (around 95 percent) 2 . The main reason is the memory restrictions of using the complicated CNN architecture presented in [55], [56]. To validate the proposed WKAFL for scenarios with large scale distributed devices, the number of total clients P was set as 3000. In AFL, 3000 duplicates of CNN are needed to be stored and computed because of the asynchronous manner. However, this requires a computer with large memory size. Therefore, we adopted a light model LeNet whose prediction accuracy under centralized machine learning is only around 67 percent 3 . Based on it, the achieved decentralized accuracy affected by both non-IID data and high staleness is acceptable and our main 2. https://github.com/junyuseu/pytorch-cifar-models 3. https://github.com/icpm/pytorch-cifar10 purpose is to validate the performance of WKAFL for any given model and dataset.
Different Degrees of Non-IID Data
Non-IID data is a basic characteristic of FL, however, there is no exact metrics of it. As adopted in [9], we also use the number of label classes L num to measure non-IID level. For example, for CIFAR10 which has labels 0, · · · , 9, L num = 1 means that each client only owns data with one label, such as 5. Obviously, a small value L num means high level of non-IID data. We set three levels of data heterogeneity on both EMNIST MNIST and CIFAR10, L num = 1, 5, 10. The experimental results of test accuracy on EMNIST MNIST and CIFAR10 are shown in Fig.9 and Fig.10 respectively. Table 3 shows the final prediction accuracy and training stability.
Comparisons on EMNIST MNIST. Four conclusions are drawn from Fig.9 and Table 3.
Firstly, Fig.9 shows that the proposed WKAFL converges faster than other three algorithms for all three levels of data heterogeneity, especially for the highest level (Fig.9a with L num = 1). This advantage benefits from full use of historical gradients (as discussion in Section 4.3.1). As the level of non-IID data increases, the direction of uploaded gradients will seriously deviate from the globally unbiased gradient and accumulated gradients can narrow the gap between uploaded gradients and consistent gradients. Therefore, the advantage of WKAFL is more significant for the high level of non-IID data.
Secondly, the proposed WKAFL achieves the highest prediction accuracy for all three levels of data heterogeneity as shown in columns 3-6 of Table 3 (rows 3-5) which illustrates that exploiting the historical gradients can alleviate the impact of non-IID data (as explained in Section 4.3.1), leading to a higher accuracy.
Thirdly, the training stability decreases as the level of non-IID data increases, i.e., L num from 10, 5 to 1, as shown in columns 7-10 of Table 3 (rows [3][4][5]. This is because the gradient's direction of non-IID data is likely to be deviated from the globally unbiased gradient. At different iterations, the direction of aggregated gradients based on K participants will differently deviate from the globally unbiased gradient. Therefore, higher heterogeneity means more uncertainty. Experiment results validate our analysis (Section 5).
Fourthly, WKAFL has a better training stability as shown in columns 7-10 of Table 3 (row 3-5). A small value of training stability means a stable model. Although WKAFL does not have the best training stability, the comprehensive advantage of WKAFL is still significant, which can be described from two aspects. On the one hand, compared to GSGM with the best training stability, its prediction accuracy (row 3-5 in Table 3) and convergence rate ( Fig. 9) is significantly lower than WKAFL. For example, when L num = 1, the prediction accuracy for GSGM is only 85.53 percent, significantly lower than 97.28 percent for WKAFL. However, the training stability of WKAFL, i.e., the fluctuation degree in Fig.9b and Fig.9c, is almost the same as GSGM. On the other hand, compared to TWAFL who has a slight prediction accuracy reduction, its training stability is significantly weaker than WKAFL. For example, when L num = 1, the training stability for TWAFL is 0.0107, almost with 44 percent increment than 0.006 for WKAFL. Besides, Fig.9a shows that the fluctuation of TWAFL is more obvious than WKAFL.
Comparisons on CIFAR10. Similar conclusions are drawn from Fig.10 and Table 3, based on the same analysis on EMNIST MNIST. The only difference is that TWAFL achieves a slight improvement of prediction accuracy than the proposed WKAFL when L num = 5, 10 (columns 2-3) because of the randomness in the experiments. In fact, TWAFL and WKAFL have almost the same training process in low level of data heterogeneity as shown in Fig.10b and Fig.10c.
CONCLUSION
In this paper, we propose a two-stage weighted K-async FL (WKAFL) algorithm to improve the model utility of AFL These improvements are achieved from three aspects.
Firstly, WKAFL estimates the globally unbiased gradient by accumulating historical gradients to alleviate the impact of non-IID data and aggregating K gradients based on the staleness. Secondly, WKAFL picks gradients consistent with the estimated gradient and assigns them with a high weight and vice versa to improve the effect for mitigating non-IID data while preventing the stale gradients to bring down model utility. Thirdly, by further clipping the stale gradient in the second stage and adjusting the learning rate based on staleness, WKAFL improves the final prediction accuracy. The experiment results on four FL datasets validate that WKAFL can accelerate the training process and improve final prediction accuracy while guaranteeing a stable model, especially in settings with high staleness or high level of non-IID data. His research interests focus on data privacy protection, federated learning and privacy-preserving machine learning. He is a member of the IEEE and the ACM. Shusen Yang received his Ph.D. degree in Computing from Imperial College London in 2014. He is a professor and director of the National Engineering Laboratory for Big Data Analytics, and deputy director of Ministry of Education(MoE) Key Lab for Intelligent Networks and Network Security, both at Xi'an Jiaotong University (XJTU), Xi'an, China. Before joining XJTU, He worked as a lecturer (assistant professor) at the University of Liverpool from 2015 to 2016, and a research associate at Intel Collaborative Research Institute (ICRI) on sustainable connected cities from 2013 to 2014. Shusen is a DAMO Academy Young Fellow, and an honorary research fellow at Imperial College London. He is a senior member of IEEE and a member of ACM. His research focuses on distributed systems and data sciences, and their applications in industrial scenarios, including data-driven network algorithms, distributed machine learning, Edge-Cloud intelligence, industrial internet and industrial intelligence.
APPENDIX A COMPARISON EXPERIMENT OF WKAFL AND ZENO++
In this section, we conducted experiments to compare the performance of WKAFL to that of ZENO++. Note that, ZENO++ is proposed to prevent byzantine workers for distributed ML from diverging the model, instead of a federated learning approach for privacy-preserving distributed ML. It works as follows. Before training, the server collects some non-private data and derives the benchmark gradients by pre-training. Then, the server scores every received candidate gradient by comparing with the benchmark gradients, and picks those with high scores to update the global model. To alleviate the computational burden on the server, ZENO++ adopts a lazy update mode, in which, the server estimates the benchmark gradients every k (k ∈ N + ) iterations.
A.1 Experiment Setup
We also used EMNIST MNIST dataset in the comparison experiment. We varied the number of classes of collected data L num ∈ {1, 10} and the frequency of lazy update k ∈ {2, 5} on the server for ZENO++. Table 4 shows the detailed parameter settings. Particularly, since ZENO++ is a fully asynchronous distributed ML algorithm, and the number of participant clients K in each iteration is set to one. However, WKAFL adopts K-async FL and gradients selection and K must be set to larger than one. Therefore, the number of total clients and participant clients for ZENO++ and WKAFL are set different in Table 4. To guarantee the approximately equivalent staleness, the ratio P/K is set to the same value for WKAFL and ZENO++. The parameter of score function 0.1 ρ
The parameter of score function 0.02 ε
The parameter of score function 0.1
A.2 Experiment Results and Analysis
Fig . 11 shows the relationship between the test accuracy and the number of iterations, for ZENO++ and WKAFL on EMNIST MNIST dataset. On the one hand, ZENO++ and WKAFL have similar performance only when the collected data can represent the global data distribution and the server updates the benchmark gradients frequently. For L num = 10 and k = 2, ZENO++ converges faster than WKAFL when the iteration number is less than 300. However, WKAFL converges to a higher final accuracy. Both ZENO++ and WKAFL can converge stably. On the other hand, WKAFL behaves much better than ZENO++ when the collected data of ZENO++ can not represent the overall data or the server has infrequent lazy update. When k = 5, the model utility decreases due to the fact that the benchmark gradients have larger staleness and become inaccuracy. When L num = 1, the model will greatly decrease and fluctuate heavily around a low accuracy. It should note that, the server has to collect some highquality data before the collaborative training for ZENO++. However, this is infeasible in federated learning scenarios, because clients are prohibited to upload raw data due to privacy in design. Therefore, ZENO++ cannot be directly applied in federated learning scenarios. p j,i (g(w j,i , ξ j,i ) − ∇F i (w τ (j) )))( K k=1 p j,k ∇F k (w τ (j) ))] = K i=1 K k=1 p j,i p j,k E[(g(w j,i , ξ j,i ) − ∇F i (w τ (j) ))∇F k (w τ (j) )] = K i=1 K k=1 p j,i p j,k E[(E ξj,i [g(w j,i , ξ j,i )] − ∇F i (w τ (j) )) ∇F k (w τ (j) )] = 0 Equation (18) is derived from Equation (16) and Equation (19) is derived based on Assumption 3.3.
Based on Lemma B.1, we provide a proof of Theorem 5.1. Proof of Theorem 5.1: Based on Assumption 3.1 and update rule w j+1 = w j − η j j l=1 α j−l K i=1 p l,i g(w l,i , ξ l,i ), we have
F (w j+1 ) − F (w j ) ≤ ∇F (w j )(w j+1 − w j ) + L 2 ||w j+1 − w j || 2 2 = −∇F (w j )η j j l=1 α j−l K i=1 p l,
Define s j = j l=1 α j−l = 1−α j 1−α and take expectation on both sides of Equation (20):
E[F (w j+1 )] − F (w j ) = −η j j l=1 α j−l E[ K i=1 p l,i ∇F (w j )
Fig. 1 .
1K-async FL with K=2 and P =4. The green shadow arrows indicate the gradients selected for updating. Red arrows indicate the gradients which fail to update the global model in current iteration.
Fig. 2 .
2Motivation experiments. (a) Exploration of the effect of non-IIDness; (b) Relation of non-IID data (number of aggregated gradients) and staleness (average staleness).
Fig. 2 (
2a) depicts the histogram over the number of predominated gradients in all aggregations during training.
Fig. 3 .
3WKAFL with five main components, one for clients (x) and four for the server (y -|). Gradients computation (x). Improvement of stability on non-IID data (y). Estimation of globally unbiased gradient (z). Selection and aggregation of consistent gradients ({). Learning rate adaptation (|).
2 :
2Initialize the estimated gradientḡ(w 0 ) = 0. 3: Initialize the stage state stage = 1; 4: Broadcast w 0 and j to all clients.5: while the stopping criteria is not satisfied do6:
Fig. 4 .
4Estimated gradient before and after clipping the gradients respectively.
Theorem 5.3 (Stage Two). Assume Assumption 3.1, 3.2, 3.3 hold.
Fig. 5 .
5Relation between test accuracy and iteration on CelebA and EMNIST ByClass.
Fig. 6 .
6Relation between test accuracy and iteration on CelebA and EMNIST ByClass.
Fig. 7 .
7Prediction accuracy on EMNIST MNIST with different levels of staleness measured by P/K. Here P is the number of total clients and K is number of participants at each iteration.
Fig. 8 .
8Prediction accuracy on CIFAR10 with different levels of staleness measured by P/K. Here P is the number of total clients and K is number of participants at each iteration.100, 3000/20 = 150, 3000/10 = 300 on EMNIST MNIST and 1000/20 = 50, 3000/20 = 150, 3000/10 = 300 on CIFAR10. The experimental results of test accuracy under the three levels on EMNIST MNIST and CIFAR10 is shown in
Fig. 9 .Fig. 10 .
910Relation between test accuracy and iteration on EMNIST MNIST with different levels of Non-IID degrees (Lnum). A smaller Lnum means a higher level of non-IID data. The level of staleness was fixed as 300 (P/K = 3000/10). Relation between test accuracy and iteration on CIFAR10 with different levels of Non-IID degrees (Lnum). A smaller Lnum means a higher level of non-IID data. The level of staleness was fixed as 150 (P/K = 3000/20).
Fig. 11 .
11Comparison between WKAFL and ZENO++.
i g(w l,i , ξ l,
l,i g(w l,i , ξ l,i )|| 2 2 .
and S. Yang are with School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China. E-mails: {wszzh15139520600, gogll2,shusenyang}@stu.xjtu.edu.cn. • X. Ren is with School of Computer Science and Technology, Xi'an Jiaotong University , Xi'an, Shaanxi 710049, China. E-mail: [email protected].
TABLE 1 Notations
1Variables
Meaning
η j
Learning rate at the j-th iteration
P
Number of total clients
K
Number of gradients needed at each iteration
m
Mini-batch size
L
Lipschitz constant
J
TABLE 2 Prediction
2Accuracy with Different Levels of Staleness.Datasets
Staleness
Prediction Accuracy
WKAFL TWAFL SASGD
EMNIST MNIST
100
0.9734
0.962
0.9319
150
0.9756
0.9649
0.8753
300
0.9728
0.9572
0.8553
CIFAR10
50
0.4433
0.4246
0.3308
150
0.4368
0.4241
0.2560
300
0.3970
0.3468
0.2029
TABLE 3 Prediction
3Accuracy and Training Stability with Three Levels of Non-IID Degree Measured by Lnum.Dataset
Lnum
Final Accuracy
Model Stability
WKAFL TWAFL GSGM SASGD
WKAFL TWAFL GSGM SASGD
EMNIST MNIST
1
0.9728
0.9572
0.9057
0.8553
0.0060
0.0107
0.0032
0.0834
5
0.9754
0.9649
0.908
0.8753
0.0068
0.0095
0.0027
0.0237
10
0.9658
0.9623
0.9075
0.8619
0.0036
0.0060
0.0018
0.0206
CIFAR10
1
0.4368
0.4141
0.3568
0.2560
0.0284
0.0464
0.0135
0.0226
5
0.4443
0.4454
0.3568
0.2855
0.0133
0.0300
0.0083
0.0141
10
0.4489
0.4525
0.3430
0.2874
0.0180
0.0260
0.0144
0.0189
0
2000
4000
Iteration
0.2
0.4
0.6
0.8
1
Accuracy
WKAFL
TWAFL
SASGD
GSGM
Yanan Li received his Bachelor and Master degree from Henan Normal University of China in 2004 and 2007, respectively. He is currently working towards the PhD degree in the School of Mathematics and Statistics at Xi'an Jiaotong University. Before that, he worked as a lecturer in Henan Polytechnic University from 2007 to 2017. His research interests include machine learning, federated learning, and edge-cloud intelligence. Xuebin Ren received his PhD degree in the Department of Computer Science and Technology from Xi'an Jiaotong University (XJTU), China in 2017. Currently, he is an associate professor in the School of Computer Science and Technology and a member of National Engineering Laboratory for Big Data Analytics (NEL-BDA), both at XJTU, Xi'an, China. He has been a visiting PhD student in the Department of Computing at Imperial College London, UK from 2016 to 2017.
TABLE 4
4Notations and parameters.Algorithm Variables
Meaning
Values
WKAFL
P
Number of total clients
3000
K
Number of participant gradients
20
J
Number of iterations
1500
ZENO++
P
Number of total clients
100
K
Number of participant gradients
1
J
Number of iterations
1500
γ
t t k=1 α t−j I l,k,t .
APPENDIX B PROOF OF THEOREM 5.1In this section, we provide a proof of Theorem 5.1. To convey the proof clearly, it would be necessary to prove one certain useful lemma.Lemma B.1. Assume Assumption 3.3 holds. Then, we have:Proof of Lemma B.1: Observe that,0≤i,k≤K p j,i p j,k (||∇F i (w τ (j) )|| 2 2 + ||∇F k (w τ (j) )||22 )] = E[ 0≤i,k≤K p j,i p j,k ||∇F i (w τ (j) )||2 2]Based on K i=1 p j,i = 1, we haveEquation(17)holds since the cross term is 0 as derived below.Equation(21)Equation(23)is derived because learning rate η j satisfies. Now, we aim to bound the term A.With respect to term A,Equation24is derived based on Assumptions 3.1 and 3.3. Define τ l = max(1, l − τ max ). With respect to term B,Equation(25)is derived from Lemma B.1. By replacing Equations(24),(25)into Equation(23), we haveSince the learning rate satisfies thatThen, we haveTaking summation with respect to j ∈ {1, 2, · · · , J} on both sides, we obtain the following result:APPENDIX C PROOF OF THEOREM 5.3Proof of Theorem 5.3: Based on Assumption 3.1 andDefine s j = j l=1 α j−l = 1−α j 1−α and take expectation on both sides of Equation(26):Equations(27)and(28)are derived based on Jensen's inequality. Equation(29)is derived because the learning rate satisfies that η j ≤ 1Lsj . With respect to term C,By replacing Equation(31)into Equation(29), we haveTaking summation with respect to j on both sides, we obtainEquation (32) amounts to:APPENDIX D MINIMAL ESTIMATION OF AVERAGED STALENESSTheorem D.1. In K-async FL, assume that there are totally P clients participating in the training process and K clients uploading their gradients at one iteration. Then, after P/K iterations, the averaged staleness of gradients is more than Proof of Theorem D.1: At j-th iteration, define a series of numbers from small to large τ j,1 , τ j,2 , · · · , τ j,P as the staleness of P clients' gradients. After P/K iterations, the first K numbers τ j,1 , τ j,2 , · · · , τ j,K are 1. And the second K numbers τ j,K+1 , τ j,K+2 , · · · , τ j,2K cannot be smaller than 2 because the gradients have been calculated for at least two iterations. Then, we have: τ j,1 +τ j,2 + · · · + τ j,P = K + τ j,K+1 + τ j,K+2 + · · · + τ j,P ≥ K + 2 + 2 + · · · + 2 K +τ j,2K+1 + · · · + τ j,P ≥ K + 2K + · · · + [P/K] * K +([P/K] + 1) * (P − [P/K] * K) ≥ K + 2K + · · · + [P/K] * K Therefore, the least averaged staleness is:(τ j,1 +τ j,2 + · · · + τ j,P )/P ≥ K 2P(1 + [P/K]) * [P/K].Remark D.2. When P is much higher than K, the lower bound of averaged staleness is around P 2K .
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] | [] | A crucial open problem inS large-scale quantum networks is how to efficiently transmit quantum data among many pairs of users via a common data-transmission medium. We propose a solution by developing a quantum code division multiple access (q-CDMA) approach in which quantum information is chaotically encoded to spread its spectral content, and then decoded via chaos synchronization to separate different sender-receiver pairs. In comparison to other existing approaches, such as frequency division multiple access (FDMA), the proposed q-CDMA can greatly increase the information rates per channel used, especially for very noisy quantum channels.QUantum networks for long distance communication and distributed computing require nodes which are capable of storing and processing quantum information and connected to each other via photonic channels 1-3 . Recent achievements in quantum information 4-10 have brought quantum networking much closer to realization. Quantum networks exhibit advantages when transmitting classical and quantum information with proper encoding into and decoding from quantum states 11-17 . However, the efficient transfer of quantum information among many nodes has remained as a problem yet to be solved 18-24 , which becomes more crucial for the limited-resource scenarios in large-scale networks. Multiple access, which allows simultaneous transmission of multiple quantum data streams in a shared channel, may provide a remedy to this problem.Popular multiple-access methods in classical communication networks include time-division multiple-access (TDMA), frequency-division multiple-access (FDMA), and code-division multiple-access (CDMA). SeeFig. 1for an illustration of different multiple-access methods. In TDMA, different users share the same frequency but transmit on different time slots, but timing synchronization and delays become serious problems in large-scale networks. In FDMA, different users share the same time slots but operate on different frequency bands. However, only a narrow band of the data transmission line has a low leakage rate and the bands assigned to different users should be sufficiently separated to suppress interference. Unlike TDMA and FDMA, CDMA utilizes the entire spectrum and time slots to encode the information for all users, while distinguishes different users with their own unique codes. Therefore, CDMA is adopted as the key technology of the currently-used third generation mobile communication systems, and can accommodate more bits per channel use 25 compared with TDMA and FDMA. It has achieved great success in commercial applications of classical communications.Although FDMA has already been used in quantum key distribution networks[26][27][28][29][30], to the best of our knowledge, CDMA has not yet been applied in quantum networks and internet 1 . A q-CDMA network would require that the states sent by each transmitting node of the quantum network are encoded into their coherent superposition before being sent to the common channel, and the quantum information for each of the intended receiving node is coherently and faithfully extracted by proper decoding at the end of the common channel. This, however, is not a trivial task but rather a difficult one.In this paper, we propose a q-CDMA method via chaotic encoding and chaos synchronization among senders and receivers, which require a quantum channel to transmit quantum superposition states and N classical channels for chaos synchronization to decode the quantum signals at the receiver nodes. It can be seen that the proposed q-CDMA provides higher transmission rates for both classical and quantum information, especially in very noisy channels. | 10.1038/srep02211 | null | 10,651,314 | 1204.1742 | cd075c582095c389c8d7cbd68ff48126337c5aaf |
Quantum internet using code division multiple access
Jing Zhang
CEMS
RIKEN
351-0198SaitamaJapan
Department of Automation
Tsinghua University
100084BeijingP. R. China
Center for Quantum Information Science and Technology
TNList
100084BeijingP. R. China
Yu-Xi Liu
CEMS
RIKEN
351-0198SaitamaJapan
Center for Quantum Information Science and Technology
TNList
100084BeijingP. R. China
Institute of Microelectronics
Tsinghua University
100084BeijingP. R. China
Kaya Ö Zdemir
CEMS
RIKEN
351-0198SaitamaJapan
Electrical and Systems Engineering
Washington University
63130St. LouisMissouriUSA
Re-Bing Wu
CEMS
RIKEN
351-0198SaitamaJapan
Department of Automation
Tsinghua University
100084BeijingP. R. China
Center for Quantum Information Science and Technology
TNList
100084BeijingP. R. China
Feifei Gao
Department of Automation
Tsinghua University
100084BeijingP. R. China
Center for Quantum Information Science and Technology
TNList
100084BeijingP. R. China
Xiang-Bin Wang
CEMS
RIKEN
351-0198SaitamaJapan
Department of Physics
Tsinghua University
100084BeijingP. R. China
Lan Yang
Electrical and Systems Engineering
Washington University
63130St. LouisMissouriUSA
Franco Nori
CEMS
RIKEN
351-0198SaitamaJapan
Physics Department
The University of Michigan
48109-1040Ann ArborMIUSA
Quantum internet using code division multiple access
10.1038/srep02211
A crucial open problem inS large-scale quantum networks is how to efficiently transmit quantum data among many pairs of users via a common data-transmission medium. We propose a solution by developing a quantum code division multiple access (q-CDMA) approach in which quantum information is chaotically encoded to spread its spectral content, and then decoded via chaos synchronization to separate different sender-receiver pairs. In comparison to other existing approaches, such as frequency division multiple access (FDMA), the proposed q-CDMA can greatly increase the information rates per channel used, especially for very noisy quantum channels.QUantum networks for long distance communication and distributed computing require nodes which are capable of storing and processing quantum information and connected to each other via photonic channels 1-3 . Recent achievements in quantum information 4-10 have brought quantum networking much closer to realization. Quantum networks exhibit advantages when transmitting classical and quantum information with proper encoding into and decoding from quantum states 11-17 . However, the efficient transfer of quantum information among many nodes has remained as a problem yet to be solved 18-24 , which becomes more crucial for the limited-resource scenarios in large-scale networks. Multiple access, which allows simultaneous transmission of multiple quantum data streams in a shared channel, may provide a remedy to this problem.Popular multiple-access methods in classical communication networks include time-division multiple-access (TDMA), frequency-division multiple-access (FDMA), and code-division multiple-access (CDMA). SeeFig. 1for an illustration of different multiple-access methods. In TDMA, different users share the same frequency but transmit on different time slots, but timing synchronization and delays become serious problems in large-scale networks. In FDMA, different users share the same time slots but operate on different frequency bands. However, only a narrow band of the data transmission line has a low leakage rate and the bands assigned to different users should be sufficiently separated to suppress interference. Unlike TDMA and FDMA, CDMA utilizes the entire spectrum and time slots to encode the information for all users, while distinguishes different users with their own unique codes. Therefore, CDMA is adopted as the key technology of the currently-used third generation mobile communication systems, and can accommodate more bits per channel use 25 compared with TDMA and FDMA. It has achieved great success in commercial applications of classical communications.Although FDMA has already been used in quantum key distribution networks[26][27][28][29][30], to the best of our knowledge, CDMA has not yet been applied in quantum networks and internet 1 . A q-CDMA network would require that the states sent by each transmitting node of the quantum network are encoded into their coherent superposition before being sent to the common channel, and the quantum information for each of the intended receiving node is coherently and faithfully extracted by proper decoding at the end of the common channel. This, however, is not a trivial task but rather a difficult one.In this paper, we propose a q-CDMA method via chaotic encoding and chaos synchronization among senders and receivers, which require a quantum channel to transmit quantum superposition states and N classical channels for chaos synchronization to decode the quantum signals at the receiver nodes. It can be seen that the proposed q-CDMA provides higher transmission rates for both classical and quantum information, especially in very noisy channels.
Results
To present the underlying principle of our method, we consider the simplest case, where two pairs of sender and receiver nodes communicate quantum information, encoded into quantized electromagnetic fields with the same frequencies, via a single quantum channel [see Fig. 2(a)].
The schematic diagram of our strategy is shown in Fig. 2(b). The quantum information sent by the nodes 1 and 2 is first encoded by two chaotic phase shifters CPS 1 and CPS 2 , whose operation can be modelled by the effective Hamiltonian d i t ð Þ a { i a i , with d i (t) being time dependent classical chaotic signals and i 5 1, 2. This encoding spreads the spectral content of the quantum information across the entire spectrum. The two beams are then combined at the 50550 beamsplitter BS 1 and transmitted via a common channel to the receivers. At the end of the channel, the quantum signal is first amplified by a phase-insensitive linear amplifier (LA), then divided into two branches by a second 50550 beamsplitter BS 2 , and finally sent to nodes 3 and 4 through two chaotic phase shifters CPS 3 and CPS 4 , which are introduced to decode the information by applying the effective Hamiltonian {d j t ð Þ a { j a j , with j 5 3, 4. Amplifier gain is set as G 5 4 to compensate the losses induced by the beamsplitters.
The actions of the chaotic devices CPS i 5 1,2,3,4 induce the phase shifts exp [2ih i (t)], where h i t ð Þ~Ð t 0 d i t ð Þ dt. Thus, to achieve faithful transmission between the senders and the receivers, the effects of d 1 (t) and d 2 (t) on the quantum signals should be minimized in the fields received by the nodes 3 and 4. Intuitively, this could be done by simply adjusting the system parameters such that d 1 (t) 5 d 3 (t) and d 2 (t) 5 d 4 (t). However, such an approach is impractical, because any small deviation in the system parameters can be greatly amplified by the chaotic motion, making it impossible to keep two chaotic circuits with the same exact parameters and initial conditions. Instead, auxiliary classical channels between senders and the intended receivers can be used to synchronize the chaotic circuit as shown in Fig. 2(b). This classical chaotic synchronization helps to reduce the parameter differences between the chaotic phase shifters and to extract the quantum information faithfully.
Modelling of quantum CDMA network. Hereafter, for the sake of simplicity, we assume that CPS 1 (CPS 2 ) and CPS 3 (CPS 4 ) have been synchronized before the start of the transmission of quantum (a) Quantum information transmission between two pairs of nodes via a single quantum channel. Quantum states from two senders are combined to form a superposition state and input to the channel. At the receiver side, they are coherently split into two and sent to the targeted receivers. (b) Schematic diagram of the q-CDMA network by chaotic synchronization. Wave packets from the sender nodes are first spectrally broadened by using the chaotic phase shifters CPS 1 and CPS 2 , and then mixed at a beamsplitter (BS 1 ) and input to the channel. After linear amplification (LA) and splitting at the second beamsplitter (BS 2 ), individual signals are recovered at the receiver end with the help of CPS 3 and CPS 4 , which are synchronized with CPS 1 and CPS 2 , respectively. www.nature.com/scientificreports SCIENTIFIC REPORTS | 3 : 2211 | DOI: 10.1038/srep02211 information, i.e., h 1 (t) 5 h 3 (t) [h 2 (t) 5 h 4 (t)]. The whole information transmission process in this quantum network can be described by the input-output relationship
a 3~a1 za 2 e i h1{h2 ð Þ z ffiffi ffi 6 p 2 e ih1 a { LA z 1 ffiffi ffi 2 p e ih1 a BS , a 4~a2 za 1 e i h2{h1 ð Þ z ffiffi ffi 6 p 2 e ih2 a { LA { 1 ffiffi ffi 2 p e ih2 a BS ,ð1Þ
where a { LA and a BS are the creation and annihilation operators of the auxiliary vacuum fields entering the linear amplifier LA and the second beamsplitter BS 2 . For the pseudo-noise chaotic phase-shift h i (t), we should take an average over this broadband random signal, which leads to exp
+ih i t ð Þ ð Þ < ffiffiffiffiffi ffi M i p With M i~e xp {p ð v ui v li dv S di v ð Þ v 2 :ð2Þ
In Eq. (2), S di v ð Þis the power spectrum density of the signal d i (t), and v li and w ui are the lower and upper bounds of the frequency band of d i (t), respectively. Equation (1) can then be reduced to
a 3~a1 z ffiffiffiffiffiffiffiffiffiffiffiffi ffi M 1 M 2 p a 2 z ffiffiffiffiffiffiffiffi ffi 3M 1 2 r a { LA z ffiffiffiffiffiffi ffi M 1 2 r a BS , a 4~a2 z ffiffiffiffiffiffiffiffiffiffiffiffi ffi M 1 M 2 p a 1 z ffiffiffiffiffiffiffiffi ffi 3M 1 2 r a { LA { ffiffiffiffiffiffi ffi M 2 2 r a BS :ð3Þ
For a chaotic signal with broadband frequency spectrum, the factor M i is extremely small, and can be neglected in Eq. (3). This leads to a 3 < a 1 and a 4 < a 2 , implying efficient and faithful transmission of quantum information from nodes 1 and 2 to nodes 3 and 4, respectively.
In our q-CDMA network, the information-bearing fields a 1 and a 2 , having the same frequency v c , are modulated by two different pseudo-noise signals, which not only broaden them in the frequency domain but also change the shape of their wavepackets [see Fig. 2
(b)].
Thus, the energies of the fields a 1 and a 2 are distributed over a very broad frequency span, in which the contribution of v c is extremely small and impossible to extract without coherent sharpening of the v c components. This, on the other hand, is possible only via chaos synchronization which effectively eliminates the pseudo-noises in the fields and enables the recovery of a 1 (a 2 ) at the output a 3 (a 4 ) with almost no disturbance from a 2 (a 1 ). This is similar to the classical CDMA. Thus, we name our protocol as q-CDMA.
Quantum state transmission. Let us further study the transmission of qubit states over the proposed q-CDMA network using a concrete model. The qubit states { w 1 j i~ffi ffiffiffi ffi
p 1 p g 1 j iz ffiffiffiffiffiffiffiffiffiffiffi ffi 1{p 1 p e 1 j i, and w 2 j i~ffi ffiffiffi ffi p 2 p g 2 j iz ffiffiffiffiffiffiffiffiffiffiffi ffi 1{p 2 p e 2 j i, with p 1 , p 2 g [0,
1]}, to be transmitted are encoded in the dark states of two L-type threelevel atoms; i.e., atom 1 in cavity 1 and atom 2, in cavity 2, as shown in Fig. 3(a). The qubit states are transferred to the cavities by Raman transitions and are transmitted over the q-CDMA network. At the receiver nodes, the quantum states are transferred and stored in two L-type atoms; i.e., atom 3 in cavity 3, and atom 4 in cavity 4. We assume that the four coupled atom-cavity systems have the same parameters. Let jg i ae, je i ae, and jr i ae be the three energy levels of atom i. As shown in Fig. 3(a), the jg i ae R jr i ae and je i ae R jr i ae transitions are coupled with a classical control field and a quantized cavity field with coupling strengths V i (t) and g i (t). By adiabatically eliminating the highest energy level jr i ae, the Hamiltonian of the atom-cavity system can be expressed as
H i~gi t ð Þ c i e i j i g i h jzc { i g i j i e i h j ,ð4Þ
where c i is the annihilation operator of the i-th cavity mode; g i (t) 5 gV i (t)/D is the coupling strength which can be tuned by the classical control field V i (t); and D is the atom-cavity detuning. The cavity fields c i are related to the travelling fields a i by
a 1~ffi ffiffi k p c 1 za 1,in , a 2~ffi ffiffi k p c 2 za 2,in , a 3,out~ffi ffiffi k p c 3 za 3 , a 4,out~ffi ffiffi k p c 4 za 4 ,ð5Þ
where k is the decay rate of the cavity field; and a 1,in , a 2,in (both in vacuum states) and a 3,out , a 4,out are the input and output fields of the whole system, respectively. The chaotic phase shifters CPS i 5 1,2,3,4 are realized by coupling the optical fields to four driven Duffing oscillators, with damping rates c, described by the Hamiltonian
H Duf f ,i~v o 2 p 2 i z v o 2 x 2 i {mx 4 i {f t ð Þx i ,ð6Þ
where x i and p i are the normalized position and momentum of the nonlinear Duffing oscillators, v 0 /2p is the frequency of the fundamental mode, m is a nonlinear constant, and
f(t) 5 f d cos (v d t)
is the driving force. The interaction between the field a i and the i-th Duffing oscillator is given by the Hamiltonian
H i~gf {o x i a { i a i ,ð7Þ, x 3 ) 5 k I (x 1 2 x 3 ) 2 /2.
The nonlinear coupling between the optical fields and the Duffing oscillators and the chaos synchronization to achieve the chaotic encoding and decoding could be realized using different physical platforms. For example, in optomechanical systems, the interaction Hamiltonian (7) can be realized by coupling the optical field via the radiation pressure to a moving mirror connected to a nonlinear spring (see Fig. 3(b)). Chaotic mechanical resonators can provide a frequency-spreading of several hundreds of MHz for a quantum signal, and this is broad enough, compared to the final recovered quantum signal, to realize the q-CDMA and noise suppression. Chaos synchronization between different nonlinear mechanical oscillators can be realized by coupling the two oscillators via a linear spring. This kind of synchronization of mechanical oscillators have been realized in experiments 32 , but it is not suitable or practical for long-distance quantum communication. Chaos synchronization with a mediating optical field, similar to that used to synchronize chaotic semiconductor lasers for high speed secure communication 33 , would be the method of choice for long-distance quantum communication. The main difficulty in this method, however, will be the coupling between the classical chaotic light and the information-bearing quantum light. This, on the other hand, can be achieved via Kerr interactions. There is a recent report 34 that proposes to use Kerr nonlinearity in whispering gallery mode resonators to solve this problem. Another approach for chaotic encoding and chaos synchronization between distant nodes of the network could be the use of electro-optic modulators (EOMs). See, e.g., Fig. 3(c). In this case, the input information-bearing quantum signal is modulated by the EOM driven by a chaotic electrical signal 35 . The EOM can prepare the needed broadband signal, and there have been various proven techniques of chaotic signal generation and synchronization in electrical circuits. Indeed, recently experimental demonstration of chaos synchronization in a four-node optoelectronic network was reported 36 .
To show the efficiency of state transmission in q-CDMA, let us calculate the fidelities F 1 5 AEw 1 jr 3 jw 1 ae and F 2 5 AEw 2 jr 3 jw 2 ae, where r 3 and r 4 are the quantum states received by atoms 3 and 4, and w 1 j i~ffi ffiffiffi ffi p 0 p g 1 j iz ffiffiffiffiffiffiffiffiffiffiffi ffi 1{p 0 p e 1 j i and w 2 j i~ffi ffiffiffiffiffiffiffiffiffiffi ffi 1{p 0 p g 2 j iz ffiffiffiffi ffi p 0 p e 2 j i are the two quantum states to be transmitted. By designing the control parameters g i (t), using the Raman transition technique 18 , we find for the particular chosen quantum states that the fidelities F 1 and F 2 can be approximated as F 1 5 F 2 < 1 2 M. When the Duffing oscillator enters the chaotic regime, we have M < 0, leading to fidelities F 1 , F 2 < 1, which means that the qubit states can be faithfully transmitted over the q-CDMA network.
We show the feasibility of the q-CDMA method using numerical simulations with the system parameters v d /v 0 5 5, g f-o /v 0 5 0.03, m/v 0 5 0.25, c/v 0 5 0.05, k I /v 0 5 0.1, and p 0 5 0.6. In Fig. 4(a), it is seen that there are three distinct regions representing how the chaotic motion affects the fidelity of the quantum state transmission. In the periodic regime characterized by 0 , f d /v 0 , 15, both F 1 and F 2 experience slight increases with increasing f d /v 0 , with 0.4 , F 1 , 0.5 and 0.6 # F 2 # 0.64. At f d /v 0 5 15, the Duffing oscillator enters the soft chaotic regime which is indicated by a positive Lyapunov exponential l < 0.038 and a sudden jump in fidelities. In this regime, delineated by 15 # f d /v 0 # 33, both F 1 and F 2 are still below 0.7. The dynamics of the Duffing oscillator enters the hard-chaos regime at f d / v 0 < 33, where both F 1 and F 2 suddenly jump to 1, which corresponds to an almost 100% faithful state transmission. In Fig. 4(b), we plot the trajectories of F 1 and F 2 as a function of p 0 in the hardchaotic regime f d /v 0 5 36, corresponding to M < 0.0103. It is seen that F 1 and F 2 are very close to 1 2 M < 0.9897 and almost constant regardless of the value of p 0 . There are small deviations from 1 2 M, because here M 2 terms are not neglected. The average fidelity F~F 1 zF 2 ð Þ =2 is maximum at p 0 5 1/2, which corresponds to an equally-weighted superposition of the quantum states jw 1 ae and jw 2 ae. In such a case, the crosstalk between the channels becomes minimum, inducing only a very slight disturbance on these indistinguishable states.
Information transmission rates. Next we consider the maximum transmission rates of classical and quantum information over the proposed q-CDMA network, and compare them, under certain energy constraints, with the achievable bounds of transmission rates in a q-FDMA network and in quantum networks without any multiple access method (i.e., single user-pair network). Here the classical information transmission rates are calculated in terms of the Holevo information 37,38 and the quantum information transmission rates are defined by the coherent information [39][40][41] . We assume that the frequencies allocated to different user pairs in the FDMA network are equally spaced such that the number of users is maximized and cross-talks between adjacent channels are suppressed. Moreover, we restrict our discussion to Gaussian channels and Bosonic channels, respectively for the transmissions of quantum and classical information.
We briefly summarize the main results here and in Fig. 5(a)-(c). (i) For lossless channels (i.e., g 5 1 where g denotes the transmissivity of the central frequency of the information-bearing field), upper bounds of classical and the quantum information transmission rates for the proposed q-CDMA network are higher than those of the quantum FDMA and the single user-pair networks if the crosstalk in the q-CDMA is suppressed by setting M=1. (ii) With the increasing number N of user-pairs in the networks, q-CDMA increasingly performs better than the q-FDMA for classical and quantum information. (iii) Information transmission rates for the q-CDMA is more robust to noise. For fixed N, quantum information transmission rates of the q-FDMA and the single user-pair networks degrades very fast to zero as the loss 1 2 g increases from zero (ideal channel) to 1/2, whereas the q-CDMA network retains its non-zero rate even for very noisy channels. For the classical information transmission, the situation is similar except that the transmission rates of q-FDMA and the single user-pair network drops to zero when g 5 0 which corresponds to a completely lossy channel.
The robustness of the proposed q-CDMA network for noisy channel can be explained as follows. The chaotic phase shifters in the q-CDMA network spread the information-bearing field across a broad spectral band. Thus, the energy distributed in a particular mode is almost negligible, and thus the photon loss is also almost negligible. Therefore, increasing g has very small effect on the transmission rates. In Fig. 5(b)-(c), we consider the noise to be broadband, and shows that the transmission rates of classical and quantum information over the q-CDMA network change only slightly.
Discussion
We have introduced a q-CDMA network based on chaotic synchronization where quantum information can be faithfully transmitted with fidelities as high as 0.99 between multiple pairs of nodes sharing a single quantum channel. The proposed quantum multiple-access network is robust against channel noises, and attains higher transmission rates for both classical and quantum information when compared to other approaches. A q-CDMA network based on our proposal requires the realization of two important issues. First, quantum interference of signals from different chaotic sources. This has recently been demonstrated by Nevet et. al 42 . Second is the implementation of chaotic phase shifters and their synchronization. These could be implemented in various systems, including but not limited to optomechanical, optoelectrical 35 , and all-optical systems 33 . In particular, whispering-gallery-mode (WGM) optical resonators are possible platforms as chaotic behavior in a WGM microtoroid resonator has been reported in Ref 43 . Although synchronization of self-sustaining oscillations in directly coupled microring resonators have been demonstrated 44 , and mechanical mode synchronization in two distant resonators coupled via waveguides has been proposed 45 , demonstration of chaos synchronization in such optomechanical resonators are yet to be demonstrated. Although the tasks to be fulfilled are not trivial, we believe that we are not far away from such realizations due to the rapid pace of experimental and theoretical developments we have seen in the field in the past few years. We think that our proposal will pave the way for long distance q-CDMA networks, and will give new perspectives for the optimization of quantum networks.
Methods
Averaging over the chaotic phase shift. A chaotic signal d i (t) can be expressed as a combination of many high-frequency components, i.e.,
d i t ð Þ~X a A ia cos v ia tzw ia ð Þ ,ð8Þ
where A ia , v ia , w ia are the amplitude, frequency, and phase of each component, respectively. Then the phase of the signal at any given time t can be written as
h i t ð Þ~ð t 0 d i t ð Þdt~X a A ia v ia sin v ia tzw ia ð Þ :
Using the Fourier-Bessel series identity 31 :
exp ix sin y ð Þ X n J n x ð Þexp iny ð Þ,
with J n (x) as the n-th Bessel function of the first kind, we can write
exp {ih i t ð Þ ½ P a X na J na A ia v ia e {inaviat{inaw ia " # :
If we take average over the ''random'' phase h i (t), the components related to the frequencies v ia should appear as fast-oscillating terms and thus can be averaged out. This treatment corresponds to averaging out the components that are far offresonance with the information-bearing field, and keeping only the near-resonance components. Hence, only the lowest-frequency terms, with n a 5 0, dominate the dynamical evolution. Thus, we have
exp {ih i t ð Þ ½ P a J 0 A ia v ia :ð9Þ
Since the chaotic signal d i (t) is mainly distributed in the high-frequency regime, we have A ia =v ia . Using the expressions J 0 (x) < 1 2 x 2 /4, log(1 1 x) < x for x=1, it is
Figure 1 |
1Illustration for different multiple-access methods. (a) TDMA: the users share the same frequency at different time slots. (b) FDMA: different frequency bands are assigned to different data-streams. (c) CDMA: the entire spectrum is utilized to encode the information from all users, and different users are distinguished with their own unique codes. Each user in the network is represented by a different color.
Figure 2 |
2Diagrams of the quantum multiple access networks.
Figure 3 |
3Quantum state transmission over q-CDMA network. (a) The broom-shaped or shovel-shaped purple symbols denote photon detectors. The red arrow inside each (green) cavity denotes the classical driving field with amplitude V i (t) (i 5 1, 2, 3, 4). The green circles denote L-type threelevel atoms. (b) Schematic diagram of the chaotic synchronization realized by the moving mirrors. (c) Chaotic encoding and decoding by electrooptic modulators.
Figure 4 |
4Fidelities of quantum state transmission. (a) Fidelities F 1 andF 2 versus the strength f d of the driving force acting on the Duffing oscillator with p 0 5 0.6, and t 5 2p/v 0 as the unit of time. (b) F 1 , F 2 and their average (F 1 1 F 2 )/2 versus p 0 in the hard-chaotic region, with f d /v 0 5 36. The average fidelity is maximized at p 0 5 0.5, which corresponds to | w 1 ae 5 | w 2 ae.
Figure 5 |Figure 6 |
56Quantum information transmission rates. (a) Fidelities F 1 and F 2 versus the strength f d of the driving force acting on the Duffing oscillator with p 0 5 0.6, and t 5 2p/v 0 as the unit of time. (b) Upper bounds of the classical and quantum information transmission rates of different methods for ideal channel with g 5 1 versus the number of the user pairs N. (c) and (d) Upper bounds of classical (quantum) information transmission rates of different methods for noisy channel with 0 , g , 1. The correction factor in the q-CDMA network is M 5 0.01. FDMA is constrained by the frequency bandwidth dv/v 5 0.2. All the methods are constrained with the total energy E=v~1. C g c q ð Þ,CDMA FDMA ð Þ denote the classical (c) and quantum (q) information transmission rates in q-CDMA and q-FDMA networks with transmissivity g. The rates for the single user-pair channel are C g c,sig and C g q,sig . Input-output structure of quantum CDMA network. The black dashed lines denote the desired chaotic synchronization channel. The red lines show the quantum optical channels. ''LA'' refers to linear amplifier. ''BS'' refers to beamsplitter. ''CPS'' denotes chaotic phase shifter.
www.nature.com/scientificreports SCIENTIFIC REPORTS | 3 : 2211 | DOI: 10.1038/srep02211
SCIENTIFIC REPORTS | 3 : 2211 | DOI: 10.1038/srep02211
whereConsequently, from Eqs.(9)and(10), we obtain the equationInput-output relationship of the quantum CDMA network. Here we calculate the input-output relationship of the quantum CDMA network shown inFig. 6, we can express the input-output relationships of the chaotic phase shifters CPS i 5 1,2,3,4 asand those of the two beam splitters BS 1 and BS 2 and the linear quantum amplifier ''LA'', respectively, asandThen, using Eqs.(12)(13)(14)(15), we obtain the total input-output relationship of the quantum network aswhere h 1 and h 2 are independent chaotic ''noises'' as we have not considered chaos synchronization yet.Author contributionsJ.Z. proposed the main idea. J.Z., S.K.O., Y.X.L. and F.N. wrote the main manuscript text. R.B.W., F.F.G., X.B.W. and L.Y. participated in discussing the results and contributed to the findings of this paper.Additional informationCompeting financial interests: The authors declare no competing financial interests.
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Grant-in-Aid for Scientific Research (S), MEXT Kakenhi on Quantum Cybernetics, and the JSPS via its FIRST program. J.Z. and R.B.W. are supported by the NSFC under Grant Nos.61174084, 61134008, 60904034, and project supported by State Key Laboratory of Robotics. J Z Acknowledgements, X Y Dr, Lü, M Prof, 12-02-92100MURI Center for Dynamic Magneto-Optics. 0726909Shenyang Institute of Automation Chinese Academy of SciencesAll authors thank F. Monifi for the illustrations in Fig. 1. F.N. is partially supported by the ARO, RIKEN iTHES Project. is supported by the NSFC under Grant Nos. 10975080, 61025022, 60836001. L.Y. is supported by ARO grant No. W911NF-12-1-0026Acknowledgements J.Z. thanks Dr. X.Y.Lü. and Prof M. Tsang for helpful discussions, and also thanks Dr. G. Y. Chen for providing related materials. All authors thank F. Monifi for the illustrations in Fig. 1. F.N. is partially supported by the ARO, RIKEN iTHES Project, MURI Center for Dynamic Magneto-Optics, NSF grant No.0726909, JSPS-RFBR contract No. 12-02-92100, Grant-in-Aid for Scientific Research (S), MEXT Kakenhi on Quantum Cybernetics, and the JSPS via its FIRST program. J.Z. and R.B.W. are supported by the NSFC under Grant Nos.61174084, 61134008, 60904034, and project supported by State Key Laboratory of Robotics, Shenyang Institute of Automation Chinese Academy of Sciences, China. Y.X.L. is supported by the NSFC under Grant Nos. 10975080, 61025022, 60836001. L.Y. is supported by ARO grant No. W911NF-12-1-0026.
| [] |
[
"Reaction dynamics on amorphous solid water surfaces using interatomic machine learned potentials Microscopic energy partition revealed from the P + H − −− → PH reaction",
"Reaction dynamics on amorphous solid water surfaces using interatomic machine learned potentials Microscopic energy partition revealed from the P + H − −− → PH reaction"
] | [
"G Molpeceres [email protected] \nDepartment of Astronomy\nGraduate School of Science\nThe University of Tokyo\n113-0033TokyoJapan\n",
"V Zaverkin [email protected] \nFaculty of Chemistry, Institute for Theoretical Chemistry\nUniversity of Stuttgart\n\n",
"K Furuya \nNational Astronomical Observatory of Japan\n181-8588TokyoJapan\n",
"Y Aikawa \nDepartment of Astronomy\nGraduate School of Science\nThe University of Tokyo\n113-0033TokyoJapan\n",
"J Kästner \nFaculty of Chemistry, Institute for Theoretical Chemistry\nUniversity of Stuttgart\n\n"
] | [
"Department of Astronomy\nGraduate School of Science\nThe University of Tokyo\n113-0033TokyoJapan",
"Faculty of Chemistry, Institute for Theoretical Chemistry\nUniversity of Stuttgart\n",
"National Astronomical Observatory of Japan\n181-8588TokyoJapan",
"Department of Astronomy\nGraduate School of Science\nThe University of Tokyo\n113-0033TokyoJapan",
"Faculty of Chemistry, Institute for Theoretical Chemistry\nUniversity of Stuttgart\n"
] | [] | Context. Energy redistribution after a chemical reaction is one of the few mechanisms to explain the diffusion and desorption of molecules which require more energy than the thermal energy available in quiescent molecular clouds (10 K). This energy distribution can be important in phosphorous hydrides, elusive yet fundamental molecules for interstellar prebiotic chemistry. Aims. Our goal with this study is to use state-of-the-art methods to determine the fate of the chemical energy in the simplest phosphorous hydride reaction. Methods. We studied the reaction dynamics of the P + H − −− → PH reaction on amorphous solid water, a reaction of astrophysical interest, using ab-initio molecular dynamics with atomic forces evaluated by a neural network interatomic potential. Results. We found that the exact nature of the initial phosphorous binding sites is less relevant for the energy dissipation process because the nascent PH molecule rapidly migrates to sites with higher binding energy after the reaction. Non-thermal diffusion and desorption-after-reaction were observed and occurred early in the dynamics, essentially decoupled from the dissipation of the chemical reaction energy. From an extensive sampling of reactions on sites, we constrained the average dissipated reaction energy within the simulation time (50 ps) to be between 50 and 70 %. Most importantly, the fraction of translational energy acquired by the formed molecule was found to be mostly between 1 and 5 %. Conclusions. Including these values, specifically for the test cases of 2% and 5% of translational energy conversion, in astrochemical models, reveals very low gas-phase abundances of PH x molecules and reflects that considering binding energy distributions is paramount for correctly merging microscopic and macroscopic modelling of non-thermal surface astrochemical processes. Finally, we found that PD molecules dissipate more of the reaction energy. This effect can be relevant for the deuterium fractionation and preferential distillation of molecules in the interstellar medium. | 10.1051/0004-6361/202346073 | [
"https://export.arxiv.org/pdf/2303.03059v1.pdf"
] | 257,365,845 | 2303.03059 | 8b6dbc115a23659cb3c30b5291eb463009beba17 |
Reaction dynamics on amorphous solid water surfaces using interatomic machine learned potentials Microscopic energy partition revealed from the P + H − −− → PH reaction
March 7, 2023
G Molpeceres [email protected]
Department of Astronomy
Graduate School of Science
The University of Tokyo
113-0033TokyoJapan
V Zaverkin [email protected]
Faculty of Chemistry, Institute for Theoretical Chemistry
University of Stuttgart
K Furuya
National Astronomical Observatory of Japan
181-8588TokyoJapan
Y Aikawa
Department of Astronomy
Graduate School of Science
The University of Tokyo
113-0033TokyoJapan
J Kästner
Faculty of Chemistry, Institute for Theoretical Chemistry
University of Stuttgart
Reaction dynamics on amorphous solid water surfaces using interatomic machine learned potentials Microscopic energy partition revealed from the P + H − −− → PH reaction
March 7, 2023Received March 7, 2023; accepted March 7, 2023Astronomy & Astrophysics manuscript no. aandaISM: molecules -Molecular Data -Astrochemistry -methods: numerical
Context. Energy redistribution after a chemical reaction is one of the few mechanisms to explain the diffusion and desorption of molecules which require more energy than the thermal energy available in quiescent molecular clouds (10 K). This energy distribution can be important in phosphorous hydrides, elusive yet fundamental molecules for interstellar prebiotic chemistry. Aims. Our goal with this study is to use state-of-the-art methods to determine the fate of the chemical energy in the simplest phosphorous hydride reaction. Methods. We studied the reaction dynamics of the P + H − −− → PH reaction on amorphous solid water, a reaction of astrophysical interest, using ab-initio molecular dynamics with atomic forces evaluated by a neural network interatomic potential. Results. We found that the exact nature of the initial phosphorous binding sites is less relevant for the energy dissipation process because the nascent PH molecule rapidly migrates to sites with higher binding energy after the reaction. Non-thermal diffusion and desorption-after-reaction were observed and occurred early in the dynamics, essentially decoupled from the dissipation of the chemical reaction energy. From an extensive sampling of reactions on sites, we constrained the average dissipated reaction energy within the simulation time (50 ps) to be between 50 and 70 %. Most importantly, the fraction of translational energy acquired by the formed molecule was found to be mostly between 1 and 5 %. Conclusions. Including these values, specifically for the test cases of 2% and 5% of translational energy conversion, in astrochemical models, reveals very low gas-phase abundances of PH x molecules and reflects that considering binding energy distributions is paramount for correctly merging microscopic and macroscopic modelling of non-thermal surface astrochemical processes. Finally, we found that PD molecules dissipate more of the reaction energy. This effect can be relevant for the deuterium fractionation and preferential distillation of molecules in the interstellar medium.
Introduction
Phosphorous is an element of great relevance for the chemistry of the interstellar medium (ISM). This is due to the intrinsic connection between abiotic and prebiotic phosphorous, with abiotic molecules such as phosphine (PH 3 ) serving as an indicator of the potential presence of more complex oxoacids (Turner et al. 2018). Biocompatible phosphorous can be found in several essential biomolecules, such as DNA, RNA, or phospholipids. So far, phosphorous has been unambiguously detected in different regions of the ISM in the form of CP, HCP, PN, PO, or HPO (Turner & Bally 1987;Guelin et al. 1990;Agúndez et al. 2007;Agúndez et al. 2014;Fontani et al. 2016;Lefloch et al. 2016;Rivilla et al. 2016Rivilla et al. , 2018Ziurys et al. 2018). However, the detection of phosphorous-bearing molecules in cold star-forming regions is limited to PN, PO, and recently, PO + (Yamaguchi et al. 2011;Rivilla et al. 2016;Lefloch et al. 2016;Rivilla et al. 2018;Rivilla et al. 2020Rivilla et al. , 2022. Despite laboratory experiments and computational simulations show that PH 3 must be easily hydrogenated on dust grains (Nguyen et al. 2020;Molpeceres & Käst-ner 2021;Nguyen et al. 2021), no single phosphorous hydride has been detected in the cold ISM (Chantzos et al. 2020).
The sequence of hydrogenation of phosphorous atoms in the ISM is known to finalize with the formation of phosphine, PH 3 Nguyen et al. 2021) and proceeds on interstellar dust grains through a sequence P +H − −− → PH +H − −− → PH 2 +H − −− → PH 3 . Competing with this sequence, there are reactions mediated by quantum tunnelling in the backward direction PH 3 + H − −− → PH 2 + H 2 , PH 2 + H − −− → PH + H 2 , PH + H − −− → P + H 2 that proceed with very low activation barriers and are very rapid, even at 10 K . These establish a pseudo-equilibrium that favours addition (PH 3 as end molecule) in the absence of other mechanisms that reduce the intermediate species' abundance. We find chemical desorption and non-thermal diffusion among this sequence's possible mechanisms. The low binding energies (BE) of the Pbearing molecules combined with the confirmation of chemical desorption of phosphine under astrophysical conditions (Nguyen et al. 2020;Furuya et al. 2022b) Article number, page 1 of 14 arXiv:2303.03059v1 [astro-ph.GA] 6 Mar 2023 A&A proofs: manuscript no. aanda indicate that the hydrogenation sequence may be altered nonthermally such as the release of intermediates to the gas phase or by fast diffusive reactions with other atoms or radicals, promoted by the release of chemical energy.
Regrettably, there is no possibility of determining the efficiency of non-thermal effects from static quantum chemical calculations, not only in phosphorous chemistry but also in interstellar surface chemistry. This is because the appearance of non-thermal effects requires the interplay between energy dissipation to the bulk and the conversion of the reaction energy to kinetic energy. Classical molecular dynamics simulations have been used to that end, targeting, for example, HCO formation on H 2 O (Pantaleone et al. 2020), H 2 formation on H 2 O (Pantaleone et al. 2021), or energy dissipation of CO 2 , H 2 O and CH 4 on water ice after inoculation of a significant amount of energy (Fredon et al. 2017;Fredon & Cuppen 2018;Fredon et al. 2021;Upadhyay et al. 2021;Upadhyay & Meuwly 2022). Finally, very recently, the hydrogenation sequence of the nitrogen atom was studied on H 2 O (Ferrero et al. 2022b). The interaction potentials for these studies come in different flavours, such as empirical pair potentials or on-the-fly ab-initio molecular dynamics simulations. The former's low computational cost is the latter's main drawback. Likewise, ab-initio molecular dynamics' main advantage is that it is, in principle, general, can treat reactive systems, and new potentials do not need to be generated for each system.
In this work, we study the energy dissipation following the first hydrogenation step of the phosphorous atom (P) with a twofold aim. First of all, we study the energy dissipation dynamics of a relatively heavy radical PH after its formation via P + H − −− → PH. The energy dissipation and reaction energy redistribution to the different degrees of freedom of the molecule is the crucial quantity behind non-thermal events and, ultimately, in the fate of phosphorous hydrides in the ISM. The second aim of this paper is to serve as a proof-of-concept for applying interatomic neural network potentials to study reactions in the context of surface astrochemistry. These potentials permit an extensive sampling of reaction trajectories while maintaining a reasonable computational cost, yet somewhat higher than empirical potentials. Using neural-network potentials opens new avenues for studying any adsorbate on any substrate, provided that the chosen reference method can reproduce the electronic structure of the system. The PH radical is especially suited as a test case for several reasons. First, its formation reaction is barrierless and exothermic (∼ 305-315 kJ mol −1 ). Second, the binding energy of the PH radical is relatively low, especially compared to the reaction energy, permitting the study of various outcomes after the reaction. Third, the reaction occurs on a triplet potential energy surface (PES) which facilitates the construction of the training set compared with more complex radical-radical recombinations occurring in the singlet channel. Fourth and last, PH is a sufficiently heavy molecule in which the drawbacks of using classical dynamics not accounting for nuclear quantum effects can be partially palliated. This paper's structure is as follows. In Section 2 we briefly introduce the methodology employed in the paper and the particularities of the training set used to train the interatomic neural network potential. In Section 3, we present the results derived from the present study, starting with the adsorption energetics of the P atom at our level of theory and continuing with exploratory molecular dynamics simulations of different binding sites. Later, we focus on extensive sampling, extracting statistical values on energy dissipation and translational kinetic energy of the newly formed PH radical. Finally, we discuss our results under an as-trochemical prism in Section 4, and include our results in an astrochemical model of a molecular cloud.
Methods
Interatomic Potential Model
In this work, we have developed an entirely reactive machineleaned interatomic potential (MLIP) to study the reaction dynamics for forming the PH molecule on top of amorphous solid water (ASW). Our approach resembles Born-Oppenheimer molecular dynamics at a fraction of its cost. In our simulations, we must sample many trajectories with different reaction outcomes for sufficiently long time scales. Moreover, we need sufficient energy conservation to obtain correct dynamics, especially during the initial step of the reaction. Small integration time steps ensure the desired energy conservation, vide infra. We have achieved all these prerequisites by combining ample sampling with a Gaussian-Moment Neural Network (GMNN) potential Zaverkin et al. 2021a) trained on energies and atomic forces obtained from density functional theory (DFT). The GMNN source code is available free-of-charge at gitlab.com/zaverkin_v/gmnn.
To calculate reference energy and atomic force values, we use the DFT method PBE-D3BJ/def2-TZVP (Perdew et al. 1996;Weigend & Ahlrichs 2005;Grimme et al. 2011) as it provides similar energetics to those obtained by ; Nguyen et al. (2021). The P+H − −− → PH reaction energy (without zero-point vibrational energy, ZPVE) in the gas phase is within 9 kJ mol −1 of reference UCCSD(T)-F12/cc-pVTZ-F12 calculations, −314 vs. −305 kJ mol −1 . We studied the reaction in the triplet channel since the quintet channel was found to be repulsive in an earlier publication . The training data set generated to train the GMNN model comprises 11,249 structures. The training set size required to achieve the accuracy of 1 kcal mol −1 with respect to the reference electronic structure method is smaller than in our previous works Zaverkin et al. 2021b) owing to the recent improvements in our model (Zaverkin et al. 2021a). The electronic structure calculations were carried out using the ORCA (v 5.0.3) code (Neese 2012;Neese et al. 2020).
A summary of the structures which comprise the training set can be found in Table 1. Like in our previous work , we generated the training structures by running ab-initio molecular dynamics at exploratory, computationally cheaper levels of theory. Here, we used a combination of HF-3c (Sure & Grimme 2013) for the exploration of reactions on clusters and GFN2-xTB (Bannwarth et al. 2019), as well as GFNFF (Spicher & Grimme 2020) for studying water clusters and isolated adsorbates on water clusters. The training set includes sub-sets tailored to reproduce the water-water interaction, for which molecular dynamics in the canonical (NVT) ensemble was carried out, and other sub-sets simulating collision and reactions, where a microcanonical ensemble (NVE) was used.
Three independent MLIP models were then trained for 1000 epochs. All models have been trained on 9000 structures from the data set in Table 1, while other 1500 structures have been reserved for the early stopping technique (Prechelt 2012). The achieved mean absolute error with respect to the reference DFT level is about 0.5 kcal mol −1 and 0.3 kcal mol −1 Å −1 for total energies and atomic forces, respectively. The provided errors have been evaluated on the test data set consisting of the remaining 749 structures. We then used an ensemble of the three models to predict energies and atomic forces while running our simula- Table 1. Composition of the data set used for training the machine-learned interatomic potentials. Note that the propagation method is only used for sampling geometries. For these structures, energies and forces are later calculated at the PBE-D3BJ/def2-TZVP level of theory. tions. The cutoff radius is set to 5.5 Å because we are interested in short and medium-range interactions of the adsorbates with the surface. All other relevant hyper-parameters used for training our NN potentials can be found in the original work (Zaverkin et al. 2021a). The advantage of using an ensemble is two-fold. First, lower error values in the potential energy can be achieved, and the predicted atomic forces are much smoother than for a single model. The latter is crucial for energy conservation. Second, the variance or disagreement between the three independent models estimates the uncertainty in predicted energy and atomic force values. Thus, using an ensemble of at least three NN potentials provides the means of running a more stable molecular dynamics simulation and allows for estimating uncertainties in predicted values such as binding energy, reaction energy and reaction profiles. Finally, a detailed analysis of the accuracy of the employed NN ensemble is provided in Appendix A.
Water ice surface model
With the MLIP trained, we created a slab model for an amorphous solid water surface by running heating and cooling cycles of molecular dynamics simulations. Very briefly, from a randomly, pre-optimized and fully periodic packed simulation cell with 500 water molecules at ρ=0.998 g cm −3 we run NVT dynamics at 300 K for 100 ps using a Langevin thermostat to control the temperature (friction coefficients of 0.02 ps −1 and 0.5 fs timestep). Five different snapshots along the trajectory are chosen and later quenched to 10 K for 10.0 ps (friction coefficients of 0.05 ps −1 and 0.5 fs timestep), serving as initial models for the subsequent reactivity studies. This system is periodic in the X and Y directions but not in the Z-direction. We propagate our dynamics in periodic systems while having trained on cluster data. We refer to our recent publication for details on this procedure (Zaverkin et al. 2022).
We fixed the bottom layers of the simulation cell in the Z direction to simulate a bulk structure. Once we had the structural models for the ice ready, we sampled the binding energy distribution of the P atom by placing P atoms atop the surface and relaxing the structure. Finally, from representative configu-rations of the binding energy distribution (see Section 3.1), we start NVE simulations to trace the reaction dynamics. The chemical reaction under investigation is P+H − −− → PH. The incoming H atoms are placed around the pre-adsorbed P atom in different binding sites at a distance of 3.5 Å. For all simulations, we run NVE dynamics for 50 ps using an integration timestep of 0.25 fs. All molecular dynamics calculations were performed with the ASE simulation package (Hjorth Larsen et al. 2017).
Results
Binding energy distributions calculation
The binding energy distribution of the P atom and the PH molecule on ASW, computed with the MLIP, is very similar to our previous work (Molpeceres & Kästner 2021) using more sophisticated density functionals explicit calculations. To ease the discussion in Fig. 1, we identify several segments. One corresponds to high binding (HB), at values of > 120% of the average binding energy (which for P is at 1371 K, equivalent to 1241 K of our previous work and other works (Nguyen et al. 2021)). Medium binding (MB) sites, around the average binding energy, weak binding (WB) sites at 70-40% of the average binding energy, and very weak binding (VW) sites at < 40% of the average binding energy. Overall, the binding energy of P on ASW is relatively small for an atom as heavy as phosphorus. The average binding energy for PH is 1843 K, also in good agreement with our previous work (1616 K; ), especially considering that we omitted ZPVE in the distributions because we cannot capture the ZPVE during the reaction dynamics.
As mentioned above, we selected four initial configurations to study the reaction from the distribution of binding energies for the P atom. The local environments for each configuration in one of our ice models are depicted in Fig. 2. The figure illustrates that more oxygen neighbours increase the binding energy, as predicted previously (Cuppen et al. 2013). On the contrary, interactions with H atoms lead to weaker binding. The first part of our study evaluated the binding site's dependence on the dynamics, which was sampled for 19 trajectories because we faced difficulties finding a VW situation for one of the ice models. As We sampled 245 binding sites to obtain these distributions.
mentioned, the P-H incoming distance was set to 3.5 Å. We run a short (∼ 2 ps) NVT simulation to initialize the velocities of the water molecules to 10 K before the production run. During this short pre-equilibration, both P and H positions can relax, but we enforce a constraint in the internuclear distance to the abovementioned value.
Exploratory PH reaction dynamics
An example of the onset of the NVE dynamics for a binding site is presented in Fig. 3. From each of the trajectories, we monitored the evolution of the kinetic energy of the nascent PH molecule along the trajectory and the kinetic energy of all the water molecules within the ice. Two exemplary kinetic energy profiles for the reaction can be found in Fig. 4, left panels. For comparison, we have selected two trajectories with very similar binding energies but different energy dissipation behaviour. In the case shown on top, the ice structure accommodates the excess reaction energy into the ice lattice quite fast, whereas, in the one shown at the bottom, most of the reaction energy remains longer as vibrational and rotational energy within the molecule. While this does not mean that energy dissipation does not occur, it hints at different dissipation timescales but without a significant dependence on the binding site.
To check the dependence of the energy dissipation process on the initial binding site, we have gathered the outcome of all these trajectories and compared their initial P binding energy with their final PH binding energy. The results can be found in Table 2. We represent the population of P atoms in a given binding site before and after the reaction, along with the fraction of dissipated energy for each binding site in each ice model. The fraction of dissipated energy is calculated as: where T final PH is the final kinetic energy averaged over the last 2.5 ps (to avoid oscillations) of the PH molecule and T max PH is the maximum of the kinetic energy of PH along the trajectory. Both quantities are calculated from the running averages with a window size of 25 fs. We found different values of F for different binding sites and ice models. However, the dependence on the different binding sites is unclear, indicating that further sampling is required. We increased the sampling for MB and HB sites in Section 3.3.
F = T final PH − T max PH T max PH ,(1)
The lack of correlation between F and the binding site can be found by looking at the different populations of P binding sites before the reaction and PH binding sites after the reaction shown by the second and the third column in Table 2. While we start our sequential simulations with P atoms in VW and WB binding sites along with MB and HB, we find that PH populates only HB and MB sites at the end. This is an essential conclusion of our work, pointing to the fact that nascent molecules arising from exothermic radical-radical recombination are unlikely to be found in the weak binding sites on a pristine surface and affecting the (effective) distributions of binding energies recently reported in the literature for low-temperature studies (Ferrero et al. 2020;Duflot et al. 2021;Bovolenta et al. 2022). The population of HB and MB sites comes as a result of non-thermal diffusion of the nascent PH molecule that uses part of the reaction energy to freely roam the surface until landing on an HB or MB site.
We also evaluated the translational energy (part of the kinetic energy corresponding to the translational motion of the whole PH molecule) depicted in the right panels of Fig. 4 as the instantaneous kinetic energy (no running average). It is calculated as the kinetic energy of the centre of mass (T COM ). From the right panels of Fig. 4, we observe that, despite having similar P binding energies, these PH molecules acquire different translational energies. Both roam the surface before thermally equilibrating and land at a site where they remain for the rest of the simulation. The maximum of the translational energy is, on average, only between 1 and 5 % of the reaction energy (-314 kJ mol−1, as a univocal value), with significant variability. Later we incorporate these values into astrochemical models.
The key to energy dissipation is the binding site where the particles land rather than the binding site where the molecule is formed, in contrast to non-thermal events (like non-thermal diffusion) that depend on the initial site. That conclusion highlights the importance of the ice morphology in the whole energy dissipation vs. non-thermal processes interplay. Non-thermal events are characteristic of VW, and WB sites, where P (and PH) interact with fewer neighbours. Hence, a molecule formed in a region with a low density of water molecules will be more likely to experience non-thermal diffusion or desorption. It is essential to mention that even starting the reaction from HB and MB sites, the newly formed PH molecule still may non-thermally diffuse to other (different) HB and MB sites. The fraction of PH molecules diffusing from MB and HB sites can be as low as 7% for HB, indicating the correlation between the binding energy and nonthermal events. In one out of the 19 trajectories, we observed chemical desorption. We can attribute the mechanism of desorption to the interaction of a highly rotationally excited PH molecule with a dangling O-H bond during an instantaneous exploration of a very repulsive part of the potential (short H-H distance of 1.51 Å), counteracted by momentum conservation and desorp-tion, see Fig. 5. Another important observation of the present work is that the type of non-thermal process, either diffusion or desorption, is determined early in the dynamics (1-10 ps, inferred from the time at which peaks in the translational energy of Fig. 4 and Fig. 5 take place) indicating essentially different timescales between non-thermal processes and energy dissipation. The latter, for example, can take up to a few nanoseconds,e.g., for CO 2 molecules (Upadhyay & Meuwly 2022). In Upadhyay & Meuwly (2022) the authors also found that there are two components for the energy dissipation, one rapid, occurring in tens of picoseconds, which is coherent with our observed behaviour in Table 2 for F . The second one, occurring in the nanosecond scale is not captured in our simulations because of the simulated time-window. We note that the PH molecule of the desorbing trajectory possesses a translational kinetic energy three times lower than the average binding energy for the PH radical on ASW. This shows that the likelihood of chemical desorption, like adsorption or diffusion, depends on the whole distribution of binding energies, rather than only on its average.
Our results are also in line with recent results on the N + 3 H − −− → NH 3 system, with dissipation fractions between 58 and 90% within the first picoseconds of the reaction dynamics (Ferrero et al. 2022b). We observe a similar qualitative behaviour at longer time scales (e.g. Figure 4, left panels and Table 2), which can be attributed to the different masses and vibrational frequencies of PH and NH.
Extended sampling
In the previous section, we determined that the timescales for non-thermal diffusion and desorption are different compared to the energy dissipation. Based on that, we also conclude that, except for the trajectories leading to immediate desorption of PH, the MB and HB are especially interesting for investigating/evaluating energy dissipation because, after the few picoseconds of a simulation, they are the only sites with a PH population different from zero (Table 2), whereas WB sites are the most interesting from a chemical desorption perspective. P is a physisorbed species with relatively low binding energy, so VW binding sites are most likely transient.
To extract reliable statistics on F and related quantities, namely the maximum translational energy (T COM ), the linear displacement (d) of the PH molecule after formation, and the number of trajectories with significant non-thermal diffusion (N diff , defined as the fraction of trajectories with d > 2 Å), we have carried out an extensive sampling of MD trajectories (358 trajectories, similar to the ones presented before) starting only from MB, HB, WB situations. We observed explicit chemical desorption in three of 358 trajectories, all of which have been initialized from a WB site. These results correspond to about 1% and the 3 % of total chemical desorption probability and chemical desorption probability coming from WB sites, respectively. The latter number is similar to the one found for PH 3 (a similar system with more degrees of freedom) in experiments (Nguyen et al. 2020), suggesting that the experimental observations may also come from WB sites. Note, however, that the sampling is not enough to guarantee a statistical match between theory and experiments, summed to the inherent differences between the PH and PH 3 system. That is why we refer to our discussion in terms of translational energy in Section 4. The significant quantities for these trajectories are shown in Table 3. There, we also include statistics of further 233 trajectories, in which the reactive H atom was replaced by D for MB and Table 3. Number of trajectories, average dissipation fraction (F ), average translational kinetic energy (T COM , in kJ mol −1 ), the fraction of trajectories exhibiting non-thermal diffusion (N diff , see text) and average diffusion distance for the diffusing trajectories (d, in Å, the standard deviation in parentheses). HB sites (i.e. MB-D and HB-D). We did not include the study with deuterium in WB sites because of the computational expense of the respective calculations. With the deuterium substitution, we want to observe whether the different vibrational properties of PH and PD affect the dynamics of the nascent molecule. For example, PD has its fundamental vibrational mode in the range of water bendings (∼ 1600 cm −1 ), unlike PH that appears in a clean region of the spectra at 2218 cm −1 , which can facilitate the energy dissipation through the bending modes of water. The analysis of the trajectories for the MD runs reveals several important conclusions. First, we have found that the type of binding site significantly affects the energy dissipation in the considered timescales. This fact could be inferred from previous studies (Garrod, R. T. et al. 2007;Minissale, M. et al. 2016;Fredon & Cuppen 2018;Fredon et al. 2021) where the efficiency of chemical desorption is a function of BE. However, as previously indicated for thermal desorption studies Ferrero et al. 2022a;Molpeceres, G. et al. 2022), we must emphasize that considering the whole distribution of binding sites is essential to unravel the role of non-thermal mechanisms. Here we found that the binding energy anticorrelates with the fraction of energy dissipated into the lattice and the acquired translational energy of the nascent molecule. On average, the reaction energy dissipates a 12% more in high-binding sites. Likewise, T COM , N diff , and d are significantly higher for MB sites than HB sites. This indicates the importance of different factors in energy dissipation and related properties.
It is essential to mention that, even though at the end of the dynamics, the reaction energy is not entirely dissipated (see Figure 4), the translational component T COM is already wholly dissipated. For WB sites, we observe the highest T COM , N diff , and d which reinforces the idea that non-thermal events are maximized on WB sites. It is worth mentioning that any trajectory starting from WB, PH migrates to a deep binding site. Moreover, F for WB does not follow the trend that we saw for MB and HB, with F values in between HB and MB. As we mentioned in Section 3.2, WB sites are not populated after the reaction, and the reaction energy dissipation takes place either in MB and HB sites following rapid diffusion after formation, and this observation is further reinforced by the values of F starting from WB sites, appearing in between MB and HB values. For T COM , we keep finding values corresponding to 1-5% of the total reaction energy, rarely going as high as 6.8% in just a single case, in a case starting from a WB situation.
The efficiency of energy dissipation also depends on the hydrogen isotope under consideration. We found that, on average, the energy dissipates 6-7 % faster when the incoming particle is deuterium, a fact that could be attributed to the coupling of PD with the bending vibration modes of water, as mentioned above. Similarly, we observed that the average diffusion length of the nascent deuterated molecule is lower than for hydrogen additions, by ∼0.5 Å on average (which corresponds to a 10% of the total traversed distance). At this point, it is important to stress that our classical MD simulations disregard the differences that arise from the quantum nature of H and D. For example, ZPVE is not included in our simulations. We do not expect significant changes in the dynamics at short timescales because of how energetic the reaction is. However, quantum effects may be significant during thermalization, and are vital in supporting quantitative claims in the deuterations.
Astrochemical Implications
Non-thermal effects of relevance to interstellar chemistry: Astrochemical models
The findings presented in this paper carry significant implications for the chemistry of phosphorous-bearing species in particular and interstellar chemistry in general. As we presented in the introduction, there is no single detection of phosphorous hydrides, including phosphine (PH 3 ), in cold astronomical environments. One common explanation for the absence of a molecule (or family of molecules) in the cold ISM relies on chemical conversions and the appearance of proxy species, a hypothesis applicable to the gas and solid phase (Shingledecker et al. 2019(Shingledecker et al. , 2020Cooke et al. 2020;Shingledecker et al. 2022)). The PH molecule is the most simple phosphorous hydride, and it is thought to be the current onset in the formation of interstellar PH 3 (Chantzos et al. 2020;, but an early release to the gas-phase or efficient non-thermal diffusion to encounter other radicals and react may decrease its abundance. Our simulations reveal that between 1 and 5% of the reaction energy is transferred to translation, a quantity labelled χ in the literature (Fredon et al. 2021). This number is in agreement with the current formulation of chemical desorption in terms of RRKM theories (Garrod, R. T. et al. 2007;Minissale, M. et al. 2016), where a value between 0.5 and 1.0 % is assumed for the fraction of energy going to the desorption mode (e.g., oscillation of the z coordinate of the molecule's centre of mass). Since we are considering three translational degrees of freedom instead of the single one used in the statistical study of chemical desorption, our values are in the range predicted by those models. Our results also fall within the lower bounds considered in the chemical models of Fredon et al. (2021) and specifically with their low-conversion energy model of assumed 5% of energy conversion. For the specific P + H − −− → PH reaction, 1-5 % of energy conversion corresponds to 3.14-15.7 kJ mol −1 energy available for diffusion or desorption, assuming a univocal value of the reaction energy corresponding to -314 kJ mol−1 (the value in the gas phase).
Regarding non-thermal diffusion, the 1-5 % of the reaction energy corresponds to more or less the average binding energy of the PH molecule and assuming a diffusion energy is a fraction of that energy (even though such an assumption must be done with care under astrophysical conditions, see Furuya et al. (2022a)), the PH molecule can visit several binding sites, reacting in the process with other pre-adsorbed molecules. However, the translation energy may not be enough to overcome any diffusion barriers for reactions starting in high-energy binding sites. This is illustrated by the value of d, Table 3. From an astrochemical per-spective, one important factor is missing in our simulations: the H 2 molecule coverage on the surface. At 10 K, most high-energy binding sites will be populated by H 2 molecules (10 orders of magnitude more abundant than the most abundant phosphorous bearing molecule (Chantzos et al. 2020)), and PH would likely not be locked on high binding sites. Our average diffusing distance (d) is, therefore, a lower bound of the real one. Hence PH can perform a few hops on the grain, meeting reaction partners and converting to more complex P-bearing molecules. This contrasts with the classical picture of PH x forming under thermal conditions by hydrogenation in which only hydrogen can move. Thus, we suggest that the effective binding energy applicable to non-thermal events should be lower than the average binding energy computed from distributions, which we tested later in our chemical models (see below).
The other critical non-thermal process is chemical desorption, a phenomenon vital for explaining the return of interstellar adsorbates to the gas phase, where they are ultimately detected. Chemical desorption has been treated analytically in the astrochemical literature. Corresponding approaches are briefly reviewed here. The most straightforward scheme assumes a constant desorption probability of normally 1% for every reaction. The second Garrod, R. T. et al. (2007) propose a scheme based on RRKM theory explicitly introducing the binding energy and reaction enthalpy in the formalism. Improving on that, Minissale, M. et al. (2016) also included a collisional parameter ( ) where a mass term is introduced, allowing for good agreement for rigid surfaces (graphite) under the assumption of a purely elastic collision. The last attempt to analytically formulate this phenomenon can be found in Fredon et al. (2021), where the authors explicitly consider the fraction of energy inoculated into translational degrees of freedom (χ), which is a step beyond the equipartition of the reaction energy predicted by RRKM theory. In the latter study, the authors indicate that the least known part of their model is the determination of χ, which is most likely not constant for every reaction. Here, we have found χ to be between 1 and 5 % with average values of around 2% for the P + H − −− → PH reaction (Table 3), depending on the binding site under consideration. The probability of chemical desorption (p) as a function of χ, ∆H r (the enthalpy of the reaction), and EB (the binding energy) proposed in Fredon et al. (2021) for one-product reactions is:
p = 0.5 1 − exp − χ∆H r − EB 3EB .(2)
We have explicitly addressed the impact of our derived χ values in a 3-phase, e.g. gas, grain and (inert) mantle astrochemical model of a molecular cloud. The model is a recent modification of the model reported by Furuya et al. (2022a) using the chemical network of Garrod (2013) and adding the hydrogenation sequence of the P atom based on our recent study and also the H 2 abstraction reactions for these species. The physical conditions of the model are gathered in Table 4 and are the standard conditions of a pseudo-timedependent molecular cloud model. Reaction-diffusion competition is explicitly considered in our models with diffusion energy assumed to be 0.4 of the binding energy. Quantum tunnelling of atomic H is taken into account in the evaluation of the diffusionreaction competition (Chang et al. 2007). Non-thermal mechanisms are accounted for, including photodesorption (with a constant photodesorption yield per photon of 1×10 −3 ), desorption induced by cosmic rays in its Hasegawa and Herbst formulation χ: 5 % Fig. 6. Astrochemical models for the evolution of PH x species in the gas (solid lines) and in the ice (dashed lines) for models with different treatment of chemical desorption. Labels in the topmost graph apply to all graphs. (Hasegawa & Herbst 1993) with a cosmic-ray-ionization-rate (ζ) of 1.3×10 −17 s −1 and, of most relevance for this paper, chemical desorption.
To see the impact on the phosphorous hydrides chemistry, we evaluated two chemical desorption schemes, one assuming a constant desorption probability for all phosphorous hydrogenation reactions of p = 3% following the recent experimental and modelling studies (Nguyen et al. 2021;Furuya et al. 2022b) (and 1% for all the other reactions) and another one using the Fredon et al. (2021) approach and explicitly including the χ values derived here for the P + H − −− → PH. Specifically, we took χ values Table 4. Initial physical conditions utilized in the molecular cloud model.
Parameter
Value n gas 2x10 4 cm −3 A v 10 mag ζ 1.3x10 −17 s −1 T g 10 K T d 10 K Table 5. Gas-phase maximum fractional (with respect to H nuclei) abundances and fractional abundances at t=1×10 7 yr of PH x molecules in the models shown in Figure 6. The notation A(-B) is equivalent to A×10 −B .
Peak Abundances t=1×10 7 yr Species p=3% χ=2% χ=5% p=3% χ=2% χ=5% P 7.3(-10) 7.1(-10) 7.2(-10) 2.4(-10) 7.3(-14) 3.3(-10) PH 3.7(-11) 4.1(-12) 2.7(-11) 1.2(-11) 1.6(-15) 1.5(-11) PH 2 3.9(-11) 2.7(-13) 4.4(-11) 1.6(-11) 1.6(-15) 3.2(-11) PH 3 6.7(-11) 3.5(-16) 1.2(-16) 1.4(-11) 1.5(-16) 4.4(-17) of 2.0 and 5.0 %. The former corresponds to the average value obtained in this work, while the latter represents an upper value of it. For simplicity, we assumed that χ is equal throughout the whole hydrogenation sequence. The χ value for all the other reactions in the reaction network is set to 1%. The binding energies for P and PHx molecules are taken from . The elemental abundances for the model are taken from Aikawa & Herbst (1999); e.g. corresponding to the low metallicity values, with initial P fractional abundances of 1×10 −9 with respect to H nuclei, and where all the elements are in atomic or ionic form but H, that is contained in H 2 . Enthalpies of formation used to compute ∆H r are taken from the original from the original Ruaud et al. (2016) reaction network for P + H − −− → PH and PH + H − −− → PH 2 , based on the KIDA database data (Wakelam et al. 2012). In contrast, for PH 3 , we took it from the NIST database because it was unavailable in the original reaction network.
The results for the model are presented in Figure 6 with peak abundances, and abundances at t=1×10 7 yr gathered in Table 5. From the figure, we extract several conclusions: First, we observe that the model with χ = 2.0 % yields gas abundances of P hydrides that are very low, which is partially palliated for χ = 5.0 %. This trend is inverted for ice abundance. In all models explicitly using χ, the abundance of PH 3 is very low, in contrast with the model assuming a constant desorption probability that is supported by recent experiments (Nguyen et al. 2021). We believe that the reduced efficiency for chemical desorption that we found for χ=2% and 5% stems from considering a single BE and a single χ parameter for our systems. From Table 3, we learned that there is an anti-correlation between BE and T COM (a proxy of χ) with a significant variability for χ.
We ran the same chemical models, with χ = 5% (as an upper value), and using an average BE for PH x that does not includes the high binding sites (Modified Value A), or considering the BE 0.7 of the real binding energy, to consider only weak binding sites (Modified Value B). Hence, the original values of are re-weighted according to these rules and included in the model. Table 6 enumerates these changes, where for MOdified Value A, we excluded the binding sites with energies higher than 120 % the average BE. The abundances for this chemical models are shown in Figure Table 6. Update of the PH x binding energies to exclude high-binding energy sites (Modified Value A), and to only consider weak-binding sites (0.70 of EB Modified Value B) Value B P 1241 963 869 PH 1616 1411 1128 PH 2 1808 1523 1265 PH 3 2189 1930 1532 Table 7. Same as Table 5 but with the modified binding energies scheme of Table 6 and corresponding to the models shown in Figure 7.
Molecule Molpeceres & Kästner (2021) Modified Value A Modified
Peak Abundances t=1×10 7 yr Species Model A Model B Model A Model B P 7.2(-10) 7.2(-10) 4.2(-10) 4.5(-10) PH 3.5(-11) 3.6(-11) 1.2(-11) 2.1(-11) PH 2 4.6(-11) 4.0(-11) 3.4(-11) 3.1(-11) PH 3 4.9(-12) 3.4(-11) 2.1(-12) 1.4(-11) 7 with peak fractional abundances and abundances at t=1×10 7 collated in Table 7. Abundance [H] χ: 5 %, Modified Value B Fig. 7. Astrochemical models equivalent to bottom panel in Figure 6, but updating the binding energies according to Table 6. Labels in the topmost graph apply to all graphs.
Even though the changes portrayed in Table 6, Modified Values A, are minimal and fall within the variability for binding energies as presented by various methods for other adsorbates (Das et al. 2018;Ferrero et al. 2020;Bovolenta et al. 2022), we ob-serve a significant increase in the abundance of gas-phase PH 3 by four orders of magnitude. The changes for the Modified Values B are even more drastic because they disregard average and deep binding sites (MB and HB), and we equally observe variations of 5 orders of magnitude in the P-bearing abundances with respect to the values using the original average BE, approaching the values predicted by a constant probability of chemical desorption. This is the key finding of our work because even with an average BE lower than reaction energies by two orders of magnitude, the binding site is still defining the dynamics of chemical desorption of the nascent molecule. For all of our portrayed molecules, both in Figure 6 and 7, we can calculate the analytic chemical desorption probability (p) as a function of χ. This is shown in Figure 8, and is helpful to rationalize our observations. We see that for the model with the BE and χ=2% ( Figure 6, middle panel), the chemical desorption probability is null for all reactions. The molecules' release to the gas phase is driven only by cosmic ray-induced desorption and photodesorption, mechanisms that are insufficient to release a significant portion of the phosphorous hydrides contained in the ice. For χ=5% (Figure 6, bottom panel) and PH and PH 2 , there is a real value of the chemical desorption probabilities that is higher for PH 2 owing to its higher ∆H r . Reducing the effective binding energy with the Modified Values B of Table 6 finally enables desorption of PH 3 for χ=5 % (Figure 7, bottom panel). A model based on translational energy gained by the nascent molecule, such as the one presented in Fredon et al. (2021), is a significant step forward in understanding chemical desorption. Furthermore, it is grounded on a physically sound basis, and allow to go beyond equipartition into vibration, rotation and translational degrees of freedom, something that we also do in our simulations. However, in this work, we found that such a model is susceptible to the χ parameter, which, in addition, is not only species-dependent but also binding-site-dependent. In line with our results, to reproduce the experimental abundances for the chemical desorption probability of PH 3 , Furuya et al. (2022b) derived a χ value for chemical desorption of 7% to account for a chemical desorption probability per reactive event of 3% in the PH 2 +H − −− → PH 3 reaction, the same value obtained in this work (Continue green line in Figure 8). However, based on our simulations on PH, we cannot explain a value of 7% for χ, where our upper bound for the distribution is 5% with the absolute maximum value found for a single trajectory is 6.7%, occurring only once, and in a system (PH) with fewer degrees of freedom to distribute the reaction energy into the molecule. The difference between 1-5 % of our simulations and 7% obtained from the fit of experimental data stems from the range of values in EB, ∆H r , and χ, making it challenging to establish a clear and univocal relation between our microscopically derived data with macroscopical models of chemical desorption. We will continue investigating a way of improving on the basis provided by equation 2 and encourage further work in this subject.
Finally, from an astrochemical point of view, we cannot confidently determine the abundances of PH 3 in molecular clouds, but it is not our intention with this paper. We refer the reader to Chantzos et al. (2020) for a much more in-depth analysis of interstellar phosphorous chemistry. The purpose of the here presented chemical models is to evaluate the sensitivity of the models to the parameters governing chemical desorption, reiterating the extreme variability induced by the BE distributions.
Deuteration reactions and PD dynamics
Finally, we dedicate a few words to our study of the energy dissipation of PD vs PH. Although we are aware that the main caveat of our simulations remains the classical nature of the nuclear motion, we found that energy dissipation is, on average, more efficient for PD than for PH. It remains to establish if such an effect is a particularity of PD because of a coupling between the fundamental stretching mode of PD and the water bending modes or if, by contrast, is a consequence of the different mass terms during the dynamics. Suppose this effect is due to the coupling between vibrational frequencies. In that case, we can expect a preferential distillation of molecules whose vibrational modes are compatible with the substrate's composition, which will be important for desorption of molecules as a function of the surface composition. In contrast, if the greater energy dissipation is general to all deuterations, that would imply that molecules on grains would be deuterium enriched compared to their gas counterparts.
Although the limitation of our method does not allow for strong conclusions in this regard, we encourage further work in this direction because of the crucial implications that our simulations carry for the deuterium fractionation in the ISM. We postpone the respective study to our future work.
Conclusions
In this work, we have used a novel methodology based on neuralnetwork interatomic potentials to study the chemical energy dissipation in the surface reaction P + H − −− → PH on ASW. The main finding of our work, that is extremely counterintuitive, is that, even with the reaction energy being two orders of magnitude higher than the binding energy, the initial binding energy of the adsorbate prior to reaction is absolutely crucial, to the point that in our study, only adsorbates on WB sites are susceptible of experience chemical desorption. From our study, we draw several conclusions, which we list below: 2. The binding energies of P and PH on ASW, obtained by neural-network potentials, are, on average, 1317 and 1843 K, values in very good agreement with our previous study (1241 and 1616 K).
Our exploratory simulations of four types of binding sites
show that below-average binding energy sites are not likely to be populated after reaction on a pristine surface. By contrast, medium and high energy binding sites are likely to be the endpoint of a nascent molecule, where the reaction energy is dissipated. 4. Non-thermal effects such as diffusion and desorption are determined early in the dynamics, in the first ∼ 10 ps. Energy dissipation and non-thermal events seem to be decoupled processes with a marked dependence on the binding energy of the adsorbate. 5. From a more extensive pool of simulations on high, medium and weak binding energy sites, we could constrain the fraction of dissipated energy and, most importantly, the translational kinetic energy acquired by the PH molecule after the reaction. We showed that on high-energy binding sites, energy dissipates 12% more than on average binding sites. Concerning transfer to translational energies, we found that it is a rather inefficient process, and we constrained it to a yield of 1-5 %. Molecules formed on medium-energy binding sites acquire, on average, nearly double the translational energy as molecules formed on high-energy binding sites. In weak binding sites, the energy dissipation fraction is meaningless because the population of weak binding sites is null after the reaction, with a consequential maximum of translational energy. 6. We found the differences between the deuteration reaction and the hydrogenation, P + D − −− → PD. On average, energy dissipates 6-7 % more for the same binding site for the deuteration reaction, and PD acquires around 10% less translational energy. 7. We incorporated our derived values into astrochemical models accounting for chemical desorption. We found that the latest literature model for chemical desorption (eq. 2) is extremely sensitive to the molecule's binding energy and the fraction of energy redistributed to translational degrees of freedom. Using our average value, χ=2%, we obtained that PH x gas abundances must be very low, which confronts recent studies (Nguyen et al. 2020;Furuya et al. 2022b). Using an upper limit of χ along with updating binding energies to exclude high-energy binding sites (a situation expected in the ISM for P-bearing molecules), we found changes in between 1-5 orders of magnitude in the abundances of PH x compounds, reconciling our values with those of recent studies mentioned above. This strongly suggests that chemical desorption, like thermal diffusion or desorption, is sensitive to the binding site under consideration.
Continuations to this work will deepen the connection between the atomistic simulations and the analytical formulas for chemical desorption (and diffusion), ideally tackling the problem of using a single binding energy/translational energy value in contrast to the distribution of such values.
G. Molpeceres et al.: Reaction dynamics on amorphous solid water surfaces using interatomic machine learned potentials
Appendix A: Quality assessment for employed MLIPs
In this section, we assess the quality of the GMNN-based MLIPs employed to study the reaction dynamics of the PH formation. For this purpose, first, we analyse the errors obtained on the test data, i.e., the data drawn from the original data set but not seen during training, including the early stopping technique. Fig. A.1 shows the correlation of predicted energies and atomic forces with the respective reference values. All values lie tightly on the diagonal, indicating an excellent performance of developed MLIPs. We obtained a mean absolute error (MAE) of 0.01 kcal/mol/atom and 0.28 kcal/mol/Å for predicted energies and atomic forces, respectively. The respective root-mean-square errors (RMSE) are 0.01 kcal/mol/atom and 0.42 kcal/mol/Å. The MAE and RMSE values in predicted energies and atomic forces are abstract quality measures for MLIPs. To assess the potential's robustness and reliability, we investigate their uncertainty during real-time applications. For this purpose, we consider two possible outcomes, namely a reaction with subsequent desorption (1) and diffusion (2) of created PH species, and (1), obtained by disagreement between an ensemble of three models. The figure shows that most uncertainties are of values less than 1 kcal/mol/Å, an empirical threshold used to make statements on the model's reliability. Thus, the model performs overall robustly and leads to correct dynamics. However, at least one of the atoms participating in the reaction requires more rigorous analysis. Fig. A.2 (right) shows the norm of the force acting on the hydrogen atom evaluated along a molecular dynamics trajectory. Additionally, we show the uncertainty of the force norm in Fig. A.2 (right). The figure shows that the maximal uncertainty does not exceed 9.6 % of the value of the respective force norm. However, additional inspection has shown that those uncertainties correspond to high-energy regions where, e.g., the vibrations stimulated by the reaction should be accounted for. In this case, the distance between P and H reaches values smaller than those sampled by the training set, leading to a higher model's uncertainty. As this part of the potential energy surface is irrelevant to the performed study and its results, the uncertainty in those high-energy regions could not influence them. Similar results have been obtained for the case where PH species diffuses on the surface after the reaction (2). The respective uncertainties are shown in Fig. A.3. Finally, we consider a geometry optimization profile on a 20 (H 2 O 20 ) using explicit DFT calculations at the reference level and using the MLIP. Fig. A.4 shows that both energy profiles almost coincide with an MAE of 0.002 kcal/mol/atom and an RMSE of 0.01 kcal/mol/atom. Moreover, atomic forces correlate very well with the reference values which corresponds to low MAE and RMSE values of 0.21 kcal/mol/Å and 0.69 kcal/mol/Å, respectively. This indicates that the MLIP reproduces exactly the reaction profile predicted by DFT.
Fig. 1 .
1Distribution of the binding energies of P (Top) and PH (Bottom) on ASW. Each bar is coloured according to the approximated attributed binding site. The colour code is high binding (HB) red, medium binding (MB) blue, weak binding (WB) purple, very weak binding (VW) green.
Fig. 2 .
2Local environments of the P atom prior to the reaction with H. The models portrayed in this figure come from different ice models.
Fig. 3 .
3Onset for the title reaction in our simulations. The figure represents around 50 fs of the total dynamics (50 ps). The distance of 3.5 Å represents the P-H initial distance.
Fig. 4 .
4Evolution of the total kinetic energy of water ice and PH molecules (left panels) and PH translational energy (right panels) for two P + H − −− → PH reaction trajectories, one with significant energy dissipation in the timelapse of the simulation (BE=739 K, top panel) and without significant energy dissipation (BE=805 K, bottom panel). The pale blue colour in the left panels shows the instantaneous kinetic energy of the PH molecule. The dark blue colour shows the running average evaluated over a window size of 25 fs. The black dashed line in both panels depicts the average PH binding energy. X, Y, and Z indicate the component along the translational energy is distributed, with Z being the normal to the surface.
Fig. 5 .
5Top panel: Molecular dynamics frame prior to chemical desorption. Note that even in such repulsive parts of the PES, our potential only has an uncertainty of around 10 % between models, indicating high-quality predictions of the PES also during/throughout the reaction. Bottom panel: Translational kinetic energy of the respective trajectory.
Fig. 8 .
8Chemical desorption probability (p) in theFredon et al. (2021) formulation as a function of the translational energy conversion fraction χ.
Fig. A. 1 .
1Correlation of the predicted potential energies (Top) and atomic forces (Bottom) with the corresponding reference values for all structures in the test data. Mean absolute (MAE) and root-mean-square (RMSE) errors are shown as an inset.
Fig
. A.2. (Top) Norm uncertainty of the atomic forces predicted by an ensemble of three GM-NN models. (Bottom) Comparison of the force norm and the respective uncertainty for the hydrogen atom during first steps of an MD simulation leading to the desorption of the created PH species. The time scale is restricted to the reaction of P and H and the desorption of the resulting PH molecule. The uncertainty of the ensemble does not exceed 9.6 % of the respective force norm. the distributions of uncertainties corresponding to them. Fig. A.2 (left) shows the distribution of uncertainties for the case where PH desorbs after the reaction
Fig
. A.3. (Top) Norm uncertainty of the atomic forces predicted by an ensemble of three GM-NN models. (Bottom) Comparison of the force norm and the respective uncertainty for the hydrogen atom during first steps of an MD simulation leading to the diffusion of the created PH species. The time scale is restricted to the reaction of P and H and the diffusion of the resulting PH molecule. The uncertainty of the ensemble does not exceed 12 % of the respective force norm.
Fig
. A.4. (Top) Comparison of the reference and predicted potential energy profiles obtained during a geometry optimization. (Bottom) Correlation of the atomic forces predicted by the developed MLIP with the corresponding reference values.
Table 2 .
2Initial and final binding sites of our exploratory simulations for the P + H − −− → PH reaction on ASW, including all types of binding sites along with the fraction of the reaction energy dissipated into the ice bulk for each trajectory. One trajectory is desorbing. Not included in the counting.2 No VW binding site was found in our sampling. 3 Desorbing trajectory.Pop. Number
F for Ice Model
Site P (Initial) PH (Final) 1
001
002 003
004
005
VW
4
0
N/A 2 0.55 0.93 0.27 3 0.52
WB
5
0
0.78 0.64 0.35 0.43 0.33
MB
5
6
0.85 0.48 0.51 0.51 0.74
HB
5
12
0.49 0.65 0.53 0.66 0.27
1
A spherical wall potential is applied to ensure the structural integrity of the cluster. 2 Interaction of P with the cluster at the distance of the cutoff radius (5.5 Å, required for binding energies).3 The reaction part comprises reactions in weak, medium and high binding sites at different P-H internuclear distances (3.0-4.5 Å). 4 398 structures have been removed from the training set (some distances between neighboring atoms are larger than the selected cutoff-radius).
. Our MLIP achieves high accuracy, comparable to the reference level of theory, even in reactivity studies, where very abrupt changes in the potential occur in very short time scales.Article number, page 10 of 14 G.Molpeceres et al.: Reaction dynamics on amorphous solid water surfaces using interatomic machine learned potentials
Acknowledgements. G.M. thanks the Japan Society for the Promotion of Science (Grant P22013) for its support. V.Z. and J.K. acknowledge the support by the Stuttgart Center for Simulation Science (SimTech).
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| [] |
[
"Perturbative Gluon Shadowing in Heavy Nuclei *",
"Perturbative Gluon Shadowing in Heavy Nuclei *"
] | [
"K J Eskola \nNuclear Science Division\nUniversity of California\n70A-3307, 94720Mailstop, Lawrence Berkeley Laboratory, BerkeleyCAUSA\n\nLaboratory of High Energy Physics\nUniversity of Helsinki\nP.O. Box 920C) SF-00014SiltavuorenpengerFinland\n",
"Jianwei Qiu \nDepartment of Physics and Astronomy\nIowa State University\n50011AmesIAUSA\n",
"Xin-Nian Wang \nNuclear Science Division\nUniversity of California\n70A-3307, 94720Mailstop, Lawrence Berkeley Laboratory, BerkeleyCAUSA\n"
] | [
"Nuclear Science Division\nUniversity of California\n70A-3307, 94720Mailstop, Lawrence Berkeley Laboratory, BerkeleyCAUSA",
"Laboratory of High Energy Physics\nUniversity of Helsinki\nP.O. Box 920C) SF-00014SiltavuorenpengerFinland",
"Department of Physics and Astronomy\nIowa State University\n50011AmesIAUSA",
"Nuclear Science Division\nUniversity of California\n70A-3307, 94720Mailstop, Lawrence Berkeley Laboratory, BerkeleyCAUSA"
] | [] | We study how much gluon shadowing can be perturbatively generated through the modified QCD evolution in heavy nuclei. The evolution of small-x gluons is investigated within the semiclassical approximation. The method of characteristics is used to evaluate the shadowed distributions in low-Q and small-x region. In solving the modified evolution equation, we model in simultaneously fusions from independent constituents and from the same constituent, both in a proton and in a large loosely bound nucleus of A ∼ 200. In addition to the actual distributions at small x, we study the ratios of the distributions at an initial scale Q 0 = 2 GeV, and show that a strong nuclear shadowing can follow from the modified QCD evolution. | 10.1103/physrevlett.72.36 | [
"https://arxiv.org/pdf/nucl-th/9307025v1.pdf"
] | 19,009,924 | nucl-th/9307025 | f855c6ea663a407644f45f7ce7bfdca84d8870c2 |
Perturbative Gluon Shadowing in Heavy Nuclei *
Jul 1993 July 23, 1993
K J Eskola
Nuclear Science Division
University of California
70A-3307, 94720Mailstop, Lawrence Berkeley Laboratory, BerkeleyCAUSA
Laboratory of High Energy Physics
University of Helsinki
P.O. Box 920C) SF-00014SiltavuorenpengerFinland
Jianwei Qiu
Department of Physics and Astronomy
Iowa State University
50011AmesIAUSA
Xin-Nian Wang
Nuclear Science Division
University of California
70A-3307, 94720Mailstop, Lawrence Berkeley Laboratory, BerkeleyCAUSA
Perturbative Gluon Shadowing in Heavy Nuclei *
Jul 1993 July 23, 1993arXiv:nucl-th/9307025v1 26
We study how much gluon shadowing can be perturbatively generated through the modified QCD evolution in heavy nuclei. The evolution of small-x gluons is investigated within the semiclassical approximation. The method of characteristics is used to evaluate the shadowed distributions in low-Q and small-x region. In solving the modified evolution equation, we model in simultaneously fusions from independent constituents and from the same constituent, both in a proton and in a large loosely bound nucleus of A ∼ 200. In addition to the actual distributions at small x, we study the ratios of the distributions at an initial scale Q 0 = 2 GeV, and show that a strong nuclear shadowing can follow from the modified QCD evolution.
The semihard gluonic subprocesses are expected to play an essential role in the formation of high energy densities in heavy ion collisions at collider energies [1,2,3,4].
However, there are many theoretical uncertainties in modeling these QCD processes. One of the major ones, nuclear gluon shadowing, comes from the unknown initial gluon distributions at small x. Unlike for the quark and antiquark distributions, there are no direct experimental data for the gluons in nuclei. Getting theoretical control over the nuclear gluon shadowing is therefore a very urgent and important issue. The purpose of this Letter is to study how much gluonic shadowing is generated perturbatively through the modified QCD evolution [5,6] in heavy nuclei.
"Shadowing" in the context of the deep inelastic lA-scattering refers to the measured depletion of the nuclear structure function F A 2 at small x Bj , as compared to F 2 of unbound nucleons [7]. The same kind of depletion at small x is expected to happen also in the nuclear gluon distributions. During the recent years there have been many efforts to explain the measured nuclear shadowing of quarks and antiquarks [8]- [15] but for gluons the situation is still inconclusive. Once the nuclear parton distributions are known at an initial scale Q 0 , the QCD-evolution to larger Q can be computed [6,16,17]. The problem is how to get input, theoretically or experimentally, for the nuclear gluon distributions at Q 0 , and to understand the reliability of QCD-evolution for the proper range of x-and Q-values.
Shadowing-phenomenon is also predicted to happen in protons. In this case, "shadowing" refers to the depletion of the actual parton distributions, and is caused by the fusions of overcrowding gluons at very small x. This mechanism proceeds through perturbative QCD-evolution as formulated in [5,6]. It has been shown by Collins and Kwieciński that the singular gluon distributions actually saturate due to the fusions [18]. In this Letter our basic idea is quite straightforward. We first compute the gluon shadowing in a proton, by using techniques introduced in [18] for solving the smallx evolution including gluon recombination. Then we apply the same mechanism of recombining gluons to heavy nuclei and study to what extent nuclear shadowing is generated perturbatively through the QCD-evolution at Q 0 = 2 GeV.
At small values of x, leading order QCD evolution equation predicts that the number of gluons becomes extremely large. It has been known [5,6] that for sufficiently small values of x and/or of Q 2 , the total transverse area occupied by the gluons will be larger than the transverse area of a hadron, so that the interaction between gluons can no longer be neglected. Such gluon recombination results in a modification of the QCD evolution equations. In the limit of small-x the modified QCD evolution equation can be cast in the form [5,6]
∂ y ∂ t G(y, t) = cG(y, t) − λ exp(−t − e t )[G(y, t)] 2 ,(1)
where y = ln(1/x), t = ln[ln(Q 2 /Λ 2 QCD )], G(y, t) = xg(x, Q 2 ) and c = 12/(11−2N f /3) with N f the number of quark flavors.
Strength of the gluon recombination is controlled by the factor λ, originating from two possible sources. The two fusing gluon ladders, which couple 4 gluons to 2 gluons, can arise either from independent constituents of proton/nucleus or from the same one, as discussed in [6,18,21]. We will refer to the former case as "independent" and to the latter as "non-independent" fusion. Since recombinations from both sources happen simultaneously, we divide the parameter λ into two parts:
λ = λ I + λ II ,(2)
where λ I corresponds to the independent recombination and λ II to the non-independent one. Let us first study the two sources of recombination within the models for twogluon densities given in ref. [6]. In a proton, the strength of the independent fusion then takes the form
λ I = 2 3 1 πR 2 p · π 3 c 2 2Λ 2 QCD ,(3)
where R p ∼ 1 fm is the radius of a proton. The magnitude of the non-independent fusion of the gluon ladders can be estimated as λ II ≈ 16 81
1 π(2/Q i ) 2 · π 3 c 2 2Λ 2 QCD ,(4)
where we have made a simplification by fixing the initial x of the valence quark to
x i ∼ 1.
We also approximate the scale of the initial valence quark by Q i ∼ 2 GeV. Let us then consider a large loosely bound nucleus. Naturally, both types of fusions are still there but only for the independent one an A 1/3 -scaling arises. In this case
λ A I = 9 8 A πR 2 A · π 3 c 2 2Λ 2 QCD ,(5)
where the nucleus is taken to be a sphere with a sharp surface at R A = 1.12A 1/3 fm. The strength of the non-independent fusion remains the same as in the case of a free proton: λ A II = λ II . It is interesting to notice how the relative contributions of the two types of recombination will change when going from a proton to a nucleus of A ∼ 200: λ II /λ I ≈ 7.6 and λ A II /λ A I ≈ 1.0. Thus the non-independent fusion is clearly dominant in a free proton while in a large nucleus the contributions from both types are of the same order. As a result, parton recombination is strongly enhanced in a heavy nucleus.
In order to solve Eq. 1 exactly by integration, one would need the initial distribution either at fixed y 0 or t 0 and the derivatives along a boundary line (y, t 0 ) or (y 0 , t), respectively. However, since the expression for the non-linear term in Eq. 1 is not valid for the regions where x is large, or where both x and Q are very small, the natural boundary condition at x = 1 (or y 0 = 0) is not suitable here. In addition, since we do not have sufficient information on other boundary lines, we cannot solve Eq. 1 by direct integration. Instead, with the semiclassical approximation [5], we are going to adopt the idea introduced in [18] to use the method of characteristics, so that we can avoid the region III (see discussion later). The semiclassical approximation corresponds to neglecting the second order derivative term, ∂ y ∂ t ln(G), which leaves us with the evolution equation as
∂ y z(y, t)∂ t z(y, t) = c − λ exp[−t − e t + z(y, t)],(6)
where z(y, t) = ln[G(y, t)]. The above equation can then be cast and solved in the form of a set of characteristic equations as shown in detail in [18].
The evolution of gluon distribution in a proton and in a nucleus is similar, so let us first consider the general idea, as illustrated in Fig. 1. We divide the (y, t)-plane into three regions. In region I for y ≤ y 0 (x ≥ x 0 ), we expect the traditional, non-corrected Altarelli-Parisi (AP) evolution [19] to hold down to scales Q ∼ 1 GeV. In region II with y ≥ y 0 (x ≤ x 0 ), the evolution of the gluon distribution is then approximately given by Eq. 6. The initial values of the gluon distribution and its t-and y-derivatives are determined numerically at (y 0 , t min ≤ t ≤ t 0 ) from the singular gluon distribution of the CTEQ-collaboration [20], which is obtained with a lower initial Q-value to provide the necessary t-dependence of the boundary condition at y 0 , and a larger
x-cut to ensure the validity of AP evolution in region I. The evolution in the region II is finally terminated at t = t 0 , corresponding to Q 0 = 2 GeV. Notice that the characteristics approach the t 0 -line from below; the lowest scale we have to go down to is about Q = 1.25 GeV, corresponding to t min . At these scales QCD perturbation theory should be still valid in region I. In region III with extremely large y (small x) and/or small t we do not expect our analysis to be valid anymore, since the higher order terms in the evolution equation will become important.
Before performing the actual evolutions, we have to consider how to choose the boundary y 0 (= − ln(x 0 )) for a proton and a nucleus, and how to conserve momentum.
For a proton, we assume that the recombinations start to be effective at x ∼ x 0 ∼ 0.01, which is consistent with [18,21]. We can use the results from global fitting, like CTEQ, to constrain x 0 . In fact, we will see that with x 0 = 0.01 the shadowed gluons deviate considerably from the CTEQ gluons only after x < 0.001, so the choice for
x 0 seems to be reasonable, and we do not expect the results to be very sensitive to small changes of x 0 .
As explained above, the gluon recombination is strongly enhanced in heavy nuclei and it starts at somewhat larger values of x than in protons. The corresponding boundary line x A 0 for a nucleus is approximately determined by the relative magnitude of the evolution terms in Eq. 1: G A (x A 0 ) ∼ G(x 0 )λ A /λ, so that the relative contribution from the gluon fusion in a nucleus is about the same as in a nucleon.
This gives x A 0 ∼ 0.05-0.1. This range of x A 0 is also supported by other studies [17]. Let the original fraction of momentum in gluons be f 0 = 1 0 dx xg CTEQ (x, Q 2 0 ). In the case of a proton, shadowing in the region II changes the gluonic momentum typically by less than a per cent, which we can clearly neglect as a small overall change.
Perturbative shadowing reduces the gluonic momentum more in a nucleus than in a proton. Assumed that the momentum fraction of gluons is conserved, there must be a corresponding enhancement in the region II. In addition to this, we also take into account a possible momentum transfer from quarks and antiquarks to the gluons.
Here we consider nuclei with A ∼ 200, for which we expect an overall increase in the fraction of the momentum, ǫ A , to be only about 4% [9,10,17]. We combine these two sources of the momentum flow, which results in solving a A iteratively from
x A 0 0 dx xg(x, Q 2 0 ) C + a A 1 x A 0 dx xg CTEQ (x, Q 2 0 ) = f 0 (1 + ǫ A ),(7)
with the condition C :
g A (x A 0 , Q 2 0 ) = a A g CTEQ (x A 0 , Q 2 0 ) on the boundary. Typically, a A ∼ 10% for A ∼ 200.
Let us now turn to the results, presented in Figs. 2. In Fig. 2a, nucleon and effective nuclear gluon distributions for a nucleus of A = 200 are compared with the input CTEQ gluon distribution at Q 0 = 2 GeV. Notice the ∼20 % uncertainty in the nuclear case resulting from varying x A 0 from 0.05 to 0.1. To demonstrate the formation of strong perturbative nuclear shadowing, corresponding to the relative depletion of gluon distributions in a nucleus, we plot the ratio G A (x, Q 2 0 )/G(x, Q 2 0 ) in Fig. 2b. Notice also that as x decreases, gluon distribution in a proton increases much faster, or shows the sign of saturation at a much smaller x than that in a nucleus. Therefore, as shown in Fig. 2b the ratio saturates only when the gluons in a proton do so. Thus, saturation of the perturbative nuclear shadowing reflects actually the behavior of the gluons in a proton. As a main result, we conclude that, due to the enhanced gluon recombination in a heavy nucleus, a ∼50% nuclear shadowing in small-x region is generated perturbatively through the modified QCD evolution, accompanied by a ∼10 % antishadowing from the momentum conservation.
As seen more clearly in Fig. 2b, with x A 0 = 0.1 there is a slight deviation in the initial derivatives of the gluon distributions determined from the region I as compared to what can be determined from Eq. 6 from the region II. This in turn is a reflection of an apparent fact that eventually one cannot apply the small-x approximation at too large x. Letting x A 0 ∼ 0.1, we are really pushing the small-x evolution equation to its limit; surely beyond this point the Eqs. 1 and 6 cannot be applied without additional correction terms. However, taking the initial conditions from the "known" region I, as we do, should improve the analysis and reduce the uncertainty in the small-x region. From Fig. 2b it is seen that we cannot make conclusive claims about the "beginning" of nuclear shadowing. However, since the result with x A 0 = 0.1 does not differ considerably from the result with x A 0 = 0.05, we believe our result shows the correct order of magnitude of the perturbative shadowing at very small x.
It is clear that the absolute strengths of the λ's depend on the models assumed for the two-gluon densities in a proton and in a nucleus. However, we do not expect our qualitative results for the nuclear shadowing to change very much with different details. One may also question what happens to nuclear shadowing, if the initial gluons diverge more strongly(weakly) when x → 0 than CTEQ gluon distribution used here. In that case, the recombinations would be enhanced(suppressed) both in a proton and in a heavy nucleus. However, the result for nuclear shadowing,
G A (x, Q 2 0 )/G(x, Q 2 0 )
, is not extremely sensitive to the small-x behavior of the input gluon distribution because we use only the part with x > 0.01, which has been relatively well-tested experimentally. This question will be studied in more detail elsewhere [23].
We would like to comment briefly on the general consequences of our result for perturbative nuclear shadowing. The semihard processes with typical scales Q ∼ a few GeV involve x ∼ Q/ √ s. In heavy ion collisions at √ s = 200 GeV, the x's in the semihard processes will be larger than 0.01, so these processes rather probe the onset of perturbative nuclear shadowing than the region of saturation. On the contrary, in collisions with √ s in the TeV range, the semihard processes will happen at x's typically smaller than 10 −3 , and are therefore affected considerably more by the perturbative shadowing. Examples of the possible effects on minijet production can be found in [2,17,22]. Other processes clearly suppressed by the nuclear gluon shadowing at very high energies are heavy quark and their bound state production.
Also the production of total transverse energy and energy density will be suppressed, as compared to the predictions with non-shadowed gluons [2,22]. Through this, the thermalization of the possibly formed quark-gluon plasma is also slowed down, in which case the thermal electromagnetic signals are suppressed. In order to make precise predictions for these processes, nuclear shadowing has to be studied at scales Q > 2 GeV. The scale dependence of nuclear gluon shadowing is an interesting question to which we will return in the future [23].
To conclude, we have considered the perturbative aspects of the nuclear modifications to the gluon distributions. As we have shown here, a strong nuclear shadowing is generated through the modified QCD evolution, and it may well be the dominant mechanism for the small-x modifications. We emphasize that the use of the method of characteristics is necessary to avoid the region III where even the modified evolution equation is not expected to be valid. We feel we now have more quantitative control over the nuclear gluon distribution at small x, based on perturbative QCD. We believe this study could serve as an interesting starting point for more detailed calculations of nuclear gluon shadowing and its consequences in ultra-relativistic heavy ion collisions.
Figure Captions
Fig. 1 .
1The evolution plane. In the region I the traditional AP-equations are expected to be valid. Both x-and Q-dependence in this region form the initial conditions for the evolution in the small-x (large y) region II. Examples of the characteristics of the Eq. 6 in the region II are shown.
Fig. 2 .
2a. The gluon distributions xg(x, Q 2 0 ) at Q 0 = 2 GeV vs. x. The result for proton is labeled by x 0 , and the results for A ∼ 200 by x A 0 , respectively. The CTEQ gluon distribution[20] is labeled by "CTEQ". b. The ratio xg A (x, Q 2 0 )/xg(x, Q 2 0 ) of the shadowed gluon distributions vs. x,
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| [] |
[
"Semi-annual, annual and Universal Time variations in the magnetosphere and in geomagnetic activity: 4. Polar Cap motions and origins of the Universal Time effect",
"Semi-annual, annual and Universal Time variations in the magnetosphere and in geomagnetic activity: 4. Polar Cap motions and origins of the Universal Time effect"
] | [
"Mike Lockwood \nDepartment of Meteorology\nUniversity of Reading\nReadingUK\n",
"Carl Haines \nDepartment of Meteorology\nUniversity of Reading\nReadingUK\n",
"Luke A Barnard \nDepartment of Meteorology\nUniversity of Reading\nReadingUK\n",
"Mathew J Owens \nDepartment of Meteorology\nUniversity of Reading\nReadingUK\n",
"Chris J Scott \nDepartment of Meteorology\nUniversity of Reading\nReadingUK\n",
"Aude Chambodut \nÉcole et Observatoire des Sciences de la Terre\nUniversité de Strasbourg\nCNRS\nFrance\n",
"Kathryn A Mcwilliams \nInstitute of Space and Atmospheric Studies\nUniversity of Saskatchewan\nSaskatoonSaskatchewanCanada\n"
] | [
"Department of Meteorology\nUniversity of Reading\nReadingUK",
"Department of Meteorology\nUniversity of Reading\nReadingUK",
"Department of Meteorology\nUniversity of Reading\nReadingUK",
"Department of Meteorology\nUniversity of Reading\nReadingUK",
"Department of Meteorology\nUniversity of Reading\nReadingUK",
"École et Observatoire des Sciences de la Terre\nUniversité de Strasbourg\nCNRS\nFrance",
"Institute of Space and Atmospheric Studies\nUniversity of Saskatchewan\nSaskatoonSaskatchewanCanada"
] | [] | and collaborating institutes for the compilation and data basing of the and a indices which were downloaded from http://isgi.unistra.fr/data_download.php and to the staff of the Space Physics Data Facility (SPDF) at NASA's Goddard Space Flight Center for the Omni composite of interplanetary observations (made available by SPDF from https://omniweb.gsfc.nasa.gov/ow_min.html ). | 10.1051/swsc/2020077 | [
"https://arxiv.org/pdf/2012.13324v1.pdf"
] | 229,371,562 | 2012.13324 | 414eb36d7280266a7e89788eba8b04870515306f |
Semi-annual, annual and Universal Time variations in the magnetosphere and in geomagnetic activity: 4. Polar Cap motions and origins of the Universal Time effect
Mike Lockwood
Department of Meteorology
University of Reading
ReadingUK
Carl Haines
Department of Meteorology
University of Reading
ReadingUK
Luke A Barnard
Department of Meteorology
University of Reading
ReadingUK
Mathew J Owens
Department of Meteorology
University of Reading
ReadingUK
Chris J Scott
Department of Meteorology
University of Reading
ReadingUK
Aude Chambodut
École et Observatoire des Sciences de la Terre
Université de Strasbourg
CNRS
France
Kathryn A Mcwilliams
Institute of Space and Atmospheric Studies
University of Saskatchewan
SaskatoonSaskatchewanCanada
Semi-annual, annual and Universal Time variations in the magnetosphere and in geomagnetic activity: 4. Polar Cap motions and origins of the Universal Time effect
Accepted for publication in J. Space Weather Space Clim., 23 rd December 20201 File: Lockwood_Paper4n_F.doc
and collaborating institutes for the compilation and data basing of the and a indices which were downloaded from http://isgi.unistra.fr/data_download.php and to the staff of the Space Physics Data Facility (SPDF) at NASA's Goddard Space Flight Center for the Omni composite of interplanetary observations (made available by SPDF from https://omniweb.gsfc.nasa.gov/ow_min.html ).
Abstract. We use the am, an, as and the a geomagnetic indices to the explore a previously overlooked factor in magnetospheric electrodynamics, namely the inductive effect of diurnal motions of the Earth's magnetic poles toward and away from the Sun caused by Earth's rotation. Because the offset of the (eccentric dipole) geomagnetic pole from the rotational axis is roughly twice as large in the southern hemisphere compared to the northern, the effects there are predicted to be roughly twice the amplitude of those in the northern hemisphere. Hemispheric differences have previously been discussed in terms of polar ionospheric conductivities generated by solar photoionization, effects which we allow for by looking at the dipole tilt effect on the time-of-year variations of the indices. The electric field induced in a geocentric frame is shown to also be a significant factor and gives a modulation of the voltage applied by the solar wind flow in the southern hemisphere that is typically a 30% diurnal modulation for disturbed intervals rising to 76% in quiet times.
For the northern hemisphere these are 15% and 38% modulations. Motion towards/away from the Sun reduces/enhances the directly-driven ionospheric voltages and reduces/enhances the magnetic energy stored in the tail and we estimate that approximately 10% of the effect appears in directly driven ionospheric voltages and 90% in changes of the rate of energy storage or release in the near-Earth tail. The hemispheric asymmetry in the geomagnetic pole offsets from the rotational axis is shown to be the dominant factor in driving Universal Time (UT) variations and hemispheric differences in geomagnetic activity. Combined with the effect of solar wind dynamic pressure and dipole tilt on the pressure balance in the near-Earth tail, the effect provides an excellent explanation of how the observed Russell-McPherron pattern with time-of-year F and UT in the driving power input into the magnetosphere is converted into the equinoctial F -UT pattern in average geomagnetic activity (after correction is made for dipole tilt effects on ionospheric conductivity), added to a pronounced UT variation with minimum at 02-10 . In addition, we show that the predicted and observed UT variations in average geomagnetic activity has implications for the occurrence of the largest events that also show the nett UT variation.
Introduction
The first well-informed description of a Universal Time ( ) variation in global geomagnetic activity, that we know of, was by Bartels (1925Bartels ( , 1928. Bartels postulated that it was linked to the angle of tilt of Earth's magnetic axis relative to the sunward direction ( ) from studying the "U index" which commenced in 1835 and was continued until the 1930s: until 1871 it was based on declination readings from two observatories, after which it was based on seven stations Nevanlinna, 2004). The U index is equivalent to the magnitude of the difference between successive daily averages of the modern Dst index. In their seminal book Chapman and Bartels (1940) commented (section XI.20, page 391), "since the local time of Batavia and Potsdam differ by 5-6 hours, the identity of the hours of maximum or minimum suggests the existence of a 'Universal Time' variation of . Such a variation might depend, for example, on the varying angle between the Earth's magnetic axis and the line connecting the Sun and the Earth". This idea, now usually referred to as "dipole tilt effects" or the "equinoctial hypothesis", was employed by Waldo- Lewis and McIntosh (1953), McIntosh (1959) and many authors since (e.g., Aoki, 1977). The characteristic equinoctial pattern of variation with time-of-year and that this generates is also sometimes said to be caused by the "McIntosh Effect" (e.g., Berthelier, 1990). The dipole tilt angle varies with because of the rotation of the Earth and the offset of the geomagnetic dipole axis, ⃗⃗⃗ , relative to Earth's rotational axis ⃗ ⃗⃗ . The tilt angle also varies with time-of-year because of Earth's motion around the Sun (the rotation axis ⃗⃗⃗ being fixed in the inertial frame but the direction toward the Sun rotating through 360
every year in that frame). Hence the equinoctial hypothesis links a variation with annual and semi-annual variations in geomagnetic activity by the precessions of the ⃗⃗⃗ and ⃗ ⃗⃗ axes.
For a geocentric, symmetric dipole, most effects of the dipole tilt vary in amplitude with | | and if this fully applies, effects in one hemisphere are equal and opposite to those in the other hemisphere and the net global effect is zero when averaged over intervals of a whole number of years. However, the geomagnetic field is not a symmetric, Earth-centred dipole (e.g., Koochak and Fraser-Smith, 2017) and this will cause variations even in global data and even when averaged over many years. The third mechanism discussed in relation to the semiannual variation is the "axial effect" (see review in Paper 1, Lockwood et al., 2020a) which depends on the variation of the heliographic latitude of Earth over the year because the ecliptic is inclined at about 7 with respect to the solar equator. This effect alters the probability of Earth intersecting faster solar wind and so can introduces a time-of-year variation but no UT effect because it does not invoke variations caused by the orientation of Earth's magnetic field.
As described in Papers 1 and 2 of this series (Lockwood et al., 2020a;b), the semi-annual variation is well explained by the "Russell-McPherron" (R-M) effect which is due to the effect of the orientation of the interplanetary magnetic field (IMF) on magnetic reconnection in the dayside magnetopause and hence on solar-wind magnetosphere coupling. However, this predicts a pattern of response with fraction of a calendar year ( ) and that is very different from the equinoctial pattern that is seen in geomagnetic activity using the best indices that have responses to solar wind forcing that do not vary with either or . Several studies have tried to explain the observed equinoctial pattern by amending the R-M theory to include a dipole tilt effect in solar-wind magnetosphere coupling (see review in Paper 1). However, Finch et al. (2008) showed that the equinoctial pattern is not found in data from dayside auroral and polar cap magnetometer stations responding to directly-driven currents, and only in data from nightside stations responding to the substorm current wedge. The conclusion is supported by the work Chambodut et al. (2013) who showed that in the mid-latitude indices, the equinoctial pattern is strongest in the midnight sector and weakest in the noon sector. This shows that the equinoctial pattern is an internal response of the magnetosphere-ionosphere system and not present in solar-wind/magnetosphere coupling (Lockwood, 2013). There has been much debate as to whether the R-M and equinoctial effects are separate phenomena (e.g., Berthelier, 1990;Russell and Scurry, 1990;de la Sayette and Berthelier, 1976); however, as reviewed in Papers 1 and 2, the fact that the equinoctial pattern splits into a March and a September peak when the data are sorted by the polarity of the prevailing -component of the IMF shows that the R-M effect is at the heart of the equinoctial effect (Berthelier, 1976;Nakai, 1990;Zhao and Zong, 2012;Lockwood et al., 2016), it being unique in predicting this division. As reviewed in Paper 1, there have been a large number of theories proposed, but we have not yet developed an understanding of how the characteristic Russell-McPherron -pattern in solar wind-magnetosphere coupling evolves into an equinoctial pattern in geomagnetic response. Lastly, it must be remembered that, as demonstrated by Lockwood et al. (2016), any activity indices (such as the geomagnetic index) that depend on the prior solar wind conditions integrated over timescales longer than about 12 hrs will necessarily
show an "axial" pattern (with no clear dependence) rather than either an R-M or equinoctial pattern.
On averaging over a full year, both the R-M and equinoctial patterns predict that there would be no residual UT variation if there is symmetry between the two hemispheres in terms of geomagnetic field and seasonal variations in ionospheric conductivities. However, analysis of geomagnetic data strongly suggests that this is not the case with reports of a persistent minimum at about 3-9 hr UT. This was first noted in the Auroral Electrojet indices AE and AL (Davis and Siguira, 1966;Allen and Kroehl, 1975, Basu, 1975, Aoki, 1977 and has been reported many times since (Hajkowicz, 1992(Hajkowicz, , 1998Ahn et al., 2000;Ahn and Moon, 2003).
However, these are northern-hemisphere indices based on a ring of observing stations around the northern-hemisphere auroral oval and the main limitation of these studies is that without a southern-hemisphere equivalent, variations could be seasonal effects that are cancelled on a global scale by anti-phase seasonal effects in the southern hemisphere. There have been attempts to construct southern hemisphere indices (Maclennan et al., 1991;Weygand et al., 2014) but large parts of the southern auroral oval are over sea or ocean, giving large gaps in longitudinal coverage and so detection of variations is particularly limited. Any differences in the longitudinal spacing of the stations could introduce a spurious UT variation, as could longitudinal variations in the difference in latitude between the average auroral oval location and the stations. Initially there were just 5 stations in the northern hemisphere ring, and longitudinal coverage was indeed a concern but his was soon increased to 12. As will be discussed below, we now know that the UT variation is still present in the equivalent SME and SML SuperMAG indices (Newell and Gjerloev, 2011) derived from of order 100 stations in the northern hemisphere (Singh et al., 2013;Wang et al., 2014). Hence, like Singh et al. (2013), we eliminate the positioning of the stations as the cause of the variation; however, the fact that these observing networks are in just one hemisphere remains a relevant factor.
A commonly-used planetary index is (equivalent to ) but this is unsuitable for studying UT variations as the data are mapped back via conversion tables to the observations made at one station (Niemegk) and Lockwood et al. (2019a) (Mayaud, 1967(Mayaud, , 1980 has an extremely flat F-UT response, especially at higher activity levels. This means that studies that reported a persistent UT variation in the index (Berthelier, 1976;Russell, 1989;de La Sayette and Berthelier, 1996;Cliver et al., 2000) are particularly significant. There are other global UT variations in the magnetosphere that have been remotely sensed. For example, Morioka et al. (2013) have shown that Auroral Kilometric Radiation (AKR) data has a UT variation in frequency and amplitude that is the same in both hemispheres and that this is not related to the visibility of the magnetosphere for the observing GEOTAIL spacecraft that was outside the magnetosphere. The authors infer it is generated by the effect on the auroral acceleration regions of the bending of the tail with the dipole tilt although the precise mechanism remains unclear. In addition, Luan et al. (2016) have studied the variation and hemispheric asymmetry in auroral power deposition using observations by the TIMED (Thermosphere Ionosphere Mesosphere Energetics and Dynamics) satellite. This is the fourth in a series of papers investigating semi-annual, annual, and UT variations in the magnetosphere in which we study variations in the magnetosphere making use of the global and geomagnetic indices. The indices are generated in almost the same way as , but employ only data from one of four 6-hour sectors of Magnetic Local Time (MLT) around dawn, noon, dusk and midnight (Chambodut et al., 2013). We here also study for the first time, the hemispheric sub-indices that are averaged together to generate these global indices: this mean that in addition to northern and southern hemisphere indices ), for the 6-hour MLT sectors around 06hrs, 12hrs, 18hrs and 00hrs, respectively. The first paper in this series (Lockwood et al., 2020a) compared the semi-annual variations in the four indices to those in other geomagnetic indices and showed that they revealed a great many of the same characteristics as . The amplification of the semi-annual variation, with respect to that in the estimated power input into the magnetosphere, , was shown to increase with distance away from noon, being minimal for the index for the 6-hour sector around magnetic noon, ( ), by a factor of near 2 for the ( ) and ( ) (and for the equivalent overall global index ) and by a factor of near 3 for ( ℎ ).
Throughout this paper we estimate power input into the magnetosphere, , using the theoretical estimate devised by Vasyluinas et al. (1982). This coupling function is explained, discussed and its use justified at the start of section 3-ii.
1-i. Universal Time variations in different geomagnetic indices
As shown in Paper 1, the UT variation in geomagnetic data is a highly persistent phenomenon. Figure 1 shows average values of various geomagnetic indices in a -year spectrogram format. The longest data sequence is the homogenised index, , generated by Lockwood et al. (2018a;b). This index is based on data from just two stations, roughly 180 degrees apart in longitude and so is far from ideal for detecting a variation.
However, has been compiled using the same model of the stations' sensitivity that was used by Lockwood et al. (2019a) and this allows to capture both the equinoctial pattern and the UT variation seen simultaneously in the index after 1959. Lockwood et al. (2018b) show that the equinoctial pattern is present in the data before the start of the data in 1959, right back to the start of the data in 1868. Figure 1f shows that the UT variation is also present in all years before 1959 and appears it was even of larger amplitude before 1930 than in recent decades, although the use of just two stations means that we must use these data with caution in this respect. The index ( Figure 1e) shows a very similar UT variation. For we have hemispheric sub-indices and (shown in Figures 1c and Figure 1d) and they both show UT variations, but these are almost in antiphase with the peak around 12 UT in when is a minimum. Figure 1b shows the well-known strong UT variation in the AL index. This peaks around the same time as the northern hemisphere mid-latitude indices. Figure 1a shows that the same variation is seen in the northern-hemisphere SML index which demonstrates that the UT variation in AL is not caused by the longitudinal distribution of station locations. There have been attempts to construct an equivalent network to construct southern hemisphere AE indices, with limited success because much of the southern hemisphere auroral oval is over sea or ocean and because only relatively short data sequences are available (Maclennan et al., 1991;Weygand, 2014). The results of Maclennan et al. (1991) clearly showed the antiphase UT variation in the southern hemisphere index that is seen in Figure 1 for . The results of Weygand (2014) show the same feature, but the amplitudes of the north-south differences are considerably smaller than found by Maclennan et al. (1991). Note that both of these studies lacked stations at the key longitudes in the southern hemisphere: between Mawson (MAW) at 62.9E and W Antarctic Ice Sheet Divide (WSD) at 247.1E, the only available station is Macquarie Island (MCQ), south of New Zealand at 159.0E. Also, both studies used the rather than AL (and so reflect some influence of the dayside directly-driven currents detected by ). Not shown in Figure 1 are the polar cap indices and compiled from magnetometer data from single stations at Thule and Vostock, respectively (Troshichev et al., 2006). The northern hemisphere polar cap index (available from 1975 onwards) persistently shows the same UT variation as the other northern hemisphere indices shown in Figure 1; however, the southern hemisphere index (available for most years after 1995 but only in provisional form) does not show any persistent UT variation. The , , , and indices all come from just one station and so even for the polar cap indices from stations near the centre of the polar cap, the UT variation is convolved with local time variations and so variations in photoionization-induced ionospheric conductivity variations. Later in the current paper, we will present, for the first time, the variations in the hemispheric subindices of the four indices.
Because they have the most regular network of observing stations, the most reliable data on the UT variations in the northern and southern hemisphere are undoubtedly from and , the northern and southern hemisphere sub-indices of am (shown in Figure 1c and 1d). The top panel of Figure 2 shows the average UT variations of , and for 1995-2017 (dot-dash lines) and for all the available data (for 1959-2019, solid lines). The shorter interval is chosen here because it gives an availability of simultaneous interplanetary data which results in near-continuous estimates of the power into the magnetosphere (values that are accurate to within 5% are available for over 90% of the time: see Figure 3 of Lockwood et al., 2019b). It can be seen that the values of all three indices for the whole interval are considerably higher than for the post-1995 data, which is due to the long-term decline in solar activity that began around 1985. However, the form of the variations is very similar for the two intervals: this is stressed in the lower panel of Figure 2 which shows the variations of the values normalised to the overall mean for the interval (i.e. < > ⁄ in red, < > ⁄ in blue and < > ⁄ in black. It can be seen that the variations for the two intervals in these normalised values are not identical, but they are similar. Figure 2 shows that the hemispheric differences are more complex than a simple antiphase variation with a persistent minimum in both hemispheric indices (and therefore also in ) at around 05 UT.
Paper 1 and Paper 2 of this series (Lockwood et al., 2020a;b) showed that sorting the and Figure 3 are often attributed to hemispheric conductivity differences, and these are indeed a factor, but there is a much larger and more significant factor that is discussed in this paper for the first time in the next section.
1-ii. Motions of the poles and polar caps
Ionospheric polar cap phenomena are usually ordered and plotted in a geomagnetic coordinate system, for example a geomagnetic latitude and magnetic local time ( MLT) system. The pole of these coordinate systems is based on a model of the geomagnetic field and different models assign different geomagnetic coordinates to a given geographic coordinates. There are also different definitions of the magnetic poles to consider: for example, one can use the geomagnetic poles from a fitted dipole (which could be a geocentric dipole for which the dipole axis passes through the centre of the Earth or an eccentric dipole for which, in general, it does not), or the dip pole (where the surface field is vertical). The dip poles in the northern and southern hemisphere have behaved very differently over the last century (Thébault et al., 2015). As shown by the orange points in Figure 4, the northern dip pole has migrated toward the rotational pole such that their separation in geographic latitude of 20 in 1900 has reduced to just 4 in 2020, whereas the southern dip pole has migrated away from the rotational pole such that their separation increased from 18 to 26 in the same interval. Furthermore, the very high latitude of the northern dip pole has allowed the geographic longitudinal separation of the two dip poles to drop from 165 to just 65. On the other hand, a geocentric dipole model forces the two poles to be 180 apart in longitude and, as shown by the blue points in Figure 4, the (poleward) migration of the two magnetic poles in the same interval is the same and relatively minor (by of order 2.5). In this paper, we are interested in asymmetries between the two hemispheres and so use an eccentric dipole field model, which again employs a dipole field but does not constrain the dipole's axis to pass through the centre of the Earth (Fraser- Smith, 1987;Koochak and Fraser-Smith, 2017). This introduces a third type of magnetic pole, namely the eccentric axial poles which is where the fitted eccentric dipole axis threads the Earth's surface. These poles are not, in general, at same latitude nor are they axiomatically 180 apart in longitude. We compute the position of these poles using the equations and coefficients of Koochak and Fraser-Smith (2017) which are available for after 1980-2015 and are plotted as the mauve points for 1980, 2000 and 2020 in Figure 4. The northern eccentric dipole axial pole migrated from 8.2 to 5.4 (by 2.8) from the rotational pole over 1980-2020: in the same interval the corresponding values for the dip pole were 13.1 to 3.5 (by 9.6) and for the geocentric dipole geomagnetic pole were 11.1 to 9.3 (by 1.8). In the same interval, the southern eccentric dipole axial pole migrated from 15.3 to 14.4 (by 0.9) from the rotational pole and the corresponding values for the dip pole were 24.6 to 25.9 (by -1.3) and for the geocentric dipole geomagnetic pole were 11.1 to 9.3 (by 1.8, same as for the northern hemisphere). Hence the eccentric poles reflect some of the behaviour of the dip poles, but the changes were considerable smaller, as for the geocentric dipole. The key point that we focus on here is that the offset from the geographic pole for the southern hemisphere exceeds that for the northern hemisphere by a factor of about 2 or more, except for the geocentric dipole for which it is necessarily unity.
This ratio of the offsets increases from 1.9 to 2.6 for the eccentric dipole and from 1.9 to 7.4
for the dip poles (largely because the dip pole in the Northern hemisphere has moved so close to the rotational pole). Koochak and Fraser-Smith point out that eccentric dipoles have not been exploited in magnetospheric physics despite the obvious importance to hemispheric effects and hence effects.
A key point about the offset of the magnetic and geographic poles is that it causes motion of the ionospheric footpoints of magnetic field lines toward and away from the Sun. The geographic poles only move very slowly towards or away from the Sun: the orbital motion and elliptical nature of Earth's orbit means that over each the year the poles move together toward and the away from the Sun but with a peak velocity of only 0.5 ms -1 . However, the diurnal rotation of the magnetic poles around the geographic poles yields a considerably faster velocity toward and away from the Sun, which increases linearly with the offset in geographic latitude of the magnetic pole from the rotational pole. The expected Universal
Time effect of this on the auroral oval, and on the open field line polar cap inside it, has been described using an empirical model by Tsyganenko (2019) who showed that there is only minor distortion of the shape of the oval such that the circular motion of the oval in GSEQ (Geocentric Solar Equatorial) XY plane largely reflects that of the geomagnetic pole.
Observationally, Newell and Meng (1989) surveyed 3 years' data from the DMSP (Defense Meteorological Satellite Program) F7 satellite and showed that the region of cusp precipitation migrated in geomagnetic latitude by about 0.06 for each 1 shift in dipole tilt angle. That means that 94% of the motion of the magnetic pole in the GSEQ frame is reflected in the cusp location and only 0.6% in the geomagnetic frame. The cusp precipitation is on newly-opened field lines generated by magnetopause reconnection (Smith and Lockwood, 1996) and hence the motion of the dayside open-closed boundary in GSEQ largely reflects that in the magnetic pole. Vorobjev and Yagodkina (2010) Oznovich et al. (1993) showed that during low auroral activity the auroral oval as a whole was shifted by 1 degree in geomagnetic coordinates for every 10-degree change in the dipole tilt angle. This yields an estimate that 90% of the motion of the geomagnetic pole in the GSEQ frame induced by the diurnal motion of the pole is reflected in the auroral oval as a whole. Being at large longitudinal separations (if not exactly the 180 for a geocentric dipole model) the motion of the auroral ovals induced by the magnetic pole motions would be in close to, but not exactly, in antiphase in the GSEQ frame with the southern pole moving antisunward when the northern is moving sunward, and vice-versa. This has been directly observed by Stubbs et al. (2005) using full and simultaneous auroral images of the northern and southern auroral ovals made by the IMAGE and Polar satellites. These images are here reproduced in Figure 5, where the auroral intensity is plotted in a geomagnetic latitudemagnetic local time (MLT) frame: the altitude-adjusted corrected geomagnetic (AACGM) coordinate system was used (Baker and Wing, 1989). The white dots show the geographic poles which are points that are essentially fixed the GSEQ frame, their motion due to Earth's annual orbit being very slow. (Details of the mapping procedure are given in Section 2). Also shown as a cross is the corresponding location of the eccentric dipole axial pole. The oval and pole are shown for three times half an hour apart (11:20 in green, 11:50 in orange and 12:20 in mauve). We use this interval because Stubbs et al. show that during it the radius of the two polar caps was increasing but only very slightly, which makes the migration of the polar cap easier to discern because it is not complicated by expansion or contraction as it moves. Part (a) is for the northern hemisphere oval and shows both the pole and the oval moving toward the Sun; (b) is for the southern hemisphere oval and shows both the pole and the oval moving away from the Sun and in the − direction. The oval moves as a whole with motion that closely corresponds to that of the eccentric dipole pole, as expected from the above discussion.
1-ii. Effect of solar wind dynamic pressure
Paper 2 in this series (Lockwood et al., 2020b) reviews past studies revealing an independent effect of solar wind dynamic pressure on geomagnetic activity. Paper 2 shows that the geomagnetic response to a given injected power into the magnetosphere is increased if the solar wind dynamic pressure is increased. Furthermore, Paper 2 also shows that the amplitude of the equinoctial pattern increases with increased dynamic pressure as does the amplitude of the UT variation. Using models, Paper 3 (Lockwood et al., 2020c) has shown that a good explanation of this was the effect of dynamic pressure squeezing the tail and increasing both the energy stored in the near-Earth tail and the current in the cross-tail current sheet. This idea had been proposed by Lockwood (2013) as an explanation of why the equinoctial pattern was seen in association with the substorm current wedge and why it has a dependence on the square of the solar wind velocity (Finch et al., 2008). The modelling in Paper 3 indicates that the dipole tilt changes the ability of the solar wind dynamic pressure to modulate both the energy stored and the cross-tail current and that hemispheric asymmetry in the field means that positive dipole tilts have different effects to negative dipole tilts, thereby introducing a UT variation. (see Figure 19 of Paper 2). The lower panels in each pair show the level of the response for unit power input to the magnetosphere in the same bins of and ⁄ as the upper panels and so show the amplification factor of . This is greatest around the equinoxes and increases with . Hence there is a clear amplification of at the equinoxes that depends on but is an independent effect from the R-M effect. Appendix A shows two new plots that summarise findings presented in paper 1 and two that stress how important the effect of is to the generation of the semi-annual variation. which show that during and after the peak in < > (and hence also in < >) the variation in < | | > is identical for the two equinoxes and hence no asymmetry is introduced between them and there is no net variation when they are averaged together. This conclusion was found to hold for both geocentric and eccentric dipole fields and all epochs.
1-iii. The equinoctial and Russell-McPherron time-of-year/time-of-day patterns
This means that asymmetry between the March and September peaks, and hence a UT variation, is not introduced into either predicted patterns (nor into the relationship between the two), by the magnetic field model as long as the field has a single dipole axis, even if it is an eccentric one that does not pass through the Earth's centre.
1-iv. Aims of this paper
In Section 2 of this paper, we investigate the effect of the motion of the open polar caps on the dependence of geomagnetic activity using the eccentric dipole model of the geomagnetic field of Koochak and Fraser-Smith (2017). In section 3 we discuss how we model averages of mid-latitude "range" geomagnetic indices studied in this paper ( and its which, in itself, raises interesting questions as to why and how. Section 8 contains a summary discussion and conclusions.
Effect of diurnal pole motions
In this section we consider the effect of the daily motions of the magnetic poles due to Figure 11 is a schematic based on that by Lockwood and Cowley (1992) and Lockwood and Morley (2004)
∮ ⃗ . ⃗⃗⃗ = + = − = − ∫ ⃗ . ⃗⃗⃗⃗⃗(1)
hence the decoupling is caused by a change in the total magnetic flux threading the loop. The same applies to the nightside reconnection voltage (the integral of the reconnection rate) and the loop DEed and the magnetopause reconnection voltage and the loop ABba.
When using equation (1) it is important that ⃗ . ⃗⃗⃗ is evaluated by moving around the loop in a common, right-hand sense and that for all parts of the loop the electric field is quantified in a common frame of reference (i.e., it is a "fixed loop"). We here use the geocentric GSEQ frame which shares the same sunward X axis as all the frames in which we measure the solar
= , where 0 1(2)
There are two reasons for this. The first is the "flywheel effect" of thermospheric inertia whereby collisions between ionospheric ions, particularly in the E-region, and the (much) more numerous neutral thermospheric atoms and molecules tend to keep ions moving at the same speed even if the solar wind forcing of ionospheric convection changes (Deng et al., 1993). However, a more intrinsic cause of the factor is predicted by the ECPC model of ionospheric convection excitation . The ionospheric polar motions have no effect on the conditions at the magnetopause and cross-tail reconnection sites and so do not directly modulate the voltages and with which open field lines are opened and closed, respectively. In the ECPC model of ionospheric convection excitation, opening and closing of field lines perturbs the location of the open-closed field line boundary in the ionosphere and ionospheric convection is the response of the ionospheric flows as the boundary tends towards the new equilibrium configuration. This might appear to argue that the pole motions gave no effect on ionospheric flows (and hence = 0) but this overlooks the fact that the motion of the polar cap will also influence the equilibrium configuration that the system relaxing back toward.
Hence, in the limit = 1 the pole motions induce only a directly driven response in polar cap flows and in the limit = 0 they have no effect of the ionospheric flows and only generate inductive changes in the tail lobe field, i.e., the response is of a purely storage/release nature. The general value of between these two limits is a mixture of both effects. For general , the change in observed transpolar voltage is less than but is not zero. This means that, as well as modulating ionospheric voltages directly, the pole motions would modulate the rate of energy storage or release in the tail lobe:
additional energy would be stored when the pole is moving sunward, and this would be released again 12 hours later when it is moving antisunward. Because the poles are close to being 180 of geographic longitude apart, motions in the two hemispheres are close to being in antiphase and hence the energy in one tail lobe grows while the other declines but because the of hemispheric asymmetry in the geomagnetic field, these variations do not cancel. Figure 10a predicts the pole motion effect will be roughly double the size in the southern hemisphere to the northern and will peak at 12 in the northern hemisphere (with a minimum at 0 ) and at 22 in the southern hemisphere (with a minimum at 10 ).
From Figure 10b a global effect that is the average of that for both hemispheres will peak near 21 with a minimum near 9 . There are strong elements of these predicted variations seen in the , , and indices shown in Figure 2, but they are clearly also modulated by other factors. This is not surprising as we know that there are other consistent variations with (at a given time of year), such as the effect of ionospheric conductivities, the Russell-McPherron effect and the squeezing of the tail by solar wind dynamic pressure.
These all occur concurrently and in the remainder of this paper we investigate how the pole motion effects described in this section interact with other factors.
Note that in the modelling of geomagnetic indices presented in the following section, all parameters can be quantified using approximations and/or averages of observationswith one exception: there is no way to quantify the parameter . The closest we could get would be a global MHD numerical model, but this would be far from ideal because the lower boundary is not in the ionosphere and tends to be at an altitude of about 3 for computational reasons. Hence including ionospheric pole motions in a self-consistent way will not be a straightforward task. As a result, we have to treat as a free fit parameter.
There are other parameters that are quantified by fitting to data. A factor is needed to relate the effects of the magnetic shear across the near-Earth tail current sheet (i.e., the current in that sheet) to its effect on the geomagnetic index in question, and this can be quantified directly by studying the index response to modelled changes in that current.
Similarly, we use a factor to scale the theoretical R-M forcing pattern to the observed pattern of power input into the magnetosphere, ( , )⁄ : again, this is done by a direct fit to the data.
Modelling geomagnetic indices.
In the modelling presented here we use 3 multiplicative normalised factors to generate a simulation of a given geomagnetic index (here given the generic name ), 1 (where H signifies the hemisphere it applies to, i.e., N for north and S for south:
1 ( , ) = < >. ( , ). ( , ). [ ( , )](3)
where the terms . This was done for the hemispheric indices, and , and the index was then modelled as the average of the two. There is one more factor that is not included in equation (3) namely an allowance for the effect of ionospheric conductivities ( , ).
This factor is different for each index and in this paper we are dealing with 15 indices. Rather than include ( , ) in an equation (3) for each index we here adopt a procedure to remove the conductivity effects first to generate a conductivity-corrected -pattern
( , ) = ( , ) ( , ) ⁄(4)
and we then model the conductivity-corrected index using equation (3). Because in this procedure the (hemisphere-specific) conductivity correction has already been made (using equation (4) with a factor ( , ) , the derivation of which is explained in the next section), hemispheric differences between and will only be due to the polemotion term that is the main focus of the present paper.
3-i. Allowance for conductivity effects
With the terms ( , ) and ( , ), we allow for the effects of both the ionospheric Hall and Pedersen conductivities generated by photoionization. Note that this excludes enhancement of the conductivities over a background level (associated with the quiet auroral oval) by enhanced particle precipitations which will depend on the location (especially in relation to the auroral oval) and the activity level. We regard these enhanced particle precipitation effects on conductivity as an intrinsic part of the activity index that we are modelling. The ionospheric Hall and Pedersen conductivities generated by photoionization both depend on the solar zenith angle (e.g., Ieda et al., 2014) and so, at any fixed geomagnetic location, on the tilt angle . In theory, the conductivities could be evaluated for every location in the polar regions using empirical relationships for a given solar zenith angle, and sunspot activity level. However, this leaves the problem because we do not know which locations most influence the index under consideration, it could be the auroral oval, over the stations or a mix of the two (Lockwood et al. 2018b(Lockwood et al. , 2019a. Hence, we take an empirical approach using means over several days of the deviations of the northern hemisphere index and simultaneous southern hemisphere index. We then study their variation with time of year and compare with the means of the dipole tilt angle . It is assumed as a first-order approximation that on these timescales the hemispheric differences are due to conductivity effects alone and the deduced variation with caused by changes in will also apply to the variations with caused by changes in . is:
⁄ = 1.06410 −8 ( ) 4 + 2.84010 −6 ( ) 3 + 1.91010 −5 ( ) 2 +1.69510 −3 ( ) − 0.88410 −2(6)
The choice of polynomial order was made by measuring the fit residuals as a function of .
The r.m.s. fit error decreased with but going from = 4 to = 5 only decreased it by 0.2%. The concern is that use of too high a value for would render unrealistic the extrapolations from the largest/smallest datapoints to the largest/smallest possible values. It was found that although = 4 gave the largest gradients ( ⁄ ) at these extremes it was the largest value of that gave a 2 nd -order derivative ( 2 ⁄ 2 ) that varied close to linearly with at all . For a first-order correction we take to be a good estimate of on the approximately 10-day timescales considered in Figure 12, then
= − = () ⁄(7)
hence the northern hemisphere conductivity factor is
() = (1 − ⁄ ) −1(8)
The corresponding best 4 th -order polynomial fit for the southern hemisphere index is
⁄ = −0.94810 −8 ( ) 4 − 3.13710 −6 ( ) 3 −2.23810 −5 ( ) 2 − 1.13810 −3 ( ) − 0.98810 −2(9)
and as for the northern hemisphere, the conductivity-corrected index is
= − = () ⁄(10)
and the southern hemisphere conductivity factor is
() = (1 − ⁄ ) −1(11)
The corrected indices,
= ⁄ , = ⁄ and = ( + ) 2 ⁄ are
shown in Figure 12b. Note that the corrections make and very similar indeed and also that the resulting is not exactly the same as : the semi-annual variation in is slightly larger in amplitude and there is different structure around the peaks (which is also seen in both and ). This indicates that the conductivity effects in the two hemispheres do not exactly cancel in . The residuals for the polynomial fits give a percentage root mean square (r.m.s.) fit residual error in and of just 0.21%. ). It can see that the photoionization conductivity correction is greatest for the noon sector and very small for the midnight sector. Figure 14 shows the resulting time-of-year ( ) variations of the conductivity-corrected indices and corresponds to Figure 12b. In each case, the corrected index for the northern hemisphere is very similar to that for the southern and the semi-annual variation is clearly seen. Furthermore, the variations for each MLT sector are very similar indeed and similar to that for , and shown in Figure 12b: even the small-scale structure around the equinox peaks is the same in each case. The amplitude of the fractional semi-annual variation (as a ratio of the overall mean) is similar in each case, but still smallest for noon and greatest for midnight.
The polynomial fits giving the () factors can be used with the computed ( , )
pattern shown in Figure 8c to compute the ( , ) conductivity correction factors for all the hemispheric indices. The results are shown in the Figure 15 and are used to correct the indices for conductivity effects using equations (7) and (10) and the corresponding equations for the indices.
3-ii. The Russell-McPherron factor,
The top row in Figure 16 gives the -patterns of the power input into the magnetosphere, (colour pixels) estimated from 1-minute interplanetary parameters for 1980-2019 (inclusive) and here normalised by dividing by its mean value for all data, . This normalisation has the advantage of cancelling various constants in the equation for and also removes the need to be repeatedly quoting large absolute power values. This is based on the dimensional analysis theory by Vasyluinas et al. (1982)
and the derivation is described in
Lockwood (2019) but has only one free fit parameter, the coupling exponent, . This is important because ascribing a free fit parameter exponent to each variable considerably increases the danger of "overfitting" whereby a good fit is obtained to the training data that is not sustained in test data because the noise has been fitted. There are considerable sources of noise in solarwind/magnetosphere coupling studies including measurement errors, propagation lag uncertainties, the fact that solar wind seen by the upstream monitoring spacecraft may not actually hit the Earth and, most of all, data gaps in the data series, which are a particular problem especially if any data from before 1995 are employed in the fitting and training. We here use the criteria to define a valid value of (to a given required accuracy) that were derived by Lockwood et al. (2019b). These were derived in a study that introduced synthetic data gaps at random into almost continuous interplanetary data and studied how much they changed the derived values from the known correct value. In the present paper, we only use a ⁄ value if the uncertainty due to missing data is estimated to be less than 5%. All subsequent patterns as a function of fraction of a calendar year ( ) and Universal Time ( ) are generated by averaging the hourly data for each in 36 equal width bins of (each just over 10 days in width). This yields 864 -bins and we are applying a 2-dimensonal 1-3-1 triangular weighting smooth in both the and dimensions. For the observations, the geomagnetic index data are linearly interpolated to hourly values from the 3-hourly indices using linear interpolation. Figure 16 are the contours of 1 + 4 ( 2 ⁄ ) of 1.28 and 1.31 and it can be seen that agreement is very close. However, the colour pixels in Figure 16a and 16b highlight an important point made by Lockwood et al. (2020b) namely that while the "favoured" equinox/ shows a marked enhancement in ⁄ for a given polarity of [ ] the "unfavoured" equinox/ shows a decrease in ⁄ that is almost as large. This is true for both polarities of [ ] and Lockwood et al. (2020b) point out that the same thing is truebut, crucially, much less so for geomagnetic indices. Hence rather than it being the enhancement for the favoured equinox/ being the cause of the semi-annual variation (the traditional view of the R-M effect), it is the lack of a corresponding decrease for the unfavoured equinox/ that really generates the semi-annual variation. It is this fact that makes the semi-annual variation in geomagnetic indices a much larger fractional amplitude than that in ⁄ . The effect of this is seen when we look at the R-M pattern in all data in Figure 16c. The R-M pattern is still seen in the data for both [ ] polarities but it is not nearly as clear-cut and is made noisy by individual events of large negative [ ] , as pointed out by Lockwood et al. (2020a;b). Furthermore, the amplitude of the pattern is much reduced compared to that in the cases for the two [ ] polarities separately: in Figures 16a and 16b, the amplitude of the ⁄ pattern is close to 50%, whereas in Figure 16c is close to 5%. This is why R-M effect is so much clearer when we sort data according to the [ ] IMF component polarity than when we consider all data. This was one of the many valid points made by Berthelier (1976Berthelier ( , 1990. We here scale the theoretical pattern of R-M forcing,
based on the IMF orientation factor 4 ( 2 ⁄ ) (shown by the contour lines in Figure 8c) to match the amplitude of the pattern in the observed ⁄ which yields
( , ) = 4 ( 2 ⁄ ) < 4 ( 2 ⁄ ) > ⁄(12)
and = 0.1613 gives the best least-square fit to the observations and the contours shown in black in Figure 16c for 1.045 and 1.050. The normalisation is needed to ensure the mean value of ( , ) over all and is unity.
3-iii. The dipole tilt and solar wind dynamic pressure factor,
Paper 2 (Lockwood et al., 2020b) showed that the equinoctial pattern arises in the fit residuals of normalised power input to the magnetosphere, ⁄ , to the index. This is not surprising, given that the equinoctial pattern is present in the data but not present in the ⁄ estimates. However, what is significant is that Paper 2 showed the amplitude of that equinoctial pattern in the fit residuals increases linearly with the solar wind dynamic pressure, . The effect of dynamic pressure was also clearly shown to be an independent effect to that of because increases with at all fixed values of . Finch et al. (2008) show the equinoctial pattern in data from high-latitude and auroral magnetometers arises on the nightside, increases with and is associated with the auroral electrojet and substorm current wedge. In addition, Chambodut et al. (2013) showed that the amplitude of the equinoctial pattern in data from mid-latitude magnetometers is greatest at midnight and smallest at noon. Together these factors strongly indicate that geomagnetic activity is enhanced by increased squeezing the near-Earth tail, as discussed by Lockwood (2013) and consistent with the effect of on the lobe field and energy content of the near-Earth tail found by Caan et al. (1973) and Karlsson et al. (2000). The modelling presented in which is plotted as a function of and in Figure 16d. Lockwood et al. (2020b) show that the equinoctial pattern in the index increases linearly in amplitude with solar wind dynamic pressure we fit the above functional form () to the pattern amplitude the find for the conductivity-corrected observations ( ) for the mode value of of 1.50 nPa using
() = { () − } < { () − } > ⁄(14)
where the mean value is over all and . The value of is the same for the northern and southern hemisphere sub-indices because () is a measure of the whole tail and not just one tail lobe. Using the known pattern of ( , ) this yield ( , ) which is, as required, normalised to be unity. Note that increased means that the amplitude of the normalised ( , ) pattern is increased.
3-iv. The north and south pole motion factors, [ ( , )] and [ ( , )]
The sunward motion (in the + direction) in the GSEQ frame of the axial poles at speed [ ] (where H is N for the northern hemisphere and S for the southern) generates a modulation to the transpolar voltage in that hemisphere (in the GSEQ frame):
= [ ] = 0.9 . . < >. [ ](16)
where is the polar cap diameter and < > is the ionospheric magnetic field normal to the direction. The factor 0.9 allows for the fact that there is some dipole tilt motion in a geomagnetic frame, as discussed in Section (1-ii). The modulation of the transpolar voltage is by a factor (1+ / ). The factor needed (that averages unity over all and ) is
Unlike for the constants and we have no a-priori way to compute and it is derived by matching the modelled -pattern to the (conductivity-corrected) observedpattern.
Analysis
Equations (3), (12), (14), (15), (16) and (17)
Geomagnetic response by hemisphere: Results for the am, an and as indices
The results from the model are shown in Figure 17, the right-hand column of which gives the pattern for the conductivity-corrected observed index and the left-hand column gives the corresponding model prediction. The top row is for (b) the conductivity-corrected index, , and (a) its modelled equivalent . The middle row is correspondingly for and and the bottom row is for and . Figure 17 shows that the modelled patterns match the observed ones very closely. The largest disagreement is for , for which the predicted peak in the variation at both equinoxes is near 13 hrs whereas in it is near 16 hrs. This probably arises from the closeness of the northern geomagnetic dip pole from the rotation pole in recent decades (Thébault, et al., 2015) which makes it likely that the eccentric dipole fit to the field is underestimating how far the longitude separation of the two poles has fallen from 180. is not as great as in Figure 17 after we have averaged over all .
4-ii. Analysis of geomagnetic response by MLT sector: Results for the indices
The Figure 19 shows the patterns for all 12 of the conductivity-corrected indices (hemispheric and global), corrected using the -conductivity factor patterns shown in Figure 15. Figure 20 shows the corresponding modelled variations. To model the indices we use the same procedure as used for the , and indices described above. The best-fit constants and are given in Table 1. In interpreting the various values of and we should remember that the definitions used mean that larger values of and increase the amplitudes of the ( , ) and ( , ) patterns, respectively. For the only significant trend is that it is lower for noon, which is expected given that the equinoctial pattern arises on the nightside. The values for noon are also slightly larger which means a slightly larger fraction of the voltage induced by poleward motion is appearing as a directly-driven ionospheric voltage. Values for the southern hemisphere are systematically a little larger than for the northern which may well be another symptom that the eccentric dipole is underestimating the north-south asymmetry. These constants are then used to model ( ) and ( ) in the same way that they were used to model and in the previous section.
The modelled ( ) is then the average of the two.
As for the indices, the agreement is good in all cases. Figure 21 shows the variations of the modelled (dot-dash lines) and conductivity-corrected variations (solid lines) in the same format as Figure 17d. Parts (a), (b), (c) and (d) of Figure 21 are for the dawn, noon, dusk and midnight indices, respectively, and in all panels red, blue and black lines are for the north, south and global images. In general, the southern hemisphere indices are modelled more closely than the northern, but agreement is good in all cases.
4-iii. Analysis of geomagnetic response by IMF Y-component polarity.
We ). It can be seen that agreement is good, and the model successfully predicts both sub-divisions of the data. Lockwood et al. (2019b) showed that the -patterns for average were matched by corresponding patterns in the occurrence of large events. Specifically, they studied the occurrence of events exceeding the 90%, 95% and 99% quantiles (q(0.9), q(0.95) and q(0.99), respectively). The data are continuously available for 1959-2019, inclusive (61 years) which is a total of 178250 3-hourly data samples and 534720 interpolated hourly samples. Each mean value in the 864 -bins used are therefore based on 618.9 samples per bin on average. Therefore, there are 61.89 samples above the 90% quantile and just 6.19 samples in each bin above the 99% quantile and hence derived -patterns become noisy because sample numbers are low, even for these data covering 61 years. In this section, we integrate over all and studying the average nett variation with and the increases the above sample numbers by a factor of 36.
4-iv. Large events and the effect of average levels of activity.
In the above analysis we used the overall mean of for the interval 1959-2019 to estimate the overall mean of the transpolar voltage, < > and so evaluate the fractional perturbation < > ⁄ . The above sample numbers for the variation in are sufficient to allow us to divide the dataset into three 20-year intervals. To investigate the role of < >, we here break the data into three 20-year intervals, 1960-1979, 1980-1999 and 2000-2019 for which average values are 21, 25 and 17 nT (by Equation (17) (q(0.9), cyan lines) and the 99.99% quantile (q(0.9999), blue lines). The events that meet the > q(0.9) criteria last for a total of 17532 hours in total in the 20-year intervals, whereas those that meet the > q(0.9999) criteria last for a total of just 17.532 hours. It can be seen that, despite their extreme rarity, even the latter are showing variation consistent with the effect of pole motions that have been introduced here for the first time.
Discussion and Conclusions
We have identified for the first time a factor that introduces a systematic Universal Time variation into global geomagnetic activity but has been previously overlooked, namely the electric fields induced voltage Given that the largest disturbances occur in the midnight sector of Magnetic Local Time (MLT), this variation means that in geographic coordinates there is a longitudinal variation in geomagnetic activity which may be a useful fact in forecasting space weather and quantifying space weather risks. This also offers a potential explanation of some aspects of long-standing reports of longitudinal structure in auroral phenomena (e.g., Berkey, 1973;Stenbaek-Nielsen, 1974;Luan et al., 2011;Liou et al., 2018) although other aspects may be due to longitudinal structure in the field, such as the south-Atlantic anomaly.
If Earth's field were a symmetric, Earth-centred dipole, the nett effect would be relatively straightforward. The two polar caps and tail lobes would undergo matching diurnal cycles in which the pole moved against, and then with, the solar wind flow and these cycles would be in antiphase in the two hemispheres. In the first half of this cycle, the directly-driven component would mean the voltage appearing across one polar cap would be increased while that in the other would be correspondingly decreased, a situation that would be symmetrically reversed in the second half of the cycle. The storage/release response would mean that the energy being stored in the tail lobe of one hemisphere would also be increased and then decreased whilst the variation for the other lobe would be of equal magnitude and in antiphase. Hence for this effect in isolation, the average transpolar voltage for the two poles would be constant, as would the total rate of storage of tail energy, as while the northern hemisphere lobe was gaining additional stored energy the southern would be losing it (or at least gaining it more slowly) and vice versa. Hence the effect would be to alternately pump additional energy into one tail lobe and then the other, in each case recovering it during the other half of the cycle.
However, Earth's magnetic field is increasingly unlike a symmetric, Earth-centred dipole and we have here used an eccentric dipole model to investigate the effects of the (growing) hemispheric asymmetry. Because the offset from the rotational pole of the southern magnetic pole is roughly double that for the northern pole, these effects are roughly twice as large in the southern hemisphere than in the northern. Because both follow regular diurnal cycles the effects are averaged out over a full day, but they do not cancel on timescales below a day, leaving a strong variation. This asymmetric loading and unloading of one lobe relative to the other is likely to be associated with short-lived asymmetries in auroral precipitation Østgaard, 2009, Laundal et al., 2010) given that asymmetric lobe flux content caused by a strong -component of the IMF (Cowley et al, 1991) has been shown to be the cause of non-conjugate auroral behaviour (Reistad et al., 2013).
The new proposed mechanism is different to longitudinal variations in auroral ionospheric dynamics associated with hemispheric asymmetries in local magnetic field strength and direction (Gasda and Richmond, 1998) or due to ionospheric conductivities (Lyatsky et al., 2001;Newell et al., 2002). In the present paper, we do not attempt to make detailed comparisons of the mechanism we propose with other proposals; however, we do note that the possibility exists that various phenomena that have in the past been attributed to ionospheric conductivity effects and/or ionosphere-thermosphere momentum exchange and/or longitudinal structure in the geomagnetic field may have been associated with the effects of polar cap motions that were not considered.
We pressure the hemispheric asymmetry in the geomagnetic field. The last of these factors acts in two ways. The first is the effect discussed in Paper 3 on the tail squeeze, but the new effect introduced here is that it also generates a hemispheric asymmetry in the diurnal cycles of sunward motion of the two poles.
Given that the incidence of the most geoeffective solar wind impacting on the magnetosphere should be random in both and , and that the Russell-McPherron effect (that could introduce such variations) has little, or even the inverse, effect when the field is most strongly southward in the GSEQ frame (Lockwood et al., 2020a;b), the variations in the occurrence of the largest storms with both and is a puzzle. [ ] GSEQ at the equinoxes, nor is there any proposed reason why there might be. This was discussed specifically in the context of the semi-annual variation of large storms in the previous papers in this series, but Figure 23 shows it is an issue in relation to the variation as well. More work is needed to understand why a model that is aimed at predicting average geomagnetic activity levels is generating a variation that even approximates to the variation in the occurrence of the largest storms. There are a number of possibilities, including the effect of "pre-conditioning" of the magnetosphere-ionosphere system by average conditions ahead of the arrival of an event at Earth. A key part of the variation modelled here is the sinusoidal sunward motion of the poles that is here introduced for the first time and hence we need to study how this mechanism influences large events as well as the average conditions considered in this paper.
Lastly, we can now look back with the benefit of hindsight at the debate in the literature about the R-M and equinoctial patterns. Annick Berthelier was correct in the main point of her comment (Berthelier, 1990) on the paper by Russell (1990) that there was no element of the latter paper that showed the equinoctial effect (what she called the "McIntosh effect") was not operating alongside the R-M effect. In that comment, and in her 1976 paper (Berthelier, 1976), she demonstrated that she fully understood the R-M effect was active because of the observed influence of the IMF -component. It is not clear why in their response Russell and Scurry (1990) were so adamant that the "McIntosh effect" was not a factor; however, one interesting point to note here is the influence of semantics on the debate and what was meant by the term "McIntosh effect". Russell and Scurry (1990) In the present paper and in Paper 3 we have shown how the pressure equilibrium in the near-Earth tail for a hemispherically-asymmetric geomagnetic field explains how the energy input into the magnetosphere, controlled by the R-M mechanism, results in a "McIntosh pattern"
and an additional UT variation. There is no need to invoke the K-H instability, nor any other "viscous-like" (meaning anything that is not reconnection) interaction across the magnetopause. Indeed, the results of Finch et al. (2008) indicate that the equinoctial/McIntosh pattern has nothing to do with solar wind-magnetosphere coupling at all. In this context, the Expanding-Contracting Polar Cap (ECPC) model also indicates that viscous-like interaction voltages have, in the past been greatly overestimated in studies that overlook the delayed response of the tail. Lockwood et al (1990) pointed out that the ECPC model predicts that ongoing reconnection in the tail contributes to transpolar voltage which can therefore have large values even after the IMF has turned northward. Wygant et al. (1983) showed (their Figure 6) that one hour after a northward turning of the IMF the transpolar voltage could range between 10kV and 100kV, but that the upper limit decayed with time elapsed since the northward turning, such that after about 10 hours little more than open field lines of the two tail lobes and so, almost certainly, ongoing tail reconnection at some level. This means that even much of the 10kV seen more than 10 hours after the IMF was last southward is likely to be reconnection-driven and not caused by any viscous-like interaction across the magnetopause. Hence the ECPC model shows that residual voltage seen when the IMF is northward cannot be used as evidence for a viscous-like transfer of momentum across the magnetopause. Similarly, the results presented in this paper show that a McIntosh/equinoctial pattern in geomagnetic activity cannot be used as evidence for a viscous-like interaction.
Dedication. This paper is dedicated to the memory of Annick Berthelier who, as well as highlighting many issues around the time-of-year/time-of-day patterns of solar wind forcing and geomagnetic response, did much to help preserve historical geomagnetic data and to maintain the continuation of long sequences of indices. In 1987, what became the International Service of Geomagnetic Indices (I.S.G.I.) was moved from the Netherlands to France and placed under the joint responsibility of her and her colleague Michel Menvielle. After she died in 1997, at the age of just 54, he noted in her obituary: "We shall all remember Annick's profound sense for the good of the science community and of the stringent necessity for anyone to contribute. This is what pushed her to take charge of I.S.G.I., a work in which she deeply invested herself in her last years, when she understood that a definite action was needed to ensure the durability of the magnetic indices data base. Her energy, conviction and enthusiasm all decisively contributed to maintain the interest and quality of these data which are absolutely critical for geomagnetism and sun-earth connection studies". (International Association of Geomagnetism and Aeronomy (IAGA) News # 38, October 1998)
Appendix A. Summary of the role of solar wind dynamic pressure in generating the semi-annual variation
In this Appendix we present some alternative graphics to make the same points as were made using other plots in Papers 1, 2, and 3 (and brought together in the present paper) about how solar wind dynamic pressure contributes to the semi-annual variation. Figure 1 of Paper 3 (Lockwood et al., 2020c) shows that geomagnetic activity (both on average and in the occurrence of large events) is enhanced at constant power input to the magnetosphere (which depends on IMF orientation) by enhanced solar wind dynamic pressure. This is also supported by the modelling shown in Paper 3 which shows that, for a given magnetospheric open flux, energy stored in the tail and cross-tail current is increased by enhanced solar wind dynamic pressure, a result first reported from observations by Caan et al. (1973). Another plot which explains the relationship of the effects of solar wind dynamic pressure and of power input to the magnetosphere is Figure 19 of Paper 2 (Lockwood et al., 2020b). . Figure A1b shows that, in reality, a full range of GSEQ values are seen with zero and 180
(respectively, purely northward and southward IMF in the GSEQ frame) being roughly half as common as GSEQ = 90. Appendix B of Paper 1 studied the implications of the fact that GSEQ is not always 90 (Lockwood et al., 2020a). Figure A4 shows how solar wind dynamic pressure allows for most of the remaining anomaly. This plot has the same format as Figure A3, and the black lines and symbols reproduce the variations shown in Figure A3. The red, green and blue lines show the variations of the averages of the am index for simultaneous in its three tercile ranges.
The red line is for in its lower tercile of values, the green line is for in its middle
tercile of values and the blue line is for in its upper tercile. As in Figure A3, the peaks for the favoured equinox (for a given [ ] GSEQ polarity) are very similar indeed in shape (remember the variations have been normalised so that they all have the same magnitude), but the minima are very different. Figure A4 clearly shows that enhanced increases values of during the unfavoured equinox and it is this that gives the larger semi-annual variation in am than in (and a very much larger semi-annual variation than that in [ ] GSM ). Figure A4 is showing the effect on the variations through the year of the effect that is noted in Figure 7 of the main paper.
This Appendix has used different graphical plots (of the same data) to those used before in this series of papers to emphasise the key point that the half-wave rectification of solar-windmagnetosphere coupling is not the main cause of the semi-annual variation and that previous studies, which implied or stated that it was, were actually getting good agreement through the simplifying assumption that the IMF always lay in the solar equatorial frame. In reality, the key element introducing the asymmetry between the increased geomagnetic activity for the favourable IMF component polarity and the unfavourable one is the solar wind dynamic pressure not a half-wave rectified coupling function. Solar wind dynamic pressure increases geomagnetic activity, for a given level of open flux production, at both equinoxes (favoured and unfavoured) -as is shown in Figure 7 of the main paper. Hence it increases the peak for the favourable IMF component polarity whilst reducing the minimum for the unfavourable IMF component polarity, causing the asymmetry required to generate the semi-annual variation. The modelling in Paper 3 (Lockwood et al., 2020c) explains how the squeezing of the tail by solar wind dynamic pressure is less effective at the solstices than at the equinoxes because at the solstices a smaller fraction of the open flux present at any one time has migrated into the tail in the summer hemisphere (i.e., a larger fraction threads the dayside summer magnetopause). This is because, initially, after reconnection the sheath flow and curvature forces act in opposite directions on newly-opened field lines in the summer hemisphere, whereas they act in the same direction in the winter hemisphere. This difference and the solar wind dynamic pressure effect also explains the observed equinoctial (a.k.a.
McIntosh) pattern in geomagnetic activity. The hourly values were obtained by linearly interpolating the three-hour index values. Stubbs et al., 2005) transformed into the GSEQ frame (where points towards the Sun). The crosses show the location of the eccentric dipole axial pole mapped in the same way and displayed using the corresponding colour. The mapping is for an assumed emission altitude of 130 km: (a) is for the northern hemisphere oval and shows both the pole and the oval moving toward the Sun; (b) is for the southern hemisphere oval and shows both the pole and the oval moving away from the Sun and in the and in the − direction. Note that the direction is into the plane of the diagram and so the XY plane is here viewed from the southern side and so dawn is the to the left and dusk to the right and noon to the top, (in the + direction) and hence the motion of the poles with is clockwise. in each case. Note that the direction is into the plane of the diagram and so the XY plane is here viewed from the southern side and so dawn is the to the left and dusk to the right and noon to the top, (in the + direction) and hence the motion of the poles with is clockwise for this viewpoint. , which for these variations with are taken to be due to conductivity effects alone. (d) The variations of (red points) and (blue points) as a function of the mean dipole tilt angle, for the same -. The black lines in (d) are 4 th -order polynomial fits to the points that are used to correct to and to , using equations (7) and (10), respectively. Figure A3 for the am index for: (red lines) the lower tercile of the distribution of solar wind solar wind dynamic pressure, ; (green lines) for the middle tercile of ; and (blue lines) the upper tercile of . The black lines are for , as plotted in Figure A3, shown here for comparison.
⁄
), we also employ: ( ), ( ) and ( ) = { ( )+ ( )}/2); ( ), ( ) and ( ); ( ), ( ) and ( ); and ( ), ( ) and (
Stubbs et al. fitted circular polar cap boundaries to the poleward edge of the northern and southern auroral oval in the geomagnetic latitude-MLT frame by varying the radius and centre location in the noon-midnight and dawn-dusk directions and Figure 6 shows those boundaries mapped into the Geocentric Solar Equatorial frame (GSEQ) where points from the centre of the Earth to the centre of the Sun and lies parallel to the solar equatorial plane and points broadly from dusk to dawn.
Figure 1 of
1Paper 3(Lockwood et al., 2020c) shows that geomagnetic activity (quantified by the am index) is enhanced at constant power input to the magnetosphere (which depends on IMF orientation) by enhanced solar wind pressure, (which does not depend on IMF orientation). Another plot showing the relationship of the effects of dynamic pressure and power input on am isFigure 19of Paper 2(Lockwood et al., 2020b). The independent effect of also supported by the modelling shown in Paper 3 which shows that, for a given magnetospheric open flux (that depends on the prior history of the IMF orientation), energy stored in the near-Earth tail lobes and cross-tail current are both increased by enhanced solar wind dynamic pressure, and both of have the potential to enhance geomagnetic activity. This evidence is brought together, using different plots, and summarised in Appendix A to the present paper.Figure7 presents further details of the effect of solar wind dynamic pressure using all am index and solar wind data for 1995-2019 (inclusive). The data are interpolated linearly to 1-minute values from the 3-hourly raw index data and compared to 1minute interplanetary data allowing for the 60 minute response lag found in Paper 2. There are 3 groups of four panels in Figure 7: the top 4 (a-d) are for the simultaneous (allowing for the 60 minute lag) in the lower tercile of its overall occurrence distribution, < (0.33), where (0.33) is the 1/3 quantile of the cumulative distribution of all 1minute values, the middle 4 panels (e-h) are for the middle tercile of this nearsimultaneous solar wind dynamic pressure, q(0.33) < (0.67) and the lower 4 panels (i-l) are for the upper tercile of the near-simultaneous solar wind dynamic pressure, (0.67). The data are also sorted according to the polarity of the Y-component (in the GSEQ frame) of the IMF, with the left-hand panels being for IMF [ ] GSEQ < 0 , and the right-hand panels being for IMF [ ] GSEQ > 0. The panels are organized in pairs with the upper plot of each pair showing the probability distribution function (p.d.f.s) of normalised power input to the magnetosphere, ⁄ , as a function of fraction of the year F and the lower of each pair showing the corresponding plot of the normalized am amplification factor, [ ⁄ ] = ( /< > )/( ⁄ ), as a function of F and in the same and ⁄ bins as the p.d.f.s in the plot above it. The upper plots all show the Russell-McPherron (R-M) is at work, with normalised power input to the magnetosphere, ⁄ , increased around the "favoured" equinox, which is the March equinox (around = 0.22) for [ ] GSEQ < 0 and the September equinox (around = 0.73) for [ ] GSEQ > 0. Because there are some common factors in the expressions for and (specifically, the solar wind speed , mean ion mass , and number density ) larger values of ⁄ are more common for larger
Figure 8c
8ccompares the theoretical Russell-McPherron and equinoctial effects by overlaying the -patterns, both derived using the eccentric dipole geomagnetic field model of Koochak and Fraser-Smith (2017) with constants interpolated to the year 2007 (the mid-point of 1995-2019, the interval for which 1-minute resolution interplanetary data are available. The dipole tilt angle was computed as a function of and , being the angle between the Earth's (eccentric) dipole axis ⃗⃗ and the geocentric position vector of the subsolar point, , computed using the SUBSOL routine of the LOWTRAN7 Sun and Moon Models Matlab software package. The colour contours in Figure 8c give the absolute value of the dipole tilt angle, ||, superposed on which are contours showing the IMF orientation factor used by , , where is the clock angle of the IMF in the GSM frame, computed from a given IMF orientation in the GSEQ frame using the CXFORM Coordinate transformation package described above. These predictions are the average for an equal mix of [ ] GSEQ = −| | < 0 and [ ] GSEQ = +| | > 0 and contours are shown for < > of 0.28 (in black) and 0.31 (in mauve). The < > = 0.31 contours in Figure 8c define the two Russell-McPherron peaks and are used in the derivation of parts a and b. These show the mean values of | | over the range of F defined by the maximum extent in of the mauve < > = 0.31 contour as a function of : Figure 8a is for the March equinox and Figure 8bfor the September equinox. The area shaded pink is the extent of the peak defined by the corresponding mauve contour inFigure 8c. Time increases up the plots and parts a and b
hemispheric sub-indices, and , the four indices and the two hemispheric sub-indices of each) as a function of and . This involves developing factors that allow for the polar cap motions discussed in section 2, for ionospheric conductivities, for the R-M effect in solar wind forcing and for the effect of dynamic pressure and dipole tilt on the near-Earth tail. In Section 4 we compare the modelled -patterns with those for conductivity-corrected versions of the , and indices, and in section 5 we do the same for all 12 of the hemispheric and global indices to study how the model performs in the four 6-hour MLT sectors. In section 6 we also apply the model to one example (the midnight index) broken down into two subsets of the prevailing IMF Y-component. Sections 3-6 all deal with modelling the average values of the indices (at a given and ) and in section 7 we look at large and near-extreme events in the index and show the modelling has implications,
Earth's rotation. For an axisymmetric, geocentric dipole field, the motions of the two poles would be equal and opposite and any nett global effect would not show any variation. To allow for the large (and currently increasing) asymmetry in the geomagnetic field, we here use the eccentric dipole field ofKoochak and Fraser-Smith (2017). Results are presented for the interpolated eccentric dipole coefficients that apply to the years 1989 (the midpoint of the1959-2019 interval of all data), and 2007 (the midpoint of the 1995-2019 interval of quasi-continuous interplanetary data) but were also generated for 1980 and 2015 and are only different in small details for our purposes. As for Figure 6, we mapped the location of the eccentric axial poles in geographic coordinates into the GSEQ frame using the CXFORM Coordinate transformation package, initially written by Ed Santiago of Los Alamos National Laboratory and Ryan Boller of NASA's Goddard Space Flight Centre and re-coded for Matlab by Patrik Forssén (SatStar Ltd & Karlstad University) in 2017. This software package is based on the equations by Mike Hapgood of RAL Space, Rutherford Appleton Laboratory (Hapgood, 1992). The loci of the eccentric dipole axial poles in the frame of the GSEQ reference frame are shown in Figure 9, in which the rotation of the Earth makes the poles rotate clockwise. The points show the location of the poles at 12 . The greater offset in the axial eccentric dipole poles from the rotational pole in the southern hemisphere makes the radius of the orbits larger for the southern hemisphere. Figure 10 shows the -component (sunward) velocity of the two poles in the GSEQ frame as a function of . The two are in close to antiphase (but not exactly as the longitudinal separation of the axial poles is not 180) and the larger offset of the pole in the southern hemisphere means that the amplitude of its sinusoidal variation in the velocity in the southern hemisphere, , is larger than that for the northern hemisphere, . The open field line region in one hemisphere moves as a whole in GSEQ because the geomagnetic field, both open and closed field lines, moves as a whole. There appears to be some -dependent distortion of the region of open flux, presumably induced by changing pressure balance between open and closed field lines, because observations of dipole tilt effects on locations of the open-closed boundary in geomagnetic reference frames, as discussed in section 1-i, reveal that up to 10% of the dipole tilt variation is reflected in the motion in the inferred or modelled OCB boundaries in a geomagnetic frame. This means that at least 90% of the diurnal motion of the magnetic pole and the geomagnetic frame in the GSEQ frame, and in particular its motion toward and away from the Sun, must be reflected in motion of the open field line region, as a whole, in GSEQ.
Figure 11
11illustrates why this has an influence. The Expanding-Contracting Polar Cap (ECPC) model of the excitation of ionospheric convection in non-steady-state situations is based on the fact that only in steady-state (or for averagesover sufficient timescales that steady state applies) does the solar wind electric field map down open field lines into the polar ionosphere: by Faraday's law (in differential form) a non-zero rate of change of magnetic field in the tail lobe, on open field lines between the solar wind and the ionosphere gives a curl of the electric field (whereas in steady state ⃗ = − ⃗ = 0 ⁄ ) and, integrated down the open field lines, this decouples the electric field in the polar cap from that in interplanetary space (i.e., there are induction effects).
that shows open field lines mapping from the ends of reconnection X-lines AB in the dayside magnetopause and DE in the cross-tail current sheet, mapping down open field lines on the open-closed boundary to their ionospheric footpoints, the ends of the "merging gaps", ab and de, respectively. To look at the total decoupling of, for example, the voltage across the "Stern Gap" CF (the region of open magnetospheric field lines in interplanetary space such that is the integral of the interplanetary electric field along the line CF) and that along the ionospheric polar cap diameter (the "transpolar voltage" or "crosscap potential drop", ) we use Faraday's law in integral form by considering the loop CFfc and neglecting any field-parallel voltages along the field lines Cc and Ff (we know these to be comparatively very small from the minimum energies of primary precipitating electrons or ions):
wind speed and hence interplanetary electric fields and voltages (such as GSE and GSM), and in which have shown (Figures 9 and 10) that the footpoints ab, cf and de have a sinusoidal velocity variation in the direction. If we use the estimate that 90% of the variation for the axial pole is reflected in the open-closed boundary (see section 1-ii), from Figure 10 we find this polar motion velocity in the X-direction [ ] has a sinusoidal variation of amplitude of about 100 ms -1 for the southern polar cap and about 50 ms -1 for the northern at all times of year. Using an ionospheric magnetic field strength of ⃗ of 4.510 -5 T in the topside ionosphere in the − direction, this gives an electric field ⃗ = − ⃗ ⃗ with a dawn-dusk component [ ] that has sinusoidalvariations of amplitudes 5 mVm -1 and 2.5 mVm -1 for the southern and northern hemispheres, respectively. This also applies to the merging gaps ab and de as well as the polar cap diameter cf because, to a first approximation, the polar cap moves as a whole. A useful comparison is with the electric field in interplanetary space: in the GSEQ frame the solar wind speed is typically 400 kms -1 which for a flow-transverse IMF component of 5nT is an interplanetary electric field of 2 mVm -1 . However, the best way to put these variations into context is to consider the magnitude of their effect, relative to the transpolar voltage : a circular polar cap of angular radius 15 gives a polar cap area of10 13 m 2 , an open flux of 510 8 Wb and a polar cap diameter at an altitude of 850 km of 3780 km, for which the southern hemisphere pole motion gives a sinusoidal voltage modulation of amplitude = 19.0 kV and for the northern hemisphere pole motion gives 9.5 kV. These are not negligible fractions of typical values of : for example, Lockwood et al. (2009) find that average values of during quiet times (when the polar cap flux of 510 8 Wb is appropriate) is 25kV. This rises to 52 kV during substorm growth phases, 64 kV between substorm onset and peak expansion, 72 kV between peak expansion and the start of recovery, 67kV in substorm recovery phases and 83 kV during steady convection events. Lockwood et al. (1990) find the polar cap flux increases to about 10 9 Wb during steady convection events which increases the polar cap radius and hence the predicted variations due to pole motion by a factor 2 1/2 1.4. Hence for the northern hemisphere the percentage southern hemisphere induction effect is, on average, 100 ⁄ 76% in quiet times, falling to about 32% during steady convection events. Values for the northern polar cap are roughly half of these. When a polar cap is moving sunward it is effectively adding to the effect on open field lines of the (antisunward) solar wind flow and in the other half of the diurnal cycle, when is moving away from the Sun, it is reducing the effect of the solar wind flow. From equation (1) this will have a mixture of two effects. The direct effect would be the modulation of the transpolar voltage by the diurnal motion of the magnetic pole for given solar wind electric field and Stern Gap voltage . If this were the only effect, then the change in observed transpolar voltage would equal the voltage induced by the polar cap motion so = . However in general we should not expect the full effect of the pole motion to appear in the transpolar voltage and in general
, ), ( , ) and [ ( , )] account for the effects on of, respectively, the Russell-McPherron effect in solar wind-magnetosphere coupling, dynamic pressure and dipole tilt effects on the tail lobe, and the pole motions, as described in the following subsections. Each of these factors is normalised so its average value over all and is unity and so the modelled pattern is then scaled by multiplying by the average value of the index in question, < > for the full period of in integer number of years (i.e., over all and )
study the dependence of = ( − ) and = ( − ) with and compare with the corresponding variations of the mean dipole tilt angle, . Figure 12 presents the results. Figure 12a shows the variations with of the normalised observed indices blue). These are averages for all available data that are for 1959-2019, inclusive and hence there are 4951 3-hour samples in each bin. All three indices show the semi-annual variation clearly, but and show the clear effect of photoionization conductivity enhancement with enhanced index values in summer and reduced values in winter in both cases.Figure 12cshows the variations with of blue) which are close to being mirror images of each other. The same is true of the variations of these ratios with shown inFigure 12d. The best 4 th -
Figure 13
13Figure 13 shows the corresponding plots to Figure 12d (which is for the index) for the 4 pairs of hemispheric indices: (a) ( ); (b) ( ); (c) (
who shows that ⁄ correlates very highly with the index and that the one major limitation in the theoretical formulation of (the omission of the relatively small solar wind Poynting flux) causes only very small errors.Finch and Lockwood (2007) have shown performs better than (or as well as) all other simple coupling functions on all timescales between 3 hours and 1 year (on timescales approaching one year, IMF orientation factors average out and simpler coupling functions perform as well). One feature unique to the Vasyluinas et al.(1982) formulation is it employs all the relevant interplanetary variables
Figure 16a
16ashows the -pattern of ⁄ for the subset of the data when the mean IMF Y component in the GSEQ frame, [ ] was negative over the prior hour. Figure 16b shows the corresponding plot for [ ] > 0. Both plots show the behaviour expected of the R-M effect on solar-wind magnetosphere coupling (dominated by magnetopause reconnection) with [ ] < 0, giving enhanced ⁄ at the March equinox ( 0.22) and of about 22hrs, whereas [ ] > 0 gives enhanced ⁄ at the September equinox ( 0.73) and of about 10hrs. The CXFORM Coordinate transformation package was also used to compute the GSEQ to GSM transformation of unit IMF vectors in the + anddirections of GSEQ and to give the IMF clock angle and hence the Russell-McPherron predictions of the 4 ( 2 ⁄ ) IMF orientation factor in and hence the factor ( , ). Note that in the original paper, Russell and McPherron (1973) used a half-wave rectified southward component IMF orientation factor ( ⁄ ) whereas we employ = 4 ( 2 ⁄ ): these two have been compared and discussed by Lockwood et al. (2020b). The black lines in the top row of
Paper 3 of the series supports this idea and shows that the effectiveness of the squeeze depends on the dipole tilt, .The factor () was modelled in Paper 3(Lockwood et al., 2020c) using the asymmetric magnetopause location model of Lin et al. (2017) by assuming that the tail is in equilibrium with a solar wind of dynamic pressure . These authors modelled the magnetic shear across the cross tail current sheet for various values of the geoeffective southward component of the IMF [ ] , and the solar wind dynamic pressure, that are the inputs to the magnetopause model. The shape of the variation of with shown in Figure 6c of Lockwood et al. (2020c) does not vary with [ ] nor and a 4 th order polynomial that fits (with appropriate scaling) all the variations of B that is accurate to within an r.m.s. error of 0.02% is () = 3.64510 −8 4 − 1.05910 −7 3 − 1.11510 −4 2 + 4.76810 −4 + 1 (13)
[(
, )] = 1+ / = 1+ (0.9 . . < >. [ ] )/ (16) where we compute from the average of the interval in question from the regression equation given by equation (A4) in Appendix A of Lockwood et al. (2020b). = ( 6.6810 −5 ) 3 − ( 1.6610 −2 ) 2 + 1.89 + 6.17
provide a full recipe for computing a modelpattern for the conductivity-corrected indices in the two hemispheres (generically ) and the corresponding global index is the average of the two, one remaining un-quantified and free variable, in equation (15) by minimising the r.m.s. fit residuals for each index {< ( − ) 2 >} 1/2 for the 864 -averaging bins using the Nelder-Mead search method. In section 4-i we compare the model predictions with the am, an and as indices and in section 4-ii, the same procedure is used to model all 8 hemispheric indices (and hence the 4 global indices). The best-fit constants and required for each index are given in Table 1. For both and (indeed all the indices, see next section) is of order 0.1-0.2 which implies roughly 10-20% of the induced voltage caused by pole motion goes in to directly-driven changes in the transpolar voltage and 80-90% goes into the rate of change of field accumulation/loss in the near-Earth tail lobe. In Section 4-iii we study the effect of IMF component and in Section 4-iv we study the relationship of average values to the occurrence of large events.
Figure 18 shows
18the nett variations obtained by integrating over all times of year, . In each panel the colour scheme is red/blue/black lines are for the northern/southern/global index.Figure 18ashows the variations for the raw observed indices, and 18b shows the advantage of generating and modelling the conductivity-corrected indices. The variations in Figure 18a are quite complex but those in 18b are considerably simpler and we infer that the conductivity effects considerably complicate other hemisphere-and -dependent effects in the raw indices. The variation shown by the dot dash lines in Figures 18c and 18d are the variations for the modelled indices , , and and in Figure 18d are plotted on the same axes as , , and to aid comparison. The agreement is very good, and the model is capturing the major features of the variations of all three indices and even the difference in the peak times for and
)
indices give us an opportunity to study the geomagnetic response in different MLT sectors in the same way. These indices are available for 1959-2013. The 4 global indices, ( ), ( ), ( ), and ( ), are available from the International Service of Geomagnetic Indices (ISGI) website, the 8 hemispheric sub-indices ( … etc.) are available from ISGI on request.
Figure 22 .
22have also used the model to also simulate the results for the [ ] GSEQ > 0 and [ ] GSEQ < 0 data subsets. This further subdivision means there are 30 IMF polarity-index permutations and we here show the result for just the one case, the ( ) index, in The left-hand column is for IMF [ ] < 0, the middle column for [ ] > 0 and the right-hand column for all data. The top row shows the R-M solar wind forcing pattern ( , ) in each case. The middle panel shows the model predictions of the ( ) index and the bottom panel the corresponding averages of the (
, corresponding to ~ 39, 43 and 34 kV) and we again compute [ ] and [ ] for the eccentric dipole axial pole locations at the middle of each interval. The observed (black) and modelled (mauve) variations (averaged over all ) are shown in parts d-f of Figure 23. These have not been conductivity-corrected because averaged of all and for the global index these corrections are negligibly small. The higher activity levels for 1980-1999 mean that < > ⁄ is smaller and the pole motion terms [ ( , )] and [ ( , )] are less important and the variation is dominated by the other two factors, ( , ) and, in particular, ( , ). For 2000-2019 the lower activity level means that the pole motion terms have a much greater effect. For 1960-1979 the effect is halfway between that for the other two intervals. Parts a-c of Figure 23 compare the variations in average fields (reproduced as dashed black lines) with the occurrence of large geomagnetic storms by giving the 90% quantile of the distribution of 3-hourly
have compared our first-order model of the combined effects with observed mid-latitude range indices, using the , , and indices that are available for 61 years now. Using 36 equal-sized bins of fraction of calendar years, , this means we have 619 interpolated 1-hour samples in each -averaged bin. The twelve indices are only available for 1959-2013 (55 years) and so this number drops to 558 samples in each bin. These large sample numbers are important. Lockwood et al. (2020a; 2020b) show that the largest geomagnetic events are not caused by the R-M mechanism but rather by events of strong southward IMF in the GSEQ frame: indeed, these authors show that the R-M effect actually reduces the geoeffectiveness for the most southward-pointing fields in GSEQ. These large events are mainly driven by field inside, and ahead of, coronal mass ejections and the impact of such an event on Earth must be completely random in and there is no evidence that the expectation that it is also random in is incorrect. The large sample numbers in each bin are important to average out the random occurrence of the most geo-effective solar wind hitting Earth. We have demonstrated how the four factors discussed in this paper can explain why average geomagnetic activity displays equinoctial (McIntosh) time-of-year/time-of-day patterns, with an additional UT variation, instead of the Russell-McPherron pattern. This is despite the fact that sorting the data by the -component of the IMF reveals the Russell-McPherron effect is the fundamental cause of the semi-annual variation, and of the variation for each equinox separately, as demonstrated in Papers 1 and 2 (Lockwood et al., 2020a; b). The factors included in our initial modelling of the patterns are the Russell-McPherron effect; ionospheric conductivity variations; the dependence of tail squeezing on dipole tilt angle and dynamic
Cooker et al. (1992) proposed a solution to the dichotomy of the R-M effect giving the semi-annual variation, and yet that large storms are driven by CME impacts, by proposing that the IMF [ ] GSEQ component is enhanced by compression in the sheath ahead of the impacting CME and this enhanced [ ] GSEQ is converted into enhanced [ ] GSM at the favoured equinox. However, this is not the answer to the puzzle because we here show that enhanced [ ] GSM is associated with enhanced negative [ ] GSEQ and there is no enhancement in either average values or events of large negative
ascribe to the McIntosh effect an invocation of the Kelvin-Helmholtz (K-H) instability on the magnetopause. However,Berthelier makes no statements specifically invoking this mechanism and indeed the term never appears in the original paper by McIntosh (1959) who simply, and correctly, pointed out a dependence of geomagnetic activity on the dipole tilt angle. The K-H mechanism was introduced into the story 11 years after the McIntosh paper byBoller and Stolov (1970) and it is this that seems to be the main objection of Russell and Scurry, and the results of the present series of papers confirm that this is a fully valid objection. In her 1976 paper, Berthelier mentions that the K-H mechanism had been postulated but never attempts to quantify its effectiveness and never specifically invokes it. It is not mentioned in her 1990 comment at all. Hence, for example, by postulating dipole tilt effects on magnetopause reconnection,Russell et al. (2003) were adding precisely what Berthelier meant by the "McIntosh effect" to the R-M effect, i.e., a dependence on the dipole tilt angle. In fact, the results ofFinch et al. (2008) subsequently showed that the McIntosh effect is a nightside phenomenon and does not influence dayside high-latitude flows and currents in the way that a dipole tilt effect on magnetopause reconnection would.Finch et al. (2008) andLockwood (2013) ascribe the effect to the substorm current wedge in the near-Earth tail. This is fully consistent with Berthelier's assertion that the R-M and McIntosh/equinoctial effect could both be operating simultaneously.
10kV was observed. This was explained by Lockwood et al. as the transpolar voltage being enhanced in some cases by substorm expansion phases after the IMF had turned northward as energy stored in the tail is released, but such substorms became weaker as the interval of continuous northward IMF progressed because open flux lost was not replenished, a conclusion supported by the analysis of Milan (2004). Lockwood (2019) has pointed out that the geomagnetic tail never disappears and so there is always some magnetic shear between
Figure
A1 shows the occurrence distributions of two IMF parameters for a full Hale cycle of near-continuous IMF observations for the interval 1996-2018. Figure A1a is the distribution of values of the IMF component in the GSEQ frame, [ ] GSEQ and shows that negative values are very slightly more common than positive ones, the asymmetry being mainly around the peaks that are at small |[ ] GSEQ |.
Figure
A1b shows the distribution of the IMF clock angle in GSEQ , GSEQ = −1 ([ ] GSEQ [ ] GSEQ ⁄ ). The average value and the mode value of this distribution is 90 (the vertical red dashed line), which is the value assumed to apply all of the time in the demonstration of the R-M effect in the original paper
Figure
A2 makes the point that this roughly symmetric distribution of the two IMF [ ] GSEQ polarities is present all the time during the interval covered. The data are means in half-year intervals centred on the two equinoxes. The top panel shows the sunspot number, the second panel the solar polar fields, the vertical grey lines marking the solar polar field polarity reversals. The third panel shows the fraction of the time, , that the two [ ] GSEQ polarities are present. It can be seen that is close to 0.5 for the two polarities all the time, there being largest deviations in the rising phases of the solar cycles. The senses of these deviations are different in the two solar cycles and reflects the solar polar field polarity, an indication of the Rosenberg-Coleman effect at work on [ ] GSEQ (along with the fact that [ ] GSEQ and [ ] GSEQ most often have opposite polarity because of the Parker spiral configuration of the IMF). Note that the two solar cycles differ in amplitude and this will leave some net bias in the distribution for the whole Hale cycle, shown in Figure A1a. The bottom panel of Figure A2, shows the number of 1-minute [ ] GSEQ samples, (on a logarithmic scale), in the six-month intervals and in bins of [ ] GSEQ that are 1nT wide. For large |[ ] GSEQ | the distributions are highly symmetrical, but there are weak asymmetries at small |[ ] GSEQ |. The point relevant to the R-M effect is that for either equinox the distribution of |[ ] GSEQ | values for the two polarities is broadly the same: this means that in order for there to be a semi-annual variation, with equinox peaks, the increase in geomagnetic activity for the favoured IMF [ ] GSEQ polarity (for that equinox) must exceed the decrease in geomagnetic activity for the unfavoured [ ] GSEQ polarity.
Figure
A3 studies variations with fraction of a calendar year, , for the two polarities of [ ]GSEQ . In every case, the plots have been normalised to the maximum value to help reveal the differences in the behaviour of the minimum values. The blue lines and symbols are for the am geomagnetic index and clearly show that the decrease for the unfavoured polarity at a given equinox is smaller in magnitude than the increase for the favoured polarity. It is this fact that gives the large semi-annual variation in am.The orange lines and symbols give the variation predicted using the Russell-McPherron paradigm. It shows the half-wave rectified southward field in GSM, [ ] RM , computed by adopting the assumption that [ ] GSEQ = 0 (so the clock angle in GSEQ, [] GSEQ = 90), resulting in a northward field in GSM of [ ] RM = [ ] GSEQ sin ( GSEQ ), where GSEQ is the rotation angle between the GSEQ and GSM frames. This yields half-wave rectified southward field of [ ] RM = − [ ] RM for [ ] RM < 0 and [ ] RM = 0 for [ ] RM 0. A baselevel value b has been added to match the peaks of am and of the other parameters for the favoured equinox at that polarity. This R-M prediction matches the observed variations for am rather well. However, if we do not make the simplifying assumption that [] GSEQ = 90, and instead use the actual values shown in Figure A1b, we obtain the average variations for the half-wave rectified southward field in GSM, [ ] GSM , that are shown by the mauve lines and symbols.In this case, the decrease for the unfavourable polarity of [ ] GSEQ is almost equal in magnitude to the increase for the favourable [ ] GSEQ and when we add them together with the roughly matching probability distributions shown inFigure A2, we will get only a very small semiannual variation in the average value. Hence it is not just the half-wave rectification that is giving the good agreement to the semi-annual variation for the R-M effect demonstration, the simplifying assumption of [] GSEQ = 90 is a vital component, and without it the semi-annual variation predicted is very small. The explanation of why this occurs for general [] GSEQ was given in the Appendix B to Paper 1.(Lockwood et al., 2020a).The black lines and symbols inFigure A3are for the estimated power input into the magnetosphere , . In this case the decrease for the unfavourable polarity of [ ] GSEQ is somewhat smaller in magnitude than the increase for the favourable [ ] GSEQ , but not by as much as for the am index. Hence using (with its 4 ([] GSEQ /2) IMF orientation factor)issolving a small part of the anomaly introduced by using all values of [] GSEQ (i.e. allowing for the non-zero IMF [ ] GSEQ component) and not assuming [] GSEQ = 90, as in the original demonstration of the R-M effect.
Figure 1 .
1Universal Time-year spectrograms of normalised geomagnetic activity indices. In each panel the mean value in 3-hour bins of UT for a given calendar year are shown as a ratio of the overall mean for that year (generically < > ⁄ where and < > are, respectively, 3-hour and 1-year means of the index in question), colour-coded as a function of and year. (a) the SuperMAG index. (b) the auroral electrojet index. (c) the northern hemisphere component of the index, . (d) the southern hemisphere component of the index, .
Figure 2 .
2The variations of means of the (black lines) index and its two hemispheric sub-indices (for the northern hemisphere, red lines) and (for the southern hemisphere, blue line). The solid lines are for 1959-2017, the dot-dash lines for 1995-2019. The top panel shows the mean values absolute values of the index (generically termed ) in one-hour bins. The lower panel shows the mean values as a fraction of the overall mean for the interval, < > ⁄
Figure 3 .
3Universal Time variations sorted by the prevailing polarity of the IMF in the GSEQ frame ([ ] GSEQ , averaged over the prior hour). (a) The normalised estimated power input into the magnetosphere, ⁄ , for (green line) [ ] GSEQ > 0, and (mauve line) [ ] GSEQ < 0. (b) The fractional variation of the geomagnetic indices for [ ] GSEQ > 0. (c) The fractional variation of the geomagnetic indices for [ ] GSEQ < 0. Black lines are for the index, the red lines for the index and the blue lines are for .
Figure 4 .Figure 5 .Figure 6 .
456Maps of the geographic locations of (left) northern and (right) southern magnetic poles for various years. The orange and blue points are dip and geocentric dipole pole locations from the 12th generation of the International Geomagnetic Reference Field (IGRF) for 1900 to 2020 in steps of 20 years (fromThébault et al., 2015). The mauve points are the axial poles for the years 1980, 2000 and 2020 from the eccentric dipole model fits (for which the dipole axis is not constrained to pass through the centre of the Earth) ofKoochak and Fraser-Smith (2017) Two series of nearsimultaneous images of the two auroral ovals observed between 11:24 and 12:10 UT on 23 October 2002. The left column shows observations of the northern hemisphere oval made by the FUV-SI13 instrument on the IMAGE satellite and the right column shows the series of near-simultaneous images of the southern hemisphere auroral oval made by the VIS-EC, instrument on the Polar satellite. respectively. Each image is shown in the geomagnetic latitude-magnetic local time (MLT) frame (using AACGM coordinate system) and white dot gives the location of the geographic pole. (fromStubbs et al., 2005) The fitted circular poleward edges of the aurora in the AACGM geomagnetic latitude-MLT frame at three times, half an hour apart (11:20 in green, 11:50 in orange and 12:20 in mauve) on 23 October 2002 (as fitted by
Figure 7 .
7Parts a, b, e, f, i and j are plots of probability distribution functions of ( ⁄ ) as a function of F and beneath each (parts c, d, g, h, k and l) is the corresponding plot of the normalized am amplification factor, [ ⁄ ] = ( /< > )/( ⁄ ),as a function of F and in the same ( ⁄ ) bins as the p.d.f.s. The left hand panels are for IMF [BY]GSEQ < 0, the right hand panels are for IMF [BY]GSEQ > 0. The plots are in 3 groups of 4: parts a-d are for the lower tercile of the simultaneous (allowing for the propagation lag) solar wind dynamic pressure, < (0.33) ; parts e-h are for the middle tercile of the simultaneous solar wind dynamic pressure, (0.33) < (0.67); parts i-l are for the upper tercile of the simultaneous solar wind dynamic pressure, (0.67), where ( ) is the ℎ quantile of the overall distribution of values.
Figure 8 .
8Comparison of the Russell-McPherron and equinoctial patterns, derived using an eccentric dipole geomagnetic field for the year 2002. In the F-UT plot in part c, the colour contours give the absolute value of the dipole tilt angle, | |, superposed on which are contours showing the IMF orientation factor used by , namely = 4 ( 2 ⁄ ), where is the clock angle of the IMF in the GSM frame. These predictions are the average for an equal mix of [ ] GSEQ = −| | < 0 and [ ] GSEQ = +| | > 0 and contours are shown for of 0.28 (in black) and 0.31 (in mauve). Parts a and b show the mean values of | | over the range of F defined by the maximum extent in of the mauve contour as a function of : Part a is for the March equinox and Part b for the September equinox. The area shaded peak is the extent of the peak defined by the mauve contour in c. The point of parts a and b is to show that the two equinoxes go through exactly the same sequence of variations in both and w , but with a phase difference of 12 hours.
Figure 9 .
9Locations of the geomagnetic eccentric dipole axial poles mapped into the XY plane of the GSEQ frame, viewed looking northward from the south of the solar equator (so that the Z GSEQ axis that makes up the right hand set points into the page) : (a) for the northern hemisphere pole, (b) for the southern hemisphere pole. The loci are shown for: (green) the March equinox (day of year, doy, 79, = 0.21; (mauve) the June solstice (doy 172, = 0.47); (orange) the September equinox (doy 266, = 0.73); and (blue) the December solstice (doy 356, = 0.98). The dots show the location at 12
Figure 10 .
10(a) Variations with for 2002 of the sunward velocity of the northern eccentric dipole axial pole, (solid lines) and of the southern eccentric dipole axial pole northern , (dot-dash lines). The variations are shown for the two solstices and the two equinoxes, using the same color scheme as Figure 9. (b) the average of the two, .
Figure 11 .Figure 12 .
1112Schematic illustrating inductive decoupling of solar wind and ionospheric electric field and flows that is a key part of the Expanding-Contracting Polar Cap (ECPC) model of non-steady convection. In (a), the , , and axes of the Geocentric Solar Magnetospheric frame are shown. The points a, b, c, d, e and f are the ionospheric field line footpoints of the points on the magnetopause or cross tail current sheet A, B, C, D, E and F, respectively and all lie on the open-closed field line boundary, bounding the green area showing the open field line polar cap. AB is the dayside magnetopause reconnection X-line (across which the voltage is applied by the magnetic reconnection that opens field lines) and DE is the reconnection X-line in the cross tail current sheet (where the voltage is caused by reconnection that recloses open field lines). FC is the "Stern Gap" in interplanetary space, the ionospheric footprint of which is the polar cap of diameter, fc, and in which the open field lines are frozen-in to the solar wind flow, . (b) A view of the ionospheric polar cap (with noon at the top), with the green area again showing the open field line region, the blue lines showing "adiaroic" (non-reconnecting) segments of the open-closed boundary and red segments being "merging gaps" that map to the reconnection X-lines. (From Lockwood and Morley, 2004) (a) The observed variations of the geomagnetic indices with fraction of year, , shown as means in 36 equal-sized bins in as a fraction of their overall mean: (red)
Figure 13 .
13Plots corresponding to Figure 12d for the four indices: (a) ( ); (b) ( ); (c) ( ); and (). In each case the red/blue dots are for the northern/southern hemisphere component and the black lines are fourth-order polynomial fits.
Figure 14 .
14(a) The observed variations of the conductivity-corrected geomagnetic indices with fraction of year, , shown as means in 36 equal-sized bins in as a fraction of their overall mean. In all panels, red lines are for the northern hemisphere index, blue for the south and black for the average of the two. (a)
Figure 15 .
15of conductivity factors for the hemispheric indices. The top row is for northern hemisphere indices, the bottom row for southern hemisphere indices. The columns from left to right are for: (a) and (f) for the hemispheric and indices, and ; (b) and (g) for the ( ) and ( ) indices, ( ) and ( ) ; (c) and (h) for the ( ) and ( ) indices, ( ) and ( ); (d) and (i) for the ( ) and ( ) indices, ( ) and ( ); and (e) and (j) for the ( ) and ( ) indices, ( ) and ( ).
Figure 16 .
16(a)-(c): F-UT plots of mean normalised power input into the magnetosphere, ⁄ , average into 1-hour bins of and 36 equal-width bins of . The data are 1-minute values averaged into 1-hour intervals using the criteria for handling data gaps that limits the errors they cause to 5%, as defined byLockwood et al. (2019b) and are sorted by the polarity of the component of the IMF over the prior hour: (a) is for [ ] > 0 and (b) is for [ ] < 0. The black contours are the locations of the peaks predicted for the Russell-McPherron effect, being the two contour lines for = 0.28 and = 0.3 plotted in Figure 8c. (d) is the F-UT plot for the () factor (see text) and (e) and (f) are the F-UT plots sunward velocity in the GSEQ frame of the axial geomagnetic poles at an altitude of 800 km in the northern and southern hemisphere, computed for each and hourlyin the same way as inFigure 10.
Figure 17 .
17A comparison of -patterns of (right column) conductivity-corrected observed indices and (left column) the corresponding modelled pattern, the top row is for the northern hemisphere index, In all three cases the free fit parameter used is = 0.27, derived by minimising the r.m.s. fit residual for the index case, {< (
Figure 18 ..
18Observed, conductivity-corrected and modelled variations for the (red) , (blue) , and (black) indices. (a) shows the values, , , and linearly interpolated from three hourly observations to 1-hour resolution and then averaged for the 24 1-hour values for all data (for 1959-2019, inclusive). (b) shows the conductivity-(d) compares the conductivity corrected values (solid lines) and the modelled values (dot-dash lines) on the same plot.
Figure 19 .
19plots of the conductivity-corrected indices. The top row is for the northern hemisphere sub-indices, the middle row for the southern hemisphere sub-indices and the bottom row for the global indices. The columns are for the four 6-hour MLT sectors of the indices and from left to right are for dusk, noon, dawn and midnight.
Figure 20 .
20The same asFigure 19, for modelled values of the (conductivity-corrected) indices. The top row is for the northern hemisphere sub-indices, the middle row for the southern hemisphere sub-indices and the bottom row for the global indices. The columns are for the four 6-hour MLT sectors of the indices and from left to right are for dusk, noon, dawn and midnight.
Figure 21 .
21Comparison of conductivity-corrected and modelled variations for (red) northern hemisphere , (blue) southern hemisphere, and (black) global indices. Each panel compares the conductivity corrected values (solid lines) and the modelled values (dot-dash lines) on the same plot, as in Figure 18d. (a) for dawn, ( ), ( ) and ( ) compared with ( ), ( ) and ( ); (b) for noon, ( ), ( ) and ( ) compared with ( ), ( ) and ( ); (c) for dusk, ( ), ( ) and ( ) compared with ( ), ( ) and ( ); and (d) for midnight,
Figure 22 .
22Simulations of -UT patterns sorted by the polarity of the average IMF [ ] component during the prior hour. The top row shows the Russell-McPherron patterns ( , ) in normalised power input into the magnetosphere simulated using the eccentric dipole geomagnetic field model for 2002 with (a) [ ] = +| |, (b) [ ] = −| | , and (c) an equal mix of [ ] = +| | and [ ] = −| |. The middle panels show the corresponding modelled patterns of ( , ) = ( , ). ( , ). ( ). The observed patterns in ( ) data shown in the bottom panels are for (g) [ ] < 0 , (h) [ ] > 0 and (i) all data.
Figure 23 .
23Universal time variations in averaged over all for: (a) and (d) 1960-1979; (b) and (e) 1980-1999; and (c) and (f) 2000-2019. The top panels show the variation in the 90% quantile of the distribution of values (q(0.9), cyan lines) and the 99.99% quantile(q(0.9999), blue lines). The dashed lines show the variation for the mean . Bottom panels show the variations for mean observed (black lines) and modelled , (mauve lines).
Figure A1 .
A1Distributions of IMF parameters for the full 22-year Hale cycle between 1996 and 2018: (a) the IMF component, [ ] GSEQ , the IMF clock angle in GSEQ , GSEQ = −1 ([ ] GSEQ [ ] GSEQ ⁄ ). In (a) the bins of [ ] GSEQ are 0.2nT wide and the vertical red dash line marks [ ] GSEQ = 0; in (b) the bins of GSEQ are 2 wide and the vertical dashed red line marks GSEQ = 0.
Figure A2 .Figure A3 .
A2A3Variations with time of values over half-year intervals (centred on the equinoxes) for the 1996-2018 interval used in Figure A1. (a). The sunspot number, . (b) The polar field strength from Wilcox Solar Observatory (WSO) magnetograms, , where red/blue is for the north/south solar pole, respectively. (c) The fraction of time f that the IMF has [ ] GSEQ > 0 polarity (mauve line with open circles) and [ ] GSEQ < 0 polarity (green line with solid circles). (d) The number of one-minute samples of [ ] GSEQ in bins 1nT wide and 6-month intervals centred on the equinoxes , to as a function of date and [ ] GSEQ (plotted on a logarithmic colour scale to reveal the tails of the distributions as well as the peaks). The vertical grey lines mark the polarity reversals of the solar polar fields Variations with fraction of a calendar year, , for the two polarities of the IMF in the GSEQ frame. Normalised mean values, < > ⁄ , are shown in bins 1/36 yr wide, with open circles being for [ ] GSEQ > 0 and solid triangles are for [ ] GSEQ < 0. The data are for the interval 1996-2018. < > is the mean in each bin of and is the largest value of < > for a generic parameter . The blue lines are for the index; the mauve lines are for the half-wave rectified southward field, ; the black lines are for the power input to the magnetosphere, ; the orange line is the half-wave rectified southward field predicted for the R-M effect with IMF clock angle [] GSEQ = 90, [ ] RM : [ ] RM = −[ ] RM for [ ] RM < 0 and [ ] RM = for [ ] RM > 0: and where b is the best-fit baselevel value for northward IMF in GSM and [ ] RM = [ ] GSEQ sin ( GSEQ ) is the northward field in GSM obtained by assuming [] GSEQ = 90, GSEQ being the rotation angle between the GSEQ and GSM frames.
Figure A4 .
A4The same as
indices by the prevailing polarity of the IMF Y-component in the GSEQ (Geocentric Solar Equatorial) reference frame ([ ] GSEQ ) revealed that the R-M effect is at work, even though these indices show the equinoctial pattern with time of year F and UT, rather the pattern predicted for the R-M effect. Figures 3b and 3c show the UT variations for the 1995-2019 interval, but with the data sorted according to the polarity of [ ] GSEQ , averaged over the prior hour to be consistent with the optimum lag found by Lockwood et al. (2019b). The green and mauve lines in Figure 3a show the UT variations of normalised power input to the magnetosphere, ⁄ and reveal the average UT variations predicted by the R-M effect (note that the [ ] GSEQ > 0 data are dominated by enhancements at the March equinox and the [ ] GSEQ < 0 data are dominated by enhancements at the September equinox as also predicted by the R-M effect). The normalisation is achieved by dividing by , the average of for the whole interval (1995-2018) which cancels various constants in the expressionfor . It can be see that all three indices ( , , and
) reflect the UT variation predicted
by the R-M effect in ⁄ but there are additional effects, with the
index enhanced
around 12 UT for both IMF [ ] GSEQ polarities and the
index enhanced around 00 UT for
both IMF [ ] GSEQ polarities and the
and
indices are both somewhat lower than
expected at about 3-8
. For both IMF [ ] GSEQ polarities,
and
(and so, by
definition,
) are the same around 06 and 18 UT for both IMF [ ] GSEQ polarities. The
differences between and
and
in
Table 1 .
1Best-fit coefficients used to derive modelled patterns of geomagnetic activityIndex
for the northern
polar cap
for the southern
polar cap
0.78
0.10
0.11
(
)
0.85
0.09
0.09
(
)
0.50
0.29
0.33
(
)
0.80
0.22
0.28
(
)
0.85
0.22
0.25
Acknowledgements and data and software sources. The authors are grateful to the staff of
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| [] |
[
"ON AN ELASTIC STRAIN-LIMITING SPECIAL COSSERAT ROD MODEL",
"ON AN ELASTIC STRAIN-LIMITING SPECIAL COSSERAT ROD MODEL"
] | [
"K R Rajagopal ",
"C Rodriguez "
] | [] | [] | Motivated by recent strain-limiting models for solids and biological fibers, we introduce the first intrinsic set of nonlinear constitutive relations, between the geometrically exact strains and the components of the contact force and contact couple, describing a uniform, hyperelastic, strain-limiting special Cosserat rod. After discussing some attractive features of the constitutive relations (orientation preservation, transverse symmetry, and monotonicity), we exhibit several explicit equilibrium states under either an isolated end thrust or an isolated end couple. In particular, certain equilibrium states exhibit Poynting like effects, and we show that under mild assumptions on the material parameters, the model predicts an explicit tensile shearing bifurcation: a straight rod under a large enough tensile end thrust parallel to its center line can shear.1 Such an approach was introduced by Duhem in[22]and E. and F. Cosserat in[18,19]. | 10.1142/s021820252350001x | [
"https://export.arxiv.org/pdf/2205.10875v2.pdf"
] | 248,986,795 | 2205.10875 | a29e0f198703892c8ba9f095a830a595b1b17daa |
ON AN ELASTIC STRAIN-LIMITING SPECIAL COSSERAT ROD MODEL
K R Rajagopal
C Rodriguez
ON AN ELASTIC STRAIN-LIMITING SPECIAL COSSERAT ROD MODEL
Motivated by recent strain-limiting models for solids and biological fibers, we introduce the first intrinsic set of nonlinear constitutive relations, between the geometrically exact strains and the components of the contact force and contact couple, describing a uniform, hyperelastic, strain-limiting special Cosserat rod. After discussing some attractive features of the constitutive relations (orientation preservation, transverse symmetry, and monotonicity), we exhibit several explicit equilibrium states under either an isolated end thrust or an isolated end couple. In particular, certain equilibrium states exhibit Poynting like effects, and we show that under mild assumptions on the material parameters, the model predicts an explicit tensile shearing bifurcation: a straight rod under a large enough tensile end thrust parallel to its center line can shear.1 Such an approach was introduced by Duhem in[22]and E. and F. Cosserat in[18,19].
Introduction
Elastic rods.
A rod is a slender body that can be modeled by a deformable one-dimensional continuum, appropriately parameterized, wherein we can account for the elongation, shearing, bending, and twisting of the slender body. If the slender body in question is elastic, then we have a model for an elastic rod. A succinct history of the development of a theory for elastic rods from that for a three-dimensional elastic body can be found in pages 663-664 of the treatise [1].
The special Cosserat theory provides an alternative development of a model to describe rods. In the static setting of this alternative approach, the configuration of the deformed slender body is modeled by a one-dimensional curve in space, The six components of u(s) and v(s) = d ds r(s) in the frame {d k (s)} 3 k=1 are the measures of geometric strain in the theory. Two other vector fields along the rod, the contact force n(·) and contact couple m(·), are postulated to model how material segments exert forces and couples on other material segments. The differential 2 K. R. RAJAGOPAL AND C. RODRIGUEZ equations expressing local balance of linear and angular momentum and the specification of constitutive relations between the geometric strains and the components of the rod's contact force and contact couple yield a closed system of equations capable of prediction. See Section 2 for a review of the elements of the special Cosserat theory needed for this work. An up-to-date discussion of the results pertinent to general elastic rod theories can be found in the exhaustive book [4] on nonlinear elasticity. 1.2. Motivation for the model. In the case of the classical theory for rods developed within the context of 3-dimensional bodies, one usually assumes that the body in question belongs to the class of simple materials (see [48]), which within the confines of a purely mechanical theory leads to the stress being completely determined by the history of deformation. Elastic bodies are special cases of such simple materials. In [51] the first author generalized the notion of simple materials to include implicit relations between the history of the stress and the history of the deformation gradient (see also the work [49]). A special subclass of such materials are elastic bodies that are related through an implicit relation between the Cauchy stress T and the deformations gradient F , through
f (T , F ) = 0. (1.1)
We notice that the class of classical Cauchy elastic bodies, satisfying T = g(F ), are a special sub-class of the constitutive relations (1.1). If the body is anisotropic, then (1.1) reduces to (see [55]) f (R * T R, C) = 0, (1.2) where F = RU is the polar decomposition of the deformation gradient and C = F * F is the right Cauchy-Green tensor. If the three-dimensional body has one preferential direction a with respect to its response, then the implicit equation will take the form f (T , F , a) = 0.
A special sub-class of (1.2) are constitutive relations of the form C = h(R * T R), (1.3) and if the body is isotropic, the constitutive relation takes the form B = h(T ), (1.4) where B = F F * is the left Cauchy-Green tensor. Our reason for documenting the constitutive relations given by (1.3) and (1.4) is to motivate the constitutive relations that we propose for our special Cosserat rod in this work. However, we point out that the constitutive relations between the special Cosserat rod's geometric strains and the rod's contact force and couple that we propose are not derived from a fully 3-dimensional relation like (1.3) or (1.4). Instead, we propose intrinsic constitutive relations between the geometric strains and the components of the rod's contact force and couple (see equations (3.1), (3.2), (3.3) and (3.4)).
While it is customary and mathematically desirable to provide expressions for the components of the rod's contact force and contact couple in terms of the strains, we provide expressions for the strains in terms of the components of the rod's contact force and contact couple. There is a sound reason for our choice for specifying the constitutive relation as we do. A central notion in Newtonian mechanics, and this includes continuum mechanics, is the notion of causality. From the standpoint of causality, applied forces and applied moments are the causes and the strains are the effect, and thus it would be appropriate to express the effects in terms of the causes than vice-versa (see [53]) for a discussion of the relevant issues).
The constitutive relations that we propose in this paper are motivated by constitutive relations introduced in [52][53][54] for the response of three-dimensional elastic bodies wherein the left Cauchy-Green tensor depends nonlinearly on the stress. While the constitutive relations considered in [52] is for a 3-dimensional elastic isotropic solid, we appropriately modify it to describe an elastic special Cosserat rod that is transversely hemitropic (as defined in [29]).
The constitutive relations introduced in this work have a very special feature, namely that they are strain-limiting, that is as the applied forces and couples increase, the strains increase monotonically and reach a finite limit asymptotically (see Figure 1). This feature of the constitutive relations, we feel, is a critical feature that would make it an appropriate model to describe many rod-like materials whose final tangent stiffness greatly exceeds the initial tangent stiffness, so much so that such materials are best approximated as materials with limiting stretch (see Section 5 for a discussion). More generally, over the past decade, works on strain-limiting constitutive relations for elastic solid bodies have investigated:
• specific problems of simple shear, torsion, extension etc. [12][13][14]54], • time independent and dependent boundary value problems [6][7][8][9][10]27], • the modeling of gum metal [21,37,54], • the theory of fracture [28,32,33,36,38,62,71], • materials with density dependent moduli and damage [31, 44-47, 50, 56, 67], • viscoelasticity [23][24][25]65], • wave propagation [15,34,41,43,58], • elastic strings [63,64].
However, this paper is the first work studying a set of strain-limiting constitutive relations for special Cosserat rods.
We comment that it is not only causality that influences our choice of the constitutive relations for the strains in terms of the contact force and couple. In a real material, the limit for a measure of strain is (essentially) reached for a finite applied force, and the body seems to behave elastically (to first approximation) for a slightly larger force. But, upon applying a much larger force, the material behaves inelastically and cannot be modeled within a purely elastic framework. In the regime of elastic response, such a material could be described by a constitutive relation wherein the strain is a function of the stress and not the stress as a function of the strain. That is, the strain versus stress relation is not invertible. See Figure 1. Such constitutive relations are not Cauchy elastic and hence not Green elastic. Thus, such bodies must be described either by relations such as (1.3) or even the more general class (1.2), with these relations not being invertible. Recently, the second author studied such stretch-limiting constitutive relations for elastic strings in both static [63] and dynamic [64] settings, and the authors studied certain inelastic behavior for the quasistatic motion of an inextensible, unshearable, viscoelastic rod [57].
1.3. Main results and outline. An outline of the main results and the organization of our study are as follows. In Section 2 we briefly review the elements of the static special Cosserat theory used in this paper. In Section 3, we introduce our intrinsic set of strain-limiting constitutive relations between the geometrically exact strains and the components of the contact force and contact couple describing a uniform, special Cosserat rod capable of bending, twisting, shearing and stretching. See equations (3.1), (3.2), (3.3) and (3.4). We show that these relations arise from a complementary energy depending on the rod's contact force and contact couple. These relations can easily be specialized to enforce unshearability constraints,
r (s) = v 3 (s)d 3 (s), s ∈ [0, L],
or unshearability and inextensibility constraints,
r (s) = d 3 (s), s ∈ [0, L],
but we do not pursue this here. We then show that these relations can be inverted, expressing the components of the rod's contact couple and contact force in terms of the strains, and that these relations can be derived from a stored energy. All but one of the parameters defining these relations readily admit physical interpretations in terms of small-strain material moduli (but we emphasize that our relations are in terms of large exact strains). See Section 3.1. In the remainder of Section 3, we establish the following attractive properties of the constitutive relations, valid for all values of contact force and contact couple:
• orientation preservation 2 (Section 3.2),
• transverse hemitropy (Section 3.3),
• monotonicity of the constitutive relations (Section 3.4). To the authors' knowledge we are unaware of any other explicit shearable, extensible, special Cosserat rod model satisfying all three of these properties for all values of contact force and contact couple. In Section 4, we exhibit several explicit equilibrium states, under a prescribed end thrust, subject to either isolated contact v 3 − 1 Figure 1. Two types of strain-limiting constitutive relationships between the dilatation strain v 3 and tension n 3 . The first relation is qualitatively like those we consider in this work, and it can be expressed as n 3 =n 3 (v 3 − 1) while the latter cannot.
n 3 n 3 v 3 − 1
forces (Section 4.2) or isolated contact couples (Section 4.3). In the former case, the center line of the rod is straight, the end thrust is parallel to the center line, and we find that when a tensile end thrust for the rod exceeds a critical value, we have a bifurcation that can be associated with spontaneous shearing of the rod (see Figure 3). In the case of isolated contact couples, our constitutive relations also predict an interesting feature, namely if the chirality of the rod is opposite the chirality of the applied couple, then the applied couple elongates the rod: a Poynting type effect. 3 Finally, we exhibit an explicit 2-parameter family of equilibrium states with helical center lines subject to isolated contact couples. In general, our discussion in Section 4 is from the semi-inverse standpoint: certain aspects of the equilibrium state will be fixed at the start in order to solve for the kinematic variables and obtain simple, explicit solutions. Finally, in Section 5 we discuss potential applications for the model and future work.
Preliminaries
In this section we briefly review the theory of special Cosserat rods (see Ch. 8 of [4] for a more comprehensive introduction).
2.1. Kinematics and strains. Let E 3 be 3-dimensional Euclidean space, and let {g k } be a fixed right-handed orthonormal basis for the vector space R 3 . Let [0, L] be the reference interval parameterizing the material points of a uniform rod, with reference length L, that is straight in the reference configuration. The (deformed) configuration of a special Cosserat rod is defined by a triple:
[0, L] s → (r(s), d 1 (s), d 2 (s)) ∈ E 3 × R 3 × R 3 ,
with d 1 (s) and d 2 (s) orthonormal for each s. The curve r(·) is the center line of the rod, and {d 1 (s), d 2 (s)} are the directors at s, vectors regarded as tangent to the material cross section transversal to the center line at r(s). Let
d 3 (s) = d 1 (s) × d 2 (s).
Then {d k (s)} is a right-handed orthonormal basis for R 3 for each s, and it describes the configuration of the deformed material cross section at r(s) (see Figure 2). Figure 2. The kinematic variables defining the configuration of a special Cosserat rod.
d 1 (s) r(s) d 2 (s) d 3 (s)
Since, for each s, {d k (s)} is a positively oriented orthonormal basis for R 3 , there exists a unique vector field (the Darboux vector field )
u(s) = u k (s)d k (s) ∈ R 3 , s ∈ [0, L], such that d k (s) = u(s) × d k (s), where = d ds .
In fact, one can solve for u(s) explicitly,
u(s) = 1 2 d k (s) × d k (s).
The components u 1 and u 2 are referred to as the flexural strains, and the component u 3 is referred to as the torsional strain (or twist). We may also express the tangent vector to the center line at s in the basis {d k (s)} via
r (s) = v k (s)d k (s).
The components v 1 and v 2 are referred to as the shear strains. The component v 3 is referred to as the dilatation strain, and an orientation of the director d 3 (s) relative to the center line is fixed by requiring that for all s
∈ [0, L], v 3 (s) > 0. (2.1)
The restriction (2.1) also implies that the stretch of the rod is never zero, |r | > 0, and that the rod cannot be sheared so severely that a section becomes tangent to the center line.
m(b) + (r(b) − o) × n(b) − m(a) − (r(a) − o) × n(a).
If f (s) is an external body force per unit reference length and l(s) is an external body couple per unit reference length, then the classical equilibrium equations expressing balance of linear momentum and angular momentum are given by:
n (s) + f (s) = 0, (2.2) m (s) + r (s) × n(s) + l(s) = 0. (2.3)
At each s, the contact force and contact couple may be expressed in the basis
{d k (s)} via n(s) = n k (s)d k (s), m(s) = m k (s)d k (s).
The components n 1 and n 2 are referred to as the shear forces, and the component n 3 is referred to as the tension (or axial force). The components m 1 and m 2 are referred to as the bending couples (or bending moments), and the component m 3 is referred to as the twisting couple (or twisting moment).
Strain-limited rods
In this section we introduce our special Cosserat rod model and discuss some of its fundamental properties. We are interested in developing a fully nonlinear theory for the response of the rod. Thus, the strains that we consider below are the geometrically exact strains and are not constrained to be small. Also, as mentioned in the introduction, in keeping with causality we prescribe the geometric strains in terms of the rod's contact force and contact couple, and we show that these relations can be derived from a complementary energy that depends on the contact force and contact couple. This is analogous to the development of constitutive relations wherein the strain is expressed in terms of a Gibbs potential in the fully three-dimensional theory of continuum mechanics.
Constitutive relations.
To specify the mechanical properties of the rod and close the equations (2.2) and (2.3), we must posit relations between the components of the strains
u = u 1 u 2 u 3 , v = v 1 v 2 v 3 ,
and the components of the contact couple and force
m = m 1 m 2 m 3 , n = n 1 n 2 n 3 .
The relations we propose are inspired by a class of elastic strain-limiting models introduced by Rajagopal within the context of 3-dimensional solid mechanics [52]:
e = a −p + |bT | p −1/p T ,
where a, b, p > 0, B is the left Cauchy-Green tensor, e = 1 2 (I − B −1 ) is the Almansi-Hamel strain tensor, T is the Cauchy stress tensor and |T | 2 = tr(T 2 ). We emphasize here that the relations that we propose are between the contact force, contact couple and geometrically exact, large strains rather than linearized strains. To describe our special Cosserat rod model, let p, α, β, γ, ζ, η ∈ (0, ∞) and ι ∈ R with
α 2 β 2 − ι 2 > 0.
Then the quadratic forms are positive definite. We posit that the relations between the strains and the components of the contact couple and force are given by
Q(u, v) = α 2 (u 2 1 + u 2 2 ) + β 2 u 2 3 + +ζ 2 (v 2 1 + v 2 2 ) + η 2 (v 3 − 1) 2 + 2ιu 3 (v 3 − 1), Q * (m, n) = 1 α 2 (m 2 1 + m 2 2 ) + 1 ζ 2 (v 2 1 + v 2 2 ) + η 2 β 2 η 2 − ι 2 m 2 3 + β 2 β 2 η 2 − ι 2 n 2 3 − 2ι β 2 η 2 − ι 2 m 3 n 3 .u µ = γ p + Q * (m, n) p/2 −1/p 1 α 2 m µ , (3.1) u 3 = γ p + Q * (m, n) p/2 −1/p 1 β 2 η 2 − ι 2 (η 2 m 3 − ιn 3 ), (3.2) v µ = γ p + Q * (m, n) p/2 −1/p 1 ζ 2 n µ , (3.3) v 3 − 1 = γ p + Q * (m, n) p/2 −1/p 1 β 2 η 2 − ι 2 (−ιm 3 + β 2 n 3 ), (3.4)
where Greek letters range over {1, 2}. We note that the model is strain-limiting in the sense that u, v ∈ R 3 given by (3.1)
(u 2 1 + u 2 2 ) 1/2 < 1 α , (3.5) |u 3 | < η(β 2 η 2 − ι 2 ) −1/2 , (v 2 1 + v 2 2 ) 1/2 < 1 ζ , |v 3 − 1| < β(β 2 η 2 − ι 2 ) −1/2 . u 1 u 2 u 3 = ∂W * ∂m (m, n), v 1 v 2 v 3 − 1 = ∂W * ∂n (m, n), where W * (m, n) = 1 2ˆQ * (m,n) 0 (γ p + t p/2 ) −1/p dt. (3.7)
We observe that the relations (3.1), (3.2), (3.3) and (3.4) can be inverted to obtain the contact couple and contact force as functions of the strains,
m µ = ∂W ∂u µ = γ 1 − Q(u, v) p/2 −1/p α 2 u µ , (3.8) m 3 = ∂W ∂u 3 = γ 1 − Q(u, v) p/2 −1/p (β 2 u 3 + ι(v 3 − 1)), (3.9) n µ = ∂W ∂v µ = γ 1 − Q(u, v) p/2 −1/p ζ 2 v µ , (3.10) n 3 = ∂W ∂v 3 = γ 1 − Q(u, v) p/2 −1/p (ιu 3 + η 2 (v 3 − 1)), (3.11) for u, v ∈ R 3 satisfying Q(u, v) < 1.
The fact that the constitutive relations can be inverted is a consequence of the type of strain-limiting behavior (3.1), (3.2), (3.3) and (3.4) exhibit: the strains increase to their limiting values asymptotically (see Section 3.4 for monotonicity). If, instead, the strain-limiting behavior was of the type wherein the strains reached a finite limiting value for all sufficiently large finite values of |n| and |m|, then the relations between (u, v) and (m, n) would not be invertible (see Figure 1). We now observe that our model is hyperelastic: the relations (3.8), (3.9), (3.10) and (3.11) are derivable from a stored energy W (u, v),
m 1 m 2 m 3 = ∂W ∂u (u, v), n 1 n 2 n 3 = ∂W ∂v (u, v) where W (u, v) = γ 2ˆQ (u,v) 0 (1 − t p/2 ) −1/p dt. (3.12)
The material constants α, β and ι scale like length, γ scales like force, and ζ and η are dimensionless. In particular, our model is parameterized by seven dimensionless parameters: γ/F , α/L, β/L, ι/L, ζ, η and p, where F is the force scale. All but the constant p admit the following physical interpretation. We note that the stressstrain relations (3.8), (3.9), (3.10) and (3.11) are, to leading order in u and v − [0 0 1] * , given by
m µ = γα 2 u µ , m 3 = γβ 2 u 3 + γι(v 3 − 1), n µ = γζ 2 n µ , n 3 = γιu 3 + γη 2 (v 3 − 1).
Thus, the four constants γα 2 , γβ 2 , γζ 2 and γη 2 can be interpreted as the rod's smallstrain bending, twisting, shearing and dilatational material moduli. The constant γι couples twisting to extension and specifies the chirality of the rod.
Orientation preservation.
We recall that a condition on the strain v 3 that fixes an orientation and guarantees that the stretch of the rod is never zero, |r | > 0, is that for all s ∈ [0, L], v 3 (s) > 0. (3.13) If the material constants characterizing our rod model satisfy
β(β 2 η 2 − ι 2 ) −1/2 < 1 ⇐⇒ 1 + ι 2 β 2 < η 2 ,
then by (3.6) an arbitrary configuration of the rod satisfies (3.13).
Although mathematically 1-dimensional manifolds, special Cosserat rods are used to model slender 3-dimensional bodies. The condition (2.1) is a mild condition ensuring orientation preservation of the 1-dimensional object. However, as discussed by Antman in Section 4 of [3], consideration of the constrained deformation of the associated slender 3-dimensional body suggests requiring a much stronger condition for orientation preservation, described as follows.
For simplicity of the ensuing discussion, suppose that for each s, the material cross section A(s) located at the material point s of the slender body being modeled is circular
A(s) = (x 1 , x 2 ) | (x 1 ) 2 + (x 2 ) 2 ≤ a 2 .
We denote the reference configuration of the slender 3-dimensional body by
B = {(x 1 , x 2 , s) | s ∈ [0, L], (x 1 , x 2 ) ∈ A(s)}.
Then the constrained deformation of the slender body χ : B → E 3 determined by an arbitrary configuration r, d 1 , d 2 for a special Cosserat rod via
χ(x 1 , x 2 , s) = r(s) + x 1 d 1 (s) + x 2 d 2 (s),
preserves orientation in the sense that
det ∇χ > 0 on B, if and only if v 3 (s) > a((u 1 (s)) 2 + (u 2 (s)) 2 ) 1/2 . (3.14)
We now show that under a mild constraint on the cross sectional radii and the parameters, a slender body modeled by our special Cosserat rod model satisfies (3.14). Indeed, if
a < α(1 − β(β 2 η 2 − ι 2 ) −1/2 ),
then by (3.5) and (3.6) the strains associated to an arbitrary configuration of our strain-limiting rod satisfy
a((u 1 (s)) 2 + (u 2 (s)) 2 ) 1/2 ≤ a α < 1 − β(β 2 η 2 − ι 2 ) −1/2 < v 3 (s).
3.3. Transverse symmetry. We now discuss the transverse symmetry of our model. Let
E = 1 0 0 0 −1 0 0 0 1 , R ψ = cos ψ sin ψ 0 − sin ψ cos ψ 0 0 0 1 , ψ ∈ [0, 2π), A hyperelastic rod with stored energy density Φ is transversely hemitropic if for all θ ∈ [0, 2π), u, v ∈ R 3 , Φ(R ψ u, R ψ v) = Φ(u, v).
The rod is flip-symmetric if it is transversely hemitropic and for all u, v ∈ R 3 , Φ(Eu, Ev) = Φ(u, v), and the rod is isotropic if it is transversely hemitropic and for all u, v ∈ R 3 , Φ(Eu, −Ev) = Φ(u, v). Thus, our model with strain energy (3.12) is hemitropic and flip-symmetric. It is isotropic if and only if the twist-stretch coupling constant ι = 0. We refer the reader to [29] for more on the notion of transverse symmetry for special Cosserat rods.
3.4. Monotonicity. By using L and γ as our length and force scales, respectively, and appropriately nondimensionalizing the variables allows us to set
L = 1, γ = 1,
for the remainder of this paper. We now prove a mathematically attractive monotonicity property of our model. The following proposition implies that an increase in the bending couple m µ accompanies an increase in the flexure u µ , an increase in twisting couple m 3 accompanies an increase in the twist u 3 , an increase in shear force n µ accompanies an increase in the shear strain v µ , and an increase in tension n 3 accompanies an increase in the dilatation strain v 3 . Proposition 3.1. If u, v ∈ R 3 and Q(u, v) < 1, then the Hessian of the stored energy density,
D 2 W (u, v) = ∂m ∂u (u, v) ∂m ∂v (u, v) ∂n ∂u (u, v) ∂n ∂v (u, v)
, is positive definite.
Proof. To establish the proposition, we compute
∂m µ ∂u ν = 1 − Q(u, v) p/2 −1/p−1 × 1 − Q(u, v) p/2 α 2 δ µν + Q(u, v) p/2−1 (α 2 u µ )(α 2 u ν ) , ∂m µ ∂u 3 = ∂m 3 ∂u µ = 1 − Q(u, v) p/2 −1/p−1 Q(u, v) p/2−1 α 2 u µ (β 2 u 3 + ι(v 3 − 1)),
as well as
∂m µ ∂v ν = ∂n ν ∂u µ = 1 − Q(u, v) p/2 −1/p−1 Q(u, v) p/2−1 (α 2 u µ )(ζ 2 v ν ), ∂m µ ∂v 3 = ∂n 3 ∂u µ = 1 − Q(u, v) p/2 −1/p−1 Q(u, v) p/2−1 α 2 u µ (ιu 3 + η 2 (v 3 − 1)), and ∂m 3 ∂u 3 = 1 − Q(u, v) p/2 −1/p−1 × 1 − Q(u, v) p/2 β 2 Q(u, v) p/2−1 (β 2 u 3 + ι(v 3 − 1)) 2 ,
and
∂m 3 ∂v ν = ∂n ν ∂u 3 = 1 − Q(u, v) p/2 −1/p−1 Q(u, v) p/2−1 (β 2 u 3 + ι(v 3 − 1))ζ 2 v ν , ∂m 3 ∂v 3 = ∂n 3 ∂u 3 = 1 − Q(u, v) p/2 −1/p−1 × 1 − Q(u, v) p/2 ι + Q(u, v 3 − 1) p/2−1 (β 2 u 3 + ι(v 3 − 1))(ιu 3 + η 2 (v 3 − 1)) ,
and finally,
∂n µ ∂v ν = 1 − Q(u, v) p/2 −1/p−1 × 1 − Q(u, v) p/2 ζ 2 δ µν + Q(u, v) p/2−1 (ζ 2 v µ )(ζ 2 v ν ) , ∂n µ ∂v 3 = ∂n 3 ∂v µ = 1 − Q(u, v) p/2 −1/p−1 Q(u, v) p/2−1 ζ 2 v µ (ιu 3 + η 2 (v 3 − 1)), ∂n 3 ∂v 3 = 1 − Q(u, v) p/2 −1/p−1 × 1 − Q(u, v) p/2 η 2 + Q(u, v) p/2−1 (ιu 3 + η 2 (v 3 − 1)) 2 .
Then for all a, b ∈ R 3 , we have
1 − Q(u, v) 1/p+1 a b · D 2 W (u, v) a b = 1 − Q(u, v) p/2 )Q(a, b) + Q(u, v) p/2−1 α 2 a µ u µ + ζ 2 b µ v µ + a 3 (β 2 u 3 + ι(v 3 − 1)) + b 3 (ιu 3 + η 2 (v 3 − 1)) 2 .
Since Q(·, ·) is positive definite, the proof is concluded.
Some explicit equilibrium states
In this section we discuss some solutions to (2.2) and (2.3) with no external body force or body couple and with an end thrust n(1) prescribed. We assume that the rod is oriented so that n(1) = N g 3 , where N ∈ R. Then by (2.2), for all s ∈ [0, 1] n(s) = N g 3 .
Our discussion is from the semi-inverse standpoint rather than considering a fixed boundary value problem. Some aspects of the equilibrium state will be fixed at the start in order to solve for the kinematic variables and obtain simple, explicit solutions. In particular, we consider equilibrium states which are subject to either isolated contact forces or isolated contact couples, and in the latter case we consider only helical states. 4 4.1. Euler angles and reduced equations. As in previous works (see for example [40], [69], [2]) we compute explicit solutions by expressing the directors and equilibrium equations using Euler angles, defined as follows.
Using spherical coordinates (ϕ, θ) for the sphere S 2 , we can write d 3 = sin θ(cos ϕg 1 + sin ϕg 2 ) + cos θg 3 . 4 In this work, a helical state is an equilibrium state such that the director d 3 (s) has a constant polar angle in a fixed system of spherical coordinates. The general study of helical states was initiated by Kirchhoff [35] for uniform, inextensible, unshearable rods with linear relations between the strains and components of the contact couple. Antman [2] and Chousiab-Maddocks [17] established some qualitative properties of helical states for fairly general constitutive relations describing a uniform special Cosserat rod.
We then define a new right-handed orthonormal basis {e k } via e 3 = d 3 , e 2 = − sin ϕg 1 + cos ϕg 2 , e 1 = e 2 × e 3 = cos θ(cos ϕg 1 + sin ϕg 2 ) − sin θg 3 .
Since d 1 , d 2 are orthogonal to d 3 = e 3 , there exists ψ ∈ R such that d 1 = cos ψe 1 + sin ψe 2 ,
d 2 = − sin ψe 1 + cos ψe 2 .
The angles (ϕ, θ, ψ) are referred to as the Euler angles parameterizing the directors {d k }. We can then express n = N g 3 as n = −N sin θ cos ψd 1 + N sin θ sin ψd 2 + N cos θd 3 . Using the expressions for the directors in terms of the Euler angles, one readily verifies the following relationships between the strains and Euler angles:
u 1 = θ sin ψ − ϕ sin θ cos ψ, (4.7)
u 2 = θ s cos ψ + ϕ sin θ sin ψ, (4.8)
u 3 = ψ + ϕ cos θ. (4.9)
If we denote
u = 1 + Q * (m, n) p/2 −1/p 1 α 2 , v = 1 + Q * (m, n) p/2 −1/p 1 ζ 2 , then u µ = u · m µ , v µ = v · n µ , µ = 1,θ = 0, ψ + cos θϕ = u 3 N sin θ(v 3 − vN cos θ) = 0. (4.14)
Thus, θ is a constant and is determined by N via (4.14).
Proposition 4.1. The center line of an equilibrium state satisfying (4.14) is parallel to g 3 .
Proof. We compute the following inner products using r = v k d k along with v µ = v · n µ and (4.1):
r · g 1 = sin θ cos ϕ(v 3 − vN cos θ) = 0, r · g 2 = sin θ sin ϕ(v 3 − vN cos θ) = 0,
Thus, r is parallel to g 3 .
We consider two cases for (4.14):
sin θ = 0 or v 3 = vN cos θ.
For the first case, we consider only the sub-case θ = 0 (the sub-case θ = π can be analyzed similarly) from which it follows that d 3 = g 3 and n = N d 3 = N g 3 . We immediately conclude the following.
u 3 = 1 + β p (β 2 η 2 − ι 2 ) p/2 |N | p −1/p −ιN β 2 η 2 − ι 2 , (4.15) v 1 = v 2 = 0, v 3 − 1 = 1 + β p (β 2 η 2 − ι 2 ) p/2 |N | p −1/p β 2 N β 2 η 2 − ι 2 .
The center line of the rod is parallel to g 3 , r (s) = (v 2 1 + v 2 2 + v 2 3 ) 1/2 g 3 , and the directors are given by d 1 (s) = cos(u 3 s + ψ(0))g 1 + sin(u 3 s + ψ(0))g 2 , d 2 (s) = − sin(u 3 s + ψ(0))g 1 + cos(u 3 s + ψ(0))g 2 ,
d 3 (s) = g 3 .
We emphasize that if θ = 0, then the configuration is unsheared, v 1 = v 2 = 0, and if ι = 0, then the rod is twisted, u 3 = 0, under an isolated end thrust. In accordance with common experiences with ropes, threads, or dishtowels unwinding when stretched, and because of (4.15), we interpret ι > 0 as modeling right-handed chirality and ι < 0 as modeling left-handed chirality. As N → ±∞, we obtain the nonzero limiting strains:
lim N →±∞ u 3 = ∓ ι β (β 2 η 2 − ι 2 ) −1/2 , lim N →±∞ (v 3 − 1) = ±β(β 2 η 2 − ι 2 ) −1/2 . (4.16)
We now consider the case v 3 = vN cos θ, θ ∈ (0, π). Since v and v 3 are positive we must have n 3 = N cos θ > 0. Then the rod must be in a tensile state, v 3 − 1 > 0. We will focus on the sub-case θ ∈ (0, π/2) (the other sub-case θ ∈ (π/2, π) can be analyzed similarly). This is equivalent to assuming that N > 0.
We have that (4.17) is equivalent to
v 3 = 1 + 1 ζ 2 N 2 sin 2 θ + β 2 β 2 η 2 − ι 2 N 2 cos 2 θ p/2 −1/p n 3 ζ 2 = β 2 η 2 − ι 2 β 2 ζ 2 (v 3 − 1),
which is equivalent to
v 3 − 1 = β 2 η 2 − ι 2 β 2 ζ 2 − 1 −1 . (4.18)
Thus, a necessary condition for an equilibrium state satisfying (4.17) to exist is that the material moduli satisfy
β 2 η 2 − ι 2 β 2 ζ 2 − 1 > 0 ⇐⇒ 1 ζ 2 − β 2 β 2 η 2 − ι 2 > 0 ⇐⇒ η 2 > ζ 2 + ι 2 β 2 . (4.19)
By (4.16) we also expect that a necessary condition for (4.18) being satisfied is that the limiting value for v 3 − 1 is strictly bigger than the right hand side of (4.18):
β 2 η 2 − ι 2 β 2 ζ 2 − 1 −1 < β (β 2 η 2 − ι 2 ) 1/2 (4.20) ⇐⇒ β 2 η 2 − ι 2 β 2 ζ 2 − 1 p β 2p (β 2 η 2 − ι 2 ) p > β p (β 2 η 2 − ι 2 ) p/2 .
We now prove that (4.19) and (4.20) are sufficient to solve (4.18) uniquely for θ =θ(N ) ∈ (0, π/2), for all N > N thresh with N thresh defined in Proposition 4.3. In particular, it immediately follows that
{(N,θ(N )) | N > N thresh }
is a nontrivial branch of the set of solutions to (4.14) bifurcating from the trivial branch {(N, 0) | N ≥ 0} of tensile solutions to (4.14) at the bifurcation point (N thresh , 0). This nontrivial branch can be interpreted as a shearing instability of the rod for large tensile end thrusts. 19) and (4.20), and let N thresh > 0 be defined via
N −p thresh = β 2 η 2 − ι 2 β 2 ζ 2 − 1 p β 2p (β 2 η 2 − ι 2 ) p − β p (β 2 η 2 − ι 2 ) p/2 . (4.21)
If N > N thresh then there exists a uniqueθ(N ) ∈ (0, π/2) such that
v 3 − 1 = 1 + 1 ζ 2 N 2 sin 2 θ + β 2 β 2 η 2 − ι 2 N 2 cos 2 θ p/2 −1/p × β 2 β 2 η 2 − ι 2 N cos θ (4.22) with θ =θ(N ) satisfies (4.18). Moreover, lim N →∞θ (N ) = cos −1 1 ζ 2 − β 2 β 2 η 2 − ι 2 2 + 1 ζ 2 − β 2 β 2 η 2 − ι 2 −1/2 1 ζ . (4.23)
Proof. If p = 2, one can solve forθ(N ) ∈ (0, π/2) explicitly:
θ(N ) = cos −1 1 ζ 2 − β 2 β 2 η 2 − ι 2 2 + 1 ζ 2 − β 2 β 2 η 2 − ι 2 −1/2 1 N 2 + 1 ζ 2 1/2
In general, we write (4.18) as
f N (cos θ) = β 2 η 2 − ι 2 β 2 ζ 2 − 1 −1 where f N (x) = 1 + N 2 ζ 2 (1 − x 2 ) + β 2 β 2 η 2 − ι 2 N 2 x 2 p/2 −1/p N β 2 β 2 η 2 − ι 2 x = N −p + 1 ζ 2 (1 − x 2 ) + β 2 β 2 η 2 − ι 2 x 2 p/2 −1/p β 2 β 2 η 2 − ι 2 x,
for x ∈ [0, 1]. By (4.19) we conclude that
f N (x) = N −p + 1 ζ 2 (1 − x 2 ) + β 2 β 2 η 2 − ι 2 x 2 p/2 −1/p−1 β 2 β 2 η 2 − ι 2 × 1 + 1 ζ 2 (1 − x 2 ) + β 2 β 2 η 2 − ι 2 x 2 p/2−1 1 ζ 2 − β 2 β 2 η 2 − ι 2 x 2 > 0. Thus, f N : [0, 1] → [0, f N (1)] is an increasing bijection. We have f N (1) = N −p + β p (β 2 η 2 − ι 2 ) p/2 −1/p β 2 β 2 η 2 − ι 2 → β(β 2 η 2 − ι 2 ) −1/2
as N → ∞. We conclude that for all N > N thresh where N thresh satisfies (4.21), we have
0 < β 2 η 2 − ι 2 β 2 ζ 2 − 1 −1 < N −p + β p (β 2 η 2 − ι 2 ) p/2 −1/p β 2 β 2 η 2 − ι 2 = f N (1).
Thus, there exists a unique x N ∈ (0, 1) such that
f N (x N ) = β 2 η 2 − ι 2 β 2 ζ 2 − 1 −1
Thenθ(N ) = cos −1 x N ∈ (0, π/2) is the desired angle.
We define
θ(∞) = cos −1 1 ζ 2 − β 2 β 2 η 2 − ι 2 2 + 1 ζ 2 − β 2 β 2 η 2 − ι 2 −1/2 1 ζ .
Let {N k } be a sequence with N k → ∞ as k → ∞, and let
x N k = cos θ(N k ). Since x N k ∈ [0, 1] for all k, there exists a subsequence of {N k } denoted by {N j } and x ∈ [0, 1] such that x Nj → x as j → ∞. Then 1 ζ 2 (1 − x 2 ) + β 2 β 2 η 2 − ι 2 x 2 −1/2 β β 2 η 2 − ι 2 x = lim j→∞ f Nj (x Nj ) = β 2 η 2 − ι 2 β 2 ζ 2 − 1 −1 .
Solving for x, we conclude that x = cosθ(∞) and cos θ(N j ) = x Nj → cosθ(∞). Thus,
lim j→∞ θ(N j ) =θ(∞). Since every sequence {N k } with N k → ∞ has a subsequence {N j } with θ(N j ) → θ(∞), we conclude (4.23).
For this branch of nontrivial solutions, sin θ = 0 and (4.13) imply that ϕ = 0. Moreover, via (4.22) and (4.18) we have the identity
1 + 1 ζ 2 N 2 sin 2 θ + β 2 β 2 η 2 − ι 2 N 2 cos 2 θ p/2 −1/p = β 2 η 2 − ι 2 β 2 1 N cos θ β 2 η 2 − ι 2 β 2 ζ 2 − 1 −1 . (4.24)
Using (4.24) we now summarize the properties of the branch of sheared tensile states. See Figure 3 for a description of the two tensile branches' configurations compared to the rod's unstressed configuration.
Proposition 4.4. Assume, after a proper rotation of the plane spanned by {g 1 , g 2 } if necessary, that ϕ(0) = 0 (so then ϕ(s) = 0 for all s). Let N > N thresh and θ =θ(N ) ∈ (0, π/2) be as in Proposition 4.3. Then the strains of the associated equilibrium state are given by Figure 3. The Carolina blue vector represents r (1) and the navy vector represents d 3 (1) for each configuration. The top configuration represents the unstressed state of the rod. The second and third configurations from the top qualitatively describe the two branches of equilibrium states subject to a large enough isolated, tensile end thrust parallel to r (1).
u 1 = u 2 = 0, u 3 = β 2 η 2 − ι 2 β 2 ζ 2 − 1 −1 −ι β 2 , v 1 = − β 2 η 2 − ι 2 β 2 ζ 2 − 1 −1 β 2 η 2 − ι 2 β 2 ζ 2 tan θ cos(u 3 s + ψ(0)), v 2 = β 2 η 2 − ι 2 β 2 ζ 2 − 1 −1 β 2 η 2 − ι 2 β 2 ζ 2 tan θ sin(u 3 s + ψ(0)), v 3 − 1 = β 2 η 2 − ι 2 β 2 ζ 2 − 1 −1 . 18 K. R. RAJAGOPAL AND C. RODRIGUEZ d 3 (1) = g 3 = r (1) d 3 (1) = g 3 d 3 (1) = sin θg 1 + cos θg 3 r (1) r (1) θ
The center line of the rod is parallel to g 3 , and the directors are given by d 1 (s) = cos θ cos(u 3 s + ψ(0))g 1 + sin(u 3 s + ψ(0))g 2 − sin θ cos(u 3 s + ψ(0))g 3 , d 2 (s) = − cos θ sin(u 3 s + ψ(0))g 1 + cos(u 3 s + ψ(0))g 2 + sin θ sin(u 3 s + ψ(0))g 3 , d 3 (s) = sin θg 1 + cos θg 3 .
4.3.
Rod subject to isolated contact couples. We conclude this section by considering certain equilibrium states under isolated contact couples, N = 0. In particular, we will consider a special class of such equilibrium states, those satisfying: for all s ∈ [0, 1]
M 2 = 0 ⇐⇒ θ = 0.
As in the previous subsection, for simplicity we will assume that
θ ∈ [0, π/2].
The case θ ∈ (π/2, π] can be analyzed similarly. If θ = M 2 = 0, then by (4.11) and (4.12), we conclude that ϕ and ψ satisfy sin θϕ = −uM 1 , If M 2 = 0 and M 1 = 0 then by (4.2) and (4.3) we conclude that m µ = 0 for µ = 1, 2. By (4.25) we conclude that either: θ = 0, or θ = 0 and ϕ = 0. We then have the following.
u 3 = 1 + η p (β 2 η 2 − ι 2 ) p/2 |M 3 | p −1/p η 2 M 3 β 2 η 2 − ι 2 , v 1 = v 2 = 0, v 3 − 1 = 1 + η p (β 2 η 2 − ι 2 ) p/2 |M 3 | p −1/p −ιM 3 β 2 η 2 − ι 2 .
The director d 3 is constant, the center line of the rod is parallel to d 3 , and the directors are given by d 1 (s) = cos θ cos(u 3 s + ψ(0))g 1 + sin(u 3 s + ψ(0))g 2 − sin θ cos(u 3 s + ψ(0))g 3 , d 2 (s) = − cos θ sin(u 3 s + ψ(0))g 1 + cos(u 3 s + ψ(0))g 2 + sin θ sin(u 3 s + ψ(0))g 3 , d 3 (s) = sin θg 1 + cos θg 3 .
We comment that the length of the rod under the isolated couple m(s) = M 3 d 3 is given bŷ
1 0 |r (s)|ds − 1 = v 3 − 1 = 1 + η p (β 2 η 2 − ι 2 ) p/2 |M 3 | p −1/p −ιM 3 β 2 η 2 − ι 2 .
Thus, if ι = 0, then the rod changes length due to a isolated couple. In particular, if the chirality of the rod is opposite to the chirality of the couple, −ιM 3 > 0, then we observe a Poynting effect: the application of an isolated couple elongates the rod. As M 3 → ±∞ we obtain the nonzero limiting strains:
lim M3→±∞ u 3 = ±η(β 2 η 2 − ι 2 ) −1/2 , lim M3→±∞ (v 3 − 1) = ∓ ι η (β 2 η 2 − ι 2 ) −1/2 .ϕ(s) = − 1 + |M 1 | p 1 α 2 + η 2 β 2 η 2 − ι 2 cot 2 θ p/2 −1/p M 1 (csc θ)s, ψ(s) = − 1 − β 2 η 2 − ι 2 α 2 η 2 1 + |M 1 | p 1 α 2 + η 2 β 2 η 2 − ι 2 cot 2 θ p/2 −1/p × η 2 β 2 η 2 − ι 2 M 1 (cot θ)s + ψ(0),
Then the strains of the associated equilibrium state are given by
u 1 = 1 + |M 1 | p 1 α 2 + η 2 β 2 η 2 − ι 2 cot 2 θ p/2 −1/p 1 α 2 M 1 cos ψ(s), u 2 = − 1 + |M 1 | p 1 α 2 + η 2 β 2 η 2 − ι 2 cot 2 θ p/2 −1/p 1 α 2 M 1 sin ψ(s), u 3 = − 1 + |M 1 | p 1 α 2 + η 2 β 2 η 2 − ι 2 cot 2 θ p/2 −1/p η 2 β 2 η 2 − ι 2 M 1 cot θ, v 1 = v 2 = 0, v 3 − 1 = 1 + |M 1 | p 1 α 2 + η 2 β 2 η 2 − ι 2 cot 2 θ p/2 −1/p ι β 2 η 2 − ι 2 M 1 cot θ,
The center line is tangent to the director d 3 ,
r (s) = v 3 d 3 (s) = v 3 sin θ cos ϕ(s)g 1 + v 3 sin θ cos ϕ(s)g 2 + v 3 cos θg 3 , (4.26)
As discussed by Antman within a more general setting [2], we note that (4.26) implies that the center line r(s) is a right-handed helix of radius a = v3 sin θ ϕ and pitch b = v 3 cos θ.
In the case θ = π/2, M 3 = 0, r is a circle in the plane spanned by {g 1 , g 2 },
r(s) = 1 + |M 1 | p α −p 1/p α 2 M −1 1 sin 1 + |M 1 | p α −p −1/p α −2 M 1 s g 1 + 1 + |M 1 | p α −p 1/p α 2 M −1 1 cos 1 + |M 1 | p α −p −1/p α −2 M 1 s g 2 ,
and the rod is in a state of pure bending,
u 1 = 1 + |M 1 | p α −p −1/p 1 α 2 M 1 cos ψ(0),u 2 = − 1 + |M 1 | p α −p −1/p 1 α 2 M 1 sin ψ(0), u 3 = 0, v 1 = v 2 = v 3 − 1 = 0.
As M 1 → ±∞ we obtain the limiting strains and curvature
lim M1→±∞ u 1 = ± 1 α cos ψ(0), lim M1→±∞ u 2 = ∓ 1 α sin ψ(0), lim M1→±∞ (u 2 1 + u 2 2 ) 1/2 = 1 α .
Conclusion
In this work we introduced an intrinsic set of strain-limiting constitutive relations, between the geometrically exact strains and components of the contact couple and force, for special Cosserat rods. We then showed these relations are derivable from a complementary energy, are orientation preserving and satisfy a mathematically attractive monotonicity property. Finally, we computed some explicit equilibrium states displaying rich behavior including Poynting effects and tensile shearing bifurcations. We now discuss potential mathematical and modeling perspectives for future work. 5.1. Mathematical perspectives. In Section 4 we computed equilibrium states from the semi-inverse standpoint and not by considering a fixed boundary value problem. Upon fixing boundary conditions at the two ends, the following problems are then suggested:
• multiplicity of equilibrium states, • stability of these equilibrium states.
Needless to say, both of these problems are highly dependent on the choice of boundary conditions. Even in the case of a straight center line, the equilibrium state corresponding to the shearing bifurcation obtained in Proposition 4.3 is not present if one imposes a fixed position and orientation at the end s = 0,
r(0) = o, d k (0) = g k , k = 1, 2, 3, m(1) = 0, n(1) = N g 3 .
However, if instead one imposes that the resultant contact couple about the origin o is zero,
r(0) = o, m(1) + (r(1) − o) × n(1) − m(0) = 0, n(1) = N g 3 ,
then both branches of equilibrium states are present.
In our opinion, stability of these equilibrium states is a dynamic question and cannot even be formulated unless the time-dependent field equations and nature of dissipation are specified. For example, in the isothermal setting and interpreting the directors {d 1 , d 2 } as specifying the material cross sections' principal axes of inertia, the dynamic field equations are given by (ρA)∂ 2 t r = ∂ s n, ∂ t (ρJ )w = ∂ s m + ∂ s r × n, where (ρA)(s) is the mass density per unit reference length and (ρJ )(s, t)(w(s, t)) is the angular momentum of the of the cross section relative to r (calculable from given quantities). See Chapter 8 of [4].
However, there are infinitely many choices of dissipative mechanisms, each corresponding to a specification of a nonnegative total dissipation rate. Two distinct natural choices which reduce to the constitutive relations introduced in this paper in the static setting are
u + α∂ t u = ∂W * ∂m (m, n), v + α∂ t v = ∂W * ∂n (m, n), (5.1) and m = ∂W ∂u (u, v) + µ∂ t u, n = ∂W ∂v (u, v) + ν∂ t v, (5.2)
where α, β, µ, ν ≥ 0, W * is given by (3.7) and W is given by (3.12). The relations in (5.1) are strain-rate viscoelastic constitutive relations analogous to the smallstrain strain-rate constitutive relations studied in [6,7,58]. The relations (5.2) are standard Kelvin-Voight type constitutive relations. In the authors' opinion, developing the stability theory of the equilibrium states discussed in Section 4, for a fixed boundary value problem and either choice of dissipation (5.1) or (5.2), is an interesting and worthwhile endeavor. 5.2. Modeling perspectives. As mentioned in the introduction, the strain limiting constitutive relations introduced in this work would be ideal for modeling many rod-like materials whose final tangent stiffness greatly exceeds the initial tangent stiffness. Such materials include collagen, elastin, silk, protein, DNA, RNA and many others. 5 De Gennes [20] in describing such materials states "the elongation tends to saturate: The restoring force F which tries to make a compact chain becomes infinite if it gets completely extended". That is, to get a compact chain to its full finite length, one needs infinite force, which we interpret in our idealization as limiting extensibility. As Freed and Rajagopal [26] observe, the idea that biological fibers can be strain-limiting was first propounded by Carton et al. [16]. Later, Hunter [30] and Maksym and Bates [42] also advocated the same notion.
In particular, since DNA molecules are constantly bending, twisting and stretching inside cells during multiple biological processes, it is important to develop a simple model capturing their essential mechanical response. The small-strain special Cosserat rod model for double-stranded DNA (dsDNA) appearing in [39] suggests that for isolated tensile forces up to approximately 50 pN, our model with accurately describes the response of dsDNA (the rod is taken to be unshearable, so v 1 = v 2 = 0 always). However, as discovered by experiments on individual dsDNA molecules using optical tweezers [66], for isolated tensile forces above approximately 65 pN, a dsDNA molecule undergoes a force-dependent, rate-dependent overstretching transition wherein the molecule elongates to approximately 1.8 times its contour length and its response asymptotically approaches that of a single-stranded DNA molecule (ssDNA). After much debate over several years, its been revealed via experiments using both optical tweezers and fluorescent microscopy that there are three molecular mechanisms involved in the overstretching transition:
• the peeling of one strand away from the other, • base-pair bonds melting, • base-pair bonds remaining intact and cooperative strand unwinding, converting parts of the molecule into ladder like structures (S-form DNA).
We refer the reader to the reviews [11,70] for more on the literature, experimental techniques and results leading to these conclusions. The precise superposition of strand splitting, bond melting and conversion into S-form DNA can be quite complex and difficult to track at the molecular level during the overstretching transition. However, using the special Cosserat rod model introduced in this work, the overstretching transition can be modeled by a rod with material constants α, β, γ, ι, η and p initially given by (5.3) and converging to those of ssDNA, as an isolated tensile end thrust is applied at higher and higher forces. This process is rate-dependent and hysteretic [66]. The thermodynamic framework introduced by the first author and Srinivasa for evolving natural configurations of three-dimensional bodies [59][60][61] provides a road map for developing a thermodynamically consistent model capable of describing this phase transition at the level of continuum rods, a topic to be discussed in future work.
[ 0 ,
0L] s → r(s) ∈ E 3 , (the center line) to which a right-handed collection of orthonormal vector fields {d k (·)} 3 k=1 (the directors) is attached. 1 The directors {d 1 (s), d 2 (s)} are viewed as tangent to the material cross section transversal to the center line at r(s); see Figure 2 in Section 2.1. The directors are able to deform independently of the center line, and the Darboux vector field u(·) along the center line characterizes their deformation: d ds d k (s) = u(s) × d k (s), s ∈ [0, L], k = 1, 2, 3.
2. 2 .
2Balance laws. Let [a, b] ⊆ [0, L]. We denote the contact force by n(s) so that the resultant force on the material segment [a, b] be [0, a) ∪ (b, L] is given by n(b) − n(a). The contact couple is denoted by m(s) so that the resultant contact couple about o ∈ E 3 on the material segment [a, b] by [0, a) ∪ (b, L] is given by
are derivable from a complementary energy W * (m, n),
basis {e k }, the contact couple m = m k d k = M k e k with M 1 = m 1 cos ψ − m 2 sin ψ, (4.2) M 2 = m 1 sin ψ + m 2 cos ψ, (4.3) M 3 = m 3 , and the contact force n = N k e k with N 1 = −N sin θ, = N cos θ.
θϕ = −uM 1 , (4.10) θ = −uM 2 , ψ + cos θϕ = u 3 . Expressing (2.3) in the basis {e k } using (4.4), (4.5), (4.6) yields M 1 − M 2 cos θϕ + θ M 3 = 0, (4.11)M 2 + (M 1 cos θ + M 3 sin θ)ϕ = N v 3 sin θ − N 2 v cos θ sin θ, M 3 = 0.(4.12) The previous six equations are for the six unknowns ϕ, θ, ψ, M 1 , M 2 , M 3 . 4.2. Rod subject to isolated contact forces. In this subsection we will consider the case when only contact forces are present: for all s ∈ [0, 1] m(s) = 0 ⇐⇒ M 1 (s) = M 2 (s) = M 3 (s) = 0. By the equilibrium equations, an equilibrium state under pure contact forces exists if and only if sin θϕ = 0, (4.13)
Proposition 4 . 2 .
42Assume that θ = 0 and, after a proper rotation of the plane spanned by {g 1 , g 2 } if necessary, that ϕ(0) = 0. Then the strains of the associated equilibrium state are constant and given by u 1 = u 2 = 0,
Proposition 4 . 3 .
43Assume that the material moduli satisfy (4.
cos θϕ = u 3 , (M 1 cos θ + M 3 sin θ)ϕ = 0, where θ, M 1 , M 3 are constant.
Proposition 4. 5 .
5Assume that M 2 = 0, M 1 = 0 and, after a proper rotation of the plane spanned by {g 1 , g 2 } if necessary, that ϕ(0) = 0. Then the strains of the associated equilibrium state are constant and given byu 1 = u 2 = 0,
We conclude this study by considering the case M 1 = 0. We note that by (4.10), there exists s ∈ [0, 1] such that sin θϕ (s) = 0 if and only if M 1 = 0.
Proposition 4. 6 .
6Assume that M 2 = 0, M 1 = 0, M 3 = −M 1 cot θ with θ ∈ (0, π/2], and, after a proper rotation of the plane spanned by {g 1 , g 2 } if necessary, that ϕ(0) = 0. Define
1000 pN, γ D ι D = −70 pN nm, (5.3)
Figure 4 .
4Schematic of the overstretching transition of dsDNA with the Carolina blue curve representing the force-stretch curve for dsDNA and the navy curve representing the force-stretch curve for ssDNA.
By orientation we mean either a choice of direction for the director d 3 , relative to the center line, or a stronger condition in terms of the local invertibility of an associated constrained 3dimensional deformation. See Section 3.2 for more.
The Poynting effect was first observed by Wertheim in the 1850's; see[68] and the extended discussion in[5].
K. R. RAJAGOPAL AND C. RODRIGUEZ
Most materials such as DNA are viscoelastic, but as a first approximation can be considered as elastic bodies.
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| [] |
[
"Estimating Maximal Symmetries of Regression Functions via Subgroup Lattices",
"Estimating Maximal Symmetries of Regression Functions via Subgroup Lattices"
] | [
"Louis G Christie \nUniversity of Cambridge\nCambridgeUnited Kingdom\n",
"John A D Aston [email protected] \nUniversity of Cambridge\nCambridgeUnited Kingdom\n"
] | [
"University of Cambridge\nCambridgeUnited Kingdom",
"University of Cambridge\nCambridgeUnited Kingdom"
] | [] | We present a method for estimating the maximal symmetry of a regression function. Knowledge of such a symmetry can be used to significantly improve modelling by removing the modes of variation resulting from the symmetries. Symmetry estimation is carried out using hypothesis testing for invariance strategically over the subgroup lattice of a search group G acting on the feature space. We show that the estimation of the unique maximal invariant subgroup of G can be achieved by testing on only a finite portion of the subgroup lattice when G max is a compact connected subgroup of G, even for infinite search groups and lattices (such as for the 3D rotation group SO(3)). We then show that the estimation is consistent when G is finite. We demonstrate the performance of this estimator in low dimensional simulations, on a synthetic image classification on MNIST data, and apply the methods to an application using satellite measurements of the earth's magnetic field. | null | [
"https://export.arxiv.org/pdf/2303.13616v1.pdf"
] | 257,757,371 | 2303.13616 | 9bae20074da8029f47252abbed582f874998f661 |
Estimating Maximal Symmetries of Regression Functions via Subgroup Lattices
Louis G Christie
University of Cambridge
CambridgeUnited Kingdom
John A D Aston [email protected]
University of Cambridge
CambridgeUnited Kingdom
Estimating Maximal Symmetries of Regression Functions via Subgroup Lattices
Invariant models, non-linear dimensionality reduction
We present a method for estimating the maximal symmetry of a regression function. Knowledge of such a symmetry can be used to significantly improve modelling by removing the modes of variation resulting from the symmetries. Symmetry estimation is carried out using hypothesis testing for invariance strategically over the subgroup lattice of a search group G acting on the feature space. We show that the estimation of the unique maximal invariant subgroup of G can be achieved by testing on only a finite portion of the subgroup lattice when G max is a compact connected subgroup of G, even for infinite search groups and lattices (such as for the 3D rotation group SO(3)). We then show that the estimation is consistent when G is finite. We demonstrate the performance of this estimator in low dimensional simulations, on a synthetic image classification on MNIST data, and apply the methods to an application using satellite measurements of the earth's magnetic field.
Introduction
Many objects we wish to model in statistics obey symmetries. In biology, viral capsids can exhibit icosahedral symmetries, as shown in figure 1 below (Jiang and Tang, 2017) and these can be integrated in Cryo-EM statistical reconstruction algorithms. Time series can be seasonal (Box et al., 2015), which is a form of discrete translation symmetry. Linear models can be invariant to one or more of specific features, which is equivalent to a continuous translation symmetry along that feature's axis as in figure 2.
In statistical shape analysis (Kendall, 1984;Dryden and Mardia, 2016), we consider collections of k landmark points in R m (structured as a matrix X ∈ R m×k with m rows and k columns) as equivalent if they can be rotated, scaled, and translated into each other, or equivalently invariant to the action of Aff(R m ). Similarly, directional statistics are formulated as real angles invariant to translations by 2π radians (Mardia and Jupp, 2009). Another class of examples are found in imagery. Suppose we wish to classify images as either containing a face or not. This can be formulated as learning the indicator function 1 Face ∶ [0, 1] p → {0, 1} (here assuming that images contain p greyscale pixels). It is often assumed that this function is invariant to reflections or rotations of the images, which can be realised as invariance to a subgroup of the symmetric group on d elements, Sym d , corresponding to permutations of the appropriate pixels. (2017)). Figures (c) shows an icosahedron -a platonic solid with the same symmetries as the viruses. Figure (d) shows the symmetries of the icosahedron, where any symmetry maps a node of the same type (clue circle, red triangle, or red rhombus) are mapped to the same type. The yellow triangle is a fundamental domain.
In general, consider learning a square integrable regression function f ∶ X → R with some independent, identically distributed (iid) data D = {(X i , Y i )} n i=1 with each (X i , Y i ) ∈ X × R, E(Y i X i ) = f (X i ), and Var(Y i X i ) = σ 2 < ∞. Importantly, suppose further that f has an invariance property:
P(f (g ⋅ X i ) = f (X i )) = 1
(1.1) for all g ∈ G for some group G with an action ⋅ ∶ G × X → X . We say that such an f is G-invariant.
There are three main methods for including the information of the symmetry in an estimatorf of f . Firstly, if G is finite we can use a new dataset
D ′ = ∪ g∈G {(g ⋅ X i , Y i )} n i=1
(known in machine learning as data augmentation (Shorten and Khoshgoftaar, 2019), which is distinct from Bayesian data augmentation (Tanner and Wong, 1987;Van Dyk and Meng, 2001)). Secondly, for either finite or infinite G we can consider projecting datapoints (data projection) into the quotient space X G. Lastly, we can project the estimatef into the subspace of G-invariant functions as S Gf = E(f (g ⋅ x, D) D) (where g is a uniform † random variable on a compact group G), known as feature averaging †Uniform here means that the law of g is the Haar measure (Haar, 1933) on a locally compact group G. (Elesedy and Zaidi, 2021). Shape statistics uses data projection where we have explicit projection maps, while the Cryo-EM images are improved by feature averaging. These methods and other symmetrisation techniques have been shown to be broadly applicable in many statistical and machine learning contexts; a good overview of these contexts is Bronstein et al. (2021).
Recent studies have quantified the benefit of these methods, such as in Lyle et al. (2020); Bietti et al. (2021); Elesedy (2021). In summary, one can prove that such symmetrisation techniques outperform the base estimator, even almost surely over the data. For example, in the case of the symmetrisation operator as
f − S Gf 2 = S G f − S Gf 2 ≤ S G f −f 2 = f −f 2 (1.2)
using the G-invariance of f , which implies f = S G f , and the fact that S G is a projection (Elesedy and Zaidi, 2021) and thus has operator norm 1. This can be seen as averaging out some modes of variation in the estimatef in a flexible non-linear manner, or by reducing the entropy of the function class we are estimating over. A key example of this would be averaging a function f on R 3 with the symmetry of SO(3) acting by rotations -in this case the function S SO(3) f varies only along the radial direction at any point x ∈ R 3 and is constant in the orthogonal directions.
This of course relies heavily on the assumption that f truly is G-invariant. Otherwise we create obvious problems: we will struggle to identify images of 6s and 9s if we assume that their classification functions are invariant to any rotations. More generally, iff is a consistent estimator of f then S Gf converges in probability to S G f . This means that inappropriately used symmetries (i.e., when f is not G-invariant) can create asymptotic bias (of S G f − f ≠ 0). Therefore, we want to use as much symmetry as possible without using too much.
There have been a few efforts to develop methods that use the data to select appropriate symmetries, such as Cubuk et al. (2019); Lim et al. (2019); Benton et al. (2020). These methods lose the group structure by estimating subsets of transformations rather than subgroups.
Thus in this paper we present a methods for establishing symmetries of an object, to be used when estimating it. In particular, we present a method for estimating the best symmetry to use from a large search group G of symmetries by structuring the subgroups in a lattice (Sankappanavar and Burris, 1981). We test the hypotheses H 0 ∶ f is G-invariant for each subgroup G ≤ G in the chosen lattice. We also present two possible tests to use and the properties associated with these tests.
This method has a close analogue in variable subset selection (Mallows, 1973;Berk, 1978). Consider a linear regression f ∶ R p → R for which some features i ∈ I have β i = 0. We could estimate I by considering subsets A ⊆ {1, . . . , p} and test for whether β A = 0 (where the set subscript indicates the vector (β a ∶ a ∈ A)). These subsets can also be structured in a lattice (see section 2.3 and figure 2) where the ordering is given by subset-ting, and rejection at a node tells us about rejections higher in the lattice. Alternative algorithms from the variable selection literature (e.g. forward selection or backwards elimination) are discussed in section 7.
In the symmetry estimation case, we have the same structure, but the lattices can be much larger and are usually infinite which allows us to capture non-linear combinations of the variables to select. In figure 2, we consider the equivalence of the subset selection problem with translation invariance, showing that both problems search over similar lattices, though there are infinitely more subgroups than are shown in this sub-lattice. This generalises the notion of Allen (1974) that variable selection is a limiting case of data augmentation ‡. In contrast to variable selection, we show that we can still search over these infinite lattices by considering finite sub-lattices, which can still contain both finite and infinite subgroups of G. This allows us to adaptively smooth our estimatef non-linearly, even reducing the dimension of the estimation problem.
∅ {1} {2} {3} {1, 2} {1, 3} {2, 3} {1, 2, 3} I R 1 R 2 R 3 ⟨R 1 , R 2 ⟩ ⟨R 1 , R 3 ⟩ ⟨R 2 , R 3 ⟩ R 3 ⟨(0, 1, 1)⟩ ⟨(0, 1 2, 1 2)⟩ (a) Lattice of subsets A ⊆ {1, 2, 3}. (b)
Lattice of (some) subgroups of R p .
Fig. 2.
Variable subset selection via a lattice search, and symmetry estimation via a lattice search. The symmetry here is the translation action of R 3 on R 3 , and we have taken a finite sub-lattice of the lattice of closed subgroups K(R 3 ) where R i = {(a 1 , a 2 , a 3 ) ∈ R 3 ∶ a j = a i δ ij } translates along the i th axis. We have also included gray and dashed edges to other subgroups in K(R 3 ) such at ⟨(0, 1, 1)⟩ = {(0, a, a) ∶ a ∈ Z} and its supergroup ⟨(0, 1 2, 1 2)⟩.
The paper is organised as follows. Section 2 covers essential background material and definitions (all in bold face text), particularly that of the subgroups lattice of the search group G and the rules of how invariance behaves over this lattice. In section 3 we then describe the estimation algorithm that searches for G max (f, G) by testing the invariance of f to certain nodes of the lattice. Section 4 introduces two tests one can use in the ‡The finite sub-lattice of variable selection is contained within the lattice of all possible translation symmetries and so we can take a sequence of nodes in the translation lattice that approximates the variable selection nodes. estimation algorithm, each requiring different assumptions for the statistical problem. We then demonstrate the performance of the tests used, the symmetry estimator, and its use in non-parametric regression in simulated and synthetic situations in section 5. We then apply these methods to magnetospheric modelling in section 6. Lastly, we conclude with a discussion of these methods in section 7.
Background
Statistical Problem
Let (X , d X ) be a metric space with associated probability space (X , Σ, µ X ). Suppose we wish to estimate a regression function f ∶ X → R, where f is in some function class F ⊆ L 2 (X ). Suppose that we collect some iid training data
D = {(X i , Y i )} n i=1 in (X × R) n where X i iid ∼ µ X and with E(Y i X i = x) = f (x) and E((Y i − f (X i )) 2 ) = σ 2 < ∞ for all i ∈ N.
This nonparametric regression problem will be focus in this paper. We write
D X = {X i } n i=1
, with analogues for other random variables. We writeμ X for the empirical distribution of D X . We will often consider functions that have bounded variation, for example, the (β, L)-Hölder class of m = ⌈β⌉ − 1 times differentiable functions.
G-invariant Functions
Let f ∶ X → R be a regression function and let G be any group. We give a summary of the definition of a group and other relevant group theory results in appendix B, or a good external reference is Artin (2017). We say that f is G-invariant if P(f (g⋅X) = f (X)) = 1 for all g ∈ G. We say that a measure µ X is G-invariant if for all g ∈ G and measurable A, g ⋅ A is measurable and µ X (g ⋅ A) = µ X (A). The G-invariant square integrable functions form a subspace of L 2 (X ) (as invariance is trivially preserved by additions and scalar multiplication).
Subgroup Lattices
Our methods in section 3 are based on the idea that the subgroups of a large "search group" form a lattice in much the same way that selecting variables in a linear regression problem does. In this section we make precise the lattice we will be searching over, and show that the estimation problem is well defined. Further definitions can be found in appendix C.
It is well known that the subgroups of a group G form a lattice L(G), a partially ordered set with unique supremums (or joins, written ∨) given by A ∨ B = ⟨A, B⟩ and infimums (or meets, written ∧) given by Schmidt, 2011;Sankappanavar and Burris, 1981). In the context of variable selection, we have another lattice, where the index subsets A, B ⊆ {1, . . . , d} have joins A ∨ B = A ∪ B and meets A ∧ B = A ∩ B.
A ∧ B = A ∩ B (
One important subset of L(G) is the collection of closed subgroups, K(G). These do not necessarily form a sub-lattice for arbitrary G as we can generate non closed subgroups with the original join operator ⟨G, H⟩ (consider two copies of S 1 generating a proper dense subgroup of SO(3)). Instead, we have to adapt it by taking a new join as ⟨G, H⟩, which we establish in the following proposition. In some cases, e.g., finite G, this is equivalent to the original join.
Proposition 2.1. Let G be a topological group. The closed subgroups K(G) form a lattice with partial ordering given by subgrouping, meets given intersection, and joins given by the closure of the group generated by the two subgroups.
Unlike the variable selection problem, the lattice K(G) is often infinite both breadthwise and depth-wise. Optimising over this lattice is thus usually intractable, so instead we take a well chosen finite collection of closed subgroups A ⊆ K(G) and generate the countable or finite lattice ⟨A⟩ of K(G), which we can enumerate. Even though the sub-lattice ⟨A⟩ may be finite, it can still contain infinite groups (see Examples 2.3 and 2.4).
Example 2.2. Consider the group of symmetries of the square,
D 4 = {I, R h , R v , R , R , R π 2 , R π , R 3π 2 } (2.1)
This group has eight proper subgroups. Five are order 2 (four generated by the reflections and one generated by the order 2 rotation (labeled G 1,3 ), one is cyclic of order 4 (generated by either of the order 4 rotations), and the other two are isomorphic to C 2 ×C 2 (generated by appropriate pairs of the reflections). These can be ordered into the lattice in Figure 3, represented as a Hasse diagram (where A ≤ B if there is an ascending edge from A to B).
Example 2.3. Consider the group S 1 = R Z. The finite subgroups are those of rational rotations, and are of the form C a = ⟨[1 a] Z ⟩ of order a. These have subgroups corresponding to the factors of a, and so the lowest parts of the subgroup lattice of S 1 are the groups of prime order. Above these you have groups with two (counting with repetition) prime factors, and so on. There are also subgroups of countable infinite order, for example ⟨[1 √ 2] Z ⟩, and one subgroup of uncountable order -itself. Importantly, these infinite subgroups are all dense in S 1 (under the topology inherited from R). The lattices L(S 1 ) and K(S 1 ) are depicted in figure 4.
Example 2.4. Consider the group SO(3) of 3D Rotations. This group has 4 types of subgroups: (1) dense subgroups; (2) subgroups isomorphic to S 1 or O(2) or its dense subgroups; (3) finite subgroups that fix a plane (isomorphic to C n or D n ); (4) exceptional finite subgroups corresponding to symmetries of the platonic solids. Thus its subgroup lattice can be thought of an infinite collection of copies of S 1 and its lattices (with the copies indexed by S 1 ) and all the lattices supremums and infimums (such as the copies of O(2) generated by some S 1 and a rotation of π radians about an orthogonal axis). Some of the subgroups that are dense in SO(3) can be reached as the supremum of two non commutating high (finite) order elements (which themselves are in finite subgroups of some S 1 u ). The exceptional tetra-, octa-, and icosa-hedral symmetry groups are also generated from two subgroups. For example, the icosa-hedral group is generated from two groups of rotations ⟨S⟩ and ⟨T ⟩ of order 2 and 3 respectively, chosen such that ⟨ST ⟩ is of order 5. This means that the closed finite subgroups of SO(3) either fix a plane or have order less than 60. So if we take two rotations of order 11 around axes that correspond to neighbouring vertices of an icosahedron they must generate a group that has order greater than 60 but which doesn't fix a plane, and so is dense in SO(3). Therefore, we can take a finite sub-lattice of K(SO(3)) as depicted in figure 5.
I ⟨R h ⟩ ⟨R v ⟩ ⟨R π ⟩ ⟨R ⟩ ⟨R ⟩ ⟨R h , R π ⟩ ⟨R π 2 ⟩ ⟨R , R π ⟩ D 4
Given a chosen sub-lattice K of K(G), the following proposition shows that there is a well defined maximal object that we can look to estimate from the data.
Proposition 2.5. Suppose that G acts faithfully and continuously on X . Let K be a finite sub-lattice of K(G). Then there exists a unique node G max (f, K) in K that f is invariant to, and for which H ≤ G for all H ∈ K for which f is H-invariant. Moreover,
G max (f, K) ≤ G max (f, G).
Ideally this object would capture much of the symmetry applicable to f from within the search group G, however the lattice K does need to be well chosen. For example, if the lattice consisted of only the trivial group I and G, then our only options are total G-invariance or no invariance at all. We discuss strategies for choosing K in section 3.2.
Estimation of G max (f, G)
Suppose we have collected data D as in the statistical problem defined in section 2.1 and suppose G is a group acting faithfully and continuously on X . By proposition A.1,
G 0,1 = I C 2 = ⟨[1 2] S 1 ⟩ C 3 C 5 C 7 C p C 4 C 9 C 25 C 49 C p 2 C 6 C 15 C 35 G = S 1 ⟨[1 √ 2] S 1 ⟩ ⟨[1 √ 3] S 1 ⟩ ⟨[1 √ 5] S 1 ⟩ Infinite Finite2] S 1 ⟩ ≤ ⟨[1 √ 2] S 1 ⟩).
The lattice of closed subgroups K(S 1 ) is the same but with the only infinite group begin S 1 itself, i.e., without the blue nodes.
there is a unique closed maximal invariant subgroup G max (f, G) that we wish to estimate.
Let Test α ∶ (D, G) ↦ {−1, 1} be a hypothesis test for H 0 ∶ f is G-invariant that takes (D, G) to −1 if rejected and 1 if accepted at the significance level α. We will provide two possibilities for Test α in the next section. One could brute force a search for G max (f, G) by testing every subgroup G ≤ G, and takingĜ = ⟨G ≤ G ∶ T est α (D, G) = 1⟩. Unfortunately this is inefficient at best (for example, with finite G) and uncomputable at worst (when G is infinite).
Instead we utilise the rules for invariance above to reduce the problem. Together, these allow us to use the information about strategically chosen tests to estimate the test results at other subgroups. We first described a simple search over a finite search group G, and then a more complicated algorithm for when G is a compact connected group. (3)) that uses rotations around the axes of an icosahedron (represented by unit vectors u 1 , . . . , u 6 ).
I ⟨R u 1 2π 11 ⟩ ⟨R u 2 2π 11 ⟩ ⟨R u 3 2π 11 ⟩ ⟨R u 4 2π 11 ⟩ ⟨R u 5 2π 11 ⟩ ⟨R u 6 2π 11 ⟩ SO(3) Fig. 5. A finite sub-lattice of K(SO
Estimating Over a Finite Lattice
Let K be a finite sub-lattice of K(G) that contains the trivial group I. Consider the following algorithm for eliminating subgroups G ≤ G based a the hypothesis test Test α .
In essence, we test from the bottom up, testing each G ∈ G i and eliminating all H ≥ G that the test rejects H 0 ∶ f is G-invariant at significance level α. This is effectively a breadth first search over the lattice, formalised in algorithm 1. Let m be the height of the lattice K -the maximal length of a chain of distinct subgroups between I and max K -and let N i be the number of subgroups at height i in K. K can then be enumerated as noted in section 2.3.
Algorithm if Test α (D, G i,j ) = −1 then Delete all G ≥ G i,j from K 5:
returnG B = max K.
Note that there may not be a unique maximal element remaining in K, so as givenG B is set valued, containing all maxima. We can then choose a single estimateĜ B to use by picking from this set uniformly at random, or being more conservative and choosing the meet of all elements ofG B ,Ĝ
Alt B = ⋁ G∈G B G = ⋂ G∈G B G (3.1)
We can create a greedy version of algorithm 1. Because of lemma A.2, confidence in subgroups H 1 , . . . , H k ∈ K gives confidence in ⟨H i ∶ i ∈ [k]⟩. However we still need to be careful; as direct refinements, i.e., vertical lines in K, are not generated by just the subgroups in the level below. Thusm we can assume an acceptance of G ∈ G i if it is generated by two or more subgroups in level G i−1 that were each not rejected, and call such an amendment algorithm 1a. This does lose some specificity because we do not also test at the higher nodes (where we might find a true rejection).
Example 3.1. If we are optimising over the lattice of D 4 from example 2.2, and G 1,j pass the hypothesis test for j ∈ {1, 2, 3}, algorithm 1a will automatically assign an acceptance to G 2,1 but not G 2,2 .
Alternatively, we can look to a depth first search. This is justified by the fact that if f is G-invariant then G max ≥ G, and so if Test α (D, G) = 1 then G max will be in the portion of the lattice above G with probability at least the power of Test α . If G ≤ G then define K G = {H ∈ K ∶ G ≤ H} as the sub-lattice above G. This is the recursive algorithm 2.
Algorithm 2 Depth First Estimation 1: procedure DepthFirstEstimation(D, L, Test α , i) 2:
for j ∈ [N i ] do 3: if Test α (D, G i,j ) = 1 then return DepthFirstEstimation(D, L G i,j ,f , Test α , i + 1) 4: returnĜ D = min L
Note that at each call in the recursion we are guaranteed to increase the level by 1 so this will terminate in finite time for finite G. This algorithm is more susceptible to error with a low power test, at it relies on acceptance rather than rejection checks.
3.2. Choosing the Lattice K The choice of K will greatly affect the usefulness of this method. If we pick a trivial sub-lattice of only the identity subgroup then we are guaranteed to estimateĜ = I. A suggested strategy is to take a finite number of closed subgroups that give some level of coverage of G (if it is metrisable, as with a Lie group, then this could be the groups generated by an -net of G) and then generate a lattice from them by taking meets and joins. By the following Lemma we can then add in supergroups (of the entire lattice) to ensure that we see infinite subgroups of G if we choose.
Lemma 3.2. If K is a sub-lattice of K(G) and H ≤ G ≤ G for all H ∈ K, then K ∪ {G} is also a sub-lattice of K(G).
Theoretical Properties ofĜ
Let P f (n, α) be a lower bound of the power of Test α applied to f at the significance level α for sample size n, i.e., a bound such that P f (n, α) ≤ P(Test α (D, G) = −1) for all G ∈ K for which f is not G-invariant. We first consider simple finite sample bounds, which are not uniform over f . These avoid problems of multiple testing by taking a very loose union bound, but which is still sufficient for our purposes. Note that here wheñ G B is multiple valued we takeĜ B from this set uniformly at random.
Lemma 3.3. The probability of f beingĜ-invariant for bothĜ B andĜ D is bounded below by 1 − A (1 − P f (n, α)) where A ⊆ K are subgroups above only subgroups of G max and cover subgroups of G max .
Note that A is trivially bounded by K < ∞, but can be much smaller.
Lemma 3.4. The probability thatĜ B = G max (f, K) is at least 1 − {H ∈ K ∶ H ≤ G max (f, K)} α − A (1 − P f (n, α)).
We can use these results to a sufficient condition for the consistency ofĜ B for G max (f, K); namely the consistency of the test used and well chosen significance level used. We note here that the convergence in probability is with respect to the discrete topology on the finite set K.
Proposition 3.5 (Consistency ofĜ B ). If P f (n, α) → 1 for any α then there exists a deterministic sequence of significance levels α n such thatĜ B p → G max when using Test α n when searching over K with n samples.
Computational Complexity
A full analysis of the complexity of these algorithms is impossible without specifying the test used. Here we give a heuristic for finite G that assumes the computational time that is proportional to G for each test conducted. § The brute force approach would thus require a baseline of ∑ G∈K G computational units, which we use for comparison to these algorithms. We then discuss how the test in the previous section can further improve on this.
The finite breadth first estimator will have the same worst case runtime of ∑ G∈K G computations, if G is G-invariant (and so we test at every node in K). The greedy version is guaranteed to conclude faster unless G ≃ C p n is a cyclic group of prime power order: all other finite groups have at least two non nested subgroups and so we can remove at least one node (where those two groups meet). Usually both will cut out many other nodes through the deletion at a rejection.
The finite depth first estimator will again require ∑ G∈L G if G is cyclic of prime power order. Similarly, if G = ⟨G m−1,j ∶ j ∈ {1, . . . , N m−1 }⟩ and we happen to reject at all G m−1,j for j ∈ {1, . . . , N m−1 − 1} while accepting G m−1,N m−1 then we will still test at all nodes (even knowing that we will have to fail at G). Again, with most other structures we can rule out at least one node, for instance if two consecutive layers have more than two nodes. §This would correspond to using m ∝ G in the Asymmetric Variation Test, which is reasonable if one is sampling with a uniform µ g on G.
If G admits particular structure then we can prove even stronger results. Here we write G as an additive group when it is abelian.
Proposition 3.6. Suppose G ≃ ⊕ k i=1 C p a i i
is an abelian group. Then the worst case performance of Algorithm 1a saves at least
∑ b∶ b 0 >1 ∏ k i=1 p b i i units of computation relative to brute force, where the sum ranges over multi-indices b ∈ ∏ k i=1 F a i with more than one non-zero element. The sum ∑ b ∏ k i=1 p b i
i can be bounded simply, as ∏ p i ≥ 2 and there are ∏ a i terms in this sum, giving
b∶ b 0 >1 k i=1 p b i i ≥ 2 k i=1 a i (3.2)
Note that this doesn't account for subgroups of G not generated in this way so this bound is usually not tight. It only considers subgroups on a particular sub-lattice of K, but this is not all of K as the following example shows.
Example 3.7. Consider G = C 2 × C 2 = {(0, 0), (1, 0), (0, 1), (1, 1)}.
See that the subgroups ⟨(1, 0)⟩, ⟨(0, 1)⟩, and ⟨(1, 1)⟩ are each isomorphic to C 2 but since ⟨(1, 1)⟩ does not contain either ⟨e 1 ⟩ nor ⟨e 2 ⟩ as a subgroup it is not generated by them. Note that in this example algorithm 1a will never have to test G, saving at least
4 = ∑ b∶ b 0 >1 ∏ k i=1 p b i i > 2(1×1) units of computation.
Hypothesis Testing for a Particular Symmetry
We now turn to constructing tests for the hypothesis H 0 ∶ f is G-invariant against H 1 ∶ f is not G-invariant for use in the subgroup estimation. We present two options for Test α that make trade offs in terms of generality and power. The first requires stronger assumptions but has a theoretical guarantee of convergence, whereas the second makes far fewer assumptions but has slightly lower power and is slightly liberal.
The Asymmetric Variation Test
Suppose that F is a class of bounded variation for which the practitioner assumes f ∈ F,
and let V (x, y) = sup f ∈F f (x) − f (y) .
An example of such a class are α-Hölder con-
tinuous functions F(L, α) with f (x) − f (y) ≤ Ld X (x, y) α for α ∈ (0, 1], for which V (x, y) = Ld X (x, y) α . Let the independent mean zero additive noise i = Y i − f (X i ) be such that P( i − j > t) ≤ p t . An example of this would be p t = 2σ t exp(−t 2 4σ 2 ) √ 2π
for iid gaussian i with variance σ 2 (Proposition 2.1 of Adams (2020)). Let g ∼ µ g be any G valued random variable, for any distribution µ g on G.
We would wish to construct a statistic that captures the p-value
of H 0 . If f is G- invariant then Y i − Y j = f (X i ) − f (X j ) + i − j (4.1) = f (g ⋅ X i ) − f (X j ) + i − j (4.2) ≤ f (g ⋅ X i ) − f (X j ) + i − j (4.3) ≤ V (g ⋅ X i , X j ) + i − j , (4.4) so we know that D g ij = Y i − Y j − V (g ⋅ X i , X j ) ≤ i − j for all i, j ∈ {1, .
. . , n} and g ∈ G. In particular, this is true for j chosen such that d(g ⋅ X i , X j ) is minimised. If we then count how often D g ij is larger than some threshold t ∈ R ≥0 , we can bound the probability using the concentration of i − j . This leads to algorithm 3.
Algorithm 3 Asymmetric Variation Test
1: procedure AsymVarTest(D, V , µ g , t, p t , m) 2: for j ∈ {1, . . . , m} do 3: g j ← Sample(µ g ) 4: I(j) ← Sample({1, . . . , n}) 5: J(j) ← Index of Nearest Neighbour to g ⋅ X I(j) in {X j } n j=1 6: D g j I(j)J(j) ← Y I(j) − Y J(j) − V (g ⋅ X I(j) , X J(j) ) 7: N g t ← {D g j I(j)J(j) ≥ t} 8: p val ← ∑ m k=N g t m k p k t (1 − p t ) m−k 9: return p val
For fixed m, the D g j I(j)J(j) are asymptotically independent (because it is vanishingly unlikely that we sample the same g j ⋅X I(j) or that two g j ⋅X I(j) share a nearest neighbour). Thus under the null hypothesis, N g t is stochastically bounded by a Binom(m, p t ) variable, which allows us to bound the "true" p-value from above by the return value p val . We can thus reject the null hypothesis of G-invariance at significant level α ∈ (0, 1] if p val ≤ α.
4.1.1. Choices of t and of µ g The methodology presented here works for any choice of t and variable g, though particular choices of these will affect the power of the test. For example, if we choose t such that p t ≥ 1 then our p-value will always be 1. Similarly if g = e ∈ G almost surely then we will also only reject with probability at most α.
For the choice of t, we suggest calculating N g t from the sample of D g j I(j)J(j) at some grid of t values t 0 < t 1 < ⋯ < t k with the values of p t i spread over the interval (0, 1), and then taking the p-value of H 0 as the minimum p-value of each N g t i . This is justified as the information of the test is entirely contained in the set {D
g j I(j)J(j) } m j=1 , i.e., since the p-value is at most P(N g t ≥ k t H 0 ) for all t,
we can take an infimum over t.
For the choice of µ g , we suggest using a uniform distribution only on some set of topological generators of G. This means that we don't sample the identity or other elements that generate only subgroups of G that f may be invariant to, but if there is an element that breaks the invariance then one of the generators will too (by lemma A.5) so we should capture that chance.
Consistency of this test
Under some mild conditions on the noise distribution, and with m set at n for all n, we can prove that the asymmetric variation test is consistent. The condition on the support of µ X amounts to restricting X to the closure of the support as a practitioner would usually do. The condition that the noise admits a density is satisfied in many usual cases in regression (e.g. Gaussian noise). The condition that we can bound the concentration of the noise tightly is somewhat restrictive, but reflects the difficulty of the problem -if the asymmetry is obscured by more noise then it is much more difficult to identify it. The condition on µ g is there to ensure that we sample from enough of G to ensure we can spot points where f is not G-invariant. A sufficient condition for this is where we sample from the Haar measure on G when this exists (e.g., compact G).
Proposition 4.1. Set m = n and fix t > 0. Suppose that the law of X has a dense support on X . Suppose that admits a density f Y with respect to Lebesgue measure on Y that is decreasing in y . Suppose that µ g is chosen such that
P(f (g ⋅X) ≠ f (X) H 1 ) > 0. Then the asymmetric variation test is consistent, i.e. p val H 1 p → 0.
The proof can be found in appendix A.3 in the supplementary material.
Computational Complexity
The main computational bottleneck is in finding the nearest neighbour in step 5. A naive algorithm would search across all n samples to find J(j), which means that the algorithm would cost O(mn) operations. One improvement would be to store the feature vectors X i in a k-d tree (Bentley, 1975). Building such a tree takes O(n log n) operations, but can find nearest neighbours in O(log n) time. This speeds up the asymmetric variation to O(m log n).
Using the Asymmetric Variation Test in the Subgroup Lattice Search
Consider using the Asymmetric variation test from section 4 for Test α in the algorithms above. Since any distribution µ g on a subgroup G ≤ G is also a distribution on any supergroup H ≥ G, any test low in the lattice L(G) is technically a test for the groups above it, though one with lower -perhaps even zero -power, making this somewhat useless.
Alternatively, we could consider constructing a variable g on G by sampling a group from a finite subset A ⊆ L(G) and the from the Haar measure sampled group (recall these are all closed and so compact when G is compact). Such a variable can be used in the sampling step of the Asymmetric Variation Test for testing H 0 ∶ f is G-invariant. But it can also be used to test for invariance to the subgroups in A by including a rejection sampling step -if we subset the samples j ∈ {1, . . . , m} when g j ∈ H ∈ A we still have an iid sample we can use to count N g t and test for H-invariance of f .
This means that we can speed up doing multiple tests considerably as we only need to sample m times and compare distances once per finite block A, and then simply do the much faster subsetting and counting operation for each of the groups H ∈ A. This of course means that the test results are even more correlated than previously (given that they shared the original data D), trading some statistical power for computational power.
Permutation Variant of the Asymmetric Variation Test
The asymmetric variation test (in the previous subsection) relies on both the existence of, and the knowledge of, the bound V (x, y). In this section we show that we can remove some of the requirement of the knowledge at the cost of computational (and sometimes statistical) power with an approximation to the previous test.
This new test still has the assumption of a known order for the variation bound, but does not require any knowledge of the noise variables i (other than that they are iid).
The practitioner still has a choice of the random variable g, but now chooses a quantile q instead of threshold(s) t.
Suppose that we know only the order of the bound V , i.e., we know some V(x, y) such that for all f ∈ F there exists some (unknown)
L f with f (x) − f (y) ≤ L f V(x, y)
for all x, y ∈ X . Clearly any known V satisfies this property, but it is weaker in that we do not need to know that bound exactly. A key example of V is d X (x, y) α for α-Hölder continuous functions in any F(L, α) (whereas we would need a constant multiple of this for a class of particular α-Hölder continuous functions).
Let S g ij = Y i − Y j V(g ⋅ X i , X j ). Consider collecting S k = {S g j I(j)J(j) } m j=1 as in algorithm 3, but where J(j) ∼ U ({1, .
. . , n} instead of being the index of the nearest neighbour, and let A k be the q th -quartile of this set for some chosen q ∈ (0, 1]. Under the null hypothesis H 0 ∶ f is G-invariant, the distributions of S g ij will be the same as S e ij , so we can run a permutation test (See Good (2005)
) on the set {A k } B k=1 , comparing them to the q th -quantile A 0 of S 0 = {S e I(j)J(j) } m j=1
(sampled in the same way, but with g = e almost surely). We can approximate the p-value by the proportion of A k ≤ A 0 . This is described in algorithm 4.
Finite sample effects and choice of quantile q
Whilst it is true that S g ij D = S e ij for independent X i , X j , the finite sample estimates of these quantities are not equal. In fact, the distribution of S g ij can be biased upwards because the action on X allows us to see more points with smaller d X (g ⋅ X i , X j ) < d X (X k , X l ).
Algorithm 4 Permutation Variant of Asymmetric Variation
g j ← Sample(µ g ) 5: I(j) ← Sample({1, . . . , n}) 6: J(j) ← Sample({1, . . . , n}) 7: S g j I(j)J(j) ← Y I(j) − Y J(j) V(g j ⋅ X I(j) , X J(j) ) 8: A(k) ← quantile({S g j I(j)J(j) } m j=1 , q) ▷ R syntax 9:
for j ∈ {1, . . . , m} do
S e I(j)J(j) ← Y I(j) − Y J(j) V(X I(j) , X J(j) ) 13: A 0 ← quantile({S e I(j)J(j) } m j=1 , q) 14: p val ← {k ∶ A(k) ≤ A 0 } B 15: return p val
As m increases, we are more and more likely to see the outlying values of S g ij compared to S e ij . This can cause a bias towards rejection in this permutation test, i.e., it is liberal for small values of n. This inexactness is a known issue with permutation tests, for example when testing the variance of univariate samples (as in Good (2005), §3.7.2), and approximate permutation tests as used here are still shown to be successful and useful.
These problems are alleviated somewhat by using the quantile q ∈ (0, 1], as this is less sensitive to the outlying values of S g ij than picking q = 1 (i.e., just going with the maximum of the S g j I(j)J(j) ). This does mildly reduce the power of the test, but improves the specificity significantly. We have found in simulations that using q = 0.95 works well in practice.
Computational Complexity
Here we have made a trade in the computational complexity -we no longer compute nearest neighbours, but now have to run B extra trials. Thus the computational complexity of the sampling is of order O(Bm). We then have to compute the quantiles for each permutation, which essentially requires ordering the m samples of S g ij , which can be done in O(m log m) operations with, for example, mergesort or heapsort (Knuth, 1998). This gives an overall complexity of O(Bm log m)
Numerical Experiments
We've constructed three simulations to gauge the performance of the group estimation technique. The first is a low dimensional regression estimation where we can directly quantify the errors the group estimatorĜ(D) by the proportion of correct estimations.
The second then considers the effect of using the estimated groupfĜ to symmetrise the estimatorf where we consider the mean squared prediction error. We then consider a high dimensional (d = 784) example with the MNIST dataset (LeCun et al., 1998).
All tests were conducted locally on a 2020 13" MacBook Pro laptop with a 2 GHz Quad-Core Intel Core i5 processor and 16GB of RAM. All code is available on GitHub (HERE). Further simulations, including those explicitly for the hypothesis tests only can be found in appendix D.
Example 5.1 (Finite Group Estimation). Consider the functions f d ∶ (x 1 , . . . , x d ) ↦ exp(− x 1 ) for invariance to the subgroups of C 4 acting by rotations in the x 1 -x 2 plane. For these functions we have G max (f d , C 4 ) = C 2 , and the subgroup lattice chosen is L(C 4 ) = {I, C 2 , C 4 }. As such, in these simulations rejections of the null H
(1) 0 ∶ f d is C 2 -invariant giveĜ = I, and otherwise if H (2) 0 ∶ f is C 4 -invariant
is rejected then we get G = C 2 = G max , and if both are accepted thenĜ = C 4 . This is true for any of the three estimation algorithms.
We generate data from X i iid ∼ N (0, 4I d ) and i iid ∼ N (0, 0.05 2 ). We simulated tests at the α = 0.05 significance level of each of the null hypotheses using both tests (algorithms 3 and 4) and these are shown in figure 11 of appendix D. We ran 100 simulations for each combination of 30, 40, 50, 60, 70, 80, 90, 100, 120, 150, 200, 250, 300} (5.1)
n = m ∈ {20,
Both tests assumed that f is 1 e-Lipschitz. The asymmetric variation tests had t = 2σ = 0.1, p t = 2σ We plot the proportions of estimation in the graphs in figure 6. We see that in the lower dimensional f 2 example we very quickly converge to the true G max after only 50 samples, but in the higher dimensional f 4 we need at least 200 or so (otherwise we over estimateĜ = C 4 > G max ).
Example 5.2 (Simulation of Symmetrised Estimators). Suppose that X = R 3 and let G = SL(3, R) act on X by matrix multiplication. Let K be the sub-lattice of K(SL(3, R) from example 2.4 with SL(3, R) added. As per the statistical problem (Section 2.1), we
simulate data D = {(X i , f (X i )+ i } n i=1 and D ′ = {(X ′ i , f (X ′ i )+ ′ i } n i=1 under four scenarios: 1) (Interpolation with Symmetric f ) f 1 (x) = sin(− x 2 ) and X i , X ′ i iid ∼ N (0, 2I 3 ); 2) (Extrapolation with Symmetric f ) f 2 (x) = sin(− x 2 ) and X i iid ∼ N (0, Diag(0.1, 0.1, 2) and X ′ i iid ∼ N (0,2I
4) (Extrapolation with Asymmetric
f ) f 4 (x) = sin(− x 1 ) and X i iid ∼ N (0, Diag(0.1, 0.1, 2) and X ′ i iid ∼ N (0, 2I 3 )
The noise variables were sampled as i , ′ i iid ∼ N (0, 0.01 2 ) for all scenarios. Note that in scenarios 1 and 2 we have G max (f 1 , K) = G max (f 2 , K) = SO(3) whilst under scenarios 3 and 4 we have G max (f 3 , K) = G max (f 4 , K) = I. Scenarios 3 and 4 do contain some symmetry, rotations around either the x 3 or x 1 axes respectively, but these symmetries are not present in the lattice K used. We use the estimatorĜ(D) using the breadth first search (Algorithm 1) using the asymmetric variation test with V (x, y) = d X (x, y) and the other hyper-parameters as in example 5.1. We then consider three estimators for f : A) A standard local constant estimator (LCE)f (x, D) with bandwidths selected by the R-package NP (Hayfield and Racine, 2008);
B) A symmetrised LCEf  G πĜ(x), {(πĜ(X i ), Y i )} n i=1 whereĜ =Ĝ(D); and C) A symmetrised LCE that splits D into independent pieces half the size of D, i.e. f Î G πĜ(x), {(πĜ(X i ), Y i )} n i=⌊n 2⌋+1 whereĜ =Ĝ({(X i , Y i )} ⌊n 2⌋ i=1 ).
Here π G ∶ X → X G is the natural projection given by π I (x) = x, π SO(3) (x) = x 2 , and π SL(3) (x) = 1 x≠0 . It is possible to construct projections for the finite subgroups in K by picking a fundamental domain of the action ofĜ, but these were never needed in our simulations. The projections for the infinite groups dramatically reduce the dimension of the sample space (from 3 to 1 or even 0), which should improve the estimation when f is truly G-invariant.
The mean squared predictive errors are plotted in figures 7 and 8. In scenario 1), we see that the LCE has exactly the expected performance decaying at the rate appropriate for a 3 dimensional space. In contrast, the estimators that search for symmetry start by over smoothing (when they cannot reject SL(3, R)-invariance) but then quickly settle into the rate appropriate for the 1 dimensional quotient space which means they are both orders of magnitude more accurate than the LCE. This is even more pronounced in scenario 2), where the LCE struggles to generalise well but the symmetrised estimator learns the appropriate pattern and uses this to extrapolate accurately.
Scenarios 3) and 4) show the a similar pattern. The LCE has reasonable performance, and since G max (f, K) = I the symmetrised estimators learn that they should just give the LCE as their chosen model, so the errors converge exactly between estimators A) and B) with a lag because of the half data for estimator C.
Example 5.3 (MNIST Symmetries of the digit 5). Consider the symmetries of images of the digits 3 and 8. It is clear that rotations of both preserve the classification, however one can argue that only the 8 is invariant to horizontal reflections -a reflected 3 would be an E (in the same way that a reflected p is a q), and so can be considered oriented in a way that the 8 is not. We call the in digits O = {2, 3, 4, 5, 6, 7, 9} the oriented digits and the others {0, 1, 8} the non-oriented digits. We consider the three to be vertically asymmetric so the rotated 3 is distinct from the reflected 3, and thus 3s are rotationally invariant, and all other oriented digits are considered similarly except sixes and nines which are not rotationally symmetric.
Each digit has a function f a ∶ X → [0, 1] for which f a (X) = P(X is recognised as the digit a), and we seek to estimate the symmetries of these probabilistic regression functions using the asymmetric variation test.
We subset the MNIST dataset (LeCun et al., 1998) (Available under a CC BY-SA 3.0 licence) into particular characters, called D X (n) for n ∈ {0, 1, . . . , 9}. We then split these in half uniformly at random to form sets D 1 X (n) and D R X (n). We then apply b (the reflection through the vertical line) from the dihedral group D 4 to the elements of D R X (n) and then assign labels Y i = 1 for every for X i ∈ D 1 X (n) for all n and D R X (n) if n is non-oriented, and the label Y i = 0 for every other X i . This gives datasets
D(n) ={(X i , 1) ∶ X i ∈ D 1 X (n)} ∪ {(X i , 1 n ∈O ) ∶ X i ∈ D R X (n)} (5.2)
Here the labels Y i are the probabilities that X i will be recognised as the digit n, and so we are learning the function f n ∶ [0, 1] 784 → [0, 1] that assigns such probabilities. We assume that there is no noise, so p t = 0 for all t. Lastly we need to pick V (x, y), which we estimate with the reciprocal of the minimal distances between digits in and out of each class.
We then test each of the datasets D(n) for invariance of each f n the subgroups of D 4 (enumerated in the Hasse diagram in Figure 3) using the asymmetric variation test at significance level α = 0.05 and with V (x, y) = L n d(x, y) (where the coefficients L n were estimated separately), σ = 0 and so the concentration bound is given by p t = 0 for all t > 0.
Using the breadth first search for the digit 5, we find that tests for ⟨R v ⟩-invariance and ⟨R h ⟩-invariance are rejected, whilst ⟨R π ⟩, ⟨R ⟩ and ⟨R ∖ ⟩ are not. We then test on the second layer of L(D 4 ), but only at ⟨R π 2 ⟩ and ⟨R π , R ⟩, where again we find acceptance -thus we haveG B = {⟨R π 2 ⟩, ⟨R π , R ⟩}. This is stepped through in the diagrams of figure 9. Note that the test used lacked power because the data distribution was not ⟨R ⟩invariant -we didn't have any samples of diagonally reflected fives to compare against. We then haveĜ Alt B = ⟨R π 2 ⟩ ∩ ⟨R π , R ⟩ = ⟨R π ⟩, which in our set-up is a subset of the correct G max = ⟨R π 2 ⟩. Tests conducted for other digits are shown in appendix D.
Application to Satellite Based Magnetic Field Data
The European Space Agency (ESA) launched the SWARM mission in 2013 to measure the earth's magnetic field and other related properties. This consists of three satellites orbiting between 450km and 530km above the earths surface. These measurements complement the observations at ground based observatories to better model the field for use in navigation and geospatial modelling. This field is generated by four main components: the rotation of the inner core; the effects of magnetised material in the earth's crust; electrical currents in the upper atmosphere caused by solar winds; and the interaction of the sun's magnetic field.
We have used a subset of the data available from the SWARM satellites to estimate the field strength of the magnetic field B(x) 2 over the sphere of radius ≈ 6, 820km from the earth's core. Using the data from only Satellite α on the 25th of February 2023, we selected 220 observations over the Northern Hemisphere. Two hundred occur in a "dense" region over Europe and Asia, and 20 occur in a "sparse" region over North America and adjacent oceans. These measurements are plotted in figure 10.
We follow the same symmetry estimation procedure (Algorithm 1) for these functions as in example 5.2. We use the lattice K ⊆ K(SO(3)) given by {I, S 1 u i , SO(3) ∶ i ∈ {1, . . . , 15}} where the vectors {u i ∶ i ≤ 6} form the axes of a regular icosahedra and {u i ∶ 7 ≤ i ≤ 12} correspond to axis through an icosahedron rotated by 36 degrees. We then also add u 13 = (0, 0, 1), u 14 = (0, 1, 0), and u 15 = (1, 0, 0). We use the permutation variant of the asymmetric variation test with a Lipschitz assumption V(x, y) = d X (x, y). We use the quantile q = 0.95 and the significance level α = 0.05, and run B = 200 permutations each with m = 3000. We sample from each S 1 u with a bias towards low angles (i.e., angles sampled as θ ∼ N (0, (2π × 0.2) 2 )) and uniformly on SO(3). We rejected every symmetry except for rotations about the axis u 13 , the (Geographic) north pole. ThusĜ = S 1 u 13 .
We then estimate the function B(x) using local linear estimators (as in example 5.2. We model both on the original space S 2 parametrised by the latitudes and longitudes, and sinceĜ = S 1 u we model on the space S 2 S 1 u ≅ [0, π] via the projection x ↦ arccos(⟨u 13 , x⟩). We judge these estimations against the International Geomagnetic Reference Field (IGRF) (Thébault et al., 2015) model predictions using mean squared prediction error over a grid of latitudes and longitudes at 1 degree intervals. Over the whole northern hemisphere, the symmetrised estimator achieved an MSPE 23% lower than the local linear estimator. Over the dense region (where θ > 0) the symmetrised estimator had an error rate 2% lower than the LLE. All code and data is available HERE.
It is worth noting that the p-value of the test for S 1 u 13 symmetry was only 0.075. Whilst technically above the significance level α = 0.05, this shows that the test is giving a low confidence to the symmetry around the geographic north pole. This means that the symmetry estimator has correctly identified that the model that has this symmetry should perform better with this dataset, but just barely. With extra data one might expect this test to reject this symmetry.
Discussion
We have shown that we can generalise the idea of variable subset selection into selection of non-linear combinations of the variables by placing them within the framework of subgroup lattices. We have developed methods for testing and used these to estimate the maximal invariant symmetry of a regression function. We have demonstrated the power and applicability of these tests across several numerical experiments, showing that when the symmetry exists our methods do detect this consistently and that this can significantly improve estimates of the regression function.
The main limitations of these methods are the related to the existence of the symmetries of f . It may be the case that f is close (in L 2 distance) to an invariant function but not one itself. In this case estimation of G max (f ) is at least as hard as distinguishing between the two possible functions. However, using the symmetry can still improve the estimate of f by adding bias (smoothing from the symmetry) for much reduced variance. It would be an interesting question of further study to quantify these effects.
The other limitation is thatĜ is biased towards more symmetry because it relies on a hypothesis test. This is deliberate here -it gives a more consistent estimate ofĜ for precisely the reason above. However, it may be a detriment when usingĜ to smootĥ f (as illustrated by Scenario 1 in example 5.2). It is worth considering methods akin to forward selection and backwards elimination in the variable selection literature. One could, for example, look to minimise residual sum of squares of the smoothed modelsf G over the lattice K at each level as with forward selection.
Overall, given the use of geometric methods in statistics and machine learning which exploit symmetry, the methods presented in this paper should allow the full exploitation while also guarding against erroneous over-assumption of symmetries in the data.
References
A. Proofs and further mathematical details
A.1. Proofs in Section 2
The two propositions cited in section 2 have the following proofs.
Proof (Proof of Proposition 2.1). We need only show that K(G) contains supremums and intersections (with respect to the subgroup ordering) of arbitrary pairs G, H ∈ K(G), and that these are given by the join and meet operations in the statement. If G, H ∈ K(G), then consider that G ∩ H is still the infimum as with the subgroup lattice. For the supremum, consider that G, H ≤ ⟨G, H⟩ by definition, so this is an upper bound. If G, H ≤ A ∈ K(G) then ⟨G, H⟩ ≤ A and so ⟨G, H⟩ ≤ A = A, and so ⟨G, H⟩ is in fact the least upper bound and we are done.
Proof (Proof of Proposition 2.5). This follows from exactly the same reasoning as Proposition A.1, except here we need only consider finitely many groups H and have to take closures: i.e.
A = ⟨H ∈ K ∶ f is H-invariant⟩ (A.1)
Since I ∈ K, we know A ≥ I. Since K is a finite lattice, A must also be in K. Lastly note that Lemma A.2 implies that f is A-invariant and so A ≤ G max (f, G).
We also provide other details on related concepts for G-invariant functions.
We say that a subgroup H ≤ G is a maximal invariant subgroup of f if: f is invariant to H and if f is invariant to any other subgroup H ′ ≤ G, then H ′ ≤ H. The next proposition shows that every function f has a unique maximal invariant subgroup.
Proposition A.1. For any group G acting on X , every function f ∶ X → R has a unique maximal invariant subgroup G max .
This result requires a few supporting lemmas.
Lemma A.2. Let G be a group with subgroups G, H ≤ G. Suppose that f ∶ X → R is G-invariant. If H ≤ G then f is H-invariant. If f is H invariant then f is ⟨G, H⟩- invariant.
Proof (Proof of Lemma A.2). The first claim is clear: if f (g ⋅ X) = f (X) for all g ∈ G almost surely then it is true for all g ∈ H ⊆ G. For the second, take x ∈ X and a ∈ ⟨G, H⟩. Express a = a i 1 1 ⋯a i k k , where a j ∈ G or a j ∈ H and i j ∈ N for all j ∈ [k]. Then see that
f (a ⋅ x) = f (a i 1 1 ⋅ (a i 2 2 ⋯a i k k ⋅ X)) = f (a i 2 2 ⋯a i k k ⋅ X) (A.2)
almost surely so the second claim follows by an induction on k.
Lemma A.3. Suppose that f is not G-invariant for some subgroup G ≤ G. Then f is not H-invariant for any H with G ≤ H ≤ G. If f is H invariant then f is not G ′ invariant
for any G ′ with ⟨G ′ , H⟩ = ⟨G, H⟩.
Proof (Proof of Lemma A.3). The first result is just the contraposition of Lemma
A.2. The second part also follows as if f were G ′ invariant, then it would be G ≤ ⟨G ′ , H⟩ = ⟨G, H⟩ invariant.
We can now prove the original proposition.
Proof (Proof of Proposition A.1). Let G max = ⟨H ≤ G ∶ f is H-invariant⟩, noting that f is I-invariant and so there is at least one such H. Then the previous lemma gives the first condition of maximal invariance, and the second is trivial and also implies uniqueness.
Next, when G is a topological group acting continuously on X we can have further structure to G max (f, G).
Lemma A.4. Let G be a topological group. Suppose that f ∶ X → R is continuous and G acts continuously on X . Then f is G-invariant if and only if f is G-invariant. A.4). Since G ⊆ G the forwards direction is clear. Suppose that f is either weakly or strongly G-invariant. Take g ∈ G and a net g • in G converging to g, and note that
Proof (Proof of Lemma
f (g ⋅ X) = f ((lim g • ) ⋅ X) = f (lim g • ⋅ X) = lim f (g • ⋅ X) = lim f (X) = f (X) (A.3)
almost surely, giving the result.
Lastly we require the following result to simplify the sampling distributions when testing for symmetry in section 4.
Lemma A.5. Suppose f ∶ X → R is continuous and G acts continuously on X . If G is topologically generated by a finite set {g 1 , . . . , g k } and f is not G-invariant, then f is not invariant to at least one group generated by a topological generator ⟨g i ⟩.
Proof (Proof of Lemma A.5). Let g ∈ G be such that P(f (g ⋅ X) = f (X)) < 1, and let {h i } ∞ i=1 be a sequence in ⟨g 1 , . . . , g k ⟩ that approximates g. Suppose for the sake of contradiction that f is invariant to every ⟨g i ⟩. Then f (h i ⋅ X) = f (X) almost surely (as h i is a finite product of only g i terms) for every i ∈ N. But then the continuity of f (noting that X is a first countable metric space) implies
f (X) = lim i→∞ f (h i ⋅ X) = f ( lim i→∞ h i ⋅ X) = f (g ⋅ X) (A.4)
almost surely -a contradiction.
A.2. Proofs in Section 3
Proof (Proof of Lemma 3.3). By Lemma A.2 f isĜ B -invariant if and only if we reject all non-invariant subgroups. With the breadth-first estimator, this happens if and only if we reject all non-invariant subgroups that are adjacent to the invariant and above only invariant subgroups. Let A ⊂ L be the set of such subgroups. Then the probability of interest can be expressed as
P(f isĜ B -invariant) = P(∩ H∈A {Test α (D, H) = −1}) (A.5) = 1 − P(∪ H∈A {Test α (D, H) = 1}) (A.6) ≥ 1 − H∈A P(Test α (D, H) = 1) (A.7) ≥ 1 − H∈A (1 − P α ) = 1 − A (1 − P α ) (A.8)
The reasoning for depth first estimation is similar -f is notĜ D -invariant only if we accept an element of A.
Proof (Proof of Lemma 3.4). See thatĜ B = G max if and only if we accept all subgroups of G max and reject all in A, so the result follows from the same argument as in Proposition 3.3.
Proof (Proof of Proposition 3.5). The powers P α ↗ 1 are an increasing sequence in n for each fixed α, and decreasing as α ↘ 0. Simply take some fixed α 0 and set α i = α i−1 until P α i ≥ 1 − α i 2, then set α i+1 = α i 2. Then the result follows from Lemmas 3.3 and 3.4.
Proof (Proof of Lemma 3.2). Since G ≥ H for all H ∈ K, we know that ⟨G, H⟩ = G = G ∈ K ∪ {G}, so we have closure under joins. Similarly, we have that G ∩ H = H ∈ K ⊆ K ∪ {G} so we also have closure under meets.
Proof (Proof of Proposition 3.6). The worst case performance of algorithm 1a is when all tests are accepted. If G is abelian then G ≃ ⊕ k i=1 C p a i i for some prime powers p a i i . Let e i be the natural basis elements of G 0 , which generate subgroups isomorphic (and hence identified with) to C p a i i . Each C p a i i has a i + 1 subgroups, each of the form C p a i for a = 0, 1, . . . , a i . Then there are ∏ k i=1 (a i + 1) distinct subgroups of the form
⟨C p b i i ∶ i ∈ [k]⟩.
Algorithm 1a need only test at the subgroups C p a i , so tests only 1 + ∑ k i=1 a i of these subgroups, exactly ∏ a i fewer. Testing a subgroup of the form
H b ≃ ⟨C p b i i ∶ i ∈ [k]⟩ (using the multi-index b = (b 1 , . . . , b k ) ) requires H β = ∏ k i=1 p b i i units of computation, so testing over all b requires ∑ b ∏ k i=1 p b i i computations.
Testing only the subgroups C p b i i is this sum less the multi indices with more than one b i > 0.
A.3. Proofs in Section 4
The proof of proposition 4.1 relies on several supporting Lemmas.
Lemma A.6. In the context of proposition 4.1, if the support of µ X is dense in X then
d X (g ⋅ X I(1) , X J(1) ) P → 0 (A.9)
Proof. For any η ∈ R ≥0 , the probability that d X (g ⋅ X I(1) , X J(1) ) ≥ η given I, g is equivalent to a binomial variable K with n trials and probability of success p = µ X (X ∖ B(g ⋅ X I(1) , η)) being 0. Since the support of µ X is dense in X , µ X (B(g ⋅ X I(1) , η)) > 0, and so P(d X (g ⋅ X I(1) , X J(1) ) ≥ η) ≤ P(K = 0) = (1 − p) n (A.10)
As n → ∞, this clearly goes to 0 for all η, and for all I, g.
In the following let F X (x) be the distribution function of the real valued random variable X.
Lemma A.7. If X and Y are independent real random variables, and Y is symmetrically
distributed around 0 (so F Y (t) = 1−F Y (−t) for all t)
and admits a density f Y with respect to Lebesgue measure that is decreasing in y . Then
P( X + Y ≥ t) > P( Y ≥ t). Proof. First note P(X + Y ≥ t) = E X (P(Y ≥ t − X X)) = 1 − E X (F Y (t − X)). Thus P( X + Y ≥ t) = P(X + Y ≥ t) + P(X + Y ≤ −t) (A.11) = 2 − E X (F Y (t − X) + F Y (t + X)) (A.12) Let φ t (x) = F Y (t − x) + F Y (t + x).
Since Y is absolutely continuous (w.r.t. Lebesgue measure) with density function f Y , we can see that φ t has a critical point at
x = 0. Moreover, if x > 0 then φ ′ t (x) = f Y (t + x) − f Y (t − x) = f Y (t + x) − f Y (−t + x) < f Y (t + x) − f Y (−t − x) = 0 (A.13) as f Y is decreasing in y . Similarly φ ′ t (x) > 0 if x < 0
. Thus x = 0 is a maximum and so we have
P( X + Y ≥ t) = 2 − E X (F Y (t − X) + F Y (t + X)) ≥ 2 − E X (F Y (t) + F Y (t)) = P( Y ≥ t).
(A.14) as required.
In the following we use the notation X p → x to indicate that the random variable X converges in probability to x.
Lemma A.8. If X ∼ Binom(n, p) and Y ∼ Binom(n, q) with p < q , then F X (Y )
p → 1 Proof. First note that F X (Y ) = FX ( Y −np np(1−p) ) whereX = X−np np(1−p)
, which hasX D → N (0, 1) by the de Moivere -Laplace theorem (DLT). Then we also set:
Y = Y − np np(1 − p) = ⎛ ⎝ Y − nq nq(1 − q) + √ n q − p q(1 − q) ⎞ ⎠ q(1 − q) p(1 − p)
.
(A.15)
Again by DLT, the first term converges to N (0, q(1−q) p(1−p ), but the second term diverges to +∞ (as q > p). ThusỸ p → ∞ and so clearly FX (Ỹ ) p → 1, as required.
We can now return to the proof of Proposition 4.1.
Proof (Proof of Proposition 4.1). Since the support of µ X is dense in X , Lemma 4 gives d X (g ⋅ X I(j) , X J(j) ) p → 0. Now consider
D g j I(j)J(j) = Y I(j) − Y J(j) − V (g ⋅ X I(j) , X J(j) ) (A.16) = f (X I(j) ) + I(j) − f (X J(j) ) − J(j) − V (g ⋅ X I(j) , X J(j) ) (A.17)
We have that V (g ⋅ X I(j) , X J(j) ) → 0 as the distance goes to 0, and also f (X J(j) ) − f (g ⋅ X I(j) ) p → 0. Thus with X as a variable with the same law as each X i and g with the same law as each g j ,
D g j I(j)J(j) D → (f (X) + ) − f (g ⋅ X) − k = φ(g, X) + η (A.18)
where φ(g, X) = f (X) − f (g ⋅ X) and η = l − k . Definitionally, this means
P(D g j I(j)J(j) ≥ t) → P( φ(g, X) + η ≥ t) (A.19)
Under H 0 , φ(g, X) = 0 almost surely, so this probability is given by P 0 t = P( η ≥ t). Under H 1 , φ must take non zero values with some positive probability (using the condition on the distribution µ G ). Thus we can use Lemma A.7 to say that
P 1 t = P( φ(g, X) + η ≥ t H 1 ) > P 0 t (A.20) Let N be large enough that P(D g j I(j)J(j) ≥ t) > (P 1 t − P 0 t ) 2 > P 0 t for all n > N . Now consider that N g t H 1 ∼ Binom(m, P(D g j I(j)J(j) ≥ t H 1 ))
, which is stochastically bounded from below by A ∼ Binom(m, (P 1 t − P 0 t ) 2) for n > N . This gives, using lemma A.8, that
p val H 1 = 1 − F N g t H 0 (N g t H 1 ) ≤ 1 − F N g t H 0 (A) p → 0 (A.21)
as required.
B. Group Theory Definitions
We give here a quick primer on groups -the mathematical structure used to describe symmetries. These definitions are used frequently in this text, particularly in sections 2.2 and 2.3.
A group is set G and an associative binary operation (g, h) ↦ gh ∈ G such that G contains an identity e (such that eg = ge = g) and inverses g −1 for each g ∈ G such that g −1 g = gg −1 = e. We say that a subset H of a group G is a subgroup of G if it is itself a group.
A group encodes the structure of the symmetries of an object through a group action, a function ⋅ ∶ G × X → X that obeys the rules e ⋅ x = x and g ⋅ (h ⋅ x) = (gh) ⋅ x for all g, h ∈ G and x ∈ X . We note that every group G can act on any space X trivially, i.e., where g ⋅ x = x for all g ∈ G and x ∈ X . We say that an action is faithful if g ⋅ x = x for all x only if g = e (or equivalently, every non identity g ∈ G has some point x ∈ X such that g ⋅ x ≠ x). Note that if G acts faithfully then any subgroup of G acts faithfully too.
Let G be a group that acts on X . We have a natural quotient map
[⋅] G ∶ X → X G which takes x to its orbit [x] G = {g ⋅ x ∶ g ∈ G}.
We also sometimes write [n] = {1, . . . , n} and so keep the subscript G for clarity. We can construct a lifting map
ι G ∶ X G → X that has ι G ([x] G ) ∈ [x] G for all x ∈ X . We often write ι G (x) = ι G ([x] G ) for x ∈ X .
Ideally we would like this map to be continuous, though this is often hard to enforce in practice without strong conditions on G. We call ι G (X G) the domain of (G, ⋅) and if ι G is continuous then we call it a fundamental domain. We write ⟨A⟩ for the group generated by the elements of A, i.e., the smallest group containing A.
Let G be a group that is also equiped with a topology such that the map (g, h) ↦ g −1 h is continuous (as a function from the product G × G → G). We call such a group a topological group. When we use homomorphisms of topological groups we mean a continuous homomorphism of the groups. Isomorphisms are homeomorphic homomorphisms. Every group can be given the discrete topology, which we naturally impart to finite groups. We say that a subset S ⊆ G is a topological generator of G if the only closed subgroup of G containing S is G itself (so G = ⟨S⟩).
We say that a group action of a topological group is a continuous group action if the map ⋅ ∶ G × X → X is continuous with respect to the product topology on G × X . This implies that each map g⋅ ∶ X → X is a continuous bijection on X . Every finite group acts continuously with respect to the discrete topology. Note that if a compact G acts continuously and H is a closed subgroup of G then H acts continuously too (seen by considering composition with the continuous inclusion homomorphism φ ∶ H → G with φ(h) = h).
If G is compact then there exists a left (and right) invariant Haar measure on G (Haar, 1933) which we denote Γ G (or just Γ if G is clear from context). This measure can then be normalised to form an analogue of the uniform distribution on a bounded subset of R d (which is in fact the Haar measure on this space if it can be viewed as an additive group, e.g. by identifying [0, 1] d with R d Z d ).
C. Lattice Theory
In section 2.3 we ignored some of the algebraic properties of lattices. Here we give a few more definitions.
Joins and meets are associative, commutative, idempotent binary operations that satisfy the absorption laws a ∧ (a ∨ b) = a and a ∨ (a ∧ b) = a. Here the join is given by the groups generated by unions ⟨A, B⟩ and meet by intersections A ∩ B. We say that A covers B if A > B and there is no C ∈ L with A > C > B. We say that a subset of a lattice L is a sub-lattice of L if it is closed under the joins and meets of the original lattice. Given some set A ⊆ L, we can take ⟨A⟩ as the smallest sub-lattice of L that contains A.
When a lattice L is finite we can enumerate the lattice inductively, setting G 0 = ⋀ G∈L G = inf L and H ∈ G i if H covers some G ∈ G i−1 . Each G i is finite so we can label its elements as G i,j for j ∈ [N i ]. We can then arrange these into a Hasse diagram, with the elements of G i as nodes on level i and edges between nodes A and B if A covers B.
D. Continued Numerical Experiments
Here we give the details of the simulations for the hypothesis tests presented. These are each from the examples in section 5 simulations of both the asymmetric variation test and the permutation variant for low dimensional examples and for the orientation of digits in the MNIST dataset (d = 784).
D.1. Simulations in Low Dimensions
Let d ≥ 2 and set X = R d . Take X i iid ∼ N (0, 4I d ) and i iid ∼ N (0, σ 2 I d ). Consider the functions f d ∶ R d → R be given by (x 1 , x 2 , . . . , x d ) ↦ exp(− x 1 ). Let G = ⟨R π 2 ⟩ act on R d via the distinct actions generated by R π 2 ⋅ x = (−x 2 , x 1 , x 3 , . . . , x d ) and R π 2 ⋆ x = (−x 1 , −x 2 , x 3 , . . . , x d ) = R 2 π 2 ⋅ x. Then we know that f d is (G, ⋆)-invariant but not (G, ⋅)-invariant for all d.
We simulated tests of each of the null hypotheses H {20, 30, 40, 50, 60, 70, 80, 90, 100, 120, 150, 200, 250, 300}. The asymmetric variation tests had t = 2σ = 0.1, p t = 2σ t exp(−t 2 4σ 2 ) √ 2π and the permutation variants had B = 100 and q = 0.95. We tested with g j iid ∼ U (R π 2 , R 2 π 2 , R 3 π 2 ) for all tests.
(1) 0 ∶ f d is (G, ⋆)-invariant and H (2) 0 ∶ f d is (G, ⋅)-invariant,
We see that both tests seem to converge to an empirical power of 1, with the asymmetric variation test more powerful. Both tests have roughly (up to the expected variation from 100 simulation) the correct empirical size, though the permutation variation does appear to be slightly liberal for n ≤ 100.
D.2. MNIST digit orientation test
This is a continuation of example 5.3. We tested each of the datasets D(n) for symmetry of D 4 = ⟨a, b ∶ a 4 = b 2 = e, bab = a −1 ⟩, and for the subgroups ⟨a⟩ of rotations and ⟨b⟩ of horizontal reflections. These are done with g 1 ∼ U ({a, b}), g 2 = a almost surely and g 3 = b almost surely. The number N 0 out of m = 1000 samples that have D g I(j)J(j) > 0 is reported in table 1. Since p t = 0, N 0 > 0 if and only if the p-value is 0 and the hypothesis is rejected. All tests for the invariance of D 4 were correctly rejected, as were all tests for the subgroup ⟨b⟩. All test for the subgroup ⟨a⟩ were accepted. Whether this is correct depends somewhat on the digit and the possible classes the digits can fall into: it is clearly false to rotate the 6 by 180 degrees because that would usually be classed as a 9. However, a rotated 4 can only ever be classed as a four. In either case this is caused by the sparsity in the datasets -the class D(6) contains only 6s and no 9s to compare against, so don't see the lack of invariance. Thus the test is correct in the sense that we have not seen evidence of no ⟨a⟩-invariance.
Note that N 0 is higher for the test for ⟨b⟩-invariance than for D 4 -invariance for two reasons. Firstly, sampling from the generating set {a, b} means that f will be invariant to g 1 about half of the time, as opposed to sampling b almost surely which means that f is never invariant to g 3 . Secondly the distribution of the images µ X is invariant to ⟨b⟩ but not to D 4 . This means that often the transformed g 1 ⋅ X i is not near to any X j , and so the difference Y i − Y j − V (g ⋅ X i , X j ) is low. In contrast g 3 ⋅ X i is usually close to some X j by the construction of the dataset which makes it easier to identify if the bounded variation condition is broken.
− − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − −
Fig. 1 .
1Subfigures (a) and (b) show Cryo-EM maps of representative viruses (Figure 1 in Jiang and Tang
Fig. 3 .
3Hasse diagram of the subgroup lattice of D 4 , the dihedral group of symmetries of a square. Each dot represents a subgroup of G = D 4 , with edge from A up to B if A ≤ B. All finite groups are closed and so this is also the lattice of closed subgroups of D 4
←
Sample({1, . . . , n}) 11: J(j) ← Sample({1, . . . , n}) 12:
and the permutation variants had B = 100 and q = 0.95. We tested with g j iid ∼ U (G ∖ {1}) for all tests of G-invariance.
∼ N ( 0 ,for f 4 Fig. 6 .
046with Asymmetric f ) f 3 (x) = sin(− x 3 ) and X i iid DiagThe proportions of each possibleĜ for plotted against n for both f 2 and f 4 . The proportion ofĜ = I is in red triangles, ofĜ = C 2 = G max in black squares, and ofĜ = C 4 in red circles.
Adams, S. (2020) High-dimensional probability lecture notes.Allen, D. M. (1974) The relationship between variable selection and data agumentation and a method for prediction. technometrics, 16, 125-127.LeCun, Y., Bottou, L.,Bengio, Y. and Haffner, P. (1998) Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86, 2278-2324. Lim, S., Kim, I., Kim, T., Kim, C. and Kim, S. (2019) Fast autoaugment. Advances in Neural Information Processing Systems, 32. Lyle, C., van der Wilk, M., Kwiatkowska, M., Gal, Y. and Bloem-Reddy, B. (2020) On the benefits of invariance in neural networks. arXiv preprint arXiv:2005.00178. Mallows, C. L. (1973) Some comments on cp. Technometrics, 42, 87-94. Mardia, K. V. and Jupp, P. E. (2009) Directional statistics, vol. 494. John Wiley & Sons. Sankappanavar, H. P. and Burris, S. (1981) A course in universal algebra. Graduate Texts Math, 78, 56. Schmidt, R. (2011) Subgroup lattices of groups. In Subgroup Lattices of Groups. de Gruyter. Shorten, C. and Khoshgoftaar, T. M. (2019) A survey on image data augmentation for deep learning. Journal of big data, 6, 1-48. Tanner, M. A. and Wong, W. H. (1987) The calculation of posterior distributions by data augmentation. Journal of the American statistical Association, 82, 528-540. Thébault, E., Finlay, C. C., Beggan, C. D., Alken, P., Aubert, J., Barrois, O., Bertrand, F., Bondar, T., Boness, A., Brocco, L. et al. (2015) International geomagnetic reference field: the 12th generation. Earth, Planets and Space, 67, 1-19. Van Dyk, D. A. and Meng, X.-L. (2001) The art of data augmentation. Journal of Computational and Graphical Statistics, 10, 1-50.
using both tests (algorithms 3 and 4). The estimated power (i.e., proportion of rejections of H (1) 0 ) and empirical sizes (i.e., proportion of rejections of H (2) 0 ) are plotted in figure 11, containing rejection probabilities at significance level α = 0.05. We ran 100 simulations for each combination of n = m ∈
Fig. 7 .
7Mean squared prediction errors for the three estimators, under scenarios 1 and 2 . The average across the 100 simulations is plotted as circles for each of the estimators, with black for the LCE, blue for the model symmetrised with all the data, and red for the symmetrised estimator that split the dataset. Tick marks show the errors for each of the simulations, in the same colour. The first layer of testing. (b) Subsequently deleted nodes (c) Second layer of testing.
Fig. 9 .Fig. 10 .Fig. 11 .
91011The breadth first search algorithm illustrated to find G max (f 5 , D 4 ). Rejections are shown in red, accepted tests in green, and subsequently deleted nodes in blue and edged dotted in blue as well. The nodes are labeled as in example 2.2. Locations of the 98 stations used in the estimation of f F . Rejection proportions for f d for d ∈ {2, 4}. Plotted squares are for the asymmetric variation test, and triangles for the permutation variant. Filled shapes are the estimated power (proportions of rejections of H
Fig. 4. Hasse diagram of the subgroup lattice L(S 1 ), the group of rotational symmetries around a fixed axis. Dashed lines indicate chains of subgroups.Note that the countably infinite dense
subgroups (such as ⟨[1
√
2] S 1 ⟩) have no finite subgroups, only subgroups that are themselves
countably infinite (such as ⟨[2
√
Test 1: procedure PermVarTest(D, µ g , V, q, m, B) for k ∈ {1, . . . , B} do2:
3:
for j ∈ {1, . . . , m} do
4:
Table 1 .
1Calculated N 0 for digit orientation tests.Test for digit: 2 3
4
5
6
7
9
D 4
6 17 15 65 168 58
6
⟨a⟩
0 0
0
0
0
0
0
⟨b⟩
6 47 48 137 285 119 15
). The dashed horizontal line is the significance level α.
Scenario 4Sample SizeMean Squared Prediction
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| [] |
[
"Stroboscopic microwave spectroscopy of Voigt based optically pumped magnetometers",
"Stroboscopic microwave spectroscopy of Voigt based optically pumped magnetometers"
] | [
"Hans Marin Florez \nInstituto de Física\nUniversidade de São Paulo\n05315-970São PauloSPBrazil\n",
"Tadas Pyragius \nSchool of Physics & Astronomy\nUniversity of Nottingham\nNG7 2RDUniversity Park, NottinghamUK\n\nTokamak Energy Ltd\nOX14 4SDMilton Park, OxfordshireUK\n",
"Thomas Fernholz \nSchool of Physics & Astronomy\nUniversity of Nottingham\nNG7 2RDUniversity Park, NottinghamUK\n"
] | [
"Instituto de Física\nUniversidade de São Paulo\n05315-970São PauloSPBrazil",
"School of Physics & Astronomy\nUniversity of Nottingham\nNG7 2RDUniversity Park, NottinghamUK",
"Tokamak Energy Ltd\nOX14 4SDMilton Park, OxfordshireUK",
"School of Physics & Astronomy\nUniversity of Nottingham\nNG7 2RDUniversity Park, NottinghamUK"
] | [] | We present results of stroboscopic microwave spectroscopy of radio-frequency dressed optically pumped magnetometer. Interaction between radio-frequency dressed atoms and a synchronously pulsed microwave field followed by Voigt effect-based optical probing allows us to perform partial state tomography and assess the efficiency of the state preparation process. To theoretically describe the system, we solve the dynamical equation of the density matrix employing Floquet expansion. Our theoretical results are in good agreement with experimental measurements over a wide range of parameters and pumping conditions. Finally, the theoretical and experimental analysis presented in this work can be generalised to other systems involving complex state preparation techniques. | null | [
"https://export.arxiv.org/pdf/2305.00263v1.pdf"
] | 258,426,549 | 2305.00263 | 4d6fddb936da2d0a40707d0f1af0752461bf807b |
Stroboscopic microwave spectroscopy of Voigt based optically pumped magnetometers
Hans Marin Florez
Instituto de Física
Universidade de São Paulo
05315-970São PauloSPBrazil
Tadas Pyragius
School of Physics & Astronomy
University of Nottingham
NG7 2RDUniversity Park, NottinghamUK
Tokamak Energy Ltd
OX14 4SDMilton Park, OxfordshireUK
Thomas Fernholz
School of Physics & Astronomy
University of Nottingham
NG7 2RDUniversity Park, NottinghamUK
Stroboscopic microwave spectroscopy of Voigt based optically pumped magnetometers
(Dated: May 2, 2023)
We present results of stroboscopic microwave spectroscopy of radio-frequency dressed optically pumped magnetometer. Interaction between radio-frequency dressed atoms and a synchronously pulsed microwave field followed by Voigt effect-based optical probing allows us to perform partial state tomography and assess the efficiency of the state preparation process. To theoretically describe the system, we solve the dynamical equation of the density matrix employing Floquet expansion. Our theoretical results are in good agreement with experimental measurements over a wide range of parameters and pumping conditions. Finally, the theoretical and experimental analysis presented in this work can be generalised to other systems involving complex state preparation techniques.
We present results of stroboscopic microwave spectroscopy of radio-frequency dressed optically pumped magnetometer. Interaction between radio-frequency dressed atoms and a synchronously pulsed microwave field followed by Voigt effect-based optical probing allows us to perform partial state tomography and assess the efficiency of the state preparation process. To theoretically describe the system, we solve the dynamical equation of the density matrix employing Floquet expansion. Our theoretical results are in good agreement with experimental measurements over a wide range of parameters and pumping conditions. Finally, the theoretical and experimental analysis presented in this work can be generalised to other systems involving complex state preparation techniques.
I. INTRODUCTION
Atomic vapor based optically pumped magnetometers (OPMs) [1,2] have become the state-of-the-art devices for highly sensitive magnetic field detection reaching fT/ √ Hz sensitivities and beyond. During the last decade OPMs found applications across many domains, from fundamental physics, e.g. in the search for electric dipole moments (EDM) [3,4], in geophysical and space magnetometry, and in medical applications, such as magneto-encephalography (MEG) [5,6] and magnetocardiography [7][8][9]. A commonly used OPM architecture is based on Faraday rotation which further exploits the spin-exchange relaxation-free (SERF) regime [10,11]. In such setups, the atoms are prepared in a spin polarised state, such that in the presence of a magnetic field their precession can be detected by measuring the Faraday rotation through light matter interaction between the atoms and a probe beam. The state preparation is done by circularly polarized pump laser beams reaching a polarizability P ≈ 1 with non-thermal atoms or highly dense samples with quenching gas. In the case of Faraday rotation, the state preparation can be verified by using conventional microwave (mw) spectroscopy [12] with a circularly polarised probe beam. Alternatively, all optical state preparation detection has been demonstrated in cold atoms, but it is not suitable for atomic vapours at room at room temperature, due to the Doppler broadening [13].
A different situation is observed when atoms are driven by radio-frequency (rf) fields like in M x−magnetometers [14], double resonance magnetometer [15] or more recently, as the rf dressed state utilising the Voigt rotation [16]. In a Voigt effect based approach, sensitive field measurements are achieved by preparing an aligned state in the presence of a longitudinal magnetic field and an additional rf dressing field. Such state can, e.g., be prepared as an equal statistical mixture of two stretched states or as a clock state. By the same token, the efficacy of the state preparation process of an aligned state can be probed by standard microwave spectroscopy [17]. However, when atoms are addition-ally driven by a rf field, the Zeeman levels are dressed and the standard selection rules no longer apply. This gives rise to a rich mw spectrum which significantly complicates the inference of the prepared state [18]. Alternatively, one can employ a magneto-optical resonance signal (MORS) to probe the energy level distribution [19]. However, this requires driving the system to a non-linear magnetic regime, which would destroy the target state that is aimed to be prepared.
Inspired by previous work in [18], we propose a method of stroboscopic microwave spectroscopy which enables the probing of the population distributions of the prepared atomic states in a dressed scenario in a linear magnetic regime and via Voigt rotation. Furthermore, we demonstrate how the synchronous detection allows to measure the subtle differences in the state preparation with the use of a repump beam. Building upon our previous theoretical work employing the Floquet expansion on second order moments, we extend our theoretical analysis to the density matrix in order to calculate the microwave spectra [21]. Using this technique along with our theoretical model, we are able to determine our target state and estimate the efficiency of the state preparation process reaching more than 90%. We find that our theoretical description is effective in understanding of the experimental results.
The paper is organised as follows. In Section II we study briefly describe the spin dynamics and introduce the Floquet expansion to describe the microwave spectroscopy. Section III presents the experimental setup and the experimental sequence used to realize the synchronous microwave spectroscopy as well as the results for the synchronous microwave spectroscopy of a Voigt effect based OPM. Section IV shows the experimental and theoretical results on the synchronous microwave spectroscopy for Voigt effect based OPMs. Section V presents our conclusions.
II. MICROWAVE SPECTROSCOPY PROBED BY VOIGT ROTATION
The experiments carried out in this work are based on a Voigt effect OPM using radio-frequency dressed states (see ref. [16] for further details). The light-matter interaction for such a system is described by the Stokes parameter
Ŝ z (t) = Ŝ z (t) + G (2) F S y n F F 2 x (t) −F 2 y (t) ,(1)whereŜ z = (c/2)(â † +â+ −â † −â− ) andŜ y = (c/2)(iâ † −â+ − iâ †
+â− ) represent the photon flux differences in a circular and a 45°polarization basis; G (k) F is the rank-k coupling strength and n F are the atoms within the same F -manifold state [22]. In order to describe the dynamics of bi-linear operators of the formF 2 i (t), we adapt the Floquet expansion implemented for the Heisenberg-Langevin description [21], to the density matrix description which can incorporate the microwave interaction more conveniently at the expense of having a larger Hilbert space.
In order to theoretically describe the experimental dynamics, we consider alkali atoms in the ground state n S 1/2 , in which the orbital angular momentum is L = 0, hence one can make an approximationĴ =L +Ŝ ≈Ŝ. In particular, (without loss of generality), we consider 87 Rb atoms with hyperfine ground states F = 1 and F = 2, in which the bare atom Hamiltonian is given byĤ 0 = F E FÎF , where the partial identity operator is defined asÎ F = F m F =−F |F, m F F, m F |. In addition, we consider the interaction of the atoms in the ground state with a microwave field [18] (fast field) and a radiofrequency field (slow field).
We first consider the interaction with the fast field, neglecting the nuclear magnetic moment since it is much smaller than the electronic one, i.e. g I µ B
g S µ B . Thus, the microwave interaction Hamiltonian reduces tô
H mw (t) = µ B g S Ŝ · B mw (t),(2)
where the microwave field is described classically in the following form
B mw (t) =B mw e −iωmwt +B * mw e iωmwt ,(3)
with polarizationB mw and frequency ω mw . The microwave field can be expressed in the spherical basis as B mw = β 0 mw e 0 + β + mw e + + β − mw e − , where e 0 = e z and e ± = ∓(e x + ie y )/ √ 2, and the magnetic field amplitudes are β 0 mw = β mwz and β ± mw = ∓(β mwx ± iβ mwy ). In the Appendix A 1 we show that the dynamics of the density matrix elements interacting with the microwave field under the rotating wave approximation is given by the Liouville equation dρ
dt = i [ρ,Ĥ eff mw ], where we have definedĤ eff mw =Ĥ eff 0 +Ĥ S witĥ H eff 0 = ∆ m2 |2, m 2 2, m 2 |,(4)
where ∆ corresponds to the frequency detuning of the microwave with respect to the hyperfine clock transition, whereas the microwave field Hamiltonian can be expressed aŝ
H S = Ω πŜz + Ω σ+Ŝσ+ + Ω σ−Ŝσ− ,(5)
where the new spin operators are defined in eqs.(A13-A15). The dynamics describe the Rabi oscillations induced by the microwave field coupling the two hyperfine ground states of alkali atoms.
Let us now consider the interaction with the static field B dc and the rf dressing field B rf , the slow field interaction isĤ B = (µ B g F / )F · B, where µ B and g F correspond to the Bohr magnetron and g F -factor, respectively. The total magnetic fields is
B = (B rf cos ω rf t+B ext x )e x +B ext y e y +(B dc +B ext z )e z ,(6)
where ω rf corresponds to the frequency of the rf field and we consider the general case in which external fields B ext = (B ext
x , B ext y , B ext z ) are present. Hence, the total coherent dynamics is given byĤ T (t) =Ĥ eff MW +Ĥ B (t). Including the pumping rate Γ p (t) which allows the state preparation with an arbitrary time profile and the relaxation rate γ due to collisions, the total dynamics can be described by
dρ dt = i [ρ,Ĥ T (t)] − Γ p (t)(ρ − ρ in ) − γ(ρ − ρ 0 ). (7)
defined in Appendix A 2. Solving the dynamics in a general situation in which the pumping rate and the microwave fields are modulated with an arbitrary time profile at a frequency ω rf , is not straightforward. However, by applying the Floquet expansion to the density operator, as in ref. [21], it is possible to find a steady state solution. Considering a harmonic expansion of the density matrix elements as ρ ij (t) = n ρ (n) ij (t)e inω rf t , with n ∈ Z, it is possible to find the steady state solution of the harmonics ρ (n) ij (t), which is detailed described in Appendix A 3.
III. EXPERIMENTAL SETUP
Our magnetically unshielded experimental setup is depicted in Fig. 1 and its detailed description is presented in the Appendix B. It follows closely the shielded version of the experimental setup described in ref. [16]. A paraffin coated 87 Rb enriched vapour cell at room temperature. The atoms are dressed with a ω rf = 2π × 90 kHz rf field along the x direction and coupled to a static field in the longitudinal direction. Three pairs of Helmholtz coils are used to actively compensate and stabilise the external magnetic field. The atomic state is prepared by a pump laser beam tuned to the F = 2 to F = 1 transition of the D1 line. The atoms are stroboscopically pumped by modulating the pump amplitude with 10% duty cycle in phase with the rf field. To probe the state precession, we employ the Voigt rotation for non-destructive measurements in hyperfine ground states F = 1 and F = 2. The light is detected on a balanced photodetector where the detected signal u(t) = g el S z (t) is proportional to the ellipticity in eq. (A32) and the electronic gain g el . As it was shown in ref. [16], the ellipticity for the Voigt rotation produces a signal at the first and second harmonic of the radio-frequency dressing field such that u(t) = m 0 + m 1 e iω rf t + m 2 e 2iω rf t + c.c. This output signal is demodulated at the second harmonic, from which we can extract its mode amplitude m 2 . The microwave spectroscopy is performed with the magnetometer tuned on resonance, B z = ω rf /µ B g F using the second RF harmonic in the Voigt rotation signal, cancelling the presence of any transverse field, i.e. zeroing the first harmonic [16]. The direction of the dipole antenna is transversal to the light propagation.
IV. RESULTS
We first perform a continuous microwave spectroscopy during relaxation dynamics of the prepared state, as it is shown in Fig. 2 (a). For a microwave field composed of σ ± − and π− polarizations applied to F = 1 → F = 2 in the presence of an external static magnetic field, there are a total of 9 possible microwave transitions, 4 of which are degenerate, see Fig. 1 (a). This results in 7 distinct resonance peaks in the undressed microwave spectrum when the microwave field frequency is scanned (as- suming a thermal state in each manifold), separated by ∆ω = µ B g F B z / . If the state is polarised, e.g. we have a stretched state, then only one transition will be observed. If, as in our case, an equal statistical mixture |F = 2, m F = ±2 is prepared, then the two extreme transitions should be observed. Now, in the case of rf dressed states, the spectrum includes new transitions in the dressed basis. Figure 3 shows theoretical and experimental rf-dressed microwave spectra of the Re(m 2 ) demodulated signal amplitude for F = 2 manifolds for two differently prepared states. As described in [18], the general structure of these spectra comprises seven groups of up to seven lines. Different groups are addressed by different mw polarizations, but the complete set arises here due to the inhomogeneous field distribution from the dipole antenna. The group structure resembles that of bare transitions, alternating between π and σ-polarisations, with a spacing given the static field strength. Lines within a group are spaced by ∼ Ω 2 rf + ∆ 2 . Since in our case the rf is in resonance with the Zeeman levels, ∆ = 0, the line spacing measures the rf amplitude, Ω rf . This can be used to obtain absolute field values for both the static and rf fields which provides a method for absolute ac and dc field calibrations. Fig. 3 (a) shows data with repump light, intended to prepare a mixture of the two rf-dressed, fully stretched states. Here, population that leaks into the F = 1 manifold is redistributed into the F = 2 manifold, where it interacts with the pump beam. This results in a zero birefringence signal in the F = 1 manifold and non-zero birefringence in the F = 2 manifold, which is seen as a large offset in the data. The prepared state is close to maximal birefringence and any coupling of the F = 1 and F = 2 manifolds by resonant microwave transitions alters state populations and becomes visible mostly as spectral dips. However, the rf-dressed microwave spectrum possesses a complex and difficult to interpret structure compared to a bare spectrum. This is because the rf radiation provides additional coupling within the Zeeman states corresponding to coherences in the density matrix which contain information about the correlations between any two given populations. If the mw field is turned on all the time, then the evolution of the mw field and the state is averaged, i.e. different relative orientations between the cones end up contributing to the overall spectrum, as it is shown in Fig. 2 (b). The coupling of the mw field from one manifold to the other probes the orientation of those two cones relative to each other as the mw field frequency is continuously scanned. This allows the probing of all the possible transitions resulting in a dense forest of lines, rendering the standard selection rules obsolete. For example, coupling |F = 1, m F = −1 and |F = 2, m F = 2 is possible. Hence, addition of the rf dressing field gives rise to partially degenerate multiphoton transitions, enabling the coupling between any two given states [18]. Now, let us consider the situation in Fig. 3 (b), which shows microwave spectra of the OPM without the repump, for the Voigt rotation in F = 2. If the repump is turned off a fraction of the population is pumped into F = 1 and a another fraction remains in F = 2, where the pump beam drives the atoms to an aligned state with lower efficiency. Unlike in the previous case, the F = 1 manifold is now populated, but is in a near-thermal state with vanishing birefringence, which strongly diminishes the Re(m 2 ) offset amplitude. Nevertheless, here the microwave coupling between the two ground-state manifolds enables a 2-way population exchange, which is distinct to the previous case. The mw transitions may now change birefringence in either sense. In Appendix C, we also present mw spectra for F = 1 comparing the cases with and without repump light showing that this key difference enables one to distinguish between a completely empty and thermally populated manifolds which otherwise posses identical zero birefringence.
For the further discussion, we focus on the F = 2 manifold where we prepare the atomic state. In this case the theoretical mw spectrum in orange displays similar structure as the experimental one, with some differences. In particular, for the case of active repump in Fig. 3 (a), curve (1) shows that the two extreme groups are clearly different from the the experimental observations, whereas for the case of no repump, the same groups are properly described. Since in the first case the F = 2 population is higher than in the second case, propagation effects can be present during the state preparation process. Hence, if one assumes an effective rotation of the aligned state around the static field z-axis due to the propagation effects, curve (2) is now in agreement with experimental observations. This is not the case for spectrum in Fig. 3 (b), where applying the same rotation effect does not change the extreme group profiles but instead modifies the centre group. This indicates that for no repump, the atomic F = 2 population is low compared to the case with active repump, and no propagation effects have to be considered.
Although, from the theoretical model the prepared state used to obtain the spectrum corresponds to an aligned state, in both cases of Fig. 3, it is not straightforward to infer the population distributions in the rf dressed picture, i.e. the quasi-energy eigenstates. As a result, the need for an alternative probing scheme is required. Here we propose a rf-synchronised stroboscopic mw probing scheme in order to map the population distributions in the presence of an rf dressing field. Figure 4 (a) shows the results of pulsed microwave spectrum of Re(m 2 ) probing the F = 2 ground state manifold, for the situation analysed in Fig. 3 when the repump is active. Here the microwave field is pulsed with the same duty cycle of 10% and in phase with the radiofrequency dressing field. The experimental stroboscopic microwave spectrum shows mainly two peaks per group which lie on the two extreme edges. However, some of the manifold populations leak into the inner levels. On the other hand, the theoretical calculation shows that, when the system is prepared in an equal statistical mixture of the stretched states and then probed stroboscopically using mw radiation in phase with the dressing rf field, the atomic populations perfectly distributed into two extrema peaks in the dressed microwave spectrum repeated over the 7 groups [? ]. This is expected for an extremal aligned state, in which the majority of the populations reside in the Zeeman sublevels of the |F = 2, m F = ±2 mixed state along the x-axis. Now, the inner peaks detected on the mw scan can be described by taking into account the propagation effect as in Fig. 3. Notice that if we consider some effective rotation due to the propagation effects in the theoretical calculation, the resultant theoretical spectrum matches the experimental observations, showing the presence of the inner peaks. This indicates that without propagation effects our state preparation process can in principle reach near perfect efficiency with the atoms ending up in the target state. However, when the propagation effects are considered, the aligned state is rotated, thus reducing the preparation efficiency to ∼ 70% of the mixture of the two stretched states.
This stroboscopic mw spectrum measurement can be described as probing the states at a certain orientation in their evolution as they precess around the static and rf fields, see Fig. 5. As a result, when the mw field is pulsed and its frequency scanned, only a certain part of the cone configuration is probed and so the spectrum acquires only the shape of the corresponding bare spectrum, or rather, the spectrum is obtained based on a single orientation of the cones.
To further test the efficacy of our theoretical model, we consider the situation where the the pump couples the F = 2 → F = 2 transition which prepares the dressed "clock" state, |F = 2, m = 0 x . In this case, one can notice that the state preparation does not just produce the clock state, but a distribution within the manifold. Once again, the theoretical model shows that if the input state corresponds to a clock state in the x-direction, the stroboscopic spectrum should be mainly determined by the clock state. However, when considering the propagation effects, the model predicts the population distribution to the one observed experimentally. In addition to these observations, we show in the Appendix C that the experimental and theoretical mw spectra in the CW and stroboscopic limits are also in agreement when probing the F = 1 manifold, when the propagation effects are considered.
Details of our theoretical model are given Appendix A. Extending the model in Ref. [18], which describes isolated atomic systems, this model describes an atomic medium with pumping and relaxation rates. So far we have just discussed the mw spectra when no transverse static fields are applied, focusing on the high amplitude of the sec-FIG. 6. Theoretical dressed microwave spectra of the Re(m2) mode amplitude as a function of the microwave frequency detuning. Here the measured and calculated spectra are obtained based on the sequence in Fig. 5 where the mw interaction is stroboscopic and no propagation effect is considered. ond harmonic. However, the model can predict also the CW and the pulsed mw spectrum for the first and higher harmonics in the presence of transverse fields, contained in the magnetic interactionĤ B (t) of the total Hamiltonian in Eq. (7). Therefore, the model is quite robust in being able to describe the mw spectroscopy of the spins in different scenarios.
In order to experimentally overcome the propagation effects, a shorter cell can be used to reduce the back action of the pump for relatively high atomic populations. This would induce smaller effective rotations in the beam polarisation. Nevertheless, this theoretical model allows us to have satisfactory predictions of the expected stroboscopic spectra and their relation to the prepared input state.
An additional comparison can be made with the help of the theoretical model. A synchronously dressed oriented state in an rf magnetic field yields atomically modulated birefringence at twice the modulation frequency of the dressing field analogous to atomically modulated birefringence observed in clock or aligned states [21]. Figure 6 shows the pulsed mw spectra for the three distinct states considering the ideal case, where propagation induced rotation effects are negligible. Notice that the stroboscopic spectroscopy can give access to atomic population distributions according to each input state e.g. two extreme peaks for the mixture |F = 2, m F = ±2 , the centre peak for the clock state |F = 2, m F = 0 and one extreme peak for the oriented state |F = 2, m F = 2 . This indicates that, under negligible propagation effects, the stroboscopic mw spectroscopy can be used to measure the atomic population distributions when atoms are dressed by rf-fields.
V. CONCLUSIONS
We have presented a novel method of microwave spectroscopy of radio frequency dressed states applied to a Voigt effect based optically pumped magnetometer. In the case of continuous microwave interaction, we have shown the violation of standard selection rules manifesting in multiple microwave transitions among the rf dressed states in the hyperfine ground state manifold. We have demonstrated that it is possible to overcome the complex interpretation of the mw spectrum in CW configuration, by changing the mw interaction profile to a rf synchronised stroboscopic microwave pulsing. Applying this technique, we have shown that one can model the process of an efficient state preparation of an extremal aligned, clock and oriented state driven by a combination rf fields and as well as optical pump and repump beams. Furthermore, the method of state probing outline enables us to distinguish between thermal an empty manifold populations which otherwise posses zero birefringence. Finally, our proposed approach can be easily extended to radio-frequency dressed Faraday measurements to map the state population distributions, as well as any other dynamical state preparation processes. This work paves the way towards a new form of dressed state detection and mapping. In this section we present the theoretical description of the evolution of an atomic spin interacting with microwave and radio frequency magnetic fields with decoherence effects governed by Langevin dynamics.
VI. ACKNOWLEDGEMENTS
Spin dynamics with microwave fields
Consider 87 Rb atoms with hyperfine ground states F = 1 and F = 2. The free Hamiltonian is given bŷ
H 0 = F E FÎF ,(A1)
where the partial identity operator is defined aŝ
I F = F m F =−F |F, m F F, m F |. (A2)
In addition, we consider the interaction of the atoms in a ground state with a microwave field given by [18] H mw (t) = µ B g IÎ + g JĴ · B mw (t),
where g I and g J are the nuclear and electronic g-factors, respectively, with the corresponding angular momentum operatorsÎ andĴ. In particular, for alkali atoms, for the ground state n S 1/2 the orbital angular momentum is L = 0, hence one can considerĴ =L +Ŝ ≈Ŝ. Moreover, g I g S , which allows us to neglect the nuclear term. Thus, the microwave interaction Hamiltonian reduces tô
H mw (t) = µ B g S Ŝ · B mw (t),(A4)
where the microwave field is described classically in the following form
B mw (t) =B mw e −iωmwt +B * mw e iωmwt ,(A5)
with polarizationB mw and frequency ω mw . The microwave field can be expressed in the spherical basis as
B mw = β 0 mw e 0 + β + mw e + + β − mw e − ,(A6)
where e 0 = e z and e ± = ∓(e x + ie y )/ √ 2, and the magnetic field amplitudes are β 0 mw = β mw z and β ± mw = ∓(β mw x ± iβ mw y ). Therefore, the dynamics given by the Liouville equation is
dρ dt = i [ρ,Ĥ 0 +Ĥ mw (t)],(A7)
in which its elements are ρ F m,F m = F m|ρ|F m , where F, F ∈ {1, 2}. Transforming into the rotating frameρ F m,F m = ρ F m,F m e i(F −F )ωmwt , we show that as in manifold two level optical system [20], can be expressed in the rotating frame as
dρ αmα,βm β dt = i(α − β)∆ρ αmα,βm β + iδ F β m,q Ω q F m,F m βρ αmαF m − iδ αF m ,q Ω q F mα,F m ρ F m βm β + iδ F β m ,q Ω q F m ,F m βρ αmαF m − iδ αF m,q Ω q F mα,F mρ F mβm β ,(A8)
where we have defined the microwave detuning as ∆ = E 2 − E 1 − ω mw and the Rabi frequencies as
Ω q F m,F m = µ B g S µ q F ma,F mb β q mw ,(A9)
with µ q F m,F mF = F, m F |Ŝ q |F , m F . The dynamics can be written in terms of an effective Hamiltonian as
dρ dt = i [ρ,Ĥ eff mw ],(A10)
where we have definedĤ eff mw =Ĥ eff 0 +Ĥ S witĥ
H eff 0 = − ,(A11)
whereas the microwave field Hamiltonian can be expressed aŝ
H S = Ω πŜz + Ω σ+Ŝσ+ + Ω σ−Ŝσ− ,(A12)
where longitudinal spin operator iŝ
S z = ,(A13)
the circular positive operator iŝ
S σ+ = 0 0 0 0 0 ,(A14)
and the negative circular operator iŝ
S σ− = . (A15)
The dynamics describe the Rabi oscillations induced by the microwave field coupling the two hyperfine ground states of alkali atoms.
Spin dynamics with the radio-frequency fields
Now we add the interaction of atoms with a radiofrequency dressing field and external static fields
B = (B rf cos ω rf t + B ext x )e x + B ext y e y + (B dc + B ext z )e z ,(A16)
where B rf and ω r are the amplitude and frequency of the rf driving field and B dc is the static field along the longitudinal direction which are experimentally controlled. Additionally, we have the external fields B ext i with i = x, y and z, which originate from external sources. Therefore, the magnetic field interaction is given byĤ B = (µ B g F / )F · B, where µ B and g F correspond to the Bohr magnetron and g F -factor, respectively, such thatĤ
B (t) = (Ω rf cos(ω rf t) + Ω ext x )F x + Ω ext yFy + (Ω dc + Ω ext z )F z ,(A17)
with g F = g F / and Ω i = µ B g F B i with i = rf, dc, x, y and z. Therefore, the total interaction of the atoms with the microwave and the external magnetic fields dressed by the rf is described by the Hamiltonian
H T (t) =Ĥ eff mw +Ĥ B (t).(A18)
In addition to this term that contributes to the coherent dynamics, we need to include the relaxation term which models atom-atom collisions and atom-wall collisions, as well as the pumping rate which describes the state preparation by an pump beam. To do so, we introduce the standard input-output dynamics such that
dρ dt = i [ρ,Ĥ T (t)] − Γ p (t)(ρ − ρ in ) − γ(ρ − ρ 0 ), (A19)
where Γ p (t) correspond to the pumping rate, which in general can be time dependent (for instance the Amplitude modulation employed to do the state preparation), and γ represents the relaxation rate due to collisions. For the 8 levels of the two hyperfine ground states, one can describe the dynamics above in the Liouville space, as in the case of ref. [21], by defining X = (ρ 11 ,ρ 12 , · · · ,ρ 88 ) with dimension d ρ = 64, such that
dX(t) dt =(M(t) − Γ p (t) − γ) X(t) + Γ p (t)X in + γX 0 ,(A20)
where
M(t) =L[Ĥ eff mw +Ĥ B (t)] + R[Ĥ eff mw +Ĥ B (t)], (A21)
in which L(Ô) and R(Ô) represents the action of the operatorÔ to the left and to the right, respectively. The dynamics of the radio-frequency dressed states are dominated by the time dependence of driving rf field M(t) and the amplitude modulation of the synchronous pumping determined by Γ p (t). On one hand, since the magnetic field in eq. (A16) can be harmonically decomposed as B(t) = B (0) +B (1) e iω rf t +B (−1) e −iω rf t , therefore the magnetic interaction in eq. (A17) can be also written as
M(t) =M (0) + M (1) e iω rf t + M (−1) e −iω rf t .(A22)
Similarly, when the pump amplitude is modulated; for instance as a square-wave signal, the the pumping rate is decomposed as
Γ p (t) = Γ (0) p +Γ (1) p e iω rf t + Γ (−1) p e −iω rf t +Γ (2) p e 2iω rf t + Γ (−2) p e −2iω rf t + · · · , (A23)
such that
Γ (0) p = Γ b d, Γ (n) p = Γ (−n) = Γ b nπ sin(nπd),(A24)
where d corresponds to the duty cycle of the carrier wave. The general description in eq. (A23) can simulate a broad range of time dependent pumping rates with different spectral decomposition e.g. sine, sawtooth etc. This harmonic decomposition of the dynamics generator, leads to a Floquet expansion to obtain a stationary solutions of its harmonics [21]. In next section we present a brief description of the Floquet expansion in order to find the solution of the atomic dynamics.
Floquet expansion
Given the harmonic feature of the dynamics generator, one can assume that the solution of the atomic state can be expressed in terms of the Floquet expansion X(t) = n X (n) (t)e inω rf t so that one has to determine the amplitude of the harmonics X (n) (t) to obtain the complete solution. Substituting this expansion into eq. (A20), yields the recursive formula that can be written in the matrix form as
dX F (t) dt =[C − Γ] X F + Γ in X in + Λ rel X 0 ,(A25)
where X F = (X (−Q) , · · · , X (−n) , · · · , X (−1) , X (0) , X (1) , · · · , X (n) , · · · , X (Q) ) T correspond to the vector of harmonic amplitudes of the spin, with cutting frequency Q and the dynamics generators arẽ
C nm = C (0) − inω rf I, for n = m, C (±1) , for m = n ∓ 1 0, otherwise,(A26)
whereas the pump matrix elements (Γ) nm = Γ (n−m) p I 3×3 and
(Γ in ) nm = Γ (n) p I 3×3 , for n = m, 0, otherwise,(A27)
with the input vector (P in ) n = P in . The magnetometer microwave spectrum has three stages: synchronous pumping, microwave interaction and probing, which are shown schematically in Fig. 7. Each FIG. 7. Sequence to produce the mw spectroscopy. The sequence is composed of three separate stages. First the process is initiated by the state preparation process where the rf-dressed atoms are synchronously pumped to prepare an aligned state. This is then followed by switching all of the optical fields off and applying a short mw pulsed tuned to the ground hyperfine splitting to induce a population transfer. The atomic dynamics are then probed by an off-resonant probe beam to measure the Voigt effect.
sequence is characterised by the following equation of motion
dX F (t) dt (A28) = [C 0 − Γ T ] X F + Γ in X in + Λ rel X 0 , Pump Cycle, [C − Λ rel ] X F + Λ rel X 0 , MW Cycle, [C 0 − Λ rel ] X F + Λ rel X 0 , Probe Cycle,
where we have definedC 0 =C(H mw = 0). During the pumping stage, as before, a steady state is reached. This is given by
X Pump F = − [C 0 − Γ T ] −1 (Γ in X in + Λ rel X 0 ).(A29)
After reaching the steady state, the pump pulse is switched off, which is immediately followed by an application of a microwave pule. During microwave cycle, the time evolution is
X mw F (t) = e (Cmw−Λ rel )t X Pump F (A30) + (C mw − Λ rel ) −1 (e (Cmw−Λ rel )t − I)Λ rel X 0 .
If the microwave is continuously interacting for an integer number of rf cycles t mw = n/ω rf , then the state at a later time is simply X mw F (t mw ). After the microwave pulse, a probe pulse is applied for a given time duration t Probe . During the probing, the solution is
X Probe F (t) = e (C−Λ rel )t X mw F (t mw ) + (C − Λ rel ) −1 (e (C−Λ rel )t − I)Λ rel X 0 . (A31)
The equation above is the solution from which we can determine the slow varying envelopes for the Voigt rotation
Ŝ z (t) = Ŝ z (t) + G(2)F S y n F F 2 x (t) −F 2 y (t) ,(A32)
with its dynamics described by the second harmonic in the limit when no transverse fields are applied [16]. According to eq. (A32), when the probe has no initial ellipticity, the Voigt rotation for the ground state F = 1 and
F = 2 are Ŝ z (t) F =1 = G (2) F S y n F [ρ 13 (t) + ρ 31 (t)],(A33)Ŝ z (t) F =2 = G (2) F S y n F √ 6[ρ 46 (t) + ρ 64 (t)] + √ 6[ρ 68 (t) + ρ 86 (t)] +3[ρ 75 (t) + ρ 57 (t)] ,(A34)
where each term has a spectral expansion of the form
ρ ij (t) = ρ (0) ij (t) + ρ (1) ij (t)e iω rf t + ρ (2)
ij (t)e 2iω rf t + · · · + c.c.. Thus, by extracting the mode amplitude of the second harmonic in eq. (A31)
X (2) R = 1 2T T 0 dt (X Probe F (t )) 2 + (X Probe F (t )) −2 ,(A35)
one can express the ellipticities in eq. (A34) in terms of the solution in eq. (A35), since the second harmonic matrix elements are ρ
ij (t) = [X (2) R ] dρ×i+j .
Appendix B: Experimental setup
Here we present some more details of our magnetically unshielded experimental setup is depicted in Fig. 1 (a). A paraffin coated 87 Rb enriched vapour cell of diameter d = 26 mm and length l = 75 mm at room temperature, with a density of approximately 10 10 atoms per cubic cm. The atomic state is dressed with a radio-frequency field which is generated by cosine-theta coil along x-axis. The atoms are dressed with a ω rf = 2π × 90 kHz rf field and and coupled to a static field in the longitudinal direction. Three Helmholtz coils are used to actively compensate and stabilise the external magnetic field using an analog three channel PID controller driving a home made bipolar current source. The in-loop field is sensed by a three axis fluxgate magnetometer (Stefan-Mayer FLC3-70). The atomic state is prepared by a combination of co-propagating linearly polarized pump laser beam tuned to the F = 2 to F = 1 transition of the D1 line and a linearly polarized repump laser tuned to F = 1 to F = 2 of the D2 line. The atoms are stroboscopically pumped by modulating the pump amplitude with 10% duty cycle in phase with the rf field with the repump set in cw mode.
For a complete characterization of the atomic state, we employ the Voigt rotation as non-destructive measurement to probe the atoms in hyperfine ground state F = 1 and F = 2. To do so, the atoms are probed by a 45 • polarized laser beam relative to the pump polarization and tuned −500 MHz with respect to the F = 2 to F = 1 transition and F = 1 to F = 1 transition of the D1 line. After the interaction with the atoms, the light passes through a quarter waveplate and a polarizing beam splitter, which allows us to measure the Voigt rotation of the light polarization i.e. the ellipticity induced by the atoms. The light is detected on a balanced photodetector where the detected signal u(t) = g el S z (t) is proportional to the ellipticity in eq. (A32) and the electronic gain g el . As it was shown in ref. [16], the ellipticity for the Voigt rotation produces a signal at the first and second harmonic of the radio-frequency dressing field such that u(t) = m 0 + m 1 e iω rf t + m 2 e 2iω rf t + c.c. This output signal is demodulated at the second harmonic using a home built IQ demodulator, from which we can extract its mode amplitude m 2 .
The microwave spectroscopy is performed when the magnetometer is tuned on the resonance, B z = ω rf /µ B g F , of the second harmonic of the Voigt rotation, cancelling the presence of any transverse field i.e. zeroing the first harmonic [16]. The microwave field is generated using an rf generator (SRS SG380) with a frequency doubler (Minicircuits ZX90-2-36-S+) and an additional amplifier. The signal is coupled into a half-wave dipole antenna L = λ/2 = 21.9 mm where L is the conductor length and λ corresponds to |F = 1 → |F = 2 hyperfine microwave transition. The direction of the dipole antenna is transverse to the light propagation. Complementing the observations of the mw spectroscopy when we probed the state F = 2, in this section we present the mw spectroscopy when the F = 1 manifold is probed. Figure 8(a) shows the mw spectrum when the repump is ON during the pumping process. Figure (b) shows the case when the repump is OFF such that the birefringent signal is acquires opposite polarity, since the residual population in F = 1 can now be excited to F = 2 ground state. It is worth noting that the birefringent signal mean value is around zero since its population is empty and initially prepared in F = 2.
Similarly to the case of probing F = 2, in this case the propagation effects are also present in the pump-atom interaction , degrading the efficiency the state preparation. Notice that in the case when the repump is ON, the extreme groups describes the populations observed experimentally, whereas for the case of no repump, the correction with the rotation does not appear to change the profile. Now, implementing the synchronously mw spectroscopy, one can notice the appearance mainly of the extreme peaks of each group. However the contamination due to the propagation effects are noticeable, consistent with what is observed for F = 2.
Although the mw spectra for F = 2 have better signal-to-noise ratio, this observations show that the state characterisation can be done in both manifolds F = 1 or F = 2. Furthermore, the model allows to explore differ-ent scenarios, for instance, the mw spectroscopy of with arbitrary mw polarisation and pump amplitude modulations.
Zeeman states of 87 Rb and polarisations of mw transitions. (b) Sketch of the experimental setup.
FIG. 2 .
2Illustration of the experimental sequence and the microwave coupling of the ground state manifolds in the continuous-wave (CW) model. The sequence consists of three separate stages; state preparation, microwave interaction and probing. The states in each manifold rotate in opposite directions due to the different gF -factors. The microwave radiation transfers the state populations between the two energy levels at different cone orientations during the evolution.
FIG. 3 .
3Experimental and theoretical dressed microwave spectra of the Re(m2) mode amplitude as a function of the microwave frequency. Here the measured and calculated spectra are obtained based on the sequence inFig. 2.(a) Probe F = 2, mw CW, repump ON. (b) Probe F = 2, mw CW, repump OFF. Here the theoretical mw is applied in a CW mode over 20 rf periods, T = 2π/ω rf . In both cases we considered an effective rotation of ∼ 35 • .
FIG. 4 .
4Experimental and theoretical dressed microwave spectra of the Re(m2) mode amplitude as a function of the microwave detuning with respect to the clock transition. Here, the measured and calculated spectra are obtained based on the sequence in Fig. 5 where the mw is pulsed. (a) Extremal aligned state with repump ON. (b) Clock state with repump ON. In both cases we considered an effective rotation of ∼ 35 • .
FIG. 5 .
5Illustration of the experimental sequence and the the microwave coupling of the ground state manifolds during stroboscopic microwave probing. Here the states in each manifold are probed only at a specific time (stroboscopic microwave interaction) which means that the population transfer between the two levels happens only at a certain orientation between the two cones. The microwave pulses are in phase with the rf field.
This work was funded by Engineering and Physical Sciences Research Council (EP/M013294/1) and by Grant No. 2018/03155-9 São Paulo Research Foundation (FAPESP). Appendix A: Theoretical description of the spin dynamics with time dependent magnetic fields
FIG. 8 .
8Experimental and theoretical dressed microwave spectra of the Re(m2) mode amplitude probing the F = 1 manyfold following the sequence in Fig. 2. (a) Repump ON. (b) Repump OFF. In both cases we considered an effective rotation of 35 • .
FIG. 9 .
9Experimental and theoretical dressed microwave spectra of the Re(m2) mode amplitude for sinchronously mw interaction. (a) Aligned state. (b) Clock state. In both cases we considered an effective rotation of 35 • . Appendix C: Micro-wave spectroscopy probing F = 1 state
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| [] |
[
"Structure properties of even-even actinides",
"Structure properties of even-even actinides"
] | [
"J.-P Delaroche ",
"M Girod ",
"H Goutte ",
"J Libert \nInstitut de Physique Nucléaire\nCNRS-IN2P3\nUniversité Paris XI\n15 rue Clémenceau91406OrsayFrance\n",
"\nDPTA/Service de Physique Nucléaire\nCEA/DAM Ile de France\nBP 1291680Bruyères-le-ChâtelFrance\n"
] | [
"Institut de Physique Nucléaire\nCNRS-IN2P3\nUniversité Paris XI\n15 rue Clémenceau91406OrsayFrance",
"DPTA/Service de Physique Nucléaire\nCEA/DAM Ile de France\nBP 1291680Bruyères-le-ChâtelFrance"
] | [] | Structure properties of fifty five even-even actinides have been calculated using the Gogny D1S force and the Hartree-Fock-Bogoliubov approach as well as the configuration mixing method. Theoretical results are compared with experimental data. | 10.1063/1.2338412 | [
"https://arxiv.org/pdf/nucl-th/0606024v1.pdf"
] | 118,926,895 | nucl-th/0606024 | 99492a01ea09e2ccb4e19ef2f8961037e856ec37 |
Structure properties of even-even actinides
15 Jun 2006
J.-P Delaroche
M Girod
H Goutte
J Libert
Institut de Physique Nucléaire
CNRS-IN2P3
Université Paris XI
15 rue Clémenceau91406OrsayFrance
DPTA/Service de Physique Nucléaire
CEA/DAM Ile de France
BP 1291680Bruyères-le-ChâtelFrance
Structure properties of even-even actinides
15 Jun 2006arXiv:nucl-th/0606024v1
Structure properties of fifty five even-even actinides have been calculated using the Gogny D1S force and the Hartree-Fock-Bogoliubov approach as well as the configuration mixing method. Theoretical results are compared with experimental data.
INTRODUCTION
The existence of superheavy elements (SHEs) is a major issue in nuclear physics and many experimental and theoretical efforts have been put in that direction during the last decades. In order to achieve reliable predictions in the superheavy mass region, a good understanding of the properties of heavy mass nuclei near and at the limits of stability is essential. In a recent work we have performed large scale microscopic calculations of structure properties of fifty five even-even actinides at normal and isomeric potential deformations, namely [226][227][228][229][230][231][232][233][234][235][236][228][229][230][231][232][233][234][235][236][237][238][239][240][241][242][232][233][234][235][236][237][238][239][240][241][242][243][244][245][246][238][239][240][241][242][243][244][245][246][247][248][249][250][238][239][240][241][242][243][244][245][246][247][248][249][250][251][252][253][254][255][256][242][243][244][245][246][247][248][249][250][251][252][253][254][255][256][257][258]. Results on potential energy surfaces, barriers, multipole moments, moments of inertia, shape isomers and positive parity phonon excitations, long-lived isomers, shape isomer half-lives have been discussed. Calculations have been performed using the Gogny D1S force together with the constrained Hartree-Fock-Bogoliubov (HFB) method as well as blocking, and cranking HFB approaches and the configuration mixing method. Half-lives have been determined using the semi-classical WKB approximation. In the present paper, we recall some of the results.
FISSION BARRIERS
First, we present results from constrained-Hartree-Fock-Bogoliubov calculations using the Gogny D1S force and a sole constraint on the axial quadrupole deformation β . This deformation parameter is related to the quadrupole moments q 20 and q 22 through β = π 5 √ q 2 20 +3q 2
22
A <r 2 > , with < r 2 > = 3 5 r 0 A 1/3 2 , and r 0 = 1.2 fm. In these calculations, at low elongation the triaxial degree of freedom has been left free, whereas parity has been broken at large elongation. On Fig. 1 are plotted the potential energy curves for Nobelium isotopes as functions of β . Solid lines correspond to axial shapes, dashed lines to triaxial ones and dotted ones to asymmetric shapes.The main features are: i) triaxial inner barriers are systematically lowered by up to 4 MeV when compared to the axial ones, ii) the outer barrier is found to be asymmetric for systems with N < 152 and symmetric for more neutron-rich systems, and finally iii) super deformed minima appear to be washed out for N > 156. These features are illustrated here in the case of Nobelium isotopes, but they are common features of all the seven studied isotopic chains.
SHAPE ISOMERS
Positive parity collective states have been determined using the Generator Coordinate Method and the Gaussian Overlap Approximation. A Bohr Hamiltonian-like equation is solved for the five collective quadrupole coordinates, that is for axial and triaxial moments as well as the three Euler angles. Results for shape isomers are shown in Fig. 2. A good agreement between predictions (full symbols) and experimental data (open symbols) is found in Th, U, Pu and Cm isotopes [2], [3]. We predict a global lowering of isomer energies as A increases. Superdeformed states are even found to be lower in energy than normal deformed states in 242,244 Fm and 250 No. Nevertheless, as these states are only a few hundred keV below the octupole unstable outer barrier, they may not survive as bound states. Partial γ-back decay and fission half-lives of all the shape-isomers have been determined using the WKB method. The collective masses introduced in the calculations have been evaluated using the Adiabatic Time Dependent Hartree-Fock formalism. We have found that: i) shape isomers in Th and U decay by γ-emission, ii) fission and γ-back decay are competing for Pu and Cm, and finally iii) shape isomers in Cf, Cm and No Furthermore, longer half-lives are predicted for N = 146 Th, U, Pu and Cm isotopes, which suggests that N = 146 is a magic number at super deformation.
Inner fission barrier heights have been determined using two different methods. In the first method (method A), the barrier is calculated as the energy difference between the top of the triaxial HFB inner potential (corrected by the zero point energy) and the energy of the collective normal-deformed lowest energy 0 + state. In the second method (method B) the barrier corresponds to the energy difference between the last collective state located in the first well and the collective normal deformed lowest energy state. Theoretical results are plotted as dots (method A) and triangles (method B) for the different studied isotopes in Fig. 4. Predictions are compared to experimental data shown as open symbols. A good agreement is obtained with experimental data. A bell-shape structure is predicted in all the seven isotopic chains and N = 146 isotopes are found to have the larger inner barrier heights. The difference observed between the two sets of calculations provides a measure of the theoretical uncertainty in the determination of the barrier heights. This uncertainty is of the order of 500 keV.
THIRD WELL SPECTROSCOPY
In some nuclei, namely 246 Pu,248,250 Cm and 252,254 Cf isotopes, we predict a shallow potential minimum on top of the triaxial inner barrier.
Maxima of collective masses present in the vicinity of these minima provide stability of collective levels at such intermediate deformations, especially in 250 Cm. In Fig. 5 predicted for a standard triaxial rotor. The present band features are related to the mean deformations attached to the levels, whose values display some spread over the third potential. These levels located between 5 and 6 MeV, could qualitatively explain the broad structures observed in fission probability measurements [4] at such energies in some heavy actinides with N ≃ 154.
CONCLUSION
A rich structure information has been collected over the years in experimental studies of actinides. We have used them as playgrounds to systematically challenge meanfield based methods predictions through the actinide region, the gateway to superheavy actinides. Only a few results have been presented here. More informations, for instance concerning spin isomers, phonon states and moments of inertia can be found in [1].
FIGURE 1 .
1Potential energy curves as function of β for No isotopes. Thin solid lines are for axial barriers; dashed lines for triaxial inner barriers and dotted lines correspond to mass asymmetric outer barriers.
FIGURE 2 .
2Excitation energy (in MeV) of the shape isomers expressed with respect to normal deformed ground states for Th to No isotopes. decay through fission.
FIGURE 4 .
4Theoretical inner barriers heights for Th to No isotopes obtained using method A (dots) and method B (triangles) compared to experimental data (open symbols).
FIGURE 5 .
5are plotted the quasi-rotational K = 0 and K = 2 bands predicted at intermediate deformation in250 Cm. The quasi-rotational bands do not show regular structures typical of Excitation energies of π = + collective levels, and band structure at the 250 Cm shallow triaxial potential minimum.
. J.-P Delaroche, M Girod, H Goutte, J Libert, Nucl. Phys. A. 771103J.-P. Delaroche, M. Girod, H. Goutte, J. Libert, Nucl. Phys. A 771 (2006) 103
Table of Superdeformed Nuclear Bands and Fission Isomers. B Singh, R Zywina, R B Firestone, Nucl. Data Sheets. 97241B. Singh, R. Zywina, R.B. Firestone, Table of Superdeformed Nuclear Bands and Fission Isomers, Nucl. Data Sheets 97 (2002) 241
. S Bjørnholm, J E Lynn, Rev. Mod. Phys. 52and references therein 4.S. Bjørnholm, J.E. Lynn, Rev. Mod. Phys. 52 (1980) 725, and references therein 4.
. H C Britt, A Gavron, P D Goldstone, R Schoenmackers, J Weber, J B Wilhelmy, Phys. Rev. Lett. 401010H.C. Britt, A. Gavron, P.D. Goldstone, R. Schoenmackers, J. Weber, J.B. Wilhelmy, Phys. Rev. Lett. 40 (1978) 1010
| [] |
[
"Smart matching",
"Smart matching"
] | [
"Andrea Asperti [email protected] ",
"Enrico Tassi [email protected] ",
"\nDepartment of Computer Science\nUniversity of Bologna\n\n",
"\nMicrosoft Research-INRIA Joint Center\n\n"
] | [
"Department of Computer Science\nUniversity of Bologna\n",
"Microsoft Research-INRIA Joint Center\n"
] | [] | One of the most annoying aspects in the formalization of mathematics is the need of transforming notions to match a given, existing result. This kind of transformations, often based on a conspicuous background knowledge in the given scientific domain (mostly expressed in the form of equalities or isomorphisms), are usually implicit in the mathematical discourse, and it would be highly desirable to obtain a similar behaviour in interactive provers. The paper describes the superpositionbased implementation of this feature inside the Matita interactive theorem prover, focusing in particular on the so called smart application tactic, supporting smart matching between a goal and a given result. | 10.1007/978-3-642-14128-7_23 | [
"https://arxiv.org/pdf/1005.0349v1.pdf"
] | 9,762,604 | 1005.0349 | 8e31c925609439916aaaa8728b37f5aca657415c |
Smart matching
Andrea Asperti [email protected]
Enrico Tassi [email protected]
Department of Computer Science
University of Bologna
Microsoft Research-INRIA Joint Center
Smart matching
One of the most annoying aspects in the formalization of mathematics is the need of transforming notions to match a given, existing result. This kind of transformations, often based on a conspicuous background knowledge in the given scientific domain (mostly expressed in the form of equalities or isomorphisms), are usually implicit in the mathematical discourse, and it would be highly desirable to obtain a similar behaviour in interactive provers. The paper describes the superpositionbased implementation of this feature inside the Matita interactive theorem prover, focusing in particular on the so called smart application tactic, supporting smart matching between a goal and a given result.
Introduction
The mathematical language has a deep contextual nature, whose interpretation often presupposes not trivial skills in the given mathematical discipline. The most common and typical example of these "logical abuses" is the implicit use of equalities and isomorphisms, allowing a mathematician to freely move between different incarnations of a same entity in a completely implicit way. Equipping ITP systems with the capability of reasoning up to equality yields an essential improvement of their intelligence, making the communication between the user and the machine sensibly easier.
Techniques for equational reasoning have been broadly investigated n the realm of automated theorem proving (see eg [7,22,10]). The main deductive mechanism is a completion technique [17] attempting to transform a given set of equations into a confluent rewriting system so that two terms are equal if and only if they have identical normal forms. Not every equational theory can be presented as a confluent rewriting system, but one can progressively approximate it by means of a refutationally complete method called ordered completion. The deductive inference rule used in completion procedures is called superposition: it consists of first unifying one side of one equation with a subterm of another, and hence rewriting it with the other side. The selection of the two terms to be unified is guided by a suitable term ordering, constraining inferences and sensibly pruning the search space.
The final publication of this paper is available at www.springerlink.com arXiv:1005.0349v1 [cs.LO] 3 May 2010
Although we are not aware of any work explicitly focused on superposition techniques for interactive provers, the integration between fully automatic provers (usually covering paramodulation) and interactive ones is a major research challenge and many efforts have been already done in this direction: for instance, KIV has been integrated with the tableau prover 3T A P [1]; HOL has been integrated with various first order provers, such as Gandalf [15] and Metis; Coq has been integrated with Bliksem [8]; Isabelle was first integrated with a purpose-built prover [23] and more recently with Vampire [20]. The problems of these integrations are usually of two kinds: (a) there is a technical difficulty in the forward and backward translation of the information between systems, due to the different underlying logics (ITP systems are usually higher-order, and some of them intuitionistic); (b) there is a pragmatical problem in the management of the knowledge base to be used by the automatic solver, since it can be huge (so we cannot pass it at every invocation), and it grows dynamically (hence, it cannot be exported in advance).
A good point of the superposition calculus (and not the last reason for restricting the attention to this important fragment) is that point (a), in this context, becomes relatively trivial (and the translation particularly effective). As for point (b), its main consequence is that the communication between the Interactive Prover and the Problem Solver, in order to be efficient, cannot be stateless: the two systems must share a common knowledge base. This fact, joined with the freedom to adapt the superposition tool to any possible specific requirement of the Matita system convinced us to rewrite our own solver, instead of trying to interface Matita with some available tool. This paper discusses our experience of implementation of a (first order) superposition calculus (Section 2), its integration within the (higher-order) Matita interactive prover [5] (Section 3), and in particular its use for the implementation of a smart application tactic, supporting smart matching between a goal and a given results (Section 4). We shall conclude with a large number of examples of concrete use of this tactic.
The Matita superposition tool
One of the components of the automation support provided by the Matita interactive theorem prover is a first order, untyped superposition tool. This is a quite small and compact application (little more than 3000 lines of OCaml code), well separated by the rest of the system. It was entirely rewritten during the summer 2009 starting from a previous prototype (some of whose functionalities had been outlined in [6]), with the aim to improve both its abstraction and performance. The tool took part to the 22nd CADE ATP System Competition, in the unit equality division, scoring in fourth position, beating glorious systems such as Otter or Metis [16], and being awarded as the best new entrant tool of the competion [28].
In the rest of this section we shall give an outline, as concise as possible, of the theory and the architecture of the tool. This is important in order to understand its integration with the interactive prover.
The superposition calculus in a nutshell
Let F bet a countable alphabet of functional symbols, and V a countable alphabet of variables. We denote with T (F, V) the set of terms over F with variables in V. A term t ∈ T (F, V) is either a 0-arity element of F (constant), an element of V (variable), or an expression of the form f (t 1 , . . . , t n ) where f is a element of F of arity n and t 1 , . . . , t n are terms.
Let s and r be two terms. s| p denotes the subterm of s at position p and s[r] p denotes the term s where the subterm at position p has been replaced by r.
A substitution is a mapping from variables to terms. Two terms s and t are unifiable if there exists a substitution σ such that sσ = tσ. In the previous case, σ is called a most general unifier (mgu) of s and t if for all substitution θ such that sθ = tθ, there exists a substitution τ which satisfies θ = τ • σ.
A literal is either an abstract predicate (represented by a term), or an equality between two terms. A clause Γ ∆ is a pair of multisets of literals: the negative literals Γ , and the positive ones ∆. If Γ = ∅ (resp. ∆ = ∅), the clause is said to be positive (resp. negative).
A Horn clause is a clause with at most one positive literal. A unit clause is a clause composed of a single literal. A unit equality is a unit clause where the literal is an equality.
A strict ordering ≺ over T (F, V) is a transitive and irreflexive (possibly partial) binary relation. An ordering is stable under substitution if s ≺ t implies sσ ≺ tσ for all terms t, s and substitutions σ. A well founded monotonic ordering stable under substitution is called reduction ordering (see [11]). The intuition behind the use of reduction orderings for limiting the combinatorial explosion of new equations during inference, is to only rewrite big terms to smaller ones.
l = r t1 = t2 (t1[r]p = t2 )σ l = r t1 = t2 (t1[r]p = t2 )σ t1 = t2
if σ = mgu(l, t1|p), t1|p = x, lσ rσ and t1σ t2σ
if ∃σ = mgu(t1, t2).
Fig. 1. Inference rules
For efficiency reasons, the calculus must be integrated with a few additional optimization rules, the most important one being demodulation ( [29]).
The main algorithm
A naive implementation of the superposition calculus could just combine (superpose) all known clauses in all (admitted) ways, and repeat that process until the desired clause (called goal ) is resolved. To avoid useless duplication of work, subsumption tautology elimination demodulation it is convenient to keep clauses in two distinct sets, traditionally called active and passive, with the general invariant that clauses in the active set have been already composed together in all possible ways. At every step, some clauses are selected from the passive set and added to the active set, then superposed with the active set, and consequently with themselves (inference). Finally, the newly generated clauses are added to the passive set (possibly after a simplification). A natural selection strategy, resulting in a very predictable behaviour, would consist in selecting the whole passive set at each iteration, in the spirit of a breadth first search. Unfortunately the number of new equations generated at each step grows extremely fast, in practice preventing the iteratation of the main loop more than a few times.
S ∪ {C, D} S ∪ {C} S ∪ { t = t} S S ∪ { l = r, C} S ∪ { l = r, C[rσ]p} if ∃σ, Dσ ≡ C if lσ ≡ C|p and lσ rσ
To avoid this problem, all modern theorem provers (see e.g. [24]) adopt the opposite solution. According to some heuristics, like size and goal similarity for example, they select only one passive clause at each step. Not to loose completeness, some fairness conditions are taken into account (i.e. every passive clause will be eventually selected). This approach falls under the name given-clause ( Figure 3), and its main advantage is that the passive set grows much slower, allowing a more focused and deeper inspection of the search space that consequently allows to find proofs that require a much higher number of main loop iterations.
The main drawback of this approach is that it makes the procedure way more sensible to the selection heuristics, leading to an essentially unpredictable behaviour.
Performance issues
In order to obtain a state-of-the-art tool able to compete with the best available systems one has eventually to take into account a lot of optimizations and techniques developed for this purpose during the last thirty years.
In the following we shall shortly describe the most critical areas, and, for each of them, the approach adopted in Matita.
Orderings used to orientate rewriting rules On complex problems (e.g. problems in the TPTP library with rating greater then 0.30) the choice of a good ordering for inference rules is of critical importance. We have implemented several orderings, comprising standard Knuth-Bendix (KBO), non recursive Knuth-Bendix (NRKBO), lexicographic path ordering (LPO) and recursive path ordering (RPO). The best suited ordering heavily depends on the kind of problem, and is hard to predict: our approach for the CADE ATP System Competition was to run in parallel different processes with different orderings.
On simpler problems (of the kind required for the smart application tactic of section 5), the given-clause algorithm is less sensitive to the term-ordering, and we may indifferently choose our preferred strategy, opportunely tuning the library (we are currently relying on LPO).
Selection strategy
The selection strategy currently implemented by Matita is a based on combination of age and weight. The weight is a positive integer that provides an estimation of the "complexity" of the clause, and is tightly related to the number of occurrences of symbols in it.
Since we are not interested in generating (counter) models of false statements, we renounced to be complete, and we silently drop inferred clauses that would slow down the main loop too much due to their excessive size.
Another similar optimization we did not implement but we could consider as a future development is Limited Resource Strategy [25], which basically allows the procedure to skip some inference steps if the resulting clauses are unlikely to be processed, mainly because of a lack of time.
Data structures and code optimization We adopted relatively simple data structures (like discrimination [18] trees for term indexing), and a purely functional (in the sense of functional programming) implementation of them. After some code optimisation, we reached a point where very fast functions are the most expensive, because of the number of calls (implied by the number of clauses), even if they operate on simple data structures.
Since we are quite satisfied with the actual performance, we did not invest resources in adopting better data structures, but we believe that further optimizations will probably require implementing more elaborate data structures, such as substitution [14] or context trees [13], or even adopt an indexing technique that works modulo associativity and commutativity [12], that looks very promising when working on algebraic structures.
Demodulation Another important issue for performance is demodulation: the given clause algorithm spends most of its time (up to 80%) in simplification, hence any improvement in this part of the code has a deep impact on performance. However, while reduction strategies, sharing issues and abstract machines have been extensively investigated for lambda calculus (and in general for left linear systems) less is known for general first order rewriting systems. In particular, while an innermost (eager) reduction strategy seem to work generally better than an outermost one (especially when combined with lexicographic path ordering), one could easily create examples showing an opposite behaviour (even supposing to always reduce needed redexes).
Integrating superposition with Matita
Library management
A possible approach to the integration of superposition with Matita is to solve all goals assuming that all equations part of the library lie in the passive set, augmented on the fly with the equations in the local context of the ongoing proof.
The big drawback of this approach is that, starting essentially from the same set of passive equations at each invocation on a different goal (differing only for the local context), the given clause algorithm would mostly repeat the same selection and composition operations over and over again. It is clear that, if we wish to superpose library equations, this operation should not be done at run time but in background, once and for all. Then we have to face a dual problem, namely to understand when stopping the saturation of the library with new equations, preventing an annoying pollution with trivial results that could have very nasty effects for selection and memory occupation. We would eventually like to have mechanisms to drive the saturation process.
A natural compromise is to look at library equations not as a passive set, but as the active one. This means that every time a new (unit) equation is added to the library it also goes through one main given-clause loop, as if it was the newly selected passive equation: it is simplified, composed with all existing active equations (i.e. all other equations in the library, up to simplification), and the newly created equations are added to the passive list. At run time, we shall then strongly privilege selection of local equations or goals.
This way, we have a natural, simple but traceable syntax to drive the saturation process, by just listing in library the selected equations. As a side effect, this approach reduces the verbosity of the library by making it unnecessary to declare (and name explicitly) trivial variants of available results that are automatically generated by superposition.
Interfacing CIC and the superposition engine
Our superposition tool is first order and untyped, while the Matita interactive prover is based on a variant of the Calculus of Inductive Construction (CIC), a complex higher-order intuitionistic logical systems with dependent types. The communication between the two components is hence far from trivial.
Instead of attempting a complex, faithful encoding of CIC in first order logic (that is essentially the approach adopted for HOL in [19]) we choose to follow a more naif approach, based on a forgetful translation that remove types and just keeps the first order applicative skeleton of CIC-terms.
In the opposite direction, we try to reconstruct the missing information by just exploiting the sophisticated inference capability of the Matita refiner [3], that is the tool in charge of transforming the user input into a machine understandable low-level CIC term.
Automation is thus a best effort service, in the sense that not only it may obviously fail to produce a proof, but sometimes it could produce an argument that Matita will fail to understand, independently from the fact if the delivered proof was "correct" or less.
The choice to deal with untyped first order equations in the superposition tool was mostly done for simplicity and modularity reasons. Moving towards a typed setting would require a much tighter integration between the superposition tool and the whole system, due to the complexity of typing and unification, but does not seem to pose any major theoretical problem.
The forgetful encoding Equations r = T s of the calculus of constructions are translated to first order equations by merely following the applicative structure of r and s, and translating any other subterm into an opaque constant. The type T of the equation is recorded, but we are not supposed to be able to compute types for subterms.
In spite of the fact of neglecting types, the risk of producing "ill-typed" terms via superposition rules is moderate. Consider for instance the superposition left rule (the reasoning is similar for the other rules)
l = r t 1 = t 2 (t 1 [r] p = t 2 )σ
where σ = mgu(l, t 1 | p ) and lσ rσ. The risk is that t 1 | p has a different type from l, resulting into an illegal rewriting step. Note however that l and r are usually rigid terms, whose type is uniquely determined by the outermost symbol. Moreover, t 1 | p cannot be a variable, hence they must share this outermost symbol. If l is not rigid, it is usually a variable x and if x ∈ r (like e.g. in x = x + 0) we have (in most orderings) l r that again rules out rewriting in the wrong direction.
This leads us to the following notion of admissibility. We say that an applicative term f (x 1 , . . . , x n ) is implicitly typed if its type is uniquely determined by the type of f . We say that an equation l = r is admissible if both l and r are implicitly typed, or l r and r is implicitly typed. Non admissible equations are not taken into account by the superposition tool.
In practice, most unit equalities are admissible. A typical counter example is an equation of the kind ∀x, y : unit.x = y, where unit is a singleton type.
On the other side, non-unit equalities are often not admissible. For instance, a clause of the kind x ∧ y = true x = true could be used to rewrite any term to true, generating meaningless, ill typed clauses. Extending superposition beyond the unit equality case does eventually require to take types into consideration.
(Re)construction of the proof term
Translating a first-order resolution proof into a higher-order logic natural deduction proof is a notoriously difficult issue, even more delicate in case of intuitionistic systems, as the one supported by Matita. While resolution per se is a perfectly constructive process, skolemization and transformation into conjunctive normal forms are based on classical principles.
Our choice of focusing on the superposition calculus was also motivated by the fact it poses less difficulties, since skolemization is not needed and thus proofs have a rather simple intuitionistic interpretation.
Our technique for reconstructing a proof term relies as much as possible on the refinement capabilities of Matita, in particular for inferring implicit types. In the superposition module, each proof step is encoded as a tuple
Step of rule * int * int * direction * position * substitution where rule is the kind of rule which has been applied, the two integers are the two id s of the composing equations (referring to a "bag" of unit clauses), direction is the direction the second equation is applied to the first one, position is a path inside the rewritten term and finally substitution is the mgu required for the rewriting step.
Every superposition step is encoded by one of the following terms: where left ( l ) and right ( r) must be understood w.r.t. backward application, and where P is the one hole context that represents the position in which the superposition occurred.
A more liberal, but also slightly more expensive solution consists in indexing any equation and systematically try to read back each result of a superposition step in CIC, dropping it if it is not understood by the refiner.
At the end of the superposition procedure, if a proof is found, either a trivial goal has been generated, or a fact subsumes one of the active goals. In that latter case, we perform a rewriting step on the subsumed goal, so that we fall back into the previous case. Thus, when the procedure successfully stops, the selected clause is of the form s = t where s and t are unifiable. We call it the meeting point, because forward steps (superposition right) and backward steps (superposition left) meet together when this trivial clause is generated, to compose the resulting proof. To generate a CIC proof term, the clauses are topologically sorted, their free variables are explicitly quantified, and nested letin patterns are used to build the proof.
The most delicate point of the translation is closing each clause w.r.t. its free variables, since we should infer a type for them, and since CIC is an explicitly polymorphic language it is often the case that the order of abstractions does matter (e.g. variables standing for types must in general be abstracted before polymorphic variables).
The simplest solution is to generate so called "implicit" arguments leaving to the Matita refiner the burden of guessing them.
For instance, superposing lencat : len A x + len A y = len A (x@y) with catA : x@(y@z)
¦ ¥
Note that w must be abstracted first, since it occurs in the (to be inferred) types for x, y and z. Also note the one hole context expressed as an anonymous function whose abstracted variable is named hole, corresponding to the position of x@y in the statement of lencat. The interesting point is that refining is a complex operation, using e.g. hints, and possibly calling back the automation itself: the interpretation of the proof becomes hence a dialog between the system and its automation components, aimed to figure out a correct interpretation out of a rough initial trace.
A more sophisticated translation, aimed to produce a really nice, humanreadable output in the form of a chain of equations, is described in [6].
Smart application
The most interesting application of superposition (apart from its use for solving equational goals) is the implementation of a more flexible application tactic. As a matter of fact, one of the most annoying aspects of formal development is the need of transforming notions to match a given, existing result. As explained in the introduction, most of these transformations are completely transparent to the typical mathematical discourse, and we would like to obtain a similar behaviour in interactive provers.
Given a goal B and a theorem t: A → B, the goal is to try to match B with B up to the available equational knowledge base, in order to apply t. We call it, the smart application of t to G. We use superposition in the most direct way, exploiting on one side the higher-order features of CIC, and on the other the fact that the translation to first order terms does not make any difference between predicates and functions: we simply generate a goal B = B and pass it to the superposition tool (actually, it was precisely this kind of operation that motivated our original interest in superposition). If a proof is found, B is transformed into B by rewriting and t is then normally applied. Superposition, addressing a typically undecidable problem, can easily diverge, while we would like to have a reasonably fast answer to the smart application invocation, as for any other tactic of the system. We could simply add a timeout, but we prefer to take a different, more predictable approach. As we already said, the overall idea is that superposition right steps -realising the saturation of the equational theory -should be thought of as background operations. Hence, at run time, we should conceptually work as if we had a confluent rewriting system, and the only operation worth to do is narrowing (that is, left superposition steps). Narrowing too can be undecidable, hence we fix a given number of narrowing operations to apply to each goal (where the new goal instances generated at each step are treated in parallel). The number of narrowing steps can be fixed by the user, but a really small number is usually enough to solve the problem if a solution exists.
Examples
Example 1. Suppose we wish to prove that the successor function is le-reflecting, namely ( * ) ∀n, m.Sn ≤ Sm → n ≤ m
Suppose we already proved that the predecessor function is monotonic:
monotonic pred : ∀n, m.n ≤ m → pred n ≤ pred m
We would like to merely "apply" the latter to prove the former. Just relying on unification, this would not be possible, since there is no way to match pred X ≤ pred Y versus n ≤ m unless narrowing the former. By superposing twice with the equation ∀n.pred(Sn) = n we immediately solve our matching problem via the substitution {X := Sn, Y := Sm}. Hence, the smart application of monotonic pred to the goal n ≤ m succeeds, opening the new goal Sn ≤ Sm that is the assumption in ( * ).
Example 2. Suppose we wish to prove n ≤ m * n for all natural numbers n, m.
Suppose we already proved that multiplication is left-monotonic, namely monotonic le times l : ∀n, a, b.a ≤ b → a * n ≤ b * n
In order to apply this result, the system has to find a suitable ? a such that ? a * n = n, that is easily provided by the identity law for times.
Example 3. In many cases, we just have local equational variants of the needed results. Suppose for instance we proved that multiplication in le-reflecting in its right parameter:
le times to le times r : ∀a, n, m.a * n ≤ a * m → n ≤ m
Since times is commutative, this also trivially implies the left version:
monotonic le times l : ∀a, n, m.n * a ≤ m * a → n ≤ m
Formally, suppose to have the goal n ≤ m under the assumption (H) n * a ≤ m * a. By applying le times to le times r we obtain a new goal ? a * n ≤? a * m that is a smart variant of H.
Example 4. Suppose we wish to prove that (H) a * (Sn) ≤ a * (Sm) implies a * n ≤ a * m, where S is the successor function (this is a subcase in the inductive proof that the product by a positive constant a is le-reflecting). Suppose we already proved that the sum is le-reflecting in its second argument:
le plus to le plus r : ∀a, n, m.a + n ≤ a + m → n ≤ m
By applying this result we obtain the new goal ? a + a * n ≤? a + a * m, and if we have the expected equations for times, we can close the proof by a smart application of H.
≤ m → a ≤ b → n + a ≤ m + b
The smart application of this term to the goal n < 2 * m succeeds, generating the two subgoals n ≤ m and 1 ≤ m + 0. The former one is the assumption H1, while the latter is a smart variant of H.
The precise shape depends by the specific equations available on times.
Example 6. Let us make an example inspired by the theory of programming languages. Suppose to have a typing relation Γ M : N stating that in the environment Γ the term M has type N . If we work in De Bruijn notation, the weakening rule requires lifting
weak : Γ M : N → Γ, A ↑ 1 (M ) : ↑ 1 (N )
Suppose now we have an axiom stating that * : where * and are two given sorts. We would like to generalize the previous result to an arbitrary (legal) context Γ . To prove this, we have just to apply weakenings (reasoning by induction on Γ ). However, the normal application of weak would fail, since the system should be able to guess two terms M and N such ↑ 1 (M ) = * and ↑ 1 (N ) = .
If we know that for any constant c, ↑ 1 (c) = c (that comes from the definition of lifting) we may use such an equation to enable the smart application of weak.
Performance In Figure 5 we give the execution times for the examples of smart applications discussed in the previous section (in bytecode). Considering these times, it is important to stress again that the smart application tactics does not take any hint about the equations it is supposed to use to solve the matching problem, but exploits all the equations available in the (imported sections of the) library.
The important point is that smart application is fast enough to not disturb the interactive dialog with the proof assistant, while providing a much higher degree of flexibility than the traditional application.
Related works and systems
Matita was essentially conceived as a light version of Coq [9], sharing the same foundational logic (the Calculus of Inductive Constructions) and being partially compatible with it (see [4] for a discussion of the main differences between the two systems at kernel level). Hence, Coq is also the most natural touchstone for our work. The auto tactic of Coq does not perform rewriting; this is only done by a couple of specialized tactics, called auto rewrite and congruence.
The first tactic carries out rewritings according to sets of oriented equational rules explicitly passed as arguments to the tactic (and previously build by the user with suitable vernacular commands). Each rewriting rule in some base is applied to the goal until no further reduction is possible. The tactic does not perform narrowing, nor any form of completion. The congruence tactic implements the standard Nelson and Oppen congruence closure algorithm [21], which is a decision procedure for ground equalities with uninterpreted symbols; the Coq tactic only deals with equalities in the local context. Both Coq tactics are sensibly weaker than superposition that seems to provide a good surrogate for several decision procedures for various theories, as well as a simple framework for composing them (see e.g [2]).
Comparing the integration of superposition in Matita with similar functionalities provided by Isabelle is twofold complex, due not only to the different approaches, but also to the different underlying logics.
In Isabelle, equational reasoning can be both delegated to external tools or dealt with internally by the so called simplifier. Some of the the external tools Isabelle is interfaced with provide full support to paramodulation (and hence superposition), but the integration with them is stateless, possibly requiring to pass hundreads of theorems (all the current visible environment) at each invocation. In Matita, the active set is persistent, and grows as the user proves new equations. Of more interest is the comparison with Isabelle's internal simplifier. The integration of this tool with the library is manual: only lemmas explicitly labelled and oriented by the user are taken into account by the simplifier. Moreover, these lemmas are only used to demodulate and are not combined together to infer new rewriting rules. Nevertheless, a pre-processing phase allows the user to label theorems whose shape is not an equation. For example a conjunction of two equations is interpreted as two distinct rewriting rules, or a negative statement ¬A is understood as A = F alse. The simplifier is also able to take into account guarded equations as long as their premises can be solved by the simplifier itself. Finally it detects equations that cannot be oriented by the user, like commutativity, and restricts their application according to the demodulation rule using a predefined lexicographic order. Anyway, the main difference from the user's perspective comes from a deep reason that has little to do with the simplifier or any other implemented machinery. Since Isabelle is based on classical logic, co-implication can be expressed as an equality. Hence, in Isabelle we can prove much more equations at the prositional level and use them for rewriting. Any concrete comparison between the two provers with respect to equational reasoning is thus inherently biased, since many problems encountered in one system would look meaningless, artificial or trivial when transposed into the other one.
Conclusions
We described in this paper the "smart" application tactic of the Matita interactive theorem prover. The tactics allow the backward application of a theorem to a goal, where matching is done up to the data base of all equations available in the library. The implementation of the tactics relies on a compact superposition tool, whose architecture and integration within Matita have been described in the first sections. The tool is already performant (it was awarded best new entrant tool at the 22nd CADE ATP System Competition) but many improvements can still be done for efficiency, such as the implementation of more sophisticated data structures for indexes (we currently use discrimination trees).
Another interesting research direction is to extend the management of equality to setoid rewriting [27]. Indeed, the current version of the superposition tool just works with an intensional equality, and it would be interesting to try to figure out how to handle more general binary relations. The hard problem is proof reconstruction, but again it seems possible to exploit the sophisticated capabilities of the Matita refiner [3] to automatically check the legality of the rewriting operation (i.e. the monotonicity of the context inside which rewriting has to be performed), exploiting some of the ideas outlined in [26].
One of the most promising uses of smart application is inside the backwardbased automation tactic of Matita. In fact, smart application allows a smooth integration of equational reasoning with the prolog-like backward applicative mechanisms that, according to our first experimentations looks extremely promising. As a matter of fact, the weakest point of smart application is that it does not relieve the user form the effort of finding the "right" theorems in the library or of guessing/remembering their names (although it allows to sensibly reduce the need of variants of a given statement in the repository). A suitably constrained automation tactic could entirely replace the user in the quest of candidates for the smart application tactic. Since searching is a relatively expensive operation, the idea is to ask the automation tactic to return an explicit trace of the resulting proof (essentially, a sequence of smart applications) to speed-up its re-execution during script development.
Fig. 2 .
2Simplification rules
Fig. 3 .
3given-clause loopNumbers in parentheses reflect the steps order.
eq ind l : ∀A : Type.∀x : A.∀P : A → Prop.P x → ∀y : A.x = y → P y eq ind r : ∀A : Type.∀x : A.∀P : A → Prop.P x → ∀y : A.y = x → P y
)@z at the underlined position and in the given direction gives rise to the following piece of code, where question marks stand for implicit arguments: § ¤ let clause 59 : ∀w :?.∀x :?.∀y :?.∀z :?. len w (x@y) + len w z = len w (x@(y@z)) := λw :?.λz :?.λx :?.λy :?.eq ind r (List w) ((x@y)@z)) (λhole : List w.len w (x@y) + len w z = len w hole) (lencat w (x@y) z) (x@(y@z)) (catA w x y z) in
Fig. 4 .
4Smart application
Example 5 .
5Consider the goal n < 2 * m under the assumptions (H) 0 < m and (H1) n ≤ m. Suppose that we defined x < y as x + 1 ≤ y. Morevoer, by the defining equation of times we should know something like 2 * m = m + (m + 0). Hence the goal is equal to n + 1 ≤ m + (m + 0), and the idea is to use again the monotonicity of plus (in both arguments): le plus n m : ∀a, b.n
le times l 0.23s. 3 H : a * n ≤ a * m 0.22s. 4H : a * (Sn) ≤ a * (Sm)
Fig. 5 .
5Smart application execution times
lifting operation ↑ n (M ) is meant to relocate the term M under n additional levels of bindings: in other words, it increases by n all free variables in M .
Acknowledgements We would like to thank Alberto Griggio and Maxime Dénès for their contribution to the implementation of the superposition tool.
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| [] |
[
"Quantifying inner-outer interactions in non-canonical wall-bounded flows",
"Quantifying inner-outer interactions in non-canonical wall-bounded flows"
] | [
"Mogeng Li \nFaculty of Aerospace Engineering\nDelft University of Technology\n2629 HSThe Netherlands\n",
"Woutijn J Baars \nFaculty of Aerospace Engineering\nDelft University of Technology\n2629 HSThe Netherlands\n",
"Ivan Marusic \nDepartment of Mechanical Engineering\nUniversity of Melbourne\n3010VictoriaAustralia\n",
"Nicholas Hutchins \nDepartment of Mechanical Engineering\nUniversity of Melbourne\n3010VictoriaAustralia\n"
] | [
"Faculty of Aerospace Engineering\nDelft University of Technology\n2629 HSThe Netherlands",
"Faculty of Aerospace Engineering\nDelft University of Technology\n2629 HSThe Netherlands",
"Department of Mechanical Engineering\nUniversity of Melbourne\n3010VictoriaAustralia",
"Department of Mechanical Engineering\nUniversity of Melbourne\n3010VictoriaAustralia"
] | [] | We investigate the underlying physics behind the change in amplitude modulation coefficient in noncanonical wall-bounded flows in the framework of the inner-outer interaction model (IOIM) (Baars et al.,Phys. Rev. Fluids 1 (5), 054406). The IOIM captures the amplitude modulation effect, and here we focus on extending the model to non-canonical flows. An analytical relationship between the amplitude modulation coefficient and IOIM parameters is derived, which is shown to capture the increasing trend of the amplitude modulation coefficient with an increasing Reynolds number in a smooth-wall dataset. This relationship is then applied to classify and interpret the non-canonical turbulent boundary layer results reported in previous works. We further present the case study of a turbulent boundary layer after a rough-to-smooth change. Both single-probe and two-probe hotwire measurements are performed to acquire streamwise velocity time series in the recovering flow on the downstream smooth wall. An increased coherence between the large-scale motions and the small-scale envelope in the near-wall region is attributed to the stronger footprints of the over-energetic large-scale motions in the outer layer, whereas the near-wall cycle and its amplitude sensitivity to the superposed structures are similar to that of a canonical smooth-wall flow. These results indicate that the rough-wall structures above the internal layer interact with the near-wall cycle in a similar manner as the increasingly energetic structures in a high-Reynolds number smooth-wall boundary layer. | null | [
"https://export.arxiv.org/pdf/2304.13707v1.pdf"
] | 258,331,637 | 2304.13707 | 5251440bfd68299d77ff842f09e0f1741e63232a |
Quantifying inner-outer interactions in non-canonical wall-bounded flows
26 Apr 2023
Mogeng Li
Faculty of Aerospace Engineering
Delft University of Technology
2629 HSThe Netherlands
Woutijn J Baars
Faculty of Aerospace Engineering
Delft University of Technology
2629 HSThe Netherlands
Ivan Marusic
Department of Mechanical Engineering
University of Melbourne
3010VictoriaAustralia
Nicholas Hutchins
Department of Mechanical Engineering
University of Melbourne
3010VictoriaAustralia
Quantifying inner-outer interactions in non-canonical wall-bounded flows
26 Apr 2023(Dated: April 27, 2023)arXiv:2304.13707v1 [physics.flu-dyn]
We investigate the underlying physics behind the change in amplitude modulation coefficient in noncanonical wall-bounded flows in the framework of the inner-outer interaction model (IOIM) (Baars et al.,Phys. Rev. Fluids 1 (5), 054406). The IOIM captures the amplitude modulation effect, and here we focus on extending the model to non-canonical flows. An analytical relationship between the amplitude modulation coefficient and IOIM parameters is derived, which is shown to capture the increasing trend of the amplitude modulation coefficient with an increasing Reynolds number in a smooth-wall dataset. This relationship is then applied to classify and interpret the non-canonical turbulent boundary layer results reported in previous works. We further present the case study of a turbulent boundary layer after a rough-to-smooth change. Both single-probe and two-probe hotwire measurements are performed to acquire streamwise velocity time series in the recovering flow on the downstream smooth wall. An increased coherence between the large-scale motions and the small-scale envelope in the near-wall region is attributed to the stronger footprints of the over-energetic large-scale motions in the outer layer, whereas the near-wall cycle and its amplitude sensitivity to the superposed structures are similar to that of a canonical smooth-wall flow. These results indicate that the rough-wall structures above the internal layer interact with the near-wall cycle in a similar manner as the increasingly energetic structures in a high-Reynolds number smooth-wall boundary layer.
I. INTRODUCTION
In turbulent boundary layers, large coherent structures are found in the logarithmic region.
They carry a high level of turbulent kinetic energy, and make a significant contribution to Reynolds stress production [1]. These structures can be further classified as large-scale motions (LSMs) and very-large-scale motions (VLSMs). The former are associated with the vortex packets formed by aligned hairpin vortices [2,3], and typically have a streamwise length of ∼ 3 ( is the boundary layer thickness), while the latter may be related to the merging of multiple LSMs [2], and can reach a streamwise extent up to 20 with spanwise meandering [4,5]. These structures are observed to leave a footprint in the near-wall region [4][5][6]. As the Reynolds number of a turbulent boundary layer increases, the separation between the outer-scaled motions and viscous-scaled near-wall cycle becomes more distinct, and the strength of these large-scale footprints also intensifies, resulting in the growth of the 'inner-peak' magnitude in the broadband streamwise turbulence intensity [5,7].
In addition to the direct superposition effect manifested as large-scale footprints, it has also been found that the near-wall small scales are modulated by the large scales in the outer layer [5,8,9]. In the near-wall region, the amplitude and frequency of small-scale fluctuations show a decrease when co-existing with large-scale low-speed regions, and vice versa in the case of large-scale high-speed regions. These observations were harnessed by the 'two-scale' framework, where a local (near-wall) fine-mesh solution is coupled to the global coarse-mesh solution, in order to reduce computational costs at high Reynolds numbers [10,11]. In terms of the modelling efforts, quasisteady quasihomogeneous theory [12,13] provides an axiomatic description of the scale interaction in near-wall turbulence. The modulation effect was also quantified in recent works [14][15][16][17][18], and a predictive model, termed the inner-outer interaction model (IOIM), which outputs representative turbulence statistics in the near-wall region based on an input signal in the logarithmic region, was developed by Marusic and coworkers [19,20] and by Agostini and Leschziner [21]. The former was later revised by Baars et al. [22] using spectral linear stochastic estimation. The IOIM provides an opportunity to push the boundary of large-eddy simulations of wall-bounded flows to very high Reynolds numbers at affordable costs, thanks to its ability to provide representative real-time small-scale signals in the viscous-scaled near-wall region based only on large-scale outer-layer information [23][24][25].
Although similar amplitude modulation behaviours have been observed in turbulent boundary layers, channel, and pipe flows [26], the existence of such scale interactions remains unexplored under non-canonical conditions. A sound understanding of how the inner-and outer-scale relationship is affected by these conditions is essential for generalising its application to a wider scope of flows. Enhanced amplitude modulation has been observed in various non-canonical flows, including boundary layers over rough walls [27][28][29][30][31] or permeable surfaces [32,33], and boundary layers with modified outer structures, such as energetic large-scale motions injected into the flow via freestream turbulence [34,35], upstream dynamic roughness [36] and synthetic large-scale signals generated by plasma actuators [37], to name a few. It is not yet well understood how these seemingly different flow conditions all lead to a common increase in the amplitude modulation coefficient, and we aim to bridge this gap by establishing a physics-based quantitative relationship between the amplitude modulation coefficient and IOIM parameters.
Furthermore, here we look into another scenario where the introduced large scales in the outer layer have an energy distribution across scales similar to that of a canonical boundary layer, and only the amplitude is intensified. This is achieved by a sudden rough-to-smooth surface transition occurring in the streamwise direction, as depicted in Fig. 1. Upstream of the transition, a turbulent boundary layer develops on a rough wall with equivalent sand grain roughness height . Here, is the streamwise direction, 0 is the streamwise location of the surface transition andˆ ≡ − 0 is the distance downstream of the transition. At = 0 , the surface switches to a smooth wall, while the boundary layer continues to evolve and gradually adjusts to the new surface. The effect of the new surface condition is first felt in the near-wall region of the boundary layer and then gradually propagates to the interior of the flow [39]. The layer that separates the modified near-wall region from the unaffected oncoming flow farther away from the wall is generally referred to as the internal boundary layer (IBL) with a thickness denoted by . For more details on the observation and modelling of the flow recovery, we refer to the works by Elliott [40], Antonia and Luxton [41], Hanson and Ganapathisubramani [42], Rouhi et al. [43], and Li et al. [44,45].
Turbulent boundary layers over a rough-to-smooth change in the streamwise direction offer a new perspective to further understand the physics of inner-outer interactions. When normalised by the friction velocity at the wall, LSMs and VLSMs above are similar to their smooth-wall counterparts [46,47], but they are over-energised compared to the near-wall small scales. The former retain a memory of the upstream rough wall friction velocity whereas the latter scale on the much lower local smooth-wall friction velocity. How these structurally similar but more energised large-scale motions interact with the near-wall cycle will be investigated in this study.
The remainder of the paper is organised as follows: in Sec. II, we first present a summary of the definition of the amplitude modulation coefficient and the IOIM framework, as well as a quantitative relation between the two. We then review the previous studies on the amplitude modulation in non-canonical flows in Sec. III. In Sec. IV, we present a case study on the flow downstream of a rough-to-smooth change in the surface condition.
II. PHYSICAL UNDERPINNING OF AMPLITUDE MODULATION
In this section, we first briefly summarise the definition of the amplitude modulation coefficient and the IOIM formulation, and then provide a quantitative description of the relation between the two, supported by results computed from synthetic signals.
A. Amplitude modulation coefficient and IOIM formulation
The amplitude modulation coefficient is a commonly reported diagnostics, largely due to the fact that it provides a straightforward quantification of the degree of amplitude modulation within a single-point time series. An example of the profile is shown in Fig. 2(a). It is defined as the correlation between the low-pass-filtered envelope of small-scale fluctuations and the large-scale fluctuations at the same location [14]:
( + ) = + ( + , + ) + ( + , + ) 2 + ( + , + ) +2 ( + , + )(1)
Here, the superscript (·) + indicates inner scaling with the local friction velocity as the velocity scale, the angle brackets · denotes time average, represents the streamwise velocity, + is the Mathis et al. [20] to separate large and small scales.
zero-mean large-scale superposition signal, which is usually obtained by low-pass filtering the time series with a threshold of + = 7000 and + ≡ + − + is the detrended signal.
[·] denotes a low-pass-filtered envelope of the signal
+ + , + = L +2 ( + , + ) + H 2 + ( + , + ) ,(2)
where H [·] is the Hilbert transform, +2 ( + , + ) + H 2 + ( + , + ) is the analytic signal and L [·] denotes a low-pass filter.
The definition can be further generalised to a time-shifted amplitude modulation coefficient
( + , + ) = + ( + , + ) + ( + , + − + ) 2 + ( + , + ) +2 ( + , + ) .(3)
By definition, ( + ) ≡ ( + , 0) (see Fig. 2b). The relative shift + is the lag between the superposition imprint + and the low-pass filtered envelope + such that reaches its maximum, i.e. ( + , + ) = max [ ( + , + )], as marked by the solid circles in Fig. 2(b).
According to the IOIM [22], the statistical prediction of the fluctuating velocity + can be constructed by considering a superposition effect of large-scale content with additively a universal signal * that is subject to an amplitude modulation + + , + = * + , + 1 + Γ + + + , + − + amplitude modulation
+ + + , + superposition .(4)
Here, Γ is the amplitude sensitivity, and the large-scale imprints + can be found from a given outer-layer large-scale signal + ( + ) via
+ + , + = F −1 + , + F + + ,(5)
where F [·] and F −1 [·] represent a Fourier transform and inverse Fourier transform, respectively, and ( + ) is the linear transfer kernel incorporating the large-scale coherence between the near-wall and outer regions.
The gain of the linear kernel | | is related to the linear coherence spectrum 2 ( + ) via
| ( + )| = 2 ( + ) |F [ + ]| 2 F + 2 ,(6)
where 2 ( + ) is given by Bendat and Piersol [49] 2
( + ) = F + F [ + ] 2 F + 2 |F [ + ]| 2 ,(7)
with (·) denoting the complex conjugate.
B. Quantitative relationship between and IOIM parameters
In this subsection, we derive a quantitative relationship of the amplitude modulation coefficient by expressing in the framework of IOIM. Following Duvvuri and McKeon [36], we work with a modified expression of the amplitude modulation coefficient
2 ( + ) = 2 + ( + , + ) + ( + , + ) 2 2 + ( + , + ) +2 ( + , + ) ,(8)where 2 + = L 2 + + H 2 +(9)
for an arbitrary time series ( + ). The modified coefficient 2 uses the square of the analytic signal to avoid the difficulty in dealing with the square root in Eq. (2), leading to a simpler mathematical expression, and given that the amplitude modulation coefficient is a normalised measure, no significant difference is expected in the values of and 2 . This can be easily verified using experimental data.
By expressing the velocity signals + and + in a series of Fourier modes, and with some trigonometric manipulations, Duvvuri and McKeon [36] showed that
2 = 2 +2 + 2 2 + +2 .(10)
We substitute the detrended signal expressed using the notations of IOIM (i.e. + = * 1 + Γ + + − + ) into Eq. (10) and noting that * 2 + = 0 because * and + are uncorrelated by definition, 2 can then be expressed as
2 = 4Γ * 2 + ( + ) + + − + + 2Γ 2 * 2 + ( + ) +2 + − + 2 [ * ] + 2ΓL * 2 + + H [ * ] H * + + Γ 2 2 * + 2 +2(11)
Comparing the orders of * , + and Γ in the terms in the denominator of Eq. (11) and noting that both L [·] and H [·] are linear operators give rise to
2 [ * ] ∼ * 2 ,(12a)ΓL * 2 + + H [ * ] H * + ∼ Γ +2 1/2 * 2 ,(12b)Γ 2 2 * + ∼ Γ 2 +2 * 2 .(12c)
Typically, Γ is a relatively small number ranging from O(0.01) to O(0.10) [22], and +2 < * 2 especially in the near-wall region of + 200 which is the current focus. Therefore, terms containing higher orders of Γ +2 1/2 can be neglected, and Eq. (11) is then reduced to
2 = 4Γ * 2 + ( + ) + + − + 2 2 [ * ] +2(13)
We identify the following 3 parameters that can contribute to a change in 2 :
1. The relative shift + .
2. The amplitude sensitivity Γ.
3. The amplitude of large-scale imprints +2 .
Interestingly, 2 is not affected by changes in the amplitude of universal small scales * 2 . This is because both the numerator and the denominator in Eq. (11) (and Eq. (13) as well) contain the same order of * 2 , which eventually cancel out. Note that changes in the energy distribution across scales in + or * have more complicated consequences: for + , it will affect the auto-correlation term + ( + ) + + − + with a given + , and for * , it will affect how much energy remains in 2 [ * ] after a low-pass filter is applied. Therefore, we limit the quantitative analysis to changes in the amplitude of + and * fluctuations.
In canonical smooth-wall turbulent boundary layers, the IOIM parameters ( * , Γ, + , and ) are Reynolds number invariant over the range of ≈ 7350−13300 tested in the calibrations [50], and only a change in the large-scale imprint amplitude +2 was observed when varying the Reynolds number. However, + , Γ and +2 can all be modified under non-canonical conditions, and it is important to understand how they individually contribute to the overall amplitude modulation coefficient .
Effect of the relative shift +
We can conclude from Eq. (13) that 2 would increase with a decreasing + , as a result of reduced time lag, or improved alignment between modulation and superposition signals. Typically, the relative shift + is much smaller than the time period of + below the centre of the log region, where the predictive model is applied. Therefore, + ( + ) and + + − + are largely in phase.
The auto-correlation term + ( + ) + + − + is positive, and increases with a decreasing + . Fig. 3(a) shows (Eq. 1) computed from the velocity signals constructed from * and + with a range of different + values following (4). The IOIM parameters and the baseline values of + 0 , Γ 0 and +2 0 are taken from the dataset at = 13300 [22]. The decreasing trend of with an increasing + confirms the conclusion based on Eq. (13).
Effect of the amplitude sensitivity Γ
According to Eq. (13), 2 will increase with an increasing Γ. This trend is corroborated by the increasing computed from the signals constructed following Eq. (4) with an increasing Γ/Γ 0 (see Fig. 3b).
Effect of the superposition intensity +2
Based on Eq. (13), 2 will increase with an increasing +2 . In fact, a closer examination of the Eqs. (13) and (11) reveals that Γ and +2 can be grouped into a single variable Γ +2 1/2 . This is confirmed by the same trends of with Γ and +2 1/2 in Figs. 3(b) and (c), respectively.
We note that the monotonic increase of 2 with Γ +2 1/2 breaks down when Γ +2 1/2 becomes comparable with or larger than * 2 1/2 (not shown in the figure), and the full expression of Eq.
(11) can introduce non-monotonic dependence on Γ +2 1/2 . However, given that Γ is typically small, and +2 is much smaller than * 2 in the near-wall region at the Reynolds number range investigated in the experimental dataset ( 2 × 10 4 ), the non-monotonic is less likely to occur.
To summarise, we have shown analytically that the amplitude modulation coefficient will increase with (i) a reducing + , (ii) an increasing Γ, and (iii) an increasing +2 , provided
Γ +2 1/2 ≪ * 2 1/2 .
C. Verification with a smooth-wall turbulent boundary layer dataset
The analysis above is verified using an experimental smooth-wall turbulent boundary layer dataset with ranging from 2800 to 13400 [48]. In this series of canonical smooth-wall boundary layer profiles, both + and Γ are expected to remain constant, and only +2 increases with an increasing as a consequence of more energetic large-scale motions in the outer layer, providing an ideal test ground to examine the dependence of on +2 . Fig. 4 shows the and (c) +2 values. The baseline parameters are denoted by + 0 , Γ 0 and +2 0 , respectively, and they are computed from the dataset at = 13300 in Baars et al. [22]. The arrows in the panels indicate the direction of increasing the ratio between the varied parameter and the baseline. amplitude modulation coefficients and 2 computed directly from the velocity time series. The large-scale imprint intensity +2 is approximated by +2 − * 2 , where * 2 is the intensity of the universal small-scale signal from the calibration data of Baars et al. [22]. The large-scale imprint intensity +2 increases with an increasing as expected. The two coefficients and Assuming a perfect auto-correlation (which is essentially Eq. (13) with a zero time lag, + = 0) results in an overestimated 2 as shown by the black line. The auto-correlation term at + ≈ 50 computed from the experimental data increases slightly from 0.56 to 0.69 from the lowest to the highest measurements, because the relative phase shift + + / + reduces with increasing , with + remaining constant and + , the most energetic wavelengths of + , growing with the boundary layer thickness + (≡ ). After taking this into account, the prediction of Eq. (13) (dot-dashed blue line) captures the trend of data points very well. In summary, the increase in and 2 with increasing for a canonical smooth-wall turbulent boundary layer is primarily originated from the growing +2 , and the slight increase in the auto-correlation term also makes a small contribution to the growth.
III. REVISITING SCALE INTERACTIONS IN PREVIOUS STUDIES
In this section, we present a summary of data from the literature and the current work, focusing on how the IOIM parameters change in various flow types and with increasing , and discuss their commonalities and differences in the scale interaction mechanism.
Based on whether the near-wall cycle is modified from that of an impermeable smooth wall, various flow types collected in Table I can be further classified as 'top-down' and 'bottom-up' categories, which indicate whether the deviation from a canonical flow is introduced in the outer layer ('top-down') or near to the wall ('bottom-up'). We note that although Duvvuri and McKeon [36] used a dynamic roughness element upstream to generate the synthetic signal, the wall remains smooth at the location where the scale interaction is quantified, and therefore, this study is also classified as a 'top-down' type.
Notably, a higher amplitude modulation coefficient is found in all non-canonical flows collected in the table. A closer examination of the IOIM parameters would reveal that this increase in is caused by different mechanisms in each category. For rough-wall flows, although the strength of +2 is similar to that of a smooth-wall boundary layer, the coherence between the inner and outer layer is reduced, presumably due to the interruption of the shedded vortices in the roughness sublayer. The higher value is contributed by the increased Γ, which can be explained as follows [28]. In smooth-wall flows, small-scale fluctuations scale with viscous units. Considering that the large-scale variation of friction velocity is quasi-steady from the viewpoint of small scales, the small-scale velocity fluctuations will then scale with . By definition, ∝ √ , where is the instantaneous wall-shear stress and it is caused by and proportional to the large-scale velocity signature [12,51]. Thus, the small-scale velocity fluctuations are proportional to √ . In fullyrough flows, however, the small-scale fluctuations are the wake vortices originated from roughness elements, the intensity of which is proportional to . Therefore, the modulation sensitivity in rough-wall flows is expected to be stronger than that in smooth-wall flows. Following a similar argument, the near-wall region of a permeable substrate is dominated by upwelling/downwelling associated with the large-scale streamwise motions (∝ ) [32,33], leading to an increase in as well.
Regarding the 'top-down' flow category, more energetic large scales can be introduced in the outer layer by increasing in a canonical smooth-wall boundary layer, through FST, or an upstream dynamic roughness element. Another example in this category, to be discussed in Sec.
IV, is that of a rough-to-smooth change in the wall condition, where the introduced large scales are the remnant of upstream rough-wall large-scale structures. The near-wall cycle (represented by the intensity of * ) is little modified in all cases. The coherence between large scales in the inner and outer regions, which manifests in | |, either remains the same (canonical smooth wall with increasing ), or increases (FST), and they all lead to stronger footprints +2 close to the wall.
The stronger +2 is primarily responsible for the increased in the 'top-down' flow cases.
IV. CASE STUDY: TURBULENT BOUNDARY LAYER FOLLOWING A STEP CHANGE IN
SURFACE ROUGHNESS
In this section, we present the new experimental results of a turbulent boundary layer downstream of a rough-to-smooth change in the surface conditions, which is another non-canonical flow configuration in the 'top-down' category. We will briefly present the experimental conditions in Sec. IV A, and then explore the detailed modulation behaviours using both single-probe and two-probe results in Secs. IV B-IV C. The results are briefly discussed in Sec. IV D.
A. Experimental setup
Simultaneous two-probe hotwire anemometry measurements are performed in the High Abrasives) from the inlet to 0 = 7.2 m, while the remaining length is a smooth aluminium surface.
The peak-to-trough roughness height is ≈ 1.2 mm, and the equivalent sand grain roughness is ≈ 2.43 mm. A nominal zero-pressure gradient is achieved by adjusting the bleeding slots on the tunnel roof. More details of the facility can be found in [44] and [54].
Two-probe hotwire measurements are performed at two freestream velocities, 22.5 m s −1 and 31.0 m s −1 , and these two cases are named as R-to-S1 and R-to-S2, respectively. The flow conditions of these cases correspond to the single-point dataset of cases Re10ks16 and Re14ks22
in [44], and the local friction velocity of the current cases is interpolated from the skin-friction versusˆ trajectory in the same study, which was measured directly at the wall using oil-film interferometry. The procedures of the two-probe hotwire anemometry measurements are similar to that described by Mathis et al. [20] and Baars et al. [22]. The outer probe is fixed at + = 3.9 √ , which is the geometric centre of the logarithmic region where the large-scale motions are highly active. The inner probe is traversed from the wall to below the outer probe with approximately 20 logarithmically spaced points in between. Both probes are conventional single-wire hotwire probes with a Wollaston wire etched to expose the sensing element. The length-to-diameter ratio of the exposed filament is / ≥ 200 [55]. For the case R-to-S1, both wires have a diameter of = 2.5µm, leading to a viscous-scaled filament length of + , + ≈ 21. For the case R-to-S2, = 1.5µm wire is selected for the inner probe to maintain a similar spatial resolution with + ≈ 19 with an increased freestream (and friction) velocity, while the outer probe filament diameter remains = 2.5µm
because only the large-scale signal at this location is of the interest. Both probes are conventional single-wire hotwire probes operated by an in-house Melbourne University Constant Temperature using the corresponding smooth-wall friction velocity [48], and dashed black line is a rough-wall reference acquired just upstream of the rough-to-smooth change and normalised by the rough-wall friction velocity.
The white circles represent the edge of the IBL, determined from the variance profile [44]. The profile highlighted by triangular symbols is close to the streamwise location where the two-probe measurements in this study are performed.
Anemometer (MUCTA). The hotwire sampling time is more than 20000 boundary layer turnover time ( 99 / ∞ ) to achieve a good convergence of the statistics. A two-probe smooth-wall dataset from Mathis et al. [20] and Baars et al. [22] is also included for comparison in this study.
Parameters of the two-point measurements are summarised in Table II.
B. Single-probe results
We first present the evolution of the mean velocity and turbulence statistics downstream of the rough-to-smooth change in Fig. 5. Generally speaking, immediately after the rough-to-smooth as evidenced by the increased inner-peak magnitude in wall units shown in Fig. 5(a). The case R-to-S2 has a higher 0 and + 0 compared to the case R-to-S1, thus, stronger large-scale motions (which scale on 0 ) above the internal boundary layer are expected in the former.
The amplitude modulation coefficient (Eq. 1) at a range of downstream locations from Legends are the same as in Fig. 6. The solid black line is a smooth-wall reference with = 1.3 × 10 4 [48].
/ 0 = 0.08 to 53.1 is shown in Fig. 6. The cut-off wavelength of the large-scale filter L [·]
is computed using the local friction velocity. Only the results of R-to-S1 are presented here for brevity, while the R-to-S2 case shows similar behaviour. At smallˆ / 0 , a high is observed in the near-wall region. The coefficient decreases with an increasingˆ / 0 , and beyondˆ / 0 > 20, it becomes very similar to that of a smooth-wall boundary layer.
Isocontours of (Eq. 3) atˆ / 0 = 0.2 (orange) and 0.9 (green) are shown in Fig. 7. In addition to the difference in the magnitude, at small fetches, the contours of positive correlations also shifts to the positive + direction. In other words, close to the rough-to-smooth change, there is a smaller time lag between the envelope of the small-scale fluctuations and large-scale motions.
The optimal positive correlation achieved at the time shift + is shown in Fig. 7(b). Similar to the zero-time-shift (shown in Fig. 6), the maximum values are also higher at smaller fetches, implying that the high magnitudes of observed in Fig. 6 is more than the consequence of a smaller lag between the envelope and large-scale motions.
A comparison of + (time shift required for the optimum positive correlation) can be found in Fig. 8(a), where + becomes more negative with an increasing fetch. Fig. 8(b) shows the argument of the complex co-spectrum at + = 50, premultiplied by the time period 1/ + , following Jacobi and McKeon [56,57] and Deshpande et al. [58]. The co-spectrum is defined
as ≡ F + F [ + ]
, and it can be viewed as the spectral equivalent of the amplitude modulation coefficient . The premultiplied argument + ( + ) ≡ arg( )/(2 + ) is essentially the time shift between + and [ + ] in each Fourier mode, and it is reasonable that the overall time shift + falls in the same range as + ( + ) at each corresponding measurement location. In addition, the increase of the time lag with increasing fetch is also apparent here, confirming the trend of + in Fig. 8(a). Further, the absolute time lag + ( + ) is smaller at higher frequencies, meaning that the amplitude of small scales are more in-phase with the higher-frequency modes of the large scales. The increased maximum values at smaller fetches can partially be explained by the smaller scatter of + ( + ) across a range of frequencies, because a single time shift + can better align all Fourier modes in the + and [ + ] signals in these cases.
The increase in compared to the smooth-wall reference has been previously observed in rough-wall flows and attributed to the stronger correlation between the amplitude of the small-scale turbulence associated with the roughness elements and the large-scale motions [28]. Compared to the most downstream location, in the logarithmic region is still noticeably higher atˆ / 0 = 0.9, which is equivalent toˆ / ≈ 110, a fetch where we might expect a large portion of the small-scale motions directly generated from the flow interaction with the roughness elements to diminish.
However, small scales may form through the shear between the surviving rough-wall structures, and exhibit a stronger amplitude modulation effect with those structures from which they originate.
In summary, based on the single-point measurements of the rough-to-smooth cases, the increase in is contributed by both reduced | + | and increased +2 , which are two out of the three factors with a spacing of 0.2. Line colours black, green and magenta represent the smooth, R-to-S1 and R-to-S2 cases, respectively. (b) and (c) are the difference in + + between the R-to-S1 and R-to-S2 cases and the smooth-wall reference, and (e) and (f ) are the corresponding difference in + + * * .
identified in the analysis in Sec. II. Further two-probe measurements are required to quantify the effect of the modulation sensitivity Γ.
C. Two-probe results
To further understand the origin of the enhanced modulation effect observed in Sec. IV B, IOIM calibration following the procedure detailed in Baars et al. [22] is performed on the three cases listed in Table II. In the results below, we will be using black for the smooth-wall reference, green and magenta for the R-to-S1 and R-to-S2 cases, respectively. . Line colours black, green and magenta represent the smooth, R-to-S1
and R-to-S2 cases, respectively. moving inner probe at + = 100. Both quantities are filtered by a 25% bandwidth moving filter. Line colours black, green and magenta represent the smooth, R-to-S1 and R-to-S2 cases, respectively. The premultiplied energy spectrum + + is also shown in (b) by thin lines of corresponding colours on the left vertical axis for reference.
interpreted as the large-scale footprints in the near-wall region. The universal small scale spectra of the three cases (Fig. 9d) are very similar, indicating a re-establishment of the near-wall cycle after the rough-to-smooth change.
The sensitivity of small scales to amplitude modulation is indicated by Γ: for a higher Γ, the universal small-scale signal * will be multiplied by a higher fraction of the superposition signal + to generate the prediction. Two calibrations of a smooth-wall turbulent boundary layer at ≈ 7350 and 13300 result in very similar Γ [22]. For rough-to-smooth cases, as shown in Fig. 10, Γ from the three calibrations reach a good overall agreement, suggesting that the over-energised rough-wall structures in the outer layer do not seem to alter the amplitude modulation mechanism in the near-wall region. Fig. 11(a) shows | |, the gain of the linear kernel (which relates the superposition + to the outer-layer large-scale signal + via Eq. 5), at a wall-normal position of + = 100. Similar trends are also seen in other wall-normal positions, and are not shown here for brevity. The magnitude of the linear transfer kernel | | is found to increase in the rough-to-smooth cases compared to the smooth-wall reference. The linear coherence spectra are shown in Fig. 11(b). In the rough-tosmooth cases, 2 deviates from 0 at a smaller 1/ + , and remains higher than that of the smooth-wall reference. A higher 2 indicates a stronger correlation between the velocity fluctuations obtained by the inner and outer probes. These results suggest that for a given structure in the outer layer, it will leave a stronger footprint (superposition) in the near-wall region in the rough-to-smooth case as a result of the enhanced inner-outer coherence.
D. Discussion
Overall, the amplitude modulation mechanism appears to be little modified after introducing a roughness heterogeneity (at least atˆ / 0 = 2.3 as examined here). However, there is a stronger correlation between the large-scale velocity fluctuations obtained by the inner and outer probes, and the gain in the linear kernel is also higher in rough-to-smooth cases. The large-scale fluctuation +2 is already stronger in the rough-to-smooth cases, and a larger fraction of it will contribute to the near-wall superposition signal through the increased gain | |. In light of the analysis in Sec. II, the increase of +2 is primarily responsible for the higher value downstream of a rough-to-smooth change.
It is interesting to draw a direct comparison between the FST [34,35] and the current rough-tosmooth cases, as both are in the 'top-down' category with broadband energetic large-scale motions imposed in the outer region. The increased strength in the outer large scales makes them less susceptible to the interruptions from near-wall motions, leading to an increase in the correlation between outer-and near-wall large-scale signals, which is eventually reflected in an higher | | in both. The major difference between the two is in the coefficient Γ: the former has a lower Γ, while the value is unchanged in the latter, similar to the independence of Γ on values observed in canonical smooth-wall boundary layers [22]. We speculate that such difference is rooted in the manner in which the outer large scales are organised, as well as the energetic wavelengths. In the FST case, the outer structures are created by an active grid, which are inherently different from the structures organised by hairpin vortices in a developing boundary layer. The dissimilarity in the generation mechanism between large-and small-scale structures in the FST might be the reason for the reduction in Γ. The highly energetic large scales arranged in a manner similar to that of a naturally developed boundary layer in the current configuration makes it a good mimetic of a smooth-wall turbulent boundary layer at very high Reynolds numbers.
V. CONCLUDING REMARKS
In this work, we first show the underlying physics behind the increased amplitude modulation coefficient as reported in many previous studies, utilising the framework of IOIM. An analytical relationship between the amplitude modulation coefficient 2 and IOIM parameters is derived and verified using a smooth-wall turbulent boundary layer dataset. This framework is then applied to classify and interpret the reported amplitude modulation behaviours in previous works. We then present the case study of a turbulent boundary layer downstream of a rough-to-smooth change with both single probe and simultaneous two-probe measurements. A stronger amplitude modulation effect evidenced by an increased is observed. Further analysis of the two-probe data reveals that the modulation strength Γ is similar to that of a canonical smooth-wall turbulent boundary layer, and it remains the same for the two Reynolds numbers tested. The increase in is primarily attributed to the stronger large-scale footprints +2 in the near-wall region, which is contributed by both the over-energetic outer layer motions +2 and stronger coherence between the inner and outer layers. These results and analyses offer a new perspective to interpret the abundant literature on the the scale interactions of non-canonical turbulent boundary layers, which can be meaningful for incorporating the IOIM in the numerical simulation of a wide range of flow conditions.
FIG. 1 .
1Schematic of a turbulent boundary layer over a rough-to-smooth change in surface condition. The roughness transition occurs at 0 , andˆ = − 0 denotes the fetch downstream of the transition. Reproduced from Li et al.[38].
FIG. 2 .
2(a) Amplitude modulation coefficient ( + ) (Eq. 1) of a smooth-wall turbulent boundary layer with = 1.3 × 10 5 [48]. (b) Isocontour of the time-shifted amplitude modulation coefficient ( + , + ) (Eq. 3) of the same boundary layer profile. The solid contour lines are from 0.1 to 0.5 with a step of 0.1, and the dashed contour lines are from −0.3 to −0.1 with a step of 0.1. The solid and filled circles mark the + values where reaches its maximum and minimum at each wall-normal location, respectively. (c) Premultiplied energy spectra of the measured velocity + , universal small scales * and superposition + at + ≈ 10, and + , the velocity measured by the outer probe at + = 469. (d) The gain of the linear kernel | | and the linear coherence spectrum 2 ( + ) at + ≈ 10. The vertical dotted line in (c) and (d) marks the cut-off wavelength + = 7000 (with the mean velocity at the outer probe as the convective velocity) used by
As shown in Figs. 2(c) and (d), only the large-scale energy of the outer velocity signal + is retained in the superposition signal + , which contributes to the large-scale end of the energy spectrum of + . The small-scale end of the spectrum, on the other hand, is mainly from the universal small scale, * . The linear transfer kernel | | enables a smooth roll-off of the coherence from large to small scales, and the scale separation here is around the commonly used cut-off threshold of + = 7000.
FIG. 3 .
3Amplitude modulation coefficient of the velocity signal constructed with various (a) + , (b) Γ
FIG. 4 .
4Amplitude modulation coefficients (grey circles) and 2 (blue triangles) of a canonical smoothwall turbulent boundary layer plotted against the corresponding superposition signal intensity +2 at + = 50. From pale to dark blue, the shade of blue triangles indicates increasing from 2800 to 13400.The solid black line is Eq. (13) with a zero time lag ( + = 0), and the dot-dashed blue line is Eq. (13) with the auto-correlation term + ( + ) + + − + estimated from the experimental data.
FIG. 5 .
5Profiles of mean streamwise velocity and turbulence intensity corresponding to the flow conditions of R-to-S1. (a) is inner scaled using the local smooth-wall , while (b) is outer scaled, but using 0 , the friction velocity measured just upstream of the rough-to-smooth transition. Line colours indicate the fetch, from orange to green to blue corresponds toˆ / 0 = 0.2, 0.5, 0.9, 1.9, 4.2, 7.4, 14.8, 29.2 and 53.1. The solid black line is a smooth-wall reference with = 1.0 × 10 4 acquired in the same facility and normalised
FIG. 6 .
6Amplitude modulation coefficient at various streamwise locations downstream of a rough-tosmooth change. Line colours indicate the fetch, from red to blue corresponds toˆ / 0 = 0.53.1. The white circles represent the edge of the IBL. The solid black line is a smooth-wall reference with = 1.0 × 10 4 [48]. change, the rough-wall turbulent boundary layer starts adapting to the new surface conditions first at the wall, and as the modified region enlarges, to the interior of the flow. The skin-friction coefficient experiences an undershoot before gradually increasing to the smooth-wall value. Fig. 5(a) shows the inner-scaled mean velocity profiles, where the velocity scale is selected as the friction velocity measured locally with oil-film interferometry over the smooth surface. Note that in this paper, the superscript (·) + indicates inner scaling with the local friction velocity as the velocity scale. Fig. 5(b) shows the outer-scaled profiles, where the velocity scale is chosen as the friction velocity 0 measured on the rough wall just upstream of the rough-to-smooth transition atˆ → 0 − . The inner-scaled mean velocity profiles of Fig. 5(a) collapse with the smooth-wall reference first close to the wall, while the outer-scaled profiles in Fig. 5(b) agree well with the rough-wall reference above the IBL (marked by the open circles). In the turbulence intensity profiles, a strong 'outer-peak' manifests at the IBL, which is a result of the remaining energetic rough-wall structures. These structures leave a strong footprint in the near-wall region,
atˆ / 0 = 0.2 (orange) and 0.9 (green) and the smooth-wall reference (black). The solid contour lines are from 0.1 to 0.5 with a step of 0.1, and the dashed contour lines are from −0.3 to −0.1 with a step of 0.1. The solid and empty circles mark the + values where reaches its maximum and minimum at each wall-normal location, respectively. (b) Maximum values, which are essentially at the locations marked by solid circles in (a).
FIG. 8 .
8(a) Time shift required for to reach its maximum. (b) Argument of the complex co-spectrum of + and [ + ] at + = 50 (marked in (a) by the vertical dashed line), premultiplied by the time period 1/ + .
FIG. 9 .
9Premultiplied spectra of (a) + and (d) * . The contour levels in both plots are from 0.2 to 2
Premultiplied energy spectra of the measured velocity fluctuation and the universal small-scale signal are shown in Figs. 9(a) and (d), respectively. Figs. 9(b) and (c) are the difference between the rough-to-smooth and smooth-wall reference. A band with excess energy at 1/ + ≈ 2000 is
FIG. 10 .
10Amplitude sensitivity Γ( + )
11. (a) | |, gain of the linear kernel and (b) 2 , linear coherence spectra between the fixed outer and
TABLE I. Summary of IOIM parameters in various flow types. Quantities that cannot be inferred from the reported results are marked by '?'. Changes in the parameters are relative to a smooth-wall turbulent boundary layer at comparable Reynolds numbers. In the category of smooth-wall boundary layers with increasing Reynolds numbers, the comparison is made with regard to a lower Reynolds number. Results of the current study will be presented in details in Sec. IV.Reynolds Number Boundary Layer Wind Tunnel (HRNBLWT) at the University of Melbourne. An
upstream portion of the 27 m working section floor is covered by P24 sandpaper (SP40F, Awuko
Flow type
Reference
+2
+2
| |
* 2
Γ
(near wall)
top-down
R-to-S change
present study
higher
higher
higher
similar
similar
higher
Smooth wall,
increasing
Mathis et al. [14, 20]
Baars et al. [22]
higher
higher
similar
similar
similar
higher
FST
Dogan et al. [34, 35]
higher
higher
higher
similar
lower
higher
Single frequency
large-scale input
Duvvuri and McKeon [36]
higher
higher
?
similar
?
?
bottom-up
Rough wall
Squire et al. [28]
Anderson [29]
Pathikonda and Christensen [52]
Blackman et al. [31]
similar
lower
lower
dependent on
roughness
morphorlogy
higher
higher
Permeable wall
Efstathiou and Luhar [53]
Kim et al. [32]
similar/lower
?
?
dependent on
substrate
morphorlogy
?
higher
Case
Colourˆ / 0
3.9
√
∞
+
+
+
+
(ms −1 )
(ms −1 )
Smooth
black
-
7350
334
10.0
0.34
441
-
22
22
R-to-S1
green
2.3
7200
330
22.5
0.69
331
890
21
21
R-to-S2
magenta
2.3
9600
382
31.0
0.92
388
1180
19
31
TABLE II. Experimental parameters of the two-point hotwire measurements, where the outer probe remains
at a fixed location + and the inner probe is traversed between [0, + ]. + and + are the viscous-scaled
hotwire filament length of the inner and outer probes, respectively.
have very similar values, and they exhibit an increasing trend with +2 , which is well captured by Eq.(13). The auto-correlation term + ( + ) + + − + +2 in Eq. (13) is smaller than 1 for a finite time shift + , but the exact value need to be determined from the time series of + .
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| [] |
[
"Simulations of Protoplanetary Disk Dispersal: Stellar Mass Dependence of the Disk Lifetime",
"Simulations of Protoplanetary Disk Dispersal: Stellar Mass Dependence of the Disk Lifetime"
] | [
"Ayano Komaki \nDepartment of Physics\nThe University of Tokyo\n7-3-1 Hongo113-0033BunkyoTokyoJapan\n",
"Shuhei Fukuhara \nDepartment of Multi-Disciplinary Sciences\nGraduate School of Arts and Sciences\nThe University of Tokyo\n3-8-1 Komaba153-8902MeguroTokyoJapan\n",
"Takeru K Suzuki \nDepartment of Multi-Disciplinary Sciences\nGraduate School of Arts and Sciences\nThe University of Tokyo\n3-8-1 Komaba153-8902MeguroTokyoJapan\n\nKomaba Institute for Science\nThe University of Tokyo\n3-8-1 Komaba153-8902MeguroTokyoJapan\n\nDepartment of Astronomy\nGraduate School of Science\nThe University of Tokyo\n7-3-1 Hongo113-0033BunkyoTokyoJapan\n",
"Naoki Yoshida \nDepartment of Physics\nThe University of Tokyo\n7-3-1 Hongo113-0033BunkyoTokyoJapan\n\nKavli Institute for the Physics and Mathematics of the Universe (WPI)\nUT Institute for Advanced Study\nThe University of Tokyo\n277-8583KashiwaChibaJapan\n\nSchool of Science\nResearch Center for the Early Universe (RESCEU)\nThe University of Tokyo\n7-3-1 Hongo113-0033BunkyoTokyoJapan\n"
] | [
"Department of Physics\nThe University of Tokyo\n7-3-1 Hongo113-0033BunkyoTokyoJapan",
"Department of Multi-Disciplinary Sciences\nGraduate School of Arts and Sciences\nThe University of Tokyo\n3-8-1 Komaba153-8902MeguroTokyoJapan",
"Department of Multi-Disciplinary Sciences\nGraduate School of Arts and Sciences\nThe University of Tokyo\n3-8-1 Komaba153-8902MeguroTokyoJapan",
"Komaba Institute for Science\nThe University of Tokyo\n3-8-1 Komaba153-8902MeguroTokyoJapan",
"Department of Astronomy\nGraduate School of Science\nThe University of Tokyo\n7-3-1 Hongo113-0033BunkyoTokyoJapan",
"Department of Physics\nThe University of Tokyo\n7-3-1 Hongo113-0033BunkyoTokyoJapan",
"Kavli Institute for the Physics and Mathematics of the Universe (WPI)\nUT Institute for Advanced Study\nThe University of Tokyo\n277-8583KashiwaChibaJapan",
"School of Science\nResearch Center for the Early Universe (RESCEU)\nThe University of Tokyo\n7-3-1 Hongo113-0033BunkyoTokyoJapan"
] | [] | Recent infrared and submillimeter observations suggest that the protoplanetary disk lifetime depends on the central stellar mass. The disk dispersal is thought to be driven by viscous accretion, magnetohydrodynamics (MHD) winds, and photoevaporation by the central star. We perform a set of onedimensional simulations of long-term disk evolution that include all the three processes. We vary the stellar mass in the range of 0.5-7 M , and study the mass dependence of the disk evolution. We show that a significant fraction of the disk gas is lost by MHD winds in the early stage, but the later disk evolution is mainly governed by photoevaporation. The disk radius decreases as photoevaporation clears out the gas in the outer disk efficiently. The qualitative evolutionary trends of the disk mass are remarkably similar for the wide range of the central stellar mass we consider, and the time evolution of the disk mass can be well fitted by a simple function. The dispersal time is approximately ten million years for low mass stars with weak mass dependence, but gets as short as two million years around a 7 M star. In the latter case, a prominent inner hole is formed by the combined effect of accretion and MHD winds within about one million years. The strength of the MHD wind and viscous accretion controls the overall mass-loss rate, but does not alter the dependence of the dispersal timescale on the central stellar mass. | null | [
"https://export.arxiv.org/pdf/2304.13316v1.pdf"
] | 258,332,034 | 2304.13316 | f28092d3da13f05c74ac41336b6522d88f4531f2 |
Simulations of Protoplanetary Disk Dispersal: Stellar Mass Dependence of the Disk Lifetime
April 27, 2023
Ayano Komaki
Department of Physics
The University of Tokyo
7-3-1 Hongo113-0033BunkyoTokyoJapan
Shuhei Fukuhara
Department of Multi-Disciplinary Sciences
Graduate School of Arts and Sciences
The University of Tokyo
3-8-1 Komaba153-8902MeguroTokyoJapan
Takeru K Suzuki
Department of Multi-Disciplinary Sciences
Graduate School of Arts and Sciences
The University of Tokyo
3-8-1 Komaba153-8902MeguroTokyoJapan
Komaba Institute for Science
The University of Tokyo
3-8-1 Komaba153-8902MeguroTokyoJapan
Department of Astronomy
Graduate School of Science
The University of Tokyo
7-3-1 Hongo113-0033BunkyoTokyoJapan
Naoki Yoshida
Department of Physics
The University of Tokyo
7-3-1 Hongo113-0033BunkyoTokyoJapan
Kavli Institute for the Physics and Mathematics of the Universe (WPI)
UT Institute for Advanced Study
The University of Tokyo
277-8583KashiwaChibaJapan
School of Science
Research Center for the Early Universe (RESCEU)
The University of Tokyo
7-3-1 Hongo113-0033BunkyoTokyoJapan
Simulations of Protoplanetary Disk Dispersal: Stellar Mass Dependence of the Disk Lifetime
April 27, 2023(Received; Revised; Accepted) Submitted to ApJDraft version Typeset using L A T E X twocolumn style in AASTeX63
Recent infrared and submillimeter observations suggest that the protoplanetary disk lifetime depends on the central stellar mass. The disk dispersal is thought to be driven by viscous accretion, magnetohydrodynamics (MHD) winds, and photoevaporation by the central star. We perform a set of onedimensional simulations of long-term disk evolution that include all the three processes. We vary the stellar mass in the range of 0.5-7 M , and study the mass dependence of the disk evolution. We show that a significant fraction of the disk gas is lost by MHD winds in the early stage, but the later disk evolution is mainly governed by photoevaporation. The disk radius decreases as photoevaporation clears out the gas in the outer disk efficiently. The qualitative evolutionary trends of the disk mass are remarkably similar for the wide range of the central stellar mass we consider, and the time evolution of the disk mass can be well fitted by a simple function. The dispersal time is approximately ten million years for low mass stars with weak mass dependence, but gets as short as two million years around a 7 M star. In the latter case, a prominent inner hole is formed by the combined effect of accretion and MHD winds within about one million years. The strength of the MHD wind and viscous accretion controls the overall mass-loss rate, but does not alter the dependence of the dispersal timescale on the central stellar mass.
INTRODUCTION
Recent observations discovered diverse planetary architectures around various types of stars (Fulton et al. 2017;Zhang et al. 2018). Planets are formed in protoplanetary disks (PPDs) out of disk materials within some limited time, and thus the dynamical evolution of a PPD affects significantly planet formation. In particular, disk dispersal process and its characteristic timescale are thought to be critically important for setting the scene for the formation of diverse planets.
Observations of several star forming regions suggest that PPDs have a lifetime of a few million years (e.g., Haisch et al. 2001;Meyer et al. 2007;Hernández et al. [email protected] 2007;Mamajek 2009;Bayo et al. 2012;Ribas et al. 2014). It is also found that the disk lifetime may depend on the central stellar mass (e.g., Carpenter et al. 2006;Lada et al. 2006;Allers et al. 2007;Dahm & Hillenbrand 2007;Kennedy & Kenyon 2009;Fang et al. 2012;Yasui et al. 2014;Ribas et al. 2015). For example, Ribas et al. (2015) sorted disks by the system age and the central stellar mass, and calculated the disk fraction of each sample. Based on the finding that the disk fraction decreased as the stellar mass increased, they concluded that disks around high-mass stars had shorter lifetimes.
The three major disk dispersal mechanisms proposed so far are viscous accretion onto the central star, magnetohydrodynamic (MHD) winds, and photoevaporation. In a PPD, efficient angular momentum transfer causes the gas of the inner disk to fall onto the star. The ac-cretion rate is characterised by α parameter (Shakura & Sunyaev 1973;Lynden-Bell & Pringle 1974), which can be estimated in several ways based on observations (Calvet et al. 2005;Fedele et al. 2010;Mathews et al. 2012). Interestingly, a positive correlation is suggested between the stellar mass and the accretion rate (Muzerolle et al. 2003;Hartmann et al. 2006;Herczeg & Hillenbrand 2008;Fairlamb et al. 2015;Hartmann et al. 2016).
The role of MHD disk winds in the evolution of PPDs has attracted considerable attention in recent years (e.g., Pascucci et al. 2022). In addition to the direct mass loss (Suzuki & Inutsuka 2009), MHD disk winds remove angular momentum from the disk, which promotes mass accretion (Blandford & Payne 1982;Bai 2016;Béthune et al. 2017;Gressel et al. 2020). As a result, the radial profile of surface density is significantly altered from the one without the effect of MHD disk winds (Suzuki et al. 2016;Hasegawa et al. 2017;Tabone et al. 2022). MHDdriven winds also cause a great impact on the evolution of solid particles in PPDs (Taki et al. 2021;Arakawa et al. 2021) and the formation and migration of planets (Ogihara et al. 2018;Kimmig et al. 2020).
Photoevaporation is driven by high-energy radiation such as far-ultraviolet (FUV; 6 eV hν < 13.6 eV), extreme-ultraviolet (EUV; 13.6 eV < hν 100 eV), and X-ray (100 eV hν 10 keV) photons, which heat the gas on the disk surface to launch photoevaporative flows. The effect of photoevaporation on the disk evolution has been studied by a number of authors (Hollenbach et al. 1994;Richling & Yorke 1997;Gorti & Hollenbach 2009;Ercolano et al. 2009;Owen et al. 2010;Tanaka et al. 2013;Wang & Goodman 2017;Nakatani et al. 2018a,b;Picogna et al. 2019;Komaki et al. 2021;Picogna et al. 2021). Gorti & Hollenbach (2009) performed 1+1 dimensional simulations of disk photoevaporation by varying the central stellar mass in the range of 0.3-7.0 M . They incorporated both accretion and photoevaporation as major disk dispersal processes. Komaki et al. (2021) used radiation hydrodynamics simulations to show that the mass-loss rate by photoevaporation increased as the central stellar mass.
It is important to notice that disk dispersal may likely be caused by a combination of multiple physical mechanisms, between which there can be complicated interplay, and that the whole dispersal process can last over a long time of several to ten million years. Clearly, it is necessary to study the long-term disk evolution with incorporating all the proposed dispersal mechanisms in a consistent manner. This motivated several theoretical and numerical studies (Clarke et al. 2001;Gorti et al. 2015;Kunitomo et al. 2020Kunitomo et al. , 2021, but none of them con-sider accurate radial profiles of photoevaporation that are derived from detailed radiation hydrodynamics calculation with non-equilibrium chemistry.
In the present paper, we perform a set of long-term one-dimensional simulations of PPD evolution considering realistic photoevaporation profiles. Our calculations incorporate, for the first time, physical models of viscous accretion, MHD winds, and photoevaporation for PPDs around a wide range of the central stellar mass. We follow the disk evolution until the disk is dispersed nearly completely, and derive the disk lifetime accurately. We also study in detail the mass-loss processes at a variety of evolutionary phases.
The rest of the paper is organized as follows. In Section 2, we explain the methods we apply. In Section 3, we show the main results. In Section 4, we discuss detailed properties of our model calculations. Finally in Section 5, we summarise the paper.
NUMERICAL METHODS
We perform long-term disk evolution simulations varying the central stellar mass in the range of 0.5-7.0 M . We adopt the cylindrical coordinates (r, φ, z) assuming the disk is axisymmetric about the z-axis. We follow the evolution of the disk surface density, Σ = ρ dz, which is the integrated gas density in z-direction. We calculate the time evolution of the gas temperature consistently until the disk disperses. We incorporate accretion, MHD winds and photoevaporation in the following manner. The governing equations are
∂Σ ∂t + 1 r ∂ ∂r (rΣv r ) +Σ w +Σ pe = 0, rΣv r = − 2 rΩ ∂ ∂r r 2 Σα rφ c 2 s + r 2 α φz ρc 2 s mid ,
whereΣ w andΣ pe are the surface mass-loss rates by MHD winds and photoevaporation, respectively. Details of the mass-loss profiles by MHD winds and by photoevaporation are described later in this section. We incorporate the two mass-loss mechanisms as a simple sum ofΣ w andΣ pe . Although this might overestimate the total mass-loss rates, we have confirmed that the disk evolution does not vary significantly if we use a conservative setup with taking the larger value ofΣ w andΣ pe , in which the only dominant mechanism operates at a given time. This is simply because only one effect dominates for most of the time during the disk evolution.
In the above equations, v r , Ω and c s are the velocity in r-direction, the angular velocity and the sound speed respectively. The subscript 'mid' expresses the value at the mid-plane. The dimensionless parameters α rφ and α φz express the efficiency of the viscous and wind-driven accretion, respectively. The mass-weighted averages are denoted as α rφ and α φz (see Suzuki et al. (2016) for the definition.) We calculate the disk gas temperature by considering irradiation from the central star and viscous heating following Suzuki et al. (2016). The z-averaged temperature is calculated from the balance equation
T 4 = T 4
irr + T 4 vis , where T irr and T vis are the temperatures corresponding to the energy by irradiation and by viscous heating. The irradiation temperature is given by
T irr = 280 K L * L 1/4 r 1 au −1/2 ,
where L * is the stellar bolometric luminosity and L is the solar bolometric luminosity. Here we assume that the temperature reaches equilibrium quickly so that the gas temperature is equal to the dust temperature. The above equation effectively approximates that the dust temperature is determined by the incident radiation from the central star. The bolometric luminosity of the central star at the chracteristic age of 1 Myr is adopted from Gorti & Hollenbach (2009). There are several stellar evolution models for pre-main sequence stars, but there remains a discrepancy especially for stars younger than 1 Myr (Tout et al. 1999;Siess et al. 2000). For completeness of our model, we use the stellar luminosity at the age of 1 Myr (Table 1).
The viscous heating temperature is determined by the equation
2σ SB T 4 vis = 3 8 τ R + 1 2τ P F rad ,
where σ SB , τ R , τ P , and F rad are Stefan-Boltzman constant, the Rosseland mean optical depth, the Planck mean optical depth, and the radiation flux from the disk surface (Nakamoto & Nakagawa 1994). This equation expresses the energy transfer in z-direction under the assumption of thermodynamical equilibrium in a geometrically thin disk. The disk gas is heated by viscous dissipation, and the deposited thermal energy diffuses out to the disk surface. A part of the energy liberated by accretion is finally released from the disk surface as radiation. The Rosseland mean optical depth is expressed as
τ R = κ R Σ,
where κ R is the opacity of dust grains and given by Hueso & Guillot (2005) as
κ R = 4.5 T 150 K 2 cm 2 g −1 (T < 150 K)
4.5 cm 2 g −1 (150 K ≤ T ≤ 1500 K) 0 cm 2 g −1 (T > 1500 K)
, assuming silicate and water ice as the dominant opacity agents (Pollack et al. 1985). Note that the temperature dependence changes at around 150 K because of water ice depletion. In our calculation, we adopt a smooth function κ R = 2.25 cm 2 g −1 1 − tanh T − 1500 K 150 K × min 1, T 150 K 2 which approximates the above values well. For lowtemperature dust, the Rosseland and Planck mean opacities can be expressed by a power-law, and the relation τ P = 2.4τ R holds at low temperature (Nakamoto & Nakagawa 1994). We thus set the Planck mean optical depth as τ P = max(2.4τ R , 0.5),
where a lower limit of 0.5 is adopted to reproduce the optically thin limit. We consider a wide range of the effective viscosity α as a parameter. Recent ALMA observations toward starforming regions suggest that the α parameter varies over two orders of magnitude (Rafikov 2017;Ansdell et al. 2018). Ansdell et al. (2018) estimated the gas radius of each disk from 12 CO line emission. They simulated the evolution of gas radius with varying α parameter in the range of 10 −4 -10 −2 assuming an initial disk model by Facchini et al. (2017) and compared the result with the observations. They found a wide distribution of the value of α over a few orders of magnitudes. Several observations have been conducted to estimate the disk accretion rate using the Hα equivalent width (Fedele et al. 2010;Mathews et al. 2012;Hartmann et al. 2016). These studies also have shown that there is a large variation in the accretion rate. The optical-UV observations toward T Tauri stars have shown that there is a positive relationship between the accretion rate and the stellar mass given asṀ acc ∝ M 2 * (Muzerolle et al. 2003;Calvet et al. 2004;Hartmann et al. 2006). We thus assume α rφ ∝ M * to match the observed trend. In general, viscous accretion is effective only in the radial portions of the disk where magnetorotational instability (MRI hereafter) (Velikhov 1959;Chandrasekhar 1961;Balbus & Hawley 1991) operates to generate strong turbulence. While MRI actively induces MHD turbulence in the inner region < 1 au and the regions near the disk surfaces owing to sufficient ionization, an MRI-inactive region called a dead zone (e.g., Gammie 1996) is supposed to occupy 1 r a few tens au of a PPD because of insufficient ionization near the midplane (Mori & Okuzumi 2016;Pinte et al. 2016;Flaherty et al. 2017). On the other hand, a low but finite value of α rφ can still be sustained by purely hydrodynamical processes such as vertical shear instability (Urpin & Brandenburg 1998;Nelson et al. 2013;Lin & Youdin 2015;Flock et al. 2020;Manger et al. 2020). Although the ionization degree is expected to be varied over a large distance from the central star, the detailed radial extent of the dead zone is not still well understood quantitatively. Considering viscous accretion is dominant in the inner region, we assume a constant α rφ throughout the disk. We adopt α rφ = 1.0 × 10 −4 (M * /1 M ) as our fiducial value. In order to examine the impact of enhanced accretion in MRI-active disks, we also perform simulations
with α rφ = 1.0×10 −2 (M * /1 M ), 1.0×10 −3 (M * /1 M ).
We parametrise the wind loss termΣ w , following Suzuki et al. (2010):
Σ w = (ρc s ) mid C w = ΣΩ √ 2π C w ,(0)
where C w expresses a dimensionless mass flux given by
C w = min (C w,0 , C w,e ) .
Here, the constant maximum value C w,0 is estimated from local shearing box MHD simulations of Suzuki & Inutsuka (2009). We adopt C w,0 = 2.0 × 10 −5 for the MRI-active case and C w,0 = 1.0 × 10 −5 for the MRIinactive case. We calculate the energetics-constrained mass flux following Suzuki et al. (2016). We consider two cases with strong and weak winds. In the strong wind case, all the liberated gravitational energy is used to drive MHD winds. In this case, the mass flux C w,e and the energy flux F rad are given by
C w,e = max 2 r 3 Ω(ρc s ) mid ∂ ∂r r 2 Σα rφ c 2 s + 2c s rΩ α φz , 0 F rad = max − 1 r ∂ ∂r r 2 ΣΩα rφ c 2 s , 0 .
In the weak wind case, a relatively small fraction of the sum of the liberated gravitational energy and the energy by viscous heating is spent to drive MHD winds, and the rest is emitted as radiation. We define the fractional ratio of energy used to launch winds as rad . We then set
C w = (1 − rad ) 3 √ 2πc 2 s r 2 Ω 2 α rφ + 2c s rΩ α φz F rad = rad 3 √ 2π(ρc 3 s ) mid 2 α rφ + rΩα φz (ρc 2 s ) mid .
For the main results presented in the following, we adopt the strong disk wind case as our fiducial model. We discuss the effect of weak wind separately in Section 4, where we set rad = 0.9 for the weak wind case. Note that this is a limiting case in which the radiation loss is maximally evaluated. Local shearing-box simulations show that the MHD wind torque satisfies α φz ∼ 10 −5 -10 −3 and has a positive dependence on the strength of the net vertical magnetic field B z (Bai 2013). The dependence is well approximated as
α φz ∝ B 2 z 8π(ρc 2 s ) mid 0.66 .
The variation of α φz can be described by the ratio of the surface density with respect to the initial value on the assumption that the vertical magnetic flux stays constant during the disk evolution. Then α φz is expressed as
α φz = min 10 −5 Σ Σ int −0.66 , 1 ,(0)
where Σ int is the initial surface density. Since the evolution of the magnetic field is still poorly understood, we place a conservative upper limit on the value of α φz . We should note that Armitage et al. (2013) reported that the vertical magnetic field also diffuses away as the surface density decreases; in this case, the dependence of α φ,z on Σ would be weaker than in Eq. 2. We adopt the profile ofΣ pe based on the calculations by Komaki et al. (2021). In practice, we fit the results of the radiation hydrodynamics simulations by a function that combines a quadratic function and a function with negative power. The radiation hydrodynamics simulations incorporate EUV, FUV and X-ray radiation from the central star. Komaki et al. (2021) show that strong FUV radiation heats the disk gas effectively and drives rapid mass loss around a high mass star. It is also found that the gas in the outer disk is heated to generate flared structure.
Previous studies have shown that the mass-loss rate by photoevaporation,Ṁ pe , does not strongly depend on the initial disk mass by performing hydrodynamics simulations (Wölfer et al. 2019;Nakatani et al. 2021).Ṁ pe is primarily determined by the density at the disk surface rather than the density at the mid-plane, which is directly connected to the total mass. For the same reason, we also assume thatΣ pe is constant throughout the calculation.
The initial surface density is configured following the minimum solar disk model (Hayashi 1981), which is de-fined as
Σ int = Σ 1 au r 1 au −3/2 exp − r r cut ,
where r cut is a cut-off radius. Motivated by the observation that the disk radius increases proportionally to the stellar mass (Andrews et al. 2018), we assume r cut ∝ M * . We list the physical parameters of our simulations in Table 1. Note that we normalize the disk gas density (and hence mass) by Σ 1 au , the surface density at r = 1 au. We determine the value so that the total disk mass satisfies M disk = 0.117(M * /1 M ) M as Hayashi (1981) suggests. We set the computational domain at r in = 10 −2 au < r < r out = 10 4 au. We impose the zero-torque condition on the inner and outer boundaries (Suzuki et al. 2016;Kunitomo et al. 2020). We calculate the evolution until the disk mass decreases to an extremely small value of M disk < 10 −10 M . Since the disk mass decreases rapidly in the last stage, the calculation time does not change even if a different threshold mass is adopted, provided that it is M disk < 10 −5 M .
3. RESULTS Figure 1 shows the snapshots from 0 to 8 Myr for the run with M * = 1, 3, 7 M . In this section, we focus on the M * = 1 M case and we explain the other cases in Section 3.2. The inner disk in the vicinity of the central star loses quickly a significant amount of mass by MHD winds in a few million years. After 2 Myr, the mass-loss rate by MHD winds,Ṁ w , decreases, and a steep density profile is found near the central star. This is partly a numerical effect caused by the upper limit of α φz defined in Eq. 2; the mass-loss rate by MHD winds is effectively limited. To examine the numerical effect, we perform a set of simulations varying the upper limit of α φz by a factor of 0.1 and 10. We have confirmed that the massloss rate and the disk lifetime are not affected by this change. Figure 2 showsΣ w andΣ pe at the age of t = 1, 3 and 5 Myr. We also indicate by triangles the disk radii measured from our simulation outputs. Photoevaporation is the main dispersal process in the outer disk with r 20 au from the initial stage, while the inner disk is mainly dispersed by MHD winds. As shown in Figure 2, Σ w decreases with time because it is proportional to the surface density as given in Eq. 2. These features are consistent with the results of Kunitomo et al. (2020).
Around a solar-mass star
An important result is that the disk radius decreases with time as indicated by the triangles in Figure 2. Photoevaporation is the dominant dispersal process in the outer region, which we expect to cause the disk radius to decrease. We have run test simulations without photoevaporation to study its effect clearly. We have found that viscous accretion re-distributes the angular momentum and can indeed cause bloating of the disk even if α rφ is set to a very low value of 10 −4 . We thus conclude that photoevaporation plays a crucial role in shaping the disk morphology and its size through the effective mass-loss in the outer region.
At the age of ∼ 5 Myr, the major mass-loss is caused by photoevaporation in r > 1 au. We explain the details ofΣ pe and investigate the impact of distinct photoevaporation profile on disk evolution later in Section 4.3. Figure 3 shows the mass-loss rate as a function of time. The mass-loss rate by accretion is calculated aṡ
M acc = −2π(rv r Σ),
using the values at the inner boundary. The mass-loss rates by MHD winds and photoevaporation are calculated as follows.Ṁ
w = r out rinΣ w dṙ M pe = r out rinΣ pe dr,
where r out is the disk radius. Initially, strong MHD winds disperse a large amount of mass, and photoevaporation becomes the dominant dispersal process after a few Myrs, as has been discussed in the above. In contrast,Ṁ pe is almost constant with time. The slight decrease in the photoevaporative mass-loss ( Figure 3) is explained by the decreasing outer radius of the disk.
We integrate the mass-loss rate over time to derive the total mass lost. We calculate the fractional contributions from the three dispersal processes. The fraction of accretion, MHD winds and photoevaporation are 24%, 62% and 13%, respectively. Clearly, more than a half of the disk mass is dispersed by MHD winds. SinceΣ w scales proportional to the surface density, the winds are blown with high densities from near the central star in the initial phase. We note that previous studies suggest thatṀ w may often be overestimated. The 3D MHD simulations of an accretion disk show that a part of the gas launched as MHD winds falls back onto the central star through funnel-walls (Takasao et al. 2018(Takasao et al. , 2020. In practice, the partially re-accreted gas should be included inṀ acc rather than inṀ w . For a similar reason, the upper limit of C w should be appropriately calibrated in our future work.
Protostellar jets can be another mass dispersal process. The mass-loss rate due to jets is estimated to be about 10% of the accreted mass around low-mass stars (Calvet et al. 2004;Natta et al. 2014). Since accretion itself is a relatively minor mass-loss process according to our calculations, we expect that the influence of jets on the disk dispersal would be limited.
Simulations with different stellar masses
We perform disk evolution simulations with varying the stellar mass in the range of 0.5-7 M . We compute the mass-loss rate by each process in the same manner as in the 1 M case. In all cases, the disk is dispersed dominantly by MHD winds in the early phase, and then photoevaporation replaces to become the dominant disk dispersal process in the last stage.
We find very similar disk evolution in the runs with lower mass stars (< 1 M ). Initially, MHD winds disperse the disk gas, and photoevaporation becomes the main dispersal process at t ∼ 4-5 Myr. Similarly to our 1 M run, about 60% of the total mass-loss is due to MHD winds.
The case with 3 M shows a noticeable difference. The evolution of the surface density is shown in the middle panel of Figure 1. The main disk dispersal process is MHD winds and later switches to photoevaporation as in our 1 M run, but the transition occurs later at the age of ∼ 6 Myr. Around a 3 M star,Ṁ pe is relatively low, because an intermediate-mass star does not have a well developed convective layer and the magnetic activity on the stellar surface is weak and hence generates lower Xray emission. The low X-ray luminosity results in the overall low mass-loss rate by photoevaporation (Komaki et al. 2021).
The run with M * = 7 M shows another difference; the disk is dispersed efficiently by both MHD winds and photoevaporation. While MHD winds dominate the mass-loss in the very initial phase, the primary mechanism shifts to photoevaporation already at t = 1 Myr because the inner disk is completely dispersed as shown in Figure 1. The contribution of accretion, MHD winds and photoevaporation to the total mass loss is 23%, 45% and 32%, respectively. The relative contribution from the photoevaporation is larger than the other lower mass cases.
The disk structure and its evolution can be seen more clearly by performing the following visualization. We reconstruct two-dimensional density maps using the surface density obtained in the simulation with M * = 7 M . Figure 4 shows the snapshots of the reconstructed disk density maps. The vertical scale height throughout the disk is calculated by assuming hydrostatic equilibrium in the z-direction. We see clearly that the diffuse gas in the outer disk disperses quickly in less than 1 Myrs. Photoevaporation critically affects the overall disk shape even though its contribution to the total mass loss is small in the early epoch. The inner disk has a small scale height, and the gas density is high on and near the mid-plane. An inner "hole" opens at ∼ 10 au at t ∼ 1.4 Myr by the combined effect of accretion and MHD winds, but the disk gas remains at 10 au r 200 au where the effect is relatively weak.
stellar mass dependence of the disk lifetime
Based on our simulations, we propose an objective way of determining the disk lifetime. Figure 5 shows the evolution of the disk mass in the case of M * = 1 M . We find that the time evolution is accurately fitted with a simple vertical sigmoid function given by
log 10 M disk = a log 1 − x x + d x = (t/1 Myr) − b c .
We treat a, b, c, d in the equation as fitting parameters.
With this functional form, the characteristic dispersal time is given by (b + c) Myr. We do the fitting and estimate the disk dispersal times for all the cases we simulated. In Figure 6 we compare the results of our simulations with available observational estimates based on the disk fractions (Bayo et al. 2012;Ribas et al. 2015). We assume that the disk fraction decreases exponentially, following exp(−t/t dis ), where t dis is a disk dispersal timescale. Overall our results show that the disk lifetime is shorter for higher stellar mass, except in the case with 3 M which we discuss separately in Section 4.2. The trend is consistent with the observations. It is worth noting here that observations toward protoplanetary disks are conducted often in infrared wavelength, and thus thermal emission from dust grains is observed primarily (Ribas et al. 2015;Andrews et al. 2018). Both theoretical studies and observations suggest that the gas component and dust are distributed differently (de Gregorio-Monsalvo et al. 2013;Ansdell et al. 2018;Toci et al. 2021;Long et al. 2022). Recent high-resolution observations have made it possible to observe the gas disk at the 10 au scale using a variety of molecular lines (Öberg et al. 2021), to show clear images of disk morphology around stars with 1-2 M in the nearby star-forming regions. We expect that future observations toward a number of disks will reveal evolution of disk morphology around stars with different stellar masses. In our future work, we compare our simulation results directly with the gas observations to study if a disk loses its mass from outside to inside around low-mass stars and also whether or a disk around a high-mass star has an inner hole at the later stage.
disk parameter
We have performed disk evolution simulations incorporating accretion, MHD winds and photoevaporation with several assumptions and approximations. For instance, we assume that all the energy liberated by ac- The disk dispersal timescale of each stellar mass case. The blue circles are the simulation data. The black points and green horizontal bars represent inner-disk lifetimes estimated from observational disk fractions in Bayo et al. (2012) and Ribas et al. (2015), respectively. See the main text for more detail. run. Here we discuss possible variations of the results due to our choice of various model parameters.
First we study cases when only a part of gravitational energy liberated by accretion is used for winds. In order to examine the effects quantitatively, we run a series of simulations assuming 10% of liberated gravitational energy is converted to launch MHD winds. Figure 7 shows the mass-loss rate by each disk dispersal process. The contribution by MHD winds is lower than in the fiducial case. At the early phase, accretion and MHD winds are the main dispersal processes. At the later stage, photoevaporation becomes the dominant process. In the case of M * = 7 M , the disk is dispersed mainly by photoevaporation. We estimate the disk lifetime in the same way as the fiducial case. In the case of 1 M , the disk lifetime is longer than that of the fiducial case only by ∼ 1.4 Myr, which corresponds to 1.2 times of the fiducial case. Considering the weak wind case is an extreme limit of the reduced mass-loss by MHD winds, we conclude that the mass dependence of the disk lifetime is not affected by the choice of the strong wind or weak wind setups.
Recent observations show that the viscous accretion efficiency, α parameter, varies by a few orders of magnitudes (Hartmann et al. 2016;Ansdell et al. 2018). In order to investigate the evolution of disk with high viscous accretion, we also run simulations with α rφ = 10 −2 (M * /1 M ) and 10 −3 (M * /1 M ). BothṀ acc anḋ M w increases accordingly. Figure 8 shows the resulting disk lifetime with different values of α rφ . The lifetime is reduced by 1/5 with an increase in α rφ by a factor of 100. Interestingly the stellar mass dependence remains essentially the same. As α rφ increases, bothṀ acc andṀ w become larger. At the same time,Ṁ pe also increases because the disk radius increases by the strong angu- lar momentum transport caused by the high viscosity. These effects roughly cancell out, and the overall disk dispersal time is not significantly affected by the choice of α rφ . Kunitomo et al. (2021) performed long-term simulations of disk evolution considering accretion, photoevaporation and stellar evolution. They adopted the photoevaporation mass-loss rate depending on the time varying stellar luminosity in order to study the effect of stellar evolution on disk evolution. In particular, the X-ray luminosity of a M * = 3 M star decreases by a factor of ∼ 3 to t ∼ 1.5 Myr. They clarified that the reduction of X-ray luminosity resulted in the decrease of the mass-loss rate. They also showed that a disk around a ∼ 3 M star had a long lifetime because of the low X-ray luminosity as we have also found in our simulation.
star and disk evolution
In our calculations, we useΣ pe derived from the numerical simulations of Komaki et al. (2021). EUV, FUV and X-ray photons are considered as heating sources, and it is shown that FUV radiation efficiently heats the gas at the disk surface. According to Kunitomo et al. (2021), FUV luminosity increases by at least a few orders of magnitudes for a star with intermediate mass of 1.5-4 M . The FUV luminosity increases because the stellar surface becomes hot enough to radiate FUV photons directly. In the lower-mass side of M * < 2.5 M , the luminosity increases after the age of several Myr, which is comparable to the disk dispersal timescale. On the other hand, FUV luminosity increases earlier at the age of ∼ 1 Myr around a 3 M star. This is possibly early enough to affect the disk evolution by photoevap-oration. Komaki et al. (2021) also run a series of photoevaporation simulations varying the stellar luminosity. In all the cases of M * = 0.5, 1, 3 M , the mass-loss rates follow the relationship,Ṁ pe ∝ L 0.5 FUV . In the light of this, we perform a disk evolution calculation around a 3 M star by incorporating the time-dependent FUV luminosity. We approximate the evolution of FUV luminosity given by Kunitomo et al. (2021) with a sigmoid function. We obtain the surface mass-loss profile by multiplying the same factor asṀ pe from the fiducial value. The disk around a 3 M star has a lifetime of ∼ 5.1 Myr, which is half of the fiducial value. Photoevaporation plays a crucial role in determining the disk lifetime. It is important to consider evolution of the stellar luminosity around an intermediate-mass star. Kunitomo et al. (2021) have shown that the FUV luminosity of a 1 M star decreases gradually by a factor of 10 from t = 1 Myr to 3 Myr when the main source of the FUV radiation changes from accretion to radiation from the stellar chromosphere because of the rapid decrease of the accretion rate. SinceṀ pe decreases as the FUV luminosity decreases (Komaki et al. 2021), our calculation may overestimate the effect of photoevaporation, and thus may underestimate the disk lifetime. We expect that the stellar dependence of disk lifetime is steeper if we consider evolution of stellar luminosity.
photoevaporation model
We constructedΣ pe based on Komaki et al. (2021). While previous theoretical studies incorporated EUV and X-ray radiation as a heating source, Komaki et al. (2021) also considered FUV radiation. EUV photons are absorbed by the gas near the central star and ionize hydrogen atoms. As a result, Hii regions are formed in the vicinity of the central star, and EUV photons do not reach the outer disk. FUV photons contribute to the disk heating by photoelectric heating on dust grains. X-ray photons penetrate into the deeper region of a disk and launch a dense gas flow. Previous disk evolution simulations take the maximum value among the mass-loss rates by each radiation and incorporate asΣ pe . The photoevaporation simulations by Nakatani et al. (2018a,b) and Komaki et al. (2021) showed that X-ray photons promoted photoelectric heating by ionization of various species and the decrease of the positive charges of dust grains. This suggests that it is necessary to perform disk photoevaporation simulations considering FUV and X-ray radiation at the same time.
Previous calculations of disk evolution (e.g., Kunitomo et al. 2020Kunitomo et al. , 2021, and more if any) often useΣ pe given by Owen et al. (2010). Figure 9 shows the difference in photoevaporation mass-loss profile by Owen Komaki et al. (2021) in the case of M * = 1 M . We multiply by 3.1 × 10 −10 the profile of Owen et al. (2010) by 3.1 × 10 −10 which is given in an arbitrary unit. We determine the coefficient so that the mass-loss rate inside 100 au becomes equal to that of our photoevaporation profile.
The profile of Komaki et al. (2021) has two peaks generated by EUV and FUV heating, while the profile by Owen et al. (2010) has one peak. However, this interesting difference at the inner region of several au does not affect significantly the results of the simulations because MHD winds are the dominant disk dispersal process in the vicinity of the central star.
The computational region of Komaki et al. (2021) is [0.89 au, 178 au], while that of Owen et al. (2010) is [0.82 au, 100 au]. In the present paper, we extrapolate the mass-loss profile for the outer disk. Since photoevaporation becomes the dominant dispersal process at r > 20 au, the high mass-loss profile results in the fast dispersal in the outer region. In order to examine the dependence on the mass-loss profile of disk dispersal process, we conduct a disk evolution simulation with a profile given bẏ Σ pe,s = 3.2×10 −12 g cm −2 s −1 L X−ray 10 30 erg s −1 r 2.5 au −2 , which is a simple fit for the profile by Owen et al. (2010). This simple profile is shown by an orange line in Figure 9. Even though Owen et al. (2010) only provides a profile within 100 au, we extend the fit to r > 100 au so that we incorporate photoevaporation in the outer disk. We set the inner boundary to 0.14r g = 1.2 au fol-lowing Liffman (2003), which calculate the gravitational radius theoretically. The dominant disk dispersal process changes from MHD winds to photoevaporation at the age of ∼ 2 Myr. Figure 10 shows the reconstructed 2D snapshots of the simulation. As shown in Figure 9, Σ pe,s decreases with increasing distance from the central star following r −2 , while the profile by Komaki et al. (2021) maintains ∼ 3.0 × 10 −14 g cm −2 s −1 in the outer region. The disk at a several tens of au remains at the last stage of disk dissipation. Following these, we need to apply realistic photoevaporation profile to obtain the disk density distribution.
Other effects
We assumeṀ pe is constant throughout our calculations. It has been suggested that photoevaporation can be enhanced when an inner hole opens (Alexander et al. 2014;Owen et al. 2010;Nakatani et al. 2021). In our study, only the simulation with M * = 7 M shows the formation of a hole at ∼ 1 Myr, before the disk disperses. The time evolution ofṀ pe , especially when a hole or a gap is formed, may need to be taken into account to evaluate the disk lifetime more accurately.
Disk observations have shown that disks in a strong radiation field have specific shapes with long tails (O'dell et al. 1993;McCaughrean & O'dell 1996;Winter et al. 2018). Theoretical studies suggested that the dense gas was eroded by high radiation from the nearby high-mass stars (Richling & Yorke 1998;Haworth & Clarke 2019). Haworth & Clarke (2019) performed 2D hydrodynamics simulations. They assumed that the disk was exposed to the high FUV radiation fields and solved radiation hydrodynamics to calculate the mass-loss rate. As s result, the outer disk is effectively removed because the thin disk is heated efficiently by the radiation from the nearby stars. The present study does not consider radiation by nearby stars. We expect that the outer disk would be dispersed more efficiently under an external radiation field, and the disk radius would shrink rapidly.
SUMMARY
Recent PPD observations in star forming regions suggested that the lifetime decreases with increasing central stellar mass. Theoretical studies proposed three disk dispersal mechanisms: accretion, MHD winds and photoevaporation. In order to understand both the typical lifetime of a few million years and the stellar mass dependence, it is necessary to conduct disk evolution simulations throughout the disk lifetime considering all the disk dispersal mechanisms. We performed 1D disk evolution simulations varying the central stellar mass in the range of 0.5-7 M . We showed the disk loses its mass mainly by MHD winds in the early stage, and later by photoevaporation. Especially, in the case with M * = 7 M , photoevaporation by the high radiation contributes more than other cases, and a gap opens at ∼ 1 Myr. We find that photoevaporation shapes the disk morphology by clearing out the gas in the outer disk efficiently. The time evolution of disk mass can be described by a simple function that is given by transforming the sigmoid function, which yields an accurate estimate of the disk dispersal time. The dispersal time around a high-mass star is ∼ 2 Myr, which is shorter than the other cases by the factor of ∼ 5. The trend is consistent with recent observations. Finally, the mass dependence of the dispersal timescale does not vary by the choice of strong wind or weak wind or the choice of viscous parameter.
Figure 1 .
1Snapshots of surface density in simulations with M * = 1, 3, 7 M . The initial surface density is shown in dark blue and as time goes on, the surface density is shown in more yellowish line.
Figure 2 .
2The surface mass-loss profiles by the two disk dispersal processes; MHD winds(orange), photoevaporation(green) in the case of M * = 1 M . The triangles express the disk radius at the age of 0
Figure 3 .
3The mass-loss rates by the three dispersal processes; accretion (blue), MHD winds (orange), and photoevaporation (green) in our fiducial run with M * = 1 M . The instantaneous mass-loss rates are calculated using Eq. 3.1 and Eq. 3.1.
Figure 4 .
4Snapshots of the reconstructed gas density in a simulation with M * = 7.0 M at t = 0, 0.8, 1.0, 1.8 Myr. The color map shows the density and the black line expresses the surface density.
Figure 5 .
5The disk mass evolution around a 1 M star. The blue line shows the simulation data and the orange dotted line expresses the fit.
Figure 6 .
6Figure 6. The disk dispersal timescale of each stellar mass case. The blue circles are the simulation data. The black points and green horizontal bars represent inner-disk lifetimes estimated from observational disk fractions in Bayo et al. (2012) and Ribas et al. (2015), respectively. See the main text for more detail.
Figure 7 .
7cretion is transferred to drive MHD winds in our fiducial Same asFigure 3but for the weak wind case.
Figure 8 .
8= 10 −2 (M * /1M ) α rφ = 10 −3 (M * /1M ) α rφ = 10 −4 (M * /1M ) The disk dispersal timescale in simulations varying α rφ parameter. The blue, brown and red circles are the simulation data with α rφ = 10 −4 (M * /1 M ), 10 −3 (M * /1 M ) and 10 −2 (M * /1 M ).
Figure 9 .
9The photoevaporation mass-loss profiles: fitted based on Komaki et al. (2021)(blue), Owen et al. (2010)(orange) and a simple approximation(green). Since the fitted data by Owen et al. (2010) is given in an arbitrary unit, the mass-loss profile in the figure is multiplied by 3.1 × 10 −10 . et al. (2010) and
Figure 10 .
10Snapshots of the reconstructed gas density in a simulation with M * = 1 M at t = 1, 4, 7 Myr. The color map shows the density and the black line expresses the surface density. The top row shows the density distribution in the fiducial simulation and the bottom row shows the density distribution in the simulation with the simple photoevaporation profile .
Table 1 .
1Fiducial stellar parameters in the model (adopted from Gorti & Hollenbach (2009)) M * ( M ) L bol ( L ) M disk ( M ) rcut ( au) logLFUV ( erg s −1 ) logφEUV ( s −1 ) logLX-ray ( erg s −1)
. DISCUSSION
ACKNOWLEDGMENTSThis research was supported by Forefront Physics and Mathematics Program to Drive Transformation (FoPM), a World-leading Innovative Graduate Study (WINGS) Program, the University of Tokyo. NY acknowledges financial support from MEXT/JSPS KAK-ENHI 20H05844. T.K.S. is supported in part by Grantsin-Aid for Scientific Research from the MEXT/JSPS of Japan, 17H01105, 21H00033 and 22H01263 and by Program for Promoting Research on the Supercomputer Fugaku by the RIKEN Center for Computational Science (Toward a unified view of the universe: from large-scale structures to planets; grant 20351188-PI J. Makino) from the MEXT of Japan.
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| [] |
[
"Quantum battery based on superabsorption",
"Quantum battery based on superabsorption",
"Quantum battery based on superabsorption",
"Quantum battery based on superabsorption"
] | [
"Yudai Ueki \nFaculty of Pure and Applied Sciences\nUniversity of Tsukuba\n305-8571TsukubaJapan\n",
"Shunsuke Kamimura \nFaculty of Pure and Applied Sciences\nUniversity of Tsukuba\n305-8571TsukubaJapan\n\nResearch Center for Emerging Computing Technologies\nNational Institute of Advanced Industrial Science and Technology (AIST)\n1-1-1 Umezono305-8568TsukubaIbarakiJapan\n",
"Yuichiro Matsuzaki \nResearch Center for Emerging Computing Technologies\nNational Institute of Advanced Industrial Science and Technology (AIST)\n1-1-1 Umezono305-8568TsukubaIbarakiJapan\n\nNEC-AIST Quantum Technology Cooperative Research Laboratory\nNational Institute of Advanced Industrial Science and Technology (AIST)\n305-8568TsukubaIbarakiJapan\n",
"Kyo Yoshida \nFaculty of Pure and Applied Sciences\nUniversity of Tsukuba\n305-8571TsukubaJapan\n",
"Yasuhiro Tokura \nFaculty of Pure and Applied Sciences\nUniversity of Tsukuba\n305-8571TsukubaJapan\n",
"Yudai Ueki \nFaculty of Pure and Applied Sciences\nUniversity of Tsukuba\n305-8571TsukubaJapan\n",
"Shunsuke Kamimura \nFaculty of Pure and Applied Sciences\nUniversity of Tsukuba\n305-8571TsukubaJapan\n\nResearch Center for Emerging Computing Technologies\nNational Institute of Advanced Industrial Science and Technology (AIST)\n1-1-1 Umezono305-8568TsukubaIbarakiJapan\n",
"Yuichiro Matsuzaki \nResearch Center for Emerging Computing Technologies\nNational Institute of Advanced Industrial Science and Technology (AIST)\n1-1-1 Umezono305-8568TsukubaIbarakiJapan\n\nNEC-AIST Quantum Technology Cooperative Research Laboratory\nNational Institute of Advanced Industrial Science and Technology (AIST)\n305-8568TsukubaIbarakiJapan\n",
"Kyo Yoshida \nFaculty of Pure and Applied Sciences\nUniversity of Tsukuba\n305-8571TsukubaJapan\n",
"Yasuhiro Tokura \nFaculty of Pure and Applied Sciences\nUniversity of Tsukuba\n305-8571TsukubaJapan\n"
] | [
"Faculty of Pure and Applied Sciences\nUniversity of Tsukuba\n305-8571TsukubaJapan",
"Faculty of Pure and Applied Sciences\nUniversity of Tsukuba\n305-8571TsukubaJapan",
"Research Center for Emerging Computing Technologies\nNational Institute of Advanced Industrial Science and Technology (AIST)\n1-1-1 Umezono305-8568TsukubaIbarakiJapan",
"Research Center for Emerging Computing Technologies\nNational Institute of Advanced Industrial Science and Technology (AIST)\n1-1-1 Umezono305-8568TsukubaIbarakiJapan",
"NEC-AIST Quantum Technology Cooperative Research Laboratory\nNational Institute of Advanced Industrial Science and Technology (AIST)\n305-8568TsukubaIbarakiJapan",
"Faculty of Pure and Applied Sciences\nUniversity of Tsukuba\n305-8571TsukubaJapan",
"Faculty of Pure and Applied Sciences\nUniversity of Tsukuba\n305-8571TsukubaJapan",
"Faculty of Pure and Applied Sciences\nUniversity of Tsukuba\n305-8571TsukubaJapan",
"Faculty of Pure and Applied Sciences\nUniversity of Tsukuba\n305-8571TsukubaJapan",
"Research Center for Emerging Computing Technologies\nNational Institute of Advanced Industrial Science and Technology (AIST)\n1-1-1 Umezono305-8568TsukubaIbarakiJapan",
"Research Center for Emerging Computing Technologies\nNational Institute of Advanced Industrial Science and Technology (AIST)\n1-1-1 Umezono305-8568TsukubaIbarakiJapan",
"NEC-AIST Quantum Technology Cooperative Research Laboratory\nNational Institute of Advanced Industrial Science and Technology (AIST)\n305-8568TsukubaIbarakiJapan",
"Faculty of Pure and Applied Sciences\nUniversity of Tsukuba\n305-8571TsukubaJapan",
"Faculty of Pure and Applied Sciences\nUniversity of Tsukuba\n305-8571TsukubaJapan"
] | [] | A quantum battery is a device where an energy is charged by using a quantum effect. Here, we propose a quantum battery with a charger system composed of N qubits by utilizing a collective effect called a superabsorption. Importantly, the coupling strength between the quantum battery and charger system can be enhanced due to an entanglement. While the charger time scales as Θ N −1/2 by applying a conventional scheme, we can achieve a charging time Θ N −1 in our scheme. Our results open the path to ultra-fast charging of a quantum battery. | 10.7566/jpsj.91.124002 | [
"https://export.arxiv.org/pdf/2205.03823v1.pdf"
] | 248,572,279 | 2205.03823 | e7f599a7a0d5926f0f0d6853cd5af97c3e9a0f00 |
Quantum battery based on superabsorption
(Dated: May 10, 2022)
Yudai Ueki
Faculty of Pure and Applied Sciences
University of Tsukuba
305-8571TsukubaJapan
Shunsuke Kamimura
Faculty of Pure and Applied Sciences
University of Tsukuba
305-8571TsukubaJapan
Research Center for Emerging Computing Technologies
National Institute of Advanced Industrial Science and Technology (AIST)
1-1-1 Umezono305-8568TsukubaIbarakiJapan
Yuichiro Matsuzaki
Research Center for Emerging Computing Technologies
National Institute of Advanced Industrial Science and Technology (AIST)
1-1-1 Umezono305-8568TsukubaIbarakiJapan
NEC-AIST Quantum Technology Cooperative Research Laboratory
National Institute of Advanced Industrial Science and Technology (AIST)
305-8568TsukubaIbarakiJapan
Kyo Yoshida
Faculty of Pure and Applied Sciences
University of Tsukuba
305-8571TsukubaJapan
Yasuhiro Tokura
Faculty of Pure and Applied Sciences
University of Tsukuba
305-8571TsukubaJapan
Quantum battery based on superabsorption
(Dated: May 10, 2022)
A quantum battery is a device where an energy is charged by using a quantum effect. Here, we propose a quantum battery with a charger system composed of N qubits by utilizing a collective effect called a superabsorption. Importantly, the coupling strength between the quantum battery and charger system can be enhanced due to an entanglement. While the charger time scales as Θ N −1/2 by applying a conventional scheme, we can achieve a charging time Θ N −1 in our scheme. Our results open the path to ultra-fast charging of a quantum battery.
I. INTRODUCTION
Quantum thermodynamics is an emerging field to extend the conventional thermodynamics to microscopic systems where not only thermal but also quantum fluctuations should be taken into account [1][2][3][4][5][6][7]. One of the aims of quantum thermodynamics is to investigate whether quantum devices provide enhancement of performance over classical devices.
A quantum heat engine is one of the promising devices with the enhancement over the classical ones by using the quantum property [8][9][10][11][12][13][14][15][16]. It is possible to obtain a quadratically scaling power P = Θ(N 2 ) by using the quantum enhanced heat engine with an entanglement while a conventional separable engine shows a power of P = Θ(N ) where N denotes the number of qubits [15]. Here, the key feature of this scheme is to adopt a collective effect called a superabsorption that was proposed in Ref. [17], and its proof-of-concept experiment was recently demonstrated using barium atoms in an optical cavity [18].
A quantum battery is also a prominent research subject in quantum thermodynamics to charge an energy of quantum systems. As is the case for conventional batteries (such as lithium-ion batteries) using electrochemical reactions to convert chemical energy into electrical energy, the main issue of the quantum battery is to increase a performance of charging and discharging processes [19][20][21][22]. Such a quantum battery was first proposed in Ref. [19]. In Ref. [20], it was found that the use of entangling operations can improve the performance of the quantum battery compared with the one using only separable operations. Here, the performance is defined as a storable energy in a quantum battery per a unit time. Also, in Ref. [22], it has been found that Dicke-quantum batteries composed of collective N -qubit systems give us a scaling √ N times larger compared to independent N -qubit batteries. In these studies, external pulses are applied to charge the isolated quantum battery.
On the other hand, the battery can also be charged by using an interaction with an environment [23,24]. In Ref. [24], when N quantum batteries interact with N charger systems that are prepared in a steady state with a population inversion, the charging time scales as Θ(N ) by using a collective charging process. Here, the charging time is defined as how long it takes for the quantum battery to be a steady state. In Ref. [25], the improvement of the quantum battery performance due to the collective effects has been experimentally confirmed, and the charging time scales as Θ N −1/2 where N denotes the number of atoms.
Here, we propose a quantum battery using a charger system composed of an entangled N -qubit system. Our quantum battery provides a charging time scaling as Θ(N −1 ). This is stark contrast to the conventional scheme where the charging time scales as Θ(N −1/2 ) by using N three-level systems as a charger. The key factor for the enhanced charging time is utilizing the superabsorption for the charging process. The charger systems are prepared in an entangled N -qubit state called Dicke state via an interaction with the environment, and the quantum battery can strongly interact with the charger system in a collective way. An energy exchange between the charger and the battery occurs, and the battery, initially prepared in a ground state, can be eventually raised in an excited state with a necessary time scaling as Θ(N −1 ).
Our paper is organized as follows. In Sec. II, we review a charging model with one three-level system and one qubit battery. Also, we explain a charging model using N -qubit charger initially prepared in a separable state, where the charging time scales as Θ(N −1/2 ). In Sec. III, we explain our charging scheme with a charging time scaling as Θ(N −1 ) by using an N -qubit charger initially prepared in an entangled state. In Sec. IV, we conclude our discussion.
II. BATTERY CHARGING WITH SEPARABLE N THREE-LEVEL CHARGERS
Let us review the conventional charging model with separable states. We consider the charger system with a three-level system and the quantum battery system with a qubit. The total arXiv:2205.03823v1 [quant-ph] 8 May 2022
Hamiltonian H 1 tot is given as follows.
H 1 tot = H 3−level + H 1 B + H 1 I ,(1)H 3−level = 2 i=0 E i |i C i| ,(2)H 1 B = ∆ 2 σ z , ∆ ≡ E 1 − E 0 ,(3)H 1 I = g (|0 C 1| ⊗ |1 B 0| + h.c.) ,(4)
where E i (i = 0, 1, 2, E 0 < E 1 < E 2 ) is the eigenenergy of the charger system, ∆ is the energy of the quantum battery system, g is a coupling strength between the charger and quantum battery. We prepare the initial system ψ 1 (0) = |1 C |0 B where the charger state is prepared as |1 C and the battery state is prepared as |0 B . A steady state of the charger system after a coupling with two thermal baths can be |1 C by adjusting the parameters where one of the thermal baths induces a transition between |0 C and |2 C while the other thermal baths induces a transition between |2 C and |1 C , and this is called a population inversion where the population of the first excited state becomes higher than that of the ground state [14,17,24]. The purpose of the charging is to obtain a battery state of |1 B from the initial state. The battery state ρ 1 B (t) can be described as
ρ 1 B (t) = Tr C e −iH 1 tot t ψ 1 (0) ψ 1 (0) e iH 1 tot t = cos 2 (gt) |0 B 0| + sin 2 (gt) |1 B 1| ,(5)
where Tr C denotes partial trace of the charger system. When we turn off the interaction at gt = π/2, we obtain a state of|1 B , and this means that the charging is done. Next, let us explain a scheme to use N three-level systems as a charger. We consider the charger system with N threelevel systems and the quantum battery system with a qubit. Strictly speaking, we need three-level systems for the charger systems, because this allows us to use a population inversion when the charger system becomes a steady state after the coupling with thermal baths, as we mentioned above. However, once we successfully obtain the population inversion for the charger system, the dynamics between the quantum battery and charger system is confined in a subspace spanned by |0 C and |1 C , and so we consider only this subspace for simplicity.
We define the collective operators J z and J ± as J z =
1 2 N i=1 σ i z and J ± = N i=1 σ i ± .
The total Hamiltonian H sep tot is given by follows.
H sep tot = H sep N + H sep B + H sep I ,(6)H N = ω A J z ,(7)H sep B = ω A 2 σ z ,(8)H sep I = g (σ + ⊗ J − + σ − ⊗ J + ) ,(9)
where ω A denotes a frequency of the qubits for the charger system and a quantum battery and g is coupling strength between the charger and quantum battery. This Hamiltonian was experimentally realized by using a superconducting qubit and an electron-spin ensemble [26][27][28][29]. We prepare the initial system ψ N (0) = |11 · · · 1 C |0 B where the charger state is prepared as all excited states, |11 · · · 1 C , and the battery state is prepared as |0 B . The purpose of the charging is to obtain a battery state of |1 B from the initial state. The battery state ρ N B (t) can be described as
ρ N B (t) = Tr C e −iH sep N t ψ N (0) ψ N (0) e iH sep N t = cos 2 √ N gt |11 · · · 1 C |0 B + sin 2 √ N gt |W C |1 B ,(10)|W C = 1 √ N (|111 · · · 0 C + · · · + |011 · · · 1 C ) . (11)
From this analysis the necessary time to obtain |1 B from |0 B is t = π/2 √ N g. This means that the charging time for the battery scale as Θ(N −1/2 ) in this model. Such a behavior was theoretically predicted in [30,31].
III. BATTERY CHARGING WITH SUPERABSORPTION
Here, we introduce our scheme to charge the quantum battery with a charging time to scale as Θ(N −1 ) by using N charger qubits.
A. Hamiltonian
We consider the charger system and the quantum battery system. The former one is composed of N -qubits while the latter one is composed of two-qubits. The total Hamiltonian H tot is given as follows.
H tot = H N + H B + H I ,(12)H N = ω A J z + ΩJ 2 z ,(13)H B = ω A + δ 2 σ (1) z + |ω A + 2Ω| + δ 2 σ (2) z ,(14)H I = 2g σ (1) x + σ (2) x ⊗ J x ,(15)
where H N (H B ) denotes the Hamiltonian for N -qubit system (2-qubit system), H I denotes the interaction Hamiltonian between charger (N -qubit) and quantum battery (2-qubit), ω A denotes a frequency of the each N -qubits, Ω denotes a total coupling constant between the N -qubits, g denotes a coupling constant between charger and battery. Here, let us introduce J ±1 and J ±2 , which are the part of the ladder operator. They are defined as follows.
J + = J +1 + J +2 , J − = J −1 + J −2 ,(16)J +1 = N/2 M =3/2 √ a M |M C M − 1| , J −1 = (J +1 ) † ,(17)J +2 = 1/2 M =1−N/2 √ a M |M C M − 1| , J −2 = (J +2 ) † ,(18)a M ≡ N 2 + M N 2 − M + 1 .(19)
Here, we introduced the Dicke states, which are the simultaneous eigenstates of J 2 and J z . These can be written as |J, M , and the corresponding eigenvalues are J(J + 1) and M . In this paper, we take Dicke states as |M = |N/2, M in the subspace with total angular momentum J = N/2 and assume N is odd. We assume conditions of strong coupling, i.e., |Ω| > ω A . Also, we assume ω A > 0 and Ω < 0. Theses conditions allow us to construct a Λ-type structure for the Dicke states between |3/2 C , |1/2 C , and |−1/2 C as shown in the FIG 3. In this case, |1/2 C has the highest energy in the charger system. This means that J +1 and J −2 play a role in inducing a transition from a higher energy state to a lower energy state in the charger system. On the other hand, J −1 and J +2 induce a transition from a lower energy state to a higher energy state in the charger system. We are going to use a rotating wave approximation (RWA) for gN ω A . In the RWA, we typically ignore terms that oscillate with a high frequency. In our case, terms such as (σ
(1) − + σ (2) − )J +1 , (σ (1) − + σ (2) − )J −2 , (σ (1) + + σ (2) + )J −1 , (σ (1) + + σ(2)
+ )J +2 will be dropped. So, by using the RWA, we obtain
H I g(A + A † ) where A ≡ σ (1) + + σ (2) + (J +1 + J −2 ).
In this case, the Hamiltonian of the N -qubit system can be diagonalized as follows. hand, |ω A + 2Ω|+δ denotes the frequency of qubit 2, and this is detuned from ∆ 3/2 by δ.
H N = N/2 M =−N/2 E M |M C M | , E M = ω A M + ΩM 2 ,(20)M ∈ − N 2 , − N 2 + 1, · · · , N 2 .(21)
B. Perturbation analysis
Let us derive an effective Hamiltonian. We move into an interaction picture. We consider a unitary operator U (t, 0) = T e −i t
0 H i I (t)dt where H i I (t) ≡ e i(H N +H B )t H I e −i(H N +H B )
t denotes an interaction Hamiltonian in the interaction picture. We expand U (t, 0) to the second order term with respect to the coupling constant g.
U (t, 0) = I − i t 0 H i I (t )dt − t 0 H i I (t )dt t 0 H i I (t )dt + O(g 3 ).(22)
Under the assumptions of gN δ ω A < |Ω| and 1 gN t, the first order of the interaction term induces transitions between states with a large energy difference, and these have terms to oscilalte with a high frequency. So the second order term, which includes a resonant transition, becomes the relevant term. Then we obtain
U (t, 0) = I − itH eff e −itH eff .(23)H eff = g 2 δ √ a 1/2 √ a 3/2 L + +L − .(24)L + = σ (2) + σ (1) − |3/2 C −1/2| ,L − = L + † .
Here, when the initial state of the battery is confined in a subspace spanned by {|01 B , |10 B }, the dynamics of the battery state is also confined in this subspace as long as the second order term is relevant. (See the appendix for the details of the derivation). Although a similar approximation has been used in quantum optics [32], we firstly applied this technique to the model of the quantum battery with a superabsorption. We discuss the dynamics of the system with the effective Hamiltonian. We prepare the initial state |ψ(0) = |−1/2 C |10 B (see FIG 4). The purpose of the charging is to obtain a battery state of |01 B from the initial state. Considering the charging process using the effective Hamiltonian H eff , the battery state ρ eff B (t) can be described as ρ eff B (t) = Tr N e −iH eff t |ψ(0) ψ(0)| e iH eff t = cos 2 (λt) |10 C 10| + sin 2 (λt) |01 C 01|. (25) Here, λ = g 2 δ √ a 1/2 √ a 3/2 and Tr N denotes a partial trace of the charger system. We set δ = 10gN to satisfy condition gN δ. Since √ a 1/2 √ a 3/2 ∝ O(N 2 ) and δ ∝ O(N ), the necessary time to obtain |01 B from the initial state of |10 B scales as O(N −1 ). This means that the charging time for the battery scales as O(N −1 ). We use a phenomena called super-absorption where the coupling strength becomes collectively enhanced around the middle of the Dicke ladder structure. However, we use approximations to derive the effective Hamiltonian. To check the validity of the approximation, we will perform numerical simulations in the next subsection.
C. Numerical analysis
The total Hamiltonian for numerical simulations is given by
H tot = H 0 + H I (26) H 0 = H N + H B(27)H I = 2g σ (1) x + σ (2) x ⊗ J x ,(28)
where we use the interaction Hamiltonian H I without the rotating wave approximation. We consider the battery state 7). From the numerical results, we confirm that there is an oscillation between {|10 B } and {|01 B }, while a population leakage to the other states is negligible. Secondly, we numerically calculate how the charging time depends on the number of qubit N as shown in FIG 8. We introduce a ergotropy
A =10 A =5 A =1EffectiveE(t) = Tr B [H B ρ B (t)] − min U :unitary Tr B H B U ρ B (t)U † [33]
, a measure of extractable energy from battery. We assume that the charging is done when the ergotropy of the quantum battery can be stored up to 80% of its maximum value, and we call the necessary time for this a charging timeτ N . In FIG 8, we numerically confirm thatτ N scale as O(N −1 ) in gN δ ω A < |Ω|. However, in the region δ ω A ⇔ N N * = ω A δ where δ = 10gN , we cannot finish the charging process because the target ergotropy cannot be 80% of its maximum value. This comes from the fact that the detuning δ is not strong enough to confine the dynamics into the subspace spanned by {|10 B , |01 B }. In other words, the first order term in the Eq. (A9) has a non-negligible contribution to the dynamics. Here, we define N * = ωA/δ, and our effective Hamiltonian becomes invalid when the number of the qubits becomes more than N * . We set the parameters as g = 10 −3 , δ = 10gN and Ω = −2.3ωA.
IV. CONCLUSION
In conclusion, we propose a quantum battery with a charger system composed of N -entangled qubits. We ulitize a superabsoption for an entanglement-enhanced energy exchange between the charger system and quantum battery. Our scheme provides a scaling of a charging time Θ (N ) while a conventional scheme provides a scaling of a charging time Θ N −1/2 . Our results pave the way for the realization of a ultra-fast quantum battery. In this section, we derive the effective Hamiltonian
H eff = g 2 δ √ a 1/2 √ a 3/2 L + +L − , L + = σ (2) + σ (1) − |3/2 C −1/2| ,L − = L + † from the unitary operator (A9). We define µ 1 = 1 2 2ω A +δ |Ω| + 1 , µ 2 = 1 2 δ Ω + 1 , µ 3 = 1 2 δ |Ω| + 3 , and µ 4 = 1 2 δ−2ω A Ω − 1 .
We adjust parameters to satisfy the following
µ i = M, M ∈ {−N/2, −N/2 + 1, · · · , N/2} (A1)
for all µ i , i ∈ {1, 2, 3, 4}. First, we expand the first order of interaction term
t 0 H i I (t )dt . t 0 H i I (t )dt = g t 0 N/2 M =3/2 e i(ω A +δ+∆ M )t √ a M σ 1 + |M M − 1| + 1/2 M =−N/2 e i(ω A +δ−∆ M )t √ a M σ 1 + |M − 1 M | + g t 0 N/2 M =3/2 e i(|ω A +2Ω|+δ+∆ M )t √ a M σ 2 + |M M − 1| + 1/2 M =−N/2 e i(|ω A +2Ω|+δ−∆ M )t √ a M σ 2 + |M − 1 M | + h.c.(A2)
It is worth mentioning that the first order of the interaction term induces transitions between states with a large energy difference. In this case, we have terms to oscillate with a high frequency, and these tend to be small. On the other hand, these terms also have a collective enhancement factor of √ a M . We are going to evaluate whether these can be negligible or not as a total.
(A2) = N/2 M =3/2 g i (ω A + δ + ∆ M ) e i(ω A +δ+∆ M )t − 1 √ a M σ 1 + |M M − 1| + 1/2 M =−N/2 g i (ω A + δ − ∆ M ) e i(ω A +δ−∆ M )t − 1 √ a M σ 1 + |M − 1 M | + N/2 M =3/2 g i (|ω A + 2Ω| + δ + ∆ M ) e i(|ω A +2Ω|+δ+∆ M )t − 1 √ a M σ 2 + |M M − 1| + 1/2 M =−N/2 g i (|ω A + 2Ω| + δ − ∆ M ) e i(|ω A +2Ω|+δ−∆ M )t − 1 √ a M σ 2 + |M − 1 M | + h.c.
By choosing suitable parameters, these terms are negligible as we show below.
g i (ω A + δ + ∆ M ) √ a M ≈ g 2ω A + δ + (2M − 1)Ω O(N ) = g δ 1 1 + (2M − 1) Ω δ + 2ω A δ O(N ) 1 ∵ M = 1/2, g |Ω| N 1, |Ω| δ 1 ∵ M = 1/2, g ω A N 1, ω A δ 1 , g i (ω A + δ − ∆ M ) √ a M ≈ g δ − (2M − 1)Ω O(N ) = g δ 1 1 + (2M − 1) Ω δ O(N ) 1 ∵ M = 1/2, g |Ω| N 1, |Ω| δ 1 ∵ M = 1/2, g δ N 1 , g i (|ω A + 2Ω| + δ + ∆ M ) √ a M ≈ g δ + (2M − 3)Ω O(N ) = g δ 1 1 + (2M − 3) Ω δ O(N ) 1 ∵ M = 3/2, g |Ω| N 1, |Ω| δ 1 ∵ M = 3/2, g δ N 1 , g i (|ω A + 2Ω| + δ − ∆ M ) √ a M ≈ g −2ω A + δ − (2M + 1)Ω O(N ) = g δ 1 1 − (2M + 1) Ω δ − 2 ω A δ O(N ) 1 ∵ M = −1/2, g |Ω| N 1, |Ω| δ 1 ∵ M = −1/2, g ω A N 1, ω A δ 1 .
Therefore, we can drop the term of t 0 H i I (t )dt for gN δ ω A < |Ω|. Next, we expand the second order of the
interaction term t 0 H i I (t )dt t 0 H i I (t )dt . Since µ i = M for all i, we obtain t 0 H i I (t )dt t 0 H i I (t )dt = N/2 1,M =3/2 g 2 i (ω A + δ + ∆ M ) t 0 dt A i (t ) e i(ω A +δ+∆ M )t − 1 √ a M σ 1 + |M M − 1| + 1/2 2,M =−N/2 g 2 i (ω A + δ − ∆ M ) t 0 dt A i (t ) e i(ω A +δ−∆ M )t − 1 √ a M σ 1 + |M − 1 M | + N/2 3,M =3/2 g 2 i (|ω A + 2Ω| + δ + ∆ M ) t 0 dt A i (t ) e i(|ω A +2Ω|+δ+∆ M )t − 1 √ a M σ 2 + |M M − 1| + 1/2 4,M =−N/2 g 2 i (|ω A + 2Ω| + δ − ∆ M ) t 0 dt A i (t ) e i(|ω A +2Ω|+δ−∆ M )t − 1 √ a M σ 2 + |M − 1 M | (A3) − N/2 1,M =3/2 g 2 i (ω A + δ + ∆ M ) t 0 dt A i (t ) e −i(ω A +δ+∆ M )t − 1 √ a M σ 1 − |M − 1 M | + 1/2 2,M =−N/2 g 2 i (ω A + δ − ∆ M ) t 0 dt A i (t ) e −i(ω A +δ−∆ M )t − 1 √ a M σ 1 − |M M − 1| + N/2 3,M =3/2 g 2 i (|ω A + 2Ω| + δ + ∆ M ) t 0 dt A i (t ) e −i(|ω A +2Ω|+δ+∆ M )t − 1 √ a M σ 2 − |M − 1 M | + 1/2 4,M =−N/2 g 2 i (|ω A + 2Ω| + δ − ∆ M ) t 0 dt A i (t ) e −i(|ω A +2Ω|+δ−∆ M )t − 1 √ a M σ 2 − |M M − 1| (A4) + N/2 1,M =3/2 g 2 i (ω A + δ + ∆ M ) t 0 dt A i † (t ) e i(ω A +δ+∆ M )t − 1 √ a M σ 1 + |M M − 1| + 1/2 2,M =−N/2 g 2 i (ω A + δ − ∆ M ) t 0 dt A i † (t ) e i(ω A +δ−∆ M )t − 1 √ a M σ 1 + |M − 1 M | + N/2 3,M =3/2 g 2 i (|ω A + 2Ω| + δ + ∆ M ) t 0 dt A i † (t ) e i(|ω A +2Ω|+δ+∆ M )t − 1 √ a M σ 2 + |M M − 1|g 2 i (ω A + δ + ∆ M ) t 0 dt A i † (t ) e −i(ω A +δ+∆ M )t − 1 √ a M σ 1 − |M − 1 M | + 1/2 2,M =−N/2 g 2 i (ω A + δ − ∆ M ) t 0 dt A i † (t ) e −i(ω A +δ−∆ M )t − 1 √ a M σ 1 − |M M − 1| + N/2 3,M =3/2 g 2 i (|ω A + 2Ω| + δ + ∆ M ) t 0 dt A i † (t ) e i(|ω A +2Ω|+δ+∆ M )t − 1 √ a M σ 2 − |M − 1 M | + 1/2 4,M =−N/2 g 2 i (|ω A + 2Ω| + δ − ∆ M ) t 0 dt A i † (t ) e −i(|ω A +2Ω|+δ−∆ M )t − 1 √ a M σ 2 − |M M − 1| .(A6)
We calculate the first term of the (A3).
g 2 i (ω A + δ + ∆ M ) t 0 dt e i(|ω A +2Ω|+δ)t e i(ω A +δ+∆ M )t − 1 e i∆ M +1 t √ a M +1 √ a M σ 2 + σ 1 + |M + 1 M − 1| = N/2 M =3/2 g 2 i (ω A + δ + ∆ M ) 1 2i (∆ M + δ) e 2i(∆ M +δ)t − 1 √ a M +1 √ a M σ 2 + σ 1 + |M + 1 M − 1| + N/2 M =3/2 g 2 i (ω A + δ + ∆ M ) 1 i (−ω A + ∆ M + δ) e i(−ω A +∆ M +δ)t − 1 √ a M +1 √ a M σ 2 + σ 1 + |M + 1 M − 1| .
We assume that every M satisfies ω A +δ+(2M −1)Ω = 0 and δ + (2M − 1)Ω = 0. Also, by choosing suitable parameters, L + ≡ σ 2 + σ 1 − |3/2 −1/2| . Therefore, we obtain the following.
|ψ(t) |ψ(0) − ig 2 t δ √ a 1/2 √ a 3/2 L + +L − . (A9)
We can this rewrite as |ψ(t) − |ψ(0) t = ig 2 t δ √ a 1/2 √ a 3/2 L + +L − and, we obtain
d|ψ(t) dt −iH eff |ψ(t) (A10)
where we define the effective Hamiltonian as H eff ≡ ig 2 t δ √ a 1/2 √ a 3/2 L + +L − .
By solving the (A10), we obtain a state at a time t with the effective Hamiltonian. This kind of approximation has been used in quantum optics [32], but we firstly apply this technique to the model of the quantum battery with a superabsorption.
FIG. 2 .
2Schematic of N-level charging model.Each frequency of qubits are ωA and initial state of qubits are excited state |1 C .
Also, we describe the energy eigenstates of the N -qubit system as the Dicke states. The energy differences between the Dicke states are written as ∆ M = E M − E M −1 = ω A + Ω(2M − 1). Let us denote the frequency of the qubit 1 by ω A + δ, which is detuned from ∆ 1/2 by δ. On the otherFIG. 3. Schematic of Λ-type Dicke states.
FIG. 4 .
4Schematic of battery charging with our model. The initial state of a battery is |10 B and charger is |−1/2 C .
FIG. 5 .
5Plot of the population of |01 B when we use the effective Hamiltonian or the exact Hamiltonian with ωA = 10, 5, 1. We set the other parameters as g = 10 −3 , δ = 10gN, Ω = −2.3ωA and N = 101.ρ B (t) described as ρ B (t) = Tr N e −iHtott |ψ(0) ψ(0)| e iHtott .(29)Firstly, we compare the dynamics with the effective Hamiltonian H eff with that of the exact Hamiltonian H tot . Actually, as shown in FIG 5 , the dynamics with the effective Hamiltonian is different from the exact Hamiltonian for ω A = 1 where the condition of δ ω A < |Ω| is violated. On the other hand, in Fig 6, we observe a deviation of the dynamics with the effective Hamiltonian from that of the exact Hamiltonian for δ = gN where the condition of gN δ is violated. Next, we analyze the population of {|00 B , |01 B , |10 B , |11 B } of the battery state. We choose the parameters to satisfy conditions of gN δ ω A < |Ω| (See FIG
FIG. 6 .FIG. 7 .FIG. 8 .
678Plot of the population of |01 B when we use the effective Hamiltonian or the exact Hamiltonian with δ = 10gN, 5gN, gN .We set the other parameters as g = 10 −3 , ωA = 10, Ω = −2.3ωA and N = 101. Plot of the population of the battery state against time. Pij denotes the population of |ij B , i, j ∈ {0, 1}. We choose the parameters as g = 10 −3 , ωA = 10, δ = 10gN, Ω = −2.3ωA and N = 31. Plot of the charging timeτN against the number of qubit N .
ACKNOWLEDGMENTS
This work was supported by MEXT's Leading Initiative for Excellent Young Researchers, KAKENHI (20H05661),JST PRESTO (Grant No. JPMJPR1919) and JST's Moonshot R&D (Grant No. JP-MJMS2061), Japan. Appendix A: Derivation of the effective Hamiltonian
(|ω A + 2Ω| + δ − ∆ M ) t 0 dt A i † (t ) e i(|ω A +2Ω|+δ−∆ M )t − 1
i (t ) e i(ω A +δ+∆ M )t − 1 √ a M σ 1 + |M M −
the other terms are negligible as we show below.The other three terms of (A3) can be small for gN δ ω A < |Ω|. Therefore, we can ignore the contribution from (A3). When an initial state is |ψ(0) = |−1/2 C |10 B , the dominant term in (A4) isand N/2We evaluate these two terms. First, for (A7), we obtainwhere we use 1 δt in the last line. Also, we impose a condition of 1 gN t so that the second order (gN ) 2 t δ of the interaction should be larger than the first order gN δ . Since we assume 1 gN t and gN δ, the necessary condition is written as 1 gN t δt. Second, for (A8), we obtainFrom these calculations, the second order of the interaction can be described as
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[
"A strong He ii λ1640 emitter with extremely blue UV spectral slope at z = 8.16: presence of Pop III stars?",
"A strong He ii λ1640 emitter with extremely blue UV spectral slope at z = 8.16: presence of Pop III stars?"
] | [
"Xin Wang \nSchool of Astronomy and Space Science\nUniversity of Chinese Academy of Sciences (UCAS)\n100049BeijingChina\n\nNational Astronomical Observatories\nChinese Academy of Sciences\n100101BeijingChina\n\nInstitute for Frontiers in Astronomy and Astrophysics\nBeijing Normal University\n102206BeijingChina\n",
"Cheng Cheng \nChinese Academy of Sciences South America Center for Astronomy\nNational Astronomical Ob-servatories\nCAS\n100101BeijingChina\n",
"Junqiang Ge \nNational Astronomical Observatories\nChinese Academy of Sciences\n100101BeijingChina\n",
"Xiao-Lei Meng \nNational Astronomical Observatories\nChinese Academy of Sciences\n100101BeijingChina\n",
"Emanuele Daddi \nIRFU/Service d'Astrophysique\nLaboratoire AIM\nCEA/DSM\nCNRS\nUniversité Paris Diderot\nBât. 709\n\nCEA Saclay\nF-91191Gif-sur-Yvette CedexFrance\n",
"Haojing Yan \nDepartment of Physics and Astronomy\nUniversity of Missouri-Columbia\n65211ColumbiaMOUSA\n",
"Tucker Jones \nDepartment of Physics and Astronomy\nUniversity of California Davis\n1 Shields Avenue95616DavisCAUSA\n",
"Matthew A Malkan \nDepartment of Physics and Astronomy\nUniversity of California Los Angeles\n430 Portola Plaza90095Los AngelesCAUSA\n",
"Pablo Arrabal Haro \nNSF's National Optical-Infrared Astronomy Research Laboratory\n950 N. Cherry Ave85719TucsonAZUSA\n",
"Gabriel Brammer \nCosmic Dawn Center (DAWN)\nDenmark\n\nNiels Bohr Institute\nUniversity of Copenhagen\nJagtvej 128, DK2200Copenhagen NDenmark\n",
"Masamune Oguri \nCenter for Frontier Science\nChiba University\n1-33 Yayoi-cho, Inage-ku263-8522ChibaJapan\n\nDepartment of Physics\nChiba University\n1-33 Yayoi-Cho, Inage-Ku263-8522ChibaJapan\n"
] | [
"School of Astronomy and Space Science\nUniversity of Chinese Academy of Sciences (UCAS)\n100049BeijingChina",
"National Astronomical Observatories\nChinese Academy of Sciences\n100101BeijingChina",
"Institute for Frontiers in Astronomy and Astrophysics\nBeijing Normal University\n102206BeijingChina",
"Chinese Academy of Sciences South America Center for Astronomy\nNational Astronomical Ob-servatories\nCAS\n100101BeijingChina",
"National Astronomical Observatories\nChinese Academy of Sciences\n100101BeijingChina",
"National Astronomical Observatories\nChinese Academy of Sciences\n100101BeijingChina",
"IRFU/Service d'Astrophysique\nLaboratoire AIM\nCEA/DSM\nCNRS\nUniversité Paris Diderot\nBât. 709",
"CEA Saclay\nF-91191Gif-sur-Yvette CedexFrance",
"Department of Physics and Astronomy\nUniversity of Missouri-Columbia\n65211ColumbiaMOUSA",
"Department of Physics and Astronomy\nUniversity of California Davis\n1 Shields Avenue95616DavisCAUSA",
"Department of Physics and Astronomy\nUniversity of California Los Angeles\n430 Portola Plaza90095Los AngelesCAUSA",
"NSF's National Optical-Infrared Astronomy Research Laboratory\n950 N. Cherry Ave85719TucsonAZUSA",
"Cosmic Dawn Center (DAWN)\nDenmark",
"Niels Bohr Institute\nUniversity of Copenhagen\nJagtvej 128, DK2200Copenhagen NDenmark",
"Center for Frontier Science\nChiba University\n1-33 Yayoi-cho, Inage-ku263-8522ChibaJapan",
"Department of Physics\nChiba University\n1-33 Yayoi-Cho, Inage-Ku263-8522ChibaJapan"
] | [] | Cosmic hydrogen reionization and cosmic production of first metals are major phase transitions of the Universe occurring during the first billion years after the Big Bang 1-4 , but still poorly explored observationally. Using the JWST NIRSpec prism spectroscopy, we report the discovery of a sub-L * galaxy at z spec = 8.1623 +0.0007 −0.0008 , dubbed RXJ2129-z8HeII, via the detection of a series of strong rest-frame UV/optical nebular emission lines and the clear Lyman break. A strong He ii λ1640 emission is present, the highest redshift He ii line currently known. Its high rest-frame equivalent width (EW = 19 ± 3Å) and extreme flux ratios with respect to UV metal lines and Balmer lines raise the possibility that part of RXJ2129-z8HeII's stellar populations could be Pop III-like 5, 6 . RXJ2129-z8HeII also shows a pronounced UV continuum with an extremely steep (i.e. blue) spectral slope of β = −2.50 ± 0.08, the steepest amongst all spectroscopically confirmed galaxies at z 7, in support of its very hard ionizing spectrum that could lead to a significant leakage of its ionizing flux 7 . Therefore, RXJ2129-z8HeII is representative of the key galaxy population driving the cosmic reionization. To date, this is also the most compelling case where trace Pop III stars might coexist with more metal-enriched stars. 2 The JWST observations analyzed in this work were acquired by a Director's Discretionary program (DD-2767; PI: P. Kelly), targeting the field of the galaxy cluster RXJ2129.7+0005 (henceforth RXJ2129) at z = 0.234. These observations consist of an imaging component using the Near-Infrared Camera (NIRCam) and a spectroscopic component using the Near-Infrared Spectrograph (NIRSpec). The imaging exposures were taken on 6 October 2022 (UT dates quoted throughout) using the F115W, F150W, F200W, F277W, F356W, and F444W filters, covering the wavelength range of λ obs ∈ [1, 5]µm, with the 5-σ limiting depth of ∼29 mag. In Fig. 1, we show a colorcomposite image produced from these data. We also take advantage of the archival HST imaging obtained by the CLASH HST program (PI: M. Postman; 1 ), and compile a photometric catalog combining both the new NIRCam imaging and the existing HST imaging in a self-consistent manner (see the Methods section). The image stamps of our target are displayed in the top row of Fig. 2.This object in fact has two components A and B, with A dominating the total flux in all filters in the NIRCam long wavelength channel. Hereafter unless otherwise specified, we use RXJ2129-z8HeIIto stand for its component A, which is targeted by NIRSpec described below.The follow-up JWST spectroscopy was carried out on 22 October 2022, using the NIR-Spec instrument in the Multi-Object Spectroscopy (MOS) mode. RXJ2129-z8HeII was among the NIRSpec targets that were pre-selected based on their photometric redshifts 8 . The field was observed using the prism disperser, offering a very wide, continuous wavelength coverage of λ obs ∈ [0.6, 5.3]µm. In the inset ofFig. 1, we show a zoom-in view of RXJ2129-z8HeII with the position of the MOS slit superposed. We reduce the NIRSpec MOS spectroscopic data following 1 https://archive.stsci.edu/prepds/clash/ 3 | null | [
"https://export.arxiv.org/pdf/2212.04476v1.pdf"
] | 254,408,693 | 2212.04476 | 119a9ed07ac3edc9cc107f625dca89f1042319d8 |
A strong He ii λ1640 emitter with extremely blue UV spectral slope at z = 8.16: presence of Pop III stars?
8 Dec 2022
Xin Wang
School of Astronomy and Space Science
University of Chinese Academy of Sciences (UCAS)
100049BeijingChina
National Astronomical Observatories
Chinese Academy of Sciences
100101BeijingChina
Institute for Frontiers in Astronomy and Astrophysics
Beijing Normal University
102206BeijingChina
Cheng Cheng
Chinese Academy of Sciences South America Center for Astronomy
National Astronomical Ob-servatories
CAS
100101BeijingChina
Junqiang Ge
National Astronomical Observatories
Chinese Academy of Sciences
100101BeijingChina
Xiao-Lei Meng
National Astronomical Observatories
Chinese Academy of Sciences
100101BeijingChina
Emanuele Daddi
IRFU/Service d'Astrophysique
Laboratoire AIM
CEA/DSM
CNRS
Université Paris Diderot
Bât. 709
CEA Saclay
F-91191Gif-sur-Yvette CedexFrance
Haojing Yan
Department of Physics and Astronomy
University of Missouri-Columbia
65211ColumbiaMOUSA
Tucker Jones
Department of Physics and Astronomy
University of California Davis
1 Shields Avenue95616DavisCAUSA
Matthew A Malkan
Department of Physics and Astronomy
University of California Los Angeles
430 Portola Plaza90095Los AngelesCAUSA
Pablo Arrabal Haro
NSF's National Optical-Infrared Astronomy Research Laboratory
950 N. Cherry Ave85719TucsonAZUSA
Gabriel Brammer
Cosmic Dawn Center (DAWN)
Denmark
Niels Bohr Institute
University of Copenhagen
Jagtvej 128, DK2200Copenhagen NDenmark
Masamune Oguri
Center for Frontier Science
Chiba University
1-33 Yayoi-cho, Inage-ku263-8522ChibaJapan
Department of Physics
Chiba University
1-33 Yayoi-Cho, Inage-Ku263-8522ChibaJapan
A strong He ii λ1640 emitter with extremely blue UV spectral slope at z = 8.16: presence of Pop III stars?
8 Dec 20221
Cosmic hydrogen reionization and cosmic production of first metals are major phase transitions of the Universe occurring during the first billion years after the Big Bang 1-4 , but still poorly explored observationally. Using the JWST NIRSpec prism spectroscopy, we report the discovery of a sub-L * galaxy at z spec = 8.1623 +0.0007 −0.0008 , dubbed RXJ2129-z8HeII, via the detection of a series of strong rest-frame UV/optical nebular emission lines and the clear Lyman break. A strong He ii λ1640 emission is present, the highest redshift He ii line currently known. Its high rest-frame equivalent width (EW = 19 ± 3Å) and extreme flux ratios with respect to UV metal lines and Balmer lines raise the possibility that part of RXJ2129-z8HeII's stellar populations could be Pop III-like 5, 6 . RXJ2129-z8HeII also shows a pronounced UV continuum with an extremely steep (i.e. blue) spectral slope of β = −2.50 ± 0.08, the steepest amongst all spectroscopically confirmed galaxies at z 7, in support of its very hard ionizing spectrum that could lead to a significant leakage of its ionizing flux 7 . Therefore, RXJ2129-z8HeII is representative of the key galaxy population driving the cosmic reionization. To date, this is also the most compelling case where trace Pop III stars might coexist with more metal-enriched stars. 2 The JWST observations analyzed in this work were acquired by a Director's Discretionary program (DD-2767; PI: P. Kelly), targeting the field of the galaxy cluster RXJ2129.7+0005 (henceforth RXJ2129) at z = 0.234. These observations consist of an imaging component using the Near-Infrared Camera (NIRCam) and a spectroscopic component using the Near-Infrared Spectrograph (NIRSpec). The imaging exposures were taken on 6 October 2022 (UT dates quoted throughout) using the F115W, F150W, F200W, F277W, F356W, and F444W filters, covering the wavelength range of λ obs ∈ [1, 5]µm, with the 5-σ limiting depth of ∼29 mag. In Fig. 1, we show a colorcomposite image produced from these data. We also take advantage of the archival HST imaging obtained by the CLASH HST program (PI: M. Postman; 1 ), and compile a photometric catalog combining both the new NIRCam imaging and the existing HST imaging in a self-consistent manner (see the Methods section). The image stamps of our target are displayed in the top row of Fig. 2.This object in fact has two components A and B, with A dominating the total flux in all filters in the NIRCam long wavelength channel. Hereafter unless otherwise specified, we use RXJ2129-z8HeIIto stand for its component A, which is targeted by NIRSpec described below.The follow-up JWST spectroscopy was carried out on 22 October 2022, using the NIR-Spec instrument in the Multi-Object Spectroscopy (MOS) mode. RXJ2129-z8HeII was among the NIRSpec targets that were pre-selected based on their photometric redshifts 8 . The field was observed using the prism disperser, offering a very wide, continuous wavelength coverage of λ obs ∈ [0.6, 5.3]µm. In the inset ofFig. 1, we show a zoom-in view of RXJ2129-z8HeII with the position of the MOS slit superposed. We reduce the NIRSpec MOS spectroscopic data following 1 https://archive.stsci.edu/prepds/clash/ 3
We employ the BAGPIPES software 9 to conduct detailed spectro-photometric analysis of both the broad-band photometry and the full NIRSpec spectrum that we obtained for RXJ2129-z8HeII. Performing full spectrum fitting is highly critical since it is the only appropriate approach to extract the detailed information of stellar population from the low-resolution prism spectroscopy. According to the up-to-date lens model produced by the GLAFIC software 10, 11 , we estimate that the magnification for our object is µ = 2.26 ± 0.14. After correcting for magnification, we obtain the following physical picture of RXJ2129-z8HeII: it is a very low mass (log(M * /M ) ∼ 7.65), young (t age ∼ 210 Myrs) galaxy, actively forming stars (SFR S ∼ 2M /yr) with sub-solar metallicity (log(Z * /Z ) ∼ −0.86) and little dust (A S V ∼ 0.12). Its UV absolute magnitude of M UV ∼ −19.58 mag places it at the level of 40% L * at z ≈ 8 12 . Importantly, the bulk 4 of the ionizing UV photons that caused the cosmic reionization are thought to come from sub-L * systems 13,14 .
Indeed, from the strong detection of the far UV stellar continuum at λ rest ∈ [1300, 2600]Å, we measure the UV spectral slope β = −2.50 ± 0.08 using the standard formalism of f λ ∝ λ β . This ranks RXJ2129-z8HeII as the spectroscopically confirmed galaxy in the EoR having the steepest UV continuum slope -a strong implication of significant leakage of its ionizing radiation to the IGM 7,15 . The completion of reionization by z ∼ 6 requires that the absolute escape fraction of the Lyman continuum (LyC) photons from galaxies should be f LyC esc 10% on average 16,17 . In the EoR, a negative correlation between M UV and β has been seen from photometric analyses, in support of the pivotal roles played by the intrinsically faint systems in dominating the IGM-ionizing photon budget 18,19 . The lower panel of Fig. 3 summarizes all the current measurements of β and M UV of galaxies at z 7, where the spectroscopically confirmed sources are highlighted in color. Using the HST/COS observations of the LyC signals from the Lowredshift Lyman Continuum Survey (LzLCS), Ref. 7 found a strong correlation between β and f LyC esc :
f LyC esc = (1.3 ± 0.6) × 10 −4 · 10 (−1.22±0.1)·β . We therefore derive the LyC escape fraction estimate of RXJ2129-z8HeII: f LyC esc = 0.15 +0.04 −0.03 . We present the detailed measurements of these physical properties in Table 1.
We utilize the pPXF software to perform accurate emission line analyses. The emission line fluxes and equivalent widths 2 (EWs) are measured by fitting the stellar continuum and emission 2 EWs are always measured in rest frame throughout this paper. 5 lines simultaneously using the BC03 stellar population library 20 23 , in agreement with the expectation at z 8 if sub-L * galaxies are the major sources of reionization 14 .
Notably, the He ii emission is the only UV line clearly detected in RXJ2129-z8HeII. This line has the intrinsic (corrected for the magnification) flux f He ii = 64 ± 10 × 10 −20 erg s −1 cm −2 and EW He ii = 19 ± 3Å. At z = 8.1623, this is the highest redshift at which an He ii line detection has been reported in the literature. The nebular He ii line requires a hard ionizing background radiation, which is usually attributed to Wolf-Rayet stars, stripped stars, X-ray binaries, or active galactic neucli (AGN) 24,25 . The location of RXJ2129-z8HeII in the mass-excitation diagram 26,27 and the low limit of f C iv / f He ii (given in Table 1) disfavor a supermassive black hole as the culprit.
6
The lack of variability estimated from archival HST imaging disfavors the possibility of X-ray binaries. In the right panel of Fig. 4 RXJ2129-z8HeII is an intriguing galaxy in the EoR. It has a sub-L * M UV , a large O 32 flux ratio, and high-EW rest-frame optical lines. In particular, its UV spectral slope (β = −2.50 ± 0.08)
is the steepest amongst all spectroscopically confirmed galaxies at z 7 reported to date, implying that it has a large LyC escape fraction f LyC esc ∼ 15%, the largest among its cohort. RXJ2129-z8HeII
is a promising representative of the predominant galaxy populations that are capable of producing and leaking their extreme UV photons to the IGM, causing the bulk of the neutral hydrogen in the IGM ionized. Importantly, its prominent He ii line and the large flux ratios of He ii versus restframe UV metal lines and Balmer lines suggests that it is the best candidate by far to find possible coexistence of Pop III and normal stars in its stellar populations. This work showcases the power of the JWST NIRSpec prism spectroscopy to characterize the stellar continuum as well as nebular emission properties of high-z galaxies simultaneously, thanks to its exceedingly wide wavelength coverage and exquisite sensitivity. Bottom: UV spectral slope (β) as a function of absolute UV magnitude (M UV ). The color-coded circles represent currently existing galaxies with spectroscopic redshifts which also have β measured from Refs. He ii/C iii] and He ii/C iv flux ratios, as compared to local galaxies 28,30 , galaxies at the cosmic noon epoch 24 , and CR7 at z = 6.6 29 . This indicates a different channel for He+ ionization that is unique in the reion- Table 1: Physical properties of RXJ2129-z8HeII. The parameter values are given in median posterior constraints followed by 1-σ confidence interval (CI). The lensing magnification is predicted from the up-to-date GLAFIC lens model 10,11 , and accounted for in the estimates of these physical properties, if necessary. All reported line flux ratios have been corrected for nebular dust extinction.
Methods
Cosmological model. Throughout this paper, we adopt the standard concordance ΛCDM model
with Ω m = 0.3, Ω Λ = 0.7, H 0 = 70 km s −1 Mpc −1 , and the AB magnitude system 48 .
JWST/NIRCam data reduction and mosaicing. The NIRCam imaging data were acquired by a Director's Discretionary program (DD-2767; PI: P. Kelly), whose primary goal is to measure the light curves and spectra of a strongly lensed supernova SN 2022riv at z = 1.52 49 and calwebb_image3 to calibrate each image and build the mosaic image of each band. Furthermore, we remove the "snowball" 3 and 1/f noise are using the scripts from https://github. com/chriswillott/jwst, and the "wisps" 4 are removed using the default wisps templates 5 .
We produce the imaging mosaics on 60 and 30 mas pixel scales, astrometrically aligned to the GAIA DR2 astrometry frame. The image reduction script is uploaded to https://github.com/ chengchengcode/jwst-pipeline-note/blob/main/F150W-snowballflag-wisps-1overf.
ipynb. The data doi is 10.17909/2dxj-z303.
JWST/NIRCam and HST/ACS-WFC3 photometry. For comprehensive photometry of RXJ2129-z8HeII, we not only use the 6-band imaging mosaics produced above, but also include the publicly released HST imaging mosaics from the Cluster Lensing and Supernova Survey (CLASH; Ref. 51 ). CLASH provides the imaging data in 6 bands taken with ACS/WFC (F775W and F814W) and
WFC3/IR (F105W, F125W, F140W, and F160W), highly relevant to the work here.
We first transform the CLASH 0.065" imaging mosaics to mosaics on 0.06" plate scale, and presented in Table 2. For filters with no detections, we report the 2-σ upper limits. The NIRCam photometry normalized at their individual F200W flux is shown in Fig. 5.
Photometric redshift estimates. We use the EAzY software 53 to estimate the photometric redshift from the broad-band photometry presented in We also perform extensive tests to verify the photometric calibration of NIRSpec prism spectra. On one hand, we cross-check the flux levels of our extracted spectra and that of our broad-band photometry from NIRCam imaging in the similar wavelength regime. They show good agreement with each other within 10% (see the upper panel of Fig. 3). On the other hand, we conduct detailed investigation of the wavelength calibration by performing line identifications in low-z galaxies showing multiple emission lines across their entire prism spectra, observed in the same MSA setting whose spectra are extracted in the same fashion. We find that the wavelength offset between the best-fit centroid and that expected for the mean redshift shows a scatter <0.004µm, much smaller than the instrument resolution. So we conclude that our flux and wavelength calibrations are sufficiently accurate.
Spectro-photometric analyses. We employ the BAGPIPES software 55 to perform the spectral energy distribution (SED) fitting of both our spectroscopic and photometric observations simultaneously. We adopt a double power law star-formation history model which is in widespread use to describe the evolution of the cosmic star-formation rate (SFR) density 56 , and has great flexibility 21 to account for both rising and declining star-formation activities, i.e., BAGPIPES utilizes the nested sampling algorithm to perform efficient Bayesian inference and obtain posterior distribution of parameters. We take the [16th, 50th, 84th] percentiles of the parameter posteriors as the median and 1-σ CI, and show the 1D posterior distributions of some key parameters in Fig. 7. For each set of sampled parameters, BAGPIPES produces realistic galaxy SED through its model_galaxy module. We thus overlay the median and 1-σ CI of such model realizations in orange curve and band in the upper panel of Fig. 3, which shows that they are faithful fits to the observations. We also present the median constraints and 1-σ uncertainties of some key physical parameters in Table 1. Here, M * stands for the stellar mass of RXJ2129-z8HeII. SFR S represents the SFR averaged over a few 100 Myrs time scale. t age is the mass-weighted age of the stellar populations.
SFR(t) ∝ t τ a + t τ −b −1(1)
We follow the prescription in Ref. 58 to estimate the absolute UV magnitude, where µ is the magnification in magnitudes. At z = 8.16, the WFC3/F140W filter covers the restframe λ rest ∈ [1300, 1800]Å and therefore appropriate to use. We follow the methodology in Ref. 7 to derive the UV spectral slope (β). We fit f λ ∝ λ Bayesian inference of ISM properties. We apply a well-established Bayesian forward-modeling inference framework to the measured line fluxes derived from our dedicated emission line profile fitting presented in Table 3. We constrain jointly three key ISM properties: gas-phase metallicity
M UV = m F140W + µ − 5 log(d L /10 pc) + 2.5 log(1 + z)(2) − 1 2 · i f EL i − R i · f Hβ 2 σ EL i 2 + f Hβ 2 · σ R i 2 .(3)
Here Strong lensing models. We adopt the cluster lens model of RXJ2129 constructed using the GLAFIC software 10,11 , updated from the initial work of Ref. 66
our customized procedures, elaborated in the Methods section. The middle and bottom rows of Fig. 2 show the resultant 2D and 1D prism spectra of RXJ2129-z8HeII. We detect pronounced emission features of the [O iii] doublets and Hβ line complex at λ obs ∼ 4.5µm, and a clear continuum break at λ obs ∼ 1.1µm with prominent continuum flux redward of the break. The identification of this continuum break as the Lyman break occurring at the Lyα wavelength (λ rest = 1216Å) is in excellent agreement with both the photometric redshift of RXJ2129-z8HeII measured from broadband photometry and the strong [O iii] +Hβ lines in the prism spectrum. Fitting simultaneously the entire spectroscopic (continuum + lines) and the photometric data sets, we confirm RXJ2129-z8HeII as an extreme emission line galaxy with strong continuum emission at z spec = 8.1623 +0.0007 −0.0008 (see Fig. 3), when the Universe is ∼613 Myr years old. This puts RXJ2129-z8HeII deep in the epoch of reionization (EoR), when the IGM was mostly neutral hydrogen 3, 4 .
AcknowledgementsFigure 1 :
1This work is based on observations made with the NASA/ESA/CSA James Webb Space Telescope, associated with the program JWST-DD-2767. The specific observations analyzed can be accessed via https://doi:10.17909/2dxj-z303. The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. XW is supported by CAS Project for Young Scientists in Basic Research, Grant No. YSBR-062. JG acknowledges support from the Youth Innovation Promotion Association of the Chinese Academy of Sciences (No. 2022056). MO acknowledges support by JSPS KAKENHI grants JP20H00181, JP20H05856, and JP22H01260. Author Contributions XW wrote the paper and developed the main interpretation of the results, aided by ED, HY, TJ, and MM. XW also performed the spectro-photometry analyses and NIRSpec data reduction with the help from GB and PAH. CC conducted the NIRCam data reduction and image mosaicing. JG performed the emission line profile fitting. XLM led the photometry and the morphological analyses. MO modeled the galaxy cluster. All authors contribute to the writing and discussion of the main science results. A color-composite NIRCam image of the RXJ2129 galaxy cluster. The regions of formally infinite magnification at z = 8.16 predicted by the up-to-date GLAFIC lens model 10, 11 , are represented by the magenta curves. The location of RXJ2129-z8HeII is highlighted by the red circle. It is estimated to have a magnification factor of µ = 2.26 ± 0.14 (see Methods section). The inset corresponds to a zoom-in view (3 ×3 ) of RXJ2129-z8HeII. In this stamp, we also show the orientation of the NIRSpec slit (0. 2 × 1. 4), constructed by three MSA slitlets, through which the prism spectroscopy of RXJ2129-z8HeII is taken by the JWST Director's Discretionary program (DD-2767; PI: P. Kelly). Here, we use the F115W+F150W, F200W+F277W, and F356W+F444W imaging as the blue, green, and red colors, respectively.
Figure 2 :
2JWST-HST imaging and JWST/NIRSpec spectroscopy of RXJ2129-z8HeII. Top: RXJ2129-z8HeII's postage thumbnails in multiple filters, cut from the HST CLASH (PI: M. Postman) and JWST/NIRCam (DD-2767; PI: P. Kelly) imaging mosaics. These stamps follow the same sky orientation as shown in Fig. 1. The strong flux deficit in ACS/F814W, WFC3/F105W and NIRCam/F115W bands, as compared to the other NIRCam filters indicates z 8. Middle and Bottom: NIRSpec prism spectroscopy of RXJ2129-z8HeII. In the middle trace, we show the unsmoothed 2D prism spectra combined from the three individual slit-dithering sequence, using the up-to-date reduction software msaexp (see Methods section). We detect prominent emission features of [O iii] and Hβ at λ obs ∼ 4.5µm, and strong continuum break blueward of Lyα at λ obs ∼ 1.1µm. On the bottom row, we show the 1D spectrum optimally extracted from the 2D spectral trace in gray histograms with the 1-σ uncertainty in cyan. We mark the locations of multiple rest-frame UV and optical emission lines using the magenta dotted lines, and the Lyman break in the blue dashed line. The strong detections of [O iii] +Hβ and the Lyman break pinpoints a secure spectroscopic redshift of RXJ2129-z8HeII to be z spec = 8.1623 +0.0007 −0.0008 .
Figure 3 :
3Spectro-photometric analyses of the full JWST-HST observations of RXJ2129-z8HeII. Top:We employ the BAGPIPES software to perform full spectrum fitting to the data of JWST/NIRSpec prism spectrum (observed flux in blue and 1-σ uncertainty in cyan band), and the broad-band photometry observed with JWST/NIRCam and HST/WFC3+ACS. Our best-fit SED model is represented by the orange curve.
Figure 4 :
4Emission line analyses for RXJ2129-z8HeII. Left: We adopt the pPXF code to conduct dedicated emission line profile fitting to the observed NIRSpec prism spectra of RXJ2129-z8HeII. In the top panel, the stellar continuum (in red) is fitted using the BC03 stellar population templates 20 , dust corrected with the Calzetti reddening curve 46 . The emission lines (in blue) are fitted assuming Gaussian profiles. The inset zooms in on the strong detection of the He ii line. The bottom panel displays the residual with continuum + emission lines removed, as compared to the 1-σ error spectrum (in yellow), demonstrating that the noise properties are well defined. Right: UV line ratios determined from individual galaxies at various redshifts, which show affirmative He ii detections (SNR 2.5). These measurements are color-coded in EW He ii reported in the respective work. The arrows denote 2-σ lower limits. RXJ2129-z8HeII shows greatly elevated
Figure 5 :
5then PSF-match the 6 HST ACS/WFC3 filters and 6 JWST/NIRCam filters to the F444W resolution (FWHM = 0.14"). We utilize a window function to remove the high-frequency noise in the Fourier domain. We modify the rms images in order to match the F444W resolution for every other filtersusing RMS F444W,matched = ΣW 2 i,F444W * RMS 2 i , where RMS F444W,matched and RMS i are the rms after PSF-matching and the original rms in filter i respectively, W i,F444W is the kernel used to match all PSFs to that of the F444W filter through PSF F444W,matched = PSF i * W i,F444W . In the JWST NIRCam F200W imaging, RXJ2129-z8HeII displays two components A and B, marked by the arrows in the F200W stamp shown in Fig. 2. The JWST NIRSpec MOS slit falls on component A, which is also predominant in the total flux in the long-wavelength NIRCam filters. For the sake of accurate photometry of the two components, we use the F200W image as the detection filter and rerun our photometry routine to segment them. To maximize the detection SNRs and thus obtain more accurate SED fitting, fluxes in each filter are measured within fixed apertures of 2×FWHM of the F444W image (0.28 ) in diameter. Furthermore, the flux is dust corrected for galactic extinction through F i,cor. = F i × 10 (A i /2.5) , where A i is the extinction in the i-th band 52 . The resultant photometric measurements of the two components of RXJ2129-z8HeII is We compare the photometry for both components of RXJ2129-z8HeII, normalized at their individual F200W flux. In each filter, the data points are slightly offset in wavelength for clearer visualization. The photometry suggests that component B has comparably blue UV continuum, yet possibly with substantially fainter oxygen lines in the rest-frame optical (probed by F444W), indicating that component B is more metal poor than A. Also, component B's SED shape might accommodate higher He ii EW, albeit at lower confidence.
Figure 6 :
6EAzY fit to the broad-band photometry of RXJ2129-z8HeII, performed on the components A and B separately. The inset panel shows the P(z) of the photometric redshift. The photometric redshift constraints of the two components are both in good agreement with z spec = 8.1623 determined from the NIRSpec prism spectroscopy of component A. This strongly implies that component B is also part of the galaxy RXJ2129-z8HeII. galaxies with strong emission lines. The resultant photometric redshifts of RXJ2129-z8HeII component A and B are z phot = 8.63 ± 0.40 and z phot = 8.44 ± 0.63 respectively (see Fig. 6), in good agreement with the precise spectroscopic redshift we derived from NIRSpec prism spectroscopy. JWST/NIRSpec prism spectroscopy and data reduction. The JWST NIRSpec observation of RXJ2129-z8HeII was carried out on 22 October 2022 using the prism spectroscopy mode with the resolution of R = λ/∆λ ∼ 50 − 400. The relative position of the slit to the galaxy is shown in the inset of Fig. 1, primarily covering component A dominating the total flux. The slit is composed of three shutters in the micro-shutter array (MSA), accounting for a total size of 0.2"x1.4", approximately. The observation adopts the standard 3-point nodding pattern accumulating a total exposure time of 4464.2 seconds, to facilitate background subtraction.We reduce all the individual exposures with the latest version of the reference files for NIR-Spec jwst_1014.pmap. First of all, the level-1 calwebb_detector1 calibration pipeline is employed to reduce the raw exposures (uncal.fits) into count rate maps (rate.fits), during which step the detector artefacts and cosmic rays are removed and flagged. The resultant count rate maps are visually inspected to manually mask any remaining artefacts and bad pixels. We then rely on the custom reduction software msaexp 6 to perform the remaining steps of the reduction. It first preprocesses the rate images to equalize the pedestal of each exposure, identify and correct the "snowball" and 1/f noise features. Then it calls some specific modules from the standard level-2 calwebb_spec2 calibration pipeline to carry out the the bulk of the data reduction. These modules include AssignWcs,Extract2dStep, FlatFieldStep, PathLossStep, PhotomStep, which perform WCS initialization, 2D extraction of spectra, slit-level flat-fielding, path-loss correction, wavelength and flux calibration of each science exposure. msaexp subsequently subtracts background from each exposure using the two associated exposures at the other dithered positions from the nodding sequence, and drizzles the background-subtracted science exposures onto a common wavelength grid via inverse-variance weighting, with outlier rejection and bad pixel masking. We adopt an oversampling rate of 2 for the output wavelength grid, to Nyquist sample the line spread function in order to improve the sampling of the emission line profiles. Finally msaexp obtains the 1D spectrum from the 2D combined spectral traces following an optimal extraction methodology 54 , which uses the actual light profile of our target as the optimal aperture for spectral extraction. The object light profile along 6 https://github.com/gbrammer/msaexp 20 the cross-dispersion direction is modeled as a Gaussian function fit to the collapsed 2D spectra (summed along the dispersion direction). As a result, we obtain the 2D/1D prism spectra, as shown in in the middle/lower panels inFig. 2, covering an uninterruptedly wide wavelength range of λ obs ∈ [0.6, 5.3]µm. For z ∼ 8 sources, this corresponds to a contiguous coverage of the restframe UV and optical galaxy SED at λ rest ∈ [700, 6000]Å.
with a Jeffery's prior on the exponents: a, b ∈ (0.01, 1000) and a flat prior on the peak time of star formation: τ ∈ (0, t z ), where t z is the age of the Universe at the observed redshift.BAGPIPES relies on the BC03 20 stellar population synthesis model and the nebular emission model created by the Cloudy photoionization code. For other key assumptions, we choose the Chabrier initial mass function 57 , the Calzetti dust attenuation law 46 with A S V ∈ (0, 3), a stellar metallicity range of Z/Z ∈ (0, 2), and a conservative Gaussian redshift prior of z ∼ N(8.16, 0.01).
Figure 7 :
7The 1D posterior distributions of some key parameters fit from our spectro-photometric analyses: spectroscopic redshift (z spec ), stellar mass (M * ), star-formation rate (SFR S ), UV spectral slope (β), and stellar-phase metallicity (Z * ).
β in the wavelength range of λ rest ∈ [1300, 2600] to the model galaxy SEDs produced by BAGPIPES with all possible emission features properly masked. Note that throughout this work, β stands for β obs in the nomenclature used by Ref. 7 without dust correction. After correcting for dust, Ref. 7 finds out that the intrinsic UV spectral slopes (β int ) of the LzLCS galaxies observed with HST/COS are clustered in the range of [Emission line fitting and measurements. We perform the stellar population analysis and emission line fittings to the prism data by using the pPXF code 59 and BC03 stellar population library that generated based on the Chabrier IMF 20 , by assuming the CAL reddening curve for dust extinction correction 46 . During the spectral analyses, the range of emission-line intrinsic width is set to [0, 300] km/s. The flux ratio of [O iii] λλ4959,5007 doublet is fixed to the theoretical value of 0.33. For the weak emission lines in the UV bands, we bound their shift of line centers to those 23 stronger emission lines (e.g. He II) to improve the fitting robustness. As shown in the left panel of Fig. 4, the residual of the spectral fitting is consistent with the flux error at the wavelength space. The error of emission-line fluxes and EWs are hence obtained from the spectral fitting of 50 mock spectra that generated based on the best-fitting model spectrum and flux error. The measured emission line fluxes and EWs are listed in Table 3. The inset in the left panel of Fig. 4 shows a zoom-in view of our fitting result to the source spectrum at λ rest ∈ [1600, 1700], where He ii and the oxygen auroral lines O iii]λλ1661,1666 are marked in vertical dashed lines. We verify that the detection of the He ii line is robust against negligible contamination from the oxygen auroral lines.
( 12+log (
12+logO/H)), nebular dust extinction (A N V ), and de-reddened Hβ flux ( f Hβ ), through the following likelihood function following Refs. 60, 61 , i.e., L ∝ exp
26 Figure 8 :Figure 9 :
2689. In total, 22 individual images of 7 multiply lensed background galaxies spectroscopically confirmed by Ref67 are used as the strong lensing constraints. The macroscopic mass model consists of one cluster-scale dark matter halo in the elliptical Navarro-Frenk-White profile68 , combined with galaxy-scale halos modeled using the pseudo-Jaffe profile according to the scaling relation of the velocity dispersion and luminosity of cluster member galaxies69 . The best-fit model is derived using the Markov chain Monte Carlo The corner plot showing the 1D/2D posterior distributions of the fitting parameters, i.e., gas-phase metallicity (12 + log(O/H)), nebular dust content (A N V ), and intrinsic Hβ line flux ( f Hβ ) of RXJ2129-z8HeII, constrained using our Bayesian forward-modeling method. The panel in the upper right shows the −χ 2 /2 values for all the 80,000 parameter sampling iterations. They clearly reach a global minimum of χ 2 , implying a good convergence of our Bayesian inference. 27 sampling process with a χ 2 minimization assuming a positional uncertainty of 0.4". A hundred additional realizations are also created to bootstrap the 1-σ statistical uncertainties for magnifications. As a consequence, we obtain the best-fit and 1-σ CI of the magnification estimates of RXJ2129-z8HeII to be 2.26 and [2.12, 2.40]. We also double check with an independent lens model built by the Lenstool software 67 , and derive consistent results. Source morphology. As shown in the multiple image stamps displayed in the upper panels of Fig. 2, the entire RXJ2129-z8HeII galaxy consists of two components: A and B, with the former dominating the flux in rest-frame optical probed by the NIRCam long wavelength channels, and the latter clearly manifesting in the rest-frame UV covered by the short wavelength channels, especially F200W. Therefore, our morphology analyses are performed in two NIRCam bands, i.e., F200W and F444W, representing the rest-frame UV and optical light profiles of our galaxy, respectively. We rely on Galfit ( 70 ) to model both components of RXJ2129-z8HeII observed in F200W. First, two models are chosen to fit component A. One is a Sérsic profile describing the underlying extended structure with a Sérsic index (n) of 4, an effective radius (R e ) of 3 pixels, an axis ratio (b/a) of 0.68 and a position angle (θ) of 48 deg. Pure PSF model is adopted to fit the nucleated emission in the center of component A. Component B is well reconstructed with a Sérsic profile (n = 4, R e = 2 pixels, b/a = 0.8, θ = 45 deg) and a pure PSF model. The resultant model residuals removing component A and both components are shown in the lower panels of Fig. 9. The inset of Fig. 1 shows the relative location of the NIRSpec MOS slit, covering primarily component A of RXJ2129-z8HeII. As implied by the observed prism spectra shown in the upper panel of Fig. 2, the broad-band flux of the NIRCam F444W filter is likely dominated by the high The rest-frame optical and UV morphology of the entire RXJ2129-z8HeII galaxy, comprising two components: A and B. We use Galfit to perform morphological modeling of both components in JWST NIRcam imaging taken at filters F444W (upper panels) and F200W (lower panels). At z spec = 8.16, F444W and F200W correspond to rest-frame optical and UV, respectively. In each row, we show the original observed image, Galfit models of both components, residual removing models of both components, Galfit model of component A on which the JWST prism spectroscopy is acquired, and residual removing only the models of component A. As indicated by the residual panels, we achieve reasonable morphological models of the entire galaxy. equivalent width [O iii] +Hβ nebular emission lines (also see Table 3). Indeed we observe a more extended light profile in the rest-frame optical than that in the UV. We thus model the image of component A in F444W with an exponential disk and a Sérsic profile. The diffuse and extended structure is well reproduced by an exponential disk model with disk scale-length of 4 pixels, b/a of 0.36 and θ of 43 deg. The nucleated structure is modeled by a Sérsic profile (n=4) with R e = 7 pixels, b/a = 0.75 and θ = 65 deg. Component B in F444W is very faint and can be well fitted by an exponential disk model with disk scale-length of 1.5 pixels, b/a of 0.86 and θ of 40 deg. The model residuals are shown in the upper panels of Fig. 9.
and the Gaussian component describing each line (see the left panel of Fig. 4). From the NIRSpec prism data, we detect 5 limits of EW Lyα < 7Å. The absence of Lyα line is likely due to the damping wing opacity caused by the diffuse neutral IGM at z ∼ 8 and/or dense self-shielding HI gas clouds inside the large ionized HII bubbles 21 . After correcting for dust attenuation, we derive a high intrinsic flux ratio of [O iii]λ5008 (:= [O iii]) and [O ii] to be O 32 = 11.3 ± 3.9. This high O 32 value provides further confirmation of a hard ionizing spectrum and a large LyC escape fraction 22 , in accord with our estimate based on β. From our measurements of EW [O iii] , we estimate the ionizing photon production efficiency to be log(ξ ion [erg −1 Hz]) ∼ 25.50 following Ref.emission lines -[O iii] λλ5008,4960, Hβ, Hγ, [O ii]λλ3727, 3730 (:= [O ii]), and He iiλ1640 (:=
He ii) -with signal-to-noise ratios (SNRs) 2. In addition, we give 2-σ upper limits for other
lines in Table 3, including He ii λ4686, [O iii] λ4363, Hδ, C iii]λλ1907, 1909 (:= C iii]), and
C ivλλ1548, 1551 (:= C iv). We do not see any signatures of Lyα emission and put a 2-σ upper
, we show the flux ratios of He ii against rest-frame UV carbon lines measured in galaxies at various redshifts with He ii detections at SNR 2.5 24, 28-30 . As opposed to He ii, these carbon lines of RXJ2129-z8HeII are much weaker. We also do not detectany other rest-frame UV metal lines (e.g. O iii]λλ1661,1666, N iv]λλ1483,1487, N vλ1240),
inconsistent with the signatures of Wolf-Rayet and stripped stars that are frequently seen in local
galaxies 31, 32 .
Alternatively, the He ii line could be due to high-mass, metal-free population III (Pop III)
stars that have strong He + ionizing flux 1 . While the interstellar medium (ISM) of RXJ2129-
z8HeII is already metal-enriched to roughly one-twelfth solar (12 + log(O/H) ∼7.62), its measured
flux ratio of f He ii / f Hβ = 2.1 ± 0.4 is much larger than the fiducial value for metal-poor Hii regions
( f He ii / f Hβ
0.15) 1 . All this, together with the large EW He ii ∼ 20Å, suggests that RXJ2129-
z8HeII likely has a mixture of "normal" and Pop III stars, the latter of which are the energy source
of the He ii line emission 5, 6 . The putative popIII contribution might be intermixed to enriched
populations, or spatially segregated. Intrinuingly, the prism spectroscopy is taken on component A
of RXJ2129-z8HeII only, with component B residing ∼0. 2 (i.e. ∼1kpc proper) to the North East.
We rederive photometry for A and B separately, and find that they have comparable photometric
estimates: z A
phot ∼ 8.63±0.40 and z B
phot ∼ 8.44±0.63, both in good agreement with the spectroscopic
redshift measured for A. Our separate photometry for component B shows a comparably blue UV
continuum similar to A's, but with much smaller ratio of the F444W/F200W flux (see Fig. 5 in
the Methods section). Assuming that the different color is driven by EW [O iii] , we estimate that B's
7
EW [O iii] is lower by a factor of ∼ 2 than A's, indicating that B's ISM is more chemically pristine.
The SED shape might also allow for an higher EW HeII line in component B. Simulations predict
that Pop III stars can continue to form if there exist pockets of primordial gas not yet polluted by
metal-enriched outflows from nearby star-forming galaxies 2 . A deep NIRSpec/IFU observation
with high spectral resolution covering both components can provide a more comprehensive picture
of the origin of this prominent He ii emission observed in RXJ2129-z8HeII.
8,[33][34][35][36][37][38][39][40] . Among this cohort, RXJ2129-z8HeII has by far the steepest slope, highlighted by the star. The gray diamonds show the high-z galaxy candidates selected with z phot 8, sourced from Refs.[41][42][43][44][45] . The dotted and dashed lines denote the population-averaged correlation of M UV and β at z ∼ 718 and z ∼ 819 . The shaded regions correspond to the parameter space where the escape fraction of the ionizing radiation ( f LyC esc ) is above 5, 10, and 20 percent, respectively, derived by Ref.7 from the LzLCS dataset. This strongly suggests that RXJ2129-z8HeII is hitherto the most promising spectroscopically confirmed galaxy that enables significant LyC leakage into the IGM to actually contribute to the cosmic reionization.
ization epoch, likely associated with Pop III-like stellar populations, giving rise to strong He ii emission with no metal lines in the rest-frame UV spectrum of RXJ2129-z8HeII. The red shaded region marks the criterion for strong LyC leakers ( f LyC esc > 10%) found by Ref.47 .Parameters
Values
R.A. [deg]
322.416266
Decl. [deg]
0.099675
z spec
8.1623 +0.0007
−0.0008
µ
2.26 +0.14
−0.14
Spectro-photometric analyses
log(M * /M )
7.65 +0.10
−0.08
SFR S [M /yr]
2.00 +0.66
−0.38
A S
V [mag]
0.12 +0.05
−0.05
log(Z * /Z )
-0.86 +0.05
−0.05
t age [Myr]
210.59 +93.73
−138.85
M UV [mag]
-19.58 +0.03
−0.03
β
-2.50 +0.08
−0.08
f LyC
esc
0.15 +0.04
−0.03
Emission line analyses
EW He ii [Å]
19 ± 3
f He ii / f C iv
>2.5 (2-σ)
f He ii / f C iii]
>3.8 (2-σ)
f He ii / f Hβ
2.1 ± 0.4
O 32
11.3 ± 3.9
12+log(O/H)
7.62 +0.13
−0.09
SFR N [M /yr]
6.64 +3.20
−1.36
A N
V [mag]
0.29 +0.36
−0.21
log(ξ ion [erg −1 Hz])
25.50 +0.05
−0.06
F277W, and F444W). We reduce these NIRCam images using the standard JWST pipeline version 1.8.1 and making use of the jwst_1013.pmap context. We follow the standard three stages of the image data reduction pipeline calwebb_detector1, calwebb_image2, 50 discovered
in the RXJ2129 field, by a HST SNAP program (GO-16729; PI P. Kelly). The exposure times
are 4982 seconds for the F150W/F356W filters, and are 2061 seconds for the other four NIRCam
filters (F115W, F200W,
Table 2 .
2All together, we use the HST filtersACS/F775W, ACS/F814W, WFC3/F105W, WFC3/F125W, WFC3/F140W, WFC3/F160W, as well
as the JWST NIRCam filters F115W, F150W, F200W, F277W, F356W and F444W. We adopt the
standard set of galaxy SED templates eazy_v1.1_lines.spectra.param, which includes star-forming
Table 2 :
2Detailed photometry of RXJ2129-z8HeII separating components A and B from HST ACS optical, WFC3 infrared, and JWST NIRCam infrared imaging. All uncertainties refer to the 1-σ standard deviation unless specified otherwise.
Table 3 :
3Rest-frame UV and optical line flux properties measured from our emission line analyses of the NIRSpec/MSA prism spectrum with the pPXF software (see Methods section for details). We list line fluxes for 2σ detections, or the 2-σ upper limits. The error bars correspond to 1-σ standard deviations. Recall that intrinsic fluxes are a factor of ∼ 2.2 smaller, due to magnification.ing case B recombination conditions. We also estimate the instantaneous star formation rate (SFR N ) from the Balmer line luminosity with the Kennicutt calibration 64 and the Chabrier IMF57 : SFR N = 1.6 × 10 −42 L(Hβ) [erg s −1 ] [M yr −1 ]. The value of SFR N for strong emission line galaxies is usually much higher than that of SFR S derived from SED fitting, as the former tightly reflects the burstiness in star formation65 .We adopt the Emcee software to perform the Markov Chain Monte Carlo Bayesian parameter sampling, with 100 of walkers each sampling 1000 iterations. After a burn-in of 200 for each walker, we have sampled the parameter space 80,000 times. We show the resultant 1D and 2D parameter constraints inFig. 8. The median constraints and 1-σ CI are shown inTable 1. To understand the energy source that power the photoionization of the nebular emission observed in RXJ2129-z8HeII, we rely on the mass-excitation diagram26,27 , which is an empirical relation between stellar mass and the [O iii]/Hβ flux ratio with demarcation schemes separating Hii region and and AGN. We verify that RXJ2129-z8HeII can be safely classified as a star-forming galaxy with negligible contamination from AGN ionization.25
https://jwst-docs.stsci.edu/data-artifacts-and-features/snowballs-artifact 4 https://jwst-docs.stsci.edu/jwst-near-infrared-camera/nircam-features-and-caveats/ nircam-claws-and-wisps 5 https://stsci.app.box.com/s/1bymvf1lkrqbdn9rnkluzqk30e8o2bne
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"Dipartimento di Fisica dell'Universit a\nINFN Sezione di Perugia\nvia A. PascoliI{06100PerugiaItaly"
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"Mon. Not. R. Astron. Soc"
] | We re-estimate the angular 2-point galaxy correlation function from the Lick galaxy catalogue. We argue that the large-scale gradients observed in the Lick catalogue are dominated by real clustering and therefore they should not be subtracted prior to the estimation of w( ). We nd that if no such correction is introduced the galaxy correlations are perfectly consistent with the those found in the APM survey. Thus, the long standing discrepancy between the Lick and APM angular correlations is lifted. | null | [
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Lick Galaxy Correlation Function Revised
1994
Manolis Plionis
Scuola Internazionale Superiore di Studi Avanzati
SISSA
via Beirut 2{4I{34013TriesteItaly
International Center for Theoretical Physics
ICTP
I{34014TriesteStrada CostieraItaly
Stefano Borgani
Scuola Internazionale Superiore di Studi Avanzati
SISSA
via Beirut 2{4I{34013TriesteItaly
Dipartimento di Fisica dell'Universit a
INFN Sezione di Perugia
via A. PascoliI{06100PerugiaItaly
Lick Galaxy Correlation Function Revised
Mon. Not. R. Astron. Soc
0001994galaxies: clustering { galaxies: formation { large{scale structure of Uni- verse
We re-estimate the angular 2-point galaxy correlation function from the Lick galaxy catalogue. We argue that the large-scale gradients observed in the Lick catalogue are dominated by real clustering and therefore they should not be subtracted prior to the estimation of w( ). We nd that if no such correction is introduced the galaxy correlations are perfectly consistent with the those found in the APM survey. Thus, the long standing discrepancy between the Lick and APM angular correlations is lifted.
INTRODUCTION
The Lick galaxy counts (Shane & Wirtanen 1967;Seldner et al. 1977, SSGP hereafter) was the rst large wide-angle catalogue to give rise to a lot of statistical studies of galaxy clustering (cf. Limber 1954;Peebles & Hauser 1974;Groth & Peebles 1977, GP77 hereafter). The latter authors calculated, among other things, the two-point angular correlation function of the Lick counts and derived the well known power law form w( ) = A 1 (1) for angular separations < 2:5 , with A 0:0684 and 1:741 :
(2) Probably the most impressive and much debated feature of w( ) is its break at 2:5 corresponding to a spatial scale, at the characteristic depth of the Lick catalogue, of 10 h 1 Mpc. Geller, de Lapparent & Kurtz (1984) and de Lapparent, Kurtz & Geller (1986) argued for an arti cial origin of the break, relating it to systematic e ects due to plate-toplate variations of the limiting magnitude and to observer dependent e ects. Groth & Peebles (1986a,b, GP86, hereafter) reanalysed the Lick counts, in view of this criticism and concluded that the shape of the 2-point correlation function is not signi cantly a ected by the e ects discussed by de Lapparent et al. (1986). Nevertheless, due to this interesting debate, many issues related to plate matching procedures and to observer{dependent e ects came to light, which help guide new attempts to construct wide-angle catalogues of extragalactic objects.
With the advent of new automated galaxy surveys (APM, COSMOS), the existence of the break in w( ) was tested and has been con rmed. Earlier, Hewett (1982) and Stevenson et al. (1985) had presented this evidence although at that time the scale on which it occurs was in dispute. In the APM galaxy catalogue, the break in the two-point angular correlation function occurs at 1:5 which corresponds to 2:5 3 at the depth of the shallower Lick catalogue (Maddox et al. 1990). Furthermore Collins et al. (1989) and using the COSMOS galaxy survey in the South Galactic hemisphere nd consistent results supporting the reality of the break in w( ).
Although the issue of the reality of the break in w( ) has been clari ed, an inconsistency between the Lick and APM correlation functions has emerged. This is the behaviour of w( ) at large angular scales, beyond the break. The APM catalogue shows a smoother break which implies more largescale power. This issue is of great importance since it places severe constrains in models of galaxy formation. In particular the quite popular standard CDM model successfully predicts the Lick map 2-point correlation function, but cannot accommodate the excess power seen in the APM w( ) at large 's (Bond & Couchman 1988;Moscardini et al. 1993;Baugh & Efstathiou 1993).
Since the volumes probed by both catalogues are large enough to be considered fair samples of the Universe (V 10 7 10 8 h 3 Mpc 3 ) the above discrepancy should probably be attributed to errors related to the reduction-estimation and/or plate matching procedure. Nichol & Collins (1993) investigated the e ects of galactic extinction and plate{to{ plate matching errors on the COSMOS survey. They concluded, however, that the estimated w( ) is only marginally a ected by such e ects, a result pointing in favour of a reliable large-scale power detection. However, Fong, Hale-Sutton & Shanks (1992), discussing the e ects of possible unaccounted photometric zero-point errors on the APM w( ) showed that removing the e ects of such errors re-moves the excess power, at large angular scales, seen in the APM results. In this paper we attack this issue from the opposite side. We argue that one of the procedures used by GP77 to estimate the w( ) (ie., large-scale density gradients subtraction) eliminates real clustering, which then results in the sharp break of the correlation function. If all other corrections are included (GP77, GP86) the resulting Lick correlation function is identical to that of the APM survey.
ESTIMATING THE TWO-POINT CORRELATION FUNCTION
The Lick galaxy counts are contained in 1246 plates covering 8.8 steradians of sky, north of declination 23 . Each plate covers an area of 6 6 and are spaced along lines of constant declination such that they overlap by at least 1 in declination and right ascension. We use the`free-of-overlap' SSGP corrected counts (Plionis 1988). Note that GP77 and GP86 used the inner 5 5 of each plate not taking into account the stretching of the plates with varying declination. Thus they used double counted cells ( 8% of total) while we have not. The 2-point galaxy correlation function has been evaluated using the estimator west( ) = hninji h 1 2 (ni + nj)i 2 1 ;
( 3) where =2 < + =2 and ni is the galaxy count of the i th cell. Using the above local normalization (Peebles & Hauser 1974) has the advantage of making some correction for inhomogeneities arising from patchy obscuration and from intra-plate gradients (cf. Sharp 1979).
We have estimated w( ) using two di erent cell sizes at two di erent separation ranges:
The original SSGP 10 0 cells with i and j cells from the same plate (intra-plate estimate) for angular separations 0:2 3:6 in linear bins of amplitude = 0:2 . 60 0 cells in the separation range 1:6 52 . For < 6 we have used linear bins with = 0:625 , while for > 6 we have used logarithmic bins with (log ) 0:06. We have estimated the intra-plate w( ) for separations < 4:8 and the inter-plate w( ) for the whole range of angular separations. Using the notation of GP86, the correction factor error model is C e i = Ci + Cisi + (Ci C ) ;
(4) where C e i and Ci are the estimated and real plate correction factors, is the amplitude of systematic errors in plate correction factors and si is the random plate correction factor error (with ij = hsisji their covariance). According to this model the intrinsic correlation function w , is: w = (west A)=B ;
(5) with A = 2 ij ;
B = hC =3 1 i Cji(1 + 2 ij ) :(6)
where we have dropped the dependance, since GP86 found = 0 0:15, while hC =3 1 i Cji 1 (with a very weak dependance on ). For our intra-plate estimates of w( ) we use ii = 0:006 (as found by GP86) and not the original 0.004 value of GP77 while for inter-plate estimates we use ij = 0 (for neighbouring plates having a 5 overlap, GP77 used -0.001, a value that would enhance w( ) at the corresponding scales, while GP86 found a bij value between -0.0015 and 0.001).
The only correction used by GP77, which we have neglected, is the subtraction of the large-scale galaxy density gradients from the map, prior to the estimation of w( ). The reasons are the following: Plionis (1988) found that the large-scale number count anisotropies, seen in the Lick map, add up to a robust dipole pointing within 30 of the CMB dipole direction. In fact, this result was the rst indication that there is a contribution to the local peculiar motion from scales 50 h 1 Mpc, something which is veri ed by a number of independent studies (cf. Following GP77, we evaluated the 2{point Lick galaxy correlation function by dividing the whole sample into four zones, corresponding to four galactic latitude intervals: (A) 40 b 55 ; (B) b > 55 ; (C) 55 b 40 ; (D) b < 55 . We present our estimates of the Lick correlation function in Figure 1. The open symbols denote the corrected intra-plate estimates of the correlation function, based on the 10 0 cells, while the solid circles denote the inter-plate estimates (based on the 60 0 cells). These results are obtained by averaging between the four zones while the errorbars are the corresponding 1 scatter. We also plot w( ) for jbj 50 (solid line), to test the possible e ects of galactic absorption, and it is evident that these estimates are consistent with our main ones.
Note, that the intra-and inter-plate estimates are in close agreement in the overlapping range of angular separations and comparing our 60 0 -cell based estimates of the intra{ and inter{plate w( ) (in the range 1:6 < < 4:8 ),
we nd a mean o set h wi < 0:004 (see insert of Figure 1) and a mean uctuation h w=wi 0:1. The value of h wi corresponds to an rms number count variation among plates of 0.06 which in turn corresponds, assuming log N / 0:6m, to a magnitude o set among plates, m 0:04, which is of the same order as that found in the COSMOS survey (Collins et al. 1992) and 40% larger than that found in the APM survey (Maddox et al. 1990). The di erence between the intra{ and inter{plate estimates of w( ) is probably the result of the plate{to{plate variation of the limiting apparent Figure 1. The Lick 2{point correlation function, compared with the APM result. Open and lled circles refer to the intra{ and inter{plate estimates, respectively. Results are the average between the four subsamples, corresponding to four di erent galactic latitude intervals (see text). For clarity reasons we plot only a few errorbars (1 scatter between these subsamples). The solid line is the w( ) estimate for jbj 50 and the dashed line is the best{ t to the GP77 analysis. Crosses are the APM correlation data, scaled to the Lick depth (Maddox et al. 1990). The insert shows the di erence of the 60 0 cell based correlation estimates between inter{ and intra{plate analysis, in the angular range of overlap. The errorbars correspond to di erent values of b ij (in the range -0.001, 0.001]). magnitude which introduce correlations between nearby intrinsically faint galaxies, revealed in 'deep' plates, and the bright galaxies at similar distances seen in a neighbouring shallow' plate. GP77 discussed this problem and estimated it to have an amplitude of 0.15w( ) which is quite close to the value we nd. In accordance with GP77 we do not correct for this e ect, because its amplitude is uncertain and model dependent note however that if we correct the interplate estimates by w inter = (1 f) winter, using f = 0:1, we nd qualitatively the same results as those before applying this correction].
In Figure 1 we plot, as crosses, the APM correlation function scaled at the depth of the Lick counts (Maddox et al. 1990). As can be seen the two correlation function are perfectly consistent with each other, even at scales larger than 2:5 , where the power{law shape of w( ) breaks down. This represents our main result, which resolves the long{standing discrepancy between APM and Lick galaxy correlation functions.
We have also tted the power law of eq.(1) to our results in the range 0 2:5 , using a 2 minimization in which each value of w( ) is weighted by 1= , and we obtain: = 1:753 0:004 and A = 0:0746 0:0004 This value of is consistent with the original GP77 one (see eq.2) while the amplitude A is 9% higher than the corresponding GP77 one, most probably due to the fact that eliminating the long-wavelength component of the correlations a ects the amplitude of w( ) even on scales 1:5 2:5 .
CONCLUSIONS
We re{analyzed the Lick galaxy angular correlation function with the aim of explaining the reason for the discrepancy found between previous determinations (GP77, GP86) and more recent estimates based on the APM and COS-MOS automated surveys (Maddox et al. 1990;Collins et al. 1992). We nd that, after including all the corrections in the Lick w( ) estimate, as in GP77 and GP86, except that for large{scale gradients, a perfect consistency exists between Lick and APM results, even at scales larger than 2.5 , at which the w( ) power{law shape breaks. Therefore, under the valid, according to our previous discussion, assumption that the major part of the observed large-scale galaxy-density gradients are intrinsic to the galaxy distribution, the long standing discrepancy between the APM and Lick angular correlation functions, at angular scales > 2:5 is lifted. Inverting this argument we can also conclude that the consistency of the APM and Lick angular correlation functions favours the physical origin of the large{scale gradients observed in the Lick galaxy counts, as advocated by Plionis (1988) and Brown & Groth (1989), and supports the reliability of clustering analyses based on the Lick sample.
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could introduce large-scale gradients, but by a factor of
more than 2
AcknowledgmentsWe thank Steve Maddox for supplying the APM galaxy correlation results.
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| [] |
[
"Non-Gaussian CMB and LSS statistics beyond polyspectra",
"Non-Gaussian CMB and LSS statistics beyond polyspectra"
] | [
"Gonzalo A Palma \nGrupo de Cosmología y Astrofísica Teórica\nDepartamento de Física\nFCFM\nUniversidad de Chile\nBlanco Encalada 2008SantiagoChile\n",
"Bruno Scheihing \nGrupo de Cosmología y Astrofísica Teórica\nDepartamento de Física\nFCFM\nUniversidad de Chile\nBlanco Encalada 2008SantiagoChile\n",
"H ",
"Spyros Sypsas \nGrupo de Cosmología y Astrofísica Teórica\nDepartamento de Física\nFCFM\nUniversidad de Chile\nBlanco Encalada 2008SantiagoChile\n\nDepartment of Physics\nFaculty of Science\nPhayathai Rd\nChulalongkorn University\n10330BangkokThailand\n"
] | [
"Grupo de Cosmología y Astrofísica Teórica\nDepartamento de Física\nFCFM\nUniversidad de Chile\nBlanco Encalada 2008SantiagoChile",
"Grupo de Cosmología y Astrofísica Teórica\nDepartamento de Física\nFCFM\nUniversidad de Chile\nBlanco Encalada 2008SantiagoChile",
"Grupo de Cosmología y Astrofísica Teórica\nDepartamento de Física\nFCFM\nUniversidad de Chile\nBlanco Encalada 2008SantiagoChile",
"Department of Physics\nFaculty of Science\nPhayathai Rd\nChulalongkorn University\n10330BangkokThailand"
] | [] | Cosmic inflation may have led to non-Gaussian initial conditions that cannot be fully parametrised by 3-and/or 4-point functions. In this work, we discuss various strategies to search for primordial non-Gaussianity beyond polyspectra with the help of cosmological data. Our starting point is a generalised local ansatz for the primordial curvature perturbation ζ of the form ζ = ζ G + F NG (ζ G ), where ζ G is a Gaussian random field and F NG is an arbitrary function parametrising non-Gaussianity that, in principle, could be reconstructed from data. Noteworthily, in the case of multi-field inflation, the function F NG can be shown to be determined by the shape of tomographic sections of the landscape potential responsible for driving inflation. We discuss how this generalised local ansatz leads to a probability distribution functional that may be used to extract information about inflation from current and future observations. In particular, we derive various classes of probability distribution functions suitable for the statistical analysis of the cosmic microwave background and largescale structure. | 10.1088/1475-7516/2020/02/027 | [
"https://arxiv.org/pdf/1907.05332v2.pdf"
] | 195,886,378 | 1907.05332 | f30fc5c158755b4b4c58626002d66de7d57fbadd |
Non-Gaussian CMB and LSS statistics beyond polyspectra
Gonzalo A Palma
Grupo de Cosmología y Astrofísica Teórica
Departamento de Física
FCFM
Universidad de Chile
Blanco Encalada 2008SantiagoChile
Bruno Scheihing
Grupo de Cosmología y Astrofísica Teórica
Departamento de Física
FCFM
Universidad de Chile
Blanco Encalada 2008SantiagoChile
H
Spyros Sypsas
Grupo de Cosmología y Astrofísica Teórica
Departamento de Física
FCFM
Universidad de Chile
Blanco Encalada 2008SantiagoChile
Department of Physics
Faculty of Science
Phayathai Rd
Chulalongkorn University
10330BangkokThailand
Non-Gaussian CMB and LSS statistics beyond polyspectra
arXiv:1907.05332v2 [astro-ph.CO] 3 Sep 2019
Cosmic inflation may have led to non-Gaussian initial conditions that cannot be fully parametrised by 3-and/or 4-point functions. In this work, we discuss various strategies to search for primordial non-Gaussianity beyond polyspectra with the help of cosmological data. Our starting point is a generalised local ansatz for the primordial curvature perturbation ζ of the form ζ = ζ G + F NG (ζ G ), where ζ G is a Gaussian random field and F NG is an arbitrary function parametrising non-Gaussianity that, in principle, could be reconstructed from data. Noteworthily, in the case of multi-field inflation, the function F NG can be shown to be determined by the shape of tomographic sections of the landscape potential responsible for driving inflation. We discuss how this generalised local ansatz leads to a probability distribution functional that may be used to extract information about inflation from current and future observations. In particular, we derive various classes of probability distribution functions suitable for the statistical analysis of the cosmic microwave background and largescale structure.
Introduction
Our observable universe is consistent with an extremely simple set of initial conditions. For all practical purposes, the observed cosmological inhomogeneities were seeded by a primordial curvature fluctuation ζ distributed according to a Gaussian profile, parametrised by an almost scale invariant power spectrum [1][2][3][4][5]. The confirmation of this state of affairs by future surveys would reinforce our confidence in the single-field slow-roll inflation paradigm, that is, the idea that ζ was the consequence of quantum perturbations of a single scalar fluid (the inflaton) that evolved adiabatically during inflation [6][7][8][9][10]. Tiny deviations from Gaussianity, due to small nonlinear self-interactions affecting ζ, are known to emerge in single-field inflation but these are predicted to be too small to be observed in the near future. On the other hand, large non-Gaussianity (within current bounds) may emerge in models of inflation beyond the canonical single-field paradigm, resulting from possible nongravitational self-interactions of ζ and/or interactions between ζ and other degrees of freedom. The observation of non-Gaussianity (NG) would therefore offer a unique opportunity to characterise the class of interactions that affected ζ during inflation, allowing us to pin down certain fundamental aspects about the period of inflation and, consequently, have a glimpse on the structure of the ultra-violet (UV) framework where it is realised.
While current cosmic microwave background (CMB) observations show no evidence of primordial non-Gaussianity, future large-scale structure (LSS) surveys, such as Lsst [11], Euclid [12], Spherex [13] and Ska [14], promise to revitalise its search. The proliferation of modes due to the three-dimensional volume probed by galaxy surveys is expected to yield constraints on primordial non-Gaussianity that might not only complement current bounds from CMB observations, but even surpass them. Among the most prominent effects of non-Gaussianity on the matter distribution is the celebrated observation that a nonzero skewness of the probability distribution function of ζ leads to an enhanced abundance of collapsed structures and a scale dependent correction in the halo bias [15], a result which has brought LSS surveys in the front line of the search for non-Gaussianity. Furthermore, there has been an intense effort to study how UV physics can show up in the matter power spectrum and bispectrum [16][17][18][19].
Given that inflation might not be unique 1 in explaining the observed inhomogeneities [24], in order to confront future data with theory, we may adopt an agnostic perspective about the details involved in the description of the pre-Big-Bang dynamics. In that case, we should agree that the main outcome from inflation, or any other framework intending to explain the initial conditions of our universe, consists of a relation giving us back the profile of ζ(x) written in terms of a purely Gaussian random field ζ G (x). Such a relation must be of the form
ζ(x) = ζ G (x) + F NG (ζ G (x), ∇) ,(1.1)
where F NG represents a nonlinear function of the field ζ G (x) and spatial gradients ∇ acting on it. In Fourier space, the previous relation may be reexpressed as the following expansion in powers of ζ G , starting at quadratic order:
ζ k = ζ G k + (2π) 3 n=2 1 n! k 1 ... kn δ (3) k − n i=1 k i F n (k 1 , ..., k n ) ζ G k 1 ...ζ G kn , (1.2)
where k ≡ (2π) −3 d 3 k, and where F n (k 1 , ..., k n ) are functions of the momenta, symmetric under their permutations. This relation is sufficiently general to describe any form of primordial non-Gaussianity, and may be formally obtained in a generic manner from a quantum field theoretical framework (see App. A for details). The F n functions parametrise the non-Gaussian deviations generated by nonlinear interactions to which ζ were subject and, in the case of inflation, may be deduced from a particular model by studying the evolution of ζ k from sub-horizon scales up until the end of inflation (e.g., using the in-in formalism). In fact, any n-point correlation function for ζ may be computed out of (1.2). For instance, at tree-level, the bispectrum parametrising the amplitude of the 3-point function is found to be given by B(k 1 , k 2 , k 3 ) = [P ζ (k 1 )P ζ (k 2 )F 2 (k 1 , k 2 ) + perm], where P ζ is the power spectrum of the Gaussian field ζ G k . In the particular case of single-field slow-roll inflation, up to first order in the slow-roll parameters, the bispectrum is recovered as long as F 2 is given by (up to terms that vanish upon imposing momentum conservation)
F 2 (k 1 , k 2 ) = 1 2 (η − ǫ) + ǫ 2 k 1 + k 2 |k 1 + k 2 | + 2ǫ k 2 1 + k 2 2 |k 1 + k 2 |(k 1 + k 2 + |k 1 + k 2 |) ,
where ǫ and η are the usual slow-roll parameters describing the steady evolution of the FLRW background during inflation. The effective field theory (EFT) of inflation approach [25] to study models beyond the canonical single-field paradigm will also yield a specific form of F 2 (k 1 , k 2 ), in which the sound speed of ζ plays an important role. Moreover, the well-known local ansatz [26][27][28][29][30][31][32][33], related to the presence of multi-field dynamics, corresponds to another particular instance of this relation, where F 2 = 6 5 f local NL . One could thus take upon the challenge of directly reconstructing the form of F NG in Eq. (1.1) -or equivalently, the functions F n appearing in (1.2)-out of cosmological data. This would constitute a bottom-up approach to determine the properties of the model that gave origin to the initial conditions. To guide such a reconstruction, one could consider restricting the functions F n according to certain rules dictated by the symmetries of the alleged bulk model that led to (1.2) at the end of the pre-Big Bang period. For instance, scale invariance of the spectra is equivalent to the invariance of F n under the simultaneous rescaling of all the momenta: F n (λk 1 , ..., λk n ) = F n (k 1 , ..., k n ). Furthermore, the validity of soft theorems [29,[34][35][36][37][38][39][40][41][42][43] (under certain circumstances) would require some relations among F n of different order in the limit where one or more of the momenta vanishes.
An objection to this program (the direct reconstruction of F NG ) is that perturbation theory applied to the study of the evolution of ζ implies that the expansion involved in the writing of Eq. (1.2) is hierarchical. That is, given a small dimensionless coupling constant g parametrising the self-interactions experienced by ζ during inflation, the F n functions are naturally expected to satisfy F n ∝ g n−1 .
(1. 3) For instance, in the case of single-field slow-roll inflation g happens to be of order ǫ and η and hence, non-Gaussianity is expected to be well-parametrised by the bispectrum. In noncanonical single-field models described by the EFT of inflation, where the ζ fluctuations propagate with a reduced sound speed c s < 1, the coupling g is enhanced by a factor c −2 s , but in order to trust perturbation theory, one still requires that it stay sufficiently suppressed. Based on this argument, we could say that most efforts to characterise non-Gaussianity so far have focused on a truncated version of (1.2), where only F 2 (k 1 , k 2 ) and F 3 (k 1 , k 2 , k 3 ) (which at tree-level give the bispectrum and trisprectrum) are taken under consideration, with the Planck data implying weak constraints on the form of F 2 (k 1 , k 2 ), with the help of the so-called local, equilateral, orthogonal and folded templates [4].
In the present article, we wish to argue in favour of reconstructing the full function F NG of Eq. (1.1) from CMB and LSS observations without necessarily assuming a hierarchical dependence of the functions F n on a given coupling constant g. We posit that the search for non-Gaussianity focused on low n-point correlation functions may miss the existence of richer types of statistics [44][45][46][47][48] that we may be unable to predict by following standard perturbation theory techniques. For example, in App. A we show that in certain classes of multi-field models, the function F NG is found to be proportional to the gradient of the landscape potential ∆V (ψ) controlling the dynamics of the isocurvature field ψ. More to the point, we show that
F NG (ζ) ∝ ∂ ∂ζ ∆V (ψ(ζ)) , with ψ(ζ) ≡ H 2πA 1/2 s ζ, (1.4)
where H is the Hubble expansion rate during inflation and A s is the amplitude of the power spectrum of ζ (constrained by Planck to be 10 9 A s = 2.105 ± 0.030 at the CMB pivot scale k = 0.05 Mpc −1 [2]). Hence, the shape of F NG provides a tomographic view of the landscape potential away from the inflationary trajectory. Certainly, one cannot discard the possibility of having potentials ∆V (ψ) characterised by a rich structure (i.e. with consecutive minima separated by field distances smaller than the Hubble expansion rate during inflation) in which case the bispectrum would not constitute an efficient tool to parametrise non-Gaussianity [46,47]. A simple example of this, related to the presence of an axionic isocurvature direction, is
F NG (ζ G ) ∝ sin(ζ G /f ζ ) with f 2 ζ < σ 2 ζ ,
where σ 2 ζ is the variance of the Gaussian field ζ G . In this case, the function F NG (ζ G ) remains bounded, but varies vigorously within the relevant range determined by σ ζ .
Despite of having a well motivated construction justifying the need of studying non-Gaussianity beyond polyspectra, we wish to keep the discussion as general as possible and, thus, in the main part of this work we will not refer to any specific origin of this parametrisation. Instead, we will focus our attention on the consequences -for various cosmological observables-implied by having initial conditions parametrised by a function F NG (ζ) not necessarily respecting a hierarchical structure. In order to make the discussion more tractable, we will focus on the specific case of local NG, for which F NG in Eq. (1.1) does not involve gradients and hence the coefficients F n in the expansion (1.2) are independent of momenta 2 . This implies a simplified version of (1.1) given by ζ(x) = ζ G (x) + F NG (ζ G (x)). However, since in practice observations are restricted to a finite range of wavelengths, we will consider a version of (1.1) recovered from (1.2) but with every momenta restricted to a finite range (including those appearing in the integrals). This relation is given by
ζ(x) = ζ G (x) + F NG [ζ G ] (x), with F NG [ζ G ] (x) = k y e ik·(x−y) F (ζ G (y)) ,(1.5)
where y ≡ d 3 y. In the previous expression, k ≡ (2π) −3 d 3 k is restricted to a given range of scales specified by convenience 3 (with ζ(x) and ζ G (x) also restricted in that manner). In this version of the ansatz, F (ζ G ) is the function of the Gaussian random field ζ G that determines the values of the F n coefficients in (1.2) as F n = ∂ n ζ F | ζ=0 .
Main results
The object that determines the statistics of fluctuations, either in the temperature or in the density field, is the probability distribution function (PDF) from which one can compute moments, cumulative functions and several other objects such as mass functions, bias factors, etc. Since both the temperature and density fluctuations are sourced by curvature perturbations, the parametrisation (1.5) of ζ in terms of Gaussian random fields allows us to construct various classes of such functions characterising the statistics 4 of CMB and LSS, hence, enabling us to use their distributions as a probe of the primordial universe. In this work we derive several classes of PDFs for the curvature perturbation ζ, the CMB temperature fluctuation Θ ≡ ∆T /T and the matter overdensity δ ≡ δρ/ρ, all following from the non-Gaussian ansatz (1.5) by using appropriate transfer functions. Specifically, we achieve the following goals:
⋆ We derive the full probability distribution functional implied by the generalised ansatz (1.5). The result is shown in Eq. (2.10) and it explicitly depends on the function F (which at the same time, is related to the landscape potential in the case where the primordial non-Gaussianity is produced during inflation). can in principle be observationally distinguished from an ansatz where the function F2 satisfies the consistency relation. The methods used in [49] should be valid to analyse our proposed generalised version of the local ansatz. 3 For now, we do not need to specify the range of momenta and the reader may take the range as the entire momentum space, in which case FNG = F . In Sec. 2.2, we will consider the problem of reducing the range of momenta (in a Wilsonian manner) and explore how the function F appearing in Eq. (1.5) runs with the IR and UV cutoffs employed to restrict the momenta. 4 There are several nonprimordial sources of non-Gaussianity, especially in late-time fields. Here, we always refer to its primordial component.
⋆ We derive the NG halo mass function expressed in terms of the function F . This result is shown in Eqs. (4.8)-(4.10). We further comment on how statistical estimators probing the PDF can be used in LSS datasets in connection to primordial NG via Eq. (4.13).
⋆ We derive a generalised version of the linear halo bias resulting from the non-Gaussian deformation F NG . The result is written in Eq. (4.27), where we show that the scale dependent bias is sensitive to the full function F and not in any specific truncated version of it. In other words, all nonlinearity parameters appear in a degenerate manner.
To summarise, we wish to point out that since non-Gaussianity has not shown up in low n-point functions and given that its presence has solid foundations in our understanding of UV physics and quantum field theory/gravity, extending our study towards other estimators is part of the next step. The PDF is the object that fully encodes the statistical information of fluctuations enclosing all possible cases and as such it may serve as a central tool in the search for non-Gaussianity.
Structure of the paper
The article is organised as follows: In Sec. 2, we write down the probability functional describing the distribution of the curvature fluctuation ζ, its Fourier dual -the partition function-as well as the 1-point and 2-point PDFs. We discuss several aspects regarding the use of window functions either related to renormalisation or survey sky coverage. In Sec. 3, we derive the temperature distribution and focus on the 2-point PDF commenting on its use with CMB data. In Sec. 4, we derive the halo mass function and the halo bias for arbitrary local NG. We conclude in Sec. 5, while in the Appendices we present various details regarding the NG ansatz and the functionals emanating thereof presented in the main text.
ζ Statistics
In this section we derive the probability distribution functional (and other classes of distributions) for the primordial curvature perturbation ζ. Our starting point is the generalised local ansatz shown in Eq. (1.5). The resulting PDF's will set the stage for later sections, where we explore the consequences their non-Gaussian deviations may have on both CMB and LSS datasets.
2.1 Probability distribution functional: "bare" theory
The first task we can perform, and the most readily available to us thus far, is to attempt to use Eq. (1.5) to its full potential and derive directly the complete functional distribution that governs the ζ statistics: given that we know how a Gaussian random field is distributed, we may simply perform a change of variables (as in Ref. [50]) to obtain the probability distribution functional P[ζ] for ζ(x) such that
ζ(x 1 ) · · · ζ(x n ) = Dζ P[ζ] ζ(x 1 ) · · · ζ(x n ), (2.1)
where Dζ is the functional measure of the PDF. The resulting distribution must account, to leading order in the perturbation F NG , for every conceivable correlation function that may be constructed from the field ζ(x) and for every expectation value of function(al)s of ζ(x). To start with, the Gaussian random field ζ G is drawn from the following functional distribution:
P G [ζ G ] = N exp − 1 2 x y ζ G (x)Σ −1 (x, y)ζ G (y) = N exp − 1 2 k ζ G (k)ζ G (−k) P ζ (k) , (2.2)
where N is an overall normalisation constant, while Σ −1 (x, y) and P ζ (k) are the covariance matrix and the power spectrum respectively, related as
Σ −1 (x, y) ≡ k e ik·(x−y) P ζ (k) . (2.3) In implementing the transformation ζ(x) = ζ G (x)+F NG [ζ G ](x)
, there are two contributions that emerge: one is composed by the terms that come out of the Taylor expansion of the Gaussian distribution by regarding the perturbation F NG as small, and the other arises from the functional determinant of the transformation. The latter is given by
det δζ(y) δζ G (x) = exp tr ln δζ(y) δζ G (x) = exp tr k e ik·(x−y) ln 1 + dF dζ (ζ G (x)) = exp x k ln 1 + dF dζ (ζ G (x)) . (2.4)
As long as the determinant exists (concretely, dF/dζ > −1) and is nonzero, we may, in principle, find an inverse to the relation ζ = ζ G + F NG [ζ G ] and denote it with ζ G (x) = G[ζ](x). Then we may change variables from the PDF associated to ζ G , to find
P[ζ] = N exp − 1 2 x y G[ζ](x)Σ −1 (x, y)G[ζ](y) − x k ln 1 + dF dζ (ζ G (x)) . (2.5)
This is an exact result. However, if F is small in comparison to the typical scales of the background theory on which the fluctuation field ζ lies, and so is dF/dζ, we can approximate the logarithm in the exponential with the first term in its power series expansion, and furthermore, we may approximate the inverse mapping by
G[ζ] ≈ ζ − F NG [ζ]
. This yields a non-Gaussian exponential correction to the PDF, the exponent of which, to first order in F NG , reads
P[ζ] = P G [ζ] × exp x y ζ(x)Σ −1 (x, y)F NG (ζ(y)) − x δ δζ(x) k y e ik·(x−y) F NG (ζ(y)) .
(2.6) Using the definition of the covariance (2.3), we may finally write the result as
P[ζ] = P G [ζ] × exp x k y e ik·(x−y) ζ(x) P ζ (k) − δ δζ(x) F NG [ζ](y) . (2.7)
This functional will serve as the guiding principle for all subsequent results. Note that even in a first order approximation, the probability distribution is always positive. However, to make computations tractable, we find it convenient to also perform a power series expansion of the non-Gaussian exponential factor. Then, to first order in F NG , we find that
P F [ζ] = P G [ζ] × 1 − x y k e ik·(x−y) δ δζ(x) − ζ(x) P ζ (k) F NG [ζ](y) = P G [ζ] × 1 − y k e −ik·y δ δζ(−k) − ζ(k) P ζ (k) F NG [ζ](y) . (2.8)
The functional distribution (2.8) has support at all the scales where the underlying theory does, or at least, at the scales where the corresponding EFT is presumed to hold true. However, observable quantities do not typically involve all of the scales, and therefore it may be that the "bare" departure from Gaussianity F NG is not the most adequate quantity to describe them. We thus now turn to the discussion of using window functions and how to integrate scales out.
Probability distribution functional: running and renormalising
When making predictions, any EFT will force us to recognise certain scales at which the theory is no longer well-suited to describe physical observables. This typically implies a high-energy scale, where the theory has to be cut off. Thus, we will set k UV as the maximum possible wavenumber the mode expansion of the curvature perturbation can have. Similarly, while it is not always introduced, one can make the same assertion with the very long wavelengths.
As much as our theory may have predictions concerning phenomena happening beyond the present Hubble radius, they are currently unobservable. Therefore, in establishing predictions for observable quantities it seems natural to integrate out those scales, so that they are properly incorporated into the final, effective result. Because of this, we will take a conservative attitude and also define an infrared cutoff k IR , which can be thought of as the inverse of the current Hubble radius, thus bounding the domain of the theories we will be studying to k ∈ (k IR , k UV ).
Moreover, in realistic situations the experiment at hand may not allow us to access every value for the momentum scale k evenly. In those cases we may wish to introduce a window function W (k) to filter our results and give more weight to some scales. Accordingly, one would be interested in the statistics of the filtered field
ζ W (x) ≡ [W ⋆ ζ](x) = k y e ik·(x−y) W (k)ζ(y). (2.9)
To derive the probability distribution functional of the field ζ W (x) it is enough to perform the change of variables ζ → ζ W (k) = W (k)ζ(k) in (2.8). One obtains
P W [ζ W ] =N W exp − 1 2 k ζ W (k)ζ W (−k) P W (k) × 1 − y k e −ik·y W (k) δ δζ W (−k) − ζ W (k) P W (k) F NG q e −iq·y ζ W (q) W (q) (y) ,(2.10)
where P W (k) ≡ W 2 (k)P ζ (k). Leaving aside the argument of the deviation from Gaussianity F NG for a moment, this PDF has the same structure as the unfiltered PDF of Eq. (2.8).
This expression for P W poses an interesting question: what if the window function of choice is defined (as usual) with hard cutoffs, just as if we were redefining the limits of our EFT? That is to say, how does P W look if we have
W (k) = 1 if k ∈ (k L , k S ) 0 if k ∈ (k L , k S ), (2.11)
as the window function? It turns out that, for the functional integral to be well-defined, we have to integrate out of the theory all the modes which will not take part in our observable quantities. Upon doing so, we can avoid dealing with ζ W (q)/W (q), which from the perspective of the theory with the window function would be ill-defined for the scales where W = 0, but is perfectly finite (and equal to ζ(q)) from the perspective of the original theory. To this end, let us take the original functional distribution (2.8) (with k-space variables) and integrate out the modes outside the support of the window function W . We may write this as
P W [ζ] = Dζ |k| / ∈(k L ,k S ) P[ζ]. (2.12)
Upon integration over the prescribed range of modes, the purely Gaussian term in (2.8) gives a reduced Gaussian measure that considers only k ∈ (k L , k S ). To integrate the term containing F NG it is convenient to separate the integral over momentum space in (2.8) into two contributions:
k e −ik·y δ δζ(−k) − ζ(k) P ζ (k) F NG [ζ](y) = |k|∈(k L ,k S ) e −ik·y δ δζ(−k) − ζ(k) P ζ (k) F NG [ζ](y) + |k| ∈(k L ,k S ) e −ik·y δ δζ(−k) − ζ(k) P ζ (k) F NG [ζ](y).
(2.13)
In this form, it is easy to see that upon integrating over ζ(k), the term in the second line of (2.13) vanishes after performing a functional partial integration. Thus, we will only need to know how to deal with the quantity 5
δ δζ(−k) − ζ(k) P ζ (k) Dζ |k| / ∈(k L ,k S ) exp − 1 2 k ζ(k)ζ(−k) P ζ (k) y |k|∈(k L ,k S ) e −ik·y F NG [ζ](y),
(2.14) so as to see if, and how, the interaction is renormalised. It turns out that if we definē involves scales |k| ∈ (kL, kS), while the functional integration goes over modes with |k| / ∈ (kL, kS). We may thus pull it out of the integral.
F (ζ W (x)) ≡ Dζ k / ∈(k L ,k S ) exp − 1 2 |k| / ∈(k L ,k S ) ζ(k)ζ(−k) P ζ (k) F (ζ(x)) = ∞ −∞ dζ e − (ζ W (x)−ζ) 2 2σ 2 out
then we may identifyF
NG [ζ](x) = y |k|∈(k L ,k S )
e ik·(x−y)F (ζ(y)), (2.16) as the effective self-interaction, because we would have integrated out all the scales that are outside the range of interest and still leave the other scales within the measure P of the PDF, while maintaining its analytic structure.
The last equality of (2.15), the Weierstrass transform of F (ζ), can be obtained in numerous manners. If one were to follow standard diagrammatic perturbation theory, it arises from summing back every "loop" contraction performed by the Gaussian measure of F NG with itself. Since the momenta flowing through those loops are bounded by the range being integrated out, and there is no "external" momenta flowing through the diagrams, we have that their numerical value is the same for every loop and equal to the variance
σ 2 out = |k| ∈(k L ,k S ) P ζ (k),(2.17)
that was subtracted from the Gaussian field statistics when the modes |k| ∈ (k L , k S ) were integrated out of the theory. Therefore, by using these results in Eq. (2.12), we obtain
P F NG ,W [ζ] = PF NG [ζ |k|∈(k L ,k S ) ],(2.18)
i.e., that P W [ζ] for the restricted variable (2.9) has the same functional form as the original PDF, with the only modification that now the departure from Gaussianity is given by a "filtered" interactionF NG instead of the "bare" interaction F NG .
In practice, there is more than one way of how to represent the running of F depending on the scales one wants to include in the theory. Perhaps the most ethereal representation, but at the same time the most revealing of the theory's structure, is through the differential expression of the Weierstrass transform, as inF
(ζ) = exp σ 2 out 2 ∂ 2 ∂ζ 2 F (ζ). (2.19)
This expression makes clear how the theory runs by removing more or less scales, as well as the fact that the transformation rule between F at different scales follows an adequate composition property: integrating out ranges of momenta A and B is implemented via σ 2 out,A and σ 2 out,B , and doing so yields the same result independently of the order in which one subtracts the modes from the theory. Furthermore, this shows that the functional form of F NG andF NG is the same, in the sense that the quantities that determine its concrete expression are exactly the same; the only thing that the window does is to restrict the range of modes entering in the observables.
Conversely, just as the PDF may be recast in an analogous manner to that of the original theory, the field with modes between k L and k S may also be written down as a local departure from Gaussianity
ζ W (x) = ζ G W (x) +F NG ζ G W (x),(2.20)
merely because it follows statistics analogous to ζ. Here we have to remind the reader that this is only so for W (k) of the form (2.11). Other window functions still give rise to an explicit PDF, namely equation (2.10), but the deviations from Gaussianity may no longer be written as concisely as in (2.20). The difference lies in that a general window function does not render irrelevant some degrees of freedom of the theory; it only gives them dissimilar weights in the final result. However, in order to obtain the functionF NG it is crucial that we reduce the number of independent variables in our theory by integrating out their effects, as all of them will leave their signature, if small, in any given correlation function. Having made this point explicitly, from now on, and for the rest of this paper, we will go back to considering general window functions.
Thus far, we have established how we may write the probability distribution functional of our theory depending on the scales under consideration and also how we may incorporate window functions into the distribution functional. Before passing to simpler statistical estimators stemming from this functional, we now discuss the partition function.
Partition function and n-point correlators
In practice, using the full probability distribution functional directly on cosmological data proves to be difficult, as there is only one realisation of our universe to probe and conduct measurements in, so it is not possible to take a frequentist approach to its statistics. While this suggests the use of Bayesian analysis to find the most probable F NG given the data by the means of the functional (2.8), it also reveals why one typically chooses to work with correlation functions to probe departures from Gaussianity: they can be computed from many Fourier modes on the sky, whose past history is presumably independent (at least if the nonlinearities are turned off), and therefore averages may be performed and compared with the theoretical predictions for the expectation values or correlations.
Fittingly, there is an object that encapsulates the information of all the correlation functions in a perhaps clearer way than the full probability density functional P. This is the partition function Z[J], which is the object that generates the n-point functions via functional differentiation:
ζ(k 1 ) · · · ζ(k n ) = δ n Z[J] (iδJ(−k 1 )) · · · (iδJ(−k n )) J=0
, (2.21) or equivalently, the functional Fourier transform of the PDF, which in the context of probability theory is called the characteristic function,
Z[J] = Dζ P[ζ] e i k ζ(k)J(−k) . (2.22)
Both expressions may be employed to obtain Z[J]: the first requires to know all the n-point functions beforehand and reconstruct the object that has them as its functional derivatives, while the second requires to know an explicit expression for the probability distribution functional. Since we have the latter, we may carry out this computation explicitly 6 to first order in F NG , obtaining
Z[J] = exp − 1 2 k J(k)J(−k)P ζ (k) × 1 − x k e ik·x J(−k) k e ik·x J(−k)P ζ (k) ζ exp − (ζ−i k e ik·x J(−k)P ζ (k)) 2 2σ 2 ζ √ 2πσ ζ σ 2 ζ ∂ ∂ζ −ζ F (ζ) .
(2.23)
Here we have defined σ 2 ζ ≡ k P ζ (k) as the 1-point variance associated to the power spectrum for the relevant range of momenta. Window functions are easily incorporated by substituting J(k) with J(k)W (k), as this procedure will add a factor of W (k) to every external leg in any given diagram. Now that we have equation (2.23), we may compute the n-point functions directly, without having to resort to functional integration as we would with (2.8). Moreover, the structure that will emerge in these correlations is more closely related to (2.23), as is demonstrated by their explicit expressions in position space
ζ W (x 1 ) · · · ζ W (x n ) c = f n−1 n i=1 x k W (k)e ik·(x i −x) k e ik·(x i −x) W (k)P ζ (k) n i=1 k W (k)P ζ (k)e ik·(x i −x) , (2.24)
where the subscript c indicates the result only considers the fully connected piece. For completeness, we write down their counterparts in momentum space
ζ W (k 1 ) · · · ζ W (k n ) c = f n−1 (2π) 3 δ (3) n i=1 k i n j=1 W (k j )P ζ (k j ) n i=1 1 P ζ (k i ) , (2.25)
where the coefficients 7 f n are given by (probabilists') Hermite moments of F :
f n ≡ − 1 σ n ζ ∞ −∞ dζ e − ζ 2 2σ 2 ζ √ 2πσ ζ He n ζ σ ζ F (ζ). (2.26)
The coefficients f n are quantities of mass dimension 1 − n, which are invariant under the renormalisation procedure, discussed in Sec. 2.2, in a very fitting sense: because {f n } ∞ n=2 are the coefficients of a Hermite polynomial expansion 8 , we have
F (ζ; σ 2 ζ ) = − ∞ n=2 f n n! σ n ζ He n ζ σ ζ = − ∞ n=2 f n n! exp − σ 2 ζ 2 ∂ 2 ∂ζ 2 ζ n ,(2.27)
where we have introduced the variance σ 2 ζ as an argument of F in order to emphasise that the associated field has the corresponding amplitude for its fluctuations. This means thatF in Eq. (2.15), or equivalently Eq. (2.19), takes the following form:
F (ζ) = exp σ 2 out 2 ∂ 2 ∂ζ 2 F (ζ; σ 2 ζ ) = F (ζ; σ 2 ζ − σ 2 out ), (2.28)
where the coefficients {f n } n remain unchanged; only the variance gets reduced to its new value after integrating some modes out.
Nonetheless, it is important to point out that each individual n-point function does not encapsulate all of the non-Gaussian information contained in F . Indeed, each correlation function only yields one term of an infinite series expansion of F , all of which are independent, at least in principle. Therefore, it is natural to try and find objects that are able to keep all of this information, without having to compute an infinite number of quantities. For that reason, we now turn to exploring 1-and 2-point probability density functions.
Fixed-point probability distribution functions
In the presence of a generic deviation from Gaussian statistics, involving both local and nonlocal terms, to assume that it is possible to capture all non-Gaussian information by looking at the single-point statistics of a field (i.e., correlations with all of the spatial coordinates at the same position) seems misguided, as the restriction to a single point is likely to mix local and nonlocal effects, making it difficult to disentangle them. However, if we restrict ourselves to local deviations from Gaussianity only, it is indeed possible to capture all such information.
In this subsection we write down explicit 1-point and 2-point PDFs for the curvature perturbation. In order to alleviate the discussion, we shall leave some of the details for App. C.
1-Point probability distribution function
Now we set ourselves to derive the simplest distribution that can be obtained within this framework: a density function for the 1-point statistics. It is defined as the distribution P(ζ; x) that satisfies
ζ n (x) = ∞ −∞ dζP(ζ; x)ζ n . (2.29)
Given that we assume a homogeneous universe, P(ζ; x) cannot depend on x. Thus, we write P(ζ; x) = P(ζ). From a functional perspective, it is given by Thus, in the same spirit as the probability distribution functional we obtained earlier represents a first-order correction to Gaussian statistics, the 1-point probability density function may also be written as a slight departure from Gaussianity. Moreover, this density function resembles more closely the structure of Z[J] than that of P[ζ] because marginalising over all the other positions in presence of a finite range of wavelengths induces a filtering, which is manifest in Z[J] but not so in P[ζ]. Given the various applications it may find in LSS or in primordial black hole formation [48], it is of interest to write it for an arbitrary window function.
P(ζ) = Dζ P[ζ] δ(ζ(x) −ζ),(2.
Smoothing the field and its variance as in Eq. (2.9), the resulting expression is
P(ζ W ) = 1 √ 2πσ W e − ζ 2 W 2σ 2 W [1 + ∆ W (ζ W )] , (2.31) where ∆ W (ζ W ) ≡ ∞ 0 dx 4πx 2 W (x) s 2 (x) ∞ −∞ dζ exp − (ζ−b(x)ζW ) 2 2σ 2 W (x) √ 2πσ W (x) ζ − σ 2 ζ ∂ ∂ζ F ζ . (2.32)
Here we have written W (x) = k e ik·x W (k), the position-space representation of the window function W , with x = |x|, and we have also defined the (co)variances
σ 2 W ≡ k W 2 (k)P ζ (k) , s 2 (x) ≡ k e ik·x W (k)P ζ (k) , σ 2 W (x) ≡ σ 2 ζ − b 2 (x)σ 2 W , (2.33)
with a "bias" factor 9 given by
b(x) ≡ s 2 (x) σ 2 W . (2.34)
In words, this expression means that, given a range of modes defining the cutoffs of the theory, the perturbative correction to the PDF scans for the structure of the effective interaction at those scales through the action of (ζ − σ 2 ζ ∂ζ), and then filters it, through the Weierstrass transform, according to the difference between the variance and the correlation implied by incorporating window functions. This observable can account for all the information contained within the local function F . This can be seen from the fact that all of the information concerning F is stored within the f n coefficients, which can be retrieved by looking only at the 1-point statistics (c.f. Eq. (2.24)). Indeed, an analysis aiming to constrain the departure from Gaussianity F NG by the means of the 1-point CMB temperature distribution has already been performed in [47] .
Before passing to the 2-point functional, and in order to make contact with current literature, let us comment on the relation of this PDF to the Edgeworth representation. If we use the Hermite polynomial expansion of F , given in Eq. (2.27), the effect of the Gaussian filtering of Eq. (2.32) becomes transparent:
∞ −∞ dζ exp − (ζ−b(x)ζW ) 2 2σ 2 W (x) √ 2πσ W (x) F (ζ; σ ζ ) = F bζ W ; b 2 σ 2 W = ∞ n=2 f n n! He n ζ W σ W s 2n (x) σ n W , (2.35)
that is, it replaces the field variable with the biased one. Using the fact that the Weierstrass transform commutes with derivatives, the latter now being evaluated at the biased field with its corresponding variance, and that (x − σ 2 ∂ x )He n (x/σ) = σHe n+1 (x/σ), we can rewrite the NG deviation of the 1-point PDF as
∆ W (ζ W ) = ∞ n=2 f n n! He n+1 ζ W σ W 1 σ n+1 W ∞ 0 dx 4πx 2 W (x)s 2n (x) = ∞ n=2 He n+1 ζ W σ W (n + 1)! ζ n+1 W c σ n+1 W , (2.36)
where ζ n W c is given by Eq. (2.25) integrated over momenta. This is exactly the Edgeworth expansion of a non-Gaussian PDF truncated to first order in the couplings f , since we have restricted our derivation of the PDF (2.32) to first order 10 in F .
Moreover, in this case it is possible to invert the Edgeworth expansion (2.36) in terms of the function F . Taking the mth Hermite moment of this expansion gives
ζ m W c σ m W = ∞ −∞ dζ W e − ζ 2 W 2σ 2 W √ 2πσ W He m ζ W σ W ∆ W (ζ W ),(2.37)
and by replacing the cumulant ζ m W c in terms of f m we get
f n = σ n+1 W /(n + 1) ∞ 0 dx 4πx 2 W (x)s 2n (x) ∞ −∞ dζ W e − ζ 2 W 2σ 2 W √ 2πσ W He n+1 ζ W σ W ∆ W (ζ W ),(2.
2-Point probability distribution function
Now we would like to write down an observable able to account for the non-Gaussian statistics, without integrating out the information about correlations in the sky at different scales. Therefore it cannot be a single-point PDF. Thus, we try to do the next least complicated thing: a 2-point PDF P(ζ 1 , ζ 2 ; x 1 , x 2 ). This function satisfies
ζ n W (x 1 )ζ m W (x 2 ) = dζ 1 dζ 2 P W (ζ 1 , ζ 2 ; |x 1 − x 2 |)ζ n 1 ζ m 2 , (2.39)
where we have written P(ζ 1 , ζ 2 ; x 1 , x 2 ) = P(ζ 1 , ζ 2 ; |x 1 − x 2 |) because we are assuming our universe to be statistically homogeneous. One way to obtain such an object is by conditioning in two points in a manner analogous to Eq. (2.30),
P(ζ 1 , ζ 2 ; |x 1 − x 2 |) = Dζ P[ζ] δ(ζ(x 1 ) − ζ 1 )δ(ζ(x 2 ) − ζ 2 ), (2.40)
where again, the final result can only depend on the spatial coordinates through the distance between the two positions x 1 and x 2 . With this in mind, let us define the scalar variables
r ≡ |x 1 − x 2 |, r 1 ≡ |x − x 1 |, r 2 ≡ |x − x 2 |. (2.41)
Then, by inspecting the n-point functions of the theory (2.24) evaluated at the two points of interest, we obtain a similar expression as the 1-point PDF, but with two points defining the filtering instead of one:
P W (ζ 1 , ζ 2 , r) = P G,W (ζ 1 , ζ 2 , r) 1 − x ∞ −∞ dζ exp − (ζ−ζW (r,r 1 ,r 2 )) 2 2σ 2 W (r,r 1 ,r 2 ) √ 2πσ W (r, r 1 , r 2 ) × W (r 1 ) s 2 (r 1 ) G 11 ∂ ∂ζ − G 12 + W (r 2 ) s 2 (r 2 ) G 21 ∂ ∂ζ − G 22 F ζ ,(2.42)
where P G,W (ζ 1 , ζ 2 , r) is the bivariate Gaussian measure, with a covariance matrix given by the 2 × 2 bottom right block of Σ Σ Σ, defined below in Eq. (2.43). Let us go through this expression: the first thing to notice is the presence of two points, x 1 and x 2 , defining a filtering through the same function as in the 1-point case. The second important aspect is that now the Gaussian that is convoluted with F has a different mean and variance. However, they emerge in the same manner as σ 2 W (x) and ζ W (x) emerge in the 1-point case: σ 2 W (r, r 1 , r 2 ) and ζ W (r, r 1 , r 2 ) are the variance and mean ofζ after conditioning on the values of (ζ 1 , ζ 2 ), starting from a joint Gaussian distribution for (ζ, ζ 1 , ζ 2 ) with covariance matrix
Σ Σ Σ = σ 2 ζ s 2 (r 1 ) s 2 (r 2 ) s 2 (r 1 ) σ 2 W σ 2 W,ext (r) s 2 (r 2 ) σ 2 W,ext (r) σ 2 W , (2.43)
where we have written the covariance between the two externally chosen points x 1 and x 2 as
σ 2 W,ext (r) = k e ik·(x 1 −x 2 ) W 2 (k)P ζ (k). (2.44)
The functions G ij also appear in a similar way: G i1 and G i2 are "rotated" versions of σ 2 ζ and ζ, involving combinations of the free theory covariances that make the overall expression reduce to that of the 1-point PDF as x 1 → x 2 . Their precise definitions are listed in App. C. In there, we delineate how to obtain the 2-point PDF in an alternative way: we start from correlators of the type ζ n W (x 1 )ζ m W (x 2 ) and we deduce the function from which they emanate, corresponding to (2.39).
This PDF contains all the information of the free theory, as having two points allows to scan over all the range of distances in the sky, thus probing, among others, the 2-point correlation function completely, which is the defining object of a Gaussian theory. Even though Eq. (2.42) has its non-Gaussian features encoded in a perhaps more complicated fashion than its 1-point counterpart (2.31), both contain the same information about the underlying function F NG . Indeed, one can obtain Eq. (2.31) by integrating over one of the field variables in Eq. (2.42). However, observationally, it might be more efficient to have information on the scale, since then we can, for example, disentangle different momentum shapes of correlation functions.
Θ Statistics
It is also of interest to write down testable quantities that we can obtain by looking at the primordial information that can be stored in spherical shells on the sky, such as the CMB. For instance, the probability distribution functional P[ζ] and its associated partition function Z[J] may be projected onto the celestial sphere to yield distributions of, say, the temperature fluctuations δT (n).
Consider a generic linear transfer function T (k,n) from the primordial perturbations ζ to an observable defined on the sphere Θ(n) such that Θ(n) = k T (n, k)ζ(k).
(3.1)
Of course, we are interested in the specific case where Θ(n) = (T (n) − T 0 )/T 0 (where T (n) is the CMB temperature measured in a given directionn, and T 0 its average over all directions), however, the present discussion is also valid for more general observables that depend on the line of sightn. Then, if we take Σ(n,n ′ ) to be the covariance matrix defining the observable's correlations between different directions in the sky (n,n ′ ), i.e., Θ(n)Θ(n ′ ) = Σ(n,n ′ ), (3.2) and Σ −1 (n,n ′ ) as its inverse matrix, we find that the probability distribution functional for Θ is given by
P[Θ] = N Θ e − 1 2 n n ′ Θ(n)Σ −1 (n,n ′ )Θ(n ′ ) 1 − x Ker 1 (x) ∂F Θ ∂ζ (ζ Θ (x); x) + x Ker 2 (Θ; x)F Θ (ζ Θ (x); x) , (3.3)
where now Ker 1 (x), Ker 2 (Θ; x), ζ Θ (x), and F Θ (ζ; x) depend implicitly on the transfer function and, when noted, the variable of interest Θ(n). All of these quantities may be intuitively understood as the result of projecting a field defined on three spatial dimensions over a twodimensional spherical surface. For instance, ζ Θ (x) is given by
ζ Θ (x) ≡ k n n ′ T (n, k)e ik·x P ζ (k)Σ −1 (n,n ′ )Θ(n ′ ),(3.4)
which basically amounts to saying: take the original statistics of your theory, i.e., e ik·x P ζ (k), project them onto the sphere by applying T (n, k) and then correlate it with the field of interest Θ(n ′ ) by means of its inverse covariance matrix Σ −1 (n,n ′ ). The integration kernels have similar definitions:
Ker 1 (x) ≡ k k ′ n n ′ T (n, k)e ik·x P ζ (k)Σ −1 (n,n ′ )T (n ′ , k ′ )e ik ′ ·x ,(3.5)Ker 2 (Θ; x) ≡ k n n ′ T (n, k)e ik·x Σ −1 (n,n ′ )Θ(n ′ ). (3.6)
However, the function F Θ is a slightly different object than before. As a result of the projection, it acquires a spatial dependence, whose exact nature in terms of the primordial departure from Gaussianity F is given by the Weierstrass transform:
F Θ (ζ; x) ≡ exp σ 2 ζ − σ 2 ζΘ (x) 2 ∂ 2 ∂ζ 2 F (ζ) = ζ exp − (ζ−ζ) 2 2(σ 2 ζ −σ 2 ζΘ (x)) 2π(σ 2 ζ − σ 2 ζΘ (x)) F (ζ),(3.7)
where σ 2 ζΘ (x) may be understood as a position-dependent effective variance of ζ, modified by projection effects:
σ 2 ζΘ (x) ≡ k k ′ n n ′ T (n, k)e ik·x P ζ (k)Σ −1 (n,n ′ )T (n ′ , k ′ )e ik ′ ·x P ζ (k ′ ). (3.8)
In all of the above, we have written Σ(n,n ′ ) as a general function of the direction on the sphere. However, if we take into account that our universe is homogeneous, it must be possible to write it as a function of the angle between the two vectors, or equivalently, in terms of their scalar productn ·n ′ . We will not overemphasize this in what follows, as the notation we deem natural to treatn andn ′ is with them as separate directions, because they will have to be multiplied with another vector, the integration variable x that will appear in the NG kernel that modifies the PDF, which makes usingn ·n ′ notationally heavier than just using (n,n ′ ).
Now we have to find ways of using this. One option would be to compare how likely is our present-day CMB given a certain primordial deviation from Gaussianity of the local type F by using Bayesian statistics. However, in order for this to be useful at its maximum capacity, it is likely that one would first have to establish a definitive imprint of primordial NG and be forced to introduce extra parameters into the effective description because more often than not a model comparison will favour the one with less parameters. Therefore, we will turn to the observable we outlined at the end of Sec. 2, which, to our knowledge, has been largely unexplored in this context 12 and may offer valuable constraints on the nature of primordial NG: a 2-point PDF. We present this result after briefly discussing the 1-point statistics.
1-Point temperature distribution function
Given the discussion we have followed so far, it is now straightforward to reproduce the results of [47] and compute the 1-point PDF of the temperature map. The only difference is that the transfer functions map the fluctuations to the celestial sphere instead of the three-dimensional universe, but this does not prevent us from finding the distribution that generates the correlations. In analogy to (2.31), the result is given by
P(Θ) = 1 √ 2πσ Θ e − ζ 2 Θ 2σ 2 Θ [1 + ∆ Θ (Θ)] , (3.9) where ∆ Θ (Θ) ≡ x W Θ (x,n) s 2 Θ (x,n) ∞ −∞ dζ exp − (ζ−bΘ(x,n)Θ) 2 2σ 2 Θ (x,n) √ 2πσ Θ (x,n) ζ − σ 2 ζ ∂ ∂ζ F ζ ,(3.10)
with the corresponding definitions of the window function W Θ (x,n) = k e ik·x T (k,n), (3.11) and that of the variances
σ 2 Θ ≡ k T 2 (k,n)P ζ (k) ,(3.12)s 2 Θ (x,n) ≡ k e ik·x T (k,n)P ζ (k) , (3.13) σ 2 Θ (x,n) ≡ σ 2 ζ − b 2 Θ (x,n)σ 2 Θ , (3.14) with b Θ given by b Θ (x,n) ≡ s 2 Θ (x,n) σ 2 Θ . (3.15)
The final result is independent of the angular directionn because of homogeneity. In practice this appears because the form of the transfer functions T (k,n) will imply that the only dependence onn will be through scalar products with x or k, which as integration variables span the whole space, and thus they render all directions statistically homogeneous.
In Ref. [47], an analysis of the CMB statistics was carried out to search for non-Gaussian signals by estimating the PDF (3.9) directly from the 2015 Planck dataset [53], with an inconclusive result: if primordial non-Gaussianity of this type is present in the CMB, its statistical imprints are smaller than the intrinsic noise to which the 1-point statistics are subject to. With this and the forthcoming surveys in mind, we now proceed to outline another statistical object whose reconstruction from observations shows promise to have a lower intrinsic noise but still preserve all of the information contained within the 1-point PDF: a 2-point PDF.
2-Point temperature distribution function
It is interesting to write down expressions for quantities that are not typically used directly when characterising cosmological datasets. For instance, one rarely bothers to write down the full 2-point PDF for the CMB temperature map, as all of its information (in the Gaussian case) is already specified through the power spectrum. However, the scale dependence of this PDF may be a useful tool to search for non-Gaussianities. The result, analogously to what we had in the simpler case of a spatial window function, is
P Θ (Θ 1 , Θ 2 ,n 1 ,n 2 ) = P G,W (Θ 1 , Θ 2 ,n 1 ,n 2 ) 1 − x ∞ −∞ dζ exp − (ζ−ζΘ(x,n1,n2)) 2 2σ 2 Θ (x,n 1 ,n 2 ) √ 2πσ Θ (x,n 1 ,n 2 ) × W Θ (x,n 1 ) s 2 Θ (x,n 1 ) G Θ 11 ∂ ∂ζ − G Θ 12 + W Θ (x,n 2 ) s 2 Θ (x,n 2 ) G Θ 21 ∂ ∂ζ − G Θ 22 F ζ .
(3.16)
As before, let us go through this expression: the first thing to notice is that the two filtering points x 1 and x 2 in (2.42) are now replaced by two directions in the sky,n 1 andn 2 . Secondly, the Gaussian that is convoluted with F now has a different mean and variance, obtained by conditioning the joint multivariate Gaussian PDF on the values of Θ 1 and Θ 2 . The starting point to this is a joint Gaussian distribution for (ζ, Θ 1 , Θ 2 ) with covariance matrix
Σ Σ Σ = σ 2 ζ s 2 Θ (x,n 1 ) s 2 Θ (x,n 2 ) s 2 Θ (x,n 1 ) σ 2 Θ σ 2 Θ,ext (n 1 ,n 2 ) s 2 Θ (x,n 2 ) σ 2 Θ,ext (n 1 ,n 2 ) σ 2 Θ ,(3.17)
where we have written the covariance between the two externally chosen pointsn 1 andn 2 as σ 2 Θ,ext (n 1 ,n 2 ) = k T (n 1 , k)T (n 2 , k)P ζ (k) = Σ(n 1 ,n 2 ), (3.18) and σ 2 Θ = σ 2 Θ,ext (n,n), which is independent of the directionn. The precise definitions of all the additional functions involved in this section are given in App. C. It is worth mentioning, as a reminder to the reader, that, as in the 2-point PDF for curvature fluctuations (2.42), both (3.16) (besides from the temperature variables) and Σ(n 1 ,n 2 ) depend only on the angular distance betweenn 1 andn 2 , or equivalently, onn 1 ·n 2 , and that this is a consequence of our universe's homogeneity.
From this function, i.e., from the 2-point PDF (3.16), it is possible to obtain refined constraints on the local ansatz. Given a dataset, one can construct the 2-point PDF as follows: divide the angular distance into N bins of width δϑ and the temperature in M × M bins of size δΘ × δΘ, as in the 1-point PDF but now with two axes for the temperature field. Now, for each bin associated to a given angular distance ϑ n = n · δϑ, and for each value of (i, j), count how many pairs of pixels separated by that angular distance ϑ n have the values (Θ [i] , Θ [j] ) for the temperature in their respective positions. This process would generate N two-dimensional histograms, with two temperature axes, which we label by (Θ 1 , Θ 2 ), whose value at coordinate (Θ 1[i] , Θ 2[j] ) would give the number of pairs of pixels with temperatures in the (i, j)th bin, separated by angular distance in the nth bin. A 1/2 symmetry factor must be included in the number counts for temperature bins with i = j, as the bin (
Θ 1[i] , Θ 2[j] ) is equivalent to (Θ 1[j] , Θ 2[i] ).
How does this give refined constraints on the local ansatz? Let us appreciate that this set of PDFs contains information on the scale, or more concretely, on the temperature power spectrum and of its expansion in spherical harmonics (the standard C ℓ s) through the 2-point correlation Σ(n 1 ,n 2 ). If NG is absent, then at each value of the angular distance the 2point PDF will be a 2-variable Gaussian probability density with variances σ 2 Θ and covariance σ 2 Θ,ext (n 1 ,n 2 ) = Σ(n 1 ,n 2 ). Then, in the presence of NG, each 2-point PDF will undergo a NG deviation induced by the same primordial mechanism F , but for each angular separation this deviation will be experienced differently because the covariance matrix implied by the Gaussian part is different. That means that for each angular distance, the kernel that acts upon F in (3.16) gives a different deviation from Gaussianity, and therefore, each of the 2-point PDFs gives an independent estimator on the primordial NG field. For local NG, all of the N 2-point PDFs at different angular separations in the sky should give consistent 13 constraints/estimations of F . Conversely, if NG is measured and it does not adjust to the statistics implied by (3.16) at different angular scales, then purely local NG would be ruled out. Therefore, looking towards possible future directions to be explored, this type of object (a set of 2-point PDFs) shows promise to disentangle different shapes of NG, such as equilateral or orthogonal templates, and in particular, from the local ansatz.
In order to search for non-Gaussianity within a 2-point PDF, many approaches are possible. Given a model, i.e., an explicit expression for F , and using it as a template with few adjustable parameters is usually the method that will give the best constraints. In the spirit of Eq. (2.27), however, another one is worth mentioning: one can use bivariate Hermite polynomials on the temperature variables (Θ 1 , Θ 2 ), so as to express the PDF in terms of a bivariate Edgeworth expansion [54,55]. One then looks for any statistically significant nonzero coefficient in the expansion, in analogy to what was done for the 1-point case in [47]. This way, the existence of NG can be tested as a yes/no question, as any nonzero coefficient in an Edgeworth expansion implies a non-Gaussian distribution. This may be particularly useful when searching for NG in the next generation CMB surveys [56].
δ Statistics
Upcoming cosmological surveys will focus on the statistics of LSS, promising to bring precision cosmology to a new era. Indeed, the proliferation of observed modes due to the three-dimensional probe offered by LSS will enhance our statistics giving us invaluable information about the fundamental aspects of the early/late universe.
At small scales several sources of nonlinearity induce NG, like gravitational interactions and galaxy bias, obscuring the primordial contribution to the statistics. However, for relatively long modes, k 0.1 Mpc −1 and higher redshifts [57], linear perturbation theory can be trusted, which makes it easier to identify primordial signatures. Perturbative techniques pushing our analytic control towards smaller, weakly nonlinear scales include several schemes like SPT [58] and more recently EFTofLSS [59] and TSPT [60][61][62], which are set within a hydrodynamics framework, while going even further requires full Boltzmann solvers via N-body simulations. In this work, we will focus on the purely linear regime, leaving weakly nonlinear evolution with NG initial conditions [62] for future study.
The main probes of non-Gaussianity are the bispectrum and/or trispectrum, number counts and bias. The spectra retain information about the shape of the 3-and 4-point functions in momentum space, which can be linked to the mechanism responsible for generating NG. Number counts probe directly the 1-point PDF, which even though loses the shape information, it serves as a complementary and equally powerful estimator of NG [50,[63][64][65][66][67][68]. Finally, the halo bias serves as a third independent channel, which can give clear enhanced signatures of local NG at large scales.
In this section, we wish to track how primordial NG, in the form of the generic local ansatz (1.5), gets transmitted to the matter field in the linear regime. We extend our result, the non-Gaussian PDF of curvature fluctuations, in two directions: 1) we deduce a PDF for the matter density contrast δ, and hence, a halo mass distribution; 2) we compute the effect of the generalised local ansatz on the halo bias. These are complementary probes of the non-Gaussian initial condition via cluster number counts and power spectra, respectively, which should be accessible by surveys such as Lsst, Euclid, Spherex and Ska.
Halo mass function
The matter overdensity δ(x) = δρ(x)/ρ, withρ = Ω m ρ cr , is related to the primordial Newtonian potential, Φ = 3 5 ζ, as
δ(k) = α(k)Φ(k), with α(k) = 2r 2 H k 2 T (k)D(z) 3Ω m ,(4.1)
where D(z) is the linear growth rate, r H the current Hubble radius and T (k) the transfer function [69]. We smooth the density field over a radius R M = (3M/4πρ) 1/3 as in Eq. (2.9), Now, in principle, having the matter distribution function one can compute the halo number density, that is, the number density of halos of mass between M and M + dM at redshift z, and the observables derived from it. One way to do this is via the Press-Schechter (PS) scheme [70] extended to the NG case [71,72]. Let
δ W (x) = k y e ik·(x−y) W M (k)δ(k),(4.µ > (M, z) = ∞ νc(z) dν P(ν),(4.4)
(with ν ≡ δ/σ W ) be the tail distribution above some threshold value δ c (z). Then, one can assert that the total fraction of mass collapsed into bound structures will be proportional to this cumulative PDF: 1 ρ The collapse threshold, through which the z dependence arises, is taken to be given by the spherical model as δ c (z) ≃ 1.686 D(0)/D(z). The Gaussian PS function can be evaluated exactly by replacing P → P G in Eq. (4.4):
dn PS dM G (M, z) = 2ρ M e − ν 2 c (z) 2 √ 2π dν c (z) dM . (4.7)
For the NG case, the PS mass function is easy to compute from Eq. (2.31): 14 The fudge factor 2 corrects for the cloud-in-cloud problem, that is, a collapsed object of mass M1 tracing a volume V1 can be missed by the PS function if it is part of an underdense region of larger volume V2. The Gaussian value 2 is a good approximation in case of a small NG deformation [50].
dn PS dM (M, z) = dn PS dM G (M, z) 1 − 1 δ c 1 − ν 2 c + 1 (ln ν c ) ′ d dM x W (x)F bδ c ; b 2 σ 2 W , (4.8)
where a prime stands for the derivative with respect to the mass, while F bδ c ; b 2 σ 2 W is the Weierstrass transform of the local ansatz [see Eq. (2.35)] evaluated at the threshold δ c (z) (note that here, W (x) is the Fourier transform of the window function written in Eq. (4.3)). Upon using the Hermite expansion (2.27) of the function F , we may obtain a series representation of the mass function (4.8):
dn PS dM (M, z) = dn PS dM G (M, z) 1 + ∆ (ν c (z)) − 1 dν c (z)/dM ∞ n=2 He n (ν c (z)) (n + 1)! κ ′ n+1 , (4.9)
where we have defined the reduced cumulants κ n (M ) ≡ Refs. [63,73]) to arbitrary functions F . From its moment expansion (4.9), we can see that positive moments of the PDF (κ n , κ ′ n > 0) lead to overabundance of collapsed objects at the high mass end, where ν, ν ′ ≫ 1, as long as (ln κ n+1 ) ′ < νν ′ , which is satisfied since the cumulants κ depend weakly on the mass [73].
However, due to the highly nonlinear character of the collapse, one cannot fully parametrise collapsed objects by a single threshold number δ c . Indeed, it has been shown that the PS prescription does not accurately estimate the halo abundance even in the Gaussian case (for example a small non-spherical perturbation can have a considerable impact) [74]. Hence, the extension to the NG case is guaranteed to also have errors with respect to simulations. What can be done though is to characterise the deviation from Gaussianity by comparing the ratio of G-to-NG densities to that of the PS scheme [63,75], since the latter is expected to deviate equally in both cases: .
(4.10)
For the Gaussian mass function, dn dM G , we can adopt a Sheth-Tormen (ST) ansatz [74], which is better fitted to simulations than the Press-Schechter one, in which case the Gaussian ratio reads
r G [ν c (z)] = √ aA 1 + (aν 2 c ) −p edN dz (M, z) = 4π H(z) dz H(z) 2 f sky ∞ M dm dn dm (m, z), (4.12)
where f sky is the fraction of the sky covered by each survey.
This formula can serve as a template for the number density of clusters. For example, in the context of light isocurvature axions one can compute the non-Gaussian PDF deformation ∆ [46] and thus the mass function dn dM and create mock data via simulations. One can then pick an estimator and devise an overlap between the template and the data, in exactly the same manner that one uses the cosine estimator to measure the overlap of the local template with the actual bispectrum. The simplest thing to do is to take the number counts as an estimator. For example, given a model, one can count how many clusters of mass M exist at redshift z in the mock data and compare this number to the real data.
We may also try to reconstruct the PDF from data using other statistical estimators. This can be done by simply solving Eq. (4.10), together with (4.6), as an ODE for the cumulative PDF, µ > , to get
µ > [δ c (z), σ W ] = 1 2ρ ∞ M dm m r G [ν c (z)] dn dm (m, z), (4.13)
with r G given by Eq. (4.11). The left hand side now gives the tail distribution of δ, that is, the probability of δ > δ c at redshift z. On the right hand side we have the total mass of collapsed objects with m > M at redshift z, accounting for corrections to the spherical collapse model via assigning a mass value m(z) = m obs /r G [ν c (z)] to an object of observed mass m obs and multiplicity dn dm (m obs , z); in principle, this can be deduced from number counts. One can now apply statistical estimators like Minkowski functionals to the dataset m obs r G [νc(z)] , dn dm (m obs , z) , whose difference from their Gaussian estimate, as in the CMB [76,77] and LSS [78], will be a direct probe the NG deformation of the primordial PDF. Let us note, however, that measuring accurately the mass and redshift of clusters is a hard task, which might complicate a direct connection between cluster counts and primordial NG in this manner.
To summarise, in the context of tomographic NG, one can 1) use Eqs. (4.9) and (4.10) as a template for the mass function and thus the number density up to arbitrary order in the Edgeworth expansion; 2) assume a microphysical model, compute the PDF (as in e.g. Ref. [46]) and use Eqs. (4.8) and (4.10) as a template without having to refer to moments; and finally, 3) consider statistical estimators on the LSS dataset probing the primordial PDF via the counting scheme implied by Eq. (4.13).
Scale dependent halo bias
Another powerful probe of primordial non-Gaussianity is the halo bias, which enters in the latetime power spectra. As already argued, non-Gaussian initial conditions alter the halo abundance in a nontrivial way by increasing the number of rare density peaks that collapse into halos. This is easier to visualise in the case of the quadratic local ansatz δ = δ G + f NL δ 2 G [15]: a positive f NL adds positive skewness to the density distribution; thus, the same probability now corresponds to higher values of δ with respect to the Gaussian field, leading to more probable enhanced peaks. In [15], it was shown how this is encoded in a scale dependent correction to the halo bias given by
∆b(k) = 2δ c (b G − 1) f NL α(k) , (4.14)
where b G is the Eulerian Gaussian bias. The k 2 factor in α (see Eq. (4.1)) implies that the effects of non-Gaussianity will be accentuated in large scales (the transfer function goes to 1 for k → 0), which brings surveys like Ska to the frontline of NG searches, promising σ(f NL ) ∼ 1 [79]. This result was rederived and generalised in [80,81] using different approaches and has been confirmed with N-body simulations (see e.g. Refs. [82][83][84][85][86][87]), while in Refs. [51,73], the scale dependent halo bias was computed for the case of cubic g NL , τ NL -type local NG -see Refs. [88,89] for reviews.
In what follows we extend it to the generalised local ansatz (1.5) and show how in the case of an isocurvature source, future surveys can probe the landscape potential via measurements of the bias factor.
To begin with, let us rewrite the NG ansatz (1.5) for the Newtonian potential Φ = 3 5 ζ as
Φ(x) = φ(x) + 3 5 F NG [φ](x), (4.15)
where φ is a Gaussian random field with standard deviation σ 0 = φ 2 . The 3/5 factor is put explicitly to relate the primordial curvature perturbation ζ to the late-time (matter-dominated era) gravitational potential Φ. Following the peak-background split method [90,91], we now separate the Gaussian gravitational potential into long and short modes with respect to some characteristic halo scale R ⋆ ∼ R(M ) as
φ = φ L + φ S , (4.16)
which induces a similar split in the variance,
σ 2 0 = σ 2 L +σ 2 S = φ L (x) 2 + φ S (x) 2 .
With the help of the expansion in Hermite polynomials (2.27), and using known identities of these polynomials, we can write
F (φ L + φ S ; σ 2 0 ) = F (φ L , φ S ) = ∞ m=0 β m (φ L ) m! σ m S He m φ S σ S , (4.17) where β m (φ L ) ≡ − ∞ s=0 f m+s s! σ s L He s φ L σ L ,(4.18)
with f 0 = f 1 = 0. How is this expansion of F useful? If one is mainly interested in the short-wavelength dynamics (for instance, to study the gravitational collapse of the matter distribution into galaxies), this expansion allows one to identify the functions β m of long-wavelength fluctuations as effective nonlinearity parameters for the short modes, as may be seen directly from (4.17): m = 1 is a correction to the amplitude of the short modes, m = 2 a nonlinearity of f NL -type, m = 3 a nonlinearity of g NL -type, etc.
We now have to understand quantitatively how long wavelength density fluctuations affect the statistics of short modes and thus the halo number density per unit of halo mass, dn/dM , which from now on we denote by n L (x), through each term in the expansion (4.17). Generically, the halo mass function is a function of the matter contrast and the amplitude of the short modes:
n L = n L [ρ(x), ∆ φ S ]. (4.19)
Firstly, because of the dependence on ρ, irrespective of the presence of non-Gaussianity, a long wavelength perturbation δ L will induce a linear background shift in n as 20) wheren = n L [ρ]. Moving to the non-Gaussian part, the purely long wavelenght contributions -m = 0 coefficient in the expansion (4.17)-by definition will not affect the power spectra, and hence, the bias to first order in the nonlinearity parameters f m , so we may disregard them. From a short-wavelength modes' perspective, they are constant numbers that will only affect the background density through δ L in the previous expression.
n L [ρ(1 + δ L (x))] =n 1 + ∂ log n L ∂δ L δ L (x) ,(4.
Next, we may observe that the coefficient of the term linear in φ S (m = 1) depends on φ L . Therefore, a long mode will induce a shift in n(x) through this term, since short modes feel a background perturbed by the local amplitude of the long wavelength perturbation as
∆ φ S → (1 + β 1 (φ L )) ∆ φ S , leading to n L (x) =n 1 + ∂ log n L ∂δ L δ L (x) + β 1 (φ L ) ∂ log n L ∂ log ∆ φ S . (4.21)
Finally, we need to take into account that the halo density depends on the local amplitude of the long modes through all the functions β m (φ L ) for all m ≥ 2, since each β m acts as a nonlinearity parameter assuming a local value set by the long wavelength fluctuation φ L [51,81]. That is, we should consider the halo density as a function of the form
n L (x) = n L [ρ(x); {β m } m ],(4.22)
where β 1 controls the amplitude of the short modes, and the rest controls the amplitude of their nonlinearities. Thus, we can write
n L (x) =n 1 + ∂ log n L ∂δ L δ L (x) + m=1 ∂ log n L ∂β m β m (φ L (x)) . (4.23)
From this expression we can now compute the linear bias, defined as
b(k) = P mh (k) P mm (k) ,(4.24)
where the matter-halo power spectrum is defined as (4.25) and the matter-matter one as
P mh (k) = F x−y [ δ L (x)n L (y) ](k),P mm (k) = F x−y [ δ L (x)δ L (y) ](k) = |δ k | 2 . (4.26)
Thus, we will need to compute the matter-halo correlator and Fourier transform it (F). The matter contrast δ is related -in subhorizon scales-to the Newtonian potential through the Poisson equation (4.1), so that P φδ = P δδ /α. The first term in Eq. (4.23) is trivial since it just yields the δ propagator, which in Fourier space cancels the denominator in Eq. (4.24), resulting in the standard constant bias, while the second yields the scale dependent correction. Putting everything together, we get
b(k) = b G + 1 α(k) ∞ m=1 ∂ log n L ∂β m β ′ m (φ L ) = b G − 1 α(k) ∞ m=1 ∂ log n L ∂β m f m+1 . (4.27)
Evidently, the coefficient of the scale dependent correction contains a summation over all the nonlinearity parameters, or better put, the whole function F . In order to see this, we can write it down, equivalently, as
b(k) = b G + 1 α(k) ∞ −∞ N (φ L ; σ L ) dF (φ L ; σ L ) dφ L dφ L . (4.28) where N ≡ e −φ 2 L /2σ 2 L √ 2πσ L ∞ m=1 ∂ log n L ∂βm 1 σ m L He m φ L σ L .
Consequently, a scale-dependent halo bias can only signal the presence of some form of local NG but not of a specific parametrisation of it.
A downside of this is that the derivatives of n L w.r.t. β m in Eq. (4.28) would have to be computed from simulations if one is to obtain information about F , or alternatively, assume a model of collapse into haloes. In the first case, one may compute the derivatives of the halo mass function by varying the initial conditions of the simulations. However, the coefficients (4.18) of the scale dependent bias cannot be observed in real data in this manner, i.e, by repeating the collapse process as one would do in simulations, because they only come once and with the same initial condition for the curvature perturbation field ζ. Therefore, it becomes necessary that the next step be a connection between the effective bias coefficient N dF dφ dφ and observable quantities by modelling the halo mass functional n L in some way.
It turns out that the previous section 4.1 provides enough tools to accomplish this. In particular, if one models local NG at the short scales φ S using the extended PS scheme, as in Eq. (4.10), the variation with respect to the effective nonlinearity parameter d log n L /dβ m may be computed directly as d log n L /df S m , where f S m is the local nonlinearity parameter of the short-scale theory (and, to first order in the perturbation F , it also defines the corresponding nonlinearity parameter of the full theory because of Eq. (2.27)). This can be done systematically using Eqs. (2.36), (4.9) and (4.27).
The first scale dependent NG correction (m = 1), associated to cubic NG, was computed in Ref. [81] assuming a universal mass function, in agreement with the result of Ref. [15]:
∂ log n L ∂f S 1 = 2δ c (b G − 1), (4.29) where b G ≡ ∂ log n L ∂δ L
is the Gaussian bias. The second term (m = 2), corresponding to a g NL -type of NG, was computed in Ref. [73] using an Edgeworth expansion of the halo mass function and was found to be given by
∂ log n L ∂ log f S 2 = κ 3 (M ) 6 He 3 [ν c (M )] − κ ′ 3 (M ) 6ν ′ c (M ) He 2 [ν c (M )] ,(4.30)
where κ n (M ), is the reduced cumulant defined below Eq. (4.9). It turns out that when (4.27) is expanded up to f 3 = g NL , the resulting expression is in very good agreement with simulations [73]. Derivatives of the halo mass function with respect to the higher NG nonlinearity parameters f 4 , f 5 , . . . , may be easily computed from Eqs. (2.36) and (4.9). Concretely, they are given by
f S m ∂ log n L ∂β m = ∂ log n L ∂ log f S m = κ m+1 (M ) (m + 1)! He m+1 [ν c (M )] − κ ′ m+1 (M ) ν ′ c (M )(m + 1)! He m [ν c (M )] ,(4.31)
where primes denote derivatives w.r.t. the mass of the halo M . These terms can (and should) also be tested with N-body simulations for f S m>3 NG initial conditions, but such a computation is beyond the scope of this paper.
From Eq. (4.27) it is clear that the bias alone cannot differentiate between f NL , g NL or any higher order NG but it can give an answer to the question of whether Gaussianity is present or not. However, given a specific model for primordial NG, this can be a powerful probe in the sense of matching a template against data. For example, in App. A, we show that one particular realisation of this situation is the presence of an isocurvature mode with a potential ∆V (ψ). In this case, the function F of the generalised local ansatz (1.5), is related to the potential as F ∝ ∆V ′ . Thus, a measurement of or a constraint on the bias would translate into a constraint on the parameters of the landscape potential. In such a context, one can choose a well-motivated potential ∆V , which fixes the function F , depending on few parameters (two or three cover most physically motivated potentials). Then, with the help of simulations, one can use the PDF (2.10) with the replacements (4.3) to draw appropriate NG initial conditions for the density fluctuation field, and then look for a bias of the form (4.27). For example, within the axion parameter space, there are regions that lead to such cases [46] and may be probed in the near future with LSS surveys.
Concluding remarks
Non-Gaussianity, notwithstanding how small, is a robust prediction of inflation, and may be detectable if the inflaton had the chance to interact with other degrees of freedom during inflation. Up until now, NG does not show up in low n-point correlation functions of the temperature map [4], at least not in the form of a bispectrum nor a trispectrum, when their estimators are compared to the well-motivated local, equilateral, folded and orthogonal templates. However, primordial NG might be hidden in the data in the form of patterns that need to be revealed through different estimators.
In this work, we have focused on the object that contains the full information about the distribution of anisotropies of the temperature and density fields, that is, the probability density function. Our starting point has been the bottom-up parametrisation of the curvature perturbation as a Gaussian random field plus an arbitrary analytic function thereof. Instead of truncating the series expansion of this function to the first few terms, corresponding to the standard f NL , g NL parametrisation, we have kept the entire series enabling us to derive a probability functional that encodes the full local ansatz.
The various kinds of distributions derived from this functional provide us with new estimators, like the 1-point and 2-point statistics, which constitute complementary channels for the search of NG. In particular, the 2-point PDF encoding the scale dependence of the temperature distribution can serve as a full probe of the power spectrum able to disentangle local NG from other shapes. Moreover, the community has been shifting the focus towards LSS statistics as the latter will probe much more modes enhancing the statistical power of the datasets. With a view towards the near-future LSS surveys, we have computed the distribution of halos of mass M at redshift z showing that it is directly related to the primordial NG deviation. Finally, we computed the NG scale dependent correction to the linear halo bias arguing that it is the full local ansatz that contributes to the scaling. Therefore, the bias offers a unique probe of local NG in full generality, which cannot be used to distinguish the set of nonvanishing nonlinearity parameters, since these contribute as a sum to the correction.
Our results can be used with the future CMB/LSS data in various ways. For example, one can obtain refined constraints on the local ansatz from an Edgeworth expansion of the 2-point CMB temperature PDF. In addition, upon assuming well-motivated microphysical models leading to NG, one can use the formulas for the halo mass function and the bias as templates to be compared with data. Finally, we may use other types of statistical estimators like Minkowski functionals on the CMB/LSS datasets to draw conclusions on the primordial PDF.
Let us finish by pointing out that even though this work is focused on local primordial non-Gaussianity, it should be possible to extend the discussion to other cases. An "equilateral" ansatz of the form discussed in the Introduction would lead to a different scale dependence in the 2-point PDF, which could in turn serve as an equilateral NG estimator. Note that constraining such an object via e.g. an appropriate Edgeworth expansion could be less costly than that of n-point functions using the standard templates. Finally, one may be able to further generalise this by including combinations of spatial derivatives acting on the curvature fluctuation that lead to enfolded correlators. In this more general case, by the same procedure of changing variables in the Gaussian, one would be able to fully describe all the four momentum shapes employed at the level of the PDF. general statement, one can write down the field operator ζ(x) at the final time slice t as
ζ(x, t) = U † (t, t 0 )ζ I (x, t)U (t, t 0 ), (A.1)
where U is the temporal evolution operator in the interaction picture of quantum mechanics, and ζ I the interaction-picture field, which follows the dynamics of the free theory.
Naturally, the field ζ will generate a specific set of n-point functions when computing expectation values. One can then construct a PDF P[ζ] that generates these statistics through functional integration
ζ(x 1 ) · · · ζ(x n ) = DζP[ζ]ζ(x 1 ) · · · ζ(x n ), (A.2)
over the field configurations ζ(x). Quantum mechanics does provide the tools to determine P[ζ] directly, at least in principle. The operation ζ = U † ζ I U can be reframed in terms of a functional expression
ζ(x) = O[ζ I , Π I , {ψ I i , Π I i } i ](x), (A.3)
that depends on the whole spacetime evolution of the interaction-picture fields, ζ I and other degrees of freedom ψ I i , and that of their conjugate momenta, Π I and Π I i . In principle, one can compute correlations directly from this expression. However, if the dependence of O on the interaction-picture fields is known, one can determine the PDF of ζ by integrating over all possible configurations:
P[ζ] = Dζ I DΠ I i Dψ I i DΠ I i P G [{ψ I i , Π I i } i ] x δ ζ(x) − O[ζ I , Π I , {ψ I i , Π I i } i ](x) ,
(A.4) where P G is a Gaussian measure, with appropriate prescriptions to take into account the ordering of operators in O. The measure is guaranteed to be Gaussian because the free fields evolve linearly in time, and therefore the contraction of quantum field operators obeys Wick's theorem, which is equivalent to saying that the statistics are Gaussian.
Once P[ζ] is obtained, the statistics of ζ is fully determined. Computationally, however, it is useful to have a probability distribution from which one knows how to obtain expectation values. On the other hand, one knows that the observed statistics for ζ are consistent with Gaussianity, and that deviations, if any, must be small. This motivates finding a functional map
ζ G (x) = G[ζ](x), with inverse ζ(x) = ζ G (x) + F[ζ G ](x)
, such that ζ G has Gaussian statistics, i.e., such that
P G [ζ G ] = P[ζ G + F NG [ζ G ]] × det δζ δζ G . (A.5)
The difficulty, of course, lies in finding such a mapping. Afterward, one can include another mapping, that makes the power spectrum of ζ G to be consistent with current observations, if this is not the case already.
Thus, we have justified writing ζ in terms of a Gaussian field ζ G ,
ζ(x) = ζ G (x) + F[ζ G ](x), (A.6)
where the functional F NG is, in principle, arbitrary, and should be determined from the specifics of the inflationary model at hand. We now give an example of a model that motivates searching for non-Gaussianity wherein
F NG [ζ G ](x) = y k e ik·(x−y) F (ζ(y)), (A.7)
which is the main focus of our work.
A.1 A concrete example: multi-field inflation
Our current understanding of fundamental theories, such as string theory and supergravity, requires us to take into consideration the existence of many scalar fields. In such a framework, it does not make much sense to talk about an inflaton field; instead, one has an inflationary path meandering through a landscape potential. Consequently, curvature fluctuations may have been coupled to (many) other dynamically active degrees of freedom inevitably sourcing departures from Gaussianity. Even in the simplest extension -from single-field to multi-field inflationone finds a rich phenomenology that can be tested by near-future surveys. The presence of multiple fields during inflation leaves unique imprints in the CMB, a subject that has gained a lot of attention in the last years with quasi-single field inflation [92,93] and the Cosmological Collider program [94]. The "smoking gun" signature of massive degrees of freedom active at the inflationary energy scale is a bispectrum of the local shape that peaks for triplets of modes with one momentum going to zero.
In the case of two-field models of inflation, the Lagrangian describing the dynamics of the curvature fluctuation ζ interacting with an isocurvature field ψ is found to have the following generic form:
L = a 3 ǫ(ζ − αψ) 2 − ǫ a 2 (∇ζ) 2 + 1 2ψ 2 − 1 2a 2 (∇ψ) 2 − 1 2 µ 2 ψ 2 + L int . (A.8)
In the previous expression, a is the scale factor, ǫ = −Ḣ/H 2 is the first slow-roll parameter (where H =ȧ/a is the Hubble expansion rate during inflation), µ is the so-called entropy mass of the isocurvature field ψ, and α is a coupling that appears as a consequence of the shape of the background inflationary path in the multi-field space. It parametrises the bending of inflationary paths in field space [95][96][97], with α = 0 corresponding to the case of geodesic trajectories (a straight line in field space). The piece L int contains operators of cubic order (and higher) in terms of the fields ζ and ψ. A nonvanishing value of α makes ψ act as a source for the amplitude of ζ via the quadratic mixing term
L (2) mix = −2ǫa 3 αζψ. (A.9)
Crucially, this mixing allows the transfer of non-Gaussian statistics from the isocurvature field to the curvature fluctuation (but not the other way around [98]). The generated non-Gaussianity will depend on the particular form of L int .
A standard approach to deal with (A.8) consists in assuming that L int can be organised perturbatively in terms of powers of both fields ζ and ψ. Such an approach supposes that the nonlinear dynamics is dominated by cubic operators L int ⊃ L (3) int , and implies that the main non-Gaussian departures will be parametrised by the bispectrum, the amplitude of the 3-point function in momentum space [26][27][28][29][30][31][32][33]. Within this approach, one finds that the main operators leading to a nonvanishing bispectrum are given by a self-interaction term and a mixing term of the forms L
int turns out to be suppressed by powers of the slow-roll parameters. Both interaction terms lead to a non-analytic behaviour of the bispectrum: in the µ/H < 3/2 regime, with µ the entropy mass of an extra scalar fluctuation, one obtains an NG amplitude enhanced by g, with the bispectrum following an irrational power law in the squeezed limit leading to an intermediate shape between the so-called equilateral and local ones [92,93]; the other region µ/H > 3/2 leads to a bispectrum with a shape sensitive to the value of the entropy mass, displaying oscillatory patterns [99][100][101][102].
However, one may conceive regimes characterised by interaction Lagrangians L int in which one is not allowed to disregard terms of higher powers in the fields. For instance, the field ψ may have a potential ∆V (ψ) displaying a rich structure within a wide field range ∆ψ [103]. In this case, the Lagrangian (A.8) can be rewritten as
L = a 3 ǫ(ζ − αψ) 2 − ǫ a 2 (∇ζ) 2 + 1 2ψ 2 − 1 2a 2 (∇ψ) 2 − ∆V (ψ) + · · · , (A.10)
where we have absorbed the mass term in ∆V (ψ) by means of the identification µ 2 ≡ ∆V ′′ (0), where primes denote derivatives with respect to ψ. The elipses · · · denote terms that are either suppressed by slow-roll parameters or by α/H, which is assumed to be small. This Lagrangian (A.10) describes the dynamics of perturbations in those cases where the landscape potential has a rich structure in directions orthogonal to the inflationary path. If the height of the potential remains shallow ∆V /H 4 ≪ 1, the field ψ will display characteristic fluctuations ∆ψ that could traverse many minima and maxima of the potential, therefore probing the structure of the landscape. In this case, one cannot just stick to computations involving the first two terms of the expansion ∆V = µ 2 3 ψ 2 + g 3! ψ 3 + · · · ; instead, one has to perform perturbative computations where information about the entire potential ∆V is kept under control. As shown in [46,47], as long as ∆V /H 4 ≪ 1, it is indeed possible to compute every n-point correlation function for ζ and to derive its probability distribution function, P(ζ). Remarkably, this PDF turns out to be given by a Gaussian profile with small non-Gaussian corrections determined by the shape of the landscape potential ∆V . Thanks to this property, it is in principle possible to reconstruct the shape of the landscape potential ∆V out of CMB observations [47], providing information about a section of ∆V during the period of inflation when the modes of wavelengths relevant to the CMB were exiting the horizon (hence the term, tomographic non-Gaussianity) and beyond.
The generation of tomographic non-Gaussianity may be traced back to the self-interactions of an isocurvature field ψ. These self-interactions are transferred to the curvature perturbations thanks to a linear interaction coupling both fields [see Eq. (A.10)]. To appreciate how selfinteractions give rise to this form of non-Gaussianity, we may start by recalling that in the interaction picture, the evolution of the field ψ(x, τ ) is given by
ψ(x, τ ) = U (τ, τ 0 )ψ I (x, τ )U † (τ, τ 0 ). (A.11)
From a quantum-mechanical perspective in the Heisenberg-picture, one can write
ψ(x, τ ) = ψ(x, τ 0 ) + i τ τ 0 dτ ′ [H(τ ′ ), ψ(x, τ ′ )]. (A.12)
Similarly, the interaction picture gives that to first order in the interaction the field is given by
ψ(x, τ ) = ψ I (x, τ ) + i τ dτ ′ [H I (τ ′ ), ψ I (x, τ )], (A.13)
where the subscript I informs us that the corresponding operator is in the interaction picture and evolves as a free field. If, for simplicity, we take de Sitter spacetime as a background, the interaction-picture Hamiltonian reads H I (τ ) = x a 4 (τ )∆V (ψ I (x, τ )) with a(τ ) = −1/Hτ . With this, we may work on our previous equation to obtain
ψ(x, τ ) = ψ I (x, τ ) + i τ dτ ′ x a 4 (τ ′ ) [ψ I (x ′ , τ ′ ), ψ I (x, τ )] ∂∆V ∂ψ (ψ I (x ′ , τ ′ )). (A.14)
With the help of canonical commutation relations for the appropriate field v ≡ aψ [46,98], the commutator [ψ I (x ′ , τ ′ ), ψ I (x, τ )] is just a number and we can carry out the integral over τ ′ explicitly provided that:
1. The quantum field ψ I (x ′ , τ ′ ) in the argument of ∂∆V /∂ψ may be treated as a constant over time τ ′ .
2. The range of modes under consideration involves super-horizon modes with |kτ ′ | 1.
In fact, if the range of modes satisfies |kτ ′ | ≪ 1 for all k, then the first condition is implied by the second.
Let us give some comments about these conditions: The first point seems natural in the sense that the statistics of ψ I do not evolve over time: it may be seen as a Gaussian random field with a definite covariance. The second point, although appealing, is both physically and mathematically suspect, since in principle the interaction-picture Hamiltonian involves every mode (i.e. every momentum scale). Nonetheless, from an EFT perspective this is perfectly acceptable, as long as the potential ∆V is responsible for describing the physics at those scales. Moreover, this is the appropriate course of action when studying CMB or LSS modes that spent a large number of e-folds outside the horizon, because they do satisfy |kτ ′ | ≪ 1 throughout most of their history (practically for every time after horizon crossing this condition is fulfilled).
Using these considerations, one obtains ψ(x, τ ) = ψ I (x, τ ) − ∆N 3H 2 y k e ik·(x−y) ∂∆V ∂ψ (ψ I (y, τ )), (A. 15) where ∆N is the number of e-folds spent outside the horizon by the range of modes under consideration, which we take to satisfy ∆N ≫ 1. While the former may seem to be a heavy restriction, if we consider that the currently observable range of scales in the CMB satisfies ln(k S /k L ) ≃ 8 [3] and ∆N ∼ 60, we see that approximating ∆N by a single value for all modes in the considered range is justified. Of course, as was earlier suggested, a rigorous proof of this requires to go through every n-point function, performing an adequate renormalisation procedure to make the computations consistent at every bounded range of momenta. This was thoroughly dealt with in previous works [46,47], although stopping just short of writing down Eq. (A.15).
The curvature perturbation ζ(x, τ ) may be obtained in a completely analogous manner, only that we now have to consider an extra commutator to account for the quadratic mixing term (A.9):
ζ(x, τ ) = ζ I (x, τ ) + i τ dτ ′ [H α I (τ ′ ), ζ I (x, τ )] − τ dτ ′ τ ′ dτ ′′ [H V I (τ ′′ ), [H α I (τ ′ ), ζ I (x, τ )]] + · · · , (A.16) where H V
I is the term of the interaction-picture Hamiltonian that contains the ψ self-interactions and H α I contains terms associated to the quadratic mixing. The ellipsis · · · stand for higher order terms. It is of crucial importance to obtain the correct result to notice that the commutators in the last term of (A.16) only give a nonzero result when the pieces of the interaction Hamiltonian are written in that order. This, alongside the time ordering, yields an additional 1/2 factor for the statistical transfer of the nonlinear perturbation ∆V . After a calculation analogous to the one that led us to (A.15), with the same working assumptions, one obtains ζ(x, τ ) = ζ I (x, τ ) + α H ∆N ψ I (x, τ ) − 1 2 ∆N 3H 2 y k e ik·(x−y) ∆V ′ (ψ I (y, τ )) .
(A.17)
This equation can accommodate a variety of regimes. However, we choose to work in a situation wherein the linear transfer from ψ to ζ dominates. Even though this last equation is a perturbative result, ζ = α∆N H ψ + ζ 0 is an exact solution of the equations of motion on superhorizon scales [98]. This allows us to neglect the first term. Furthermore, since conventionally ζ should satisfy ζ = 0, from this point forward we will consider
ζ(x, τ ) ≈ α H ∆N ψ I (x, τ ) − 1 2 ∆N 3H 2
y k e ik·(x−y) [∆V ′ (ψ I (y, τ )) − ∆V ′ (ψ I (y, τ )) ] , (A.18) or equivalently, omitting the temporal coordinate and the projection integration y k e ik·(x−y) ,
ζ(x) = ζ G (x) − α 2 ∆N 2 2H 2 ∆N 3H 2 ∂ ∂ζ ∆V Hζ G (x) α∆N − ∂ ∂ζ ∆V Hζ G (x) α∆N , (A.19)
where we have written ζ G instead of ψ I to stress the nature of our result: ζ is made up from a Gaussian contribution plus a local non-Gaussian term. This result has the desired form ζ(x) = ζ G (x) + F NG [ζ G ](x). To identify F NG it is convenient to recognise that, because at linear order ζ ≃ α∆N H ψ, the power spectrum of ζ satisfies P ζ = α 2 ∆N 2 H 2 P ψ , where k 3 P ψ /2π 2 = H 2 /4π 2 (because in the free theory ψ behaves as a massless field). This allows one to find [47] α 2 ∆N 2 = 4π 2 A s where A s = k 3 P ζ /2π 2 is the amplitude of the power spectrum P ζ of ζ. This finally leads to
F NG (ζ) ∝ ∂ ∂ζ ∆V H 2πA 1/2 s ζ , (A.20)
which is the result reported in the Introduction.
B The partition function
The defining property of a partition function is that upon functional differentiation as in Eq. (2.21), it should give the n-point functions:
ζ(k 1 ) · · · ζ(k n ) = δ n Z[J] (iδJ(−k 1 )) · · · (iδJ(−k n )) J=0
= DζP[ζ]ζ(k 1 ) · · · ζ(k n ). over a Gaussian measure, we will do exactly that. Therefore, we will have to compute Now, using that Σ(x, z) = q P ζ (q)e iq·(x−z) , and replacing the last term into the corresponding term of (B.4), we get
x k z e ik·(x−z) P ζ (k) q P ζ (q)e iq·(x−z) e i y ζ(y)J(y) ∂F ∂ζ (ζ(z)) G .
(B.7)
Then, integrating over x gives |q| = |k|, and thus yields z k e i y ζ(y)J(y) ∂F ∂ζ (ζ(z)) G , (B.8)
which is equal but opposite in sign to the first term of (B.5). Therefore, those two cancel out, and we only have to compute There are three type of contractions in this expression: two self-contractions of fields originating from equivalent expressions and a mixed one. Performing the combinatorics gives i 2m ′ +ℓ ′ g 2n ′ +ℓ 2 m ′ m ′ !2 n ′ n ′ !ℓ ′ ! In the main text we argued that the 2-point PDF can be deduced from the probability density functional upon conditioning in two points. Here we arrive at the same result starting from the corresponding 2-point correlators and deriving the PDF. To our knowledge, this has not been derived earlier and thus we outline the procedure in some detail.
i m g n m!n! m ′ ,n ′ ,ℓ ′ 2m ′ +ℓ ′ =m 2n ′ +ℓ ′ =n m! 2 m ′ m ′ !(m − 2m ′ − ℓ ′ )!
As in the previous case, the PDF must include the non-fully connected contributions. This is a combinatorial mess, for we expect ζ n 1 (x 1 )ζ n 2 (x 2 ) = m 1 ,m 2 ,mt # n i ,n 2 ,m 1 ,m 2 ,mt σ 2
ζ | x 1 m 1 σ 2 ζ (x) mt σ 2 ζ | x 2 m 2
× ζ n 1 −2m 1 −mt (x 1 )ζ n 2 −2m 2 −mt (x 2 ) c .
(C.1)
Let us calculate # n 1 ,n 2 ,m 1 ,m 2 ,mt : we have an overall factor n 1 !n 2 ! from which we must divide out the overcounted terms. In this counting, we have m t ! redundant permutations when connecting x 1 and x 2 , plus 2 m 1 m 1 !2 m 2 m 2 ! when pairing amongst themselves. Finally, the ones that are assigned to the fully connected contribution undergo no further permutation, thus we must also divide by (n 1 − 2m 1 − m t )!(n 2 − 2m 2 − m t )!. Thus, ζ n 1 (x 1 )ζ n 2 (x 2 ) = m 1 ,m 2 ,mt
n 1 ! n 2 ! σ 2 ζ | x 1 m 1 σ 2 ζ (x) mt σ 2 ζ | x 2 m 2 2 m 1 m 1 ! (n 1 − 2m 1 − m t )! m t ! (n 2 − 2m 2 − m t )! 2 m 2 m 2 ! × ζ n 1 −2m 1 −mt (x 1 )ζ n 2 −2m 2 −mt (x 2 ) c . (C.2)
Careful inspection of this result reveals that (C.2) leads to a 2-point distribution analogous to what was obtained in [47], but with two points defining the filtering instead of one: P(ζ 1 , ζ 2 , r) = P G (ζ 1 , ζ 2 , r) 1 + √ 2πσ ζ (r, r 1 , r 2 ) × W (r 1 ) s 2 (r 1 ) G 11 ∂ ∂ζ − G 12 + W (r 2 ) s 2 (r 2 )
G 21 ∂ ∂ζ − G 22 F ζ , (C.3)
where the coefficients G ij , defined right below, depend on both the field variables and the spacetime positions x, x 1 , x 2 via the scalar variables
5
Note that when considering the functional integration of the first line of Eq. (2.13), the differential operator δ δζ(−k) − ζ(k) P ζ (k)
be evaluated in the same way as the partition function Z[J] by writing down the Dirac delta as δ(ζ(x) −ζ) = γ e iγ(ζ(x)−ζ) and noticing that what will be left over in the functional integral is exactly Z[J(−k) = γe −ik·x ]. Then the remaining integral over γ may be carried out by completing squares.
top-hat filter W M (x) = V −1 M H(R M − x), with H the Heaviside function and V M the volume enclosing a mass M. The probability distribution for the smoothed overdensity, P(δ W ), is then given by the 1-point PDF of Eq. (2.31) upon the replacements ζ → δ and W (k
(M ) is the number density of halos with masses in the range M and M + dM . The PS mass function then reads 14 dn PS dM (M, z) = −2ρ M dµ > dM (M, z). (4.6)
the f NL , g NL truncation 15 of the local ansatz (see e.g.
dn dM (M, z) = r G (M, z) dn PS dM (M, z), with r G (M, z) = dn dM G (M, z) dn PS dM G (M, z)
ST parameters a = 0.707, A = 0.322184, p = 0.3. With the mass function (4.10) at hand, which is just Eq. (4.8) with the replacement n PS | G → n ST , we may compute the number of clusters per redshift bin above some mass M as[63]
a 3 αζψ 2 , respectively. Any other term in L
accomplished by the following functional:Z[J] = Dζ P[ζ] e i k ζ(k)J(−k) = Dζ P[ζ] e i y ζ(y)J(y) . (B.2)To evaluate this expression, let us use the expression for P[ζ] as given in Eq. (2.8):P F [ζ] =P G [ζ] now evaluate (B.2) by expanding F in a power series and using Wick's theorem. Since (B.2) may be read as computing the expectation value of e i y ζ(y)J(y)
subscript G instructs to take the expectation value over a Gaussian measure. Note that the second expectation value, per Wick's theorem, can be written as the sum of
n ′ n ′ !(n − 2n ′ − ℓ ′ )! (Σ(z, z)) n ′ , (B.12)which one can sum over n, m to eliminate the constraints on the m ′ , n ′ , ℓ ′ sums.
ik·x J(−k) k e ik·x J(−k)P ζ (k) ik·x J(−k) k e ik·x J(−k)P ζ (k) ζ exp − (ζ−i k e ik·x J(−k)P ζ (k))
38 )
38which means that we have obtained the coefficients of the Hermite polynomial expansion (2.27) of F . Therefore, all one needs to write an explicit expression for F is to perform the correlation integrals∞
Other scenarios, such as the ekpyrotic[20,21] and bouncing universe[22,23], have been proposed as models able to reproduce nearly Gaussian initial conditions.
The details of this derivation are presented in App. B.
In a more standard notation, the first few terms would correspond to f2 = fNL, f3 = gNL, etc.8 We omit n = 0, 1 in the Hermite expansion because we assume ζ = 0 and ζζ to be set by the free theory matching the power spectrum of observations. To put it differently, with this definition of the local ansatz, due to orthogonality properties of the Hermite polynomials, the power spectrum of ζ is not modified to first order in the nonlinearity parameters[51].
This parameter is not the usual bias. If we think of the window as setting the scale of a tracer, we can define a bias as the ratio of field-tracer to the tracer-tracer correlation functions. b here is the ratio of the field-tracer to the 1-point tracer-tracer correlation functions.
dx 4πx 2 W (x)s 2n (x), and then sum back 11 its Hermite expansion(2.27). This allows for a direct test of NG in a given dataset and provides a way of constraining the primordial quantities involved in its generation, in a manner that is identical to what has been previously done in[47].However, this type of analysis integrates out information about the momentum dependence of the correlation functions, which can be used to place tighter constraints on the parameters of a given model. We now proceed to explore the two-point PDFs for the windowed primordial perturbations, which do contain such information.10 In principle, there is nothing stopping us from computing the Edgeworth expansion to any order in the couplings; indeed, we can expand the exact PDF of Eq. (2.5) to any order in F . We choose, however, for simplicity to truncate the series to first order.11 The precise scaling with n of the correlation integrals depends on the window function of choice W , and thus it is not always possible to give the result for F in a closed form.
The 2-point PDF has been used in LSS count-in-cells statistics[52].
That is, within the experiment's theoretical and systematic uncertainties.
For example, when truncated to n = 2, the expansion (4.9) agrees with Eq. (4.19) of Ref.[63].
During inflation, the primordial curvature perturbation field ζ is sourced by quantum fluctuations of the inflaton field, or possibly other degrees of freedom such as isocurvaton fields. As a
AcknowledgementsWe wish to thank Yashar Akrami, Diego Blas, Xingang Chen, Rolando Dunner, Nelson Padilla, Domenico Sapone and Eva Silverstein for useful discussions and comments. We are also grateful to the organisers of the workshop Inflation & Geometry at IAP, Paris, for creating a vibrant atmosphere, which helped us shape part of this work, as well as to all the participants for extensive discussions and feedback. GAP and BSH acknowledge support from the Fondecyt Regular project number 1171811 (CONICYT). BSH is supported by a CONICYT grant number CONICYT-PFCHA/MagísterNacional/2018-22181513. SS is supported by the CUniverse research promotion project (CUAASC) at Chulalongkorn University.A Generalised ansatz from quantum fluctuations during inflationIn obtaining this expression, we have defined a number of functions that depend uniquely on the structure of the Gaussian theory. The variance σ 2 ζ (r, r 1 , r 2 ) and ζ(r, r 1 , r 2 ) are the regression coefficients obtained by conditioning a Gaussian distribution of (ζ, ζ 1 , ζ 2 ) over (ζ W 1 , ζ W 2 ), with covariance matrix given by(2.43):Furthermore, the coefficients G ij are given by.(C.9)C.2 2-Point PDF: map to the sphere S 2The only difference with the previous Appendix is that the window function should be a map onto an angular coordinaten instead of a three-dimensional (flat) space. In the text, we wrote(C.10)Here we havewhile the regression coefficients are given by, (C.14)s 2 Θ (x,n 1 )σ 2 Θ − s 2 Θ (x,n 2 )σ 2 Θ,ext (n 1 ,n 2 ) σ 4 Θ − σ 4 Θ,ext (n 1 ,n 2 ), (C.16)Θ,ext (n 1 ,n 2 ).(C.17)
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| [] |
[
"New protocols for quantum key distribution with explicit upper and lower bound on secret-key rate",
"New protocols for quantum key distribution with explicit upper and lower bound on secret-key rate"
] | [
"Arindam Dutta \nDepartment of Physics\nMaterials Science & Engineering\nJaypee Institute of Information Technology\nA 10, Sector 62UP-201309NoidaIndia\n",
"Anirban Pathak \nDepartment of Physics\nMaterials Science & Engineering\nJaypee Institute of Information Technology\nA 10, Sector 62UP-201309NoidaIndia\n"
] | [
"Department of Physics\nMaterials Science & Engineering\nJaypee Institute of Information Technology\nA 10, Sector 62UP-201309NoidaIndia",
"Department of Physics\nMaterials Science & Engineering\nJaypee Institute of Information Technology\nA 10, Sector 62UP-201309NoidaIndia"
] | [] | Here we present two new schemes for quantum key distribution (QKD) which neither require entanglement nor require an ideal single photon source. Thus, the proposed protocols can be implemented using realistic single photon sources which are commercially available. The schemes are shown to be secure against multiple attacks (e.g., intercept resend attack and a class of collective attacks). Bounds on the key rate are obtained and it is shown that by applying a certain type of classical pre-processing, the tolerable error limit can be increased. A trade-off between quantum resources used and information revealed to Eve is observed and it is shown that by using slightly more quantum resources it is possible to design protocols having higher efficiency compared to a protocol of the same family that uses a relatively lesser amount of quantum resources. Specifically, in our case, SARG04 protocol is a protocol of the same family and it is clearly shown that the proposed protocols can provide higher efficiency compared to SARG04 at the cost of consumption of more quantum resources. *. At this moment we assume that the transmitted qubits have not undergone any decoherence. Further, Alice will not disclose basis information to Bob, otherwise up to this stage protocol is similar to BB84 protocol[5].Bob generates a sequence (S B1 ) of quantum states in accordance with his measurement results in ψ y J and sends it to Alice. For each qubit of the sequence (S B1 ), Alice uses the same basis (used in sequence S A ) to measure and records the outcome. Now, Alice will get the state ψ x ⊥ J with probability 1 4 as our assumption is that the sequence is very long and there is no noise in the quantum channel. If, the probability of getting the state ψ x ⊥ J is within the tolerable (threshold) limit around 1 4 , then Alice will publicly ask Bob to send the next sequence of qubit (S B2 ).Preparation and measurement of second sequence (S B2 ). After receiving Alice's request, Bob uses the other MUB (i.e., if Z (X) basis was used earlier to prepare n th qubit of the sequence S B1 , then X (Z) basis will be used to prepare the n th qubit of the sequence S B2 ) to prepare the elements of sequence S B2 with same bit value for the corresponding positions of the elements (ψ y J ) of the sequence S B1 . Bob sends the sequence S B2 to Alice. Alice measures the received qubits of the sequence S B2 using the following rule: If Alice gets the same state ψ x J after measuring the qubit sequence S B1 then she would use the other MUB (second basis) but after getting the state ψ x ⊥ J (orthogonal to the corresponding elements of the initial sequence S A ), Alice uses the same basis only. | 10.48550/arxiv.2212.13089 | [
"https://export.arxiv.org/pdf/2212.13089v1.pdf"
] | 255,125,562 | 2212.13089 | e39aaa4eb12e7946b11a5f64cd16373fc394b231 |
New protocols for quantum key distribution with explicit upper and lower bound on secret-key rate
Arindam Dutta
Department of Physics
Materials Science & Engineering
Jaypee Institute of Information Technology
A 10, Sector 62UP-201309NoidaIndia
Anirban Pathak
Department of Physics
Materials Science & Engineering
Jaypee Institute of Information Technology
A 10, Sector 62UP-201309NoidaIndia
New protocols for quantum key distribution with explicit upper and lower bound on secret-key rate
Here we present two new schemes for quantum key distribution (QKD) which neither require entanglement nor require an ideal single photon source. Thus, the proposed protocols can be implemented using realistic single photon sources which are commercially available. The schemes are shown to be secure against multiple attacks (e.g., intercept resend attack and a class of collective attacks). Bounds on the key rate are obtained and it is shown that by applying a certain type of classical pre-processing, the tolerable error limit can be increased. A trade-off between quantum resources used and information revealed to Eve is observed and it is shown that by using slightly more quantum resources it is possible to design protocols having higher efficiency compared to a protocol of the same family that uses a relatively lesser amount of quantum resources. Specifically, in our case, SARG04 protocol is a protocol of the same family and it is clearly shown that the proposed protocols can provide higher efficiency compared to SARG04 at the cost of consumption of more quantum resources. *. At this moment we assume that the transmitted qubits have not undergone any decoherence. Further, Alice will not disclose basis information to Bob, otherwise up to this stage protocol is similar to BB84 protocol[5].Bob generates a sequence (S B1 ) of quantum states in accordance with his measurement results in ψ y J and sends it to Alice. For each qubit of the sequence (S B1 ), Alice uses the same basis (used in sequence S A ) to measure and records the outcome. Now, Alice will get the state ψ x ⊥ J with probability 1 4 as our assumption is that the sequence is very long and there is no noise in the quantum channel. If, the probability of getting the state ψ x ⊥ J is within the tolerable (threshold) limit around 1 4 , then Alice will publicly ask Bob to send the next sequence of qubit (S B2 ).Preparation and measurement of second sequence (S B2 ). After receiving Alice's request, Bob uses the other MUB (i.e., if Z (X) basis was used earlier to prepare n th qubit of the sequence S B1 , then X (Z) basis will be used to prepare the n th qubit of the sequence S B2 ) to prepare the elements of sequence S B2 with same bit value for the corresponding positions of the elements (ψ y J ) of the sequence S B1 . Bob sends the sequence S B2 to Alice. Alice measures the received qubits of the sequence S B2 using the following rule: If Alice gets the same state ψ x J after measuring the qubit sequence S B1 then she would use the other MUB (second basis) but after getting the state ψ x ⊥ J (orthogonal to the corresponding elements of the initial sequence S A ), Alice uses the same basis only.
I. INTRODUCTION
Cryptography is known to be an extremely essential and useful technique for mankind. Since the historical past, cryptographic methods have been used for camouflaging secret information, but cryptanalysts often find more powerful methods to decipher the secret message. A paradigm shift in cryptography was observed in 1970s when methods of public key cryptography, like RSA [1] and Diffie Hellman (DH) [2] schemes were introduced. The security of these schemes and other similar classical schemes for key distribution arises from the complexity of the computational tasks inherently used in designing these schemes. For example, the security of the RSA scheme and DH scheme arises from the computational complexity of the factorization of an odd bi-prime problem and discrete logarithm problem, respectively [3]. Interestingly, in a seminal work, in 1994, Peter W. Shor [4] showed that both factorization of an odd bi-prime problem and discrete logarithm problem can be solved efficiently (i.e., in polynomial time) using quantum computers. This established that many of the existing classical schemes for key distribution will be vulnerable if a scalable quantum computer can be built. Thus, cryptography faced a serious challenge from quantum computers or more precisely from quantum algorithms which can be used to solve several computational tasks much faster than their classical counterparts. Interestingly, a solution to the challenge posed by quantum computers were already there in terms of the schemes for quantum key distribution (QKD) where key distribution happens with the assistance of quantum resources and security arises from laws of physics and not from the difficulty of the computational complexity of a problem. In fact, the first such scheme for QKD was proposed 10 years before the work of Shor that put classical cryptography in crisis. Specifically, in 1984, Bennett and Brassard [5] proposed the first scheme for QKD. Physical principles, like the no-cloning theorem [6], collapse on measurement postulate and Heisenberg's uncertainty principle played a crucial role in the establishment of the security of this single qubit based scheme which can be realized using polarization encoded single photons and other alternative realizations of photonic qubits. It may be noted that in an ideal situation, any effort for eavesdropping leaves a detectable trace in a QKD protocol. However, in a realistic situation, due to device imperfections, we can have a scenario where eavesdropping may happen without causing a detectable disturbance.
The BB84 protocol was followed by several protocols for QKD [7][8][9][10] and other related cryptographic tasks [10][11][12][13][14][15][16][17] (for a review see [18,19]). Each of these protocols has its own advantages and disadvantages. Most of these schemes are unconditionally secure in the ideal situation. However, in the real-life situations, devices used are not perfect and that leads to side channels for performing quantum hacking using device imperfection(s). For example, BB84 protocol (and many other protocols of similar nature, like B92 protocol [7]) ideally require a single photon source as to implement this type of protocols, Alice must be able to send single photon states to Bob. Currently, commendable experimental efforts have been devoted to construct a reliable single-photon source (see [20,21] and references therein). However, in most of the commercial products, weak coherent pulses (WCPs) produced by attenuating the output of lasers are used as an approximate single photon source. Quantum state of a WCP produced by attenuating a laser can be described as
|α = | √ µ exp(iθ) = ∞ n=0 e −µ µ n n! 1 2 exp(inθ)|n ,(1)
where |n represents a Fock state (or equivalently an n photon state) and mean photon number µ = |α| 2 1. Effectively, Alice produces a quantum state which can be viewed as a superposition of Fock states with a Poissonian photon number distribution given by p(n, µ) = e −µ µ n n! . Thus, if such a source is used Alice produces the desired onephoton state with a probability p(1, µ) and produces the multi-photon pulses with total probability 1−p(0, µ)−p (1, µ). In this scenario, Alice creates a multi-photon state with the same information and opens a window for the side channel attack that allows Eve to perform the photon number splitting (PNS) attack [22]. Further, in long-distance communication, the channel loss is a concern as it allows an Eavesdropper with superior technology to replace the lossy channel with a perfectly transparent channel and perform eavesdropping attack [23] showing that the effect of her activities is due to channel loss. To counter this, Scarani et al., proposed a QKD scheme (SARG04) in 2004 [24], which is robust against PNS attack. Here, we aim to propose a set of two new protocols for QKD which would be robust against PNS attack (like SARG04) and a family of other attacks, with some specific advantages over SARG04 and other existing protocols for QKD having a similar structure in general.
It is interesting to note that in every QKD protocol, information splitting happens. In protocols, like BB84 [5] and B92 [7] the information is split into a classical piece (information about the basis in which the transmitted qubits are prepared) and a quantum piece (transmitted qubits). A similar kind of information splitting happens in SARG04 protocol [24], whereas in some other protocols, like Goldenberg Vaidman (GV) protocol [9], information is split into two quantum pieces. The security of all these protocols arises from the inability of Eve to simultaneously access these two or more pieces of information. Here we wish to study a foundationally important question that arises from the above observation: Can we modify the efficiency of a protocol and/or the bounds on the secret-key rate of the protocol by modifying the protocol in such a way that information contained in the classical piece is reduced? In what follows, we will use SARG04 protocol as our test bed to answer this question. Specifically, we will introduce two new protocols for QKD which are quite similar to SARG04 protocol but the information content in the classical pieces in the revised protocol are lesser than that in SARG04 protocol. SARG04 protocol was designed to make PNS attack [22] highly improbable, but it was less efficient 1 compared to a set of other single photon based schemes for QKD. In this work, we aim to propose two new protocols for QKD which will be more efficient than SARG04, but at the same time would remain robust against PNS attack and a set of other well known attacks.
The rest of the paper is organized as follows. In Section II, we propose a new single photon based protocol for QKD which does not require ideal single photon source. The protocol which will be referred to as Protocol 1, is first described in a generalized way and then in a step-wise manner. It is shown that a simple modification in the sifting subprotocol of the Protocol 1 leads to a new protocol (Protocol 2) having higher efficiency. The detailed security analysis is done in Section III. To perform the security analysis, we use a depolarizing channel that represents the error introduced by Eve (or the channel itself) and allows us to calculate the tolerable error limit for the first quantum particle sequence prepared by Bob. Further, we consider the security against a set of collective attack scenario. The paper is concluded in Section IV with specific attention on the security-efficiency trade-off observed in our schemes.
II. PROPOSED QKD PROTOCOLS
We have already mentioned that many QKD schemes which use a random sequence of non-orthogonal states involve splitting of information into quantum and classical pieces enforcing Eve to leave traces of measurements performed by her in an effort to eavesdrop. In all such schemes, at some stage, Alice and Bob perform some kind of comparison of the state (basis) initially prepared (used for the preparation of the state) by Alice/Bob and obtained by (used for measurement of the state) Bob/Alice to check correlations to reveal eavesdropping. After this step, Alice and Bob keep the states which satisfy certain condition and thus can lead to the final key. This step can be viewed as a subprotocol which may be referred to as a classical key-sifting subprotocol. In what follows, we will see that in this work, we use a bi-directional quantum channel to distribute the quantum information in the form of single photon to distribute a secret key between two legitimate parties, Alice and Bob after the key-sifting subprotocol. Here, Alice has the prior information of the quantum states of her initial sequence that she prepares to send to Bob. This prior information helps her to agree with the position of the sifted key after information reconciliation. In what follows, we assume the following notation: To encode the bit value x, Alice generates the quantum state ψ x J , for different encoding using mutually unbiased bases (MUBs) in Hilbert space H of dimension d 2 , where x represents the bit value and J represents the basis used for encoding the bit value x. Without loss of generality we choose, J := {Z, X}, where the basis set Z and X correspond to {|0 , |1 } and {|+ , |− } , respectively. The basis set Z and X are often referred to as the computational and diagonal basis set, respectively. For the convenience of classical key-sifting, in what follows, we use J = 0 for Z basis and J = 1 for X basis. Now using the above notation, we may propose the basic structure of our protocol in generalized form as follows:
(1) State generation-transmission and measurement: Alice prepares and sends a sequence (S A ) of qubits to Bob which is made up of one of the four quantum states ψ x J := {ψ x Z ,ψ x X } to encode a random sequence of bit value x ∈ {0, 1}. Bob measures randomly with computational or diagonal basis and gets a sequence with one of the three quantum states ψ y J := {ψ x Z ,ψ
(2) Condition for key-sifting. To maximize the fraction of raw key after the sifting process, we propose a classical subprotocol with a lesser amount of classical information revealed in comparison to the SARG04 protocol. Alice announces the positions of the qubits for which Bob will keep the measurement results for the elements of the sequence S A to get the secret key under two conditions: (a) If Alice gets orthogonal state (ψ x ⊥ Z ) to the corresponding elements of her initial sequence (S A ) after measuring S B1 and the measurement result of the sequence S B2 is ψ
x/x ⊥ Z , Alice decodes that the Bob's measured state of sequence S A was ψ
x/x ⊥ X . (b) If Alice gets the same state (ψ x Z ) corresponding to the elements of the sequence S A after measuring S B1 and the measurement result of the second sequence sent by Bob (S B2 ) is obtained to be ψ x X , then Alice concludes that the measurement result of the sequence S A by Bob was ψ x Z if and only if the J value is announced by Bob for the measurement of each element is same with the J value for corresponding elements of Alice's initial sequence S A .
In what follows, we will first describe our main protocol (Protocol 1) in a step-wise manner. Subsequently, we will show that a small change in the condition of the key-sifting subprotocol of Protocol 1 can lead to an increase in the efficiency of the proposed QKD protocol. The modified version will be described as Protocol 2 (see Table I).
2 Let us suppose two orthonormal bases set in the d-dimensional Hilbert space are ψ j 1 := {ψ 1 , ψ 2, . . . , ψ d } and ψ j 2 := {ψ 1 , ψ 2, . . . , ψ d }, they are called mutually unbiased bases when the square of the magnitude of the inner product between two different basis elements equals the inverse of the dimension d can be expressed ψa|ψ b
2 = 1 d , ∀a, b ∈ {1, 2, . . . , d}.
If one measures the system that is prepared in one of the MUBs, then the measurement outcome using another basis will be equally probable or maximally uncertain.
|+ 1 /8 0 |0 0 |0 |0 |+ 1 /64 1 − 0 |0 |+ |0 |0 |− 1 /64 − − 0 |+ |0 |1 |0 1 /32 − |+ − |+ |0 |+ 1 /64 1 − 1 |− |− |1 |0 |− 1 /64 − − 1 |− |1 |1 1 /32 − |− − |− |1 |− |1 |− 1 /8 0 |1 1 |1 |1 |+ 1 /64 − − 0 |+ |+ |0 |1 |− 1 /64 1 − 0 |+ |1 |0 |0 1 /32 − |+ − |+ |1 |+ 1 /64 − − 1 |− |− |1 |1 |− 1 /64 1 − 1 |1 |0 |1 1 /32 − |− − |− |+ |0 |+ |0 1 /8 1 |+ 0 |+ |+ |0 1 /64 0 − 0 |+ |0 |+ |+ |1 1 /64 − − 0 |0 |+ |− |+ 1 /32 − |0 − |0 |+ |0 1 /64 0 − 1 |1 |1 |− |+ |1 1 /64 − − 1 |1 |− |− 1 /32 − |1 − |1 |− |1 |− |1 1 /8 1 |− 1 |− |− |0 1 /64 − − 0 |0 |0 |+ |− |1 1 /64 0 − 0 |0 |− |+ |+ 1 /32 − |+ − |0 |− |0 1 /64 − − 1 |1 |1 |− |− |1 1 /64 0 − 1 |− |+ |− 1 /32 − |1 − |1
Protocol 1
To describe these protocols we use the elements of the bases Z and X, and a notation that describes the basis
elements as | + z /| − z (| + x /| − x ) := |0 /|1 (|+ /|− ).
Here, we define the Z and X bases elements as
| + x = 1 √ 2 (|0 + |1 ) , | − x = 1 √ 2 (|0 − |1 ) | + z = 1 √ 2 (| + x + | − x ) , | − z = 1 √ 2 (| + x − | − x ) .(2)
Step1: Alice randomly prepares single qubit sequence S A using Z or X basis and sends it to Bob by keeping the basis information secret.
Step2: Bob measures the qubits of the sequence S A randomly with basis Z or X and records the measurement result. Bob then prepares a new qubit sequence S B1 with the same states corresponding to the measurement result of the sequence S A and sends it to Alice.
Step3: Alice measures each qubit of the sequence S B1 using the same basis which was used to prepare the qubit of the sequence S A ; for example, if Alice chooses to prepare the i th qubit of the sequence S A in Z basis (X basis), then she would measure the i th qubit of the sequence S B1 using the Z basis (X basis). Alice records the measurement outcome of sequence S B1 and asks Bob to proceed if the measurement outcomes are within the threshold limit of the expected probability distribution of the possible results.
Step4: Bob prepares a second qubit sequence S B2 for the same bit values, but using the other basis, and sends the sequence to Alice. For example, if i th qubit of the sequence S B1 is prepared in the state | ± z (| ± x ), then the i th qubit of the sequence S B2 will be prepared by bob in the state | ± x (| ± z ).
Setp5: Alice performs a measurement on each qubit of the sequence S B2 based on the measurement result for the elements of the sequence S B1 , such that, she uses X basis or Z basis (Z basis or X basis) if she gets the same state | ± z or | ± x (states orthogonal to the initial state (i.e., | ∓ z or | ∓ x )) as a measurement result of the sequence S B1 for the corresponding elements to her initial sequence S A .
Step6: Alice isolates the conclusive measurement results (measurement results which can be used to conclusively determine the measurement results of Bob) obtained by her measurement on the sequence S B1 and S B2 , such that, if Alice prepares the i th qubit of the sequence S A in | ± z (| ± x ) and gets the measurement result for the corresponding element of the sequence S B1 and S B2 as Table II).
| ∓ z (| ∓ x ) and | ± z (| ± x ) or | ∓ z (| ∓ x ), respectively. Alice then determines the Bob's measurement result of sequence S A as | ± x (| ± z ) or | ∓ x (| ∓ z ) (see
Step7: Alice retains those bits as sifted key for which the J value will be the same for both of them. For example, if Alice prepares the sequence S A in the state | ± z (| ± x ) and the measurement result for the corresponding qubit of the sequence S B1 and S B2 are | ± z (| ± x ) and | ± x (| ± z ), respectively then Alice determines Bob's measurement result of the sequence S A as | ± z (| ± x ) only when the basis used for the preparation and measurement of each element of the sequence S A by Alice and Bob are the same i.e., J value is same. It may be noted that a classical sifting process is performed in this step.
Result determined with same J value | ± z | ± z ,| ± x − | ± z | ∓ z ,| ± z | ± x − | ± x | ± x ,| ± z − | ± x | ∓ x ,| ± x | ± z − Protocol 2
We now introduce a new variable M ∈ {0, 1}, that will be useful to interpret Bob's measurement results of the sequence S A i.e., M (= 0) := {| + z , | + x } and M (= 1) := {| − z , | − x } for the classical key-sifting process. Steps 1 to 6 are the same for this second protocol with some differences in the classical sub-protocol as explained in Step 7. Using this classical sifting process, we get the sifted key with a maximum inherent error having the probability 1 /16 but having better efficiency in comparison with Protocol 1. This trade-off part will be explained later with a detailed analysis.
Step7: If Alice prepares the elements of the sequence S A in |+z /|+x (|−z /|−x ) with conditions: (1) Bob announces the value of M as 1(0), Alice determines Bob's measurement result of the sequence S A as |−x /|−z (|+x /|+z ) irrespective of the measurement result of the sequences S B1 and S B2 , (2) Bob announces the value of M is
0(1), Alice determines Bob's measurement result of the sequence S A as (i) | + x /| + z (| − x /| − z ) if the measurement result of the sequence S B1 and S B2 are | + z /| + x (| − z /| − x ) or | − z /| − x (| + z /| + x ) and | − x /| − z (| + x /| + z ) or | + z /| + x (| − z /| − x ) respectively and (ii) | + z /| + x (| − z /| − x ) if the measurement result of the sequence S B1 and S B2 are | + z /| + x (| − z /| − x ) and | + x /| + z (| − x /| − z )
respectively (see Table III).
III. SECURITY PERFORMANCE FOR THE PROPOSED PROTOCOLS
We have already mentioned that Alice first needs to approve the sequence S B1 before Bob sends the sequence S B2 . After obtaining the acceptance of S B1 from Alice, Bob sends the second sequence S B2 to her and eventually, Alice and Bob agree with the secret key if the computed percentage of error is found to be lower than the tolerable error limit after successful execution of the protocol. The main motivation behind the security analysis of the proposed protocols is to compute the maximum amount of error that can be tolerated in presence of a set of collective attacks. To perceive the attack strategy to be adopted by Eve, we use the approach followed in Ref. [26] in which the authors used a depolarizing map that transforms any two-qubit state into a Bell-diagonal state. If we wish to analyze the security of the QKD
1 − − | − x /| − z | + z /| + x | + z /| + x | − x /| − z | + x /| + z 0 | − z /| − x | + z /| + x | + z /| + x | + x /| + z | + z /| + x 0 − − | + x /| + z | − z /| − x | − z /| − x | + x /| + z | − x /| − z 1 | + z /| + x | − z /| − x | − z /| − x | − x /| − z | − z /| − x
protocols proposed here in line with Ref. [26], we have to transform/map our protocols into equivalent entanglementbased schemes. Such an entangled state based scheme equivalent to Protocol 1/2 described above may be viewed as a scheme where Alice prepares n two-qubit entangled states (say, Bell states) and performs her encoding operation on the first qubit of each two-qubit state and sends the second qubit to Bob, i.e., if Alice prepares |Φ + then she transforms it to A j ⊗I 2 |Φ + , and sends the second qubit of it to Bob, where |Φ ± = 1 √ 2 (|00 ±|11 ), and A j and I 2 are Alice's encoding operation and identity operation in two dimensions. Bob also randomly applies one of his encoding operators B j to each of the qubits that he receives. We can denote the 2n qubit state shared by Alice and Bob asρ n AB . Finally, Alice and Bob measure their qubits ofρ n AB randomly in X and Z basis and map each measurement outcome to bit value 0 or 1. Now, we may use two completely positive maps (CPMs) O 1 and O 2 , where O 1 is entirely defined by the protocol and O 2 is independent of the protocol. Specifically, these CPMs are defined as
O 1 (ρ) = 1 N j p j A j ⊗B j (ρ)A † j ⊗B † j and O 2 (ρ) = l M l ⊗M l (ρ)M † l ⊗M † l .
Here, p j ≥ 0 is the probability that Alice and Bob decide to keep the bit value during the sifting subprotocol, N is essentially the normalization factor and M l describes a quantum operation such that M l ∈ {I 2 , σ x , σ y , σ z :
I 2 = |0 0| + |1 1|, σ x = |0 1| + |1 0|, iσ y = |0 1| + |1 0|, σ z = |0 0| − |1 1|}. The structure of O 2 (ρ)
shows that the same operator is applied on both the qubits, thus M l ⊗M l ∈ {I ⊗ I, σ x ⊗ σ x , σ y ⊗ σ y , σ z ⊗ σ z } . Now, these two-qubit operators are applied with equal probability or equivalently these are applied randomly. Interestingly, the random application of these operations mimics the action of a depolarizing channel that transforms any two-qubit state to a Bell diagonal state. If Alice and Bob apply unitary operation A j ⊗ B j , they get their sifted key after the sifting phase with the normalization factor N , here j p j = 1. We use a normalized two-qubit density operator from the Eq (1) of Ref. [27] as n = 1 (see for details [26]). We use the notation P |Φ = |Φ Φ| which describes a state projection operator that projects a quantum state of the same dimension onto the state |Φ . Here,
ρ 1 [µ] = µ 1 P |Φ + + µ 2 P |Φ − + µ 3 P |Ψ + + µ 4 P |Ψ − ,(3)
where P |Φ ± and P |Ψ ± are the state projection operators onto the Bell states |Φ ± = 1 √ 2 (|00 ± |11 ) and |Ψ ± = 1 √ 2 (|01 ± |10 ) and µ 1/2 and µ 3/4 are the respective probabilities of getting the corresponding Bell states in the depolarizing channel. In what follows, to analyze the security of the sequence S B1 , we use the following key rate equation
r := I(A : B) − max ρ∈R S(ρ),(4)
where A and B are the quantum states obtained after the measurements are performed by Alice and Bob, I(A : B) is the mutual information between Alice and Bob, S(ρ) is the von Neumann entropy of the composite state of both the parties (i.e., ρ) and R 3 is the density range of the density operator ρ (for a precise definition of density range see Definition 3.16 of [28]). This equation was introduced in Ref. [28] (see Eq. (22) of [28]). Here, both parties gain the correlation for their secret key after the key sifting process depending on the sequence S B1 . When Alice knows the elements of her own initial sequence S A , we can calculate the tolerable error limit using the above key rate equation.
Let us now assume that the quantum bit error rate (QBER) is E ∈ [0, 1] for the measurements done in both X and Z bases. The outcome for the projective measurements on the system ρ can be captured through a random variable V . As the measurement in basis Z and X can lead to four different outcomes, we can have four probabilities associated with these measurement outcomes. In fact, the probabilities of the measurement outcome in basis Z and X can be defined as the probabilities (µ i : i ∈ {1, 2, 3, 4}) of obtaining different values of V . The entropy of this variable V , is
H(V ) = − V ∈µi V log 2 V ≥ S(ρ)
. These probabilities µ i s can be computed easily by taking expectation values of ρ with respect to the relevant states. For example, in our case, µ 1 = Φ + |ρ|Φ + , µ 2 = Φ − |ρ|Φ − , µ 3 = |Ψ + |ρ|Ψ + and µ 4 = |Ψ − |ρ|Ψ − . Through a long, but straightforward calculation, we obtain relations between the probabilities µ i s associated with the system described in Eq. (3) as follows: µ 3 + µ 4 = E, µ 2 + µ 4 = E, µ 1 + µ 2 = 1 − E and µ 1 + µ 3 = 1 − E. These four equations are not linearly independent. Actually, there are three linearly independent equations (for example, you may consider (i) the first three of these equations or (ii) the first two and the last one as linearly independent equations). In this situation, we cannot solve the above set of equations, but we can consider one of the probabilities as a free parameter and express the rest of the probabilities in terms of that. Here we choose µ 4 as the free parameter to express other probabilities in terms of it as µ 1 = 1 − 2E + µ 4 and µ 2 = µ 3 = E − µ 4 . It may be noted that µ 4 ∈ [0, E] as the range of any probability is [0, 1] and µ 2 + µ 4 = E. Now, the condition for the maximization of the entropy of the random variable V can be obtained by solving
1 + 1−E 2 log 2 1−E 2 + E 2 log 2 E 2 − h(E) = 0
and obtain E ≈ 0.0617 i.e., 6.17% bit error rate (also see Fig. 1 (a)). This result shows that if Y is announced instead of revealing the values of j A and j B , the maximum tolerable error limit for measuring the sequence S A and S B1 will be increased to 6.17%.
Alice first checks the security threshold for the sequence S B1 and proceeds only if it is under the expected limit. Then, Alice measures the second sequence S B2 sent by Bob and completes the sifting subprotocol. After the key sifting process, Alice and Bob need to check whether the QBER is under the security threshold or not. To determine the tolerable threshold value we follow the same methodology as described above. The entropy of Bob's final bit string b after sifting subprotocol equals 1 (i.e., H(b) = 1). Now, the conditional entropy of bit string b when Alice's bit string a is known, can be computed as
H(b|a) = h( 1 6 + 2E 3 ) with a, b ∈ {0, 1}.
In a similar manner, the positive key rate equation is obtained as
1 − h( 1 6 + 2E 3 ) − 2h(E) = 0(5)
(i.e., by considering r = 0) and solving it we can obtain the security threshold as E ≈ 0.0316 (see Fig. 1 (b)) i.e., 3.16% QBER (see Appendix B for detail). To improve the security threshold, one can introduce a random variable X = a ⊕ b that contains information about the error position. The introduction of X decreases the quantum part (last part) of the Eq. (4) but not the minimum entropy value of string b (for details see Sec. 5.1 of Ref. [28]). To elaborate on this point, we can divide the quantum system into four subsystems, each two subsystem will correspond to an error and no-error situation for each basis. For basis Z(X) the error and no-error comprise a fraction of E
(1 − E)h 1 − 2E + µ 4 1 − E + Eh E − µ 4 E = H(V ) − h(E)(6)
We can substitute this reconditioned entropy for variable V in the key rate equation of our protocol to obtain a modified key rate equation as
r = 1 − h( 1 6 + 2E 3 ) − h(E).(7)
here, the solution of the equation for positive r is the security threshold, E ≈ 0.15. Thus, the corresponding new bit error rate would be 15% (see Fig. 1 (c)).
We may now analyze the secret-key rate under the condition that protocol is secure against collective attack by Eve. Firstly, we will characterize the initial state ρ n AB in which characterization depends on the threshold QBER for which the protocol does not abort. Thus, ρ n AB is a quantum state ideally to be shared between Alice and Bob only, but is partially available to Eve for performing the collective attack. Let us define Γ as the set of all two-qubit sates σ AB which may be produced after Eve performs the collective attack on the initial state ρ n AB . The attack will be considered successful if it does not leave a detectable trace. In that case, we must have σ ⊗n AB = ρ n AB . However, the attack will not be successful in all cases. In the cases, where it iss not successful it will leave detectable traces and the protocol will be aborted. We are interested in a situation where the protocol is not aborted. To consider the possibility of such a situation, we assume that there exists a protocol (operation) that Eve can use to produce a state σ ⊗n AB = ρ n AB using ancillary qubits and that part of the initial state shared by Alice and Bob which is available to Eve in the channel accessible to her. Now, following [27], we may define a set Γ QBER as the subset of Γ containing all the states σ AB such that the protocol does not abort i.e., if σ AB ∈ Γ QBER , then the secret key is supposed to be generated by the protocol. Renner et al. [27] have proved the following relation using the above conditions to obtain a lower bond and an upper bound on the secret-key rate for any protocol with one-way post-processing.
r ≥ sup c←a inf σ AB ∈Γ QBER (S(c|E) − H(c|b)) .(8)
here r c←a is the rate that can be achieved if the channel c ← a is used for the pre-processing, S(c|E) denotes the von Neumann entropy of c conditioned on Eve's initial state i.e., S(c|E) = S(σ cE ) − S(σ E ). This state σ cE is obtained from the two-qubit state σ AB by taking a purification σ ABE of the Bell diagonal state σ diag AB := O 2 (σ AB ), state σ diag AB has the same diagonal elements as in σ AB with respect to Bell basis. Here, a, b and e are the outcomes of Alice, Bob, and Eve's after the measurement is applied to the first, second and third subsystem of σ ABE . To prove the upper bound of the rate, it is sufficient to consider the collective attack only. The composite system of Alice, Bob and Eve has the product form given by ρ n ABE := σ ⊗n ABE , where σ ABE is a tripartite state. The n−fold product state σ n abE fully specifies the situation where the single state σ abE is obtained when Alice and Bob do the measurement on the state σ ABE (for detailed proof see Sec. IV of Ref. [27]). So the upper bound on the secret key rate is, r(a, b, e) = sup c←a (H(c|e) − H(c|b)) .
This equation implies that if the supremum is taken over all the channels (including both quantum and classical channels) c ← a will be the upper bound on the secret key rate. Now, we analyze our protocol in the context of lower bound and upper bound of the secret key-rate. As before we take n = 1, σ AB = ρ 1 [µ]. It is required to consider a purification |Ψ ABE of the Bell diagonal state O 2 (σ AB ) originated from σ AB that can be written as,
|Ψ ABE := 4 i=1 µ i |ϕ i AB ⊗ |ε i E ,(10)
where |ϕ i AB denotes the Bell states which correspond to the joint system of Alice and Bob 4 and |ε i E denotes some mutually orthogonal states in Eve's system which forms the basis ε E ∈ {|ε 1 E , . . . , |ε 4 E }. It can be easily verified that Alice measures her qubit with Z(X) basis and Bob measures his qubit with Z or X basis with equal probability, resulting in the outcomes of both parties |A and |B , respectively. As an example, here we consider |A ∈ {|0 , |1 } and |B ∈ {|0 , |1 , |+ , |− }. Under this consideration, Eve's state will be |φ A,B , where
|φ 0,0 = 1 √ 2 √ µ 1 |ε 1 E + √ µ 2 |ε 2 E ,|φ 1,1 = 1 √ 2 √ µ 1 |ε 1 E − √ µ 2 |ε 2 E , |φ 0,1 = 1 √ 2 √ µ 3 |ε 3 E + √ µ 4 |ε 4 E , |φ 1,0 = 1 √ 2 √ µ 3 |ε 3 E − √ µ 4 |ε 4 E , |φ 0,+ = 1 2 √ µ 1 |ε 1 E + √ µ 2 |ε 2 E + √ µ 3 |ε 3 E + √ µ 4 |ε 4 E , |φ 0,− = 1 2 √ µ 1 |ε 1 E + √ µ 2 |ε 2 E − √ µ 3 |ε 3 E − √ µ 4 |ε 4 E , |φ 1,+ = 1 2 √ µ 1 |ε 1 E − √ µ 2 |ε 2 E + √ µ 3 |ε 3 E − √ µ 4 |ε 4 E , |φ 1,− = 1 2 − √ µ 1 |ε 1 E + √ µ 2 |ε 2 E + √ µ 3 |ε 3 E − √ µ 4 |ε 4 E .(11)
We are now equipped to compute the density operators of Eve's system for which Alice gets the outcome as 0 and 1, and denote them as σ 0 E and σ 1 E , respectively. Here, we will consider the system that will be accepted by Alice and Bob after classical pre-processing of the protocol, which is given by σ 0 E = 1 3 P |φ 0,0 + P |φ 0,1 + 1 6 P |φ 0,+ + P |φ 0,− and σ 1 E = 1 3 P |φ 1,0 + P |φ 1,1 + 1 6 P |φ 1,+ + P |φ 1,− . We can obtain the state of Eve with respect to the basis |ε i E , where i ∈ {1, · · · , 4} as
σ k E = µ 1 (−1) k √ µ 1 µ 2 0 0 (−1) k √ µ 1 µ 2 µ 2 0 0 0 0 µ 3 (−1) k √ µ 1 µ 2 0 0 (−1) k √ µ 1 µ 2 µ 4 ,(12)
where k ∈ {0, 1}} .
We have already mentioned channel c ← a which provides a noisy version of a. We may consider that Alice uses bit-flip with probability q to make c, i.e., p c|a=0 (1) = p c|a=1 (0) = q. We may now use the following standard relations to simplify the right hand side of Eq. (8)
S(c|E) = S(cE) − S(E) = [H(c) + S(E|c) − S(E)] ,(13)
and
H(c|b) = H(cb) − H(b) = [H(c) + H(b|c) − H(b)] .(14)
Substituting Eq.
4 Alice measures her qubit with Z basis and Bob measures his qubit with Z or X basis with 1 2 probability.
The above substitution will modify Eq. (8) in a manner that will allow us to compute the lower bound of the secret key rate for our protocol. If only Eve's system is used for the calculation of the entropy, there would be only two possibilities, where Alice can have 0 and 1 bit value. At the same time, getting the entropy of E conditioned on the value c, announced by Alice is dependent on bit flip probability. So we have,
S(E|c) = 1 2 S (1 − q)σ 0 E + qσ 1 E + 1 2 S qσ 0 E + (1 − q)σ 1 E , and S(E) = S 1 2 σ 0 E + 1 2 σ 1 E .
Now, we consider Bob's bit string, which he gets from the measurement result of his particle (system) B in the state |Ψ ABE . Intuitively, there must be two equal possibilities for getting the bit value 0 and 1 only when Bob's bit string is considered. In addition, if the conditional entropy of Bob's bit string is to be calculated for the situation, where the noisy version of Alice's bit (c value) the string is provided, then error and no-error probability are to be considered. So we would have
H(b) = 1 and H(b|c) = h[q(1 − E) + (1 − q)E].
Using these expressions, for an optimal choice of the parameter q, we get the positive secret key if E ≤ 0.124 (see Fig. 2 (a)). We get this tolerable limit for error rate under the classical pre-processing i.e., noise introduced by Alice.
Let us now calculate the upper bound of the secret key rate using the Eq. (9). Here again, |φ 0,0 , |φ 0,+ , |φ 1,1 , and |φ 1,− are the states of Eve based on the event that Alice and Bob get the results (0, 0), (0, 0), (1, 1) and (1, 1), respectively. Inherently, we take the best situation for the adversary, Eve when she performs a von Neumann measurement with respect to the projectors along 1 √ 2 (|φ 0,0 +|φ 1,1 ), 1 √ 2 (|φ 0,0 −|φ 1,1 ), 1 √ 2 (|φ 0,+ +|φ 0,+ ), and 1 √ 2 (|φ 0,+ −|φ 0,+ ) 5 , to obtain an outcome e. Now, we can modify Eq. (9) by appropriately applying the above condition as follows,
here
χ(E) = S 1 3 P |φ 0,0 + P |φ 1,1 + 1 6 P |φ 0,+ + P |φ 1,− − 1 2 S (1 − q) 2 3 P |φ 0,0 + 1 3 P |φ 0,+ + q 2 3 P |φ 1,1 + 1 3 P |φ 1,− − 1 2 S q 2 3 P |φ 0,0 + 1 3 P |φ 0,+ + (1 − q) 2 3 P |φ 1,1 + 1 3 P |φ 1,−
is the Holevo quantity [29] that defines the maximum value of mutual information between e and c over all possible measurement scenarios that Eve can perform. This allows to compute the upper bound for the key rate by solving r(a, b, e) = 0. The solution yields an upper bound as E ≥ 0.114 provided that optimal value of q is used (cf. Fig. 2 (c)). 5 The probability of applying the last two projectors operation is half of the first two, i.e., the probability of applying the last (first) two projectors are 1
IV. DISCUSSION
In this paper, we have proposed a new protocol for QKD and a variant of the same. The protocols consume more quantum resources compared to SARG04 or similar protocols, but transmit less classical information in public channel and thus reduces the probability of some side channel attacks. Further, rigorous security analysis of the proposed protocols is performed. We have also calculated the tolerable error limit for the upper and lower bound of the secret key rate under a set of collective attacks. It is shown that by applying a certain type of classical pre-processing, the tolerable error limit can be increased and the same is illustrated through the graphs. Now, before we conclude, we may emphasize some of our important observations. In the seminal paper [27], authors computed density operators of Eve's final state for six-state QKD protocol. Interestingly, for our protocols, we have obtained the same expressions for the density operators describing the final system of Eve in spite of the fact that in our Protocol 2 (1) neither Alice nor Bob (Alice never) discloses the results of the measurements performed by them (her) in the cases where they have (she has) used different bases for preparation and measurement. The reason behind obtaining the same density operators for the system of Eve is that the terms that appear in the density matrix in the cases where basis mismatch happens cancel each other. Further, it is explicitly established that for the proposed protocols, the tolerable error limit of QBER E ≤ 0.124 for the lower bound of key rate and E ≥ 0.114 for the upper bound of key rate if classical pre-processing is used. In our case, the tolerable error limits are expected to decrease in absence of classical pre-processing.
In the practical implementation of cryptography, different kind of errors may happen during the transmission of qubits. Now if we consider QBER > 0, then Eve can try to attack using partial cloning machines [30][31][32]. Acin et al., have shown that honest users of SARG04 can tolerate an error of up to 15% when Eve uses a best-known partial cloning machine. They further found that this value of tolerable error limit is greater than the corresponding tolerable error rate for the BB84 protocol. In our case for QBER > 0, the tolerable error limit is also computed to be 15% (cf. Section III) which is better than the BB84 protocol and its variants.
Security-efficiency trade-off for our protocol: In 2000, Cabello [25] introduced a measure of the efficiency of quantum communication protocols as η = bs qt+bt , where b s is the number of secret bits which can be exchanged by the protocol, q t is the number of qubits interchanged (by quantum channel) in each step of the protocol and b t is the classical bit information exchanged between Alice and Bob via classical channel 6 . We consider the sifting subprotocol of our first protocol for which, the values of the essential parameter are: b s = 0.75, q t = 3, b t = 0.625, and which gives the efficiency as η = 0.2069 with no inherent error. For this specific sifting condition, the basis information will be revealed at end of the protocol which may increase the chance of PNS attack by the powerful Eve. When we consider the sifting subprotocol of the second QKD protocol, the values of the essential parameters are: b s = 0.9375, q t = 3, b t = 0.75 which gives the efficiency as η = 0.25. To apply this more efficient QKD protocol, one has to consider the inherent error probability value of 0.0625, and the exchange of classical information is also more than the previous protocol. We want to stress on the point that the second protocol is more robust against PNS attack compared to the first one, as in this case, Alice and Bob have not revealed the basis information (see only two non-orthogonal state information is announced for the sifting process (M value)). Our two protocols are more efficient than the SARG04 protocol 7 and one can use one of our two protocols as per the requirement for the necessary task.
APPENDIX A
We use these relations to compute the key rate. The conditional probability, P r (B = |i |A = |j ) = P r(B=|i ,A=|j )
= 0 .
0This yields that H(V ) maximizes for µ 4 = E 2 and the corresponding value ofH(V ) becomes 2h(E), where h(E) = −E log 2 E − (1 − E) log 2 (1 − E)is the binary entropy function. The entropy of Bob's measurement result is H(B) = 2 and the conditional entropy of B when A is known is given by H(B|A) threshold (or equivalently maximum tolerable error limit) is the maximum value of E such that the key rate will be positive. Under such conditions, the solution of the Eq. we can obtain E ≈ 0.0314 i.e., 3.14% QBER (see Appendix A for detail). To improve the security threshold one can introduce a new variable Y = j A ⊕ j B , where j A and j B are the basis chosen by Alice and Bob to measure the particles of sequence S B1 and S A , respectively and j A , j B ∈ {0, 1}. Here, we compute the solution for new key rate equation
total number of qubits. After calculating the entropy of four subsystems in the error and no-error scenarios, one can obtain h for error and no error situation and after performing the statistical averaging over the four possible subsystems we obtain,
Figure 1 .
1(Color online) Plot of the secret key rate as a function of quantum bit error rate E: (a) plot to evaluate a maximum tolerable error limit (security threshold) for sequence SB1, solid (black) line and dashed (blue) line illustrate it for the situation with and without the introduction of the new variable Y, respectively, (b) plot to evaluate maximum tolerable error limit (security threshold) for sequence SB2 without introducing new variable X , and (c) plot to evaluate maximum tolerable error limit (security threshold) for sequence SB2 with the introduction of new variable X .
(13) and Eq.(14) into the right hand side of Eq.(8), we can express the entropy difference as follows S(c|E) − H(c|b) = S(E|c) − S(E) − (H(b|c) − H(b)) .
r(a, b, e) = H(c|e) − H(c|b) = H(e|c) − H(e) − [H(b|c) − H(b)] ≤ χ(E) − [H(b|c) − H(b)],
Figure 2 .
2(Color online) Variation of secret key rate with bit-flip probability (q) and QBER (E): (a) lower bound on the secret-key rate of our protocol as function of bit-flip probability and QBER, (b) contour plot for lower bound error limit; QBER vs bit-flip probability, (c) upper bound on the secret-key rate of our protocol as function of bit-flip probability and QBER, and (d) contour plot for upper bound error limit; QBER vs bit-flip probability.
B = |i |A = |j ) log 2 P r(B = |i |A = |j ), (17) P r (B = |0 |A = |0 ) = P r (B = |1 |A = |1 ) = µ1+µ2 2 , P r (B = |1 |A = |0 ) = P r (B = |0 |A = |1 ) = µ3+µ4 2 , P r (B = |+ |A = |+ ) = P r (B = |− |A = |− ) = µ1+µ3 2 , P r (B = |− |A = |+ ) = P r (B = |+ |A = |− ) i = j and i, j ∈ {|0 , |1 , |+ , |− , }. Now, using Eq. (17) we can obtain H(B|A)
Table I .
IThis table describes encoding and decoding rules for the protocol 1 and protocol 2 and also express measurement outcome after classical sifting subprotocolSA
SB1
SB2 Measurement
result of
SB1by Alice
Measurement
result of
SB2by Alice
Probability J value
for P1
Result
determine
by P1
M value
for P2
Result
determine
by P2
|0
|+
|0
Table II .
IITable for mapping between measurement result and determined result by Alice for Protocol 1SA
Measurement result of SB1,
SB2 by Alice
Result determined without J
value
Table III .
IIITable for mapping between measurement result and determined result by Alice for protocol 2SA
Value of M
Measurement result of
SB1by Alice
Measurement result of
SB2by Alice
Result determined
Efficiency is computed using Cabello's definition[25]. In this approach, the cost of transferring qubits is the same as the cost of transferring classical bits and the quantum channel is not too noisy which is not always realistic for long distance communication using present technology.
Here, R := R(a, b) is the set of density operators on the H A ⊗ H B such that the outcomes of a measurement of any ρ ∈ R.
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| [] |
[
"Skill-based Multi-objective Reinforcement Learning of Industrial Robot Tasks with Planning and Knowledge Integration",
"Skill-based Multi-objective Reinforcement Learning of Industrial Robot Tasks with Planning and Knowledge Integration"
] | [
"Matthias Mayr ",
"Faseeh Ahmad ",
"Konstantinos Chatzilygeroudis ",
"Luigi Nardi ",
"Volker Krueger "
] | [] | [] | In modern industrial settings with small batch sizes it should be easy to set up a robot system for a new task. Strategies exist, e.g. the use of skills, but when it comes to handling forces and torques, these systems often fall short. We introduce an approach that provides a combination of task-level planning with targeted learning of scenario-specific parameters for skill-based systems. We propose the following pipeline: the user provides a task goal in the planning language PDDL, then a plan (i.e., a sequence of skills) is generated and the learnable parameters of the skills are automatically identified, and, finally, an operator chooses reward functions and hyperparameters for the learning process. Two aspects of our methodology are critical: (a) learning is tightly integrated with a knowledge framework to support symbolic planning and to provide priors for learning, (b) using multi-objective optimization. This can help to balance key performance indicators (KPIs) such as safety and task performance since they can often affect each other. We adopt a multi-objective Bayesian optimization approach and learn entirely in simulation. We demonstrate the efficacy and versatility of our approach by learning skill parameters for two different contact-rich tasks. We show their successful execution on a real 7-DOF KUKA-iiwa manipulator and outperform the manual parameterization by human robot operators. | 10.1109/robio55434.2022.10011996 | [
"https://arxiv.org/pdf/2203.10033v1.pdf"
] | 247,593,911 | 2203.10033 | 7f32a65ffb1fd865c5d2edf7828817695fc9b799 |
Skill-based Multi-objective Reinforcement Learning of Industrial Robot Tasks with Planning and Knowledge Integration
Matthias Mayr
Faseeh Ahmad
Konstantinos Chatzilygeroudis
Luigi Nardi
Volker Krueger
Skill-based Multi-objective Reinforcement Learning of Industrial Robot Tasks with Planning and Knowledge Integration
In modern industrial settings with small batch sizes it should be easy to set up a robot system for a new task. Strategies exist, e.g. the use of skills, but when it comes to handling forces and torques, these systems often fall short. We introduce an approach that provides a combination of task-level planning with targeted learning of scenario-specific parameters for skill-based systems. We propose the following pipeline: the user provides a task goal in the planning language PDDL, then a plan (i.e., a sequence of skills) is generated and the learnable parameters of the skills are automatically identified, and, finally, an operator chooses reward functions and hyperparameters for the learning process. Two aspects of our methodology are critical: (a) learning is tightly integrated with a knowledge framework to support symbolic planning and to provide priors for learning, (b) using multi-objective optimization. This can help to balance key performance indicators (KPIs) such as safety and task performance since they can often affect each other. We adopt a multi-objective Bayesian optimization approach and learn entirely in simulation. We demonstrate the efficacy and versatility of our approach by learning skill parameters for two different contact-rich tasks. We show their successful execution on a real 7-DOF KUKA-iiwa manipulator and outperform the manual parameterization by human robot operators.
I. INTRODUCTION
Industrial environments with expensive and fragile equipment, safety regulations and frequently changing tasks often have special requirements for the behaviour policies that control a robot: First, the trend in industrial manufacturing is to move to smaller batch sizes and higher flexibility of work stations. Reconfiguration needs to be fast, easy and should minimize downtime. Second, it is important to be able to guarantee the performance as well as safety for material and workers. Therefore, it is crucial to be able to understand what action is performed when and why. Finally, in industrial environments digital twins provide a lot of task-relevant information such as material properties and approximate part locations that the robot behavior policies have to consider.
One way to fulfill these criteria is to use systems based on parameterized skills [1], [2], [3]. These encapsulated abilities realize semantically defined actions such as moving the robot arm, opening a gripper or localizing an object with vision. State-of-the-art skill-based software architectures can not only utilize knowledge, but also automatically generate plans (skill-sequences) for a given task [4], [5]. The skillbased approach is powerful when knowledge can be modeled and formalized explicitly [1], [2]. But it is often limited when it comes to skill parameters of contact-rich tasks that are difficult to reason about. One example are the parameters of a peg insertion search strategy where material properties (e.g. friction) and the robot behavior need to be considered. While it is possible to create a reasoner that follows a set of rules to determine such skill parameters, it is challenging to implement and to maintain.
Another way to handle this is to have operators manually specify and try values for these skill parameters. However, this is a manual process and can be cumbersome.
Finally, it is possible to allow the system to learn by interacting with the environment. However, many policy formulations that allow learning (e.g. artificial neural networks) have deficiencies which make their application in an industrial domain with the abovementioned requirements challenging. Primarily during the learning phase, dangerous behaviors can be produced and even state-of-the-art RL methods need hundreds of hours of interaction time [6]. Learning in simulation can help to reduce downtime and dangers for the real system. But many policy formulations are black boxes for operators and it can be hard to predict their behavior, which could hinder the trust to the system [7] Moreover, the simulation-to-reality gap [8], [9] is bigger in lower-level control states (i.e. torques), and policies working directly on raw control states struggle to transfer learned behaviors to the real systems [6]. Our policy formulation consisting of behavior trees (BT) with a motion generator [10] has shown to be able to learn interpretable and robust behaviors [11].
The formulation of a learning problem for a given task is often not easy and becomes more challenging if factors such as safety or impact on the workstation environment need to be considered. Multi-objective optimization techniques allow to specify multiple objectives and optimize for them concurrently. This allows operators to select from solutions that are optimal for a certain trade-off between the objectives (usually represented as a set of Pareto-optimal solutions). In order to learn sample-efficient and to support the large variety of skill implementations as well as scenarios, we use gradient-free Bayesian optimization as an optimization method. In this paper we make the following contributions: 1) We introduce a new method which seamlessly integrates symbolic planning and reinforcement learning for skill-based systems to learn interpretable policies for a given task. 2) A Bayesian multi-objective treatment of the task learning problem, which includes the operator through easy specification of problem constraints and task objectives (KPIs); the set of Pareto-optimal solutions is presented to the operator and their behavior can be inspected in simulation and executed on the real system. 3) We demonstrate our approach on two contact-rich tasks, a pushing task and a peg-in-hole task. We compare it to the outcome of the planner without reasoning, randomly sampled parameter sets from the search space and the manual real-world parameterization process of robot operators. In both tasks our approach delivered solutions that even outperform the ones found by the manual search of human robot operators.
II. RELATED WORK
A. Skill-based Systems
Skill-based systems are one way to support a quick setup of a robot system for a new task and to allow re-use of capabilities. There are multiple definitions of the term skills in the literature. Some define it as a pure motion skills [12] or "hybrid motions or tool operations" [13]. Other work has a broader skill definition [1], [2], [3], [4], [5], [14], [15]. In this formulation, skills can be arbitrary capabilities that change the state of the world and have pre-and postconditions. Their implementation can include motion skills, but also proficiencies such as vision-based localization of objects. In [16] skills are "high-level reusable robot capabilities, with the goal to reduce the complexity and time consumption of robot programming". However, compared to [3] and [14] they do not use pre-and postconditions. In [17], an integrated system for manual creation of task plans is presented. It shares the usage of BTs with our approach.
Task planners are used in [1], [2], [4], [5], [14], [18], [16], [13] while [17] lacks such a capability.
In [16] it is suggested that "Machine learning can be performed on the motion level, in terms of adaptation, or can take the form of structured learning on a task/error specification level". However, none of the reviewed work offers a combination task-level planning with learning.
B. Policy Representation and Learning
An important decision to make when working with manipulators is the type of policy representation and on which level it interfaces with the robot. The latter can strongly influence the learning speed and the quality of the obtained solutions [19], [20]. These choices also influence the form of priors that can be defined and how they are defined [6]. Not many policies combine the aforementioned properties of being a) interpretable, b) paramterizable for the task at hand and c) allow learning or improvement.
The commonly used policy representations for learning systems include radial basis function networks [21], dynamical movement primitives [22], [23] and feed-forward neural networks [21], [24]. In recent years deep artificial neural networks (ANN) seem to become a popular policy. All of them have in common that their final representation can be difficult to interpret. Even if a policy only sets a target pose for the robot to reach, it can be problematic to know how it reacts in all parts of the state space. In contrast to that, [11] suggests to learn interpretable policies based on behavior trees [10] that work explicitly in end-effector space.
C. Planning and Learning
Symbolic planning is combined with learning in [25], [26], [27], [28]. In [25], the PLANQ-Learning algorithm uses a symbolic planner to shape the reward function based on the conditions defined which are then used by the Qlearner to get an optimal policy with good results on the grid domain. [26] uses the combined symbolic planner with reinforcement learning (RL) in a hierarchical framework to solve complex visual interactive question answering tasks. PEORL [27] integrates symbolic planning and hierarchical reinforcement learning (HRL) to improve performance by achieving rapid policy search and robust symbolic planning in the taxi domain and grid world. SPOTTER [28] uses RL to allow the planning agent to discover the new operators required to complete tasks in Grid World. In contrast to all these approaches, our approach aims towards real-life robotic tasks in an Industry 4.0 setting where a digital twin is available.
In [29], the authors combine symbolic planning with behavior trees (BT) to solve blocks world tasks with a robot manipulator. They use modified Genetic Programming (GP) [30] to learn the structure of the BT. In our approach, we focus on learning the parameters of the skills in the BT and utilize a symbolic planner to obtain the structure of the BT.
III. APPROACH
Our approach consists of two main components that interact in different stages of the learning pipeline: First, SkiROS [14], a skill-based framework for ROS, which represents the implemented skills with BTs, hosts the world model (digital twin), and interacts with the planner. SkiROS is also (2) a learning scenario for the plan is created;
(3) rewards and hyperparameters are specified; (4) learning is conducted using the skills and the information in the world model; (5) after policy learning, the operator can choose which policies to execute on the real system (6).
used to execute BTs while learning and to perform tasks on the real system. Second, the learning framework that provides the simulation, the integration with the policy optimizer as well as the reward function definition and calculation. The architecture of the system and the workflow is shown in Figure 2: (1) an operator enters the task goal into a GUI;
(2) a plan with the respective learning scenario configuration is generated; (3) an operator complements the scenario with objectives and reward functions; (4) learning is conducted in simulation using the skills and information from the world model; (5) in the multi-objective optimization case, a set of Pareto-optimal solutions is generated and presented to the operator; finally, (6) the operator can select a good solution from this set given the desired trade-off between KPIs and execute it on the real system.
A. Behavior Trees
A Behavior Tree (BT) [31] is a formalism for plan representation and execution. Like [32], [33], we define it as a directed acyclic graph G(V, W ) with |V | nodes and |W | edges. It consists of control flow nodes (processors), and execution nodes. The four basic types of control flow nodes are 1) sequence, 2) selector, 3) parallel and 4) decorator [33]. A BT always has one initial node with no parents, defined as Root, and one or more nodes with no children, called leaves. When executing a BT, the Root node periodically injects a tick signal into the tree. The signal is routed through "SetMGGoalPose" "ChangeStiffness" "ApplyForce" "skiros:contain", "Gripper", "Peg" ¬ "skiros:at", "Peg", "Box" "skiros:at", "Gripper", "ApproachPose" "PegInsertion" <||FS> "skiros:at", "Peg", "Box" ¬ "skiros:at", "Gripper", "ApproachPose" "skiros:at", "Gripper", "Box" "skiros:at", "Gripper", "StartPose" "GoToLinear" <||FS> ¬ "skiros:at", "Gripper", "StartPose" "skiros:at", "Gripper", "GoalPose" "SetMGGoalPose" "EEPoseDistance" "EEPoseDistance" →* Ø "OverlayMotion" Fig. 3. The BT of the generated plan for the peg insertion task in eBT format [32]. Each node has conditions or pre-conditions shown in the upper half and effects or post-conditions shown in the lower half. The serial start control flow node (→ * ) executes in a sequence and remembers the successes. The skills have a parallel-first-success processor (<||F S>).
the branches according to the implementation of the control flow nodes and the return statements of their children. By convention, the signal propagation goes from left to right. The sequence node corresponds to a logical AND: it succeeds if all children succeed and fails if one child fails. The selector, also called fallback node, represents a logical OR: If one child succeeds, the remaining ones will not be ticked. It fails only if all children fail. The parallel control flow node forwards ticks to all children and fails if one fails. A decorator allows to define custom functions. Implementations like extended Behavior Trees (eBT) in SkiROS [32] add custom processors such as parallel-firstsuccess that succeeds if one of the parallel running children succeeds. Leaves of the BT are the execution nodes that, when ticked, execute one cycle and output one of the three signals: success, failure or running. In particular, execution nodes subdivide into 1) action and 2) condition nodes. An action performs its operation iteratively at every tick, returning running while it is not done, and success or failure otherwise. A condition performs an instantaneous operation and returns always success or failure and never running. An example of the BT for the peg insertion task is in Fig. 3.
B. Planning and Knowledge Integration
The Planning Domain Definition Language (PDDL) [34], [4] is used to formulate the planning problem. We use the SkiROS [18] framework that automatically translates a task into a PDDL planning problem by generating domain description and problem instance using the world model. We then use the semantic world model (WM) from SkiROS [14] as the knowledge integration framework.
Actions and fluents are obtained by utilizing the predicates that have pre-or post-conditions in the world model. For the problem instance, the objects (robots, arms, grippers, boxes, poses, etc.) in the scene and their initial states (as far as they are known) are used. After getting the necessary domain description and the problem instance SkiROS calls the planner. The goal of the planner is to return a sequence of skills that can achieve the goal conditions of the task. The individual skills are partially parameterized with explicit data from the WM. The WM is aware of the skill parameters that need to be learned for the task at hand and they are automatically identified in the skill sequence.
C. Policy Optimization
In order to optimize for policy parameters, we adopt the policy search formulation [21], [6], [24]. We formulate a dynamical system in the form:
x t+1 = x t + M (x t , u t , φ R ),(1)
with continuous-valued states x ∈ R E and actions u ∈ R U . The transition dynamics are modeled by a simulation of the robot and the environment M (x t , u t , φ R ). They are influenced by the domain randomization parameters φ R . The goal is to find a policy π, u = π(x|θ) with policy parameters θ such that we maximize the expected long-term reward when executing the policy for T time steps:
J(θ) = E T t=1 r(x t , u t )|θ ,(2)
where r(x t , u t ) is the immediate reward for being in state x and executing action u at time step t. The discrete switching of branches in the BT and most skills are not differentiable. Therefore, we frame the optimization in Eq. (2) as a blackbox optimization and pursue the maximization of the reward function J(θ) only by using measurements of the function. The optimal reward function to solve the task is generally unknown, and a combination of reward functions is usually used. In the RL literature, this is usually done with a weighted average, that is, r(x t , u t ) = i w i r i (x t , u t ). In this paper, we chose not to use a weighted average of reward functions that represent different objectives (as the optimal combination of weights cannot always be found [35]), but optimize for all objectives concurrently (Sec. III-E) using Bayesian Optimization.
D. Bayesian Optimization
We consider the problem of finding a global minimizer (or maximizer) of an unknown (black-box) objective function f : s * ∈ arg min s∈S f (s), where S is some input design space of interest in D dimensions. The problem addressed in this paper is the optimization of a (possibly noisy) function f : S → R with lower and upper bounds on the problem variables. The variables defining S can be real (continuous), integer, ordinal, and categorical as in [36]. We assume that the function f is in general expensive to evaluate and that the derivatives of f are in general not available. The function f is called black box because we cannot access other information than the output y given an input value s.
This problem can be tackled using Bayesian Optimization (BO) [37]. BO approximates s * with a sequence of evaluations, y 1 , y 2 , . . . , y t at s 1 , s 2 , . . . , s t ∈ S, which maximizes an utility metric, with each new s t+1 depending on the previous function values. BO achieves this by building a probabilistic surrogate model on f based on the set of evaluated points {(s i , y i )} t i=1 . At each iteration, a new point is selected and evaluated based on the surrogate model which is then updated to include the new point (s t+1 , y t+1 ). BO defines an utility metric called the acquisition function, which gives a score to each s ∈ S by balancing the predicted value and the uncertainty of the prediction for s. The maximization of the acquisition function guides the sequential decision making process and the exploration versus exploitation tradeoff: the highest score identifies the next point s t+1 to evaluate. BO is a statistically efficient black-box optimization approach when considering the number of necessary function evaluations [38]. It is, thus, especially well-suited to solve problems where we can only perform a limited number of function evaluations, such as the ones found in robotics.
We use the implementation of BO found in HyperMapper [36], [39], [40], [41]. Our implementation selects the Expected Improvement (EI) acquisition function [42] and we use uniform random samples as a warm-up strategy before starting the optimization.
E. Multi-objective Optimization
Let us consider a multiple objectives minimization (or maximization) over S in D dimensions. We define f : S → R p as our vector of objective functions f = (f 1 , . . . , f p ), taking s as input, and evaluating y = f (s) + , where is a Gaussian noise term. Our goal is to identify the Pareto frontier of f , that is, the set Γ ⊆ S of points which are not dominated by any other point, i.e., the maximally desirable s which cannot be optimized further for any single objective without making a trade-off. Formally, we consider the partial order in R p : y ≺ y iff ∀i ∈ [p], y i y i and ∃j, y j < y j , and define the induced order on S: s ≺ s iff f (s) ≺ f (s ). The set of minimal points in this order is the Pareto-optimal set Γ = {s ∈ S : s such that s ≺ s}. We aim to identify Γ with the fewest possible function evaluations using BO. For this purpose we use the HyperMapper multiobjective Bayesian optimization which is based on random scalarizations [43].
F. Motion Generator and Robot Control
The arm motions are controlled in end-effector space by a Cartesian impedance controller. The time varying reference or attractor point of the end effector x d is governed by a motion generator (MG). Given the joint configuration q, we can calculate the end-effector pose x ee using forward kinematics and obtain an error term x e = x ee −x d . Together with the joint velocitiesq, the Jacobian J(q), the configurable stiffness and damping matrices K d and D d , the task control is formulated as τ c = J T (q) (−K d x e − D d J(q)q) . Additionally, the task control can be overlayed with commanded generalized forces and torques F ext = (f x f y f z τ x τ y τ z ): τ ext = J T (q)F ext . We utilize the integration introduced in [10] and used in [11], which proposes to parameterize the MG with movement skills from the BT. The reference pose is shaped by 1) a linear trajectory to a goal point and 2) overlay motions that can be added to the reference pose as discussed in [10], [11]. E.g. an Archimedes spiral for search.
To make it compliant with the dynamical system in Eq. (1), a new reference configuration of the controller is only generated at every time step t. It includes the reference pose, stiffnesses, applied wrench and forms the action u with a dimension of U = 19. The stiffness and applied force are changed gradually at every time step t to ensure a smooth motion. The state space consists of joint positions and joint velocities and is E = 14 dimensional. Direct control of the torques of a robot arm requires high update rates and we control the robot arm at 500 Hz based on the current action u, but continuously updated values for q andq. Therefore, from the perspective of Eq. (1), the controller is to be seen as part of the model M (x t , u t ).
We assume a human-robot collaborative workspace with fragile objects. Therefore, the stiffnesses and applied forces are to be kept to a minimum and less accuracy than e.g. high-gain position-controlled solutions is to be expected.
IV. EXPERIMENTS
In our experiments we use a set of pre-defined skills that are part of a skill library. In order to solve a task, the planner determines a sequence that can achieve the goal condition of the task. This skill sequence is also automatically parameterized to the extend possible, e.g. the goal pose of a movement. We evaluate our system in two contact-rich scenarios that are shown in Fig. 1: A) pushing an object with uneven weight distribution to a goal pose and B) inserting a peg in a hole with a 1.5 mm larger radius. Pure planning-based solutions for both these tasks have a poor performance in reality (Fig. 5).
As a baseline we invited six robot operators to manually parameterize the skills for the tasks. Their main objective is to find a parameter set that robustly solves the task. As an additional objective they were asked to minimize the impact of the robot arm and its tool on the environment as long as it does not affect the first objective.
The robot arm used for the physical evaluation is a 7degree-of-freedom (DOF) KUKA iiwa arm controlled by a Cartesian impedance controller (Sec. III-F).
A. Reward Functions
For each task, we utilize a set of reward functions parameterized for the learning scenario configuration. Each configured reward has an assigned objective and can be weighted against other rewards. Each experiment uses a subset of the following reward functions: 1) Task completion: A fixed reward is assigned when the BT returns success upon task completion.
2) End-effector distance to a box: We use a localized reward to attract the end effector towards the goal location r h (x) = (2 (d(p ee,x , p h ) + d o )) −1 , where d o is the distance offset and d(p ee,x , p h ) is the shortest distance function between the end effector and the box.
3) Applied wrench: This reward calculates the cumulative forces applied by the end effector on the environment.
Reward functions 4-6 share a common operation of computing an exponential function of the calculated metric to obtain the reward as used in ( [44], [24]
) r(d m ) = exp − 1 2σ 2 w (d m + d o ) ,
where σ w is a configurable width, d o is a distance offset and d m is the input metric. 4) End-effector distance to a goal: This reward uses distance between the end effectors current pose and goal pose to calculate the input metric d ee,g = p ee,x − p g 5) End-effector-reference-position distance: This reward uses the distance between the end effectors reference pose (Sec. III-F) and its current pose to calculate the input metric d ee,d = p ee,x − x d 6) Object-pose divergence: This reward uses the translational and angular distance between the object's goal pose and its current pose.
B. Push Task
The push task starts by specifying the goal in the SkiROS Graphical User Interface (GUI) as:
(skiros:at skiros:ObjectToBePushed-1 skiros:ObjectGoalPose-1). SkiROS calls the planner to generate a plan given all the available skills. The plan consists of two skills: 1) GoT oLinear skill and 2) P ush skill. The first skill moves the end effector from its current location to the approach pose of the object. This approach pose is defined in the WM and needs to be reached before interacting with the object. The push skill then moves the end effector to the object's geometric centre with an optional offset in the horizontal (x) and (y) directions. Once the end effector reaches it, the motion generator executes a straight line to the (modified) target location.
The push task is formulated as a multi-objective task. It also has two objectives, 1) success and 2) applied force. The first objective has three associated rewards: 1) object position difference from goal position, 2) object orientation difference from goal orientation, and 3) end-effector distance to the goal location. The second objective accumulates the Cartesian distance between the end-effector reference pose and the actual end-effector pose as a measure of the force applied by the controller. The learnable parameters in this task are offsets in the horizontal (x) and (y) direction of both the push skill's start and goal locations. An offset of the start location allows the robot to push from a particular point from the side of the object. Together with the offsets on the goal position, these learnable parameters collectively define the trajectory of the push.
The object to be pushed has a height of 0.07 m and is an orthogonal triangle in the horizontal dimensions (x) and 5 deg, respectively. We learn for 400 iterations and repeat the experiment 10 times. In order to obtain solutions that are robust enough to translate to the real system, we apply domain randomization. Each parameter set is evaluated in 7 worlds. Each execution uniformly samples one out of the four start positions for the robot arm. Furthermore, we vary the location of the object and the goal in the horizontal (x) and (y) directions by sampling from a Gaussian distribution with a standard deviation of 7 mm.
We compare the learned solutions with (a) the outcome of a direct planner solution without any offset on the start and goal pose while pushing, (b) ten sets of random parameters from the search space and (c) the policies that are parameterized by the robot operators. We evaluated on the four start configurations used for learning as well as on two additional unknown ones. The results are shown in Fig. 5a.
The results of a multi-objective optimization are parameters found along a Pareto front (Sec. III-E, see Fig. 4). It contained 8.3 points on average, of which some minimize the impact on the environment to an extent that the push is not successful. An operator can choose a solution that is a good compromise between the success of the task on the real system and the force applied on the environment. The performance of one of the solutions that existed on the Pareto front is shown in Fig. 5.
Furthermore, we asked six robot operators to find values for the learnable parameters of the skill sequences. They were given the same start positions used for learning and were given a script to reset the arm to a start position of their choice. They could experiment with the system until they decided that their parameter set fulfills the criteria. Their final parameter set that was also evaluated on the known and unknown start configurations. On average the operators spent (16.3 ± 6.4) min and executed 11.1 ± 3.0 trials on the system to configure this task. Four out of the six operators found solutions that achieved the task from every start state. However, two of the operators' final parameters only achieved success rates of 50 % and 16.66 %.
C. Peg-in-Hole Task
The PDDL goal of the peg insertion task is (skiros:at skiros:Peg-1 skiros:BoxWithHole-1). The BT that is generated by the planner is shown in Fig. 3 and uses two skills: 1) GoT oLinear skill and 2) P egInsertion skill. The first skill moves the end effector from its current location to the approach pose of the hole. Once it is reached, the peg insertion procedure starts.
The P egInsertion skill starts when the end effector reaches the approach pose of the box. It uses four separate SkiROS primitive skills to 1) set the stiffness of the end effector to zero in (z) direction, 2) apply a downward force in (z) direction, 3) configure the center of the box as a goal and 4) additionally apply an overlaying circular search motion on top of the reference pose of the end effector as described in [11]. The BT returns success only if the peg is inserted into the box hole by more than 0.01 m.
We model the peg insertion as a multi-objective and multi-reward task. There are two objectives of the task, 1) successful insertion and 2) applied force. To assess the efficacy of the first objective, we use three rewards, 1) success of the BT, 2) peg distance to the hole, and 3) peg distance to the box. For the second objective, we use a single reward that measures the total force applied by the peg. There are three learnable parameters in this task, 1) downward force applied by the robot arm, 2) radius of the overlay search motion and 3) path velocity of the overlay search motion.
We learn for 400 iterations in the simulation and repeat this experiment 10 times. To increase the robustness of the solutions we use domain randomization and evaluate each parameter configuration in 7 worlds. We vary the location of the box by sampling from a Gaussian distribution with a standard deviation of 7 mm and uniformly sample one out of 5 start configurations of the robot arm. We compare the performance of the learned policies with (1) the outcome of the planner without a parameterized search motion, (2) randomly chosen parameter configurations from the parameter search space used for learning and (3) policies that are parameterized by human operators (see Fig. 5b).
The learned Pareto-optimal configurations consist of 6.1 points on average. We evaluated the insertion success using the 5 known and additional 10 unknown start configurations of the robot (Fig. 5b).
To find policies for this task, the human operators took (31.8 ± 10.9) min and executed 39 ± 14 trials on the system. However, compared to the randomly sampled policies the average insertion rate only increased from 41 % to 52.2 %. This is much lower than the average insertion rate of 96 % of the best learned policies as shown in box four, Fig. 5b. Furthermore, the average force that was chosen by the operators compared to the learned policies was 16.6 % higher. Finally, the successful insertions by the learned policies were also 18.1 % faster. Therefore, the learned policies outperformed P l a n n i n g R a n the human operators in both objectives while also producing more reliable results.
V. CONCLUSION
In this paper we proposed a method for effectively combining task-level planning with learning to solve industrial contact-rich tasks. Our method leverages prior information and planning to acquire explicit knowledge about the task, whereas it utilizes learning to capture the tacit knowledge, i.e., the knowledge that is hard to formalize and which can only be captured through actual interaction. We utilize behavior trees as an interpretable policy representation that is suitable for learning and leverage domain randomization for learning in simulation. Finally, we formulate a multiobjective optimization scheme so that (1) we handle conflicting rewards adequately, and (2) an operator can choose a policy from the Pareto front and thus actively participate in the learning process.
We evaluated our method on two scenarios using a real KUKA 7-DOF manipulator: (a) a pushing task, and (b) a peg insertion task. Both tasks are contact-rich and naïve planning fails to solve them. The approach was able to outperform the baselines including the manual parameterization by robot operators.
For future work we are looking into multi-fidelity learning that can leverage a small amount of executions on the real system to complement the learning in simulation. Furthermore, the use of parameter priors for the optimum seems a promising direction to guide the policy search and make it more efficient. This work was partially supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by Knut and Alice Wallenberg Foundation. This research was also supported in part by affiliate members and other supporters of the Stanford DAWN project-Ant Financial, Facebook, Google, InfoSys, Teradata, NEC, and VMware.
Fig. 1 .
1The robot setup used for the experiments. Wooden boards indicate the start location for the push task. The goal is the corner between the fixture and the box with the hole for the peg task.
Fig. 2 .
2The architecture of the system that depicts the pipeline: (1) The operator enters the goal state;
Fig. 4 .
4(y). It has a length of 0.15 m and 0.3 m and it weights 2.5 kg. For this task we use a square-shaped peg for pushing with a side length of 0.07 m and a height of 0.05 m. Start and goal locations are ≈0.43 m apart and are rotated by 26 deg. We define success if the translational and rotational difference of the object w.r.t the goal is less than 0.01 m and Pareto front of the push task. Each experiment has a different color and each point represents a Pareto-optimal solution. It shows that higher rewards for pushing require higher interaction forces with the environment.
Fig. 5 .
5The success rates of both experiments. The box plots show the median (black line) and interquartile range (25 th and 75 th percentile); the lines extend to the most extreme data points not considered outliers, and outliers are plotted individually. The number of stars indicates that the pvalue of the Mann-Whitney U test is less than 0.1, 0.05, 0.01 and 0.001 respectively.
Fig. 6 .
6Pareto front of the peg task. Each experiment has a different color. The goal is to maximize insertion reward while minimizing the interaction forces.APPENDIXThe implementation and the supplemental video are available at:https://sites.google.com/ulund.org/SkiREIL ACKNOWLEDGEMENT We thank Alexander Durr, Elin Anna Topp, Francesco Rovida and Jacek Malec for the interesting discussions and the constructive feedback.
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"Jianguo Zhang [email protected] \nCRIPAC&NLPR\nCASIA\n\n\nComputing\nSchool of Science and Engineering\nUniversity of Dundee\nUnited Kingdom\n",
"Kaiqi Huang [email protected] \nCRIPAC&NLPR\nCASIA\n\n\nUniversity of Chinese Academy of Sciences\n\n\nCAS Center for Excellence in Brain Science and Intelligence Technology\n\n"
] | [
"CRIPAC&NLPR\nCASIA\n",
"CRIPAC&NLPR\nCASIA\n",
"Computing\nSchool of Science and Engineering\nUniversity of Dundee\nUnited Kingdom",
"CRIPAC&NLPR\nCASIA\n",
"University of Chinese Academy of Sciences\n",
"CAS Center for Excellence in Brain Science and Intelligence Technology\n"
] | [] | Person re-identification (ReID) focuses on identifying people across different scenes in video surveillance, which is usually formulated as a binary classification task or a ranking task in current person ReID approaches. In this paper, we take both tasks into account and propose a multi-task deep network (MTDnet) that makes use of their own advantages and jointly optimize the two tasks simultaneously for person ReID. To the best of our knowledge, we are the first to integrate both tasks in one network to solve the person ReID. We show that our proposed architecture significantly boosts the performance. Furthermore, deep architecture in general requires a sufficient dataset for training, which is usually not met in person ReID. To cope with this situation, we further extend the MTDnet and propose a cross-domain architecture that is capable of using an auxiliary set to assist training on small target sets. In the experiments, our approach outperforms most of existing person ReID algorithms on representative datasets including CUHK03, CUHK01, VIPeR, iLIDS and PRID2011, which clearly demonstrates the effectiveness of the proposed approach. | 10.1609/aaai.v31i1.11201 | [
"https://arxiv.org/pdf/1607.05369v3.pdf"
] | 14,844,989 | 1607.05369 | 76ad6daa899a8657c9c17480e5fc440fda53acec |
A Multi-task Deep Network for Person Re-identification
Weihua Chen
CRIPAC&NLPR
CASIA
Xiaotang Chen
Jianguo Zhang [email protected]
CRIPAC&NLPR
CASIA
Computing
School of Science and Engineering
University of Dundee
United Kingdom
Kaiqi Huang [email protected]
CRIPAC&NLPR
CASIA
University of Chinese Academy of Sciences
CAS Center for Excellence in Brain Science and Intelligence Technology
A Multi-task Deep Network for Person Re-identification
Person re-identification (ReID) focuses on identifying people across different scenes in video surveillance, which is usually formulated as a binary classification task or a ranking task in current person ReID approaches. In this paper, we take both tasks into account and propose a multi-task deep network (MTDnet) that makes use of their own advantages and jointly optimize the two tasks simultaneously for person ReID. To the best of our knowledge, we are the first to integrate both tasks in one network to solve the person ReID. We show that our proposed architecture significantly boosts the performance. Furthermore, deep architecture in general requires a sufficient dataset for training, which is usually not met in person ReID. To cope with this situation, we further extend the MTDnet and propose a cross-domain architecture that is capable of using an auxiliary set to assist training on small target sets. In the experiments, our approach outperforms most of existing person ReID algorithms on representative datasets including CUHK03, CUHK01, VIPeR, iLIDS and PRID2011, which clearly demonstrates the effectiveness of the proposed approach.
Introduction
Person re-identification (ReID) is an important task in wide area video surveillance. The key challenge is the large appearance variations, usually caused by the significant changes in human body poses, illumination and camera views. It has many applications, such as inter-camera pedestrian tracking and human retrieval.
Recently, deep learning approaches Ahmed, Jones, and Marks 2015;) are successfully employed in person ReID with significant performance, especially on large datasets, such as CUHK03. Most deep learning methods Yi, Lei, and Li 2014;Ahmed, Jones, and Marks 2015) solve the problem as a binary classification issue and adopt a classification loss (e. g. a softmax loss) to train their models. The core behind these approaches is to learn identifiable features for each pair for classification. The binary classification loss is usually designed to require all positive pairs should hold smaller distances than all negative pairs. However, in person ReID, we don't have to require all positive pairs holding smaller Copyright c 2017, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. distances than all negative pairs regardless of query images. Instead, what we want is for each query image, its positive pairs have smaller distances than its negative ones. Therefore, in some cases 1 , the application of binary classification loss may lead the learned model to an undesired locally optimal solution, which is elaborated as below.
The example is shown in Fig. 1 (a). Case 1 and 2 illustrate two projected distributions of scores obtained by trained binary classifiers. For each pair sample, the score underneath denotes the similarity probability between its two images. Query:X indicates where an image from person X is used as a query image (the left image in a pair). For example, Query:A means an image from person A is used as a query image. Green-coloured rectangle indicates a positive pair, and red rectangle for the negative pair. In Case 1, it is evident that for each query image (w.r.t one particular person), we can get the correct rank-1 match, i. e. two images within its positive pairs always hold larger similarity score than those within its negative pairs. However, in this case it is very difficult for a classifier to determine a suitable threshold to get a low misclassification cost (e. g. less than two misclassified samples). On the contrary in Case 2, where the vertical dashed line denotes the decision threshold learned by the classifier, the classifier has a lower misclassification rate. As a result, a binary classifier will favor Case 2 rather than Case 1, as the classification loss in Case 2 will be lower than that in Case 1. But in ReID, we prefer Case 1, which outputs correct ranking results for all of the three persons, rather than Case 2 that contains a false rank-1 result (highlighted in an orange circle). Case 2 could be potentially rectified by a ranking loss.
As person ReID commonly uses the Cumulative Matching Characteristic (CMC) curve for performance evaluation which follows rank-n criteria, some deep learning approaches (Ding et al. 2015;Chen, Guo, and Lai 2016;Cheng et al. 2016) begin to treat the person ReID as a ranking task, similar to image retrieval, and apply a ranking loss (e. g. a triplet loss) to address the problem. The main purpose is to keep the positive pairs maintaining shorter relative distances in the projected space. However, the person ReID differs from image retrieval in that person ReID needs to identify the same person across different scenes (i. e. , a task of predicting positive and negative pairs, focusing on identifiable feature learning, and a positive pair is not necessarily the most similar pair in appearance). Ranking-based approaches are sensitive to their similarity measurements. The current measurements (e. g. the Euclidean distance in the triplet loss) care more about the similarity to query images in appearance. In the projection space obtained by a model trained on the triplet loss, it's very challenging to find out a true positive which holds a less similar appearance. As shown in Fig. 1 (b), there are three query images. Each has a ranking list returned by a ranking loss, and the left-most is the most similar one to the query. The green rectangle indicates the positive pair (ground truth). We can observe that the image ranked first w.r.t each query image is a mismatched image but holding a more similar appearance to the query image than the matched does.
In the person ReID, either the binary classification loss or the ranking loss has its own strengths and weaknesses. As two tasks handle the person ReID from different aspects, we take both of them into account and build a more comprehensive person ReID algorithm. In our method, two tasks are jointly optimized in one deep network simultaneously. We set the binary classification loss and the ranking loss on different layers according their own advantages. The ranking loss encourages a relative distance constraint, while the classification loss seeks to learn discriminative features for each pair during the similarity measurement. As the classification task focuses on feature of pairs, we import the joint feature maps to represent the relationships of paired person images.
Meanwhile, deep learning approaches, such as convolutional neural networks (CNN), benefit a lot from a large scale dataset (e. g. ImageNet). However, this is not the case in person ReID. Since manually labeling image pairs is tedious and time-consuming, most of current ReID datasets are often of limited sizes, e. g. CUHK01 (Li, Zhao, and Wang 2012), VIPeR (Gray, Brennan, and Tao 2007), iLIDS (Zheng, Gong, and Xiang 2009) and PRID2011 (Hirzer et al. 2011). It could hinder the attempts to maximize the learning potential of our proposed network on each of those datasets. This case can be migrated by using some auxiliary datasets. However, the variations across camera views are different from dataset to dataset. As a consequence, the data of the auxiliary dataset can't be directly used to train models on small datasets. In this paper, the problem is considered as a semi-supervised cross-domain issue (Ganin and Lempitsky 2015). The target domain is the small dataset that contains only a few samples and the source domain is an auxiliary dataset which is large enough for training CNN models. As person ReID can be considered as a binary classification problem, our purpose is to keep the samples of the same class in different domains closer. A cross-domain architecture is further proposed to minimize the difference of the joint feature maps in two datasets, which are belonged to the same class of pairs (i. e. , positive pair and negative pair), and utilize the joint feature maps of the auxiliary dataset to fine tune those of small datasets during the training process. In this case, the joint feature maps of small datasets are improved with the data of the auxiliary dataset and boost the ReID performance on smaller target datasets.
In summary, our contributions are three-fold: 1) a novel multi-task deep network for person ReID, where two tasks focuses on different layers and are jointly optimized simultaneously for person ReID; 2) a cross-domain architecture based on the joint feature maps to handle the challenge of limited training set; 3) a comprehensive evaluation of our methods on five datasets, and showing the superior performance over most of state-of-the-art methods.
Related work
Most of existing methods in person ReID focus on either feature extraction (Zhao, Ouyang, and Wang 2014;Su et al. 2015;Matsukawa et al. 2016), or similarity measurement (Li and Wang 2013;Liao and Li 2015). Person image descriptors commonly used include color histogram (Koestinger et al. 2012;Li and Wang 2013;Xiong et al. 2014), local binary patterns (Koestinger et al. 2012), Gabor features (Li and Wang 2013), and etc., which show certain robustness to the variations of poses, illumination and viewpoints. For similarity measurement, many metric learning approaches are proposed to learn a suitable metric, such as locally adaptive decision functions , local fisher discriminant analysis (Pedagadi et al. 2013), cross-view quadratic discriminant analysis (Liao et al. 2015), and etc. A few of them (Xiong et al. 2014;Paisitkriangkrai, Shen, and Hengel 2015) learn a combination of multiple metrics. However, manually crafting features and metrics require empirical knowledge, and are usually not optimal to cope with large intra-person variations.
Since feature extraction and similarity measurement are independent, the performance of the whole system is often suboptimal compared with an end-to-end system using CNN that can be globally optimized via back-propagation. With the development of deep learning and increasing availability of datasets, the handcrafted features and metrics struggle to keep top performance widely, especially on large scale datasets. Alternatively, deep learning is attempted for person ReID to automatically learn features and metrics (Li et al. 2014 each identity. VIPeR (Gray, Brennan, and Tao 2007) and PRID2011 (Hirzer et al. 2011) datasets have only two images for each person. The lack of training samples may make the multi-class classification less effective. Xiao et al. (Xiao et al. 2016) achieve a good performance, but it combines all current datasets together as its training data.
Our network considers two tasks (the classification loss and the ranking loss) simultaneously and takes both of their advantages during training. Wang et al. also discuss both classification and ranking losses, however, it trains two losses separately and combines them on the score level. In this paper, we jointly optimize two tasks simultaneously in our network.
It is worth noting that none of the works above in person ReID seeks to solve the problem of "learning a deep net on a small dataset" which is a typical case in person ReID. This paper addresses this issue by proposing a cross-domain deep architecture capable of learning across ReID datasets.
The proposed network
The multi-task network In our method, we build our architecture according to the different focuses of two tasks. As we known, the ranking task concentrates on the orders of images with the same query. Its purpose is to rank the similarities of images and obtain a good ranking list for each query. For two person images, in order to compute their similarity score, we have to compare each part of two people. We can't obtain their similarity score only based on some local parts. In other words, the global features of the whole images should be paid more attention than local parts during ranking (Tolias, Sicre, and Jegou 2016). Meanwhile, in the association, the most important purpose of the classification task is to distinguish two categories and make the learned features more identifiable. As shown in Fig. 1 (b), the possible key to distinguish the top 1 result from the query is mainly on the blue local regions, e. g. using the feature of the sleeves or the belting. So the classification loss should pay more attention on learning these local semantic features, which hold enough identifiable information. In this way, the classification loss would prefer to semantic local features instead of the global features during training.
From Wang's work , it had been shown that the higher layers in deep network capture semantic concepts, whereas lower layers encode features to capture intraclass variations. For ranking, we compare images based on a combination (global appearance oriented) of low-level features (i. e. edges, bars etc) learned in lower layers to overcome intra-class variations (as suggested by Wang's work ). Features in high layers focus on identifiable local semantic concepts, driven by the classification loss. The whole framework is shown in Fig. 2. The ranking loss provides global low-level features which could be appropriate for image similarity ranking, and the classification loss further learns the identifiable local features based on the low-level ones. Then we give the details of our multitask network.
The ranking part is a triplet-input model. For each positive pair, we produce ten triplets (a positive pair + a negative image: A 1 , A 2 , B 2 2 ). All these triplets constitute our training data. The input triplet contains three images, each of the size 3 * 224 * 224. The ranking task includes two convolutional layers at the beginning, which are used to reinforce the learning of global features. After the two convolutional layers, three sets of feature maps hold the same size of 256 * 13 * 13 and are sent to a triplet loss through a shared fully connected layer. The triplet loss being minimized is the same as FaceNet (Schroff, Kalenichenko, and Philbin 2015):
L trp = N i=1 [ f A1 − f A2 2 2 − f A1 − f B2 2 2 + α] + (1)
where α is a margin that is enforced between positive and negative pairs, N is the number of the triplets. f ∈ R 512 denotes the features input to the triplet loss from three images. Minimizing the triplet loss is to reserve the information of relative distances between input images.
In the classification part, the input of the third convolutional layer is a set of feature maps of an image pair. The three sets of feature maps with the size of 256 * 13 * 13 from the ranking task are regrouped into two types of pairs, a positive pair and a negative pair. The feature maps from the two images of the same person, i. e. (A 1 , A 2 ), are concatenated as a positive pair, while one image in the positive pair 2 A, B are the person IDs and 1, 2 mean the camera IDs.
(A 1 ) and one negative image (B 2 ) from the different camera view are stacked to form the negative pair. The size of feature maps of each pair is 512 * 13 * 13. These two pairs are fed to three convolutional layers in order, one at each time. The feature maps learned from these layers are called the joint feature maps, which come from each input pair to encode the relationship of two images. Then they are sent into the full connected layers to calculate the similarity. The joint feature maps hold the identifiable information of the input image pair that can represent the relationship of two images. We use these joint feature maps to identify whether the input image pair is from the same person. The classification loss in our network is the binary logistic regression loss, the same as the binary softmax loss in Ahmed, Jones, and Marks 2015):
L cls = − N i=1 [(1 − y)p(y = 0|x) + yp(y = 1|x)] (2)
where y ∈ {0, 1}. When the input pair is a positive pair (e. g. (A 1 , A 2 )), y = 1. On the contrary, y = 0 for a negative pair (e. g. (A 1 , B 2 )). p(y|x) is the discrete probability distribution over two categories y ∈ {0, 1}.
Our five convolutional layers are extended from the architecture of AlexNet (Krizhevsky, Sutskever, and Hinton 2012), differing in that the size of each kernel in the third convolutional layer is (512 × 3 × 3) instead of (256 × 3 × 3) used in AlexNet. In the train phase, the triplet loss optimises the first two convolutional layers while the classification loss simultaneously trained all five convolutional layers including the first two. In other words, the kernels of the first two layers are jointly optimised by two losses for extracting a global feature of each image. The left three layers are mainly trained by the classification loss to obtain an identifiable feature for image pairs to achieve the binary person identification. In the test phase, only the classification task architecture (including the first two layers) is used. The input two images are sent through five convolutional layers and three fully connected layers, with the last layer predicting the similarity probability of a test pair.
Cross-domain architecture
For most person ReID datasets, the size of data is too small to train a deep model. The common way is to crop or mirror the images, which can increase the number of samples in datasets. However, even with these augmentation processes, the total number of the samples is still far from the requirement of deep learning. This problem is considered as a semisupervised cross-domain issue in this paper. In cross-domain transfer, the assumption is that two domains share the same task but the data distributions are different. For example, in image classification, two domains would have the same category but the images contain different views or illuminations. In our issue, the corresponding assumption is that two ReID datasets should share the same similarity function while different variations caused by views or poses widely exist in images from two datasets.
In Fig.2, the relationship of two images is reflected by the joint feature maps. For two positive pairs from two different datasets, the learned similarity metrics for each of the pairs should ideally lead to the same prediction results, i. e. both of the pairs are matched pairs. To achieve such a transfer, we propose to force the learned joint feature maps of positive pairs from two datasets closer than those of negative pairs.
The proposed cross-domain architecture is also shown in Fig.2, which utilizes a contrastive loss (Chopra, Hadsell, and LeCun 2005) to keep the two sets of joint feature maps of the same class as similar as possible during the training process. The label for the two pairs is designed as following:
label p = label a label b(3)
where means the XNOR operation, label a ∈ {0, 1} is the label for a pair from source; label b ∈ {0, 1} is the label for a pair from target; label p is the result after performing the XNOR operation between the labels of those two pairs. If the labels of the two pairs are the same (i. e. label a and label b are the same), the contrastive loss will keep the two sets of the joint feature maps closer, and otherwise farther. The loss is as following:
L cts = − N i=1 [y 1 2 d 2 w + (1 − y) 1 2 max(0, m − d w ) 2 ] d w = F a − F b 2 (4)
where y is the label of two pairs after the XNOR operation, F a and F b are responses of the feature maps after the second fully connected layer from two datasets.
The training phase of the cross-domain architecture is also a multi-task process. The softmax loss and the triplet loss are to do the re-identification task, while the contrastive loss is employed to keep two sets of joint feature maps from the same class in two datasets as similar as possible. After training, only the model on the target dataset will be reserved for testing. The whole process can be considered as another kind of fine-tune operation using a cross-domain architecture. The purpose is to use the joint feature maps learned on the auxiliary source dataset to fine tune those on smaller target sets during training and boost the ReID performances.
It is worth noting that we don't force the feature maps of two completely different people, each from one of two datasets, to be similar. Instead we ensure that the way in which image pairs are compared (encoded by the learned weights on the joint feature maps) is similar and could be shared across the two datasets. That is the motivation of importing the cross-domain architecture.
Experiments
We conducts two sets of experiments: 1) to evaluate the proposed multi-task deep net (including single-task nets) and the cross-domain architecture; 2) to compare the proposed approach with state of the arts.
Setup
Implementation and protocol. Our method is implemented using the Caffe framework (Jia et al. 2014). All images are resized to 224 × 224 before being fed to network. The learning rate is set to 10 −3 consistently across all experiments. For all the datasets, we horizontally mirror each image and increase the dataset sizes fourfold. We use a pre-trained AlexNet model (trained on Imagenet dataset (Krizhevsky, Sutskever, and Hinton 2012)) to initialize the kernel weights of the first two convolutional layers. Cumulative Matching Characteristics (CMC) curves are employed to measure the ReID performance. We report the single-shot results on all the datasets.
Dataset and settings. The experiment is conducted on one large dataset and four small datasets. The large dataset is CUHK03 , containing 13164 images from 1360 persons. We randomly select 1160 persons for training, 100 persons for validation and 100 persons for testing, following exactly the same setting as and (Ahmed, Jones, and Marks 2015). The four small datasets are CUHK01 (Li, Zhao, and Wang 2012), VIPeR (Gray, Brennan, and Tao 2007), iLIDS (Zheng, Gong, and Xiang 2009) and PRID2011 (Hirzer et al. 2011). In CUHK01 dataset, we randomly choose only 100 persons for testing, and all the rest 871 persons are used for training. For three other datasets, we randomly divide the individuals into two equal parts, with one used for training and the other for testing. Specifically, in the PRID2011 dataset, besides 100 test individuals, there are another 549 people in the gallery.
Results for the multi-task network Multi vs. single task. Results of CMCs with different rank accuracies are shown in Table. 1. The proposed multi-task network (Fig. 2) is denoted by MTDnet. As MTDnet adopts the classification loss for testing, we give results using the ranking loss for testing with the same model (denoted by MTDtrp). It's obvious that the performance of MTDnet is much better than MTDtrp which implies the last three convolutional layers trained with the classification loss indeed provide a great help to increase the person ReID performance. The results of the single-task networks using the triplet ranking loss (denoted by MTDnet-rnk) and the binary classification loss (denoted by MTDnet-cls) individually are also provided. It is worth noting that, for a fair comparison, the architecture of MTDnet-rnk network is expanded into containing five convolutional layers plus three fully connected layers as AlexNet (Krizhevsky, Sutskever, and Hinton 2012) instead of the two convolutional layers shown in Fig. 2, i. e. the number of layers in two single-task networks is the same. The similarity of two images in MTDnet-rnk is computed with the Euclidean distance. On CUHK03, our multi-task network (MTDnet) achieves a rank-1 accuracy of 74.68% and is much better than either MTDnet-cls or MTDnet-rnk, which indicates the complementarity of two tasks and the effectiveness of jointly optimizing. On four small datasets, our multi-task network consistently outperforms each of two single-task nets (MTDnet-cls and MTDnet-rnk).
Cross-domain architecture. We compare the crossdomain architecture (MTDnet-cross) with the original multitask network (MTDnet) on four small datasets. In this experiment, CUHK03 is considered as the dataset from the r = 1 r = 5 r = 10 r = 1 r = 5 r = 10 r = 1 r = 5 r = 10 r = 1 r = 5 r = 10 r = 1 r = 5 r = 10 PRDC (Zheng, Gong, Table.1. It's obvious that almost all results of the cross-domain architecture are better than those of MTDnet, which demonstrates the effectiveness of the cross-domain architecture. We also import another network (MTDnet-aug) which simply adds the source data into the target dataset directly and combined them as an augmented dataset for the target dataset training. It's clear that the results of our cross-domain architecture are better than those of MTDnet-aug. The models trained with the augmented data (MTDnet-aug) are even worse compared with MTDnet, which suggests that the direct combination of the source and target datasets is not helpful but disruptive for the training in the target dataset.
Comparison with the state of the arts
We compare ours with representative ReID methods including 18 algorithms, whichever have the results reported on at least one of the five datasets. All of the results can be seen from Table. 1. We have marked all the deep learning methods in the Type column. All the non-deep learning approaches are listed as "-". Cls indicates deep methods based on the classification loss, while Rnk are on the rank-ing loss. SIRCIR method offers the results on both the classification loss and the ranking loss. But in its network, the losses are trained separately. Its combination of two losses are only on the score level, while we jointly optimize two losses in one network and train them simultaneously. Most of these deep methods are in the top performance group among all of the methods considered. It is noted that our results are better than most approaches above, which further confirms that jointly optimizing the two losses has a clear advantage over a single loss. Under the rank-1 accuracy, our multi-task network outperforms all existing person ReID algorithms on CUHK03, CUHK01 and PRID2011. ImpTrpLoss (Cheng et al. 2016) provides the best rank-1 performance on VIPeR and iLIDS. We can see our results are comparable with its, and much better on other datasets.
Conclusion
In this paper, a multi-task network has been proposed for person ReID, which integrates the classification and ranking tasks together in one network and takes the advantage of their complementarity. In the case of having small target datasets, a cross-domain architecture has been further introduced to fine tune the joint feature maps and improve the performance. The results of the proposed network have outperformed almost all state-of-the-art methods compared on both large and small datasets.
Figure 1 :
1Problems in two tasks.(a) Classification issue: the classification loss prefer to train a lower misclassification rate model like Case 2 rather than Case 1. (b) Ranking issue: the appearance of top-rank images is more similar to the query image, while the true positive presents a much less similar appearance. (Best viewed in color and see main text for detailed explanation)
Figure 2 :
2;Ahmed, Jones, and Marks 2015;). Some of them(Ding et al. 2015;Chen, Guo, and Lai 2016;Cheng et al. 2016) consider person ReID as a ranking issue. For example, Ding et al.(Ding et al. 2015) use a triplet loss to get the relative distance between images. Chen et al.(Chen, Guo, and Lai 2016) design a ranking loss which minimizes the cost corresponding to the sum of the gallery ranking disorders. Cheng et al. (Cheng et al. 2016) add a new term to the original triplet loss function to further constrain the distances of pairs. Other approaches (Li et al. 2014; Ahmed, Jones, and Marks 2015; Wu et al. 2016) tackle the person ReID problem from the classification aspect. For instance, Yi et al. (Yi, Lei, and Li 2014) utilize a siamese convolutional neural network to train a feature representation. Li et al. (Li et al. 2014) design a deep filter pairing neural network to solve the ReID problem. Ahmed et al. (Ahmed, Jones, and Marks 2015) employ a local neighborhood difference to deal with this misalignment issue. All of them employ a binary classification loss to train their models. It is worth mentioning that there are some papers (Wu et al. 2016; Xiao et al. 2016) using multi-class classification instead of binary classification. They classify identities to solve the person ReID problem, which shares a similar idea with DeepID in face recognition (Sun et al. 2014). However, in most person ReID datasets, there are few samples for The framework of the proposed multi-task deep network and the cross-domain architecture. The crossdomain architecture is only used when an auxiliary dataset is needed for training.
Table 1 :
1The CMC performance of the state-of-the-art methods and different architectures in our method on five representative datasets. The bold indicates the best performance.Method
Type
CUHK03
CUHK01
VIPeR
iLIDS
PRID2011
source domain, while each of the four small dataset is from the target domain. Therefore, the knowledge transfer is from CUHK03 to each of the four small datasets. The results of MTDnet on four small datasets is obtained by fine tuning the CUHK03 trained model on each small dataset. In the cross-domain architecture, both the target domain network and the source domain network are initialized using the model trained on CUHK03. And in test phase, only the target domain network is used to compute results. Relevant preformance are shown inand Xiang 2011)
-
-
-
-
-
-
-
15.70 38.40
53.90
37.80 63.70
75.10
4.50
12.60
19.70
SDALF (Farenzena et al. 2010)
-
5.60
23.45
36.09
9.90
41.21
56.00
19.87 38.89
49.37
-
-
-
-
-
-
ITML (Davis et al. 2007)
-
5.53
18.89
29.96
17.10 42.31
55.07
-
-
-
29.00 54.00
70.50
12.00
-
36.00
eSDC (Zhao, Ouyang, and Wang 2013)
-
8.76
24.07
38.28
22.84 43.89
57.67
26.31 46.61
58.86
-
-
-
-
-
-
KISSME (Koestinger et al. 2012)
-
14.17 48.54
52.57
29.40 57.67
62.43
19.60 48.00
62.20
28.50 55.30
68.70
15.00
-
39.00
FPNN (Li et al. 2014)
Cls
20.65 51.00
67.00
27.87 64.00
77.00
-
-
-
-
-
-
-
-
-
mFilter (Zhao, Ouyang, and Wang 2014)
-
-
-
-
34.30 55.00
65.30
29.11 52.34
65.95
-
-
-
-
-
-
kLFDA (Xiong et al. 2014)
-
48.20 59.34
66.38
42.76 69.01
79.63
32.33 65.78
79.72
39.80 65.30
77.10
22.40 46.60
58.10
DML (Yi, Lei, and Li 2014)
Cls
-
-
-
-
-
-
34.40 62.15
75.89
-
-
-
17.90 37.50
45.90
IDLA (Ahmed, Jones, and Marks 2015)
Cls
54.74 86.50
94.00
65.00 89.50
93.00
34.81 63.32
74.79
-
-
-
-
-
-
SIRCIR (Wang et al. 2016)
Cls/Rnk
52.17 85.00
92.00
72.50 91.00
95.50
35.76 67.00
82.50
-
-
-
-
-
-
DeepRanking (Chen, Guo, and Lai 2016)
Rnk
-
-
-
70.94 92.30
96.90
38.37 69.22
81.33
-
-
-
-
-
-
DeepRDC (Ding et al. 2015)
Rnk
-
-
-
-
-
-
40.50 60.80
70.40
52.10 68.20
78.00
-
-
-
NullReid (Zhang, Xiang, and Gong 2016)
-
58.90 85.60
92.45
64.98 84.96
89.92
42.28 71.46
82.94
-
-
-
29.80 52.90
66.00
SiameseLSTM (Varior et al. 2016)
Cls
57.30 80.10
88.30
-
-
-
42.40 68.70
79.40
-
-
-
-
-
-
Ensembles (Paisitkriangkrai, Shen, and Hengel 2015)
-
62.10 89.10
94.30
53.40 76.30
84.40
45.90 77.50
88.90
50.34 72.00
82.50
17.90 40.00
50.00
GatedSiamese (Varior, Haloi, and Wang 2016)
Cls
68.10 88.10
94.60
-
-
-
37.80 66.90
77.40
-
-
-
-
-
-
ImpTrpLoss (Cheng et al. 2016)
Rnk
-
-
-
53.70 84.30
91.00
47.80 74.70
84.80
60.40 82.70
90.70
22.00
-
47.00
MTDnet-rnk
Rnk
60.13 90.51
95.15
63.50 80.00
89.50
28.16 52.22
65.19
41.04 69.94
78.61
22.00 41.00
48.00
MTDnet-cls
Cls
68.35 93.46
97.47
76.50 94.00
97.00
44.30 69.94
81.96
54.34 73.41
86.13
28.00 50.00
60.00
MTDnet-trp
Cls+Rnk 66.03 84.81
89.87
66.00 84.00
91.50
34.81 60.13
72.78
46.82 72.83
81.50
26.00 49.00
57.00
MTDnet
Cls+Rnk 74.68 95.99
97.47
77.50 95.00
97.50
45.89 71.84
83.23
57.80 78.61
87.28
32.00 51.00
62.00
MTDnet-aug
Cls+Rnk
-
-
-
75.50 93.50
97.00
43.35 70.25
78.48
54.91 74.57
84.97
27.00 46.00
59.00
MTDnet-cross
Cls+Rnk
-
-
-
78.50 96.50
97.50
47.47 73.10
82.59
58.38 80.35
87.28
31.00 54.00
61.00
AcknowledgementThis work is funded by the National Natural Science Foundation of China (Grant No. 61322209, Grant No. 61673375 and Grant No. 61403383), and the International Partnership Program of Chinese Academy of Science, Grant No. 173211KYSB20160008.
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| [] |
[
"HIGH-REDSHIFT QSOS IN THE SWIRE SURVEY AND THE Z ∼ 3 QSO LUMINOSITY FUNCTION †",
"HIGH-REDSHIFT QSOS IN THE SWIRE SURVEY AND THE Z ∼ 3 QSO LUMINOSITY FUNCTION †"
] | [
"Brian Siana ",
"Maria Del Carmen Polletta ",
"Harding E Smith ",
"Carol J Lonsdale ",
"Eduardo Gonzalez-Solares ",
"Duncan Farrah ",
"Tom S R Babbedge ",
"Michael Rowan-Robinson ",
"Jason Surace ",
"David Shupe ",
"Fan Fang ",
"Alberto Franceschini ",
"Seb Oliver "
] | [] | [] | We use a simple optical/infrared (IR) photometric selection of high-redshift QSOs that identifies a Lyman Break in the optical photometry and requires a red IR color to distinguish QSOs from common interlopers. The search yields 100 z ∼ 3 (U -dropout) QSO candidates with 19 < r ′ < 22 over 11.7 deg 2 in the ELAIS-N1 (EN1) and ELAIS-N2 (EN2) fields of the Spitzer Wide-area Infrared Extragalactic (SWIRE) Legacy Survey. The z ∼ 3 selection is reliable, with spectroscopic follow-up of 10 candidates confirming they are all QSOs at 2.83 < z < 3.44. We find that our z ∼ 4 (g ′ -dropout) sample suffers from both unreliability and incompleteness but present 7 previously unidentified QSOs at 3.50 < z < 3.89. Detailed simulations show our z ∼ 3 completeness to be ∼ 80 − 90% from 3.0 < z < 3.5, significantly better than the ∼ 30 − 80% completeness of the SDSS at these redshifts. The resulting luminosity function extends two magnitudes fainter than SDSS and has a faint end slope of β = −1.42 ± 0.15, consistent with values measured at lower redshift. Therefore, we see no evidence for evolution of the faint end slope of the QSO luminosity function. Including the SDSS QSO sample, we have now directly measured the space density of QSOs responsible for ∼ 70% of the QSO UV luminosity density at z ∼ 3. We derive a maximum rate of HI photoionization from QSOs at z ∼ 3.2, Γ = 4.8 × 10 −13 s −1 , about half of the total rate inferred through studies of the Lyα forest. Therefore, star-forming galaxies and QSOs must contribute comparably to the photoionization of HI in the intergalactic medium at z ∼ 3. | 10.1086/527025 | [
"https://arxiv.org/pdf/0711.0211v1.pdf"
] | 15,245,793 | 0711.0211 | 682ec5cc6bce9924be67570b4ecb3df42deb73b5 |
HIGH-REDSHIFT QSOS IN THE SWIRE SURVEY AND THE Z ∼ 3 QSO LUMINOSITY FUNCTION †
1 Nov 2007
Brian Siana
Maria Del Carmen Polletta
Harding E Smith
Carol J Lonsdale
Eduardo Gonzalez-Solares
Duncan Farrah
Tom S R Babbedge
Michael Rowan-Robinson
Jason Surace
David Shupe
Fan Fang
Alberto Franceschini
Seb Oliver
HIGH-REDSHIFT QSOS IN THE SWIRE SURVEY AND THE Z ∼ 3 QSO LUMINOSITY FUNCTION †
1 Nov 2007Accepted for pulication in the Astrophysical Journal Preprint typeset using L A T E X style emulateapj v. 10/09/06 Accepted for pulication in the Astrophysical JournalSubject headings: quasars -general,quasars -luminosity function: intergalactic medium
We use a simple optical/infrared (IR) photometric selection of high-redshift QSOs that identifies a Lyman Break in the optical photometry and requires a red IR color to distinguish QSOs from common interlopers. The search yields 100 z ∼ 3 (U -dropout) QSO candidates with 19 < r ′ < 22 over 11.7 deg 2 in the ELAIS-N1 (EN1) and ELAIS-N2 (EN2) fields of the Spitzer Wide-area Infrared Extragalactic (SWIRE) Legacy Survey. The z ∼ 3 selection is reliable, with spectroscopic follow-up of 10 candidates confirming they are all QSOs at 2.83 < z < 3.44. We find that our z ∼ 4 (g ′ -dropout) sample suffers from both unreliability and incompleteness but present 7 previously unidentified QSOs at 3.50 < z < 3.89. Detailed simulations show our z ∼ 3 completeness to be ∼ 80 − 90% from 3.0 < z < 3.5, significantly better than the ∼ 30 − 80% completeness of the SDSS at these redshifts. The resulting luminosity function extends two magnitudes fainter than SDSS and has a faint end slope of β = −1.42 ± 0.15, consistent with values measured at lower redshift. Therefore, we see no evidence for evolution of the faint end slope of the QSO luminosity function. Including the SDSS QSO sample, we have now directly measured the space density of QSOs responsible for ∼ 70% of the QSO UV luminosity density at z ∼ 3. We derive a maximum rate of HI photoionization from QSOs at z ∼ 3.2, Γ = 4.8 × 10 −13 s −1 , about half of the total rate inferred through studies of the Lyα forest. Therefore, star-forming galaxies and QSOs must contribute comparably to the photoionization of HI in the intergalactic medium at z ∼ 3.
INTRODUCTION
The QSO luminosity function (QLF) is an observable constraint on models of galaxy formation and the corresponding growth of super-massive black holes (SMBHs) (eg. Small & Blandford 1992;Kauffmann & Haehnelt 2000;Haiman & Menou 2000). These models are useful in interpreting observed phenomena, such as the relation between a galaxy's black hole mass and bulge luminosity (Kormendy & Richstone 1995;Magorrian et al. 1998), as well as inferring specifics of QSO activity, such as initial black hole mass functions, QSO light curves, and accretion rates (see eg. Hopkins et al. 2006).
In addition to galaxy formation models, the QLF can be used to derive the QSOs' contribution to HI and HeII reionization. QSOs are responsible for HeII reionization at z ∼ 3 (Jakobsen et al. 1994;Reimers et al. 1997;Hogan et al. 1997;Sokasian et al. 2002) and are presumed to have a neglible contribution to the HI reionization at z ∼ 6 (Madau et al. 1999). However, both of these claims require assumptions about the faint end slope of the QLF, as this has not been measured accurately at z > 2.
The first QSO luminosity functions demonstrated a rapid increase in space densities toward higher redshift (Schmidt 1968;Schmidt & Green 1983) . Deeper surveys, which primarily identified QSOs by their "UV-excess" 10 , found that the faint end of the QLF was shallower than the bright end (Boyle et al. 1988;Heisler & Ostriker 1988;Koo & Kron 1988;Hartwick & Schade 1990). Recent large surveys, most notably the 2dF Quasar Redshift Survey (2QZ, Boyle et al. 2000;Croom et al. 2004) have found thousands of z < 2.5 QSOs. With these large samples, the QSOs have been placed into smaller bins in both luminosity and redshift, accurately constraining the shape and evolution of the QLF. The data are typically fit to a broken power-law (Boyle et al. 1988;Pei 1995) Φ(L, z) = Φ(L * )/L * (L/L * ) −α + (L/L * ) −β ,
with the break at L * and a bright end slope, α, steeper than the faint end slope, β. The QLF evolution with redshift is consistent with Pure Luminosity Evolution (PLE) 2 Siana et al. (Marshall et al. 1983;Marshall 1985;Boyle et al. 1988;Hartwick & Schade 1990;Boyle et al. 2000;Croom et al. 2004;Richards et al. 2005). That is, the evolution can be parametrized by a shift in the luminosity of the break, L * (z), without any change in its shape. Recently, the 2dF-SDSS LRG and QSO Survey (2SLAQ, Richards et al. 2005) has extended the "UV-excess" QSO search one magnitude fainter to more accurately measure the faint end slope of the QLF out to z = 2.1. These deeper data, when combined with the bright end slope from the 2QZ and 6QZ (Croom et al. 2004), fit a faintend slope, β = −1.45, and bright-end slope, α = −3.31, and demonstrates a ∼40-fold increase in L * from z = 0 to z = 2. At high redshift, various surveys have been conducted through grism searches for UV emission lines (Schmidt et al. 1995), spectroscopic follow-up of point sources with optical colors away from the stellar main sequence (Warren et al. 1994;Kennefick et al. 1995;Fan et al. 2001), X-ray (Hasinger et al. 2005), searches for radio-loud (Dunlop & Peacock 1990) or infrared luminous QSOs (Brown et al. 2006). These surveys all show a precipitous decrease in space densities at z > 3. Early data from the Sloan Digital Sky Survey (SDSS, York et al. 2000), Fan et al. (2001) show a factor of six decrease in M < −25.5 QSOs from z = 3.5 to z = 6.0. Recent results from SDSS suggest that the bright end slope is no longer constant at z > 3. Rather, it is getting shallower toward higher redshift (Richards et al. 2006). Unfortunately, these high redshift surveys are shallow (i 20) and can only measure the bright end of the QLF at z > 3. Therefore, little can be said about the shape of the faint end of the high-z QLF, or its integrated properties (eg. contribution to intergalactic HI, HeII ionizing radiation or black hole growth) without a census of fainter QSOs at z > 3. Hunt et al. (2004) (hereafter, H04) searched for faint AGN at z ∼ 3 with deep Keck spectroscopy over 0.43 deg 2 and found 11 QSOs. Though limited by small numbers, the fitted faint-end slope, β = −1.24 ± 0.07, is substantially shallower than low redshift measurements.
X-ray selected samples suggest a rather modest evolution in AGN space density at high-z (Barger et al. 2005), or even a luminosity dependent density evolution (LDDE) where lower luminosity AGN peak in number density at lower redshifts than more luminous QSOs (Ueda et al. 2003;Hasinger et al. 2005).
Given the importance of the QSO luminosity function in constraining models of galaxy and black hole formation as well as the contribution of AGN to the ionizing background, these initial indications that the QLF shape is evolving at z > 3 warrant deeper surveys to better constrain the high redshift QLF.
The SWIRE Legacy Survey (Lonsdale et al. 2003), a wide-area infrared survey with deep ground-based optical data, is optimal for searches of faint QSOs at high redshift as it is deep enough to detect sub-L * QSOs at z ≤ 4 and covers sufficient area to detect large numbers of them. In Section 3 we outline our method for creating a new QSO template spanning far-UV to mid-IR wavelengths. In Section 4, we present a simple optical/IR color selection for QSOs at z > 2.8 and identify areas of possible contaminations or incompleteness. Our selection results are given in Section 5. In Section 6, the reliability of the sample is assessed through spectroscopic follow-up and analysis of the infrared colors. In Section 7 we determine, through simulations and comparisons with known samples, our sample completeness as a function of redshift. In Sections 8 and 9 we present our measurement of the QLF at z ∼ 3 and compare to previous studies. In Section 10, the QSO contribution to photoionization of HI in the IGM is computed and compared to measurements of the total photoionization rate.
Although many early studies of high redshift QLFs use an Einstein-DeSitter cosmology with H 0 = 50km s −1 Mpc −1 , throughout this paper we choose to use a more recent cosmology with H 0 = 70 km s −1 Mpc −1 , Ω m = 0.3, Ω Λ = 0.7 and correct other measurements accordingly. All optical magnitudes are Vega magnitudes unless stated otherwise.
OBSERVATIONS
The SWIRE survey covers 49 deg 2 over six fields at high galactic latitude with minimum galactic cirrus emission. Most of this area has now been imaged in multiple optical filters to depths r ′ 24.5. Our analysis is conducted within the first two fields for which both optical and infrared catalogs were available, ELAIS-N1 (16h11m+55 • 00') and ELAIS-N2 (16h37+41 • 02').
Spitzer Infrared Data
SWIRE is an IR imaging survey with all four bands on the Infrared Array Camera (IRAC, Fazio et al. 2004) and all three bands on the Multiband Imaging Photometer (MIPS, Rieke et al. 2004) aboard the Spitzer Space Telescope (Bertin & Arnouts 1996) within 1.9 ′′ and 15 ′′ diameter apertures for IRAC and MIPS 24µm, respectively . Aperture corrections were derived from measurements of composite point spread functions from bright stars in the SWIRE fields.
Optical Data
The EN1 and EN2 fields were imaged as part of the Wide Field Survey (WFS, McMahon et al. 2001) 11 . Images were taken with the Wide Field Camera (WFC) on the 2.5-meter Isaac Newton Telescope (INT). Both EN1 and EN2 have been observed with the U ,g ′ ,r ′ ,i ′ ,Z filters over 9 deg 2 each, with 600 second exposures at each pointing in each filter. A fraction of the fields ∼ 30% were not observed on the same night in every filter. The filter characteristics and depths are summarized in Table 2. The median seeing is ∼1.1 ′′ and never worse than 1.6 ′′ . The optical coverage overlaps the Spitzer IR data by 7.45 and 4.29 deg 2 in EN1 and EN2, respectively . Data processing was done by the Cambridge Astronomical Survey 11 http://www.ast.cam.ac.uk/ wfcsur/ z=3 QSO Luminosity Function 3 Unit CASU) and is outlined in and Gonzalez-Solares et al. (2005).
Photometry was measured with the CASU software, requiring a source to have five continguous pixels 1.5σ above the background pixel noise. Detection and photometry were performed in each band separately and then matched between bands. Fluxes were measured within 2.3 ′′ (7 pixels) diameter apertures. Given the typical seeing of ∼ 1.1 ′′ , these apertures contain 80-90% of the total flux. Aperture corrections for each image were derived from bright stars within that image. Total and isophotal magnitudes were computed as well but only aperture photometry is used in this analysis as our objects are point sources by definition. Limiting magnitudes (5σ) for non-detections were computed from the pixel-to-pixel noise of the corresponding image. The images were not interpolated before photometry was performed and therefore do not suffer from correlated noise from projection procedures.
QSO TEMPLATE
Our optical QSO selection (defined in Section 4.1) uses three filters which typically span the rest-frame wavelengths 700Å < λ < 2000Å. For the infrared selection (see Section 4.2) we use the same two filters (IRAC1 & IRAC2) for all targeted QSO redshifts, so the photometry samples a large range in the rest-frame optical to near-IR wavelengths, 5000Å < λ < 2µm. In order to define the expected optical/mid-IR colors of QSOs at 2.7 < z < 5.0, we have created a QSO template which spans many decades in wavelength from the far-UV to the mid-IR.
UV Template
Several composite QSO spectra have been created from large QSO surveys, and agree well with each other except for minor variations due to selection effects in optical color and luminosity. Vanden Berk et al. (2001) created a composite spectrum of 2200 QSOs from the SDSS spanning the wavelengths between 800Å < λ < 8555Å. In order to obtain such a broad coverage in rest-frame wavelength, QSOs from a wide redshift range (0.044 ≤ z ≤ 4.789) were used. Since the UV spectrum is composed from only high-redshift QSOs, the mean continuum shortward of Lyα is artificially decreased by the large number of Lyman line absorbers at high redshift. Because the level of absorption is redshift dependent, we need a template which reflects the intrinsic SED, to which we can then apply a redshift dependent model of the HI absorption. Telfer et al. (2002) have created a high S/N composite UV spectrum (300Å < λ < 1275Å) using HST UV spectra of 77 radio quiet QSOs. Because these data were taken in the observed UV rather than the optical, the QSOs contributing to the critical wavelength range 700 < λ < 1216Å are at lower redshift (< z >= 1.42) than the SDSS QSOs (whose spectra cover the same rest-frame wavelengths at z > 2.4). Therefore, there is much less Lyman line and continuum absorption in this composite. Furthermore, Telfer et al. (2002) corrected for absorbers with column densities N HI < 10 16 cm 2 and statistically corrected for lower column density absorbers, resulting in a good template of the intrinsic UV spectrum of QSOs. For our UV template we have used the composite from Telfer et al. (2002) for 300Å < λ < 1250Å and the SDSS composite for 2000Å < λ < 8555Å. The mean of the two composites is used in the overlapping regions 1250 A< λ <2000Å, where they agree well with each other (Telfer et al. 2002).
Optical to Mid-Infrared Template
The SDSS composite from Vanden Berk et al. (2001) covers the optical to λ < 8555Å. However, at λ > 5000Å the composite is produced by low-z, low luminosity AGN and suffers significant contamination from host galaxy stellar light. Indeed, a comparison with the median broadband SED of luminous QSOs from Elvis et al. (1994), for which stellar contribution to luminosity should be small, shows a much redder continuum slope in the SDSS composite at λ > 5000Å. To reliably select QSOs at 2.5 < z < 6.0 with our mid-IR photometry, a new template is needed which covers the rest-frame 5000Å < λ < 2µm and resembles the high luminosity (M 1450 < −23) QSOs that we are targeting.
We matched the SDSS photometry of all spectroscopically confirmed QSOs in the SDSS Data Release 3 with the near-IR photometry from the Two Micron All Sky Survey (2MASS, Skrutskie et al. 2006) point-source catalog, including the 2MASS Deep Lockman Hole Catalog (Beichman et al. 2003), which goes ∼ 1 magnitude deeper than the typical 2MASS all-sky depths. There were 8642 2MASS objects within 2 ′′ of the SDSS positions. To extend the template to longer wavelengths, we also matched the Spitzer IR photometry of the 241 SDSS QSOs in the SWIRE fields (Lockman, EN1, & EN2).
We chose to normalize the photometry at rest frame λ norm = 2400Å for three reasons. First, the λ norm is easily measured with our ground based optical/near-IR data for a broad range of redshifts, without using extrapolations and invoking assumptions about the SED spectral slope. Second, the UV flux is dominated by the AGN rather than the host galaxy stellar light. Third, this wavelength is far from any major emission lines which would significantly affect the photometry (eg, MgII or CIII]). Only QSOs with detections in adjacent filters surrounding λ rest = 2400Å were used to minimize the interpolation of the flux to the normalization wavelength. After an F ν (2400Å) was calculated, only those QSOs with M 2400 < −24 were used to ensure the flux was dominated by the AGN. This also corresponds well with the minimum luminosity of QSOs in our search (M 1450 = −23.5) and is ∼1 mag brighter than the classical QSO/Seyfert demarcation. This cut in absolute magnitude left 3378 QSOs (39% of SDSS/2MASS matches) with which we made the template.
The template was made by averaging all of the flux measurements (SDSS/2MASS/SWIRE) in 1000 wavelength bins spaced logarithmically between 753Å and 11 µm. The result is plotted in green in Figure 1 and traces well the combined template of Telfer et al. (2002) and Vanden Berk et al. (2001) in the UV. As expected, the relative UV to optical ratio of our template is significantly less than the Vanden Berk et al. (2001) template because our sample is subject to less contamination from the host galaxies.
Broadband photometry works well in recreating Telfer et al. (2002) template (black). The broadband template at λ < 8000Å has been replaced by the corrected Telfer/Vanden Berk template but is plotted in green to demonstrate the accuracy of the spectral slope. Also plotted for comparison are the Elvis et al. (1994) Table 3. An important feature of the template is the minimum at λ ∼ 1µm and the monotonic rise (in νf ν ) toward longer wavelengths. This is produced by hot dust at varying distances from the central engine (Barvainis 1987). The minimum at λ ∼ 1µm has been shown to be prevalent in QSO SEDs (Sanders et al. 1989) and is attributed to sublimation of dust grains at T > ∼ 1500K (Barvainis 1987 ∼ 340Å, more than 70% higher. This proves to be important at z ∼ 4 when Hα redshifts into the mid-IR and significantly affects the IRAC colors.
Finally, we point out that our broadband template is significantly redder than the Elvis et al. (1994) template, with the ratio of far-UV to near-IR nearly a factor of two larger in the latter. This is not surprising since their sample was selected in the soft (0.3-2.0 keV) and "ultrasoft" (0.1-0.3 keV) X-ray bands. Also, their sample is composed of more luminous QSOs, which are known to have bluer UV-optical colors than fainter QSOs (Richards et al. 2006a).
HIGH REDSHIFT QSO SELECTION
In this section we define z > 3 QSO selection criteria requiring only three optical bands and the two most sensitive imaging bands on the Spitzer Space Telescope (IRAC1 & IRAC2) and assess its efficacy. The method consists of an optical color selection to identify a Lyman Break in the rest-frame UV, thereby isolating QSOs to a narrow redshift range (ie. z ∼ 3 for U-band dropouts). In addition, we also require a red Spitzer IR color to eliminate typical contaminants in Lyman-Break Galaxy surveys (stars and low-z galaxies).
Optical Selection
The space density of neutral Hydrogen (HI) absorbers increases rapidly with redshift (Bechtold 1994;Weymann et al. 1998). There are hundreds of absorbers with N HI > 10 12 cm −2 along any line-of-sight (LOS) to galaxies with z > 2, resulting in Lyman line absorption, or "blanketing", of the source's continuum at λ rest < 1216Å. In addition, Lyman continuum absorption by the absorbers significantly decreases the observed flux at λ rest < 912Å. At z ≥ 2.5, these features redshift into the bluest filter accessible to ground-based telescopes (U band) and can be identified by a flux decrement at wavelengths shortward of an otherwise blue continuum. This method, known as the Lyman Break technique, has been used extensively to search for both QSOs and galaxies at z > 3 (Irwin et al. 1991;Steidel et al. 1996;Madau et al. 1996).
We use the Madau (1995) prescription for the HI opacity evolution to determine average QSO colors as a function of redshift. Figure 2 shows the U − g ′ , g ′ − r ′ color track of our our QSO template as it is redshifted and the HI absorption is applied. At z = 2.3 the Lyman Break redshifts into the U filter, resulting in redder U −g ′ colors, thereby causing the QSO to move away from the locus of blue, low-redshift QSOs. At z > ∼ 2.8 the QSO moves away from the optical color space of main sequence stars and into color space unoccupied by typical stars or galaxies. We can therefore search for QSOs in this color space while minimizing contamination.
Our color selection for both the z ∼ 3 and z ∼ 4 sample, defined in Table 4 and shown in Figures 2 and 3 were defined to select QSOs with < ∼ 2σ deviation in spectral slope (see Section 7.3.1). As we expect to eliminate stars and low-z galaxies from our sample with an additional IR selection, we are able to use a color cut which is much broader (and closer to the stellar locus) than optical-only surveys such as SDSS.
In addition to these color criteria, a candidate QSO is required to be unresolved in images taken through the two redder filters (eg. g ′ and r ′ for z ∼ 3 selection). This minimizes contamination from low-z galaxies. Our sample is limited to objects brighter than r ′ < 22, corresponding to a signal-to-noise ratio SNR ≥ 20 in both g ′ and r ′ . Therefore, we have sufficient sensitivity to detect significant deviations from a point source (eg. F W HM > 0.5 ′′ ).
Infrared Selection
As seen in Figures 2 and 3, we expect some contamination from stars in both our U -dropout and g ′ -dropout samples. Fortunately, the mid IR SEDs of stars are very z=3 QSO Luminosity Function
5 Fig. 2.-U − g ′ , g ′ − r ′ color-color diagram showing the z ∼ 3 QSO selection.
The solid curve is the color track of our QSO template with IGM absorption applied. The black filled circles denote the locations in redshift increments of 0.1 from 2.9 ≤ z ≤ 3.4. The shaded regions denote the color space spanned by QSOs with ±2σ deviations in the spectral slope (see Table 7). The lines are color tracks of various galaxy templates from 0 < z < 2: Ell (age=2Gyr, dotted), Sa (dashed), Sc(dot-dashed), and Sd(multiple dot-dashed) taken from the GRASIL library of models (Silva et al. 1998), with the boxes corresponding to their respective colors at z = 0. The black asterisks are stars from the Gunn & Stryker (1983) catalog. blue in color as they lie on the Rayleigh-Jeans side of the blackbody spectrum, whereas QSO SEDs are red, rising (in νf ν ) towards longer wavelengths as seen in Figure 1. Many groups have proven Spitzer's ability to select AGN with IRAC colors Stern et al. 2005;Hatziminaoglou et al. 2005). In Figure 4, we plot the IRAC colors of all point sources with r ′ < 22. There are two clear loci of points. The objects with blue IR colors (lower left) are stars and those with red IR colors (upper right) are AGN. Both Lacy et al. (2004) and Stern et al. (2005) use all four IRAC bands to select AGN, but Figure 4 shows that the IRAC1-IRAC2 is a robust discriminator of AGN from stars when a point-source criterion is also used. Also, we seek an IR selection using only IRAC channels 1 & 2 since they are ∼ 2 magnitudes more sensitive than channels 3 & 4 given the same exposure time (see Table 1). Therefore, we apply the IR color cut of
[3.6] − [4.5] > −0.15 (AB)(2)f ν (4.5µm)/f ν (3.6µm) > 0.87.
Finally, we search only sources with f ν (4.5µm) ≥ 10µJy (∼ 7σ) so that Poisson errors in the flux measurement are minimized, thereby decreasing the risk of contamination from sources with large errors in IRAC colors. We will see in Section 7.1 that this matches well our r ′ < 22 optical cut.
In addition to stellar contamination, some low redshift galaxies may also meet our optical color criteria. This is due to the Balmer-or 4000Å-Break being mistaken for a U -or g ′ -dropout at z < 0.1 and 0.1 < z < 0.5, respectively. Our point-source criterion will remove the lowest redshift galaxies as they will have large angular diameters, but our optical images may not resolve the most compact galaxies at 0.3 < z < 0.5. Therefore, we expect to see some contamination from galaxies within this redshift range in the g ′ -dropout sample. In Figure 5, the [3.6] − [4.5] color of our QSO and galaxy templates are plotted versus redshift, as well as the IR colors of SDSS QSOs in the SWIRE fields. Unfortunately, many galaxies which may contaminate our g ′ -dropout optical selection (ie. 0.3 < z < 0.5) are expected to meet our IR color selection. This is due primarily to the presence of strong 3.3 µm PAH emission in star-forming galaxies (Imanishi & Dudley 2000). Therefore, we expect some contamination from z ∼ 0.4 galaxies within our z ∼ 4 QSO sample.
In Figure 5, we can see that both our QSO template, and all but three of the SDSS QSOs remain above our IRAC color cut for 0 < z < 3.9, including the 18 of 19 SDSS QSOs in the targeted redshift range for Udropouts. However, the QSO template dips close to our IR color cut between 3.8 < z < 4.8, as do four of the five SDSS QSOs in this redshift range. This is partly due to the change to bluer slope at λ < 1.5µm, but is caused primarily by Hα redshifting into IRAC1. At these redshifts the Hα equivalent width is W(Hα)∼ .15µm and the IRAC1 filter width is 0.7µm. Therefore, the Hα flux causes the [3.6]−[4.5] color to decrease by ∼ 0.2 mags. As a result, the g ′ -dropout selection may suffer from signifcant incompleteness between 3.9 < z < 4.5 (in addition to the unreliability discussed above).
Selection Summary
• Point source brighter than r ′ < 22 (i ′ < 21.5 for z ∼ 4 sample) to ensure proper morphological characterization and bright enough to determine a large U − g ′ (g ′ − r ′ ) limit.
• IRAC2 > 10µJy to ensure high SNR needed for accurate mid-IR colors.
• Optical colors which identify a Lyman Break in the continuum (see Table 4).
• Red mid-IR color ([3.6] − [4.5] > −0.15) to remove interlopers.
RESULTS
We performed our search in the EN1 and EN2 SWIRE fields, covering 11.7 deg 2 with both optical and IR coverage. We found 100 z ∼ 3 and 26 z ∼ 4 QSO candidates which meet both our optical and IR criteria. The z ∼ 3 candidates in EN1 and EN2 and their optical/infrared photometric data are listed in Table 5. The optical colors of the z ∼ 3 sample are plotted in Figure 6 along with the colors of all other point sources with r ′ < 22. We will show that the z ∼ 4 sample suffers from signifcant contamination and therefore do not list them. In Section 6, we assess the reliability of our sample through spectroscopic follow-up. In Section 7, we assess the completeness through Monte-Carlo simulations and derive effective volumes for use in computing the luminosity function.
6. RELIABILITY
Spectroscopic Follow-Up
Optical spectra were obtained for 10 z ∼ 3 and 10 z ∼ 4 QSO candidates. Thirteen spectra for faint candidates (6 U -dropouts and 7 g ′ -dropouts) were obtained with the Low-Resolution Imaging Spectrometer (LRIS Oke et al. 1995) on the Keck I Telescope during the nights of 03-04 March, 2005. The U -dropout sample, with the expected redshift of z ∼ 3, have the most prominent emission lines at λ < 7500Å. Therefore, only the blue channel (see Appendix in Steidel et al. 2004) was used for these sources with a 1.5 ′′ wide longslit and a 300 l/mm grism blazed at 5000Å, giving a resolution of 1.43Å pixel −1 over 3300 < λ < 7650Å. For the g ′ -dropout sample, in order to detect the CIV (1549Å) line, we used a dichroic at 6800Å and used both the blue and red LRIS channel. We used the same grism on the blue side and the 400 l/mm grating blazed at 8500Å on the red side giving a blue side resolution of 1.43Å pixel −1 from 3300 < λ < 6800 The blue arrows and circles also have red IRAC colors, but match the optical color criteria for z ∼ 3 QSOs. The arrows denote upper limits where there is no detection in U . Spectroscopically confirmed candidates are circled. The photometry of two spectroscopically confirmed QSOs were revised and are therefore slightly out of our optical selection window but are still displayed here. They are both QSOs at z ∼ 3 and their photometry is included at the end of Table 5.
A and 1.86Å pixel −1 from 7000 < λ < 8500Å. Total exposure times ranged from 5 to 15 minutes. Both nights were photometric with seeing of ∼ 1.2 ′′ . The spectra were flux calibrated with observations of the standard star G1919B2B from Massey et al. (1988).
Seven additional spectra (4 z ∼ 3 and 3 z ∼ 4 candidates) were obtained for brighter candidates with the COSMIC spectrograph on the 5-meter Hale Telescope at Palomar Observatory on the nights of 11-14 March, 2005. A 300 l/mm grism blazed at 5500Å was used with a 1.5 ′′ wide longslit, giving a dispersion of 3.1Å pixel −1 and wavelength coverage of 3800 < λ < 8500Å. The nights were photometric with poor seeing (∼ 2 − 4 ′′ ) so long exposure times of 10-60 minutes were required. The spectra were flux calibrated with observations of the standard star G191B2B from Massey et al. (1988).
All ten U -dropout candidates are QSOs with redshifts between 2.83 < z < 3.44, the expected redshift range for our sample. The spectra are shown in Figure 7, exhibiting broad Lyα and CIV (1549Å) lines and the QSOs with spectroscopic confirmation are circled in Figure 6. The six candidates observed with Keck/LRIS were chosen simply in ascending order in declination in order to sample the QSOs randomly in color space. After these six were shown to be QSOs, we decided to target QSOs with extreme optical colors on the edges of our selection region as these are more likely to be interlopers or objects with spurious photometry. In our subsequent run at Palomar, four of these QSOs with extreme colors were observed: one with blue g ′ − r ′ color, one with red U − g ′ color, and two with optical colors that lie close to stellar main sequence. All four are in fact QSOs at the expected redshift.
As expected (see Section 4.2), our g ′ -dropout sample suffers from significant contamination. Seven of ten candidates are QSOs with 3.48 < z < 3.88 and are displayed in Figure 8 and listed in Table 6. The three other spectra are plotted in Figure 9. Two of the contaminants z=3 QSO Luminosity Function Table 5 and redshifts are given and the Lyα and CIV lines are labeled. Table 6 and redshifts are given and the Lyα line is labeled.
are galaxies at low redshift (z = 0.354,0.390) and exhibit strong breaks in the continuum at λ ∼ 5500Å. At these redshifts, the interlopers' 4000Å break falls between the g ′ and r ′ filters. The third contaminant has a high S/N spectrum, but did not show any emission features, nor a break in the continuum as expected, so we have not identified a redshift for this object. Therefore, the reliability of our z ∼ 3 and z ∼ 4 QSO samples are 100% (> 69% 1σ) and 70 +16 −26 %, respectively. The spectroscopic follow-up of the g ′ -dropouts shows the expected incompleteness at the high redshift end of the targetted range (3.5 < z < 4.5). All 7 confirmed QSOs have z < 3.9. Beyond this redshift, Hα redshifts into IRAC1 and may cause the [3.6] − [4.5] color to be bluer than our color criterion.
Infrared Reliability
In Figure 6, there are a few tens of point sources (out of more than 11,000) with red IRAC colors that lie within the stellar locus where we would not expect QSOs to lie (upper-right), suggesting possible contamination where the stellar locus crosses the color space of our QSO selection. These may be red galaxies at moderate redshift, highly reddened QSOs, or stars whose IRAC colors are just redder than our color cut. Figure 10 shows that indeed there is not a distinct bimodality of IRAC colors for point sources. It's clear there are two peaks in the distribution, but there are hundreds of sources in the valley between the two peaks. We have also plotted the IRAC color distribution of the point sources which also meet our optical color criteria. In this histogram, nearly all of the sources in the valley between the two peaks have been removed, revealing a clear bimodality. There are two reasons for the this. Firstly, galaxies can have either red or blue IRAC colors depending on redshift, but should not lie in the optical color space of z ∼ 3 QSOs. Secondly, though a few stars may have IRAC colors redder than our cut, the stars must be either very bright or very red to be detected in IRAC channel 2 at all. Stars that lie in the z ∼ 3 optical color space have r ′ − [4.5] = 1.6 mags (AB). Since the IRAC channel 2 flux limit corresponds to 21.4 in AB magnitudes, this means we shouldn't detect the stars in channel 2 unless they are brighter than r ′ 19.7 (Vega). Indeed, in Figure 11 there are two sources at the bright end of our sample that have blue optical-IR colors and may be stars.
As mentioned in Section 4.2, IRAC color selection of AGN is robust when all four bands are used because the SEDs of most contaminants are not flat (or rising toward longer wavelengths) over such a broad wavelength range. Therefore, we can use all four IRAC fluxes, when available, to assess the reliability of our single color AGN selection. In our sample, 58% (58 of 100) of our objects are detected in all four IRAC channels and their IRAC colors are plotted in Figure 12. Essentially all of these objects have IR colors in the expected locus of AGN. Therefore, it appears the contamination rate is low. However, as stellar contaminants are expected to have blue IRAC colors, we do not expect to detect the fainter interlopers in IRAC3 and IRAC4 and we cannot assess the nature of these objects. Therefore, we look at only the bright end of our sample where all sources are detected in all four bands. At r ′ < 20.5, 21 of 24 (88%) sources are detected in all four IRAC bands, and all of these have IR colors within the expected AGN locus. If we make the conservative assumption that the three non-detections (in IRAC3 & IRAC4) amongst the bright sample are not QSOs, then we obtain an upper limit to the contamination rate of 13%.
In summary, the point sources which match our optical color criteria have a distinct bimodal distribution in [3.6] − [4.5] colors, suggesting minimal contamination from stars, except at the bright end (r ′ < 20). However, the four-band IRAC colors suggest that even the bright end of our sample has a contamination rate of less than 13%.
COMPLETENESS
Infrared Completeness
As mentioned in Section 4.2, we expect our IR color cut to include nearly all QSOs at z ∼ 3 (as seen with the SDSS QSOs). However, we must ensure that no QSOs are missing due to the f ν (4.5µm) ≥ 10µJy. Figure 11 plots the optical-IR colors of all U -dropout candidates as a function of the optical magnitudes. The locus of colors agree well with the infrared selected QSOs of Brown et al. (2006). Also plotted is the limit due to the IRAC2 flux cut. Except for the faintest of the six half-magnitude bins, there is is no incompleteness due to non-detections in IRAC2. In the faintest bin, 21.5 < r ′ < 22.0, it is possible that we miss a few of the bluest r ′ − [4.5] QSOs, but we estimate this to be < 15% (based on the r ′ -[4.5] distribution in Figure 11) and make no attempt to correct for it. We've also verified that increasing the counts by 15% in this bin has a neglible effect on the QLF fit.
Morphological Completeness
Our selection requires that the candidates be unresolved in the optical data, where typical resolution is ∼ 1.1 ′′ . We have assumed that the QSOs are significantly brighter than their host galaxies and, even if we could detect the host galaxy, galaxies at z > 3 should have angular diameters less than 1 ′′ (Giavalisco 2002). To test this assumption, we've matched our catalogs to all of the SDSS QSOs (which do not have to meet a morphological criterion) within the EN1 and EN2 fields. Of the 58 SDSS QSOs at high redshift (z > 1), 57 (98%) are categorized as point sources in our optical catalogs. Therefore, we do not expect any significant incompleteness due to our point-source criterion.
Optical Completeness
Our color selection should be strict enough to minimize contamination, but broad enough to encompass the majority of the targeted QSOs to minimize completeness corrections. As a first-order look at com-z=3 QSO Luminosity Function 9 pleteness, we compare with the SDSS QSO sample between 3.1 < z < 3.2 where SDSS is ∼ 80% completete (Richards et al. 2006b). SDSS uses four colors, to look at all point-sources away from the color space of stars (Richards et al. 2002). Therefore it is possible that SDSS selects redder QSOs at z ∼ 3, which would lie outside of our color selection. Converting SDSS to Vega magnitudes, and correcting for slight U /u ′ filter differences (0.06 mag correction), ∼97.5% (392/402) of SDSS QSOs at 3.1 < z < 3.2 would also be selected with our color selection. Therefore, < 3% of SDSS QSOs are redder, and selected with other colors in the SDSS filters.
Significant dispersion in spectral features may cause some QSOs to lie outside of our selection criteria. The most important factors affecting the optical color (and therefore the completeness) are redshift, intervening high column-density HI absorbers, UV continuum slope, and emission line equivalent widths, and variability. Here we present our Monte Carlo simulations to assess the combined effects of these characteristics on our completeness as a function of magnitude and redshift. We then use this completeness to derive the effective volume of our sample as a function of apparent magnitude.
Model of QSO Optical Color Distribution
Both the UV spectral slope of the continuum α ν , where f ν ∝ ν α , and the emission line equivalent widths, W , can vary about the mean and cause significant dispersion in colors at any given redshift. The mean of the spectral slope and the equivalent widths are defined by our template spectrum and are consistent with values found by Vanden Berk et al. (2001) and Hunt et al. (2004). The distribution of these attributes are assumed to be gaussian with standard deviations taken from Francis (1996), and are listed in Table 7.
Along the line-of-sight to any given QSO at high redshift there are hundreds of intervening neutral hydrogen clouds (or filaments) absorbing the UV light through Lyman line and Lyman continuum absorption. Much of this absorption is caused by the rare, high columndensity absorbers known as Lyman Limit Systems (LLSs, N HI > 1.6 × 10 17 cm 2 ) and Damped Lyman-α Systems (DLAs; N HI > 1 × 10 20 cm 2 ). Therefore, there is a large dispersion in line-of-sight HI opacity at any given redshift due to the small numbers of these high column density clouds. Bershady et al. (1999) show that simple analytic expressions for the mean and scatter in line-of-sight opacity are insufficient and, therefore, Monte-Carlo simulations are required to correctly represent the stochastic distributions of the absorbers.
The number density distribution of HI absorbers is typically given as a fit to a power-law with redshift
N (z) = N 0 (1 + z) γ(3)
and the column density distribution is given as
df dN HI ∝ N −κ HI(4)
Since absorbers of different column densities are known to evolve differently (Kim et al. 1997 with updated values for DLA number densities. The parameters used are summarized in Table 8.
For each line of sight, the number of line-of-sight absorbers is selected from a Poisson distribution with the expectation value set to < N > of the population.
< N >= zQSO 0 N (z)dz(5)
Their redshifts and column densities are chosen randomly from the distributions in Eqns. 3 and 4, respectively. Finally, we compute Voigt profiles with natural broadening and doppler widths, b, from Table 8 for the first ten Lyman lines for each absorber. We choose not to model the distribution of Doppler widths (or its evolution) as this is expected to be a small, second order effect. Continuum absorption is applied with a scattering cross-section of σ 0 = 6.3 × 10 −18 cm 2 at the Lyman Limit and decreasing as ν −3 . An example of the Lyα Forest transmission for one line of site at z = 3 is plotted in Figure 13. For 300 lines-of-sight in each redshift bin of ∆z = 0.1, we compute QSO spectra with spectral slopes and emission line equivalent widths culled from the distributions defined in Table 8 and apply the simulated IGM absorption for that line of sight.
QSOs of these luminosities and redshifts exhibit variability of order ∆m ∼ 0.15 mags (Vanden Berk et al. 2004). 29% of our fields were observed at separate times (over two years) in the U and g ′ filters, and 29% of our fields were observed at separate times in the g ′ and r ′ filters. We therefore add a variable offset selected from a gaussian distribution with ∆m ∼ 0.15 to the computed U − g ′ and g ′ − r ′ optical colors of 30% of the simulated lines of sight. The net effect of this variability is a slight ∼ 5% decrease in completeness, and a slightly broader redshift range.
After adding in variability and photometric errors, we compute the optical colors for the 300 QSOs in each redshift bin. This allows us to determine the selection completeness based on color selection alone, but the imaging depth must still be taken into consideration.
The optical magnitude limit of our QSO search, r ′ = 22.0, is set by our infrared depths and corresponds to ∼ 20σ detections in both g ′ and r ′ . Therefore, we do not expect any incompleteness due to non-detections in g ′ or r ′ . However, when looking for significant "U -dropouts", it's important to have deeper imaging in U relative to the bands at longer wavelengths. Otherwise, non-detections will result in upper limits in flux which cannot distinguish between a true "dropout" or a red source which is just below the detection limit. The WFS was not designed for finding U -dropouts and the U band is generally less sensitive due to CCD quantum efficiency (QE) and poor throughput in the telescope optics. Therefore, the Uband images in the SWIRE fields are less sensitive (to all classes of object) than the g ′ and r ′ images. Furthermore, this sensitivity varies by ∼ 0.5 magnitudes from pointing to pointing due to changing observing conditions (seeing, airmass, lunar phase). Therefore, careful measurements of the selection completeness as a function of pointing must be assessed, along with dependencies on redshift and magnitude.
Both EN1 and EN2 are comprised of 54 22.8 ′ ×11.4 ′ optical pointings. For each CCD at each pointing, a 5σ limiting magnitude is determined within our 2.3 ′′ diameter aperture and is used as an upper limit when no object is detected.
Due to overlaps in pointings and varying observing conditions, it is nontrivial to compute the limiting magnitude for a given position on the sky. For both fields we make "depth maps" by making mosaic images of the entire field with pixel values set to the 5σ limiting magnitude of the deepest pointing which covers that pixel. In this way, we can quickly compute the area over which we can detect an object of a given magnitude U test by summing the pixels with value greater than U test . These depth maps are trimmed to the area which also has Spitzer data. The U-band depth map for EN1 is plotted in Figure 14, showing the non-uniformity and complex structure of the coverage.
As shown in Figure 2, the bluer of the two optical colors, (U − g ′ ), becomes redder as the QSO goes to higher redshift, making non-detections in U increasingly likely at higher redshift. For example, a typical r ′ = 21 QSO at z = 2.9 will have g ′ ∼ 21.5 and U ∼ 21.8, bright enough to be detected in all bands by our survey. However, a QSO with the same r ′ magnitude at z = 3.4 will have g ′ ∼ 22 and U ∼ 24.5, too faint to be detected in our U images. Our upper limit in this case is insufficient in distinguishing a high-redshift QSO candidate from a low-redshift object with bluer U − g ′ . Given our simulated spectra and our depth maps, we determine an effective completeness in the following manner. For each redshift bin, we compute U − g ′ and g ′ − r ′ colors for the 100 simulated QSOs along 100 different lines of sight. Then we compute the percentage of these QSOs which would be selected by our color criteria if our imaging was sufficiently deep. This gives us a measure of our completeness based strictly on our color criteria alone, C color . The completeness cutoff at the high redshift end (z ∼ 3.5) is nearly a step-function since the QSO color track moves perpendicular to our color cuts and most of the dispersion in color is parallel to the color cut. At the low redshift end (z ∼ 2.9), the incompleteness is predominantly due to line-of-sight variations in IGM.
Of the QSOs which meet the color criteria, the fluxes are scaled to give the desired r ′ magnitude in intervals of ∆r ′ = 0.25 mags and the g ′ − r ′ color is used to determine the U -band depth required to either detect this QSO or derive a lower limit to the magnitude which is high enough to put it in the color-color selection window. The percentage of QSOs with colors in our selection window which would also be selected given the U -band depth at that pixel value is denoted as C(r ′ , z), plotted in Figure 15. The effective volume of the survey can then be calculated as
V ef f (r ′ ) = dΩ z=∞ z=0 C(r ′ , z) dV dz dz(6)
where Ω is the solid angle of the survey and dV /dz is the differential comoving volume. The effective volumes and average redshifts for each half-magnitude bin are given in Table 9. To give an idea of the scales of the incompleteness corrections, the effective volume in our faintest magnitude bin V ef f (r ′ = 21.75) is 74% of the effective volume in our brightest bin V ef f (r ′ = 19.25), requiring a relatively small correction of 35% to the number counts.
THE Z ∼ 3 QSO LUMINOSITY FUNCTION
Given the effective volume, V ef f (r ′ ), the comoving space density is then
Φ(r ′ ) = N (r ′ ) V ef f (r ′ ) 1 w bin .(7)
where N (r ′ ) is the number of QSOs in the r ′ bin and w bin is the width of the bin in magnitudes. We then z=3 QSO Luminosity Function 11 convert r ′ to the absolute AB magnitude at 1450Å since the r ′ filter covers 1450Å over our entire redshift range and this value is generally used for high redshift QSO studies.
M 1450 = r ′ +r ′ AB (V ega)−DM (z = 3.2)+2.5log(1+3.2)+K(∆z) (8) where r ′ is the Vega magnitude listed in Table 5, r ′ AB (V ega) = 0.15 is the Vega to AB conversion, DM = 47.19 is the distance modulus at z=3.2, and K(∆z) is the K-correction resulting from shifts in redshift around z = 3.2. As we don't have exact redshifts for most of our QSOs, we set K(∆z) = 0, but note that this varies by ±0.1 mags from 2.9 < z < 3.5.
The resulting QSO luminosity function at z ∼ 3.2 is plotted in Figure 16, along with previous surveys of QSOs at these redshifts. Our measurements, while using smaller magnitude bins (w bin = 0.5 mags) than previous surveys of faint high-z QSOs, have significantly reduced error bars. The space densities at the bright end of our survey match well with the faintest bins from the SDSS (Richards et al. 2006b) and show a clear transition to a shallower slope at the faint end. We use our data in combination with the Richards et al. (2006b) SDSS results at z ∼ 3.25 because it is the largest sample available, and its completeness corrections have been carefully determined.
We do a least-squares fit to the standard double powerlaw given in Eqn. 1, converted to absolute magnitudes Φ(M 1450 , z) = 0.92 × Φ(M * 1450 ) 10 0.4(α+1)(M1450−M * 1450 ) + 10 0.4(β+1)(M1450−M * 1450 ) . (9) As mentioned previously, when the error bars are reduced, the quasar luminosity function exhibits curvature over all luminosities, not just at the break (Wolf et al. 2003;Richards et al. 2005). Therefore, since both our data and the SDSS data show some curvature, and the SDSS has much smaller error bars, a least squares fit of a double power-law will force the break position (M * ) to be contained within the range of magnitudes covered by the SDSS. This produces a very steep faint end slope and is a poor fit to our faintest bins. Of the four parameters that determine the QLF, the SDSS can only measure the bright-end slope with certainty at z ∼ 3.2. Therefore, we also perform a fit with a fixed bright end slope, α = −2.85, defined by the SDSS measured relation of α with redshift at z = 3.2. This yields a more realistic values of M * = −25.6, and a faint end slope, β = −1.62±0.19, which is constrained by 4-5 bins fainter than M * . The results of the double power-law fits are given in Table 10.
Maximum Likelihood Fit
In the above fit to the binned data, we assumed that all of our QSOs are at a single redshift (z = 3.2). For most of our objects, however, we do not have spectroscopic redshifts and do not know the absolute magnitudes. This can be problematic because we do not know, for example, if the brighter QSOs are indeed more luminous or simply at the low redshift end of our redshift range (the distance modulus changes by ∼ 0.5 mags from z = 2.8 − 3.4). Furthermore, the luminosity function is known to change over these redshifs, which may skew the apparent magnitude distribution from what is expected at a single redshift.
Our only known quantity is the distribution of apparent magnitudes of QSOs in this redshift range. Therefore, we have modeled the expected distribution in QSO apparent magnitudes to compare to the observed distribution. The model distributions are computed in the following way.
1. We allow Φ * , M * , and faint-end slope, β, to vary independent of one another. We keep the brightend slope fixed at α = −2.85 as determined by SDSS QSOs (Richards et al. 2006b).
2. For each set of parameters, we compute the apparent magnitude distribution in small redshift intervals.
3. At each redshift, we apply the completeness function as a function of apparent magnitude, C(r ′ , z), computed in Section 7.3.1.
4. Finally, we sum up the apparent magnitude distribution in each redshift interval to determine the expected apparent magnitude distribution function over the entire redshift range for each set of QLF parameters.
5. We repeat the above steps three times with different QLF evolution: no evolution, pure luminosity evolution (PLE) and pure density evolution (PDE). For the two evolving models, we choose the level of evolution to fit the variation in space density of bright QSOs seen by SDSS (∼ 40% decrease from z ∼ 2.8 − 3.5, Richards et al. 2006b).
In addition to the observed apparent magnitude distribution from our sample, we also use the QSO sample of Richards et al. (2006b) from the SDSS Data Release 3. We only use the QSOs with 2.9 < z < 3.5 and use the completeness for each QSO derived in Richards et al. (2006b).
The apparent magnitude distribution function gives the relative probability of finding a QSO with a given magnitude from the whole sample. We compute the likelihood that a given parameter set adequately describes the data as the product of the values of the apparent magnitude distribution function for all of the QSO r ′ magnitudes in our sample (Marshall et al. 1983). We then find the set of parameters which maximizes this likelihood. The best fit model parameters are given in Table 10. The maximum-likelihood fit gives a somewhat shallower faint-end slope, β = −1.42 ± 0.15, and a similar location of the break, and agrees within ∼ 1σ with the binned QLF when we assumed all QSOs were at z = 3.2. The apparent magnitude distributions which included evolution in the QLF (both PLE and PDE) produce best-fit parameters which were not significantly different from the no evolution fits because there is little evolution over such a small redshift range. We choose to use the PLE model, with β = −1.42 ± 0.15, in the subsequent analysis. Figure 16 are three previously determined luminosity functions at z ∼ 3. Pei (1995) compiled several QSO samples to produce a QLF and its evolution between 0 < z < 4.5. A constant QLF shape was assumed, and fitted with Pure Luminosity Evolution. Unfortunately, most of the high redshift QSOs are very luminous, so the faint-end slope is mostly determined by lower redshift QSOs. As seen in Figure 16, the Pei (1995) QLF has a steeper faint end slope, β = −1.6 and lies above our QLF determination until M * 1450 < −26, where it passes through the SDSS data points. We believe that this discrepancy is caused by the assumption of a constant QLF shape. The Pei (1995) QLF shape was determined mostly by QSOs at z < 3, and normalized to the bright QSOs at high-redshift. Because the bright end slope at z > 3 is getting shallower (Richards et al. 2006b), this has caused a significant over-estimate of the space densities of faint QSOs. In fact, all previous QLF estimates at high redshift that assume a shape derived at low redshift and normalize using bright, QSOs will overestimate the number of faint QSOs at z > 3. At z ∼ 3, this results in a factor of ∼ 2 overestimate, but will get worse at higher redshift as the bright end slope is measured to get even shallower at 3 < z < 5.
Plotted in
The H04 QLF, though measured with a much smaller sample, shows a very shallow faint-end slope and a steep bright-end slope, resulting in a very prominent break. Our QLF has a significantly shallower bright-end slope and a steeper faint end slope. The space densities of our faintest QSOs are about twice that of H04 and predict 2-3 times more QSOs between −24 < M 1450 < 20.
H04 threw out most of their AGN sample (16 of 29) because the emission line widths were less than 2000 km/s. These narrow-line AGN are, on average, one magnitude fainter than the broad-line sample. Therefore, if these were included, they would have added significantly to the faint-end counts, resulting in a steeper faint-end slope. We cannot discriminate between different types of AGN in our sample, since we only have spectra for 10 objects (9 of 10 have F W HM > 2000 km/s). However, narrowline AGN (ie., F W HM < 2000 km s −1 ) may explain at least part of the discrepancy in our faint-end slopes.
Recent surveys of X-ray selected AGN have also concluded that the faint end slope gets shallower at higher redshift (Ueda et al. 2003;Hasinger et al. 2005). The Hasinger et al. (2005) sample overlaps our redshift range but differs from ours in the same way as H04, as it is a soft X-ray selected type-I AGN sample. If the difference between the H04 and Hasinger et al. (2005) luminosity functions and our QLF is attributed to increasing numbers of moderately obscurred AGN at fainter rest-frame UV luminosities (as suggested by Ueda et al. (2003)), then we would expect to see increasing numbers of QSOs with redder UV spectral slopes and higher L IR /L UV ratios amongst faint QSOs. Indeed, Figure 11 shows a population of r ′ > 21 QSOs which are redder, in both r ′ − [4.5] and r ′ − [24], than any at r ′ < 21, which may not be included in the H04 or Hasinger et al. (2005) LFs.
A recent optical/X-ray search for faint AGN at 3.1 < z < 5.2 by Fontanot et al. (2007) finds relatively large space densities, requiring a steep faint-end slope, β = −1.71 ± 0.41. Though the errors are large, this rules out a significantly shallower faint-end slope at z > 3 and agrees, within 1σ, with our fit. In their study, only 18% (2/11) AGN have narrow lines. Therefore, the fraction of narrow-line to broad-line AGN is too small to completely explain the discrepancy between the shallow faint-end slope of H04 and the steep faint-end slope found in this study and in Fontanot et al. (2007).
The COMBO-17 (Wolf et al. 2003) and VVDS (Bongiorno et al. 2007) QLFs Bongiorno et al. (2007) are both consistent with our QLF, within errors. Both of these surveys did not require any optical color or morphological criteria yet they still agree with our numbers. Both the (Wolf et al. 2003) and (Bongiorno et al. 2007) samples are larger than ours (192 and 130, respectively), but the QSOs are spread out over all redshifts (0 < z < 5) so they have 5-10 times fewer QSOs in this redshift range.
QSO CONTRIBUTION TO HI IONIZING FLUX AT
Z ∼ 3.2
With this new determination of the QSO Luminosity Function, we can simply integrate Φ(L) given in Equation 1 to determine the specific luminosity density (at λ = 1450Å) of QSOs at z ∼ 3.2,
ǫ = Φ(L) L dL.(10)
Integrating from −30 < M 1450 < −20, (43.96 < log(L 1450 ) < 47.96 ergs s −1 ), we derive a value for the specific luminosity density, ǫ 1450 = 7.3 × 10 24 ergs s −1 Hz −1 h Mpc −3 . This is comparable to the values derived from the Hunt et al. (2004) QLF (7.1 × 10 24 ) and significantly lower than the value from the Pei (1995) QLF (1.4×10 25 ) when correcting for different cosmologies. Although our QLF determination predicts more integrated UV flux from faint QSOs than that of Hunt et al. (2004), our M * is more than 1.5 magnitudes fainter. These two effects essentially cancel out to produce similar luminosity densities.
We can now determine the photoionization rate z=3 QSO Luminosity Function
13 Γ = ∞ ν0 dν4π J(ν) hν σ HI (ν).(11)
However, we can not assume that J(ν) is the shape of the average QSO SED, since the higher energy photons will be reprocessed by HI and HeII, resulting in a higher value for Γ. Haardt & Madau (1996) have modeled this reprocessing in a "clumpy" IGM to determine the effect on the HI photoionization by QSOs. This correctly includes HI clouds as sources of ionizing photons, as well as sinks, and increases Γ by ∼ 40% at z ∼ 3. We multiply ǫ 1450 by the ratio of f 912 /f 1450 = 0.58 in our template, and convert to a proper volume emissivity at z = 3.2 to directly compare with the ǫ Q calculated by Haardt & Madau (1996) and how it scales with the photoionization rate, Γ HI and the ionizing intensity at the Lyman Limit J 912 . It should be stated that the scaling relations from ǫ Q to J 912 and Γ HI are dependent upon the value of ǫ Q itself, as this will effect the ionization levels of the surrounding medium. However this is a secondary effect, and our ǫ Q is within ∼ 50% of the Haardt & Madau (1996) value so we don't expect this to be a large effect on the scaling relations.
We get values of Γ HI ∼ 4.5 × 10 −13 s −1 and J 912 = 1.5 × 10 −22 ergs s −1 cm −2 Hz −1 sr −1 . As pointed out by Hunt et al. (2004), these should be taken as upper limits as it assumes that all ionizing photons escape from QSOs of all luminosities, though this is not at all clear for lower luminosity AGN.
It is interesting to compare the HI photoionization rate from QSOs with that from star-forming galaxies (LBGs at these redshifts). Unfortunately, it is difficult to determine the photoionization rate from Lyman Break galaxies, since their photoionizing SEDs are difficult to directly detect and are sensitive to parameters with large uncertainties: dust reddening, initial mass function, starburst age, metallicity, and the escape fraction of ionizing photons, f esc (Steidel et al. 2001;Shapley et al. 2006;Siana et al. 2007).
It is possible, however, to address this question indirectly if the total ionizing background (QSOs+galaxies) is accurately determined. Several groups have made these measurements by measuring the mean transmission of QSO UV flux through the Lyα forest (McDonald & Miralda-Escudé 2001;Tytler et al. 2004;Bolton et al. 2005;Bolton & Haehnelt 2007;Becker et al. 2007), or by measuring the extent of the proximity effect (Carswell et al. 1982) around high redshift QSOs (Scott et al. 2000(Scott et al. , 2002. Figure 17 shows the current estimates for the total photoionization rate at high redshift, compared with our estimate of the contribution from QSOs at z ∼ 3.2. The lower limit to Γ is derived by integrating the QLF between 30 < M 1450 < 23.5, the range covered by our survey and SDSS. The upper limit is derived by integrating the QLF 3.5 magnitudes fainter and assumes a 100% escape fraction amongst these faint QSOs as well. The Scott et al. (2000) value has been scaled (by the value given in Scott et al. (2002)) to our assumed cosmology. The McDonald & Miralda-Escudé (2001) values have also been scaled to the same cosmology using Eqn. 3 in their paper.
Although the error bars are large, all of the total photoionization measurements are consistent and give a (Tytler et al. 2004), diamonds (McDonald & Miralda-Escudé 2001, and squares (Bolton et al. 2005;Bolton & Haehnelt 2007). The shaded region is the photoionization rate determined by Scott et al. (2000) adjusted to our cosmology (decrease of 31%) as stated in Scott et al. (2002). The open circle is the Hunt et al. (2004) determination of the QSO contribution at z ∼ 3. The black square encompasses the redshift range and plausible limits of the QSO contribution to HI photoionization from our sample. The lower and upper bounds are determined by integrating the QSO luminosity function to R < 22 (M 1450 < −23.5) and R < 25.5 (M 1450 < −20), respectively.
value of Γ ∼ 1.0 × 10 12 s −1 at z = 3. The contribution from QSOs is less than half of this value. Therefore, it is likely that star-forming galaxies' contribution to the HI photoionization rate is comparable to that of QSOs at z ∼ 3.2. This is consistent with measurements by Shull et al. (2004) which examine the relative rates of HeII and HI photoionization within individual Lyα absorbers at 2.3 < z < 2.9 to infer the spectral index of background radiation at each location. They conclude that the spectral index varies greatly between absorbers, with significant contribution from "soft" sources that may be starburst galaxies or dust-attenuated AGN.
Furthermore, recent measurements of the escape fraction, f esc , of photoionizing radiation from Lyman Break Galaxies also suggest that star-formation may significantly contribute to the ionizing background at z ∼ 3. Steidel et al. (2001) made a composite rest-frame ultraviolet spectrum of 29 LBGs and found that greater than 50% of the photoionizing flux that is not absorbed by dust escapes into the IGM (ie. relative escape fraction f esc,rel > 0.5). Deeper spectra of 14 z ∼ 3 LBGs give a smaller value of f esc,rel = 0.14, but this still gives an ionizing radiation field J 900 ∼ 2.6 × 10 −22 erg s −1 cm −2 Hz −1 , nearly twice the value of our upper limit from QSOs.
SUMMARY
We present our method of finding high redshift, z > 2.8, QSOs by identifying a Lyman Break in the optical photometry, and ensuring red mid-IR ([3.6]−[4.5]) colors indicative of QSOs. The use of only three optical filters allows a search over larger areas in the SWIRE fields as most of the area does not have coverage in four or more bands. The use of only IRAC1 and IRAC2 channels is emphasized as these two bands are a factor of seven times more sensitive than IRAC3 and IRAC4.
Spectroscopic follow-up of 10 z ∼ 3 (U -dropout) candidates confirms that all 10 are QSOs between 2.83 < z < 3.44. Spectroscopy of 10 z ∼ 4 (g ′ -dropout) candidates confirmed 7 QSOs with 3.48 < z < 3.88, two galaxies at low redshift (z = 0.354, 0.390) and one unconfirmed redshift. We place reliability estimates on our z ∼ 3 and z ∼ 4 samples of 100% (> 69% 1σ) and 70 +16 −26 %, respectively. Since we have not spectroscopically confirmed all of our candidates, we only use the more reliable z ∼ 3 sample for determining a luminosity function.
By using detailed models which include variations in number and column density of line-of-sight HI absorbers, UV spectral slope, emission line equivalent width, redshift, observed magnitude, and photometric errors, we assess the completeness of the optical color selection. Completeness near the center of the redshift range of our U -dropout selection is 85-90%. However, our completeness decreases significantly in our faintest magnitudes bins (∼ 75%), due to the shallow depth of the U -band imaging.
We find 100 z ∼ 3 QSO candidates with r ′ < 22 over 11.7 deg 2 . Through our models of completeness versus redshift, we derive effective volumes for each halfmagnitude bin and compute the z ∼ 3 QSO luminosity function. When combined with SDSS data, a least-squares fit to a double power-law gives a faint-end slope, β = −1.62 ± 0.19, and location of the break at M * = −25.6.
Our binned QLF assumes that all of the QSO candidates are at z = 3.2, which may skew the fitted parameters because of luminosity function evolution over our redshift range and Eddington Bias (Eddington 1913) due to large dispersions in the actual absolute magnitude distribution. Therefore, we have performed a maximum likelihood fit of the apparent magnitude distribution of our sample with that inferred from a specific QLF over this redshift range. Our results are slightly different, with a shallower faint-end slope, β = −1.43 ± 0.15, and a somewhat fainter break at M * = −24.9. This fit is more accurate as it does not assume that all of the QSOs are at the same redshift.
The fitted slope is consistent, within the errors, with values measured at low redshift (0.5 < z < 2.0), β = −1.45 ) and therefore does not require evolution in the faint end slope of the luminosity function. Our QLF predicts significantly more faint QSOs than suggested with initial measurements at z ∼ 3 . Although it is difficult to tell with our limited spectroscopic sample, some of the difference between our faint-end slope and that of Hunt et al. (2004) may be attributed to an increasing number of narrowline, moderately reddened AGN at fainter UV luminosities which were excluded from the H04 sample.
The QLF exhibits some curvature at all magnitudes and, because of this, the parameters for a double powerlaw fit are degenerate. That is, the position of the break, (M * ,Φ * ), can be fit at different locations along the binned QLF, with appropriate changes in the bright and faint end slopes (α,β). This is especially true at high redshift, where the difference between the bright and faint end slopes appears to decrease. Therefore, one must be careful when assigning physical significance to the measured values of these parameters when comparing to models. The QSOs in our sample span the break in the luminosity function (0.25L * < L < 4.0L * ) and thus we measure the space density of QSOs that comprise the majority (55%) of the QSO UV luminosity density at these redshifts. When combined with the SDSS sample this percentage is more than 70%. Therefore, large extrapolations are not required to estimate the effects of undetected QSOs. The integrated UV luminosity density at z ∼ 3.2 is ǫ 1450 = 7.3 × 10 24 ergs s −1 Hz −1 h Mpc −3 . Using the scaling relation derived by Haardt & Madau (1996), we infer a maximum HI photoionization rate by QSOs, Γ = 4.5×10 −13 s −1 . This is about 50% of the total IGM HI photionization rate at z = 3, requiring comparable ionizing flux from either starburst galaxies or redder AGN that lie outside our color criteria.
This work is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. Support for this work, part of the Spitzer Space Telescope Legacy Science Program, was provided by NASA through an award issued by JPL/Caltech, under NASA contract 1407.
Based in part on observations obtained at the Hale Telescope, Palomar Observatory as part of a continuing collaboration between the California Institute of Technology, NASA/JPL, and Cornell University.
Based in part on data made publically available through the Isaac Newton Groups' Wide Field Camera Survey Programme. The Isaac Newton Telescope is operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Canarias.
This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.
Facilities: Spitzer(IRAC,MIPS)
Sample
Color Criteria (Vega) Optical/IR Photometry of the 100 z ∼ 3 QSO candidates. Spectroscopic redshifts are given when available. U -band magnitudes in parentheses are 5σ limits. Two objects with spectroscopic confirmation were moved just outside of our color criteria (due to revised photometry) and are no longer part of our primary sample. We list these at the end of the
U − g ′ ≥ 0.33 z ∼ 3 g ′ − r ′ ≤ 1.0 U − g ′ ≥ 3.9 × (g ′ − r ′ ) − 2.0 g ′ − r ′ ≥ 1.241 z ∼ 4 r ′ − i ′ ≤ 1.146 g ′ − r ′ ≥ 2.178 × (r ′ − i ′ ) + 0.09
Fig. 1 .
1-The new optical/IR QSO template combined with
Fig. 3 .
3g ′ − r ′ , r ′ − i ′ color-color diagram showing the z ∼ 4 QSO selection. The symbols and lines are the same as inFigure 2.
Fig. 4 .
4-IRAC colors of point sources in ELAIS-N1 with r ′ < 22. The dash-dotted line is the Stern et al. (2005) AGN selection and the dashed line is our selection [3.6] − [4.5] > −0.15 (AB) for demarcation of AGN from stars.
Fig. 5 .
5-[3.6] − [4.5] color vs. redshift. The line types are the same as in Figure 2. Filled circles are SDSS QSOs in the SWIRE fields. The lighter, long dashed line is our IRAC color selection, [3.6] − [4.5] > −0.15(AB).
Fig. 6 .
6-The U − g ′ , g ′ − r ′ optical colors of objects with r ′ < 22 and classified as point sources. The solid curve is the color track of our QSO template with IGM absorption applied, and the square denotes the z = 0 point. The green contours denote the density (in color space) of objects categorized as stars by their blue IRAC colors ([3.6]−[4.5] < −0.15). Red crosses are point sources with red IRAC colors ([3.6] − [4.5] > −0.15).
7 Fig. 7 .
77-Keck/LRIS (top 6) and Palomar/COSMIC (bottom 4) spectra of z ∼ 3 QSO candidates. The ID numbers from
Fig. 8 .
8-Keck/LRIS (top 4) and Palomar/COSMIC (bottom 2) spectra of z ∼ 4 QSO candidates. The ID numbers from
Fig. 9 .
9-Keck/LRIS (top 2) and Palomar/COSMIC (bottom) spectra of interlopers in the g ′ -dropout sample. The official SWIRE names and redshifts (if known) are labeled.
Fig. 10 .
10-The IRAC ([3.6] − [4.5]) color distribution of all point sources with r ′ < 22 in our field (dashed line). The solid line show the color distribution of point sources which also meet our optical color criteria. The shaded histogram is the distribution of spectroscopically confirmed QSOs. Our QSO sample is the portion of the solid histogram to the right of the vertical dotted line.
Fig. 11 .
11-Optical-IR color distribution of the z ∼ 3 QSO sample. The x-axis is in Vega magnitudes since our magnitude bins are defined in Vega magnitudes and the y-axis is in AB magnitudes as it is simpler to interpret. The dashed line in the top plot denotes the f 4.5µm = 10µJy limit of our search. The dashed line in the bottom plot denotes the f 24µm = 250µm completeness limit with arrows giving upper limits from non-detections. The four QSOs with r ′ − [4.5] < 0.0 are possible interlopers.
Fig. 12 .
12-IRAC colors of the z ∼ 3 QSO sample. The lines are as in Figure 4 . Red arrows are lower limits to the [5.8] − [8.0] color based on non-detections in IRAC3. Blue arrows are upper limits to the 5.8] − [8.0] color based on non-detections in IRAC4. Non-detections in both IRAC3 & IRAC4 are plotted along the left hand side.
Fig. 13 .
13-A subsection of the simulated IGM transmission (e −τ ) curve for one simulated line-of sight. All aborption features are fully resolved in the simulation.
Fig. 14 .
14-The U-band depth map for EN1. The grey-scale represents U magnitude limits in intervals of 0.1 mags from 22.5 to 24.0 (Vega) with most of the area between 23.0-23.5
Fig. 15 .
15-The z ∼ 3 QSO completeness C(r ′ , z) contours as a function of apparent r ′ magnitude and redshift. The contours are spaced at ∆C = 0.1 intervals.
9 .
9COMPARISON WITH PREVIOUS LUMINOSITY FUNCTIONS Fig. 16.-The z ∼ 3.2 QSO luminosity function. The binned data are plotted from this work (black circles and black arrows for upper limits), SDSS Richards et al. (2006b, blue squares), COMBO-17 (Wolf et al. 2003, red diamonds), Hunt et al. (2004, cyan triangles),WHO (Warren et al. 1994, green asterisks), Fontanot et al. (2007, open purple circles), and Bongiorno et al. (2007, orange stars). Filled symbols were used in the fitting of the QLF. Also plotted are the fitted QLFs using the binned data (dot-dashed line), the maximum likelihood fit with Pure Luminosity Evolution (solid line), and the QLFs of Pei (1995, dotted line), Hunt et al. (2004, magenta dashed line), and Wolf et al. (2003, red long dashed line).
Fig. 17 .
17-HI photoionization rate per atom (in units of 1×10 −12 s −1 ) versus redshift. Estimates of the total photoionization rate of the IGM are plotted as asterisks
). The IR filter characteristics and SWIRE depths are summarized in Table 1. The EN1 IRAC campaign was undertaken January 14-20, 2004 and MIPS January 21-28, 2004. MIPS went into standby mode on January 25, 2004, resulting in lost AORs which were reobserved July 29, 2004. The EN2 IRAC campaign was observed July 05-06, 2004 and MIPS July 08-11, 2004. The IRAC and MIPS photometry were measured with the SExtractor program
smooth, continuous SEDs, but sharp features such as emission lines and continuum breaks are convolved with the filters through which they are observed, effectively broadening the features. Deconvolution of this broadening is difficult since our photometry comes from 13 different filters (ugrizJHK,IRAC1-4,MIPS24). The only sharp feature at λ > 5500Å is the combined emission lines of Hα (6563Å) & [NII] (6548Å and 6592Å). We therefore use the broadband photometry to get the shape of the underlying continuum at λ > 5100Å, subtract the areas affected by Hα &[NII], and interpolate the continuum along this region. We then add the Hα+[NII] profile from the Vanden Berk et al.template (blue) and the Vanden Berk et al. (2001) SDSS
composite spectrum (red). All templates are normalized at λ =
2900Å.
(2001) template (scaled to
the 2400Å continuum value to preserve the relative line
strengths).
The resulting infrared QSO template is shown in Fig-
ure 1 and tabulated in
TABLE 1 SWIRE
1IR Depths fromSurace et al. (2005) Filter Central Wavelength (µm) Depth (µJy, 5σ) Filter Central Wavelength (Å) Width (Å) m AB (V ega) a Depth (Vega, 5σ) a m AB (Vega) is the Vega to AB conversion factor where m AB = m V ega + m AB (V ega).IRAC1
3.6
6
IRAC2
4.5
7
IRAC3
5.8
42
IRAC4
8.0
50
MIPS24
24.0
250
TABLE 2
WFS Optical Depths
U
3560
600
0.78
24.3
g ′
4857
1400
-0.09
25.2
r ′
6216
1380
0.15
24.5
i ′
7671
1535
0.40
23.7
Z
9100
1370
0.54
22.1
TABLE 3
3The broadband optical/IR template combined with the Telfer et al. (2002)/Vanden Berk et al. (2001) composite spectra. Note. -[The complete version of this table is in the electronic edition of the Journal. The printed edition contains only a sample.]TABLE 4 Optical Selection Criteria.λ (µm)
f λ
0.0302
2.250
0.0303
4.086
0.0304
6.034
0.0305
5.841
0.0306
6.000
TABLE 5
5
table. SWIRE J160637.88+535008.4 241.65784 53.83568 2.943 aNo.
Name
RA
Dec
zspec
U
g ′
r ′
i ′
Z
IRAC1 IRAC2 IRAC3 IRAC4 MIPS24
[Deg]
[Deg]
[Vega] [Vega] [Vega] [Vega] [Vega]
[µJy]
[µJy]
[µJy]
[µJy]
[µJy]
1 SWIRE J155907.09+550325.5 239.77954 55.05709
· · ·
22.08
21.37
21.42
21.17
20.86
12
20
44
46
<250
2 SWIRE J160123.47+553750.3 240.34778 55.63064
· · ·
22.75
22.19
21.62
21.14
21.34
20
20
<42
<50
<250
3 SWIRE J160153.49+542356.6 240.47287 54.39906
· · ·
22.15
21.37
21.56
21.01
20.36
99
100
137
138
593
4 SWIRE J160223.85+552012.2 240.59938 55.33673
· · ·
21.92
21.48
21.09
20.59
19.95
28
29
50
57
336
5 SWIRE J160318.38+552703.2 240.82657 55.45089
· · ·
21.51
20.95
20.39
20.01
19.63
56
83
133
225
827
6 SWIRE J160345.78+542337.2 240.94077 54.39367
· · ·
20.58
20.20
19.68
19.23
19.04
12
11
<42
<50
<250
7 SWIRE J160422.50+535454.9 241.09373 53.91525
· · ·
22.01
21.23
20.60
20.18
20.03
26
35
60
118
473
8 SWIRE J160426.31+553446.9 241.10963 55.57969
· · ·
(23.30) 21.37
20.91
20.67
20.57
22
24
41
59
<250
9 SWIRE J160452.28+542758.2 241.21782 54.46617
2.98
22.17
21.15
21.30
21.13
20.86
28
34
<42
67
<250
10 SWIRE J160520.52+552704.6 241.33548 55.45128
3.30
22.88
20.51
20.26
20.19
20.13
48
50
66
73
461
11 SWIRE J160528.14+560635.6 241.36725 56.10989
· · ·
22.88
22.21
21.78
21.52
20.67
22
21
<42
45
712
12 SWIRE J160606.33+550412.3 241.52638 55.07008
· · ·
22.83
22.25
21.89
21.55
21.13
16
19
<42
35
<250
13 SWIRE J160617.56+541649.5 241.57318 54.28042
· · ·
23.60
22.00
21.21
20.74
21.19
32
37
59
116
290
14 SWIRE J160621.18+552532.7 241.58827 55.42574
· · ·
23.12
21.59
21.11
20.68
20.21
53
61
76
78
382
15 20.63
19.79
19.47
18.95
18.75
109
169
312
739
2904
16 SWIRE J160654.19+554028.4 241.72578 55.67455
2.97
22.45
21.72
21.20
20.82
20.63
21
29
<42
58
306
17 SWIRE J160656.10+535633.4 241.73373 53.94261
· · ·
(23.33) 21.53
21.12
20.66
20.57
32
33
<42
68
276
18 SWIRE J160724.00+533615.2 241.85002 53.60422
· · ·
(23.25) 21.77
21.12
20.69
20.48
15
17
<42
45
203
19 SWIRE J160733.94+554428.7 241.89142 55.74130
· · ·
21.24
20.39
19.69
19.04
18.45
112
138
228
375
995
20 SWIRE J160754.39+533916.6 241.97661 53.65462
3.01
23.29
22.39
21.93
21.49
21.37
20
23
<42
<50
202
21 SWIRE J160758.67+543137.8 241.99446 54.52716
· · ·
21.46
20.85
20.59
20.19
19.84
44
43
<42
80
<250
22 SWIRE J160824.08+542003.7 242.10033 54.33435
· · ·
22.98
22.39
21.98
21.71
21.04
17
16
<42
<50
<250
23 SWIRE J160850.55+545800.5 242.21063 54.96680
· · ·
(23.34) 22.60
21.92
21.52
21.45
11
11
<42
<50
<250
24 SWIRE J160907.33+543329.8 242.28056 54.55827
· · ·
22.85
21.16
20.57
20.33
20.15
30
36
49
113
443
25 SWIRE J160917.26+553638.2 242.32191 55.61062
· · ·
22.85
21.33
20.96
20.64
20.21
50
62
57
87
305
26 SWIRE J160947.86+552542.9 242.44943 55.42857
· · ·
20.47
19.91
19.35
19.03
18.73
71
84
82
208
690
27 SWIRE J161008.08+552944.2 242.53366 55.49561
· · ·
22.56
21.73
21.46
21.17
20.92
14
14
<42
<50
<250
28 SWIRE J161008.34+533254.9 242.53476 53.54857
3.20
22.81
21.51
20.99
20.18
19.78
63
66
82
150
494
29 SWIRE J161051.96+531004.4 242.71651 53.16788
· · ·
22.79
21.82
21.09
20.56
20.15
32
37
58
110
417
30 SWIRE J161115.37+534029.2 242.81406 53.67478
· · ·
22.89
22.28
21.77
20.75
20.05
121
117
132
135
<250
31 SWIRE J161128.36+553409.9 242.86815 55.56943
· · ·
(23.08) 22.35
21.70
21.34
21.27
15
24
<42
<50
<250
32 SWIRE J161128.47+535809.3 242.86862 53.96926
· · ·
(23.34) 22.24
21.94
21.46
21.72
22
38
58
190
754
33 SWIRE J161132.10+542312.0 242.88374 54.38667
· · ·
22.86
21.23
20.58
20.43
20.26
48
56
70
131
299
34 SWIRE J161142.40+533104.6 242.92667 53.51794
3.06
22.42
21.35
21.04
20.77
20.51
37
43
57
86
302
35 SWIRE J161202.93+532346.9 243.01221 53.39636
· · ·
(23.12) 21.23
20.51
20.20
19.97
26
29
<42
<50
195
36 SWIRE J161251.98+534608.8 243.21658 53.76910
3.44
23.80
21.68
20.93
20.55
20.59
20
22
<42
<50
<250
37 SWIRE J161300.88+544629.6 243.25368 54.77489
· · ·
23.27
21.75
21.74
21.09
21.73
27
38
<42
76
166
38 SWIRE J161311.87+542403.7 243.29944 54.40102
· · ·
22.65
22.18
21.83
21.46
21.27
29
32
53
43
237
39 SWIRE J161326.39+530923.0 243.35994 53.15638
2.83
20.34
19.88
19.47
19.25
19.04
68
94
124
217
574
40 SWIRE J161341.27+532956.4 243.42195 53.49900
· · ·
22.79
21.95
21.24
21.05
21.26
23
31
<42
78
<250
41 SWIRE J161430.84+555133.3 243.62848 55.85926
· · ·
22.74
21.85
21.53
21.20
21.41
22
28
48
73
208
42 SWIRE J161433.14+533249.6 243.63808 53.54710
· · ·
21.04
20.15
19.55
19.36
19.07
112
124
151
306
1068
43 SWIRE J161442.44+554614.2 243.67683 55.77060
· · ·
22.86
21.68
21.12
20.76
20.48
19
22
<42
56
<250
44 SWIRE J161446.30+555229.5 243.69290 55.87486
· · ·
(23.19) 22.24
21.53
20.86
20.76
28
38
<42
107
308
45 SWIRE J161508.88+555514.6 243.78700 55.92073
· · ·
20.55
19.71
19.29
18.92
18.78
55
75
122
301
924
46 SWIRE J161526.80+555217.4 243.86165 55.87150
· · ·
22.50
21.96
21.33
20.66
20.19
67
78
117
237
810
47 SWIRE J161530.88+555247.1 243.87868 55.87976
· · ·
22.91
22.37
21.97
21.49
20.75
58
58
60
<50
<250
48 SWIRE J161549.80+540834.9 243.95749 54.14304
· · ·
(23.18) 22.25
21.84
21.66
22.05
9
11
<42
<50
<250
49 SWIRE J161626.05+535132.9 244.10854 53.85915
· · ·
(23.42) 21.48
20.82
20.29
19.93
43
45
63
98
286
TABLE 5 -
5ContinuedNo.
Name
RA
Dec
zspec
U
g ′
r ′
i ′
Z
IRAC1 IRAC2 IRAC3 IRAC4 MIPS24
[Deg]
[Deg]
[Vega] [Vega] [Vega] [Vega] [Vega]
[µJy]
[µJy]
[µJy]
[µJy]
[µJy]
50 SWIRE J161634.24+553528.2 244.14265 55.59116
· · ·
20.14
19.68
19.29
19.01
18.86
55
66
89
161
705
51 SWIRE J161638.27+555701.4 244.15947 55.95039
· · ·
21.60
20.82
20.42
19.93
19.80
46
62
72
153
616
52 SWIRE J161704.50+541200.4 244.26875 54.20012
· · ·
23.24
22.13
21.48
21.15
21.08
8
10
<42
<50
265
53 SWIRE J161719.00+540154.3 244.32916 54.03175
· · ·
22.66
22.05
21.42
21.17
20.98
28
40
45
107
356
54 SWIRE J161735.03+543830.8 244.39594 54.64189
· · ·
23.66
22.42
21.84
21.64
21.32
16
21
<42
44
381
55 SWIRE J161735.16+541405.9 244.39650 54.23497
· · ·
22.68
20.08
19.57
19.22
19.11
56
64
117
185
717
56 SWIRE J161849.23+543658.3 244.70511 54.61620
· · ·
22.37
21.91
21.53
21.28
21.13
15
17
<42
38
TABLE 7
7Spectral parameter attribute for our QSO SED modeling. W (Lyα+NV)(Å) 101.7 25.Parameter
Mean
σ
αν
-0.46 0.3
TABLE 8
8Parameters for the four column density ranges used in the Lyα Forest simulations. Kim et al. (1997) b Storrie-Lombardi et al. (1994) c Storrie-Lombardi & Wolfe(2000)Name
log(N HI (cm 2 ))
N 0
κ
γ
b(km/s)
Lyα Forest #1 a
12-14
181.36 1.46 1.29
30
Lyα Forest #2 a
14-17.2
1.297
1.46 3.10
30
Lyman Limit Systems b
17.2-20
0.27
1.50 1.55
70
Damped Lyα Systems c
20-22
0.055
1.78 1.11
70
a
TABLE 9 Tabulated
9Luminosity Function. Mpc 3 ) (10 7 mag −1 Mpc −3 )r ′
M(1450Å) n Completeness z ef f
V ef f
Φ
Vega
AB
(10 7 18.25
-27.11
0
1.00
3.22
7.87
< 0.47
18.75
-26.61
0
1.00
3.22
7.87
< 0.47
19.25
-26.11
7
1.00
3.22
7.87
1.78 +1.0
−0.7
19.75
-25.61
5
1.00
3.22
7.87
1.27 +0.9
−0.5
20.25
-25.11
12
0.99
3.22
7.83
3.07 +1.2
−0.9
20.75
-24.61
20
0.98
3.21
7.73
5.38 +1.5
−1.2
21.25
-24.11
25
0.86
3.18
6.77
7.38 +1.8
−1.5
21.75
-23.61
31
0.74
3.15
5.82
10.66 +2.3
−1.9
TABLE 10
10Parameters for double power-law luminosity function. The binned QLF assumes that all QSOs are at z = 3.2, whereas the Maximum Likelihood values are from fits to the apparent magnitude distribution. Values in parentheses denote 1σ errors and values in square brackets denote fixed parameters. The maximimum-likelihood fits that include QLF evolution (both PDE and PLE) are not significantly different from the no evolution parameters. Listed at the bottom are estimates from previous work.Data
α
β
M *
1450 (AB) Φ * (10 −7 mag −1 Mpc −3 )
χ 2
ν a
Binned QLF
SWIRE only
-3.53 (2.9)
-1.66 (0.88)
-25.8 (2.3)
3.5 (11.0)
2.4
3
SWIRE only (fixed α)
[-2.85]
-1.29 (0.85)
-24.8 (1.1)
10.4 (12.0)
2.5
4
SWIRE+SDSS
-3.47 (0.58) -1.98 (0.17)
-27.1 (0.7)
0.57 (0.65)
9.9
12
SWIRE+SDSS (fixed α)
[-2.85]
-1.62 (0.19)
-25.6 (0.3)
4.53 (2.0)
11.3 13
Maximum Likelihood Fit
SWIRE only (fixed α, no evol)
[-2.85]
-1.26 (0.21) -25.0 (0.30)
9.0 (3.0)
(PDE (1 + z) −3 )
[-2.85]
-1.22 (0.22) -25.0 (0.29)
9.3 (2.9)
(PLE )
[-2.85]
-1.25 (0.21) -25.0 (0.30)
9.2 (2.9)
SWIRE+SDSS (fixed α, no evol)
[-2.85]
-1.43 (0.15) -24.9 (0.15)
8.5 (2.0)
(PDE (1 + z) −3 )
[-2.85]
-1.41 (0.15) -24.9 (0.15)
8.6 (1.9)
(PLE )
[-2.85]
-1.42 (0.15) -24.9 (0.15)
8.6 (1.8)
Previous Studies
Hunt et al. (2004)
-4.56 (0.51) -1.24 (0.07)
-26.7
2.4
Pei (1995)
-3.52 (0.11) -1.64 (0.18) -25.8 (0.25)
6.1 (2.5)
Bongiorno et al. (2007)
-3.0
-1.38
-25.7
9.8
QSOs were most easily identified as point sources with U − B colors bluer than most stars.
a Degrees of freedom in χ 2 fit.
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Typical uncertainties are ∼ 0.04 mags in the optical and ∼ 10% in infrared fluxes. a zspec from SDSS. b Selected in the SWIRE Lockman field. c Low redshift interlopers. d Could not see emission lines for redshift confirmation. Note. -Typical uncertainties are ∼ 0.04 mags in the optical and ∼ 10% in infrared fluxes. a zspec from SDSS. b Selected in the SWIRE Lockman field. c Low redshift interlopers. d Could not see emission lines for redshift confirmation.
| [] |
[
"Autofeedback scheme for preservation of macroscopic coherence in microwave cavities",
"Autofeedback scheme for preservation of macroscopic coherence in microwave cavities"
] | [
"M Fortunato \nDipartimento di Matematica e Fisica\nCamerino and INFM\nUniversità di Camerino\nvia Madonna delle Carceri I\nUnità di Camerino\n62032Italy\n",
"J M Raimond \nDépartement de Physique de l'Ecole Normale Supérieure\nLaboratoire Kastler Brossel\n24 rue LhomondF-75231, Cedex 05ParisFrance\n",
"P Tombesi \nDipartimento di Matematica e Fisica\nCamerino and INFM\nUniversità di Camerino\nvia Madonna delle Carceri I\nUnità di Camerino\n62032Italy\n",
"D Vitali \nDipartimento di Matematica e Fisica\nCamerino and INFM\nUniversità di Camerino\nvia Madonna delle Carceri I\nUnità di Camerino\n62032Italy\n"
] | [
"Dipartimento di Matematica e Fisica\nCamerino and INFM\nUniversità di Camerino\nvia Madonna delle Carceri I\nUnità di Camerino\n62032Italy",
"Département de Physique de l'Ecole Normale Supérieure\nLaboratoire Kastler Brossel\n24 rue LhomondF-75231, Cedex 05ParisFrance",
"Dipartimento di Matematica e Fisica\nCamerino and INFM\nUniversità di Camerino\nvia Madonna delle Carceri I\nUnità di Camerino\n62032Italy",
"Dipartimento di Matematica e Fisica\nCamerino and INFM\nUniversità di Camerino\nvia Madonna delle Carceri I\nUnità di Camerino\n62032Italy"
] | [] | We present a scheme for controlling the decoherence of a linear superposition of two coherent states with opposite phases in a high-Q microwave cavity, based on the injection of appropriately prepared "probe" and "feedback" Rydberg atoms, improving the one presented in [D. Vitali et al., Phys. Rev. Lett. 79, 2442(1997]. In the present scheme, the information transmission from the probe to the feedback atom is directly mediated by a second auxiliary cavity. The detection efficiency for the probe atom is no longer a critical parameter, and the decoherence time of the superposition state can be significantly increased using presently available technology. | 10.1103/physreva.60.1687 | [
"https://export.arxiv.org/pdf/quant-ph/9902071v2.pdf"
] | 119,369,907 | quant-ph/9902071 | 84ba2526f08163cff37e7313359b468ae01d7e03 |
Autofeedback scheme for preservation of macroscopic coherence in microwave cavities
(April 1, 2022)
M Fortunato
Dipartimento di Matematica e Fisica
Camerino and INFM
Università di Camerino
via Madonna delle Carceri I
Unità di Camerino
62032Italy
J M Raimond
Département de Physique de l'Ecole Normale Supérieure
Laboratoire Kastler Brossel
24 rue LhomondF-75231, Cedex 05ParisFrance
P Tombesi
Dipartimento di Matematica e Fisica
Camerino and INFM
Università di Camerino
via Madonna delle Carceri I
Unità di Camerino
62032Italy
D Vitali
Dipartimento di Matematica e Fisica
Camerino and INFM
Università di Camerino
via Madonna delle Carceri I
Unità di Camerino
62032Italy
Autofeedback scheme for preservation of macroscopic coherence in microwave cavities
(April 1, 2022)arXiv:quant-ph/9902071v2 30 Aug 1999
We present a scheme for controlling the decoherence of a linear superposition of two coherent states with opposite phases in a high-Q microwave cavity, based on the injection of appropriately prepared "probe" and "feedback" Rydberg atoms, improving the one presented in [D. Vitali et al., Phys. Rev. Lett. 79, 2442(1997]. In the present scheme, the information transmission from the probe to the feedback atom is directly mediated by a second auxiliary cavity. The detection efficiency for the probe atom is no longer a critical parameter, and the decoherence time of the superposition state can be significantly increased using presently available technology.
I. INTRODUCTION
The problem of how the classical macroscopic world emerges from the quantum substrate is an important point in the interpretation of quantum mechanics and it is still the subject of an intense debate [1,2]. This problem is well illustrated by the possibility, opened by quantum mechanics, of having linear superpositions of macroscopically distinguishable states, the so-called "Schrödinger cat" states. An explanation of why we never observe these paradoxical states is proposed by the decoherence models, i.e., the rapid transformation of these linear superpositions into the corresponding classical statistical mixture, caused by the unavoidable entanglement of the system with uncontrolled degrees of freedom of the environment [1]. The decoherence time depends on the form of system-environment interaction [3] but, in most cases, it is inversely proportional to the squared "distance" between the two states of the superposition [4]. For macroscopically distinguishable states, the decoherence process becomes thus practically instantaneous [1]. Decoherence is experimentally accessible only in the mesoscopic domain. In this case, one is able to monitor the progressive emergence of classical properties from the quantum ones. A first important achievement has been obtained by Monroe et al. [5], who prepared a trapped 9 Be + ion in a superposition of spatially separated coherent states and detected the quantum coherence between the two localized states. However, the decoherence of the superposition state has not been studied in this experiment. The progressive decoherence of a mesoscopic Schrödinger cat has been observed for the first time in the experiment of Brune et al. [6], where the linear superposition of two coherent states of the electromagnetic field in a cavity with classically distinct phases has been generated and detected.
With the impressive development of quantum information theory in the last years [7], the study of decoherence has become important not only from a fundamental, but also from a more practical point of view. All the quantum information processing applications rely on the possibility of performing unitary transformations on a system of N quantum bits, whose decoherence has to be made as small as possible. For this reason, decoherence control is now a rapidly expanding field of investigation. In this respect, quantum error correction codes [8] have been developed in which the entangled superposition state of N qubits is "encoded" in a larger number of qubits. Assuming that only a fraction of qubits decoheres, it is then possible to reconstruct the original state with a suitable decoding procedure, provided that errors affect different qubits independently. These codes always require the entanglement of a large number of qubits, and will become practical only if quantum networks of tens of qubits become available. Up to now, the polarization states of three photons have been entangled at most [9]. Entangled states of two Rydberg atoms [10] or of two trapped ions [11] at most can be generated. Therefore, in the present experimental situation, it is more realistic to study complementary and more "physical" ways to harness decoherence, based on the knowledge of the specific process causing decoherence, which could be applied with very few degrees of freedom. This is possible, in particular, in quantum optics, when information is encoded in the quantum states of an electromagnetic mode (see for example [12]). In this case decoherence is caused by photon leakage. It could therefore be possible to develop experimental schemes able to face photon leakage and the associated decoherence.
A series of papers [13][14][15][16][17] have shown that a possible way to control decoherence in optical cavities is given by appropriately designed feedback schemes. Refs. [13] show that a feedback scheme based on the continuous homodyne measurement of an optical cavity mode is able to increase the decoherence time of a Schrödinger cat state. In Ref. [16,17] a feedback scheme based on continuous photodetection and the injection of appropriately prepared atoms has been considered. This scheme, in the limit of very good detection efficiency, is able to obtain a significant "protection" of a generic quantum state in a cavity. In [15,17] this photodetection-mediated scheme has been adapted to the microwave experiment of Ref. [6] in which photodetectors cannot be used. The cavity state can only be indirectly inferred from measurements performed on probe atoms which have interacted with the cavity mode. Under ideal conditions, this adaptation to the microwave cavity case leads to a significant increase of the lifetime of the Schrödinger cat generated in [6]. It suffers however from two important limitations, making it very inefficient when applied under the actual experimental situation. It first requires the preparation of samples containing exactly one Rydberg atom sent through the apparatus. Up to now, the experimental techniques allow only to prepare a sample containing a random atom number, with a Poisson statistics. Two-atom events are excluded only at the expense of a low average atom number, lengthening the feedback loop cycletime [10]. The original scheme requires also a near unity atomic detection efficiency, which is extremely difficult to achieve even with the foreseeable improvements of the experimental apparatus. In this paper we present a significant improvement of the microwave feedback scheme described in [15,17]. This new version, using a direct transmission of the quantum information from the probe to the feedback atom, does not require a large detection efficiency, removing one of the main difficulties of the previous design. It however also requires sub-poissonian atom statistics. We show briefly how such atomic packets could be in principle prepared with standard laser techniques. Finally, our scheme improves the efficiency of the feedback photon injection in the cavity by using an adiabatic rapid passage.
The paper is organized as follows: in section II the feedback scheme of [15,17] is reviewed and critically discussed; in section III the modifications of this scheme are introduced and in section IV the map describing the feedback cycle is derived. In section V the dynamics in the presence of feedback is studied and the protection capabilities of the new proposal are illustrated, while section VI is devoted to concluding remarks.
II. THE FEEDBACK SCHEME BASED ON ATOMIC DETECTION
Let us briefly review the original "stroboscopic" feedback scheme for microwave cavities proposed in [15,17]. This proposal is based on a very simple idea: whenever the cavity looses a photon, a feedback loop supplies the cavity mode with another photon, through the injection of an appropriately prepared atom. However, since there are no good enough photodetectors for microwaves, one has to find an indirect way to check if the high-Q microwave cavity has lost a photon or not. In the experiment of Brune et al. [6], information on the cavity field state is obtained by detecting the state of a circular Rydberg atom which has dispersively interacted with the superconducting microwave cavity. This provides an "instantaneous" measurement of the cavity field and suggests that continuous photodetection can be replaced by a series of repeated measurements, performed by non-resonant atoms regularly crossing the high-Q cavity, separated by a time interval τ pr .
The experimental scheme of the stroboscopic feedback loop is a simple modification of the scheme employed in Ref. [6]. The relevant levels of the velocity-selected atoms are two adjacent circular Rydberg states with principal quantum numbers n = 50 and n = 51 (denoted by |g and |e respectively) and a very long lifetime (30 ms). The high-Q superconducting cavity is sandwiched between two low-Q cavities R 1 and R 2 , in which classical microwave fields resonant with the transition between |e and |g can be applied.
The high-Q cavity C is instead slightly off-resonance with respect to the e → g transition, with a detuning
δ = ω − ω eg ,(1)
where ω is the cavity mode frequency and ω eg = (E e − E g )/h. The Hamiltonian of the atom-microwave cavity mode system is the usual Jaynes-Cummings Hamiltonian, given by
H JC = E e |e e| + E g |g g| +hωa † a +hΩ |e g|a + |g e|a † ,(2)
where Ω is the vacuum Rabi coupling between the atomic dipole on the e → g transition and the cavity mode. In the off-resonant case and perturbative limit Ω ≪ δ, the Hamiltonian (2) (under an appropriate redefinition of level phases) assumes the dispersive form [17][18][19]
H disp =h Ω 2 δ |g g|a † a − |e e|a † a .(3)
The Schrödinger cat state is generated when the cavity mode is initially in a coherent state |α and the Rydberg atom, which is initially prepared in the state |e , is subjected to a π/2 pulse both in R 1 and in R 2 . In fact, when the atom has left the cavity R 2 , the joint state of the atom-cavity system becomes the entangled state [6,17,19]
|ψ atom+f ield = 1 √ 2 |e |αe iφ − |αe −iφ + |g |αe iφ + |αe −iφ ,(4)
where φ = Ω 2 t int /δ and t int is the interaction time in C. A cat state, i.e. a linear superposition of two coherent states with different phases, is then conditionally generated in the microwave cavity as soon as one of the two circular atomic states is detected.
As it was shown in Ref. [17], the stroboscopic feedback scheme works only for Schrödinger cat states with a definite parity, i.e. even or odd cat states, and therefore we shall restrict to φ = π/2 from now on. In fact, when the cavity field initial state is a generic density matrix ρ, the state of the probe atom-field system after the two π/2 pulses and the φ = π/2 conditional phase-shift can be written as [17]
ρ atom+f ield = |e e| ⊗ ρ e + |g g| ⊗ ρ g + |e g| ⊗ ρ + + |g e| ⊗ ρ − ,(5)
where ρ e = P odd ρP odd (6) ρ g = P even ρP even ,
are the projections of the cavity field state onto the subspace with an odd and even number of photons, respectively, and the operators ρ ± (whose expression is not relevant here) are given in [17]. Eq. (5) shows that there is a perfect correlation between the atomic state and the cavity field parity, which is the first step in an optimal quantum non demolition measurement of the photon number [20]. It is possible to prove that this perfect correlation between the atomic state and a cavity mode property holds only in the case of an exact φ = π/2-phase shift sandwiched by two classical π/2 pulses in cavities R 1 and R 2 [17]. Moreover, the entangled state of Eq. (5) allows to understand how it is possible to check if the microwave cavity C has lost a photon or not and therefore to trigger the feedback loop, using atomic state detection only. The detection of e or g determines the parity of the field and, provided that the probe atomic pulses are frequent enough, indicates whether a microwave photon has left C or not. In fact, let us consider for example the case in which an odd cat state is generated (first atom detected in e): a probe atom detected in state e means that the cavity field has remained in the odd subspace. The cavity has therefore lost an even number of photons. If the time interval τ pr between the two atomic pulses is much smaller than the cavity decay time γ −1 , γτ pr ≪ 1, the probability of loosing two or more photons is negligible and this detection of the probe atom in e means that no photon has leaked out from the high-Q cavity C. On the contrary, when the probe atom is detected in g, the cavity mode state is projected into the even subspace. The cavity has then lost an odd number of photons. Again, in the limit of enough closely spaced sequence of probe atoms, γτ pr ≪ 1, the probability of loosing three or more photons is negligible. A detection in g means that one photon has exited the cavity. Therefore, for achieving a good protection of the initial odd cat state, the feedback loop has to supply the superconducting cavity with a photon whenever the probe atom is detected in g, while feedback must not act when the atom is detected in the e state. In Ref. [17] it has been proposed to realize this feedback loop with a switch connecting the g state field-ionization detector with a second atomic injector, sending an atom in the excited state e into the high-Q cavity. The feedback atom is put in resonance with the cavity mode by another switch turning on an electric field in the cavity C when the atom enters it, so that the level e is Stark-shifted into resonance with the cavity mode.
As it is shown in Ref. [17], if the probe atomic pulses are sufficiently frequent, this stroboscopic feedback scheme becomes extremely efficient and one gets a good preservation of an initial Schrödinger cat state. However, if we consider the adaptation of this scheme to the present experimental apparatus of Ref. [6], we see that it suffers from two main limitations, which significantly decrease its efficiency. First of all the scheme is limited by the non-unit efficiency of the atomic state detectors (η det ≃ 0.4), since the feedback loop is triggered only when the g-detector clicks. Most importantly, the above scheme assumes one has perfect "atomic guns", i.e. the possibility of having probe and feedback atomic pulses with exactly one atom. This is not experimentally achieved up to now. The actual experiment [6] has been performed using atomic pulses with a probability of having exactly one atom p 1 ≃ 0.2, close to the mean atom number in the sample. This low mean atom number has been chosen to minimize two-atom events. In this experimental situation, the proposed stroboscopic feedback scheme would have an effective efficiency η ef f = η det p 2 1 ≃ 0.016, too low to get an appreciable protection of the Schrödinger cat state. We show here how this scheme may be improved and adapted to the experimental apparatus employed in Ref. [6].
III. THE NEW STROBOSCOPIC FEEDBACK LOOP
The limitations due to the non-unit efficiency of the atomic detectors could be avoided if we eliminate the measurement step in the feedback loop and replace it with an "automatized" mechanism preparing the correct feedback atom whenever needed. This mechanism can be provided by an appropriate conditional quantum dynamics. We need a "controlled-NOT" gate between the probe atom and the feedback atom, because the feedback atom has to remain in an "off"state if the probe atom exits the cavity of interest C in the e state, while the feedback atom has to be in the excited state e when the probe atom leaves C in the g state (we are still assuming the initial generation of an odd cat state). This conditional dynamics can be provided by a second high-Q microwave cavity C ′ , similar to C, replacing the atomic detectors, crossed by the probe atom first and by the feedback atom soon later. A schematic description of the new feedback scheme, with the second cavity C ′ replacing the atomic state detectors is given by Fig. 1.
The cavity C ′ is resonant with the transition between an auxiliary circular state i, which can be taken as the immediately lower circular Rydberg state n = 49, and level g. The interaction times have to be set so that both the probe and the feedback atom experience a π pulse when they cross the empty cavity C ′ in state g (or when they enter in state i with one photon in C ′ ). This interaction copies the state of the probe atom onto the cavity mode and back onto the feedback atom. C ′ acts thus as a "quantum memory" [21], transferring directly the quantum information between the two atoms without need of a detection. This removes thus any need for a unit detection efficiency.
This fine tuning of the interaction times to achieve the π-spontaneous emission pulse condition can be obtained applying through the superconducting mirrors of C ′ appropriately shaped Stark-shift electric fields which puts the atoms in resonance with the cavity mode in C ′ only for the desired time. In this way, since C ′ is initially in the vacuum state, one has
|e p |0 C ′ → |e p |0 C ′ (8) |g p |0 C ′ → |i p |1 C ′(9)
when the probe atom crosses C ′ ; soon later a feedback atom enters C ′ in the state |i f and one has
|i f |0 C ′ → |i f |0 C ′ (10) |i f |1 C ′ → |g f |0 C ′ (11)
(the cavity has a very high Q and therefore the probability of photon leakage in the meanwhile is negligible). In this way the cavity C ′ is always left disentangled in the vacuum state. The feedback atom exiting C ′ in |g can be promoted to |e before entering C, as required by the feedback scheme, by subjecting it to a π pulse in the classical cavity R 2 (see Fig. 1). The conditional dynamics provided by C ′ eliminates any limitation associated to the measurement and leads to an "automatic feedback" scheme with unit efficiency in principle. As mentioned above, an important limitation of the stroboscopic feedback scheme of [15,17] is that it requires exactly one probe and one feedback atom per loop. This condition is still needed in the new scheme with the cavity C ′ replacing the atomic detectors. With two or more probe atoms simultaneously in C and in C ′ , one gets a wrong phase shift for the field in C and also an incomplete excitation transfer from the probe atom to the field in C ′ . The same condition holds for the feedback atoms. With two or more feedback atoms in the sample, the excitation transfers in C ′ and C are incomplete.
A better control of the atom number, providing single atom events with a high probability, could be achieved by a modification of the Rydberg atoms preparation techniques. We outline here briefly the method, which could be implemented in a future version of the experimental set-up. The Rydberg atoms preparation would start from a very low-intensity velocity-selected Rubidium atomic beam. The ground state atom density is so low that the average distance between the atoms in the beam is of the order of a few millimeters. It means that a section of the beam a few millimeters long contains on the average only one atom (with a Poisson statistics). This section could be driven by a laser resonant on the 5S to 5P transition (see Fig. 2 for a schematic diagram of the relevant 85 Rb energy levels involved). The fluorescence signal should make it possible to distinguish easily the situations where the probed section of the beam contains zero, one, two or more atoms, implementing an atom counter. When the section contains zero, two or more atoms, it is discarded. The system waits then for a time τ pr (a few microseconds to twenty microseconds, depending upon the atomic velocity and the precise length of the atomic beam section) until a fresh section of the beam comes in the probe laser beam. At variance, if the fluorescence level corresponds to exactly one atom, the circular state preparation is started. Using only adiabatic rapid passages, it should be possible to promote the single ground state atom to the desired circular state with a high probability. The circular state preparation [22,23] proceeds in two steps. First, a laser excitation of an "ordinary" Rydberg state, then a transfer to the circular state. The latter step already uses adiabatic rapid passages and has a very high efficiency. The former step could also be adiabatic, by using higher laser powers readily available.
Instead of preparing a random atom number at a given time, one thus prepares with a high probability a single Rydberg atom after a random delay (since the preparation step is triggered only when the atomic counter gives a count of exactly one). The average value of this random delay is minimal when the probability to have exactly one atom is maximized. With a Poissonian statistics, the optimal mean number of atoms in the probed section is 1. The average random delay could be of the order of 25 µs in realistic experimental conditions. This is short enough at the scale of the cavity field lifetime to play no major role in the experiment. The unavoidable imperfections of the circular state preparation could be easily taken into account by assuming that the sample contains one atom with a probability p r and no atoms with a probability 1 − p r . Two-atom events are excluded, a considerable improvement compared to other preparation methods.
The timing of the whole experiment should be conditioned to the operation of the atomic counters. When the cycle starts, the system idles until a probe atom has been counted and prepared in the circular state. After it has crossed C ′ , the preparation cycle of the feedback atom is started. The system also idles until this atom is counted and prepared in the circular state. The feedback is complete when this feedback atom has crossed the cavity C.
IV. THE FEEDBACK CYCLE IN MORE DETAIL
Let us now determine the map of a generic feedback cycle, that is, the transformation connecting the states ρ m and ρ m+1 of the cavity field in C soon after the passage of two successive feedback atoms in C. From the previous section it is clear that a new cycle begins only when one is sure to have one probe atom with certainty and therefore one has to wait a random time t r before the new probe atom enters C.
The atomic counter operate with a cycle time τ pr . The average number of atoms per probed packet being one, the probability of having exactly one atom is p 1 = 1/e = 0.37. Therefore, the random waiting time can be written as t r = lτ pr , l = 0, 1, . . ., where the probability distribution of the discrete random variable l is given by
p(l) = p 1 (1 − p 1 ) l l = 0, 1, . . . .(12)
The first step of the feedback loop is simply the standard dissipative evolution with damping rate γ for a random time lτ pr [17,24]
ρ I m = ∞ k=0 A k (lτ pr )ρ m A k (lτ pr ) † ,(13)
where
A k (t) = ∞ n=0 (n + k)! n!k! e −nγt (1 − e −γt ) k |n n + k|(14)
and where ρ i m will denote the state after the i-th step of the cycle. The second step is determined by the probe atom crossing the cavity C and interacting with it via the dispersive Hamiltonian (3). Since the probe atom has been already prepared in the circular state e and it has already crossed the classical cavity R 1 (see Fig. 1), it enters C in the state (|e + |g )/ √ 2. Due to the π/2 phase shift, the cavity mode in C gets entangled with the probe atom and, after the second feedback step, one has
ρ II m = 1 2 |e e| ⊗ P ρ I m P + |g g| ⊗ ρ I m +|e g| ⊗ P ρ I m + |g e| ⊗ ρ I m P ,(15)
where P = exp iπa † a is the cavity mode parity operator.
As it can be seen from Fig. 1, the probe atom flies then from the cavity C to the second high-Q cavity C ′ . During this time of flight one has to consider the effect of standard vacuum damping on the C cavity mode and also the effect of the π/2 pulse in R 2 on the probe atom, yielding
|e → 1 √ 2 (|e + |g ) |g → 1 √ 2 (−|e + |g ) .(16)
Note that we shall always neglect the spontaneous decay of the circular levels, since the lifetime of the involved level (about 30 ms) is much larger than the mean feedback cycle duration time (of the order of 1 ms). The two actions do not interfere and therefore, rearranging the terms, one has an expression connected to Eq. (5)
ρ III m = ∞ k=0 A k (t C→C ′ ) |e e| ⊗ ρ I e + |g g| ⊗ ρ I g +|e g| ⊗ ρ I + + |g e| ⊗ ρ I − A k (t C→C ′ ) † ,(17)
where the density matrices ρ I e and ρ I g are the odd and even projections of Eqs. (6) and (7) and
ρ ± = 1 4 [P ρP − ρ ± P ρ ∓ ρP ] .(18)
The fourth step is determined by the interaction of the probe atom with the second high-Q cavity C ′ , which is described by the resonant interaction between the C ′ cavity mode and the two lower circular levels i and g
H C ′ =hΩ ′ |g i|b + |i g|b † ,(19)
where Ω ′ is the corresponding vacuum Rabi frequency and b denotes the annihilation operator of the C ′ cavity mode. The cavity C ′ is initially in the vacuum state, and therefore, using the Stark tuning mechanism described in the preceding section to determine the effective interaction time t int pr , it is possible to impose the π pulse condition for t int
pr
Ω ′ t int pr = π 2 ,(20)
so that the conditional dynamics described by Eqs. (8) and (9) is obtained. One gets therefore the following entangled state between the probe atom and the two microwave cavity modes
ρ IV m = ∞ k=0 A k (t C→C ′ ) |e p e| ⊗ ρ I e ⊗ |0 C ′ 0| + |g p g| ⊗ ρ I g ⊗ |1 C ′ 1| +|e p g| ⊗ ρ I + ⊗ |0 C ′ 1| + |g p e| ⊗ ρ I − ⊗ |1 C ′ 0| A k (t C→C ′ ) † .(21)
However the probe atom is not observed after exiting C ′ and therefore we have to trace over it; as a result, the off-diagonal terms vanish and the following correlated state between the two microwave cavity modes is left
ρ IV m = ∞ k=0 A k (t C→C ′ ) ρ I e ⊗ |0 C ′ 0| + ρ I g ⊗ |1 C ′ 1| A k (t C→C ′ ) † .(22)
During the probe atom crossing, the beam of feedback atoms continues to pass through the apparatus in the opposite direction (see Fig. 1) in their internal ground state, which is decoupled from all the microwave cavities of the experimental arrangement. Then the electronics controlling the circular state preparation of the feedback atom is set in such a way that one feedback atom can enter the cavity C ′ in the Rydberg state i soon after the probe atom has left it. However, as it happens for the probe atoms at the beginning of the cycle, one has to wait a random time until we are sure to have one feedback atom with certainty.
We assume that also the feedback atoms are sent and counted with a time cycle τ f b . The probability of having one atom in a probed sample of the beam is again equal to p 1 = 1/e = 0.37. Therefore, the random waiting time can be written as qτ f b , q = 0, 1, . . ., where q is a discrete random variable with the same probability distribution of the probe random variable l, given by Eq. (12). During this random waiting time, one has to consider standard vacuum damping for both microwave cavities C and C ′ . Photon leakage in C ′ is particularly disturbing because it transforms the one photon state |1 1| into the vacuum, according to
|1 1| → e −γ ′ qτ f b |1 1| + 1 − e −γ ′ qτ f b |0 0| ,(23)
(γ ′ is the cavity C ′ damping rate) blurring therefore any difference between the C cavity states ρ e (that does not need any correction) and ρ g (that needs a photon back) in Eq. (22). Using Eq. (22), the resulting transformation for the joint state of the two microwave cavities becomes
ρ V m = ∞ k=0 A k (t C→C ′ + qτ f b ) ρ I e + 1 − e −γ ′ qτ f b ρ I g ⊗ |0 0| +e −γ ′ qτ f b ρ I g ⊗ |1 1| A k (t C→C ′ + qτ f b ) † .(24)
The next step of the feedback cycle is given by the resonant interaction of the feedback atom with the cavity mode C ′ . The interaction is again described by the Hamiltonian (19) and one can use again the Stark-effect tuning mechanism to determine the right interaction time to get the π pulse condition of Eq. (20). The consequent transformation is described by Eqs. (10) and (11), so that, after the feedback atom passage, the cavity C ′ comes back to its initial vacuum state and the entanglement with the cavity of interest C is transferred to the feedback atom. Actually, in the preceding steps we have neglected the effect of photon leakage out of C ′ during both probe and feedback atom passages through C ′ because of its high Q value. We can partially amend this approximation by "postponing" dissipation after the interactions and adding the probe and feedback atom crossing times t cr pr and t cr f b to the random waiting time qτ f b in (23). The resulting C-field plus feedback atom joint state becomes
ρ V I m = ∞ k=0 A k (t C→C ′ + qτ f b ) ρ I e + 1 − e −γ ′ (qτ f b +t cr pr +t cr f b ) ρ I g ⊗ |i f i| +e −γ ′ (qτ f b +t cr pr +t cr f b ) ρ I g ⊗ |g f g| A k (t C→C ′ + qτ f b ) † .(25)
As we have explained in the preceding section, the odd density matrix ρ I e does not need any correction and therefore has to be correlated with |i f , while the even part ρ I g needs a correction and therefore has to be correlated with |g f . Looking at Eq. (25), it is easy to see that the factor exp{−γ ′ qτ f b + t cr pr + t cr f b } gives the probability that the feedback loop is acting correctly, i.e., this factor plays exactly the same role of the detector efficiency in the original stroboscopic feedback scheme of Refs. [15,17]. However the times τ f b , t cr pr and t cr f b are very small in the experiment (of the order of 10 µsec) and using a very high Q cavity for C ′ , i.e. γ ′ qτ f b + t cr pr + t cr f b ≪ 1, one obtains a feedback loop with an effective unit efficiency, which, as we have remarked in the preceding section, is one of the improvements of the new feedback scheme.
Then the feedback atom flies from C ′ to C and, along its path, it passes through the cavity R 2 , within which it is subjected to a π pulse on the transition g → e. The effect of this pulse is simply to transform the state |g f into |e f in Eq. (25) and it does not interfere with the effect of vacuum damping on the C cavity mode. Therefore it is easy to see that the state of Eq. (25) is simply changed to
ρ V II m = ∞ k=0 A k (t 0 + qτ f b ) ρ I e + 1 − e −γ ′ (qτ f b +t cr pr +t cr f b ) ρ I g ⊗ |i f i| +e −γ ′ (qτ f b +t cr pr +t cr f b ) ρ I g ⊗ |e f e| A k (t 0 + qτ f b ) † ,(26)
where t 0 is the overall time of flight, i.e. the sum of the probe atom time of flight from C to C ′ and the feedback atom time of flight from C ′ to C.
We finally arrive at the last step of the feedback cycle, i.e. the interaction between the feedback atom and the cavity mode we want to protect against decoherence. If the feedback atom is in state |i nothing relevant happens and the C cavity mode state is left unchanged, as it must be. If instead the feedback atom is in state |e , it has to release its excitation to the cavity mode. In Ref. [15,17] it has been proposed to realize this excitation transfer by Stark-shifting into resonance the circular levels in order to use the resonant Jaynes-Cummings interaction [Eq. (2) with zero detuning δ]. Here we propose to use the Stark tuning mechanism in a more clever way, in order to optimize the photon transfer to the microwave cavity mode. In fact, if one uses the resonant interaction, the excitation transfer to the cavity is optimal for an odd number of half Rabi oscillations, that is
Ωt int f b √ n + 1 = π(m + 1/2) m integer .(27)
The dependence of this condition on the intracavity photon number n is a limitation of the resonant interaction because the photon transfer becomes ideal in the case of a previously known Fock state only. On the contrary it would be preferable to have a way to perfectly release the photon in C whatever the state of the cavity mode is. As explained in [25,26] this possibility is provided by adiabatic transfer, which can be realized in the present context using a Stark shift electric field in C able to change adiabatically the atomic frequency ω eg through the resonant value ω.
Let us see in more detail how it is possible to use the Stark effect to realize the adiabatic transfer of the excitation. Let us consider the Hamiltonian of the Jaynes-Cummings model (2) in the interaction picture and with a timedependent detuning δ(t) because of the adiabatic time dependence of the atomic frequency ω eg , H ad =hδ(t)a † a +hΩ |e g|a + |g e|a † .
(28)
This Hamiltonian couples only states within the two-dimensional manifold with n + 1 excitations spanned by |g, n + 1 and |e, n , where n denotes a Fock state of the cavity mode. Within this manifold one has the adiabatic eigenvalues
E n ± (t) h = δ(t) n + 1 2 ± δ 2 (t) 4 + Ω 2 (n + 1)(29)
and the corresponding adiabatic eigenstates
|v n ± (t) = N ± − δ(t) 2 ± δ 2 (t) 4
+ Ω 2 (n + 1) |e, n + Ω √ n + 1|g, n + 1 .
Now, according to the adiabatic theorem [27], when the evolution from time t 0 to time t 1 is sufficiently slow, a system starting from an eigenstate of H(t 0 ) will pass into the corresponding eigenstate of H(t 1 ) that derives from it by continuity. In the present case, the interesting adiabatic eigenstate is |v n + (t) . In fact, if we assume that the detuning δ is varied adiabatically from a large negative value −δ 0 to a large positive value δ 0 , with δ 0 ≫ Ω √ n + 1, it is easy to see from Eq. (30) that |v n + (t) will consequently show the following adiabatic transformation |e, n → |g, n + 1 ∀n (31) thereby realizing the desired excitation transfer regardless of the cavity mode state, which, in terms of cavity mode density matrices can be written as
ρ → a † 1 √ aa † ρ 1 √ aa † a .(32)
To be more precise, each adiabatic eigenstate gets its own dynamical phase factor
e −iΦn = e − ī h dtEn(t)(33)
during the adiabatic evolution [27] and therefore the transformation (32) exactly holds only if this dynamical phase factor does not depend on n. Assuming a linear sweep of the Stark-shift electric field, that is, δ(t) = δ 0 t/t s , for |t| ≤ t s and using Eq. (29), one has Φ n = − δ 0 t s 2h 1 + 4Ω 2 (n + 1) δ 2 0 + 2Ω 2 (n + 1) δ 2 0 log 1 + 4Ω 2 (n + 1)/δ 2 0 + 1 1 + 4Ω 2 (n + 1)/δ 2 0 − 1 .
(34) Therefore one has in general a photon number dependent phase-shift; however in the particular adiabatic transformation we are considering, for which δ 0 ≫ Ω √ n + 1 for all the relevant values of n, this phase factor can be well approximated, at the lowest order in Ω √ n + 1/δ 0 , by the constant phase factor exp{iδ 0 t s /2h} and therefore Eq. (32) holds exactly.
Finally we have all the ingredients to determine the last step of the feedback cycle. One has to consider the transformation (32) when the feedback atom is in state e, while nothing happens when the feedback atom passes C in state i and then one has to trace over the feedback atom because it is not observed. We have therefore the map connecting the state of the C cavity mode after two successive cycles, which is
ρ m+1 = ∞ k=0 A k (t 0 + qτ f b ) ρ I e + 1 − e −γ ′ (qτ f b +t cr pr +t cr f b ) ρ I g A k (t 0 + qτ f b ) † +e −γ ′ (qτ f b +t cr pr +t cr f b ) a † 1 √ aa † ∞ k=0 A k (t 0 + qτ f b )ρ I g A k (t 0 + qτ f b ) † 1 √ aa † a ,(35)
where the projected matrices ρ I e and ρ I g are obtained from the cavity mode state after the preceding feedback cycle ρ m by inserting Eqs. (6) and (7) into Eq. (13).
In the determination of the map (35) we have assumed that the Rydberg state preparation for both probe and feedback atoms has unit efficiency. In a realistic situation, the circular state preparation will have a non-unit efficiency p r < 1. This implies that the feedback map of Eq. (35) is realized with a probability p 2 r only. In fact, when either the probe or feedback atom Rydberg state preparation fails, the feedback does not effectively take place, because either the probe or the feedback atom is not in the correct state and the photon transfer in C cannot take place. This effect can be taken into account modifying the feedback map of Eq. (35) in this way
ρ m+1 = p 2 r Φ f b q,l (ρ m ) + (1 − p 2 r )Φ diss q,l (ρ m ) = Φ q,l (ρ m ) ,(36)
where Φ f b q,l is the map operator defined in Eq. (35) and
Φ diss q,l (ρ m ) = ∞ k=0 A k (t 0 + lτ pr + qτ f b )ρ m A k (t 0 + lτ pr + qτ f b ) † (37)
describes the standard dissipation acting during the feedback cycle time t 0 + lτ pr + qτ f b .
V. STUDY OF THE DYNAMICS OF THE AUTOFEEDBACK SCHEME
As we have observed above, the triggering of the feedback cycle only when the atomic counters have counted exactly one probe and one feedback atom makes the time evolution random. In fact, the feedback cycle map (36) we have determined in the preceding section is a random map, that is, it depends upon the two discrete random variables q, l. It is evident that if we want to study the dynamics of the microwave mode within C, two different strategies are possible to determine the averaged evolution: i) repeat the experiment many times up to the same, fixed, elapsed time t; ii) repeat the experiment many times by fixing the number of feedback cycles instead of the elapsed time. We shall consider this second possibility, in order to better understand the effect of the autofeedback scheme. In fact, fixing the elapsed time would have meant averaging over experimental runs characterized by different number of feedback cycles. Using Eq. (36), we have that in a single run, the state after N feedback cycles is
ρ N = Φ qN ,lN Φ qN−1,lN−1 . . . Φ q2,l2 Φ q1,l1 ρ(0) ;(38)
in the limit of a very large number of experimental runs, one gets the average cavity mode statē
ρ N = p(l 1 )p(q 1 ) . . . p(l N )p(q N )Φ qN ,lN Φ qN−1,lN−1 . . . Φ q2,l2 Φ q1,l1 ρ(0) ,(39)
where the probability distributions p(l) are given by Eq. (12). Since q 1 , l 1 . . . q N , l N are independent random variables, it is evident that the average stateρ N after N feedback cycles can also be written as
ρ N =Φ N ρ(0) ,(40)whereΦ = p(l)p(q)Φ q,l(41)
is the averaged feedback cycle map operator, determining all the dynamics of the microwave mode. The expression of this operator can be determined using (36), but it is cumbersome and not particularly interesting. One relevant aspect of this averaged feedback cycle operator is that, since it involves only the even and odd projections ρ g and ρ e , and the cavity mode state is initially confined within the odd subspace, it never populates the Fock subspace without a definite parity, i.e., ρ n,n+p = 0, whenever p is odd, at all times. In other words, it is possible to write ρ = ρ g + ρ e at any time. Let us finally discuss the optimal values of the various experimental parameters involved. It is evident that the protection capabilities of the proposed scheme essentially depend upon the ratio between the mean feedback cycle timet cyc and the Schrödinger cat decoherence time t dec = (2γ|α| 2 ) −1 . For smaller and smaller values of this ratio, one gets a longer and longer protection of the initially generated cat state. This average cycle timet cyc is determined by the spatial dimensions of the apparatus (which cannot be too miniaturized since we are using microwaves) and by the probe and feedback atom velocities, which have to be therefore as large as possible. However, the probe atom velocity is fixed by the π/2 phase shift condition which is needed to have a cat state with a definite parity,
Ω 2 t int δ = Ω 2 L C δv pr = π 2 ,(42)
where L C is the effective transverse length of the C cavity mode. In the actual experimental situation Ω/2π = 24 kHz, L C = 0.75 cm and the smallest possible value of the detuning, compatible with the non-resonant interaction, is δ/2π ≃ 70 kHz, so that we get v pr ≃ 250 m/s. There is no similar constraint for the feedback atom which can be taken therefore as fast as possible; we choose v f b ≃ 500 m/s, since the Rydberg atoms used are thermal Rb atoms and this velocity corresponds to the fastest usable part of the Maxwellian distribution. Once we have chosen the two atom velocities, one has to check that these values are compatible with the π pulse condition of Eq. (20) for both probe and feedback atom in C ′ and also with the conditions for adiabatic transfer for the feedback atom in C. In fact it is possible to use the Stark tuning mechanism to determine the interaction times in C ′ satisfying the π pulse condition only if the cavity crossing time t cr in C ′ is larger than π/(2Ω ′ ) [see Eq. (20)]. Since the cavities C and C ′ are resonant with the two adjacent transitions g → e and i → g, they can be assumed to be of similar design, so that Ω ′ ≃ Ω = 2π × 24 kHz and L C ′ ≃ L C = 0.75 cm and this implies t pr cr ≃ 30 µsec, t f b cr ≃ 15 µsec which are in fact larger than π/(2Ω ′ ) ≃ 10 µsec. The condition for the adiabatic passage of the feedback atom in C is instead that the feedback atom crossing time in C t f b cr ≃ 15 µsec has to be larger than Ω −1 ≃ 7 µsec and therefore this condition is verified too.
The probe and feedback atom velocities determine the overall time of flight t 0 of Eq. (26); in fact a reasonable estimate of the apparatus length from C to C ′ is 10 cm and therefore one has t 0 ≃ 600 µsec. However, the duration time of a feedback cycle is determined not only by t 0 , but also by the random waiting times lτ pr and qτ f b due to the atomic counters and also by the non-unit efficiency of the Rydberg state preparation p r which we can assume to be p r ≃ 0.9. In fact the probe and feedback atoms are prepared in the correct circular Rydberg state with a probability p 2 r and therefore the photon is effectively released into the cavity C only after a random number of cycles m, with probability p A (m) = p 2 r (1 − p 2 r ) m−1 , m = 1, 2, . . .. As a consequence, the effective mean duration time of a feedback cycle, that is, the mean time between two successful photon transfers in C, is given by the mean loop time multiplied by the mean number of "attempts", m = 1/p 2 r ,
t cyc = m (t 0 + l τ pr + q τ f b ) = 1 p 2 r t 0 + 1 − p 1 p 1 (τ pr + τ f b ) .(43)
The sampling time of the probe and feedback atomic counters τ pr and τ f b corresponds to a probed section of the beam of the order of few millimeters and therefore we can assume τ pr ≃ τ f b ≃ 15 µsec, so that we havet cyc ≃ 800 µsec. This mean duration time has to be smaller than the decoherence time of the Schrödinger cat state initially generated, otherwise the correction of the autofeedback scheme would be too late to get a significant protection. However, from the above discussion it is evident that the experimental conditions put many constraints on the possible parameter values and that this value fort cyc cannot be significantly decreased. Therefore the only way to achieve a significant Schrödinger cat state preservation is to increase the decoherence time, i.e., increase the relaxation time t rel = γ −1 of the cavity C or decrease the cat state initial mean photon number |α| 2 . In fact we can say that a cat state N ± (|α ± | − α ) is protected by the present autofeedback scheme as long as
|α| 2 < t rel 2t cyc .(44)
Alternatively, if we consider a given mesoscopic value for |α| 2 , as for example |α| 2 = 3.3 as in Ref. [6], one begins to increase the "lifetime" of the generated cat state as long as t rel > 5 ms. Relaxation times of this order of magnitude will be hopefully obtained in the near future and for this reason we have plotted in Fig. 2 the Wigner functions and the density matrices describing the averaged time evolution in the presence of the autofeedback scheme for an initial odd cat state with α = √ 3.3 and for a cavity relaxation time t rel = 10 ms. The cavity C ′ is assumed to be equal to C and the values of all the other parameters are the same as discussed above, so thatt cyc /t dec ≃ 0.53. Fig. 2(a) shows the Wigner function and the density matrix elements of the initial odd cat state; Fig. 2(b) refers instead to the state of the cavity mode after 13 feedback cycles, corresponding to a mean elapsed timet ≃ t rel ≃ 6.6t dec and Fig. 2(c) refers to the state of the cavity mode after 25 feedback cycles, corresponding to a mean elapsed timet ≃ 2t rel ≃ 13t dec . This figures show the impressive preservation of all the main aspects of the initial odd cat state up to 13 decoherence times. To better appreciate the performance of the proposed scheme we show in Fig. 3 the corresponding time evolution of the same initial odd cat state in the absence of the autofeedback scheme. Fig. 3(a) shows again the initial Wigner function and density matrix, Fig. 3(b) refers to the cavity field state after one relaxation time t rel and Fig. 3(c) describes the cavity field state after two relaxation times (in absence of feedback time evolution is no more random and therefore these are actual elapsed times). In this case, after one relaxation time, the cat state has already turned into a statistical mixture of two coherent states, with no quantum aspect left, and it approaches the vacuum state after two relaxation times [ Fig. 3(c)].
Another important aspect of the feedback-induced dynamics shown by Fig. 2 is the "distortion"of the cat state which becomes more and more "rounded" as time passes. This is due to the slow unconventional phase diffusion associated to this feedback scheme and which has been discussed in detail in Ref. [17]. In fact the present autofeedback scheme is an improvement of the original scheme of Ref. [17], and the main physical aspects are essentially the same: the fed back photon has no phase relationship with the photons already present in C and this leads to the above mentioned phase diffusion. This phase diffusion turns out to be very slow; in fact the present model is essentially a stroboscopic version of the continuous photodetection feedback scheme studied in [17], which is characterized in the semiclassical limit by a diffusion term
− γ 2 √ n, √ n, ρ ↔ γ 8n ∂ 2 ∂θ 2 W (r, θ) ,(45)
for the Wigner function in polar coordinates W (r, θ) (n is the mean photon number) and which is analogous to the phase diffusion of a laser well above threshold. It is possible to see (see also Ref. [17]) that the asymptotic state of the cavity mode is the rotationally invariant mixture of the vacuum and the one photon state ρ st = P 0 |0 0| + P 1 |1 1|, which is however reached after many relaxation times.
VI. CONCLUSIONS
In this paper we have proposed a method to significantly increase the "lifetime" of a Schrödinger cat state of a microwave cavity mode. The scheme uses "probe" and "feedback" atoms and a second high-Q microwave cavity to transfer quantum information between these two atoms without need for a detection stage. This scheme avoids some of the pitfalls of previously published ones. In particular, its efficiency does not rely on a perfect Rydberg atom detection. Even though it relies on an efficient preparation of a single atom, this is not critical since standard laser techniques can be used to fulfill this requirement. We have shown that the method is quite efficient, with realistic orders of magnitude for the experimental parameters.
We have focused on the case of a Schrödinger cat state which, thanks to its well characterized quantum features, plays the role of the typical quantum state; however, as it can be easily expected, most of the techniques presented here could be applied to the case of a generic quantum state of a cavity mode (see also Ref. [17]).
This decoherence control scheme is less general than quantum error correction methods because it exploits from the beginning the specific aspects of the physical mechanism inducing decoherence. However there are similarities between the present autofeedback scheme and quantum error correction codes. The second cavity C ′ detects the error syndrome during its interaction with the probe atom and sends the necessary correction to C via the feedback atom.
After the first experimental evidences of decoherence mechanisms, decoherence control is bound to be a rapidly expanding field in quantum physics. First, it is important as an illustration of a very fundamental relaxation process. It would be extremely interesting to tailor decoherence, as spontaneous emission in the past. This should lead to a deeper insight into relaxation theory and into the border between the microscopic and the macroscopic world. Decoherence control is also important for quantum information processing schemes, since decoherence is the main problem to manipulate large quantum systems. An experimental realization of this feedback scheme, which is quite realistic, would be an important step in this direction.
VII. ACKNOWLEDGMENTS
This work has been partially supported by INFM (through the 1997 Advanced Research Project "CAT"), by the European Union in the framework of the TMR Network "Microlasers and Cavity QED" and by MURST under the "Cofinanziamento 1997". 1. Schematic diagram of the autofeedback scheme proposed in this paper. R1 and R2 are the two cavities in which classical microwave pulses can be applied, C is the microwave cavity of interest and C ′ is the cavity automatically performing the needed correction. Electric fields can be applied at the superconducting mirrors of C and C ′ to Stark shift the Rydberg levels in order to tune the interaction times in C ′ and realize adiabatic transfer in C. The transition between the 5S 1/2 (F = 2, 3) ground state and the 5P 3/2 (F = 1, 2, 3, 4) first excited state is driven by a laser diode at λ1 = 780 nm and is used for the fluorescence detection. The corresponding cycling transition is between (F = 3, mF = 3) and (F ′ = 4, m ′ F = 4). The transition between the first and the 5D 5/2 second excited state is driven by a laser diode at λ2 = 776 nm. Finally, the transition between the second excited state and the Rydberg states is driven by a laser diode at λ3 = 1.26 µm. Fig. 2; (b) Wigner function and density matrix ρn,m of the same cat state after one relaxation time t = 1/γ; (c) Wigner function and density matrix ρn,m after two relaxation times t = 2/γ. The comparison with Fig. 2 is striking: in absence of feedback the Wigner function becomes quickly positive definite, while in the presence of feedback the quantum aspects of the state remain well visible for many decoherence times.
[ 1 ]
1W.H. Zurek, Phys. Today 44(10), 36 (1991), and references therein.
FIG. 2 .
2Schematic diagram of the relevant 85 Rb energy levels involved in the atom counting and in the preparation of the circular Rydberg states.
function and density matrix in the photon number basis, ρn,m = n|ρ|m , of the initial odd cat state, |ψ = N−(|α − | − α ), |α| 2 = 3.3 (b) Wigner function and density matrix ρn,m of the same cat state after 13 feedback cycles, corresponding to a mean elapsed timet ≃ 1/γ (t ≃ 6.6t dec ); (c) Wigner function and density matrix ρn,m of the same state after 25 feedback cycles corresponding to a mean elapsed timet ≃ 2/γ (t ≃ 13t dec ). All the parameter values are given in the text (see section V).
FIG. 4 .
4Time evolution of the same initial state of Fig. 2 in absence of feedback. (a) Wigner function and density matrix in the photon number basis, ρn,m = n|ρ|m , of the initial odd cat state of
FIG.Stark-shift
electric fields
probe atom
counter
Rydberg state
preparation
feedback atom
counter
Rydberg state
preparation
e
g
i
C
C'
R 1
R 2
V
V
Decoherence and the appearance of classical world in quantum theory. D Giulini, E Joos, C Kiefer, J Kupsch, I O Stamatescu, M D Zeh, SpringerBerlinD. Giulini, E. Joos, C. Kiefer, J. Kupsch, I.O. Stamatescu, and M.D. Zeh, Decoherence and the appearance of classical world in quantum theory, (Springer, Berlin 1996).
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See also the special issue of. C H Bennett, ; V Vedral, M B Plenio, Quantum Communication, Computing and Measurement. O. Hirota, A.S. Holevo and C.M. CavesNew YorkPlenum Press11431Physics WorldC.H. Bennett, in Quantum Communication, Computing and Measurement, edited by O. Hirota, A.S. Holevo and C.M. Caves (Plenum Press, New York, 1997), p. 25. See also the special issue of Physics World 11, 35 (1998) and V. Vedral and M.B. Plenio, Contemporary Physics, 39, 431 (1998).
. E Knill, R Laflamme, Phys. Rev. A. 55900and references thereinE. Knill and R. Laflamme, Phys. Rev. A 55, 900 (1997) and references therein.
. D Bouwmeester, J W Pan, M Daniell, H Weinfurter, A Zeilinger, quant-ph/9810035LANL e-print archiveD. Bouwmeester, J.W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, LANL e-print archive quant-ph/9810035.
. E Hagley, Phys. Rev. Lett. 791E. Hagley et al., Phys. Rev. Lett. 79, 1 (1997).
. Q A Turchette, C S Wood, B E King, C J Myatt, D Leibfried, W M Itano, C Monroe, D J Wineland, Phys. Rev. Lett. 813631Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland , Phys. Rev. Lett. 81, 3631 (1998).
P T Cochrane, G J Milburn, W J Munro, LANL e-print archive quant-ph/9809037. P.T. Cochrane, G.J. Milburn, and W.J. Munro, LANL e-print archive quant-ph/9809037.
. P Tombesi, D Vitali, Phys. Rev. A. 514913P. Tombesi and D. Vitali, Phys. Rev. A 51, 4913 (1995);
. P Goetsch, P Tombesi, D Vitali, Phys. Rev. A. 544519P. Goetsch, P. Tombesi and D. Vitali, Phys. Rev. A 54, 4519 (1996).
. D B Horoshko, S Ya, Kilin, Phys. Rev. Lett. 78840D.B. Horoshko and S. Ya. Kilin, Phys. Rev. Lett. 78, 840 (1997).
. D Vitali, P Tombesi, G J Milburn, Phys. Rev. Lett. 792442D. Vitali, P. Tombesi, and G.J. Milburn, Phys. Rev. Lett. 79 2442 (1997).
. D Vitali, P Tombesi, G J Milburn, J. Mod. Opt. 442033D. Vitali, P. Tombesi, and G.J. Milburn, J. Mod. Opt. 44 2033 (1997).
. D Vitali, P Tombesi, G J Milburn, Phys. Rev. A. 574930D. Vitali, P. Tombesi, and G.J. Milburn, Phys. Rev. A 57 4930 (1998).
S Haroche, J M Raimond, Cavity Quantum Electrodynamics. Academic Press123Berman ed.S. Haroche and J.M. Raimond in Cavity Quantum Electrodynamics, P. Berman ed., Academic Press, (1994), p. 123.
. M Brune, S Haroche, J M Raimond, L Davidovich, N Zagury, Phys. Rev. A. 455193M. Brune, S. Haroche, J.M. Raimond, L. Davidovich, and N. Zagury, Phys. Rev. A 45, 5193 (1992).
. S Haroche, M Brune, J M Raimond, Journal de Physique II. 2659S. Haroche, M. Brune, and J.M. Raimond, Journal de Physique II, Paris, 2, 659 (1992).
. X Maitre, E Hagley, G Nogues, C Wunderlich, P Goy, M Brune, J M Raimond, S Haroche, Phys. Rev. Lett. 79769X. Maitre, E. Hagley, G. Nogues, C. Wunderlich, P. Goy, M. Brune, J.M. Raimond, and S. Haroche, Phys. Rev. Lett. 79, 769 (1997).
. P Nussenzveig, F Bernardot, M Brune, J Hare, J M Raimond, S Haroche, W Gawlik, Phys. Rev. A. 483991P. Nussenzveig, F. Bernardot, M. Brune, J. Hare, J.M. Raimond, S. Haroche, and W. Gawlik, Phys. Rev. A 48, 3991 (1993).
. X Maitre, E Hagley, J Dreyer, C Wunderlich, M Brune, J M Raimond, S Haroche, J. Mod. Opt. 442023X. Maitre, E. Hagley, J. Dreyer, C. Wunderlich, M. Brune, J.M. Raimond, and S. Haroche, J. Mod. Opt. 44, 2023 (1997).
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J M Raimond, Laser Spectroscopy IX, M.S. Academic PressFeld ed.J.M. Raimond et al., in Laser Spectroscopy IX, M.S. Feld ed., Academic Press (1989).
. A S Parkins, P Marte, P Zoller, O Carnal, H J Kimble, Phys. Rev. A. 511578and references thereinA.S. Parkins, P. Marte, P. Zoller, O. Carnal, and H.J. Kimble, Phys. Rev. A 51, 1578 (1995) and references therein.
A Messiah, Quantum Mechanics. North Holland, AmsterdamA. Messiah, Quantum Mechanics (North Holland, Amsterdam, 1962).
| [] |
[
"arXiv:quant-ph/0105117v4 18 Apr 2002 From Classical State-Swapping to Quantum Teleportation",
"arXiv:quant-ph/0105117v4 18 Apr 2002 From Classical State-Swapping to Quantum Teleportation"
] | [
"N David Mermin \nLaboratory of Atomic and Solid State Physics\nCornell University\n14853-2501IthacaNY\n"
] | [
"Laboratory of Atomic and Solid State Physics\nCornell University\n14853-2501IthacaNY"
] | [] | The quantum teleportation protocol is extracted directly out of a standard classical circuit that exchanges the states of two qubits using only controlled-NOT gates. This construction of teleportation from a classically transparent circuit generalizes straightforwardly to d-state systems. | 10.1103/physreva.65.012320 | [
"https://export.arxiv.org/pdf/quant-ph/0105117v4.pdf"
] | 119,371,513 | quant-ph/0105117 | 6d7450bdbdd63db65e91a3a4f8bf9c6c49043151 |
arXiv:quant-ph/0105117v4 18 Apr 2002 From Classical State-Swapping to Quantum Teleportation
N David Mermin
Laboratory of Atomic and Solid State Physics
Cornell University
14853-2501IthacaNY
arXiv:quant-ph/0105117v4 18 Apr 2002 From Classical State-Swapping to Quantum Teleportation
The quantum teleportation protocol is extracted directly out of a standard classical circuit that exchanges the states of two qubits using only controlled-NOT gates. This construction of teleportation from a classically transparent circuit generalizes straightforwardly to d-state systems.
Quantum teleportation [1] transfers the quantum state of a two-state system (Alice's qubit, the source) to another remote two-state system (Bob's qubit, the destination) without any direct dynamical coupling between the two qubits. To do this trick Alice, who in general does not herself know the form of the state to be transferred, must possess a third qubit (the ancilla) which initially is maximally entangled with Bob's qubit in the two-qubit state
1 √ 2 | 0 | 0 + | 1 | 1 .
(1)
Depending on the outcomes of appropriate measurements on the source and ancilla, Alice can send Bob instructions that enable him to transform the state of the destination into that originally possessed by the source. The term "teleportation" is apt because the measurements that provide the information to recreate the state at the destination obliterate all traces of it from the source. If two qubits are allowed to interact, however, then their states can be exchanged in a much less subtle way, with the help of three controlled-NOT gates [2]. The action of these gates can be understood in entirely classical terms. This is illustrated in Fig. 1. Fig. 1 does indeed exchange states is readily confirmed by letting it act on a general computational basis state | x | y . If x is the value (0 or 1) of the control bit and y is the value of the target bit, then the action of a single cNOT can be compactly summarized as
That the classical [3] circuit in
| x | y → | x | y ⊕ x(2)
where ⊕ denotes addition modulo 2. If | ψ = | x and | φ = | y , then the action of the three successive gates in Fig. 1 is (reading the Figure from left to right)
| x | y → | x ⊕ y | y → | x ⊕ y | x → | y | x .(3)
This process makes perfect sense for classical bits, as well as for quantum superpositions of classical bits, to which it extends by linearity.
If the state | φ in Fig. 1 is taken to be | 0 , then the cNOT gate on the left acts as the identity, so the classical state-swapping circuit simplifies to:
ψ 0 ψ 0 X X FIG. 2.
If the upper qubit (source) in Fig. 2 belongs to Alice and the lower qubit (destination), to Bob, then this special case of the general classical state-swapping circuit provides a considerably simpler version of what happens in quantum teleportation. But the classical circuit in Fig. 2 is not teleportation, because it requires direct dynamical couplings between the qubits -couplings that teleportation manages to avoid by the use of an entangled pair of qubits and the classical communication of quantum measurement outcomes.
In this note I illuminate the way in which quantum mechanics obviates the need for the direct dynamical couplings in Fig. 2, showing explicitly how this intuitive classical state-swapping circuit leads directly to the transference of a state between uncoupled qubits that constitutes quantum teleportation. It is possible to eliminate all direct couplings between the source and the destination because quantum qubits have a richer range of logical capabilities than do classical bits. Only one indirect dynamical coupling between Alice and Bob survives this process of elimination as the initial interaction necessary to entangle Alice's ancilla with the Bob's destination qubit. All other direct dynamical coupling is replaced by classical communication.
The key to relating quantum teleportation to the apparently quite different way of exchanging a general state in Fig. 2 is to replace the cNOT gate on the left of Fig. 2 with an elementary classical circuit, only slightly more elaborate than that of Fig. 1, that changes the direct coupling of the cNOT into four couplings, all acting only through the intermediary of an unaltered ancillary qubit. To confirm this identity note that the four gates on the right act as follows on the eight computational basis states | x | y | z (with | x the input state on the top left, | z on the bottom, and | y in the middle) [4]:
| x | y | z → | x | y ⊕ x | z → | x | y ⊕ x | z ⊕ y ⊕ x → | x | y | z ⊕ y ⊕ x → | x | y | z ⊕ x .(4)
Thus the circuit on the right of Fig. 3 does indeed act as indicated on the left, performing a cNOT on the qubits associated with the top (control) and bottom (target) wires, while acting as the identity on the qubit associated with the middle wire.
Quantum mechanics first appears when we interchange control and target in the cNOT gate on the right of This follows from the fact that the unitary, self-inverse, Hadamard operator H = 1 √ 2 (σ x + σ z ) takes eigenstates of X = σ x into eigenstates of Z = σ z with corresponding eigenvalues, and vice-versa:
H : | 0 ↔ 1 √ 2 (| 0 + | 1 ), | 1 ↔ 1 √ 2 (| 0 − | 1 ),(5)
together with the fact that controlled-Z has the same action regardless of which qubit is the target and which the control [5]. The utility of this interchange emerges below.
So if we introduce an ancilla in a state | χ , to be specified in a moment, we can replace the two gates in Fig. 2, with the equivalent circuits of Figs I emphasize that Fig. 5 is merely a cumbersome way of constructing the classical circuit of Fig. 2, with the direct couping on the left of Fig. 2 replaced by the four gates on the left, mediated by an ancillary qubit whose state is unaltered, and the direct coupling on the right replaced by the three gates on the right, which by exploiting the quantum-mechanical H gates make it possible to interchange control and target qubits.
To further convert the circuit of Fig. 5 into teleportation, we must first eliminate the unacceptable leftmost coupling between the source and the ancilla. This can be done by taking the state | χ of the ancilla to be H| 0 , which the magic of quantum mechanics -this is the second place where it appears -allows to be invariant under NOT. Because
XH| 0 = H| 0 ,(6)
the leftmost controlled-X in Fig. 5 always acts as the identity, and can be removed from the circuit. So To see that Fig. 6 represents quantum teleportation note that we can also remove the final Hadamard transformation on the upper wire in Fig. 6, provided we change the final state of the qubit associated with that wire from | 0 to H −1 | 0 = H| 0 = | χ . Because the remaining Hadamard on the upper wire commutes with the cNOT that immediately precedes it on the lower two wires, we may also exchange the order of these two gates. The result is
X X ψ 0 0 χ ψ χ X Z H H FIG. 7.
This is precisely the reversible quantum teleportation circuit described by Brassard, Braunstein, and Cleve (BBC) [6]. We have thus made a direct passage from the classical circuit of Fig. 2, which requires coupling between source and destination to swap their states, to the BBC quantum teleportation circuit of Fig. 7, which, as reviewed below, can be further modified to remove all remaining coupling.
I repeat BBC's description of the connection between the circuit of Fig. 7 and teleportation, to indicate what has become of the couplings originally present in Fig. 2 and to show that the four cNOT gates arising from the classical expansion in Fig. 3 of the first cNOT gate in Fig. 2 now play roles in three distinct stages of the quantum teleportation process! [7] The cNOT on the left in Fig. 7, along with the Hadamard gate immediately to its left, used to eliminate the fourth cNOT from Fig. 3, serve to turn the state of the ancilla and destination into the maximally entangled state 1 √ 2 | 0 | 0 + | 1 | 1 . After these two gates have acted Alice keeps the ancilla and Bob takes the destination to a faraway place. Only after that need Alice acquire the source, in the state | Ψ , which may or may not be known to her.
The effect of the next cNOT and Hadamard of Fig. 7 on the source and ancilla, both in Alice's possession, is to transform unitarily the four mutually orthogonal maximally entangled states of the Bell basis [8] into the four computational basis states | x | y . If Alice's two qubits were to be measured in the computational basis after the action of the first four gates, the measurement could therefore be viewed as a coherent two-qubit measurement in the Bell basis, taking place immediately after the first two gates [9].
Such measurements in the computational basis, which are the third and final place where quantum mechanics enters the process, can be introduced, though initially at the wrong stage of the process, by noting that in the final state on the right of Fig. 7 Alice's two qubits are each in the pure state | χ , completely disentangled from Bob's. As a result, the state of Bob's qubit is entirely unaffected if Alice measures each of her qubits. So we can safely add two measurements to Fig. 7 without disrupting the transfer of | ψ from Alice's qubit to Bob's: Not only do these measurements occur too late in the process, but there also remain in Fig. 8 two other interactions between Alice's qubit or her ancilla and Bob's, besides the cNOT gate that originally entangles her ancilla with his destination. The controlled-Z on the right comes directly from the controlled-X on the right of Fig. 2, and the controlled-X immediately preceding it comes from the last of the four controlled-X gates on the right of Fig. 3. Both these interactions can be replaced by classical communication of measurement results from Alice to Bob, by moving the measurements to the earlier stage of the process mentioned above, which it is possible to do for the following reason:
Quite generally the effect of a controlled unitary operation on any number of qubits followed by a measurement of the control qubit is unaltered if the measurement of the control qubit precedes the controlled operation Here the heavy horizontal wire represents N additional qubits, and U represents a unitary transformation acting on any or all of those qubits, controlled by the single qubit represented by the light wire. The measurement and the controlled-unitary operation commute because an arbitrary input state | Ψ of the N + 1 qubits is necessarily of the form
| Ψ = a| 0 | Φ 0 + b| 1 | Φ 1(7)
where |a| 2 + |b| 2 = 1, | 0 and | 1 are computational basis states of the control qubit, and | Φ 0 and | Φ 1 are normalized (but in general non-orthogonal) states of the other N qubits. An immediate measurement on the control qubit takes | Ψ into | 0 | Φ 0 with probability |a| 2 , or into | 1 | Φ 1 with probability |b| 2 y [11]. In the first case subsequent application of a controlled-U has no further effect; in the second case it produces the state | 1 U| Φ 1 .
On the other hand an immediate application of the controlled-U operation takes | Ψ into
a| 0 | Φ 0 + b| 1 U| Φ 1(8)
and a subsequent measurement of the control qubit takes this state into | 0 | Φ 0 with probability |a| 2 , or | 1 U| Φ 1 with probability |b| 2 . Thus the two output states are the same and occur with the same probabilities, regardless of the order in which the measurement and controlled-U are performed. which shifts the actual measurements to the position of the hypothetical measurements mentioned above. Since the controlled-X or controlled-Z in Fig. 10 now follow a measurement of the control bit, their action is identical to applying the X or Z to the target qubit if and only if the outcome of the corresponding measurement is 1; i.e. the controlled operation can be executed locally by Bob depending on what Alice tells him about the outcomes of the two measurements she made on her own qubits.
To summarize, we can look at the teleportation protocol of Fig. 10, and ask what became of the original three couplings in the general classical state-swapping protocol of Fig. 1. The coupling on the left of Fig. 1 vanished by virtue of the initial choice | 0 for the state of the destination (bottom wire of Fig. 10). The middle coupling of Fig. 1 survives in the three cNOT gates coupled to the ancilla (middle wire) in Fig. 10 [12]. Two of the three cNOT's that remain do indeed provide links from Alice's qubits to the destination. But one (on the left of Fig. 10) operates only to create the initial entanglement of the ancilla with the destination, while the other (on the right) operates only through Alice's telling Bob, depending on the result of her measurement on the ancilla, whether or not to apply the transformation X to the destination [13]. The coupling on the right of Fig. 1 survives as the transformation Z applied to the destination or not by Bob depending on what Alice tells him about the result of her measurement on the source.
So you can take the BBC circuit of Fig. 7 and look back to its classical ancestry (Fig. 1) or forward to conventional teleportation (Fig. 10), seeing the same cNOT gates play entirely different roles, depending on which way you want to view the circuit, rather like an optical illusion or a piece of kinetic sculpture. Depending on how you put the punctuation marks into a sequence of operations, you can get a process that is either entirely classical or deeply quantum mechanical. This view of teleportation as a quantum mechanical deconstruction of a trivial classical state-swapping circuit generalizes readily from qubits to d-state systems ("qudits"). If we are dealing with a d-valued classical register, we can generalize cNOT to the controlled bit rotation,
cX : | x | y → | x | y ⊕ x , 0 ≤ x, y < d,(9)
where ⊕ now denotes addition modulo d. This extends by linearity to a unitary operation on quantum d-state systems, which is only self-inverse when d = 2. In the general case the inverse is
cX † : | x | y → | x | y ⊖ x , 0 ≤ x, y < d,(10)
where ⊖ denotes subtraction modulo d. We generalize the Hadamard transformation H on a single qubit to the quantum Fourier transform F on a single d-state system,
F : | y → 1 √ d z e 2πizy/d | z ,(11)
and its inverse
F † : | y → 1 √ d z e −2πizy/d | z .(12)
Note that F| 0 = F † | 0 is invariant under an arbitrary bit rotation so that
(cX)(1 ⊗ F)| ψ | 0 = | ψ | 0 .(13)
A maximally entangled state is prepared by
(cX)(F ⊗ 1)| 0 | 0 = 1 √ d z | z | z .(14)
(These are the generalizations of (1) and (6) from qubits to qudits.) An appropriate generalization to d-state systems of controlled-σ z is
cZ : | x | y → e −2πixy/d | x | y ,(15)
which remains symmetric in control and target qubits and has the inverse
cZ † : | x | y → e 2πixy/d | x | y .(16)
In the above definitions of cX, cX † , cZ, cZ † the state on the left is the control, and the state on the right, the target. More generally, in the relations below, let (cX) ij denote a cX operation in which state i is the control and state j, the target, and let (F) i denote a Fourier transform acting on state i. One easily verifies that
(cX) 12 (F) 2 = (F) 2 (cZ) 12(17)
and therefore
cX 12 = (F) 2 (cZ) 12 (F † ) 2 ,(18)so (cX † ) 12 = (F) 2 (cZ † ) 12 (F † ) 2 = (F) 2 (cZ † ) 21 (F † ) 2 ,(19)
which has the circuit representation (the generalization of Fig. 4) [14]:
X Z = = F F Z F F FIG. 13.
Therefore, following the same sequence of expansions as in the case of 2-state systems, we arrive at the generalization of the BBC circuit of Fig. 7:
X X ψ 0 0 χ ψ χ X Z F F FIG. 14. where | χ = F| 0 = F † | 0 .(20)
One can go from this to the generalization of Fig since the remark [10], that measurement of several control qubits commutes with multi-qubit controlled operations, applies equally well to d state systems even when d is not a power of 2.
The teleportation circuit of Fig. 15 for d-state systems neatly encapsulates the protocol for teleporting d-state systems spelled out in the original teleportation paper [1], along with its relation to the protocol of Fig. 10 for teleporting qubits.
I thank Gilles Brassard and Igor Devetak for useful comments on an earlier version of this essay, and Chris Fuchs for asking why I found it interesting. This work is supported by the National Science Foundation, Grants PHY9722065 and PHY0098429.
[1] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, andW. K. Wootters, Phys. Rev. Lett. 70, 1895-99 (1993).
[2] The unitary cNOT gate operates on the computational basis -i.e. the basis of classically meaningful states, | 0 | 0 , | 0 | 1 , | 1 | 0 , | 1 | 1 -as the identity if the state of the control qubit (indicated by a black dot in Fig. 1) is | 0 , and flips the state (| 0 ↔ | 1 ) of the target qubit (indicated by the boxed X in Fig. 1 BBC prefer to expand Z as HXH.
[7] It is also necessary to retrace this familiar ground to confirm that it supports the generalization to d-state systems described at the end of this note.
[8] . The Bell-basis states are 1 √ 2 | 0 | 0 ± | 1 | 1 and 1 √ 2 | 0 | 1 ± | 1 | 0 . It is easiest to see that the cNOT and Hadamard gates have this affect by looking at the inverse transformation.
[9] Conventional expositions of teleportation do indeed expand the state of Alice's two qubits in the Bell basis after the entangled pair is formed, having her then make a coherent two-qubit measurement in that basis. But it is simpler analytically when algebraically tracing the progress of a general | ψ through the protocol, as well as more straightforward to implement physically, to take seriously the circuit of BBC, letting Alice explicitly apply the next cNOT and Hadamard and follow this by independent qubit measurements in the ordinary computational basis. As BBC note, there is no need to mention the Bell basis at all.
[10] This is a straightforward extension to more than two qubits of the point made by R. B. Griffiths and C. S. Niu, Phys. Rev. Lett. 76, 3228-3231 (1996), quantph/9511007, and invoked by BBC. The same situation holds for a unitary operation controlled by the 2 M different outcomes of a measurement on M control qubits.
Such an operation has the form U = i PiUi where the Pi = | Φi Φi| project onto a complete orthonormal set of states | Φi of the control bits, and Ui is the unitary transformation on the N target bits associated with the i-th measurement outcome. (Since the Ui are unitary and the Pi commute with all the Uj and give a resolution of the identity into orthogonal projections, it follows that U is indeed unitary.) Clearly performing the von Neumann measurement associated with the Pi commutes with applying U, in the sense that the same final states arise with the same probabilities.
[11] This extension of Born's probability rule to cases in which only a subsystem is measured, which is crucial in quantum computation, receives surprisingly little explicit attention in most textbook introductions to quantum mechanics.
[12] The very first of the four cNOT gates coming from the expansion in Fig. 3 of the middle coupling of Fig. 1 was crucially rendered unnecessary by the initial choice H| 0 for the state of the ancilla.
[13] The remaining cNOT in Fig. 10 links Alice's qubit only to her ancilla. It can be viewed, if one wishes, as a part of the process of "measurement in the Bell basis." [14] Note the unfortunate but firmly entrenched convention that in circuit diagrams operations on the left act first while in equations operations on the right act first.
FIG. 1.
FIG. 3.
FIG. 4.
FIG. 5.
FIG. 6.
FIG. 8.
FIG. 9.
FIG. 10.
FIG. 11.
FIG. 15.
Alternatively one can note, in the computational basis, that if the state of the top wire is | 0 then neither of the NOT operations acts on the middle wire so the two lower self-inverse cNOT operations act in direct succession, giving the identity. But if the state of the top wire is | 1 then both NOT operations act on the middle wire, leaving its state unaltered, and ensuring that exactly one cNOT operation acts on the lower wire regardless of that state.[5] In either case controlled-Z acts as the identity on the computational basis states | 0 | 0 , | 0 | 1 , | 1 | 0 and multiplies | 1 | 1 by −1.[6] Gilles Brassard, Samuel L.Braunstein, and Richard Cleve, Physica D 120, 43-47 (1998), quant-ph/9605035.) if the state of
the control qubit is | 1 .
[3] I call a quantum circuit classical if it is classically mean-
ingful when restricted to classically meaningful states -
i.e. if every unitary gate takes computational-basis states
into other computational basis states without introduc-
ing superpositions or phases. Because the circuit of Fig. 1
exchanges computational-basis states and acts linearly on
superpositions of inputs, it also, of course, exchanges ar-
bitrary quantum states.
[4]
| [] |
[
"Breitenlohner-Freedman bound on hyperbolic tilings",
"Breitenlohner-Freedman bound on hyperbolic tilings"
] | [
"Pablo Basteiro \nInstitute for Theoretical Physics and Astrophysics and Würzburg-Dresden Excellence Cluster ct.qmat\nJulius Maximilians University Würzburg\nAm Hubland97074WürzburgGermany\n",
"Felix Dusel \nInstitute for Theoretical Physics and Astrophysics and Würzburg-Dresden Excellence Cluster ct.qmat\nJulius Maximilians University Würzburg\nAm Hubland97074WürzburgGermany\n",
"Johanna Erdmenger \nInstitute for Theoretical Physics and Astrophysics and Würzburg-Dresden Excellence Cluster ct.qmat\nJulius Maximilians University Würzburg\nAm Hubland97074WürzburgGermany\n",
"Dietmar Herdt \nInstitute for Theoretical Physics and Astrophysics and Würzburg-Dresden Excellence Cluster ct.qmat\nJulius Maximilians University Würzburg\nAm Hubland97074WürzburgGermany\n",
"Haye Hinrichsen \nInstitute for Theoretical Physics and Astrophysics and Würzburg-Dresden Excellence Cluster ct.qmat\nJulius Maximilians University Würzburg\nAm Hubland97074WürzburgGermany\n",
"René Meyer \nInstitute for Theoretical Physics and Astrophysics and Würzburg-Dresden Excellence Cluster ct.qmat\nJulius Maximilians University Würzburg\nAm Hubland97074WürzburgGermany\n",
"Manuel Schrauth \nInstitute for Theoretical Physics and Astrophysics and Würzburg-Dresden Excellence Cluster ct.qmat\nJulius Maximilians University Würzburg\nAm Hubland97074WürzburgGermany\n"
] | [
"Institute for Theoretical Physics and Astrophysics and Würzburg-Dresden Excellence Cluster ct.qmat\nJulius Maximilians University Würzburg\nAm Hubland97074WürzburgGermany",
"Institute for Theoretical Physics and Astrophysics and Würzburg-Dresden Excellence Cluster ct.qmat\nJulius Maximilians University Würzburg\nAm Hubland97074WürzburgGermany",
"Institute for Theoretical Physics and Astrophysics and Würzburg-Dresden Excellence Cluster ct.qmat\nJulius Maximilians University Würzburg\nAm Hubland97074WürzburgGermany",
"Institute for Theoretical Physics and Astrophysics and Würzburg-Dresden Excellence Cluster ct.qmat\nJulius Maximilians University Würzburg\nAm Hubland97074WürzburgGermany",
"Institute for Theoretical Physics and Astrophysics and Würzburg-Dresden Excellence Cluster ct.qmat\nJulius Maximilians University Würzburg\nAm Hubland97074WürzburgGermany",
"Institute for Theoretical Physics and Astrophysics and Würzburg-Dresden Excellence Cluster ct.qmat\nJulius Maximilians University Würzburg\nAm Hubland97074WürzburgGermany",
"Institute for Theoretical Physics and Astrophysics and Würzburg-Dresden Excellence Cluster ct.qmat\nJulius Maximilians University Würzburg\nAm Hubland97074WürzburgGermany"
] | [] | We establish how the Breitenlohner-Freedman (BF) bound is realized on tilings of two-dimensional Euclidean Anti-de Sitter space. For the continuum, the BF bound states that on Anti-de Sitter spaces, fluctuation modes remain stable for small negative mass-squared 2 . This follows from a real and positive total energy of the gravitational system. For finite cutoff , we solve the Klein-Gordon equation numerically on regular hyperbolic tilings. When → 0, we find that the continuum BF bound is approached in a manner independent of the tiling. We confirm these results via simulations of a hyperbolic electric circuit. Moreover, we propose a novel circuit including active elements that allows to further scan values of 2 above the BF bound. arXiv:2205.05081v2 [hep-th] | 10.1103/physrevlett.130.091604 | [
"https://export.arxiv.org/pdf/2205.05081v2.pdf"
] | 248,693,506 | 2205.05081 | 96502a9917150d6479a5787857657fb4eba81743 |
Breitenlohner-Freedman bound on hyperbolic tilings
Pablo Basteiro
Institute for Theoretical Physics and Astrophysics and Würzburg-Dresden Excellence Cluster ct.qmat
Julius Maximilians University Würzburg
Am Hubland97074WürzburgGermany
Felix Dusel
Institute for Theoretical Physics and Astrophysics and Würzburg-Dresden Excellence Cluster ct.qmat
Julius Maximilians University Würzburg
Am Hubland97074WürzburgGermany
Johanna Erdmenger
Institute for Theoretical Physics and Astrophysics and Würzburg-Dresden Excellence Cluster ct.qmat
Julius Maximilians University Würzburg
Am Hubland97074WürzburgGermany
Dietmar Herdt
Institute for Theoretical Physics and Astrophysics and Würzburg-Dresden Excellence Cluster ct.qmat
Julius Maximilians University Würzburg
Am Hubland97074WürzburgGermany
Haye Hinrichsen
Institute for Theoretical Physics and Astrophysics and Würzburg-Dresden Excellence Cluster ct.qmat
Julius Maximilians University Würzburg
Am Hubland97074WürzburgGermany
René Meyer
Institute for Theoretical Physics and Astrophysics and Würzburg-Dresden Excellence Cluster ct.qmat
Julius Maximilians University Würzburg
Am Hubland97074WürzburgGermany
Manuel Schrauth
Institute for Theoretical Physics and Astrophysics and Würzburg-Dresden Excellence Cluster ct.qmat
Julius Maximilians University Würzburg
Am Hubland97074WürzburgGermany
Breitenlohner-Freedman bound on hyperbolic tilings
We establish how the Breitenlohner-Freedman (BF) bound is realized on tilings of two-dimensional Euclidean Anti-de Sitter space. For the continuum, the BF bound states that on Anti-de Sitter spaces, fluctuation modes remain stable for small negative mass-squared 2 . This follows from a real and positive total energy of the gravitational system. For finite cutoff , we solve the Klein-Gordon equation numerically on regular hyperbolic tilings. When → 0, we find that the continuum BF bound is approached in a manner independent of the tiling. We confirm these results via simulations of a hyperbolic electric circuit. Moreover, we propose a novel circuit including active elements that allows to further scan values of 2 above the BF bound. arXiv:2205.05081v2 [hep-th]
Introduction:
The AdS/CFT correspondence [1][2][3], also known as holography, maps gravitational theories in ( + 1)dimensional hyperbolic Anti-de Sitter (AdS) spacetimes to strongly coupled conformal field theories (CFTs) without gravity in dimensions, defined on the AdS boundary. The AdS/CFT duality provides a precise map between CFT operators and AdS gravity fields, which is of great significance both for fundamental aspects of quantum gravity [4] and for applications to strongly correlated condensed matter systems [5].
Motivated by the goal to provide a new example of holographic duality, as well as possible realizations in tabletop experiments, in this Letter we report novel insights in this direction for discretized systems. A prime candidate is a scalar field defined on discretizations of AdS space via regular hyperbolic tilings (see Fig. 1) [6,7], which have been recently investigated using methods from lattice gauge theory in [8][9][10][11]. These works consider discretization schemes for the scalar action, the Laplace operator, and lattice bulk propagators, finding good agreement of the scaling behavior of correlation functions with analytic continuum results.
The physics of hyperbolic tilings has recently been studied in the context of condensed matter physics [12], circuit quantum electrodynamics [13][14][15], and topolectric circuits [16][17][18]. These works focus on the spectrum of tight-binding Hamiltonians on hyperbolic lattices and their realization based on coupled waveguide resonators [12,14,15] or classical non-dissipative linear electric circuits (topolectric circuits). Time-resolved measurements of wave propagation in hyperbolic space have been achieved in such architectures [17].
It remains an open question, though, how to establish a duality in the sense of a map between bulk and boundary theories for hyperbolic tessellations. Steps in this direction were taken in [19,20] using modular discretizations and in [21] via tensor networks on hyperbolic buildings. In this work we focus on hyperbolic tilings as a discretization scheme instead. Starting point is one of the key results of the continuum AdS/CFT correspondence, namely the relation between the mass of a scalar field in the bulk and the scaling dimension Δ of its dual operator on the boundary, Δ(Δ − ) = 2 2 , with being the AdS radius and the boundary spacetime dimension [2]. This is determined by the asymptotic boundary behavior of the solutions of the Klein-Gordon equation in AdS space. Confor- Figure 1. Hyperbolic {7,3} tiling in the Poincare disk representation. For the central node, the stencil of the discretized Laplace-Beltrami operator is highlighted. Red sites carry constant weights (7,3) , whereas the central node (blue) is weighted by −7 (7,3) . mally transforming AdS space into flat space, the scalar field experiences a shift of its mass-squared to 2 2 + 2 ∕4. Thus, there is a stable potential minimum for the field if 2 2 > − 2 ∕4, i.e. even for small negative mass-squared. This is the Breitenlohner-Freedman (BF) bound [22,23] [24]. In this Letter, we determine how the BF bound is realized in discrete holographic setups and how it manifests itself on finite-sized architectures accessible through simulation and experiment. We establish the BF bound for hyperbolic tessellations by first analyzing the properties of the continuum analytical solutions in the presence of a finite cutoff. This cutoff can be chosen arbitrarily and is required because only finite tilings, which do not cover the entirety of hyperbolic space, can be experimentally and numerically realized. In particular, this cutoff is independent of the Schläfli parameter { , } characterizing a regular hyperbolic tiling with regular -gons meeting at each vertex. Defining a scalar field on the vertices, we numerically solve the associated equations of motion on several tilings, finding excellent agreement with results from continuum holography. We find that the stability bound of a scalar field defined on large enough { , } hyperbolic tilings coincides with the continuum BF bound, independently of and . Our analysis extends previous investigations [13,15,17] of the eigenvalue problem of the discrete Laplacian on these tilings. In particular, we use insights from holography, such as the presence of non-normalizable modes, to provide solutions for masses-squared above the BF bound, thus beyond the standard spectrum of the Laplacian. Moreover, we propose a novel electric circuit, in the spirit of topolectric circuits [25,26], to access these new mass-squared values in experiment.
Equations of Motion on EAdS 2 : In order to investigate the physics of the BF bound for hyperbolic tilings, we consider one of the simplest continuum systems admitting a holographic duality, a free massive scalar field Euclidean AdS 2 (EAdS 2 ), with the induced metric
2 = = 2 cos 2 ( ) 2 + sin 2 ( ) 2 . (1)
Here, is the curvature radius of EAdS 2 , and ∈ [0, 2 ), ∈ [0, 2 ). The asymptotic boundary of EAdS 2 is at = 2 , corresponding to an infinite geodesic distance from the origin = 0. The scalar field action
= 1 2 ∫ 2 √ Φ Φ + 2 Φ 2(2)
yields as equation of motion the Klein-Gordon equation [27]
0 = 1 √ √ Φ − 2 Φ ≡ (□ − 2 )Φ (3) = 1 2 cos cot sin Φ − 2 Φ( ) ,(4)
with □ the Laplace-Beltrami operator on EAdS 2 . The second equality in (4) holds for a purely -dependent field configuration Φ( ) with no angular dependence. Eq. (4) admits analytic solutions in terms of hypergeometric functions (S.2) in the Supplementary Material [28], parametrized by two integration constants, which can be related by the regularity boundary condition Φ ′ (0) = 0. Asymptotically near the boundary at = ∕2, the two fundamental solutions behave as
Φ( ) ≃ (cos ) 1−Δ + (cos ) Δ ,(5)
where Δ = 1 2 + √ 1 4 + 2 2 is the scaling dimension of the boundary operator holographically dual to Φ. In AdS/CFT, the terms in (5) are denoted as non-normalizable and normalizable modes, respectively. Imposing suitable boundary conditions at = 2 [29], the coefficient of the normalizable mode is identified with the vacuum expectation value of the dual operator , while the coefficient of the nonnormalizable mode determines its source . In spatial dimension = 1 and for masses of the scalar field
2 2 < − 1 4 ,(6)
the scaling dimension Δ of the dual operator becomes complex. In the CFT, this indicates a breakdown of unitarity. In the AdS bulk, it implies that the energy of the scalar field, ceases to be a real and positive quantity [28]. Here, we have used the ansatz Φ( ) = (cos ) ΔΦ ( ) [22,23,30]. After quantization, this denotes an instability of the system towards a new, true ground state [31]. The reality condition on Δ implies 2 2 ≥ − 1 4 . This is the Breitenlohner-Freedman stability bound [22,23] and is a key element of AdS/CFT.
= ∫ cos( ) 2Δ ( Φ ) 2 + Δ 2Φ2 ,(7)
We now analyze the implications of a finite cutoff for the analytical solutions. Regularity at the origin is imposed through von Neumann boundary conditions at = 0. Introducing a finite radial cutoff ≪ 1, such that Dirichlet boundary conditions are imposed at = 2 − via Φ( ) = 1, leads to a rescaling of the solutions. We solve Eq. (4) subject to these boundary conditions and find that, above the BF bound, solutions have no zeroes in the regime ∈ [0, 2 ). Below the BF bound, however, solutions develop an infinite set of zeroes [28].
At specific values of the mass-squared 2 2 and cutoff , we observe a singular behavior of the solutions, characterized by discontinuous jumps of the field amplitude, as presented in Fig. 2. These only appear below the BF bound and are a result of the cutoff coinciding with a zero of the solution associated to the given value of 2 2 , thus making the rescaling factor diverge. We denote these pairs ( 2 2 , ) as Umklapp points. The position of the first Umklapp point below 2 2 = − 1 4 , corresponding to the zero which is furthest into the bulk, can be used as an indicator for the unstable regime. More precisely, when the cutoff is removed, the value of the mass-squared for which this zero first appears corresponds to the BF stability bound. The analytical derivation of the solutions to the KG equation (4) at a finite cutoff provided in [28] allows for an exact tracking of this first Umklapp point for different cutoffs. This provides a reference behavior, shown in black in Fig. 3, to which we compare our numerical findings.
Regular hyperbolic tilings of D 2 : EAdS 2 is isomorphic to the Poincaré disk model of hyperbolic space D 2 , which can be naturally discretized by regular hyperbolic tilings [6,7]. These preserve a large subgroup, known as a Fuchsian group of the first kind, of the isometry group PSL(2, R) of EAdS 2 [32? ], making them promising candidates for setting up a discrete holographic duality. Hyperbolic tilings are characterized by their Schläfli symbol { , }, with ( − 2)( − 2) > 4, denoting a tiling with regular -gons meeting at each vertex. The {7, 3} hyperbolic tiling and its dual {3, 7} tiling are shown in Fig. 1 as an example. Since hyperbolic space introduces a length scale through its radius of curvature , the edge lengths of hyperbolic polygons are fixed quantities, depending only on the Schläfli parameters and [33]. Their geodesic length ( , ) in units of can be computed via the Poincaré metric (1) and can be interpreted as a fixed lattice spacing that cannot be tuned. We compute ( , ) for several and in [28]. In general, this makes a continuum limit of { , } tilings in the usual way impossible. Nevertheless, we provide evidence that regular hyperbolic tilings indeed preserve some properties of the continuum scalar field theory, indicating that they are a good approximation of continuum EAdS 2 .
While the entire EAdS 2 space can be filled with an infinite { , } tiling, numerical simulations and experimental setups can only be finite-sized. The truncation of the tiling to a finite number of layers is equivalent to the introduction of a finite cutoff as mentioned earlier. Given the jagged structure of the tiling's boundary at any finite layer, an effective uniform radial cutoff needs to be drawn. This allows a direct comparison of the Umklapp points observed in numerical simulations on the truncated tilings with the analytical solutions derived in [28].
Numerical Methods: The central ingredient for our numerical analysis of the Klein-Gordon equation on { , } hyperbolic tilings is a suitable discretization, denoted by□, of the Laplace-Beltrami operator. Its action on a scalar function Φ( , ), represented on the tiling by discrete values Φ = Φ( ), can be written as
□ Φ = ∑ | −2 (Φ − Φ ),(8)
where | denotes the summation over the neighboring sites of site . In order to determine the weight factors , we implement the following method devised from the established approximation of lattice operators by finite difference quotients. Given that all cells in a hyperbolic tiling are isometric, it is clear that all weights in the stencil have to be equal, i. e. , ≡ (cf. right panel of Fig. 1), which allows us to write the lattice operator formally as a matrix
□ − 2 ≡ −2 ( − ) + ,(9)
acting on a vector of function values Φ = (Φ 0 , Φ 1 , …). Here, and denote the adjacency and degree matrix of the tiling graph and = diag(− 2 ). In order to calculate the weight , recall the approximation of a 1D second derivative by finite differences̃ 2 ( ) = ( ( − ℎ) − 2 ( ) + ( + ℎ)). First, we determine a test function ( ) such that 2 ( ) = 1, in this case ( ) = 1 2 2 . Applying the discretized derivative to this function yields̃ 2 ( ) = ℎ 2 , hence = ℎ −2 . Note that unlike this case of a Cartesian hypercubic lattice, the hyperbolic lattice spacing ℎ is fixed, as discussed above.
We now apply this procedure to the stencil on the hyperbolic lattice. First, we determine a radially symmetric test function ( ) such that □ ( ) = 1 2 cos 2 cot + cos 2 2 ( ) = 1 . (10) A possible solution is
( ) = 2 ln 1 + 1 cos .(11)
Applying□ to this function on the central site of the tiling (cf. right panel of Fig. 1) yields
□ (0) = ( ( , ) ) − (0) = 1.(12)
Solving for , the weight factors can be obtained for every { , } and are listed in [28]. Numerical Results: Given the lattice Laplacian, the continuum Klein-Gordon equation on a constant time slice can be expressed on the finite hyperbolic tiling as
□ Φ − 2 Φ = 0 for < , Φ( ) = Φ for =(13)
where the boundary condition is implemented by assigning constant values to sites outside the radial cutoff . Solving the discretized boundary value problem (13) requires iterative matrix methods [34,35] already for medium Figure 4. Section of the hyperbolic electric circuit. The structure is repeated at every vertex in the lattice. system sizes. It has to be taken into account that for negative 2 exceeding a certain threshold, most standard solvers tend to be unstable due to the lattice operator□ − 2 becoming indefinite in this parameter regime [36]. A class of algorithms which can handle indefinite, sparse linear systems are so-called Krylow subspace methods [37]. In particular, we use both the GMRES (generalized minimum residual) [38] and BiCGSTAB (biconjugate gradient stabilized) [39] methods to solve Eq. (13) and extract the positions of the Umklapp points. Results from both algorithms are fully compatible and presented in Fig. 3 for various hyperbolic tilings [40] and values of the cutoff. We find that all curves nicely converge towards 2 2 = −1∕4 for → 0, thus yielding the correct infinite volume limit and marking the main result of this Letter.
We expect that the universal behavior for all and displayed in Fig. 3 originates from a group-theoretic argument as follows. For Fuchsian groups of the first kind, which describe the isometries of { , } tilings, it is known that the boundary limit set is the circle 1 [21? ]. For infinite tilings, this implies conformal invariance of the boundary theory. The BF bound is the mass-squared threshold at which the scaling dimension Δ of the CFT operator dual to the bulk scalar field becomes complex, as can be seen from the definition of Δ below (5). Thus, the asymptotic value of the first Umklapp point must be the same for all and as → 0. In addition, our numerical results of Fig. 3 indicate that even for finite cutoff, where the Fuchsian symmetry is broken, the universality of the → 0 behavior is preserved for all and .
Hyperbolic Electric Circuits: We further propose an experimental realization of the BF bound in a suitable electric circuit. We are motivated by topolectric circuits [16], which are a platform based on circuits of capacitors and inductors which are engineered to realize a plethora of models exhibiting topological states of matter [25,26,41]. Specifically, let us consider a circuit on a hyperbolic tiling as shown in Fig. 4. On the vertices of the tiling we attach grounded capacitors and connect them via identical inductors along the polygon edges. Note that our construction differs from that in [17] in that we exchange capacitors and inductors. This is necessary because only then the site voltage represents the scalar field . In this network, the voltage at site is related to the capacitor current by = ̇ while the voltage differences between neighboring sites of are related to the induced current by ( − ) = ̇ . According to Kirchhoff's laws, the time-evolution of the voltage at site is given bÿ
= 1 ∑ | ( − ),(14)
where in our case, the weights are constant = ( , ) as discussed above. The oscillatory eigenmodes ( ) = are determined by the system of equations
− 2 = 1 ∑ | ( − ) = ( , ) □ .(15)
By identifying
− 2 2 = 2 ( , ) ,(16)
the electric circuit provides a realization of the discretized Klein-Gordon equation for 2 2 < 0 [42]. The first Umklapp point corresponds to the lowest eigenfrequency of the circuit, which represents the finite gap in the negative-definite eigenspectrum of the hyperbolic Laplacian [15]. Since the circuits described above contain only passive elements, they can only realize the regime of negative masssquared. This is however precisely the regime where according to (7), solutions to (4) are unstable within the AdS/CFT correspondence. For electric circuits to access the regime of 2 above the BF bound, thus realizing non-normalizable solutions (5) essential for holography, the implementation of active electrical elements is required. Such elements were introduced in [? ] in the context of topolectric circuits. We propose to use negative impedance converters to achieve negative values of or on the r.h.s. of (16).
In order to systematically locate the eigenmodes of the passive circuit, we apply a driving alternating current at the central node and integrate the system of Eqs. (15) over time using an explicit fourth-order Runge-Kutta method. Details of our numerical analysis are presented in [28]. Once the fundamental mode of the system is found, the corresponding negative mass-squared can be extracted according to Eq. (16). These resonances (eigenfrequencies of the circuit) are a physical manifestation of the Umklapp points introduced above. Performing this analysis for several different hyperbolic tilings [40] and finite cutoff radii, we are able to locate the positions of the lowest eigenfrequency. Similarly to our analysis of the Umklapp points, we are able to find the instability threshold on the tiling by tracking the position of the first resonance frequency of the circuit as the cutoff is removed. Again, we find an excellent agreement with the continuum prediction, as shown in the inset of Fig. 3. Our analysis thus shows how the BF bound can be experimentally realized on hyperbolic electric circuits.
Conclusions: For the first time, we have identified the implications of the Breitenlohner-Freedman bound for discrete regular tilings of hyperbolic space. Notably, we find universal behavior of the instabilities for all { , } discretizations, even for finite cutoff. In particular, we find excellent agreement between the positions of the Umklapp points as obtained via numerical simulations of the scalar field on several different { , } tilings with the analytical solutions of the Klein-Gordon equation on EAdS 2 . Moreover, for a specific hyperbolic electric circuit we show how the resonance frequencies are a manifestation of the Umklapp points. Simulations of the circuit dynamics also show excellent agreement with the analytical data by yielding the same dependence of the resonances on the cutoff size. Both these results confirm the universal behavior. Furthermore, we suggest how to adapt the electrical circuits in order to realize mass-squared values above the BF bound. Such circuit realizations will make regular hyperbolic tilings excellent candidates for bringing aspects of AdS/CFT to the laboratory.
The EAdS 2 manifold considered here describes a constant time slice of the larger AdS 2+1 spacetime. It would be interesting to generalize our analysis to a Lorentzian setting involving time, for instance by adding a temporal leg to the vertices of the tilings and equipping them with radius-dependent weights (see also [11] for a first attempt in this direction). In practice, this can be implemented by locally modifying and on the hyperbolic electric circuit. We leave this for future work. This differential equation has two fundamental solutions in terms of hypergeometric functions [43], whose linear combination gives the general solution Φ( ) = (−1) (S. 19) and̃ are integration constants. Neglecting for the moment constant prefactors we can already extract the asymptotic behavior of the solutions (S.18) towards the boundary → 2 , cos( ) 2 → 0. For this, recall that the hypergeometric functions are regular around the origin = 0, with their series expansion given by [43]
Positivity of the Energy Functional
In this section we show how solutions to the Klein-Gordon equation for masses below the BF bound are unstable. This instability is characterized by what was denoted as positivity of the energy in the original work of Breitenlohner and Freedman [22,23]. In their context, positivity requires the energy to be a a positive real number. If the energy is negative or develops a non-vanishing imaginary part, the system is considered to be unstable. The arguments in this section have been provided partly in the original papers by Breitenlohner and Freedman [22,23], as well as in an extended analysis given in [30]. These works considered the Lorentzian version of AdS spacetime in higher dimensions, but the logical arguments are insensitive to a Wick rotation and the choice of = 2. The scalar action 26) with , = { , } in the coordinates of Eq. (1) of the main text. We can associate a corresponding stress-energy tensor through the variation
= 1 2 ∫ 2 √ Φ Φ + 2 Φ 2 , (S.≡ 2 √ . (S.27)
For our scalar field, we obtain explicitly
= Φ Φ + 1 2 Φ Φ + 2 Φ 2 . (S.28)
The energy functional of the full system decomposes into a contribution from gravity and a contribution from the scalar field. The positivity of the gravity contribution was proven in [44]. The contribution from the scalar field is given by the energy functional
( ) = ∫ 2 0 , (S.29)
where the integrand denotes the "temporal" component of the stress energy tensor. In Euclidean signature, this corresponds to the angular coordinate present in Eq. (1) of the main text. Inserting our stress-energy tensor (S.28), we obtain
( ) = ∫ 2 0 3 2 ( Φ) 2 + 1 2 ( Φ) 2 + 1 2 2 2 cos( ) 2 Φ 2 . (S.30)
This expression might seem manifestly positive, but the potential term can contribute negatively for negative mass-squared. Since we are interested radially symmetric solutions, we can neglect the first term. This is not a restriction, since this term by itself is indeed manifestly real and positive. We will thus not write it explicitly in the following. In order to re-write the radial derivative and the potential terms in manifestly real and positive form, we use the ansatz Φ( ) = cos( ) ΔΦ , where the exponent is a priori not specified but we denoted it with a suggestive notation in view of the scaling dimension. Plugging in this ansatz into (S.30) and integrating by parts, we find
( ) = ∫ 2 0 cos( ) 2Δ ( Φ ) 2 + Δ + Δ(1 − 2Δ) sin( ) 2 + Δ 2 + 2 2 cos( ) 2 Φ 2 − 2Δ cos( ) 2Δ−1 sin( )Φ 2 2 0 . (S.31)
The vanishing of the boundary term for normalizable modes was proven in [22,23] and for non-normalizable modes in [30]. It thus remains to guarantee the reality and positivity of the integrand. By choosing the coefficient in our ansatz to obey Δ = This expression is manifestly real and indeed positive as long as Δ is real, which in turn requires 2 2 ≥ − 1 4 . This is precisely the BF bound. For visualization purposes, the behavior of the solutions above and below the BF bound is shown in Fig. 5. We observe a drastic change of the solutions when the mass-squared goes below the stability threshold.
Hyperbolic Electric Circuits
Here we describe in detail our numerical simulations regarding a possible experimental realization of the Breitenlohner-Freedman bound, using an electric circuit network. The primary goal is to locate the fundamental oscillatory eigenmode of the circuit. The corresponding resonance frequency directly corresponds to the position of the stability bound via Equation (15) in the main text. Specifically, we implement the setup shown in Fig. 4 in the main text numerically. The equations governing the dynamics of the currents along the links and voltages on the nodes are given by
̇ = ( , ) ( − ) ̇ = , (S.33)
where the dot denotes a temporal derivative. This represents a system of 2 coupled first-order differential equations, where is the number of nodes in the tiling and , are vertex indices. The system is to be solved for the time-dependent voltages ( ), = 1, … , . In our simulations, we set = = 1. The weights ( , ) are given in Table I for a selection of ( , ) pairs. Note that in our actual technical realization, these weights are introduced as the relative strength of the inductances as compared to the capacitances.
In order to integrate Equations (S.33) over time, we use an explicit fourth-order Runge-Kutta (RK4) method [45,46]. Radial Dirichlet boundary conditions are realized by first truncating the hyperbolic lattice at a fixed radius < 2 . Then, only those nodes inside the cutoff radius are iterated according to the dynamic rules (S.33). Sites which are positioned outside of the cutoff are considered ghost nodes. They are grounded without an additional capacitor and hence effectively not updated during the simulation. Consequently, their voltages remain zero, i. e. ( ) = 0 for ( ) > . We prepare the system in a state where all voltages are set to zero initially and apply a driving alternating voltage at the central site of the tiling, ( 0 , ) ≡ 0 ( ) ∝ sin( ). Note that in order to resolve the temporal dynamics of the circuit accurately, the time increment of the integrator is chosen significantly smaller than the typical time scale of the system, i. e. Δ ≪ −1 .
Due to the external driving force in the center of the tiling, an oscillation of the voltage throughout the entire lattice is initiated. In order to locate the lowest eigenfrequency of the system, we gradually increase and detect the points ,0 < ,1 < … where the system resonates. Most conveniently these frequencies are found by monitoring the total energy, given as tot = 1 2
{ , } ( , )( ,∑ 2 + 1 2 ∑ | 2 . (S.34)
Let us stress that this energy is a physically different quantity than the gravitational energy defined in (S.30). The latter is the energy of the gravitational system with respect to the Euclidean time at the boundary of EAdS 2 , while the former is the energy of the electrical circuit obtained from Kirchhoff's laws and defined with respect to an auxiliary time parameter. While the energy in (S.30) is the quantity responsible for the BF bound, the circuit energy in (S.34) is used in the simulations to monitor the response of the voltage with respect to the driving force and thus find the resonant frequencies. The quantity defined in (S.34) is expected to exhibit a pronounced peak whenever a resonance frequency is hit. In Fig. 6 we plot the energy against the frequency of the driving force. Indeed, we observe a number of strong peaks associated to the resonance modes of the circuit. All the results shown in the figures are for the exemplary case of a {7,3} hyperbolic tiling.
In order to verify that these are in fact the eigenmodes of the system, we shut down the driving force once the system has reached a state of steady oscillation. For reasons of numerical stability, this is done smoothly over the period of a few oscillations. If the eigenmode is met, we expect an unchanged steady oscillation of the system, even without the external drive, which is indeed confirmed in our simulations. Examples are shown in Fig. 7. During this phase of a "free" oscillation, the total energy of the system must be conserved since our circuit is assumed to be "ideal" and hence dissipation free. From the technical perspective conservation of energy represents an important validation for numerical stability. As can be seen in Fig. 7, obtained through the RK4 method with an error of only ((Δ ) 4 ), the energy indeed stays approximately constant during the further evolution of the system. Finally, as a second verification, we compute the Fourier spectra of the free oscillation, as shown in Fig. 8. For the fundamental mode = ,0 (upper panel in the figure) we find only one sharp peak in an otherwise empty spectrum, as expected. Once we have located and confirmed the fundamental eigenmode, we continue to precisely pinpoint the corresponding frequency ,0 by fitting a Lorentzian profile to the peak in the energy-frequency diagram in Fig. 6 and read off the maximum. This frequency can then be translated into the corresponding Klein Gordon mass squared as described in Equation (15) in the main text.
Figure 2 .
2Field amplitude at the origin for different cutoff values . We observe Umklapp points appearing at any finite cutoff, the rightmost of which can be used as an indicator of unstable solutions. For smaller cutoffs, the Umklapp points become denser and converge towards the continuum BF bound (dotted line).
Figure 3 .
3First Umklapp point for various hyperbolic tilings and radial cutoffs . For small , the curves tend towards the continuum bound 2 2 = −1∕4, indicated by the dotted horizontal line. Inset: corresponding results from our hyperbolic electric circuit simulations.
Acknowledgements: We are grateful to R. Thomale, R. N. Das and G. Di Giulio for fruitful discussions. Moreover, we thank F. Goth for helpful discussions regarding the numerical aspects of this article. P.B., J.E., R.M. and M.S. acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy through the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter -ct.qmat (EXC 2147, project-id 390858490). J.E. and R.M. furthermore acknowledge financial support through the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), project-id 258499086 -SFB 1170 'ToCoTronics'. SUPPLEMENTARY MATERIAL Analytic Solutions on EAdS 2 We provide the full analytical derivation of the solutions to the Klein-Gordon equation in EAdS 2 . In the coordinates given in Eq. (1) of the main text, the Klein-Gordon equation in Eq. (4) can be written as cos( ) 2 2 Φ( ) + cos( ) 2 cot( ) Φ( ) − 2 2 Φ( ) = 0 . (S.17)
where the coefficients , include possible constant prefactors. The solutions (S.18) diverge at the origin → 0 due to individual logarithmic divergences of the hypergeometric functions at that singular point. Thus, to guarantee regularity of the solutions at the origin, i.e. Φ( )| =0 = 0, we need to specify a relation between the integration constants̃ and̃ such that the logarithmic divergences cancel each other. can be further simplified by exploiting the properties of hypergeometric functions with unity argument [43] denotes the Euler-Gamma function. The coefficients (S.19) obey = + for = 1, 2 and we can thus simplify (S.we wish to impose Dirichlet boundary conditions at a finite radial cutoff = = 2 − , such that the field value equals unity, i.e. Φ( ) = 1. This fixes the remaining integration constant to bẽ regularized hypergeometric function 2̃ 1 [ , , ; ] = 2 1 [ , , ; ] Γ( ) . The solutions are thus fully determined as functions of the mass 2 and the radial cutoff = 2 − .
) 2Δ ( Φ ) 2 + Δ 2Φ2 . (S.32)
Figure 5 .
5Comparison of solutions to the Klein-Gordon equation for values of the mass-squared above and below the BF bound 2 2 = −1∕4. The solutions have been rescaled for clarity. The solutions above the BF bound for 2 2 = −1∕10 and 2 2 = −1∕5 do not exhibit any zeroes, while the solutions below the BF bound 2 < −1∕4 do. For positive values of the mass-squared, the solutions contain non-normalizable modes that diverge at the boundary → ∕2 (blue curve).
Figure 6 .
6Normalized total energy tot of the circuit as defined in (S.34), as a function of the driving frequency, measured after the shut down of the drive.
Figure 7 .
7Driving voltage (blue) and oscillation of a node in the 4th layer (orange) over time (measured in units of Runge Kutta iteration steps). The driving frequency is set precisely at the fundamental resonance of the system in the left panel and away from any resonance in the right panel. The lower parts of both panels show the corresponding total energy.
Figure 8 .
8Examples of frequency response spectra of a driven oscillation with radial cutoff ≈ 1.5644098 for three different driving frequencies. In the upper panel = ,0 ≈ 0.653589, in the lower panel = ,1 ≈ 1.002416 and in the middle panel = 0.8.
Table I. Weights and edge lengths for the tilings used in this paper.)
{3, 7}
4.213387
0.5382155
{4, 5}
0.928414
0.9045569
{4, 6}
0.616576
1.0471976
{5, 4}
0.541067
1.0147126
{6, 4}
0.240449
1.2309594
{7, 3}
0.503616
0.9224420
{8, 3}
0.233732
1.1437177
. * Erdmenger@physik, uni-wuerzburg.de* [email protected]
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| [] |
[
"The Geometry of Deep Networks: Power Diagram Subdivision",
"The Geometry of Deep Networks: Power Diagram Subdivision"
] | [
"Randall Balestriero \nECE Department Rice University\nECE Department Rice University\nECE Department Rice University\nECE Department Rice University\n\n",
"Romain Cosentino \nECE Department Rice University\nECE Department Rice University\nECE Department Rice University\nECE Department Rice University\n\n",
"Behnaam Aazhang \nECE Department Rice University\nECE Department Rice University\nECE Department Rice University\nECE Department Rice University\n\n",
"Richard Baraniuk \nECE Department Rice University\nECE Department Rice University\nECE Department Rice University\nECE Department Rice University\n\n"
] | [
"ECE Department Rice University\nECE Department Rice University\nECE Department Rice University\nECE Department Rice University\n",
"ECE Department Rice University\nECE Department Rice University\nECE Department Rice University\nECE Department Rice University\n",
"ECE Department Rice University\nECE Department Rice University\nECE Department Rice University\nECE Department Rice University\n",
"ECE Department Rice University\nECE Department Rice University\nECE Department Rice University\nECE Department Rice University\n"
] | [] | We study the geometry of deep (neural) networks (DNs) with piecewise affine and convex nonlinearities. The layers of such DNs have been shown to be max-affine spline operators (MASOs) that partition their input space and apply a regiondependent affine mapping to their input to produce their output. We demonstrate that each MASO layer's input space partitioning corresponds to a power diagram (an extension of the classical Voronoi tiling) with a number of regions that grows exponentially with respect to the number of units (neurons). We further show that a composition of MASO layers (e.g., the entire DN) produces a progressively subdivided power diagram and provide its analytical form. The subdivision process constrains the affine maps on the (exponentially many) power diagram regions to greatly reduce their complexity. For classification problems, we obtain a formula for a MASO DN's decision boundary in the input space plus a measure of its curvature that depends on the DN's nonlinearities, weights, and architecture. Numerous numerical experiments support and extend our theoretical results.Preprint. Under review. | null | [
"https://arxiv.org/pdf/1905.08443v1.pdf"
] | 160,010,022 | 1905.08443 | a49e1678fff09a85456dbaacad5e2d6d17a286cd |
The Geometry of Deep Networks: Power Diagram Subdivision
Randall Balestriero
ECE Department Rice University
ECE Department Rice University
ECE Department Rice University
ECE Department Rice University
Romain Cosentino
ECE Department Rice University
ECE Department Rice University
ECE Department Rice University
ECE Department Rice University
Behnaam Aazhang
ECE Department Rice University
ECE Department Rice University
ECE Department Rice University
ECE Department Rice University
Richard Baraniuk
ECE Department Rice University
ECE Department Rice University
ECE Department Rice University
ECE Department Rice University
The Geometry of Deep Networks: Power Diagram Subdivision
We study the geometry of deep (neural) networks (DNs) with piecewise affine and convex nonlinearities. The layers of such DNs have been shown to be max-affine spline operators (MASOs) that partition their input space and apply a regiondependent affine mapping to their input to produce their output. We demonstrate that each MASO layer's input space partitioning corresponds to a power diagram (an extension of the classical Voronoi tiling) with a number of regions that grows exponentially with respect to the number of units (neurons). We further show that a composition of MASO layers (e.g., the entire DN) produces a progressively subdivided power diagram and provide its analytical form. The subdivision process constrains the affine maps on the (exponentially many) power diagram regions to greatly reduce their complexity. For classification problems, we obtain a formula for a MASO DN's decision boundary in the input space plus a measure of its curvature that depends on the DN's nonlinearities, weights, and architecture. Numerous numerical experiments support and extend our theoretical results.Preprint. Under review.
Introduction
Deep learning has significantly advanced our ability to address a wide range of difficult machine learning and signal processing problems. Today's machine learning landscape is dominated by deep (neural) networks (DNs), which are compositions of a large number of simple parameterized linear and nonlinear transformations. Deep networks perform surprisingly well in a host of applications; however, surprisingly little is known about why or how they work so well.
Recently, Balestriero and Baraniuk [2018a,b] connected a large class of DNs to a special kind of spline, which enables one to view and analyze the inner workings of a DN using tools from approximation theory and functional analysis. In particular, when the DN is constructed using convex and piecewise affine nonlinearities (such as ReLU, leaky-ReLU, max-pooling, etc.), then its layers can be written as Max-Affine Spline Operators (MASOs). An important consequence for DNs is that each layer partitions its input space into a set of regions and then processes inputs via a simple affine transformation that changes from region to region. Understanding the geometry of the layer partition regions -and how the layer partition regions combine into a global input partition for the entire DN -is thus key to understanding the operation of DNs.
There has only been limited work in the geometry of deep networks. The originating MASO work of Balestriero and Baraniuk [2018a,b] focused on the analytical form of the region-dependent affine maps and empirical statistics on the partition. The work of Wang et al. [2019] empirically studied this partitioning highlighting the fact that knowledge of the DN partitioning alone is sufficient to reach high performance. Other works have focused on the properties of the partitioning, such as upper bounding the number of regions Montufar et al. [2014], Raghu et al. [2017], Hanin and Rolnick [2019]. An explicit characterization of the input space partitioning of one hidden layer DNs with ReLU activation has been proposed in Zhang et al. [2016] by means of tropical geometry.
In this paper, we adopt a computational and combinatorial geometry Pach and Agarwal [2011], Preparata and Shamos [2012] perspective of MASO-based DNs to derive the analytical form of the input-space partition of a DN unit, a DN layer, and an entire end-to-end DN. We demonstrate that each MASO DN layer partitions its input feature map space partitioning according to a power diagram (PD) (also known as a Laguerre-Voronoi diagram) Aurenhammer and Imai [1988] with an exponentially large number of regions. Furthermore, the composition of the several MASO layers comprising a DN effects a subdivision process that creates the overall DN input-space partition.
Our complete, analytical characterization of the input-space and feature map partition of MASO DNs opens up new avenues to study the geometrical mechanisms behind their operation.
We summarize our contributions, which apply to any DN employing piecewise affine and convex nonlinearities such as fully connected, convolutional, with residual connections:
1. We demonstrate that a DN partitions its input feature map space according to a PD subdivision (Sections 4.2, 5). We derive the analytical formula for a DN's PDs and point out their most interesting geometrical properties.
2. We study the computational and combinatorial geometric properties of the layer and DN partitioning (Section 5.2). In particular, a DN can infers the PD region to which any input belongs with a computational complexity that is asymptotically logarithmic in the number of regions.
3. We demonstrate how the centroids of the layer PDs can be efficiently computed via backpropagation (Section 5.3), which permits ready visualization of a PD.
4. In the classification setting, we derive an analytical formula for the DN's decision boundary in term of the DN input space partitioning (Section 6). The analytical formula enables us to characterize certain geometrical properties of the boundary.
Additional background information plus proofs of the main results are provided in several appendices.
Background on Deep Networks and Max-Affine Spline Operators
A deep network (DN) is an operator f Θ with parameters Θ that maps an input signal x ∈ R D to the output prediction y ∈ R C . Current DNs can be written as a composition of L intermediate layer mappings f ( ) : X ( −1) → X ( ) ( = 1, . . . , L) with X ( ) ⊂ R D( ) that transform an input feature map z ( −1) into the output feature map z ( ) with the initializations z (0) (x) := x and D(0) = D.
The feature maps z ( ) can be viewed equivalently as signals, flattened vectors, or tensors.
DN layers can be constructed from a range of different linear and nonlinear operators. One important linear operator is the fully connected operator that performs an arbitrary affine transformation by multiplying its input by the dense matrix W ( ) ∈ R D( )×D( −1) and adding the arbitrary bias vector b ( )
W ∈ R D( ) as in f ( ) W z ( −1) (x) := W ( ) z ( −1) (x) + b ( ) W .
Further examples are provided in Goodfellow et al. [2016]. Given the collection of linear and nonlinear operators making up a DN, the following definition yields a single, unique layer decomposition.
Definition 1.
A DN layer f ( ) comprises a single nonlinear DN operator composed with any (if any) preceding linear operators that lie between it and the preceding nonlinear operator.
Work from Balestriero and Baraniuk [2018a,b] connects DN layers with max-affine spline operators (MASOs) . A MASO is an operator S[A, B] : R D → R K that concatenates K independent maxaffine splines Magnani andBoyd [2009], Hannah andDunson [2013], with each spline formed from R affine mappings. The MASO parameters consist of the "slopes" A ∈ R K×R×D and the "offsets/biases" B ∈ R K×R . 1 Given the input x, a MASO produces the output z via
[z] k = [S[A, B](x)] k = max r ( [A] k,r,· , x +[B] k,r ) ,(1)
where [z] k denotes the k th dimension of z. The key background result for this paper is that a very large class of DNs are constructed from MASOs layers. Theorem 1. Any DN layer f ( ) constructed from operators that are piecewise-affine and convex can be written as a MASO with parameters A ( ) , B ( ) and output dimension K = D( ). Hence, a DN is a composition of L MASOs Balestriero and Baraniuk [2018a,b].
For example, a layer made of a fully connected operator followed by a leaky-ReLU with leakiness η has parameters
[A ( ) ] k,1,· = [W ( ) ] k,· , [A ( ) ] k,2,· = η[W ( ) ] k,· for the slope parameter and [B ( ) ] k,1,· = [b ( ) ] k , [B ( ) ] k,2 = η[b ( ) ]
k for the bias. A DN comprising L MASO layers is a continuous affine spline operator with an input space partition and a partition-region-dependent affine mapping. However, little is known analytically about the input-space partition.
This paper characterizes the geometry of the MASO partitions of the input space and the feature map spaces X ( ) . We proceed by first studying the geometry of a single layer (Section 4.2) and then the composition of L layers that forms a complete DN (Section 5). Voronoi diagrams and their generalization, Power diagrams, play a key rôle in our analysis, and we turn to these next.
Background on Voronoi and Power Diagrams
A power diagram (PD), also known as a Laguerre-Voronoi diagram Aurenhammer and Imai [1988], is a generalization of the classical Voronoi diagram.
Definition 2. A PD partitions a space X into R disjoint regions Ω = {ω 1 , . . . , ω R } such that ∪ R r=1 ω r = X, where each cell is obtained via ω r = {x ∈ X : r(x) = r}, r = 1, . . . , R, with r(x) = arg min
k=1,...,R x − [µ] k,· 2 − [rad] k ,(2)
The parameter [µ] k,· is called the centroid, while [rad] k is called the radius. The PD is a generalization of the Voronoi diagram (VD) by introduction of the external radius term to the 2 distance, leading to the Laguerre distance Imai et al. [1985]. See Figure 1 for two equivalent geometric interpretations of a PD.
In general, a PD is defined with nonnegative radii to provide additional geometric interpretations (see Appendix A). However, the PD is the same under global shifting as arg min k
x−[µ] k,· 2 −[rad] k = arg min k x − [µ] k,· 2 − ([rad] k + Q
Input Space Power Diagram of a MASO Layer
Like any spline, it is the interplay between the (affine) spline mappings and the input space partition that work the magic in a MASO DN. Indeed, the partition opens up new geometric avenues to study how a MASO-based DN clusters and organizes signals in a hierarchical fashion. However, little is known analytically about the input-space partition other than in the simplest case of a single unit with a constrained bias value Balestriero and Baraniuk [2018a,b].
We now embark on a programme to fully characterize the geometry of the input space partition of a MASO-based DN. We will proceed in three steps by studying the partition induced by i) one unit of a single DN layer (Section 4.1), ii) the combination of all units in a single layer (Section 4.2), iii) the composition of L layers that forms the complete DN (Section 5).
MAS Unit Power Diagram
A MASO layer combines K max affine spline (MAS) units z k to produce the layer output z(x) = (z 1 (x), . . . , z K (x)) given an input x ∈ X. Denote each MAS computation from (1) as
z k (x) = max r=1,...,R [A] k,r,· , x + [B] k,r = max r=1,...,R E k,r (x),(3)
where E k,r (x) is the projection of x onto the hyperplane parameterized by the slope [A] k,r,· and offset [B] k,r . By defining the following half-space consisting of the set of points above the hyperplane
E + k,r = {(x, y) ∈ X × R : y ≥ E k,r (x)},(4)
we obtain the following geometric interpretation of the unit output. Proposition 1. The k th MAS unit maps its input space onto the boundary of the convex polytope
P k = ∩ R r=1 E + k,r , leading to X × z k (X) = ∂P k (5) where we remind that z k (X) = Im(z k ) = {z k (x), x ∈ X} is the image of z k .
To provide further intuition, we highlight the role of P k in term of input space partitioning. Lemma 1. The vertical projection on the input space X of the faces of the polytope P k from (5) define the cells of a PD.
Since the k th MAS unit projects an input x onto the polytope face given by r k :
X → {1, . . . , R} (recall(2)) corresponding to r k (x) = arg max r=1,...,R E k,r (x),(6)
the collection of inputs having the same face allocation, defined as ∀r ∈ {1, . . . , R} , ω r = {x ∈ X : r k (x) = r}, constitutes the r th partition cell of the unit k PD. Theorem 2. The k th MAS unit partitions its input space according to a PD with R centroids given by [µ] k,r = [A] k,r,· , and [rad] k,r = 2[B] k,r + [A] k,r,· 2 , ∀r ∈ {1, . . . , R} (recall (2)). Corollary 1. The input space partitioning of a DN unit is composed of convex polytopes.
MASO Layer Power Diagram
We study the layer case by studying the joint behavior of all the layer units. A MASO layer is a continuous, piecewise affine operator made by the concatenation of K MAS units (recall (1)
); we extend (3) to z(x) = max r=1,...,R E 1,r (x), . . . , max r=1,...,R E K,r (x) , ∀x ∈ X(7)
and the per unit face index function r k (6) into the operator r : X → {1, . . . , R} K defined as
r(x) = (r 1 (x), . . . , r K (x)).(8)
Following the geometric interpretation of the unit output from Proposition 1, we extend (4) to
E + r = {(x, y) ∈ X × R K : [y] 1 ≥ E 1,[r]1 (x), . . . , [y] K ≥ E K,[r] K (x)},(9)
where [r] k is the k th component of the vector r(x). Proposition 2. The layer operator z maps its input space into the boundary of the dim(X)
+ K dimensional convex polytope P = ∩ r∈{1,...,R} K E + r as X × z(X) = X × z 1 (X) × · · · × z K (X) = ∂P.(10)
Similarly to Proposition 1, the polytope P is bound to the layer input space partitioning. Lemma 2. The vertical projection on the input space X of the faces of the polytope P from Proposition 2 define cells of a PD.
Rank 1
Rank 2 Orthogonal Rank 2 Figure 2: Depiction of different types of weight constraints and their impact in the layer input space partitioning. On the left is depicted the case of a low rank matrix leading to colinear cuts, only the bias is responsible for shifting the cuts. In the middle orthogonal weights are used leading to orthogonal cuts. This, while not being degenerate still limits the input space partitioning. On the right, an arbitrary weight matrix is used leading the partitioning unconstrained.
The MASO layer projects an input x onto the polytope face indexed by r(x) corresponding to
r(x) = (arg max r=1,...,R E 1,r (x), . . . , arg max r=1,...,R E K,r (x)).(11)
The collection of inputs having the same face allocation jointly across the K units constitutes the r th partition cell of the layer PD. Theorem 3. DN layer partitions its input space according to a PD with {1, . . . , R} K cells, centroids (2)). Corollary 2. The input space partitioning of a DN layer is composed of convex polytopes.
µ r = K k=1 [A] k,[r] k ,· and radii rad r = 2 1, B r + µ r 2 (recall
Weight Constraints and Cell Shapes
We highlight the close relationship between the layer weights A, B from (1), the layer polytope P from Proposition 2, and the boundaries of the layer PD from Theorem 3. In particular, how one can alter or constraint the shape of the cells by constraining the weights of the layer.
Example 1: Constraining the layer weights to be such that
[A] k,r,d = 1 {d=[i] k,r } [cst] k,r for some integer [i] k,r ∈ {1, . . . , D}, D = dim(X)
, and arbitrary constant [cst] k,r leads to an input power diagram with cell boundaries parallel to the input space basis vectors see Fig. 2. For instance if the input space X is the Euclidean space R D equipped with the canonical basis, the previous Proposition translates into having PD boundaries parallel to the axes.
Example 2: Constraining the layer weights to be such that [A] k,r,d = ±[cst] k,r for some arbitrary constant [cst] k,r leads to a layer-input power diagram with diagonal cell boundaries. 2 Lemma 3. Changing the radius of a given cell shrinks or expands w.r.t. the other Aurenhammer [1987]. Theorem 4. Updating a single unit parameters (slope or offset of the affine transform and/or the nonlinearity behavior) affects multiple regions' centroids and radius.
The above result recovers weight sharing concepts and implicit bias/regularization. In fact, most regions are tied together in term of learnable parameter. Trying to modify a single region while leaving everything else the same is not possible in general.
Input Space Power Diagram of a MASO Deep Network
We consider the composition of multiple layers, as such, the input space of layer is denoted as X ( ) , with X (0) the DN input space.
Power Diagram Subdivision
We provide in this section the formula for deriving the input space partitioning of an L-layer DN by means of a recursive scheme. Recall that each layer defines its own polytope P ( ) according to Proposition 2, each with domain X ( −1) . The DN partitioning corresponds to a recursive subdivision where each per layer polytope subdivides the previously obtained partitioning, involving the representation of the considered layer polytope in the input space X (0) . This subdivision can be analytically Input space partitioning Partition polynomial Layer 1: mapping
X (0) ⊂ R 2 to X (1) ⊂ R 6
Layer 2: mapping
X (1) ⊂ R 6 to X (2) ⊂ R 6
Layer 3 (classifier): mapping
X (2) ⊂ R 6 to X (3) ⊂ R 1 Figure 3: Top:
The partition polynomial as defined in (23), whose roots define the partition boundaries in the input space and determined by each layer parameters and nonlinearities. Bottom: Evolution of the input space partitioning layer after layer (left to right: (15)) with the newly introduced boundaries in dark and previously built partitioning (being refined) in grey. Below each partitioning, one of the newly introduced cut denoted as edge X (0) (k, ) from (22) derived from the following recursion scheme. Initialization: The initialization consists of defining the part of the input space to consider X (0) ⊂ R D .
Ω (1) , Ω (1,2) , Ω (1,2,3) from
Recursion step ( = 1): The first layer subdivides X (0) into a PD from Theorem 3 with parameters A (1) , B (1) to obtain the layer 1 partitioning Ω (1) . Recursion step ( = 2): The second layer subdivides each cell of Ω (1) . Let's consider a specific cell ω
(1) r (1) ; all inputs in this cell are projected to X (1) by the first layer via A
(1)
r (1) x + B(1)
r (1) . 3 The convex cell ω (1) r (1) thus remains a convex cell in X (1) defined as the following affine transform of the cell
aff r (1) = {A (1) r (1) x + B (1) r (1) , x ∈ ω (1) r (1) } ⊂ X (1) .(12)
Since on the cell the first layer is linear; the slice of the polytope P (2) ⊂ X (1) × R D(2) (recall (10)) having for domain aff r (1) formally defined as
P (2) r (1) = P (2) ∩ (aff r (1) × R D(2) ),(13)
can thus be expressed w.r.t. X (0) . Lemma 4. The domain restricted polytope (13) can be expressed in the input space ω
(1)
r (1) ⊂ X (0) as P (1←2) r (1) = ∩ r (2) {(x, y) ∈ ω (1) r (1) × R D(1) : [y] 1 ≥ E (1←2) 1,[r (2) ]1 (x), . . . , [y] D(1) ≥ E (1←2) D(1),[r (2) ] D(1) (x)} (14) with E (1←2) k,[r (1) ] k the hyperplane with slope A (1) T r (1) A(2)
r (2) and bias [A
r (2) ] k,r,. , B
r (1) + B(1)
r (2) ,k ∈ {1, . . . , D(1)}.
3 Recall from (1) that A (1) r (1) , B(1)
r (1) are the affine parameters associated to cell r (1) From Lemma 2, P (1←2) r (1) induces an underlying PD on its domain ω (1) r (1) that subdivides the cell into a PD denoted as PD
(1←2) r (1) leading to the centroids µ (1←2) r (1) ,r (2) = A (1) r (1) µ (1←2) r (2) , and radii rad (1←2) r (1) ,r (2) = µ (1←2) r (1) ,r (2) 2 + 2 µ (2) r (2) , B
(1) r (1) + 2 1, B
(2) r (2) , ∀r (2) ∈ {1, . . . , R} D(2) . The PD parameters thus combine the affine parameters A (1) r (1) , B
(1) r (1) of the considered cell with the second layer parameters A (2) , B (2) . Repeating this subdivision process for all cells r (1) from Ω (1) form the input space partitioning Ω (1,2) = ∪ r (1) PD (1←2) r (1) . Recursion step: Consider the situation at layer knowing Ω (1,..., −1) from the previous subdivision steps. following the intuition from the = 2, layer subdivides each cell in Ω (1,..., −1) to produce Ω (1,..., ) leading to the -up to layer -layer DN partitioning defined as
Ω (1,..., ) = ∪ r (1) ,...,r ( −1) PD (1← ) r (1) ,...,r ( −1) .(15)
Theorem 5. Each cell ω
(1,..., −1)
r (1) ,...,r ( −1) ∈ Ω (1,..., −1) is subdivided into PD (1← ) r (1) ,...,r ( −1) , a PD with domain ω (1..... −1) r (1) ,...,r ( −1) and parameters µ (1← ) r (1) ,...,r ( ) =(A (1← −1) r (1) ,...,r ( −1) ) µ ( ) r ( ) (centroids) (16) rad (1← ) r (1) ,...,r ( ) = − µ (1← ) r (1) ,...,r ( ) 2 − 2 µ ( ) r ( ) , B (1→ −1) r (1) ,...,r ( −1) − 2 1, B ( ) r ( ) (radii),(17)∀r (i) ∈ {1, . . . , R} D (i) with B (1→ −1) = −1 =1 i= −1 A (i) r (i) B ( ) r ( ) forming Ω (1,.
.., ) . The described recursion construction also provides a direct result on the shape of the entire DN input space partitioning cells. Corollary 3. The cells of the DN input space partitioning are convex polygons.
Combinatorial Geometry Properties
We highlight a key computational property of DNs contributing to their success. While the actual number of cells from a layer PD varies greatly depending on the parameters, the cell inference task always search over the maximum R
z ( −1) (x) − µ ( ) r + rad ( ) r .(18)
The computational and memory complexity of this task is O(R The above practical result is crucial to the ability of DN layers to perform extremely fine grained input space partitioning without sacrificing computation time especially at test time where one needs only feed forward computations of an input to obtain a prediction.
Centroid and Radius Computation
In practice, the number of centroids and radius for each of the partitioning Ω (1,..., ) contains too many cells to compute all the centroids and radius. However, given a cell (resp. a point x) and an up for each successively refined PD subdivision of each layer Ω (1,..., ) , and in each case, the region has an associated centroid depicted here and radius. As depth increase, as the radii overtakes the centroids pushing µ (1,..., ) x away from the region.
to layer code r (1) , . . . , r ( ) (resp. r (1) (x), . . . , r ( ) (x)), computing the centroid and radius can be done as follows:
A (1→ −1) x =(J x f (1→ −1) ) = ∇ x f (1→ −1) 1 , . . . , ∇ x f (1→ −1) D( ) (19) µ (1← ) x =A (1→ −1) x D( ) k=1 [A ( ) x ] k,. = D( ) k=1 ∇ x f (1→ ) k ,(20)rad (1← ) x = − µ (1← ) x 2 − 2 1, B ( ) x − 2 f (1→ ) (x) − A (1→ −1) x x, D( ) k=1 [A ( ) x ] k,.(21)
where we remind that µ ( )
x = D( ) k=1 [A ( ) x ] k,. and B (1→ −1) x = f (1→ ) (x) − A (1→ −1) x
x from Theorem 5. Notice how centroids and biases of the current layer are mapped back to the input space X (0) via projection onto the tangent hyperplane with basis given by A
(1→ −1) x .
Proposition 3. The centroids correspond to the backward pass of DNs and thus can be computed efficiently by backpropagations.
Note how the form in (20) correspond to saliency maps. In particular, at a given layer, the centroid of the region in which x belongs is obtained by summing all the per unit saliency maps synonym of adding all the unit contributions in the input space. We provide in Fig. 4 computed centroids for a trained Resnet on CIFAR10, for each PD subdivision, see appendix ?? for details on the model, performance and additional figures. The ability to retrieve saliency maps and the form of the centroid opens the door to further use in many settings of the centroids. For example. semi supervised learning successfully leveraged the last layer centroid in Balestriero et al. [2018] by providing a loss upon them.
Empirical Region Characterization
We provide in Fig. 5 the distribution of distances from the dataset points to the nearest region boundaries of the input space partitioning for each layer (at a current subdivision step) and at different stages of the training procedure. Clearly, training slightly impacts those distances and slight increase the number of inputs with small distance to then nearest boundary. Yet, the main impact of learning resides in shaping the regions via learning of the weights. We also train a DN on various dataset and study how many inputs share the same input space partitioning region. We observed that from initialization to the end of the learning, there are never more than one image in a given region and this with standard DNs providing near or state of the art performances. Yet, drastically reducing the size of the DN allows to have more than one image per regions when considering the first steps of the subdivision.
Geometry of a Deep Network Decision Boundary
The analysis of the input space partitioning was achieved by successively expressing the layer polytope in the input space. While Sec. 5 focused on the faces of the polytope which define the cells in the input space, we now turn to the edges of the polytope which define the cells' boundaries. In particular, we demonstrate how a single unit at layer defines multiple cell boundaries in the input space, and use this finding to finally derive the analytical formula of the DN decision boundary in classification tasks. In this section we focus on DN nonlinearities using R = 2 nonlinearities such as ReLU, leaky-ReLU, and absolute value.
Partitioning Boundaries and Edges
In the case of R = 2 nonlinearities, the polytope P
edge X ( ) (k, ) = {x ∈ X ( ) : E ( −1) k,2 (z ( → ) (x)) = 0},(22)
with z ( → −1) = z ( −1) • · · · • z ( ) .
Thus the edges correspond to the level curve of the unit in X ( ) . Defining the polynomial
Pol ( ) (x) = D( ) k=1 (z ( ) k • z ( −1) • · · · • z (1) )(x),(23)
we obtain the following result where the boundaries of Ω (1,..., ) from Theorem 5 can be expressed in term of the polytope edges and roots of the polynomial.
Theorem 7. The polynomial (23) is of order L =1 D( ), its roots correspond to the partitioning boundaries:
∂Ω (1,..., ) = ∪ =1 ∪ D( ) k=1 edge X (0) (k, ) = {x ∈ X (0) : L =1 Pol ( ) (x) = 0}
(24) and the root order defines the dimension of the root (boundary, corner, ...).
Decision Boundary and Curvature
In the case of classification, the last layer typically includes a softmax nonlinearity and is thus not a MASO layer. However, demonstrated that it can be expressed as a MASO layer without any change in the model and output. As a result, this layer introduces a last subdivision of the DN partitioning. We focus on binary classification for simplicity of notations, in this case, D(L) = 1 and a single last subdivision occurs. In particular, using the previous result we obtain the following. Proposition 4. The decision boundary of a DN with L layers is the edge of the last layer polytope P (L) expressed in the input space X (0) from Def. 3 as
DecisionBoundary = {x ∈ X (0) : f (x) = 0} = edge X (0) (1, L) ⊂ ∂Ω (1,...,L) .(25)
To provide insights let consider a 3 layer DN denoted as f and the binary classification task; we have as the DN induced decision boundary the following
DecisionBoundary = ∪ r (2) ∪ r (1) {x ∈ X (0) : α r (2) ,r (1) , x + β r (2) ,r (1) = 0} ∩ ω (1,2) r (1) ,r (2) , (26) with α r (1) ,r (2) = (A (2) r (2) A (1) r (1) ) T [A (3) ] 1,1,. and β r (1) ,r (2) = [A (3) ] T 1,1,. A (2) r (2) B
(1)
r (1) + [B (3) ] 1,1 .
Studying the distribution of α r (1) ,r (2) characterizes the structure of the decision boundary and thus open the highlight the interplay between layer parameters, layer topology, and the decision boundary. For example, looking at Figure 3 and the red line demonstrates how the weight characterize the curvature and cuts position of the decision boundary.
We provide a direct application of the above finding by providing a curvature characterization of the decision boundary. First, we propose the following result stating that the form of α and β from (26) from one region to an neighbouring one only alters a single unit code at a given layer. Lemma 6. Any edge as defined in Def. 3 reaching a region boundary, must continue in this neighbouring region.
This comes directly from continuity of the involved operator. This demonstrates that the decision boundary as defined in (26) can have its curvature defined by comparing the form of the edges of adjacent regions. Theorem 8. The decision boundary curvature/angle between two adjacent regions r and r 4 is given by the following dihedral angle Kern and Bland [1938] between the adjacent hyperplanes as cos(θ(r, r )) = | α r , α r | α r α r .
The above is illustrated in the adjacent figure with one of the angle highlighted with an arrow. The hyperplane offsets β r , β r are irrelevant to the boundary curvature. Following this, the DN bias units are also irrelevant to the boundary curvature. Also, the norm of the gradient of the DN and thus the Lipschitz constant alone does not characterize the regularity of the decision boundary. In fact, the angle is invariant under scaling of the parameters. This indicates how measures based on the input-output sensitivity do not characterize alone the curvature of the decision boundary.
Finally, we highlight an intuitive result that can be derived from the above. We remind that a neighbouring regions implies a change of a single code unit. Let denote without loss of generality the changed code index by d ∈ {1, . . . , D (1) . The other D (1) − 1 codes remain the same. When dealing with R = 2 nonlinearities, this implies that [r] d changes from a 1 to a 2 for those two neighbouring regions. Let denote by r the case with a 1 and by r the case with a 2. With those notations, we can derive some special cases of the distance formula (27) for some DN topologies.
Proposition 5. In a 2-layer DN with ReLU and orthogonal first layer weights, we have
cos(θ(r, r )) = |[W (2) ] 1,d | [W 1) ] d ,. 2 D (1) d =d 1 {[r ] d =2} |[W (2) ] 1,d | [W (1) ] d,. + 1 −1 ∈ (0, 1)(28)
From the above formula it is clear that reducing the norm of the weights alone does not impact the angles. However, we have the following result. Proposition 6. Regions of the input space in which the amount of firing ReLU is small will have greater decision boundary curvature than regions with most ReLU firing simultaneously.
As a result, a ReLU based DN will have different behavior at different regions of the space. And the angle is always positive. Interestingly, the use of absolute value on the other hand leads to the following. Proposition 7. In a 2-layer DN with absolute value and orthogonal first layer weights, we have
cos(θ(r, r )) = 1 − 2 1 + D (1) d =d |[W (2) ] 1,d | 2 [W (1) ] d,. 2 −1 ∈ (0, 1).(29)
Now, not only are the angles between 90 degrees and 270 (as opposed to ReLU between 90 and 180), but the angles also do not depend on the state of the other absolute values but just on the norm of the weights of both layers.
Conclusions
We have extended the understanding of DNs by leveraging computational geometry to characterize how a DN partitions its input space via a multiscale Power Diagram subdivision. Our analytical formulation for the partitions induced by not only the entire DN but also each of its units and layers will open new avenues to study how to optimize DNs for certain tasks and classes of signals.
B Additional Figures
Layer 3 8 units at the second layer with random weights and biases. The colors are the DN input space partitioning w.r.t. the first layer. Then for each color (or region) the layer1-layer2 defines a specific Power Diagram that will sub-divide this aforementioned region (this is the first row) where the region is colored and the Power Diagram is depicted for the whole input space. Then this sub-division is applied onto the first layer region only as it only sub-divides its region (this is the second row on the right). And finally grouping together this process for each of the 4 region, we obtain the layer-layer 2 space partitioning (second row on the left).
C Proofs
C.1 Proof of Lemma 1: Single Unit Projection
Follows from Johnson [1960] that demonstrates that the boundaries in the input space X defining the regions of the unit PD are the vertical projections of the polytope (P k ) face intersections defined as E k,r (X) ∩ E k,r * (X) for neighbouring faces r and r * .
C.2 Proof of Lemma 2: Single Layer Projection
Follows from the previous section in which it is demonstrated that the boundaries of a single unit PD is obtained by vertical projection of the polytope edges. In the layer case, the edges of P correspond to all the points in the input space X s.t. z belongs to an edge of at least one of the polytopes P k , ∀k making up P. The layer PD having for boundaries the union of all the per unit PD boundaries, it follows directly that the vertical projection of the edges of P form the layer PD boundaries.
C.3 Proof Complexity
Recalling Section 2, a layer MASO produces its output by: first inferring in which cell r(z ( −1) (x)) lies in the layer PD partitioning from Theorem ??; and then affinely transforming the input via A ( ) The D units case: apply recursively Lemma 7.
r(z ( −1) (x)) z ( −1) + B ( ) r(z ( −1) (x)) .]2} = [b] 1,[t] 1 + [b] 2,[t] 2 − 2 [A] 1,[t] 1 ,· , [A] 2,[t] 2 ,· . Proof. V([t]1, [t]2) = V([t]1) ∩ V([t]2) = {x ∈ R D | arg max i x, [A]1,i,· + [B]1,i = [t]1} ∩ {x ∈ R D | arg max j x, [A]2,j,· + [B]2,j = [t]2} = {x ∈ R D | arg min i x − [A]1,i,· 2 + [b]1,i = [t]1} ∩ {x ∈ R D |
C.5 Proof of Theorem 3: Single Layer Power Diagram
We first derive a preliminary result in which a layer follows an affine transformation to then generalize by considering how each region of the previously built partitioning transforms the inputs lying in it linearly.
Input Space Partitioning of a Single Layer Following an Affine Transform Consider a layer with input an affine transformation of x ∈ X as Gx + h where G is an arbitrary matrix and h an arbitrary vector. We consider this affine transformation as a linear DN layer. We now express the layer PD w.r.t. the input space as Ω (2) (X (0) ). Define the centroids and radius
µ (1←2) r (2) =G D (2) k=1 [A (2) ] k,[r (2) ] k ,· = G µ (2) r (2) (32) rad (1←2) r (2) = − G µ (1←2) r (2) 2 − 2µ (1←2) r ( ) h − 2 D (2) k=1 [B (2) ] k,[r] k(33)
where µ
(2) r (2) is as defined in Theorem 3. Lemma 8. The input space partitioning of a 2-layer DN with first layer linear is given by Ω (1,2) (X (0) ) = PD(X; {(µ r , rad r ), ∀r}).
Figure 1 :
1Two
R D( ) number of cells as r ( ) (x) = arg min r∈{1,...,R} D( )
( − 1)). While approximations existMuja and Lowe [2009],Arya et al. [1998],Georgescu et al. [2003], we demonstrate how a MASO induced PD is constructed in such a way that it is parameter-, memory-, and computation-efficient.Lemma 5. A DN layer solves (18) with computational and memory complexity O(log R ( ) (R ( ) upper )R ( ) D( − 1)) = O(D( )R ( ) D( − 1)) as opposed to O(R ( ) upper D( − 1)) = O((R ( ) ) D( ) R ( ) D( − 1)) for an arbitrary Power Diagram. The entire DN then solves iteratively (18) for each layer. Theorem 6. An entire DN infers an input cell with computational and memory complexity O( L =1 D( )R ( ) D( − 1)) as opposed to O( L =1 (R ( ) ) D( ) R ( ) D( − 1)) for an arbitrary hierarchy of Power Diagrams.
Figure 4 :
4Depiction of the centroids of the PD regions that contain an input x. This input belongs to a specific region ω (1,..., )x
Figure 5 :
5Depiction of the distribution of the log of the distances through the epochs (during leaning) for the left column and through the layer (as the partitioning gets subdivided) on the right column. The statistics are computed on the train set (blue) and test set (red). CLear insight in the role of the deeper layer trying to refine only some regions, likely the ones hard to classify. This is shown by the tails becoming heavier. For additional figures seeFig. 7.
a single edge. This edge can be expressed in some space X ( ) , < , as a collection of continuous piecewise linear paths. Definition 3. The edges of the polytope P ( ) k in some space X ( ) , < are the collection of points defined as
Figure 6 :Figure 7 :Figure 8 :
678Additional depiction of the partitioning and subdivision happening layer after layer. Each unit also introduces a path in the input space which is depicted below the current partitioning with the highlighted path linked via a dotted line. Additional depiction of the distances distribution.Layer 1 →Each Layer 1 region q (1) leads a different PD ↓ Layer 1 and 2 ← Sub-division of each region with respective PD Visual depiction of Cor. ?? for a 2-layer DN with 3 units at the first layer (leading to 4 regions) and
A{i, j} 2 + b{i, j} = ([t]1, [t]2)}, where, b{i, j} = [A]1,i,· 2 + 2[B]1,i + [A]2,j,· 2 + 2[B]2,j + 2 [A]1,i,·, [A]2,j,· and, A{i, j} = [A]1,i,· + [A]2,j,·.
2 − 2 x, G T [A]1,i,· + G T [A]2,j,· + G T [Ab{i, j} = [b]1,i + [b]2,j − 2 G T [A]1,i,·, G T [A]2,j,· and, A{i, j} = [A]1,i,· + [A]2,j,· and [b]1,i = − G T [A]1,i,· 2 − 2 [A]1,i,·, h − 2[B]1,i. We thus have b{i, j} = −2B{i, j} − 2A{i, j} T h − G T A{i, j} 2 .
). Left: The grey circles have center [µ] k,· and radii [rad] k ; each point x is assigned to a specific region according to the Laguerre distance defined as the length of the segment tangent to and starting on the circle and reaching x. Right: A PD in R D(here D = 2) is constructed by lifting the cen-
troids [µ] k,· up into an additional dimension in
R D+1 by the distance [rad] k and then finding
the Voronoi diagram (VD) of the augmented cen-
troids ([µ] k,· , [rad] k ) in R D+1 . The intersection
of this higher-dimensional VD with the originat-
ing space R D yields the PD.
). Thus, we allow for arbitrary radius since it can always be shifted back to nonnegative by setting Q = min k [rad] k . For additional geometric insights on VDs and PDs see Preparata and Shamos[2012] and Appendix A.
is highlighted, and which, in the last layer case (right), corresponds to the decision boundary (in red) see Figures 6, ?? in Appendix B for additional examples.
The inference problem of determining in which power diagram cell an input x falls C.4 Proof of Theorem 2: Single Unit Power Diagram Let first consider the case of 2 units. Lemma 7. The layer input space of the [1 th , 2 th ]-MASO units at layer l is a weighted Voronoi Diagram with a maximum of R × R regions, centroids A{[t]1, [t]2} = [A] 1,[t] 1 ,· + [A] 2,[t] 2 ,· , and biases b{[t]1, [t
Lemma 9 .
9The layer input space of the [1 th , 2 th ]-MASO units at layer l is a weighted Voronoi Diagram with a maximum of R × R regions, centroids A{[t] 1 , [t] 2 } = G T [A] 1,[t]1,· + G T [A] 2,[t]2,· , and biases b{[t] 1 , [t] 2 } = [b] 1,[t]1 + [b] 2,[t]2 − 2 G T [A] 1,[t]1,· , G T [A] 2,[t]2,· .
The three subscripts of the slopes tensor [A] k,r,d correspond to output k, partition region r, and input signal index d. The two subscripts of the offsets/biases tensor [B] k,r correspond to output k and partition region r.
Note that while in example 1 each per unit k, per cell r weight was constrained to contain a single nonzero element s.a. (0, 0, c, 0) for D = 4, example 2 makes the weight vector filled with a single constant but varying signs such as (+c, −c, +c, −c).
For clarity, we omit the subscripts.
A Additional Geometric InsightsThe Laguerre distance corresponds to the length of the line segment that starts at x and ends at the tangent to the hypersphere with center [A ( ) ] k,r ,· and radius rad k,r (seeFigure ??).The hyperplanar boundary between two adjacent PD regions can be characterized in terms of the chordale of the corresponding hyperspheres Johnson[1960]. Doing so for all adjacent boundaries fully characterize those region boundaries in simple terms of hyperplane intersections fromAurenhammer [1987].A.1 Paraboloid U InsightsA further characterization of the polytope boundary ∂P k can be made by introducing the paraboloid U defined as U (x) = 1 2 x 2 2 . Notice that the slope of the hyperplane is ∇E k,r = [A] k,r,· and its offset is − 1 2 [A] k,r,· 2 2 . Defining the paraboloid U defined as U (x) = 1 2 x 2 2 , we see how the hyperplane E k,r is the tangent of the paraboloid U at the point [A] k,r,· . We now highlight that the hyperplane and the paraboloid intersect at an unique pointThe faces of P k are the tangent of U at the points given by [A] k,r,· , ∀r leading toConcerning the case of abitrary bias we hve the following insights. We can characterize the hypersphere as being the intersection of the hyperplanes with the paraboloid in the following result fromAurenhammer [1987]. Proposition 8.Aurenhammer [1987]There is a bijective mapping between the hyperpshere in the input domain and the intersection of the hyperplane E in R D+1 with the paraboloid U .In fact, the projection of the intersection between the hyperplane and the paraboloid onto the input space is forming a circle where the radius corresponds to the shift of the hyperplane.
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| [] |
[
"Ground state pairing correlations in the S 4 symmetric microscopic model for iron-based superconductors",
"Ground state pairing correlations in the S 4 symmetric microscopic model for iron-based superconductors"
] | [
"Yang Wu \nDepartment of Physics\nBeijing Normal University\n100875BeijingChina\n",
"Guangkun Liu \nDepartment of Physics\nBeijing Normal University\n100875BeijingChina\n",
"Tianxing Ma \nDepartment of Physics\nBeijing Normal University\n100875BeijingChina\n\nBeijing Computational Science Research Center\n100084BeijingChina\n"
] | [
"Department of Physics\nBeijing Normal University\n100875BeijingChina",
"Department of Physics\nBeijing Normal University\n100875BeijingChina",
"Department of Physics\nBeijing Normal University\n100875BeijingChina",
"Beijing Computational Science Research Center\n100084BeijingChina"
] | [] | We present the ground state pairing correlations in the S4 symmetric microscopic model for iron-based superconductors, computed with the constrained-path Monte Carlo method. For various electron fillings and interaction strengths, we find that the sxy pairing dominate over other pairing correlations and are positive when the pair separation exceeds several lattice constants, whatever for iron pnictides and iron chlcogenides. These ground state properties, especially the long range part pairing correlations re-conform the previous finite temperature results published in Phys. Rev. Lett. 110, 107002(2013). We further our study by including the nearest neighbour interaction V and it is found that the sxy pairing correlation is slightly suppressed by the increasing V . | 10.1209/0295-5075/104/27013 | [
"https://arxiv.org/pdf/1312.4777v2.pdf"
] | 119,239,417 | 1312.4777 | c99038bbb67ed651c163b85b826473c788c7a21f |
Ground state pairing correlations in the S 4 symmetric microscopic model for iron-based superconductors
26 Dec 2013 (Dated: December 30, 2013)
Yang Wu
Department of Physics
Beijing Normal University
100875BeijingChina
Guangkun Liu
Department of Physics
Beijing Normal University
100875BeijingChina
Tianxing Ma
Department of Physics
Beijing Normal University
100875BeijingChina
Beijing Computational Science Research Center
100084BeijingChina
Ground state pairing correlations in the S 4 symmetric microscopic model for iron-based superconductors
26 Dec 2013 (Dated: December 30, 2013)
We present the ground state pairing correlations in the S4 symmetric microscopic model for iron-based superconductors, computed with the constrained-path Monte Carlo method. For various electron fillings and interaction strengths, we find that the sxy pairing dominate over other pairing correlations and are positive when the pair separation exceeds several lattice constants, whatever for iron pnictides and iron chlcogenides. These ground state properties, especially the long range part pairing correlations re-conform the previous finite temperature results published in Phys. Rev. Lett. 110, 107002(2013). We further our study by including the nearest neighbour interaction V and it is found that the sxy pairing correlation is slightly suppressed by the increasing V .
INTRODUCTION
Iron-based superconductors are the major field in superconductivity now [1][2][3][4]. One of today's major challenge in the study of iron-based superconductors is how to obtain an unified microscopic understanding of the different families of these materials, in particular, iron-pnicitides and iron-chalcogenides, which distinguish themselves from each other with distinct Fermi surface topologies [5][6][7]. Theoretical studies based on models with complicated multi-d orbital band structures [8][9][10][11][12][13][14][15][16][17][18], lack of a support from more fundamental microscopic electronic physics [19][20][21][22][23][24][25][26][27]. Recently, it has been shown that the underlining electronic structure in iron-based superconductors, which is responsible for superconductivity at low energy, is essentially governed by a two-orbital model with a S 4 symmetry. The two orbital model includes two nearly degenerated single-orbital parts that can be mapped to each other under the S 4 transformation. This electronic structure stems from the fact that the dynamics of d xz and d yz orbitals are divided into two groups that are separately coupled to the top and bottom As(Se) planes in a single Fe-(As)Se trilayer structure. The two groups can thus be treated as an S 4 isospin. The dressing of other orbitals in the d xz and d yz orbitals can not alter the symmetry characters. Despite the simplicity of this description, the model not only gives very good quantitative agreement with the band structure but also supply a uniform model to mimic different iron-based material classes, especially for the ironpnictides and iron-chalcogenides [28]. Some of us have performed a finite temperature determinant quantum Monte Carlo (DQMC) study of the pairing correlation in the S 4 symmetric microscopic model on lattices mainly with 8 2 sites. It is found that the pairing with an extensive s-wave symmetry robustly dominates over other pairings at low temperature in reasonable parameter region regardless of the change of Fermi surface topologies. The pairing susceptibility, the effective pair-ing interaction and the (π, 0) antiferromagneticcorrelation strongly increase as the on-site Coulomb interaction increases, and these non-biased numerical results provide a possible unified understanding of superconducting mechanism in iron-pnictides and iron-chalcogenides [29].
Numerical approaches like DQMC, however, had its own difficulties, typically being limited to small sizes, high temperature, and experience the infamous fermion sign problem, which cases exponential growth in the variance of the computed results and hence an exponential growth in computer time as the lattice size is increased and the temperature is lowered [30,31]. In general, to determine which pairing symmetry is dominant by numerical calculation for finite size models, we have to look at the long-range part of the pair-correlation function at zero temperature, and it seems to be dangerous to extrapolate the long-range behavior of the pair-correlation function from a lattice with 8 2 sites. In order to shed light on this critical issue, it is important to discuss the results obtained from the constrained path Monte Carlo (CPMC) [32] on larger lattice. In a variety of benchmarking calculations, the CPMC method has yielded very accurate results of the ground state energy and many other ground state observables for large system [32]. In the CPMC method, the ground-state wave function |Ψ 0 > is projected from an initial wave function |Ψ T > by a branching random walk in an over-complete space of constrained Slater determinants |φ >, which have positive overlaps with a known trial wave function. In such a space, we can write |Ψ 0 >= Σφχ(φ)|φ >, where χ(φ) > 0. The random walk produces an ensemble of |φ >, called random walkers, which represent |Ψ 0 > in the sense that their distribution is a Monte Carlo sampling of χ(φ). The resulting method is free of any decay of the signal-to-noise ratio. For more technique details we refer to Refs. [32,33].
In this paper, we report the ground state results in the S 4 symmetric microscopic model for various electron fillings, interaction strength by using CPMC method. The simulations were mainly performed on a 12 2 lattices, and compared to the paring correlation on an 8 2 , a 16 2 and a 20 2 lattices. All the lattices are with periodic boundary conditions.Our unbiased numerical calculation shows that the ground state s xy pairing dominate over other pairing correlations. The s xy pairing correlations is positive when the pair separation exceeds several lattice constants, whatever for iron pnictides and iron chlcogenides. These ground state properties, especially the long range part pairing correlations re-conform our previous finite temperature results with DQMC method [29]. We further our study by including the nearest-neighbor interaction V . It is found that the s xy pairing correlation is slightly suppressed by the increasing V .
MODEL AND RESULTS
The minimum extended Hubbard model for a single S 4 iso-spin component in the iron-square lattice is described by [28,29],
H = t 1 iησ (a † iσ b i+ησ + h.c.) +t 2 [ iσ a † iσ a i±(x+ŷ),σ + iσ b † iσ b i±(x−ŷ)σ ] +t ′ 2 [ iσ a † iσ a i±(x−ŷ)σ + iσ b † iσ b i±(x+ŷ)σ ] +U i (n ai↑ n ai↓ + n bi↑ n bi↓ ) + V iησ (a † iσ b i+ησ + h.c.) +µ iσ (n aiσ + n biσ )(1)
Here, a iσ (a † iσ ) annihilates (creates) electrons at site R i with spin σ (σ=↑, ↓) on sublattice A, b iσ (b † iσ ) annihilates (creates) electrons at the site R i with spin σ (σ=↑, ↓) on sublattice B, n aiσ = a † iσ a iσ , n biσ = b † iσ b iσ , η = (±x, 0) and (0, ±ŷ). U and V denote the on-site Hubbard interaction and NN interaction, respectively. In the above model, for simplicity and clarity, we only keep a minimum set of parameters which include three key shortest hopping parameters that are responsible for the physical picture revealed by the S 4 symmetry [28]. The selection of parameters in following studies does capture the essential physics of typical cases for iron-pnictides [34][35][36][37][38] and iron-chalcogenides [5][6][7], as that shown in Ref. [29].
The pairing correlation function we computed is
C α (r = R i − R j ) = ∆ † α (i)∆ α (j) ,(2)
where α stands for the pairing symmetry. And the corresponding order parameter ∆ † α (i) is defined as with f α (δ l ) being the form factor of pairing function, and the vectors δ l (δ ′ l denote the nearest neighbour intersublattice connections (the next nearest neighbour inner sublattice connections), where l = 1, 2, 3, 4 denoting the four different direction. As that shown in Ref. [29], we focus on four kinds of pairing form, where d x 2 −y 2 -wave : f d x 2 −y 2 (δ l ) = 1 (δ l = (±x, 0)) and : f d x 2 −y 2 (δ l ) = −1 (δ l = (0, ±ŷ)) d xy -wave : f dxy (δ ′ l ) = 1 (δ ′ l = ±(x,ŷ)) and : f dxy (δ ′ l ) = −1 (δ ′ l = ±(x,−y)) s x 2 +y 2 -wave : f sxy (δ l ) = 1, l = 1, 2, 3, 4 s xy -wave : f sxy (δ ′ l ) = 1, l = 1, 2, 3, 4
∆ † α (i) = l f † α (δ l )(a i↑ b i+δ l ↓ − a i↓ b i+δ l ↑ ) † ,(3)
To facilitate contact with prior simulations, we also examined the vertex contributions to the correlations defined by
V α (r) = C α (r) − C α (r)(5)
where C α (r) is shorthand notation for the uncorrelated pairing correlation. For each term in C α (r) like a † ↑ a ↑ a † ↓ a ↓ , it has a term like a † ↑ a ↑ a † ↓ a ↓ . In Fig. 1 (a), we compare the long-range part of pairing correlations with different pairing symmetries on the 12 2 lattices at t 1 = 0.3, t 2 = 1.4, t ′ 2 = −0.6, which is a typical case for iron-pnictides [34][35][36][37][38]. Here, the electron filling is < n >= 1.0, which corresponds to a closed shell case with N ↑ =N ↓ =72. The simulations are performed at U = 3.0. One can readily see that C sxy (r) (solid red line) is larger than pairing correlations with any other pairing symmetry for almost all long-range distances between electron pairs. With the same set of parameters as that of Fig. 1 (a), Fig. 1 (b) shows the vertex contribution defined in Eq. 5. Obviously, the vertex contribution of s xy ( dash red line) pairing symmetry dominate that of any other pairing forms. The vertex contribution of s xy pairing symmetry is a finite value over the long range part, while vertex contributions of d xy , s x 2 +y 2 and d x 2 −y 2 are simply fluctuating around zero. In the numerical results, the ratio of the statistical error to the pairing correlation C α is no more than 0.5 percent, and most of the error bars are almost within the symbols. The ratio of the statistical error to the vertex contribution V α is no more than 3 percent. This remark applies to all the following figures. Fig. 2 shows the long-range part of pairing correlations with different pairing symmetries on the 12 2 lattice at t 1 = 0.3, t 2 = 1.2, t ′ 2 = −0.8. With this set of parameters, the system shows no hole packet [29]. Again we see that, both the long range part pairing correlation and the vertex contribution indicates that the s xy type dominates over that of other pairing forms. Thus, the behavior of long-range part pairing correlation re-enforces our findings on pairing susceptibility of an 8 2 lattice in Ref. [29].
In Fig. 3, we address the question of what happens to those "long-range" correlations if the system is doped away from half filling. In Fig. 3 (a), for a closed shell case with electron filling < n >= 0.83 ( N ↑ =N ↓ =60), < n >= 0.89 ( N ↑ =N ↓ =64) and < n >= 1.00 ( N ↑ =N ↓ =72), we show the CPMC results of s xy pairing correlation for U = 3.0 and t 1 = 0.3, t 2 = 1.4, t ′ 2 = −0.6. Fig. 3 (b) shows results of s xy pairing correlation for t 1 = 0.3, t 2 = 1.2, t ′ 2 = −0.8 at < n >= 1.00, < n >= 1.13 ( N ↑ =N ↓ =81) and < n >= 1.18 (N ↑ =N ↓ =85). We notice that, whatever for system with or without hole packet, the pairing correlations decrease as the system is doped away from half filled case.
We have also studied the effect of nearest neighbour interaction on the pairing correlation at a fix U = 3.0. In Fig. 4, the pairing correlations for s xy pairing symmetries are displayed as a function of distance on the 12 2 lattice with different nearest neighbour interaction V ′ s. For both systems with and without hole packet, we notice that the s xy pairing correlation is suppressed by the repulsive nearest neighbour interaction V .
Finally, we compare the pairing correlation on an 8 2 ( green triangle down ), a 12 2 ( red circle ), and a 16 2 ( pink triangle up ) lattices in Fig. 5 to exclude the size effect. Fig.5 (a) shows the pairing correlation with s xy symmetry for t 1 = 0.3, t 2 = 1.4, t ′ 2 = −0.6, and Fig.5 (b) shows the pairing correlation with s xy symmetry for t 1 = 0.3, t 2 = 1.2, t ′ 2 = −0.8. In the inset of Fig. 5, we examine the evolution of C α with increasing the lattice size up to 20 2 . The average of long-range pairing correlation, C α = 1 √ N ′ r≥3 C α (r), where N ′ is the number of electron pairs with r ≥ 3, is plotted as a function of 1 √ N at half filling. It is clear to see that C sxy (red circle) is larger than the average of long-range pairing correlations with any other pairing symmetry for whichever lattice size we investigate.
CONCLUSIONS
In summary, our unbiased numerical results show that the s xy pairing dominate in the ground state of the S 4 model, as we illustrated in previous study. And such a domination is robust in a wide range of physical region. We also find that the nearest neighbour interaction slightly suppressed the pairing correlation. The consistent behaviours of our results on different clusters suggest that S 4 model captures the essence of iron-based superconductors.
FIG. 1 :
1(Color online) (a) Pairing correlation Cα as a function of distance for different pairing symmetries on the 12 2 lattice with t1 = 0.3, t2 = 1.4, t ′ 2 = −0.6 (a typical case for ironpnictides[34][35][36][37][38]). (b) The vertex contribution Vα with the same parameters.
FIG. 2 :
2(Color online) (a) Pairing correlation Cα as a function of distance for different pairing symmetries on the 12 2 lattice with t1 = 0.3, t2 = 1.2, t ′ 2 = −0.8. (b) The vertex contribution Vα with the same parameters.
FIG. 3 :
3(Color online) (a) Pairing correlation Cα as a function of distance on the 12 2 lattice with t1 = 0.3, t2 = 1.4, t ′ 2 = −0.6 for < n >=1.00, < n >=0.89 and < n >=0.83. (b) Pairing correlation Cα as a function of distance on the 12 2 lattice with t1 = 0.3, t2 = 1.2, t ′ 2 = −0.8 for < n >=1.00, < n >=1.13 and < n >=1.18.
FIG. 4 :
4(Color online) (a) Pairing correlation Cα as a function of distance at the nearest neighbour interaction V = 0.3, 0.5 and 1.0 on the 12 2 lattice with t1 = 0.3, t2 = 1.4, t ′ 2 = −0.6. (b) The same with (a) but at t1 = 0.3, t2 = 1.2, t ′ 2 = −0.8.
FIG. 5 :
5(Color online) (a) Pairing correlation Cs xy as a function of distance on an 8 2 , a 12 2 , and a 16 2 lattices with t1 = 0.3, t2 = 1.4, t ′ 2 = −0.6. (b) The same with (a) but at t1 = 0.3, t2 = 1.2, t ′ 2 = −0.8; Inset: the average of the long-range pairing correlation Cα vs 1 √ N at half filling.
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| [] |
[
"arXiv:quant-ph/0307051v1 8 Jul 2003 On Discrete Quasiprobability Distributions",
"arXiv:quant-ph/0307051v1 8 Jul 2003 On Discrete Quasiprobability Distributions"
] | [
"C A Muñoz Villegas \nDepartamento de Física\nUniversidad de Guadalajara\nMéxico\n",
"A Chavez Chavez \nDepartamento de Física\nUniversidad de Guadalajara\nMéxico\n",
"S Chumakov \nDepartamento de Física\nUniversidad de Guadalajara\nMéxico\n",
"Yu Fofanov \nComputer Science Department\nUniversity of Houston\nTexasUSA\n",
"A B Klimov \nDepartamento de Física\nUniversidad de Guadalajara\nMéxico\n"
] | [
"Departamento de Física\nUniversidad de Guadalajara\nMéxico",
"Departamento de Física\nUniversidad de Guadalajara\nMéxico",
"Departamento de Física\nUniversidad de Guadalajara\nMéxico",
"Computer Science Department\nUniversity of Houston\nTexasUSA",
"Departamento de Física\nUniversidad de Guadalajara\nMéxico"
] | [] | We analyse quasiprobability distributions related to the discrete Heisenberg-Weyl group. In particular, we discuss the relation between the Discrete Wigner and Qfunctons. | null | [
"https://export.arxiv.org/pdf/quant-ph/0307051v1.pdf"
] | 18,486,651 | quant-ph/0307051 | 287d52663d5b2887c7e81f4961824bdfa5987815 |
arXiv:quant-ph/0307051v1 8 Jul 2003 On Discrete Quasiprobability Distributions
April 1, 2022
C A Muñoz Villegas
Departamento de Física
Universidad de Guadalajara
México
A Chavez Chavez
Departamento de Física
Universidad de Guadalajara
México
S Chumakov
Departamento de Física
Universidad de Guadalajara
México
Yu Fofanov
Computer Science Department
University of Houston
TexasUSA
A B Klimov
Departamento de Física
Universidad de Guadalajara
México
arXiv:quant-ph/0307051v1 8 Jul 2003 On Discrete Quasiprobability Distributions
April 1, 2022
We analyse quasiprobability distributions related to the discrete Heisenberg-Weyl group. In particular, we discuss the relation between the Discrete Wigner and Qfunctons.
Introduction
Due to the recent development of hardware facilities, the description of the state and evolution of the wave processes has also evolved. In addition to the configuration space description and Fourier analysis, it has now become possible to use the Quasiprobability distributions that describe the state of the wave process in phase space of the corresponding mechanical system. By wave process we mean any acoustical, optical or quantum mechanical process, where, correspondingly, time and frequency, the coordinate of the point where the ray of light intersects the screen and the ray direction, the coordinate and momentum of the quantum mechanical particle play the role the phase space coordinates (see, e.g. [1][2][3][4][5][6]). Until quite recently, the practical use of the Quasiprobability distributions was strongly limited by the speed and memory restrictions of the computers, but now the situation is changing. The possibility to use the Quasiprobability distributions is highly attractive for it allows us to visualize the process in a intuitively clear way: to analyze the signal (e.g. to discriminate between its different components, possibly originated from different sources) and to process the signal (say, removing noise) at the level which is much higher than the one that can be achieved when working with the signal itself or its Fourier transform.
The theory of Quasiprobability distributions in its most complete form was developed in the framework of quantum mechanics (cf. [1][2][3][4]6]). In its original version, it refers to continuous variables and is intrinsically related to the Heisenberg-Weyl group of translations of phase space. However, any numerical work involves the discretization, which by itself is not a trivial procedure, as is clear from the example of the approximation of the Fourier integral by the Finite Fourier transform (see, e.g. [7]). Therefore, it is desirable to develop a systematic theory of the Quasiprobability distributions in discrete phase spaces. The basis of such a theory was given in the work of Wootters, [8] (see also [9], [10], [11], [12], [13]). The Hilbert space of system states was chosen to be a space of periodic functions with a finite number of Fourier harmonics. This is obviously a finite-dimensional space. This type of quasiprobability distributions is naturally related to the discrete Heisenberg-Weyl group of translations of discrete phase space. (It should not be confused with the quasiprobability distributions for spin systems, which are related to the SU(2) group [3,14]; in this case, the finite-dimensional Hilbert space includes spherical harmonics on the sphere.)
Our goal in this work is to establish the correspondence between the two most important discrete Quasiprobability distributions: the Wigner-Wootters ("W-function") distribution and the time dependent spectrum ("Q-function"), as it exists for the continuous case, where the Q-function can be produced by smoothing the W-function with an appropriate Gaussian. A natural periodic counterpart of the Gaussian is the Jacobi Theta-function and therefore, they appear in our construction.
The paper is organized as follows. After the presentation of the discrete coherent states (which naturally involves the discrete Jacobi Theta-functions) and the Q-function, we define the discrete Wigner function (which is made in terms of the discrete Heisenberg-Wyel group and is complementary to the original work [8]). Finally, we establish the relation between them.
We would like to note that we do have in mind possible applications to signal processing. However, at this stage of theoretical development, we found it reasonable to confine ourselves within the language of Quantum mechanics.
The model
We consider periodic functions f (x + L) = f (x). Therefore, the Fourier series coefficients contain the frequencies, ω k = k2π/L, where k is an integer,
f (x) = k f k e ix2πk/L , f k = 1 L L 0 dx f (x)e −ix2πk/L .
Let us consider functions for which Fourier series contain only M coefficients different from zero: k = 0, 1, . . . , M − 1. Then all the information is stored in the values of the function in M points on the circle:
x m = Lm M , m = 0, 1, . . . , M − 1; f (m) ≡ f (x m ) = M −1 k=0 f k e i2πkm/M .
The values of the function f (x) at arbitrary points can be recovered by the sampling theorem.
We will use the quantum mechanics notation, introducing the basis |m 0 = δ mm 0 ,
|f = M −1 m=0 |m m|f , m|f = f (x m ) = f (m).
One can use the matrix representation with
|f = f 0 f 1 . . . f M −1 .
The Discrete Fourier Transform of the vector f (m) gives us the wave function in momentum representation:
f (p) = p|f = M −1 m=0 p|m m|f , p = 0, 1, . . . , M − 1, p|m = 1 √ M e −i2πpm/M .
Using the discrete orthogonality relation (1),
M −1 m=0 e i2πm(n−k)/M = Mδ n,k(mod M ) ,(1)
one finds the DFT coefficients,
f (p) = √ M f p (mod M )
Thus, the periodicity also appears in the momentum. It is natural to introduce one-step translations in coordinate,
T x f (x m ) = f (x m−1 ),T x = exp −i L Mp x ,p x = −i∂ x , (h = 1),
and in momentum,
T p = exp i 2π Lx ,x f (x) = xf (x),T p f (x m ) = r −m f (x m ), r = e i2π/M . It is clear, thatT pTx = √ rT (1, 1) = rT xTp ,T (1, 1) = exp i 2π Lx − L Mp x .
Note, that in the course of the discretization of the initial continuous model, L completely disappears from all of the formulas.
Vacuum state
The periodic analogue to Gaussian functions is the Jacobi Θ-function defined as follows (see [15] and Appendix A),
Θ 3 (z, µ) = ∞ n=−∞ e −i2zn−µn 2 = π µ ∞ k=−∞ exp − (z − πk) 2 µ .
(Here and below we often use the Poisson transformation.) Generally speaking, Re µ > 0 but we consider only real values of µ. Recall, that Θ 3 (z + π, µ) = Θ 3 (z, µ), and hence x = zL/π. Considering the Θ-function on a discrete set of points, x m = mL/M and z m = mπ/M, we come to a discrete periodic Gaussian,
Θ(m) = m|Θ µ = Θ 3 πm M µ = ∞ n=−∞ e −i2πmn/M −µn 2 = π µ ∞ k=−∞ exp − π 2 (m − kM) 2 M 2 µ .(2)
This function will play a role of the vacuum squeezed state (with squeezing parameter µ).
The choice µ = π/M
corresponds to the vacuum Coherent State. Thus, µ ≪ 1, µM 2 ≫ 1 will be of interest. Using the discrete orthogonality relation (1) one can check the normalization,
N = Θ µ |Θ µ = M −1 m=0 |Θ(φ m |µ)| 2 = M n k e −µn 2 −µ(n−kM ) 2 = M [θ 3 (2µ)θ 3 (2µM 2 ) + θ 2 (2µ)θ 2 (2µM 2 )] , M = odd θ 3 (2µ)θ 3 (µM 2 /2), M = even = 2π µ [θ 3 (2µ)θ 3 (µ ′ /2) + θ 2 (2µ)θ 4 (µ ′ /2)] /2, M = odd θ 3 (2µ)θ 3 (2µ ′ ), M = even(3)
Here µ ′ = π 2 µM 2 ; for unsqueezed vacuum CS, µ = µ ′ = π/M. We will assume, that µ = µ 0 π/M, µ 0 ∼ 1, and thus, µ, µ ′ ≪ 1 simultaneously with µM 2 , µ ′ M 2 ≫ 1.
In Eq. (3) we use the notation, θ k (µ) = Θ k (0|µ), k = 1, 2, 3, 4;
(see Appendix A). For instance,
θ 3 (µ) = e −µn 2 = π µ e −k 2 π 2 /µ = π µ + O e −π 2 /µ , θ 2 (µ) = e −µ(n+1/2) 2 = π µ (−1) k e −k 2 π 2 /µ = π µ + O e −π 2 /µ .
In passing from the second to the third line of Eq. (3) we again used the Poisson transformation, θ 3 (µM 2 /2) = 2µ ′ /π θ 3 (2µ ′ ), and θ 2 (2µM 2 ) = µ ′ /2π θ 4 (µ ′ /2). If M ≫ 1, the normalization constant has a simple asymptotic form. Indeed, in this case only the terms k = 0 are important in the Poisson-transformed expressions for θ 2,3,4 , and neglecting the terms O e −µM 2 /2 , O e −µ ′ M 2 /2 , we have,
Θ µ |Θ µ ≈ M π 2µ .(4)
The wave function of the vacuum state in the momentum representation is given by the Discrete Fourier transform:
p|Θ µ = √ M ∞ k=−∞ e −µ(p−kM ) 2 = π µM ∞ n=−∞ e −i2πpn/M −µ ′ n 2 = π µM Θ πp M µ ′ .
Discrete Heisenberg-Weyl group
The naive way to introduce the displacement operator in phase space is (compare [10], [11]),T
One can introduce the adjoint action,
T (q, p)T (m, n)T † (q, p) = r pm−qnT (m, n).(8)
The matrix elements in the coordinate basis, k|T (m, n)|l = r nl+[nm/2] δ k,l+m(modM ) .
These matrix elements are orthogonal,
M −1 m,n=0
a|T (m, n)|b d|T (m, n)|c = Mδ ad δ bc .
Finally, TrT (m, n) = Mδ m,0 δ n,0 .
Discrete CS and Q-dunction
One can generate the complete set of (squeezed) CS by the action of the HW group (6) to the vacuum state (2), |m 0 , n 0 , µ =T (m 0 , n 0 )|Θ µ ,
Θ
and in the momentum representation,
Θ m 0 n 0 (p) = p|m 0 , n 0 , µ = √ M e −i2π(p−n 0 /2)m 0 /M ∞ k=−∞ exp −µ(p − n 0 − kM) 2 .
The normalization constant m 0 , n 0 , µ|m 0 , n 0 , µ = N is the same as in (3).
The completeness relation holds (for any µ):
1 N M −1 m 0 ,n 0 =0 |m 0 , n 0 , µ m 0 , n 0 , µ| =1.(12)
Scalar products of Coherent States. It is enough to consider the vacuum matrix element of the displacement operator,
Θ µ |m 0 , n 0 , µ = Θ µ |T (m 0 , n 0 )|Θ µ . If M is even, it is equal to Θ µ |m 0 , n 0 , µ = M 2µ ′ π Θ α πm 0 M 2µ Θ β πn 0 M 2µ ′ ,
where, α = 3, n 0 − even; α = 2, n 0 − odd; β = 3, m 0 − even; β = 2, m 0 − odd.
If M is odd, n 0 is even,
Θ µ |m 0 , n 0 , µ = M µ ′ 2π Θ 3 πm 0 M 2µ Θ 3 πn 0 2M µ ′ 2 + (−1) m 0 Θ 2 πm 0 M 2µ Θ 4 πn 0 2M µ ′ 2 ,
and if M is odd, n 0 is odd,
Θ µ |m 0 , n 0 , µ = M µ ′ 2π (−1) m 0 Θ 3 πm 0 M 2µ Θ 4 πn 0 2M µ ′ 2 + Θ 2 πm 0 M 2µ Θ 3 πn 0 2M µ ′ 2 ,
Asymptotic for large M is
Θ µ |m 0 , n 0 , µ Θ µ |Θ µ ≈ exp − µ ′ m 2 0 + µn 2 0 2 .
Q-function.
It is natural to introduce the Q-function as the diagonal matrix element of the density matrix between the coherent states, Q(m, p) = m, p, µ|ρ|m, p, µ ,
or, for a pure state, ρ = |Ψ Ψ|, Q(m, p) = | Ψ|m, p, µ | 2 .
Wigner function
By analogy with the continuous case, let us introduce the Wigner operator as a two-dimensional discrete Fourier transform of the displacement operator,
W (q, p) = 1 M M −1 m,n=0
r pm−qnT (m, n), q, p = 0, 1, . . . , M − 1.
Therefore, it is an Hermitian operator valued function on the phase space,Ŵ (q, p) =Ŵ † (q, p).
One can notice that,Ŵ (q, p) =T (q, p)Ŵ (0, 0)T † (q, p).
One immediately calculates its matrix elements,
k|Ŵ (q, p)|l = exp i2π M p(k − l) δ 2q=k+l(modM ) .
These are precisely the matrix elements of the "Phase Point Operators" considered by Wootters [8], with the property
Tr Ŵ (q, p)Ŵ (p 1 , q 1 ) = Mδ q,q 1 δ p,p 1 .(15)
This property was used to define these operators in [8]. The Wigner function (quasiprobability distribution) on the discrete phase space for the state |f is the mean value of the Wigner operator in this state,
W f (p, q) = f |Ŵ (q, p)|f .
To an arbitrary Hermitian operatorÂ, corresponds the Wigner function W f (p, q) = Tr Ŵ (q, p)Â .
If → ρ is a density matrix and Tr ρ = 1, then p,qŴ (q, p) = 1.
For a pure state, Tr ρ 2 = 1, p,qŴ 2 (q, p) = 1.
The orthogonality of the Wigner operator at different phase points (15), has a simple physical sense: the Wigner function of the Wigner operator itself is a δ-function, WŴ (p 0 ,q 0 ) (p, q) = Mδ q,q 0 δ p,p 0 Covariance under HW group. Using the adjoint action of the discrete HW group, (8), one can show thatŴ (q, p) =T (p, q)W (0, 0)T † (p, q).
The Wigner operator at the origin, W (0, 0) has a simple form, which we will explain, con In the rest of the paper we will restrict ourselves with the case when M is odd. (In the opposite case the information is lost and one cannot reconstruct the state from its Wigner function.) Overlap relation. Let us consider the following operator acting in the tensor product of two spaces,σ
= 1 M p,qŴ (q, p) ×Ŵ (q, p).
Using the orthogonality of the displacement operator matrix elements (9), one can prove that σ is an exchange operator,
a| 1 c| 2σ |d 2 |b 1 = δ b,c δ a,d ,σ|f 1 |g 2 = |g 1 |f 2 .
From here the overlap relation follows directly. Let W A (p, q) and W B (p, q) be the Wigner symbol of the operators andB. Then
1 M p,q W A (p, q)W B (p, q) = TrÂB.(16)= π 2µ (−1) mn exp − µ ′ 2 m 2 − µ 2 n 2 .(19)
From here it follows that F (m, n) = F (n, m) (provided that µ = µ ′ , i.e. the coordinate and momentum have the same rights.
Conclusions
The principal results of this paper are the relations (18),(17),(19) between the discrete Qfunction (time-dependent spectrum) and the discrete Wigner function. As in the continuous case, the Q-function can be produced by smoothing the W-function by integrating it with the discrete Gaussian function. S.C. is grateful to the Department of Computer Science, University of Houston, Texas for hospitality.
Here iµ is quasiperiod, Re µ > 0. Note, that in [15] Theta functions are written as Θ k (z, q), where r = e −µ . Θ 1,2,4 are shifts of Θ 3 : Θ 4 (z|µ) = Θ 3 (z + π/2|µ), Θ 2 (z|µ) = −ie iz−µ/4 Θ 3 (z + iµ/2|µ), Θ 1 (z|µ) = Θ 2 (z − π/2|µ) = −ie iz−µ/4 Θ 3 (z + π/2 + iµ/2|µ).
Note, that Θ 1,2 (z + π|µ) = −Θ 1,2 (z|µ), Θ 3,4 (z + π|µ) = Θ 3,4 (z|µ), and Θ 1,4 (z + iµ|µ) = −gΘ 1,4 (z|µ), Θ 2,3 (z + iµ|µ) = gΘ 2,3 (z|µ),
where g = e −i2z+µ . Finally, Θ 1 (−z|µ) = −Θ 1 (z|µ), Θ 2,3,4 (−z|µ) = Θ 2,3,4 (z|µ).
us return to the one-step translations in coordinate and momentum, which are defined in the framework of the initial continuous model asT x = e −ipxL/M ,T p = e ix2π/L ,T x |m = |m + 1 ,T x f (m) = m|T x |f = f (m − 1),T x |p = r −p |p , T p |m = r m |m ,T p |p = |p + 1 ,T p f (p) = f (p − 1).Here, once again, r = e i2π/M . These operators generate a discrete group. It is clear that there is a periodicity in momentum, as well as in coordinate,T Therefore, an arbitrary group element must include the multiplication by a phase factor,
(T
However, such an operator does not belong to the group(5). Fortunately, there exists a way to improve the situation. Let us now consider the case of odd M. Then, in the set of numbers{n (modM)} = {1, 2, . . . , M − 1} there exists a unique solution to the equation 2n = m, More generally, in the case of prime M there always exists a unique solution to the equation sn = m, and the numbers {n (modM)} then form a field.) It is clear, is unitary,T † (m, n) =T −1 (m, n) =T (−m, −n), and periodic,T (m + M, n) =T (m, n + M) =T (m, n). It is easy to check the multiplication formula, T (m 2 , n 2 )T (m 1 , n 1 ) = r [(m 1 n 2 −n 1 m 2 )/2]T (m 1 + m 2 , n 1 + n 2 ).
m 0 n 0 (m) = m|m 0 , n 0 , µ = e i2πn 0 (m−m 0 /2)/M Θ(m − m 0 ) (10) = e i2πn 0 (m−m 0 /2)/M ∞ n=−∞ exp i 2π M (m 0 − m)n − µn 2 ,
sidering examples of odd and even M. For M = 5, spectrum of Wigner operator for odd M consists of M/2 eigenvalues −1 and M/2+1 eigenvalues +1. For M = 4, for even M consists of (M − 1)/2 eigenvalues −1 and (M − 1)As in the continuous case,Ŵ (0, 0) is the inversion operator with respect to the origin. In the case of odd M, there is only one stable point (m = 0) under this inversion, while for even M there are two stable points m = 0, and m = M/2.
Therefore, for any periodic function of two variables, φ(m + M, n) = φ(m, n + M) = φ(m, n), one can prove,
Reconstruction of the initial state. If we know the Wigner function W ρ (p, a), then the density matrix of this state can be reconstructed aŝRelation between W and Q-functionsWe assume now that M is odd, and µ = µ ′ = π/M. Let us start with the kernel k|Q(q, p)|l = k|q, p q, p|l = k|T (q, p)|Θ µ Θ µ |T (q, p) † |l , and find its Fourier transform,(17) In other words,To find the function f (m, n), let us note that for any periodic discrete function φ(m+M) = φ(m) the following formulas hold:and moreover:Appendix A: Theta functionsJacobi Theta functions are the integral functions defined as follows[15]:
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| [] |
[
"Sublinear-Time Probabilistic Cellular Automata",
"Sublinear-Time Probabilistic Cellular Automata"
] | [
"Augusto Modanese \nAalto University\nEspooFinland\n"
] | [
"Aalto University\nEspooFinland"
] | [] | We propose and investigate a probabilistic model of sublinear-time one-dimensional cellular automata. In particular, we modify the model of ACA (which are cellular automata that accept if and only if all cells simultaneously accept) so that every cell changes its state not only dependent on the states it sees in its neighborhood but also on an unbiased coin toss of its own. The resulting model is dubbed probabilistic ACA (PACA). We consider one-and two-sided error versions of the model (in the same spirit as the classes RP and BPP) and establish a separation between the classes of languages they can recognize all the way up to o( √ n) time. As a consequence, we have a Ω( √ n) lower bound for derandomizing constant-time two-sided error PACAs (using deterministic ACAs). We also prove that derandomization of T (n)-time PACAs (to polynomial-time deterministic cellular automata) for various regimes of T (n) = ω(log n) implies non-trivial derandomization results for the class RP (e.g., P = RP). The main contribution is an almost full characterization of the constant-time PACA classes: For one-sided error, the class equals that of the deterministic model; that is, constant-time one-sided error PACAs can be fully derandomized with only a constant multiplicative overhead in time complexity. As for two-sided error, we identify a natural class we call the linearly testable languages (LLT) and prove that the languages decidable by constant-time two-sided error PACAs are "sandwiched" in-between the closure of LLT under union and intersection and the class of locally threshold testable languages (LTT).ACM Subject Classification Theory of computation → Formal languages and automata theory | 10.4230/lipics.stacs.2023.29 | [
"https://export.arxiv.org/pdf/2203.14614v5.pdf"
] | 247,763,177 | 2203.14614 | 0896f6607ebddb7c8e7de59abbfc9285aaed5eca |
Sublinear-Time Probabilistic Cellular Automata
Augusto Modanese
Aalto University
EspooFinland
Sublinear-Time Probabilistic Cellular Automata
10.4230/LIPIcsand phrases Cellular automatalocal computationprobabilistic modelssubregular language classes
We propose and investigate a probabilistic model of sublinear-time one-dimensional cellular automata. In particular, we modify the model of ACA (which are cellular automata that accept if and only if all cells simultaneously accept) so that every cell changes its state not only dependent on the states it sees in its neighborhood but also on an unbiased coin toss of its own. The resulting model is dubbed probabilistic ACA (PACA). We consider one-and two-sided error versions of the model (in the same spirit as the classes RP and BPP) and establish a separation between the classes of languages they can recognize all the way up to o( √ n) time. As a consequence, we have a Ω( √ n) lower bound for derandomizing constant-time two-sided error PACAs (using deterministic ACAs). We also prove that derandomization of T (n)-time PACAs (to polynomial-time deterministic cellular automata) for various regimes of T (n) = ω(log n) implies non-trivial derandomization results for the class RP (e.g., P = RP). The main contribution is an almost full characterization of the constant-time PACA classes: For one-sided error, the class equals that of the deterministic model; that is, constant-time one-sided error PACAs can be fully derandomized with only a constant multiplicative overhead in time complexity. As for two-sided error, we identify a natural class we call the linearly testable languages (LLT) and prove that the languages decidable by constant-time two-sided error PACAs are "sandwiched" in-between the closure of LLT under union and intersection and the class of locally threshold testable languages (LTT).ACM Subject Classification Theory of computation → Formal languages and automata theory
Introduction
Cellular automata (CAs) have been extensively studied as a natural model of distributed computation. A one-dimensional CA is composed of a row of fairly limited computational agents-the cells-which, by interacting with their immediate neighbors, realize a global behavior and work towards a common goal. As every model of computation, CAs have been widely studied as language acceptors [11,20]. These efforts apparently were almost exclusively devoted to the linear-or real-time case-to the detriment of the sublinear-time one [15]. This is unfortunate since, as it was recently shown in [14], the study of sublinear-time CA variants might help better direct efforts in resolving outstanding problems in computational complexity theory.
In this work, we consider a probabilistic sublinear-time CA model. Our main goal is to analyze to what extent-if at all-the addition of randomness to the model is able to make up for its inherent limitations. (For instance, sublinear-time CA models are usually restricted to a local view of their input [15] and are also unable to cope with long unary subwords [14].)
Definition 1 (Cellular automaton). A cellular automaton (CA) is a triple C = (Q, $, δ)
where Q is the finite set of states, $ / ∈ Q is the boundary symbol, and δ : Q $ × Q × Q $ → Q is the local transition function, where Q $ = Q ∪ {$}. The elements in the domain of δ are the possible local configurations of the cells of C. For a fixed width n ∈ N + , the global configurations of C are the elements of Q n . The cells 0 and n − 1 are the border cells of C. The global transition function ∆ : Q n → Q n is obtained by simultaneous application of δ everywhere; that is, if s ∈ Q n is the current global configuration of C, then ∆(s) = δ($, s 0 , s 1 ) δ(s 0 , s 1 , s 2 ) · · · δ(s n−2 , s n−1 , $).
For t ∈ N 0 , ∆ t denotes the t-th iterate of ∆. For an initial configuration s ∈ Q n , the sequence s = ∆ 0 (s), ∆(s), ∆ 2 (s), . . . is the orbit of C (for s). Writing the orbit of C line for line yields its space-time diagram.
One key theme connecting CAs and models of physics is causality: If two cells i and j are t cells away from each other, then j requires at least t steps to receive any information from i. In the sublinear-time case, this means every cell only gets to see a very small section of the input. In some sense this is reminiscent of locality in circuits (e.g., [22]), though locality in the CA model carries a more literal meaning since it is connected to the notion of space (whereas in circuits there is no equivalent notion). One should keep this limitation (of every cell only seeing a portion of the input) in mind as it is central to several of our arguments.
The usual acceptance condition for CA-based language recognizers is that of a distinguished cell (usually the leftmost one) entering an accepting state [11]. This is unsuitable for sublinear-time computation since then the automaton is limited to verifying prefixes of a constant length [15]. The most widely studied [9,10,15,18] acceptance condition for sublinear-time is that of all cells simultaneously accepting, yielding the model of ACA (where the first "A" in the acronym indicates that all cells must accept). Definition 2 (DACA). A deterministic ACA (DACA) is a CA C with an input alphabet Σ ⊆ Q as well as a subset A ⊆ Q of accepting states. We say C accepts an input x ∈ Σ + if there is t ∈ N 0 such that ∆ t (x) ∈ A n , and we denote the set of all such x by L(C).
In addition, C is said to have time complexity (bounded by) T : N + → N 0 if, for every x ∈ L(C) ∩ Σ n , there is t < T (|x|) such that ∆ t (x) ∈ A n .
We propose a probabilistic version of the ACA model inspired by the stochastic automata of [2] and the definition of probabilistic Turing machines (see, e.g., [1]). In the model of probabilistic ACA (PACA), at every step, each cell tosses a fair coin c ∈ {0, 1} and then changes its state based on the outcome of c. There is a nice interplay between this form of randomness access and the overall theme of locality in CAs: Random events pertaining to a cell i depend exclusively on what occurs in the vicinity of i. Furthermore, events corresponding to distinct cells i and j can only be dependent if i and j are near each other; otherwise, they are necessarily independent (see Lemma 11).
We consider both one-and two-sided error versions of the model as natural counterparts of RP and BPP machines, respectively. Although PACAs are a conceptually simple extension of ACAs, the definition requires certain care, in particular regarding the model's time complexity. To see why, recall that, in deterministic ACA (DACA), time complexity of an automaton C is defined as the upper bound on the number of steps that C takes to accept an input in its language L(C). In contrast, in a PACA C there may be multiple computational branches (depending on the cells' coin tosses) for the same input x ∈ L(C ), and it may be the case that there is no upper bound on the number of steps for a branch starting at x to reach an accepting configuration. In non-distributed models such as Turing machines, these pathological cases can be dealt with by counting the number of steps computed and stopping if this exceeds a certain bound. In PACA, that would either require an extrinsic global agent that informs the cells when this is the case (which is undesirable since we would like a strictly distributed model) or it would need to be handled by the cells themselves, which is impossible in sublinear-time in general (since the cells cannot directly determine the input length). We refer to Section 3 for the formal definitions and further discussion.
Finally, we should also mention our model is more restricted than a stochastic CA, 1 which is a CA in which the next state of a cell is chosen according to an arbitrary distribution that depends on the cell's local configuration. For a survey on stochastic CAs, we refer to [12].
Results
Inclusion relations. As can be expected, two-sided error PACAs are more powerful than their one-sided error counterparts. Say a DACA C is equivalent to a PACA C if they accept the same language (i.e., L(C) = L(C )). We stress the first item does not follow immediately from the definitions since it requires error reduction by a constant factor, which requires a non-trivial construction. It remains open whether in the second item we can improve the separation from o( √ n) to o(n) time. Nevertheless, as it stands the result already implies a lower bound of Ω( √ n) time for the derandomization (to a DACA) of constant-time two-sided error PACA.
Another result we show is how time-efficient derandomization of PACA classes imply derandomization results for RP (with a trade-off between the PACA time complexity and the efficiency of the derandomization).
(b − 1), RTIME[n a ] ⊆ TIME[2 O(n c ) ].
We deliberately write "deterministic CA" instead of "DACA" since, for T (n) = Ω(n), a T -time DACA is equivalent to an O(T )-time deterministic CA with the usual acceptance condition [15].
Characterization of constant time.
As a first step we analyze and almost completely characterize constant-time PACA. Indeed, the constant-time case is already very rich and worth considering in and of itself. This may not come as a surprise since other local computational models (e.g., local graph algorithms [19]) also exhibit behavior in the constanttime case that is far from trivial.
In Appendix A we give an example of a one-sided error PACA that recognizes a language L strictly faster than any DACA for L. Nonetheless, as we prove, one-sided error PACA can be derandomized with only a constant multiplicative overhead in time complexity.
Theorem 5. For any constant-time one-sided error PACA C, there is a constant-time DACA C such that L(C) = L(C ).
In turn, the class of languages accepted by constant-time two-sided error PACA can be considerably narrowed down in terms of a novel subregular class LLT, dubbed the locally linearly testable languages. Below, LLT ∪∩ is the closure over LLT under union and intersection and LTT its Boolean closure (i.e., its closure under union, intersection, and complement). Theorem 6. The class of languages that can be accepted by a constant-time two-sided error PACA contains LLT ∪∩ and is strictly contained in LTT.
It is known that the constant-time class of DACA equals the closure under union SLT ∪ of the strictly local languages SLT [18]. (We refer to Section 4.3 for the definitions.) Since SLT ∪ LLT ∪ is a proper inclusion, this gives a separation of the deterministic and probabilistic classes in the case of two-sided error and starkly contrasts with Theorem 5.
The class LLT. The languages in LLT are defined based on sets of allowed prefixes and suffixes (as, e.g., the languages in SLT) together with a linear threshold condition (hence their name): For the infixes m of a fixed length ∈ N + there are coefficients α(m) ∈ R + 0 as well as a threshold θ ≥ 0 such that every word w in the language satisfies the following:
m∈Σ α(m) · |w| m ≤ θ,
where |w| m is the number of occurrences of m in w.
In Section 4.3 we show LLT lies in-between SLT ∪ and the class of locally threshold testable languages LTT. In this regard LLT is similar to the class LT of locally testable languages; however, as we can also show, both LLT and LLT ∪ are incomparable to LT. The relation between LLT ∪∩ and LT is left as a topic for future work.
As the classes SLT, LT, and LTT (see, e.g., [4]), LLT may also be characterized in terms of scanners, that is, memoryless devices that process their input by passing a sliding window of symbols over it. Namely, the class LLT corresponds to the languages that can be recognized by scanners possessing a single counter c with maximum value θ; the counter c is incremented by α(m) for every infix m ∈ Σ read, and the scanner accepts if and only if c ≤ θ holds at the end of the input (and the prefix and suffix of the input are also allowed).
A related restriction of the LTT languages that we should mention is that of the locally threshold testable languages in the strict sense (LTTSS) [6,17]. The key difference between these languages and our class LLT is that, in the former, one sets a threshold condition for each infix separately (which corresponds to using multiple counters in their characterization in terms of scanners). In turn, in LLT there is a single threshold condition (i.e., the inequality above) and in which different infixes may have distinct weights (i.e., the coefficients α(m)). For instance, this allows counting distinct infixes m 1 and m 2 towards the same threshold t, which is not possible in the LTTSS languages (as there each infix is considered separately).
Organization
The rest of the paper is organized as follows: Section 2 introduces basic concepts and notation. Following that, in Section 3 we define the PACA model and prove standard error reduction results as well as Theorem 3. In Section 4 we focus on the constant-time case and prove Theorems 5 and 6. Finally, in Section 5 we address the general sublinear-time case and prove Theorem 4. We conclude with Section 6 by mentioning a few further research directions.
Preliminaries
It is assumed the reader is familiar with the theory of cellular automata as well as with basic notions of computational complexity theory (see, e.g., the standard references [1,5,7]). All logarithms are to the base 2. The set of integers is denoted by Z, that of non-negative integers by N 0 , and that of positive integers by N + . For a set S and n, m ∈ N + , S n×m is the set of n-row, m-column matrices over S. For n ∈ N + , [n] = {i ∈ N 0 | i < n} is the set of the first n non-negative integers. Also, for a, b ∈ Z, by [a, b] = {i ∈ Z | a ≤ i ≤ b} we always refer to an interval containing only integers.
Symbols in words are indexed starting with zero. The i-th symbol of a word w is denoted by w i . For an alphabet Σ and n ∈ N 0 , Σ ≤n contains the words w ∈ Σ * for which |w| ≤ n. For an infix m ∈ Σ ≤|w| of w, |w| m is the number of occurrences of m in w. Without restriction, the empty word is not an element of any language that we consider. (This is needed for definitional reasons; see Definitions 1 and 2 below.)
We write U n (resp., U n×m ) for a random variable distributed uniformly over {0, 1} n (resp., {0, 1} n×m ). We also need the following variant of the Chernoff bound (see, e.g., [21]): Theorem 7 (Chernoff bound). Let X 1 , . . . , X n be independently and identically distributed Bernoulli variables and µ = E[X i ]. There is a constant c > 0 such that the following holds for every ε = ε(n) > 0:
Pr i X i n − µ > ε < 2 −cnε 2 .
Many of our low-level arguments make use of the notion of a lightcone. 2 For a set S and non-negative integers n ≤ m, a lightcone L = ( i,j ) of radius m and height n over S is a trapezoidal (when n < m) or triangular (when n = m) array of elements i,j ∈ S, where i ∈ [0, n] and j ∈ [−m, m]:
0,−m 0,−m+1
· · · · · · · · · 0,0 · · · · · · · · · 0,m−1 0,m 1,−m+1 · · · · · · · · · 1,0 · · · · · · · · · 1,m−1 . . . . . . . . . n,−m+n · · · n,0 · · · n,m−n
Fundamentals
In this section, we introduce the definition of PACA. Following that, we prove basic error reduction results and conclude with the proof of Theorem 3.
As customary for randomized models of computation, one may consider both online and offline views of our model. Since it gives a more natural presentation, in the definition below we first assume an online perspective and then address the definitional issue mentioned in the introduction. In the last part, we switch to an offline view that we will use for the rest of the paper; this is more comfortable to work with since we can then refer to the cells' coin tosses explicitly. Definition 9 (PACA). Let Q be a finite set of states and Σ ⊆ Q an alphabet. A probabilistic ACA (PACA) C is a CA with two local transition functions δ 0 , δ 1 : Q 3 → Q. At each step of C, each cell tosses a fair coin c ∈ {0, 1} and updates its state according to δ c ; that is, if the current configuration of C is s ∈ Q n and the cells obtain coin tosses r = r 0 · · · r n−1 ∈ {0, 1} n (where r i is the coin toss of the i-th cell), then the next configuration of C is ∆ r (s) = δ r0 ($, s 0 , s 1 ) δ r1 (s 0 , s 1 , s 2 ) · · · δ rn−1 (s n−2 , s n−1 , $).
Seeing this process as a Markov chain M over Q n , we recast the global transition function ∆ = ∆ Un as a family of random variables (∆(s)) s∈Q n parameterized by the current configuration s of C, where ∆(s) is sampled by starting in state s and performing a single transition on M (having drawn the cells' coin tosses according to U n ). Similarly, for t ∈ N 0 , ∆ t (s) is sampled by starting in s and performing t transitions on M .
A computation of C for an input x ∈ Σ n is a path in M starting at x. The computation is accepting if the path visits A n at least once. In order to be able to quantify the probability of a PACA accepting an input, we additionally require for every PACA C that there is a function T : N + → N 0 such that, for any input x ∈ Σ n , every accepting computation for x visits A n for the first time in strictly less than T (n) steps; that is, if there is t ∈ N 0 with ∆ t (x) ∈ A n , then ∆ t1 (x) ∈ A n for some t 1 < T (n). (Hence, every accepting computation for x has an initial segment with endpoint in A n and whose length is strictly less than T (n).) If this is the case for any such T , then we say C has time complexity (bounded by) T .
With this condition in place, we may now equivalently replace the coin tosses of C with a matrix R ∈ {0, 1} T (n)×n of bits with rows R 0 , . . . , R T (n)−1 and such that R j (i) corresponds to the coin toss of the i-th cell in step j. (If C accepts in step t, then the coin tosses in rows t, . . . , T (n) − 1 are ignored.) We refer to R as a random input to C. 4 Blurring the distinction between the two perspectives (i.e., online and offline randomness), we write C(x, R) = 1 if C accepts x when its coin tosses are set according to R, or C(x, R) = 0 otherwise.
As another remark, notice that in Definition 9 we opt for using binary coin tosses along with only two local transition functions. Nonetheless, this is sufficient to realize a set of 2 k local transition functions δ 0 , . . . , δ 2 k −1 for constant k with a multiplicative overhead of k. (Namely, by having each cell collect k coins in k steps, interpret these as the binary representation of i ∈ [2 k ], and then change its state according to δ i .) Definition 9 states the acceptance condition for a single computation (i.e., one fixed choice of a random input); however, we must still define acceptance based on all computations (i.e., for random inputs picked according to a uniform distribution). The two most natural candidates are the analogues of the well-studied classes RP and BPP, which we define next.
Definition 10 (p-error PACA). Let L ⊆ Σ * and p ∈ [0, 1). A one-sided p-error PACA for L is a PACA C with time complexity T = T (n) such that, for every x ∈ Σ n , x ∈ L ⇐⇒ Pr[C(x, U T ×n ) = 1] ≥ 1 − p and x / ∈ L ⇐⇒ Pr[C(x, U T ×n ) = 1] = 0.
If p = 1/2, then we simply say C is a one-sided error PACA. Similarly, for p < 1/2, a two-sided p-error PACA for L is a PACA C with time complexity T = T (n) for which
x ∈ L ⇐⇒ Pr[C(x, U T ×n ) = 1] ≥ 1 − p and x / ∈ L ⇐⇒ Pr[C(x, U T ×n ) = 1] ≤ p
hold for every x ∈ Σ * . If p = 1/3, then we simply say C is a two-sided error PACA. In both cases, we write L(C) = L and say C accepts L.
Note that, to each 0-error PACA C, one can obtain an equivalent DACA C with the same time complexity by setting the local transition function to δ 0 . In the rest of the paper, if it is not specified which of the two variants above (i.e., one-or two-sided error) is meant, then we mean both variants collectively. See Appendix A for an example of PACAs being more efficient than DACAs.
From the perspective of complexity theory, it is interesting to compare the PACA model with probabilistic circuits. It is known that every T (n)-time DACA can be simulated by an L-uniform AC circuit (i.e., a Boolean circuit with gates of unbounded fan-in) having poly(n) size and O(max{1, T (n)/ log n}) depth [15]. Using the same approach as in [15], we note the same holds for PACAs if we use probabilistic AC circuits instead. The proof in [15] bases on descriptive complexity theory, the central observation being that the state of a cell i after log n steps given its (log n)-neighborhood is a predicate that is computable in logarithmic space. Hence, for the PACA case we need only factor in the auxiliary random input into this predicate.
A key property that PACAs have but probabilistic circuits do not, however, is distance between computational units. (Indeed, in circuits, there is no such thing as the "length" of a wire.) One consequence of this is the following simple fact.
Lemma 11 (Independence of local events). Let C be a one-or two-sided error PACA, let x ∈ Σ n be an input to C, and let T ∈ N + . In addition, let i, j ∈ [n] be such that |i − j| > 2(T − 1) and E i (resp., E j ) be an event described exclusively by the states of the i-th (resp., j-th) cell of C in the time steps 0, . . . , T − 1 (e.g., the i-th cell accepts in some step t where t < T ). Then E i and E j are independent.
Proof. For any random input R, the states of k ∈ {i, j} in the time steps between 0 and T − 1 is uniquely determined by the values of
R(t, k − T + t + 1), . . . , R(t, k + T − t − 1) for t ∈ [T ]. Without loss of generality, suppose i ≤ j. Since i + T − 1 < j − T + 1, E i and E j are conditioned on disjoint sets of values of R, thus implying independence.
Note the proof still holds in case T = 1, in which case the events E i and E j occur with probability either 0 or 1, thus also (trivially) implying independence.
Robustness of the Definition
We now prove that the definition of PACA is robust with respect to the choice of p = 1/2 (resp., p = 1/3) for the error of one-sided (resp., two-sided) error PACA.
One-Sided Error
For one-sided error, we can reduce the error p to any desired constant value p .
Proposition 12.
Let p, p ∈ (0, 1) be constant and p < p. For every one-sided p-error PACA C, there is a one-sided p -error PACA C such that L(C) = L(C ). Furthermore, if C has time complexity T (n), then C has time complexity O(T (n)).
It follows that the definition of PACA is robust under the choice of p (as long as it is constant) and regardless of the time complexity (up to constant multiplicative factors).
The proof is essentially a generalization of the idea used in [15] to show that the sublineartime DACA classes are closed under union. Namely, C simulates several copies C 0 , . . . , C m−1 of C in parallel and accepting if and only if at least one C i accepts. This idea is particularly elegant because m can be chosen to be constant and we update the C i in a round-robin fashion (i.e., first C 0 , then C 1 , C 2 , etc., and finally C 0 again after C m−1 ). The alternative is to simulate each C i for T (n) steps at a time, which is not possible in general since we would have to compute T (n) first. The construction we give avoids this issue entirely.
Proof. We construct a PACA C with the desired properties. Let m = log(1/p − 1/p) . Furthermore, let Q be the state set of C and Σ its input alphabet. We set the state set of C to Q m × [m] ∪ Σ. Given an input x, every cell of C initially changes its state from x(i) to (x(i), . . . , x(i), 0). The cells of C simulate m copies of C as follows: If the last component of a cell contains the value j, then its j-th component 5 q 0 is updated to δ(q −1 , q 0 , q 1 ), where q −1 and q 1 are the j-th components of the left and right neighbors, respectively (or $ in case of a border cell); at the same time, the last component of the cell is set to j + 1 if j < m or 0 in case j = m. A cell of C is accepting if and only if its last component is equal to j and its j-th component is an accepting state of C.
Denote the i-th simulated copy of C by C i . Clearly, C accepts in step mt + i + 1 for i ∈ [m] if and only if C i accepts in step t, so we immediately have that C has O(T (n)) time complexity. For the same reason and since C never accepts in step 0, C does not accept any input x / ∈ L(C). As for x ∈ L(C), note the m copies of C are all simulated using independent coin tosses, thus implying
Pr[C does not accept x] = Pr[∀i ∈ [m] : C i does not accept x] < p m ≤ p .
Hence, C accepts x with probability at least 1 − p , as desired.
Two-Sided Error
For two-sided error, we show the same holds for every choice of p for constant-time PACA. We remark the construction is considerably more complex than in the one-sided error case.
Proposition 13. Let p, p ∈ (0, 1/2) be constant and p < p. For every two-sided p-error PACA C with constant time complexity T = O(1), there is a two-sided p -error PACA C with time complexity O(T ) = O(1) and such that L(C) = L(C ).
To reduce the error, we use the standard method based on the Chernoff bound (Theorem 7); that is, the PACA C simulates m independent copies C 0 , . . . , C m−1 of C (for an adequate choice of m) and then accepts if and only if the majority of the C i do. More precisely, C loops over every possible majority M ⊆ [m] (i.e., every set M ⊆ [m] with |M| ≥ m/2) over the C i and checks whether C i accepts for every i ∈ M (thus reducing majority over the L(C i ) to intersection over the L(C j ) where j ∈ M). In turn, to check whether every C j accepts for j ∈ M, C tries every possible combination of time steps for the C j to accept and accepts if such a combination is found. If this entire process fails, then the majority of the C i do not accept, and thus C does not accept as well. (Obviously, this idea is only feasible if m as well as the time complexities of the C i are constant.)
Proof. Let Q be the state set of C and Σ its input alphabet. We also fix a constant m depending only on p and p which will be set later and let M = m m/2 .
Construction.
The state set of C is Q ∪ Σ, where Q is a set of states consisting of the following components:
an m-tuple from Q m of states of C an m-tuple from [T ] m representing an m-digit, T -ary counter i 0 · · · i m−1 a value from [M ]
representing a counter modulo M an input symbol from Σ a (T × m)-matrix of random bits, which are all initially set to an undefined value different from 0 or 1 Given an input x, in the first step of C every cell changes its state from x(i) to a state in Q where the Q m components are all set to x(i), the numeric ones (i.e., with values in [T ] m and [M ]) set to 0, and the value x(i) is stored in the Σ component. In the first phase of C , which lasts for mT steps, the cells fill their (T × m)-matrix with random bits. This is the only part of the operation of C in which its random input is used (i.e., in all subsequent steps, C operates deterministically and the outcome of the remaining coin tosses is ignored).
In this next phase, C simulates m copies C 0 , . . . , C m−1 of C in its Q m components (as in the proof of Proposition 12). The random bits for the simulation are taken from the previously filled (T × m)-matrix, where the T entries in the i-th column are used as the coin tosses in the simulation of C i (with the entry in the respective j-th row being used in the j-th step of the simulation). Meanwhile, the i 0 · · · i m−1 counter is taken to represent that the simulation of C j is in its i j -th step.
The cells update their states as follows: At each step, the counter is incremented and the respective simulations are updated accordingly; more precisely, if C j is in step i j and i j was incremented (as a result of the counter being incremented), then the simulation of C j is advanced by one step; if the value i j is reset to 0, then the simulation of C j is restarted by setting the respective state to x(i). Finally, we turn to setting the parameter m so that C only errs with probability at most p . Let X i be the random variable that indicates whether C i accepts conditioned on its coin tosses. Then the probability that C errs is upper-bounded by the probability that ( i X i )/m deviates from the mean µ = Pr[C(x, r) = 1] ≥ 1 − p by more than ε = 1/2 − p.
One-vs. Two-Sided Error
The results of Section 3.1 are also useful in obtaining the following:
A. Modanese
XX:11
x i ∈ L 0 0 0 0 0 0 0 0 0 0 0 0 1 · · · · · · · · · · · · x * / ∈ L 0 0 0 0 0 0 0 0 0 0 0 1 1 · · · · · · · · · · · · x j ∈ L 0 0 0 0 0 0 0 0 0 0 0 0 1 · · · · · · · · · · · · 0 i j − t j n − 1 ≥ t = t = = Figure 2 Constructing x * / ∈ L from xi, xj ∈ L.
The numbers above the cells indicate their respective indices. Since every t-neighborhood of x * appears in either xi or xj and both xi and xj are accepted in (exactly) t steps, it follows that C accepts x * in t steps.
2.
There is a language L recognizable by constant-time two-sided error PACA but not by
any o( √ n)-time one-sided error PACA.
Proof. The first item follows from Proposition 12: Transform C into a one-sided error PACA C with error at most 1/3 and then notice that C also qualifies as a two-sided error PACA (as it simply never errs on "no" instances). For the second item, consider the language
L = {x ∈ {0, 1} + | |x| 1 ≤ 1}.
We obtain a constant-time two-sided error PACA for L as follows: If a cell receives a 0 as input, then it immediately accepts; otherwise, it collects two random bits r 0 and r 1 in the first two steps and then, seeing r 0 r 1 as the binary representation of an integer 1 ≤ t ≤ 4, it accepts (only) in the subsequent t-th step. Hence, if the input x is such that |x| 1 ≤ 1, the PACA always accepts; conversely, if |x| 1 ≥ 2, then the PACA only accepts if all 1 cells pick the same value for t, which occurs with probability at most 1/4. It remains to show L(C) = L for any T -time one-sided error
PACA C where T = o( √ n). Let n be large enough so that T = T (n) ≤ √ n/2. Observe that L ∩ {0, 1} n = {0 n , x 1 , . . . , x n } where x i = 0 i−1 10 n−i .
Let us now assume that x i ∈ L(C) holds for every i. Since C accepts with probability at least 1/2, by the pigeonhole principle there is R such that C(x i , R) = 1 for at least a 1/2 fraction of the x i . In addition, by averaging there is a step t < T such that at least a 1/2T fraction of the x i is accepted by C in step t. Since there are n/2T ≥ 2T ≥ 2t + 2 such x i , we can find i, j ∈ [n] with j ≥ i + 2t + 1 and x i , x j ∈ L(C). Consider now the input
x * = 0 i−1 10 j−i−1 10 n−j ,
which is not in L. We argue C(x * , R) = 1, thus implying L(C) = L and completing the proof.
We can see this by comparing the local views of the "bad" word x * with the "good" ones x i and x j (see Figure 2): Let k ∈ [n] be any cell of C. If k < j − t, then the t-neighborhood of k on input x * is identical to that when the input is x i , so k must be accepting in step t. Similarly, if k ≥ j − t, then the t-neighborhood of k in x * is the same as in x j , so k is accepting as well. It follows all cells of C are accepting in step t for inputs x * and R.
The Constant-Time Case
In this section we now focus on constant-time PACA. Our goal will be to characterize the constant-time classes of both one-and two-sided error PACA (i.e., Theorems 5 and 6). First, we introduce the concept of critical cells, which is central to our analysis.
Critical Cells
Definition 14 (Critical cell). Let C be a one-or two-sided error PACA, and let x ∈ L(C)∩Σ n . We say a cell i ∈ [n] is critical for x in step t ∈ N 0 if there are random inputs R, R ∈ {0, 1} t×n such that i is accepting in step t of C(x, R) but not in step t of C(x, R ).
In other words, if E is the event of i being accepting in step t of C on input x, then 0 < Pr[E] < 1 (where the probability is taken over the coin tosses of C). We should stress that whether a cell is critical or not may be highly dependent on x and t; for instance, there may be inputs x 1 = x 2 where the cell i is critical for x 1 but not for x 2 .
As it turns out, the number of critical cells of a constant-time PACA is also constant. Proof. For i ∈ [n], let E i denote the event in which the i-th cell accepts in step t. Assume towards a contradiction that there is x ∈ L(C) in which strictly more than 2T ·(1+log T )·2 T 2 = 2 O(T 2 ) cells are critical for x (in step t). By inspection, this implies there is a set K ⊆ [n] of |K| ≥ (1 + log T ) · 2 T 2 cells such that every k ∈ K is critical for x and, for every distinct
i, j ∈ K, |i − j| ≥ 2T . (For instance, in the extreme case where every 0 ≤ i < 2T · 2 T 2 is critical, pick K = {2jT | j ∈ [2 T 2 ]}.)
Since there are at most T 2 coin tosses that determine whether a cell i ∈ K accepts or not (i.e., those in the T -lightcone of i),
Pr[E i ] < 1 if and only if Pr[E i ] ≤ 1 − 2 −T 2 . By Lemma 11, the events E i are all independent, implying Pr[C accepts x in step t] ≤ i∈K Pr[E i ] ≤ 1 − 1 2 T 2 |K| < 1 e 1+log T < 1 2T
.
Since t was arbitrary, by a union bound it follows that the probability that C accepts x in step t is strictly less than 1/2. This contradicts x ∈ L(C) both when C is a one-and a two-sided error PACA. Lemma 15 implies that, given any x ∈ L(C), the decision of C accepting can be traced back to a set K of critical cells where |K| is constant. To illustrate the idea, suppose that, if C accepts, then it always does so in a fixed time step t < T ; in addition, assume the cells in K are all far apart (e.g., more than 2(T − 1) cells away from each other as in Lemma 11).
Characterization of One-Sided Error
Let us locally inspect the space-time diagram of C for x, that is, by looking at the t-lightcone of each cell i. Then we notice that, if i ∈ K, there is a choice of random bits in the t-lightcone of i that causes i to accept; conversely, if i / ∈ K, then any setting of random bits results in i accepting. Consider how this changes when x / ∈ L(C) while assuming K remains unchanged. Every cell i / ∈ K still behave the same; that is, it accepts regardless of the random input it sees. As for the cells in K, however, since they are all far apart, it cannot be the case that we still find random bits for every i ∈ K that cause i to accept; otherwise C would accept x with non-zero probability, contradicting the definition of one-sided error PACA. Hence, there must be at least some i ∈ K that never accepts. In summary, (under these assumptions) we can locally distinguish x ∈ L(C) from x / ∈ L(C) by looking at the cells of K and checking whether, for every i ∈ K, there is at least one setting of the random bits in the t-lightcone of i that causes it to accept.
To obtain the proof, we must generalize this idea to handle the case where the cells in K are not necessarily far from each other-which in particular means we can no longer assume that the events of them accepting are independent-as well as of K varying with the input. Since Lemma 15 only gives an upper bound for critical cells when the input is in L(C), we must also account for the case where the input is not in L(C) and |K| exceeds said bound.
Proof. Let T ∈ N + be the time complexity of C, and let M be the upper bound on the number of critical cells (for inputs in L(C)) from Lemma 15. Without restriction, we may assume T > 1. We first give the construction for C and then prove its correctness.
Construction.
Given an input x ∈ Σ n , the automaton C operates in two phases. In the first one, the cells communicate so that, in the end, each cell is aware of the inputs in its r-neighborhood, where r = (2M −1)(T −1). (Note this is possible because r is constant.) The second phase proceeds in T steps, with the cells assuming accepting states or not depending on a condition we shall describe next. After the second phase is over (and C has not yet accepted), all cells unconditionally enter a non-accepting state and maintain it indefinitely. Hence, C only ever accepts during the second phase.
We now describe when a cell i ∈ [n] is accepting in the t-th step of the second phase, where t ∈ [T ]. Let K be the set of critical cells of x in step t. The decision process is as follows:
1. First the cell checks whether there are strictly more than M cells in its r-neighborhood N that are critical in step t of C. If i cannot determine this for any cell j ∈ N (since it did not receive the entire t-neighborhood of j during the first phase), then j is simply ignored. If this is the case, then i assumes a non-accepting state (in the t-th step of the second phase). This process repeats itself in every step t of the second phase. Note that it can be performed by i instantaneously (i.e., without requiring any additional time steps of C ) since it can be hardcoded into the local transition function δ.
Correctness. It is evident that C is a constant-time PACA, so all that remains is to verify its correctness. To that end, fix an input x and consider the two cases:
x ∈ L(C). Then there is a random input R such that C accepts x in step t ∈ [T ]. This means that, for every critical cell i ∈ K, if we set the random bits in L i according to R, then every cell in B i accepts in step t of C. Likewise, every cell i / ∈ K is accepting in step t of C by definition. In both cases we have that i also accepts in the t-th step of the second phase of C , thus implying x ∈ L(C ). x / ∈ L(C). Then, for every random input R and every step t ∈ [T ], there is at least one cell i ∈ [n] that is not accepting in the t-th step of C(x, R). If i is not critical, then i is also not accepting in the t-th step of the second phase of C (regardless of the random input), and thus C also does not accept x. Hence, assume that every such i (i.e., every i such that there is a random input R for which i is not accepting in the t-th step of C(x, R)) is a critical cell. Let J ⊆ [n] denote the set of all such cells and, for i ∈ J, let D i ⊆ J be the subset that contains every j ∈ J such that the events of i and j accepting in step t are not independent (conditioned on the random input to C). In addition, let A i denote the event in which every cell of D i is accepting in step t of C(x, U T ×n ). We show the following:
Claim. There is an i ∈ J such that Pr[A i ] = 0; that is, for every R, there is at least one cell in D i that is not accepting in step t of C(x, R).
This will complete the proof since then i is also not accepting in the t-th step of the second phase of C (since any cell in D i is necessarily at most 2(M − 1)(T − 1) cells away from i), thus implying x / ∈ L(C ). To see the claim is true, suppose towards a contradiction that, for every i ∈ J, we have Pr[A i ] > 0. If there are i, j ∈ J such that j / ∈ D i (and similarly i / ∈ D j ), then by definition
Pr[C(x, U T ×n ) = 1] ≥ Pr[A i ∧ A j ] = Pr[A i ] Pr[A j ] > 0,
contradicting x / ∈ L(C). Thus, there must be i ∈ J such that J = D i ; however, this then implies
Pr[C(x, U T ×n ) = 1] = Pr[A i ] > 0.
As this is also a contradiction, the claim (and hence the theorem) follows.
Characterization of Two-Sided Error PACA
This section in divided into two parts. In the first, we introduce the class LLT of locally linearly testable languages and relate it to other classes of subregular languages. The second part covers the proof of Theorem 6 proper.
Local Languages
We introduce some notation. For ∈ N 0 and a word w ∈ Σ * , p (w) is the prefix of w of length if |w| ≥ , or w otherwise; similarly, s (w) is the suffix of length if |w| ≥ , or w otherwise. The set of infixes of w of length (exactly) is denoted by I (w).
The subregular language classes from the next definition are due to McNaughton and Papert [13] and Beauquier and Pin [3].
Definition 16 (SLT, LT, LTT).
A language L ⊆ Σ * is strictly locally testable if there is ∈ N + and sets π, σ ⊆ Σ ≤ and µ ⊆ Σ such that, for every w ∈ Σ * , w ∈ L if and only if p −1 (w) ∈ π, I (w) ⊆ µ, and s −1 (w) ∈ σ. The class of all such languages is denoted by SLT.
A language L ⊆ Σ * is locally testable if there is ∈ N + such that, for every w 1 , w 2 ∈ Σ * with p −1 (w 1 ) = p −1 (w 2 ), I (w 1 ) = I (w 2 ), and s −1 (w 1 ) = s −1 (w 2 ), we have that w 1 ∈ L if and only if w 2 ∈ L. The class of locally testable languages is denoted by LT.
A language L ⊆ Σ * is locally threshold testable if there are θ, ∈ N + such that, for any two words w 1 , w 2 ∈ Σ * for which the following conditions hold, w 1 ∈ L if and only if w 2 ∈ L:
1. p −1 (w 1 ) = p −1 (w 2 ) and s −1 (w 1 ) = s −1 (w 2 ). 2. For every m ∈ Σ , if |w i | m < θ for any i ∈ {1, 2}, then |w 1 | m = |w 2 | m .
The class of locally threshold testable languages is denoted by LTT.
The class LT equals the closure of SLT under Boolean operations (i.e., union, intersection, and complement) and the inclusion SLT LT is proper. As for LTT, it is well-known that it contains every language Th(m, θ) = {w ∈ Σ * | |w| m ≤ θ} where m ∈ Σ * and θ ∈ N 0 . Also, we have that LT LTT and that LTT is closed under Boolean operations. We write SLT ∪ for the closure of SLT.
Definition 17 (LLT).
For ∈ N 0 , θ ∈ R + 0 , π, σ ⊆ Σ ≤ −1 , and α : Σ → R + 0 , LLin (π, σ, α, θ) denotes the language of all w ∈ Σ + that satisfy p −1 (w) ∈ π, s −1 (w) ∈ σ, and m∈Σ α(m) · |w| m ≤ θ.
A language L ⊆ Σ + is said to be locally linearly testable if there are , π, σ, α, and θ as above such that L = LLin (π, σ, α, θ). We denote the class of all such languages by LLT.
We write LLT ∪ for the closure of LLT under union and LLT ∪∩ for its closure under both union and intersection. Proof. For the first item, note the inclusions LLT ⊆ LLT ∪ and LLT ∪ ⊆ LLT ∪∩ are trivial, so we need only prove SLT ⊆ LLT ⊆ LTT. The inclusion LLT ∪∩ ⊆ LTT then follows from LTT being closed under union and intersection, while SLT ∪ ⊆ LLT ∪ follows directly from SLT ⊆ LLT. The strictness of the inclusions follow all from the second item in the claim (since LLT then contains some language L / ∈ LT, which is certainly not in SLT ∪ ⊆ LT); the only exceptions are LLT LLT ∪ , which we address further below, and LLT ∪∩ LTT, which is proved in Theorem 6 (and does not depend on the results here).
The first inclusion SLT ⊆ LLT is easiest. Given L ∈ SLT, we know there is ∈ N + so that L is defined based on sets of allowed prefixes π ⊆ Σ ≤ , infixes µ ⊆ Σ , and suffixes σ ⊆ Σ ≤ . Clearly we have then L = LLin (π, σ, α, θ) for θ = 1/2 and
α(m) = 0, m ∈ µ 1, m / ∈ µ.
For the inclusion LLT ⊆ LTT, let L = LLin (π, σ, α, θ) be given. The proof is by induction on the number k of words m ∈ Σ such that α(m) = 0. If k = 0, then the linear condition of L is always satisfied, directly implying L ∈ LTT (or, better yet, L ∈ SLT). For the induction step, suppose the claim has been proven up to a number k, and let there be k + 1 words m ∈ Σ with α(m) = 0. In addition, let µ ∈ Σ with α(µ) = 0 be arbitrary, and let r ∈ N 0 be maximal with rα(µ) ≤ θ. Consider the languages
L i = LTT (π, σ, α i , θ i ) for i ∈ [0, r], θ i = θ − iα(µ), and α i such that, for every m ∈ Σ , α i (m) = α(m), m = µ; 0, m = µ.
By the induction hypothesis, the L i are all in LTT. On the other hand, we have
L = r i=0 (Th(µ, i) ∩ L i ) \ Th(µ, i − 1),
which implies L ∈ LTT since LTT is closed under Boolean operations. For the strictness of the inclusion LLT LLT ∪ , consider the language L = Th(1, 1) ∪ Th(2, 1) over the ternary alphabet Σ = {0, 1, 2}. Clearly we have L ∈ LLT ∪ , so assume that L = LLin (π, σ, α, θ) holds for some choice of , π, σ, α, and θ. Then necessarily α(w) > 0 for some w ∈ {0, 1} with |w| 1 = 1 since otherwise we would have a contradiction to 0 10 (20 ) 2 ∈ L and 0 (10 ) 2 (20 ) 2 / ∈ L. However, we then get that 0 (10 ) 1/α(w) 20 / ∈ L, which is a contradiction.
For the second item in the claim, we show there are L 1 and L 2 such that L 1 ∈ LLT \ LT and L 2 ∈ LT \ LLT ∪ . For the first one, probably the simplest is to set L 1 = Th(1, 1) (where the underlying alphabet is Σ = {0, 1}). The language is clearly not in LT (because otherwise either both 0 n 10 n and 0 n 10 n 10 n would be in L 1 or not (for sufficiently large n)); meanwhile, we have L 1 ∈ LLT since = 1, π = σ = {ε}, α(0) = 0, α(1) = 1, and θ = 1 satisfy L 1 = LLin (π, σ, α, θ). As for L 2 , consider
L 2 = {w ∈ {0, 1} * | w / ∈ {0} * }, which is in LT because L 2 = {0, 1} * \ {0} * (
and LT is closed under Boolean operations).
To argue that L 2 / ∈ LLT ∪ , suppose towards a contradiction that
L 2 = k i=1 L i for L i = LLin i (π i , σ i , α i , θ i ).
Since k is finite, there must be at least one L i that contains infinitely many words of the form 0 j 10 j for j ∈ N 0 . This implies 0 i−1 ∈ π j , σ j as well as α(0 i ) = 0, from which it follows that
m∈{0,1} i α i (m) · |0 k | m = α i (0 i ) · |0 k | 0 i = 0 ≤ θ i
for any k ∈ N 0 , thus contradicting 0 k / ∈ L i . Finally, the third item directly follows from the following well-known characterization of LTT: A language L is in LTT if and only if it can be expressed as the Boolean combination of languages of the form Th(m, θ), πΣ * , and Σ * σ where m, π, σ ∈ Σ * and θ ∈ N 0 . Obviously these three types of languages are all contained in LLT; since LLT ⊆ LTT, it follows that LTT equals the Boolean closure of LLT.
The Proof
With the terminology of Section 4.3.1 in place, we now turn to:
Theorem 6. The class of languages that can be accepted by a constant-time two-sided error PACA contains LLT ∪∩ and is strictly contained in LTT.
The theorem is proven by showing the two inclusions. We first give a brief overview of the ideas involved.
First inclusion. The first step is showing that we can "tweak" the components of the LLT condition so that it is more amenable to being tested by a PACA (Lemma 19). In particular, we prove we can assume the α(m) weights are such that 2 −α(m) can be represented using a constant number of bits. Having done so, the construction is more or less straightforward: We collect the subwords of length in every cell and then accept with the "correct" probabilities; that is, if a cell sees a subword m, then it accepts with probability 2 −α(m) . To lift the inclusion from LLT to LLT ∪∩ , we show that the class of languages that are recognizable by constant-time two-sided error PACAs is closed under union and intersection (Proposition 20).
Second inclusion.
The second inclusion (i.e., showing that L(C) ∈ LTT for every constanttime two-sided error PACA C) is considerably more complex. Let C be a PACA with time complexity at most T . The proof again bases on the class LLT and uses the fact that LTT equals the Boolean closure over LLT (which we show as a separate result): 1. As a warm-up, we consider the case where the cells of C accept all independently from one another. In addition, we assume that, if C accepts, then it does so in a fixed time step t < T . The argument is then relatively straightforward since we need only consider subwords of length = 2T + 1 and set their LLT weight according to the acceptance probability that the cell in the middle of the subword would have in C. 2. Next we relax the requirement on independence between the cells (but still assume a fixed time step for acceptance). The situation then requires quite a bit of care since the LLT condition does not account for subword overlaps at all. For instance, there may be cells c 1 and c 2 that are further than T cells apart and that both accept with non-zero probability but where c i accepts if and only if c 3−i does not (see Appendix A for a concrete example). We solve this issue by blowing up so that a subword covers not only a single cell's neighborhood but that of an entire group of cells whose behavior may be correlated with one other. Here we once more resort to Lemmas 11 and 15 to upper-bound the size of this neighborhood by a constant. 3. Finally, we generalize what we have shown so that it also holds in the case where C may accept in any step t < T . This is the only part in the proof where closure under complement is required. The argument bases on generalizing the ideas of the previous step to the case where the automaton may accept in multiple time steps and then applying the inclusion-exclusion principle.
We now elaborate on these ideas. As discussed above, we first need to make the linear condition of LLT a bit more manageable: LLin (π, σ, α, θ), there is α such that L = LLin (π, σ, α , θ ) and: 1. The threshold θ is equal to 1.
Lemma 19. For any L =
2.
There is a constant ε > 0 such that, for every w ∈ Σ * , we have either
f (w) < 1 − ε or f (w) > 1 + ε, where f (w) = m∈Σ α (m) · |w| m . 3.
There is k ∈ N 0 such that, for every m, there is n ∈ [2 k ] such that α (m) = k − log(n + 1).
The first two items ensure the sum f (w) is always "far away" from 1. In turn, the third item enables us to represent 2 −α (m) using at most k (and, in particular, O(1) many) bits.
Proof. The case where α(m) = 0 for every m is trivial, so suppose there is some m for which α(m) > 0. Because |w| m ∈ N 0 for every w and every m, f (w) = m∈Σ α(m) · |w| m is such that, given any b ∈ R + 0 , there are only finitely many values of f (w) ≤ b (i.e., the set {f (w) | f (w) ≤ b} is finite). Hence, by setting
r = 1 2 max f (w)≤θ f (w) + min f (w)>θ f (w)
and α (m) = α(m)/r, we have that f (w) = m∈Σ α (m) · |w| m satisfies the second condition from the claim (i.e., for every w ∈ Σ * , either f (w) < 1 − ε or f (w) > 1 + ε) for some adequate choice of ε > 0. Now we show how to satisfy the last condition without violating the first two. Essentially, we will choose k to be sufficiently large and then "round up" each α (m) to the nearest value of the form k − log(n + 1). To that end, let a be the minimal α (m) for which α (m) > 0. In addition, let k ∈ N 0 be such that α (m) < k/2 for every m and that log(2 k/2 + 1) − log(2 k/2 ) = log(2 k/2
+ 1) − k 2 ≤ aε 2|Σ|
.
(This is possible because log(n + 1) − log n tends to zero as n → ∞ and the right-hand side is constant.) Then, for every m, we set α (m) = k − log(n + 1) where n ∈ [2 k ] is maximal such that α (m) ≥ α (m). By construction, f (w) ≥ f (w), so we need only argue that there is ε > 0 such that, for every w for which f (w) < 1 − ε, we also have f (w) < 1 − ε .
In particular every said w must be such that, for every m, |w| m ≤ 1/a (otherwise we would have f (w) > 1). Noting that |α (m) − α (m)| is maximal when α (m) = k/2 and α (m) = k − log(2 k/2 + 1) + δ for very small δ > 0, we observe that
f (w) = m∈Σ α (m) · |w| m < f (w) + |Σ| a log(2 k/2 + 1) − k 2 < 1 − ε + ε 2 = 1 − ε 2 ,
that is, f (w) < 1 − ε for ε = ε/2, as desired.
We also need the following result: Proposition 20. Let C 1 and C 2 be constant-time two-sided error PACA. Then there are constant-time two-sided error PACA C ∪ and C ∩ such that L(C ∪ ) = L(C 1 ) ∪ L(C 2 ) and
C ∩ = L(C 1 ) ∩ L(C 2 ).
Proof. We first address the closure under union. Using Proposition 13, we may assume that C 1 and C 2 are two-sided ε-error PACA for some ε < 1/6. As we show further below, it suffices to have C ∪ simulate both C 1 and C 2 simultaneously (using independent random bits) and accept if either of them does. To realize this, we can simply adapt the construction from Proposition 12 using m = 2 to simulate one copy of C 1 and C 2 each (instead of two independent copies of the same PACA).
The probability that C ∪ accepts an x ∈ L i \ L j for i, j ∈ {1, 2} and i = j is at least 1 − ε, and the probability that C ∪ accepts x ∈ L 1 ∩ L 2 is even larger (i.e., at least 1 − ε 2 ≥ 1 − ε). Conversely, the probability that C ∪ accepts an x / ∈ L 1 ∪ L 2 is
Pr[C ∪ accepts x] ≤ Pr[C 1 accepts x] + Pr[C 2 accepts x] = 2ε < 1 3 .
Since C ∪ has time complexity 2T = O(T ), the claim follows. For the closure under intersection, we will use a similar strategy. This time suppose that the error probability ε of C 1 and C 2 is so that (1 − ε) 2 ≥ 2/3 (e.g., ε = 1/10 suffices). Again we let C ∩ simulate two copies of C 1 of C 2 simultaneously; however, this time we use a simplified version of the construction from the proof of Proposition 13. More specifically, we set m = 2 and leave out the [M ] component from the construction. That is, C ∩ randomly (and independently) picks random inputs r 1 and r 2 for C 1 and C 2 , respectively, and accepts if and only if C 1 accepts with coin tosses from r 1 and also C 2 accepts with coin tosses from r 2 . (One must also be careful since C 1 and C 2 may accept in different time steps, but this is already accounted for in the construction from Proposition 13.)
Since the copies of C 1 and C 2 are simulated independently from one another and C ∩ accepts if and only both do (in the simulation), we have
Pr[C ∩ accepts x] = Pr[C 1 accepts x] Pr[C 2 accepts x].
In case x ∈ L(C 1 ) ∩ L(C 2 ), this probability is at least (1 − ε) 2 ≥ 2/3; otherwise it is upper-bounded by ε < 1/3. Since C has time complexity O(T 2 ), which is constant, the claim follows.
We are now in position to prove Theorem 6.
Proof of Theorem 6. We prove the two inclusions from the theorem's statement. The first one we address is that of LLT ∪∩ in the class of constant-time two-sided error PACA.
First inclusion. Given L = LLin (π, σ, α, θ), we construct a constant-time two-sided error PACA C with L(C) = L. This suffices since by the closure properties shown in Proposition 20. We apply Lemma 19 and assume θ = 1 and that there are k and ε as in the statement of Lemma 19. For simplicity, we also assume = 2k + 1.
The automaton C operates in k steps as follows: Every cell sends its input symbol in both directions as a signal and, at the same time, aggregates the symbols it sees, thus allowing it to determine the initial configuration m ∈ Σ of its k-neighborhood. Meanwhile, every cell also collects k random bits r ∈ {0, 1} k . The decision to accept is then simultaneously made in the k-th step, where a cell with k-neighborhood m accepts with probability 2 −α(m) (independently of other cells). (This can be realized, for instance, by seeing r as the representation of a k-bit integer in [2 k ] and accepting if and only if r ≤ n, where n is such that 2 −α(m) = (n + 1)/2 k .) In the case of the first (resp., last) cell of C, it also checks that the prefix (resp., suffix) of the input is in π (resp., σ), rejecting unconditionally if this is not the case.
Hence, for an input word w ∈ Σ + , the probability that C accepts is
m∈Σ 1 2 α(m) |w|m = 2 −f (w) ,
where f (w) = m∈Σ α(m) · |w| m . Thus, if w ∈ L, then C accepts with probability 2 −f (w) > (1/2) 1−ε ; conversely, if w / ∈ L, the probability that C accepts is 2 −f (w) < (1/2) 1+ε . Since ε is constant, we may apply Proposition 13 and reduce the error to 1/3.
Second inclusion.
The proof of the second inclusion is more involved. Let C be a T -time two-sided error PACA for some T ∈ N + . We shall obtain L(C) ∈ LTT in three steps: 1. The first step is a warm-up where the cells of C accept all independently from one another and that, if C accepts, then it does so in a fixed time step t < T . 2. Next we relax the requirement on independence between the cells by considering groups of cells (of maximal size) that may be correlated with one another regarding their acceptance. 3. Finally, we generalize what we have shown so that it also holds in the case where C may accept in any step t < T . This is the only part in the proof where closure under complement is required. (Here we use item 3 of Proposition 18.)
Step 1. Suppose that C only accepts in a fixed time step t < T and that the events of any two cells accepting are independent from one another. We show that L(C) = LLin(π, σ, α, θ) for an adequate choice of parameters. Set = 2t+1 and let p m be the probability that a cell with t-neighborhood m ∈ Σ accepts in step t. In addition, let π = {p −1 (w) | w ∈ L(C)} and σ = {s −1 (w) | w ∈ L(C)} as well as θ = log(3/2) and α(m) = log(1/p m ) for m ∈ Σ . The probability that C accepts a word w ∈ Σ + is
m∈Σ p |w|m m = m∈Σ 1 2 α(m) |w|m = 2 −f (w) ,
which is at least 2/3 if and only if f (w) ≤ θ. It follows that L(C) = LLin (π, σ, α, θ).
Step 2. We now relax the requirements from the previous step so that the events of any two cells accepting need no longer be independent from one another. (C still only accepts in the fixed time step t.) Let K = 2 O(T 2 ) be the upper bound from Lemma 15 and = 2(K + 2)T . Again, we set π = {p −1 (w) | w ∈ L(C)}, σ = {s −1 (w) | w ∈ L(C)}, and θ = log(3/2). As for α(m), we set α(m) = 0 unless m is such that there is d ≤ K with m = abr 1 s 1 r 2 s 2 · · · r d s d c
where a ∈ Σ T is arbitrary, b ∈ Σ 2T contains no critical cells, each r j ∈ Σ is a critical cell (for step t), the s j ∈ Σ ≤2T −1 are arbitrary, and c ∈ Σ * has length |c| ≥ T and, if c contains any critical cell, then this cell accepts independently from r d . In addition, we require c to be of maximal length with this property. Note we need a as context to ensure that b indeed does not contain critical cells (since determining this requires knowledge of the states in the T -neighborhood of the respective cell); the same holds for r d and c. By construction and Lemma 11, the group of cells r 1 , . . . , r d is such that (although its cells are not necessarily independent from one another) its cells accepts independently from any other critical cell in C. Furthermore, b ensures m aligns properly with the group and that the group does not appear in any other infix. Letting p m be the probability that every one of the r j accept, for m as above we set α(m) = log(1/p m ). Then, as before, the probability that C accepts a word w ∈ Σ + is 2 −f (w) , which is at least 2/3 if and only if f (w) ≤ θ, thus implying L(C) = LLin (π, σ, α, θ).
Step 3. In this final step we generalize the argument so it also applies to the case where C may accept in any time step t < T . First note that, given any p > 0, if we set θ = log(1/p) in the second step above (instead of log(3/2)), then we have actually shown that σ, α, θ).
{w ∈ Σ + | Pr[C accepts w in step t] ≥ p} = LLin (π,
In fact, we can generalize this even further: Given any ∅ = τ ⊆ [T ], by setting α adequately we can consider the acceptance probability for the steps in τ altogether: 7
L(τ, p) = {w ∈ Σ + | Pr[C accepts w in every step t ∈ τ ] ≥ p} = LLin (π, σ, α, θ).
This is because the bound on critical cells of Lemma 15 holds for all steps where C accepts with non-zero probability and, in addition, as defined above m already gives enough context to check if the respective critical cells also accept in any previous step.
(That is, we construct m as above by using t = max τ ; however, since the sets of critical cells for different time steps may not be identical, we must also relax the condition for the r i so that r i need only be a critical cell in at least one of the time steps of τ .) Since LTT is closed under complement, we then also have
L(τ, p) = {w ∈ Σ + | Pr[C accepts w in every step t ∈ τ ] < p} ∈ LTT.
Fix some input word w ∈ Σ + to C. For ∅ = τ ⊆ [T ], let Z τ denote the event where C accepts w in every step t ∈ τ . By the inclusion-exclusion principle, we have
Pr[C accepts w] = Pr ∃t ∈ [T ] : Z {t} = τ ⊆[T ] |τ |=1 Pr[Z τ ]− τ ⊆[T ] |τ |=2 Pr[Z τ ]+· · ·+(−1) T +1 Pr[Z [T ] ]
(where the probabilities are taken over the coin tosses of C). This means that, if we are somehow given values for p(τ ) = Pr[Z τ ] so that the sum above is at least 2/3, then we can intersect a finite number of L(τ, p(τ )) languages and their complements and obtain some language that is guaranteed to contain only words in L(C). Concretely, let p(τ ) ≥ 0 for every ∅ = τ ⊆ [T ] be given so that
∅ =τ ⊆[T ] |τ | odd p(τ ) − ∅ =τ ⊆[T ] |τ | even p(τ ) ≥ 2 3 . Let T odd = {τ ⊆ [T ] | τ = ∅, |τ | odd, p(τ ) > 0}
and similarly
T even = {τ ⊆ [T ] | τ = ∅, |τ | even, p(τ ) > 0}.
Then necessarily 7 Of course we are being a bit sloppy here since Definition 9 demands that a PACA should halt whenever it accepts. What is actually meant is that, having fixed some random input, if we extend the space-time diagram of C on input w so that it spans all of its first T steps (simply by applying the transition function of C), then, for every t ∈ τ , the t-th line in the diagram contains only accepting cells.
L(p) = τ ∈T odd L(τ, p(τ )) ∩ τ ∈Teven L(τ, p(τ )) ⊆ L(C)
contains every w ∈ L(C) for which Pr[Z τ ] ≥ p(τ ) for τ ∈ T odd and Pr[Z τ ] ≤ p(τ ) for τ ∈ T even .
The key observation is that there are only finitely many values the Pr[Z τ ] may assume. This is because Z τ only depends on a finite number of coin tosses, namely the ones in the lightcones of the cells that are critical in at least one of the steps in τ (which, again, is finite due to Lemma 15). Hence, letting P denote the set of all possible mappings of the τ subsets to these values, we may write
L(C) = p∈P L(p) ∈ LTT.
Strictness of inclusion. The final statement left to prove is that the inclusion just proven is proper. This is comparatively much simpler to prove. We show that the language
L = {w ∈ {0, 1} + | |w| 1 ≥ 2} ∈ LTT
cannot be accepted by two-sided error PACA in constant time.
For the sake of argument, assume there is such a PACA C with time complexity T ∈ N + . Consider which cells in C are critical based on their initial local configuration. Certainly a cell with an all-zeroes configuration 0 2T −1 cannot be critical. Since 0 n 10 n / ∈ L(C) for any n (but 0 n 10 n 1 ∈ L(C)), there must be m 1 , m 2 so that m 1 + m 2 = 2T − 2 and c = 0 m1 10 m2 is the initial local configuration of a critical cell. This means that in
x = 0 2T (c0 2T ) T 2 T ∈ L
we have at least 2 T cells in x that are critical for the same time step t ∈ [T ] (by an averaging argument) and that are also all independent from one another (by Lemma 11). In turn, this implies the following, which contradicts x ∈ L(C):
Pr[C accepts x] ≤ 1 − 2 −T 2 T < 1 e < 2 3 .
The General Sublinear-Time Case
Recall we say a DACA C is equivalent to a PACA C if L(C) = L(C ). In this section, we recall and briefly discuss: To obtain Theorem 4, we prove the following more general result:
Proposition 21.
There is a constant c > 0 such that the following holds: Let monotone functions T, T , h, p : N + → N + be given with h(n) ≤ 2 n , p(n) = poly(n), and p(n) ≥ n and such that, for every n and p (n) = Θ(p(n) log h(n)),
In addition, for n given in unary, let the binary representation of h(n) and p (n) be computable in O(p (n)) time by a Turing machine and, for N given in unary, let T (N ) be computable in O(T (N )) time by a Turing machine. Furthermore, suppose that, for every T -time one-sided error PACA C, there is a T -time DACA C such that L(C) = L(C ). Then RTIME[p(n)] ⊆ TIME[h(n) · T (h(n)·poly(n)) · poly(n)].
The first item of Theorem 4 is obtained by setting (say) T (n) = n ε , T (n) = n d , and h(n) = p(n) 2(1/ε−1) (assuming ε < 1). For the second one, letting p(n) = n a where a > 0 is arbitrary and (again) T (n) = n d , set T (n) = (log n) 2+a/ε and h(n) = 2 n ε /(d+1) . Finally, for the last one, letting again p(n) = n a and T (n) = n d , set T (n) = (log n) b and h(n) = 2 n c .
At the core of the proof of Proposition 21 is a padding argument. Nevertheless, we cannot stress enough that the padding itself is highly nontrivial. In particular, it requires a clever implementation that ensures it can be verified in parallel and also without initial knowledge of the input length. To see why this is so, observe that, if we simply use a "standard" form of padding where we map x ∈ {0, 1} + to x = x0 p(|x|) (where p : N + → N 0 gives the desired padding length), then it is impossible for the automaton to distinguish between this and, say, x = x0 p(|x|)/2 in o(p) time (assuming, e.g., p(|x|) = Ω(|x|)). The reason for this is that, since cells are initially completely unaware of their position in the input, the cells with an all-zeroes neighborhood must behave exactly the same. More specifically, we can use an argument as in the proof of Theorem 3 to show that the automaton must behave the same on both x and x (in the sense that it accepts the one if and only if it accepts the other) unless it "looks at the whole input" (i.e., unless it has Ω(p(|x|) time complexity).
The padding technique we use can be traced back to [9]. In a nutshell, we split the input into blocks of the same size that redundantly encode the input length in a locally verifiable way. More importantly, the blocks are numbered from left to right in ascending order, which also allows us to verify that we have the number of blocks that we need. This is crucial in order to ensure the input is "long enough" and the PACA achieves the time complexity that we desire (as a function of the input length).
Proof. Let L ∈ RP be decided by an RP machine R whose running time is upper-bounded by p. Without restriction, we assume p(n) ≥ n. Using standard error reduction in RP, there is then an RP machine R with running time p (n) = Θ(p(n) log h(n)) (i.e., polynomial in n), space complexity at most p(n), and which errs on x ∈ L with probability strictly less than 1/2h(n). Based on L we define the language
L = {bin n (0)#x 0 #0 p(n) % · · · % bin n (h(n) − 1)#x h(n)−1 #0 p(n) | n ∈ N + , x i ∈ L ∩ Σ n },
where bin n (i) denotes the n-bit representation of i < 2 n . Note the length of an instance of L is N ≤ 6h(n)p(n) = O(h(n)poly(n)).
We claim there is c > 0 such that L can be accepted in at most cp (n) = O(p (n)) (and in particular less than T (N )) time by a one-sided error PACA C. The construction is relatively straightforward: We refer to each group of cells bin n (i)#x i #0 p (n) separated by the % symbols as a block and the three binary strings in each block (separated by the # symbols) as its components. First each block a 1 #a 2 #a 3 checks that its components have correct sizes, that is, that |a 1 | = |a 2 | and |a 3 | = p(|a 1 |). Then the block communicates with its right neighbor b 1 #b 2 #b 3 (if it exists) and checks that |a i | = |b i | for every i and that, if a 1 = bin n (j), then b 1 = bin n (j + 1). In addition, the leftmost block checks that its first component is equal to bin n (0); similarly, the rightmost block computes h(n) (in O(p (n)) time) and checks that its first component is equal to bin n (h(n) − 1). Following these initial checks, each block then simulates R (using bits from its random input as needed) on the input given in its second component using its third component as the tape. If R accepts, then all cells in the respective block turn accepting. In addition, the delimiter % is always accepting unless it is a border cell.
Clearly C accepts if and only if its input is correctly formatted and R accepts every one of the x i (conditioned on the coin tosses that are chosen for it by the respective cells of C). Using a union bound, the probability that C errs on an input x ∈ L is
Pr[C(x, U T ×n ) = 0] ≤ h(n)−1 i=0 Pr[R(x i ) = 0] < h(n)−1 i=0 1 2h(n) = 1 2 .
In addition, the total running time of C is the time needed for the syntactic checks (requiring O(p (n)) time), plus the time spent simulating R (again, O(p (n)) time using standard simulation techniques). Hence, we can implement C so that it runs in at most cp (n) time for some constant c > 0, as desired. Now suppose there is a DACA C equivalent to C as in the statement of the theorem. We shall show there is a deterministic (single-tape) Turing machine that decides L with the purported time complexity. Consider namely the machine S which, on an input x ∈ {0, 1} n of L, produces the input x = bin n (0)#x#0 p(n) % · · · % bin n (h(n) − 1)#x#0 p(n) of L and then simulates C on x for T (N ) steps, accepting if and only if C does. Producing x from x requires O(N · poly(n)) time since we need only copy O(n) bits from each block separated by the % delimiters to the next (namely the string x and the number of the previous block). Using the standard simulation of cellular automata by Turing machines, the subsequent simulation of C requires O(N · T (N )) time. Checking whether C accepts or not can be performed in parallel to the simulation and requires no additional time. Hence, the time complexity of S is O(N · poly(n) + N · T (N )) = O(h(n) · poly(n) · T (h(n) · poly(n))).
Further Directions
LLT and two-sided error PACA. Besides giving a separation between one-and two-sided error, Theorem 6 considerably narrows down the position of the class of languages accepted by constant-time two-sided error PACA in the subregular hierarchy. Nevertheless, even though we now know the class is "sandwiched" in-between LLT ∪∩ and LTT, we still do not have a precise characterization for it. It is challenging to tighten the inclusion from Theorem 6 because the strategy we follow relies on closure under complement, but (as we also prove) the class of two-sided error PACA is not closed under complement. It appears that clarifying the relation between said class and LLT ∪∩ as well as LLT ∪∩ itself and LLT ∪ or also LT may give a "hint" on how to proceed.
The general sublinear-time case. Theorem 4 indicates that even polylogarithmic-time PACA can recognize languages for which no deterministic polynomial-time algorithm is currently known. Although the proof of Proposition 21 does yield explicit examples of such languages, they are rather unsatisfactory since in order to accept them we do not need the full capabilities of the PACA model. (In particular, communication between blocks of cells is only required to check certain syntactic properties of the input; once this is done, the blocks operate independently from one another.) It would be very interesting to identify languages where the capabilities of the PACA model are put to more extensive use.
Pseudorandom generators. From the opposite direction, to investigate the limitations of the PACA model, one possibility would be to construct pseudorandom generators (PRGs) that fool sublinear-time PACAs. Informally, such a PRG is a function G : {0, 1} s(n) → {0, 1} r(n) with s(n) r(n) and having the property that a PACA (under given time constraints) is incapable of distinguishing G(x) from uniform when the seed x is chosen uniformly at random. PRGs have found several applications in complexity theory (see, e.g., [21] for an introduction).
Theorem 4 suggests that an unconditional time-efficient derandomization of PACAs is beyond reach of current techniques, so perhaps space-efficient derandomization should be considered instead. Indeed, as a PACA can be simulated by a space-efficient machine (e.g., by adapting the algorithm from [14]), it is possible to recast PRGs that fool space-bounded machines (e.g., [8,16]) as PRGs that fool PACAs. Nevertheless, we may expect to obtain even better constructions by exploiting the locality of PACAs (which space-bounded machines do not suffer from). Figure 4 Comparing the words x = 0 5 1 3 2 2 3 5 ∈ L, y = 0 5 1 2 2 3 3 5 ∈ L, and z = 0 5 1 2 2 2 3 5 / ∈ L, we notice that every infix of length 5 of z appears in either x or y. This implies there is no DACA that accepts L with time complexity 3 or less.
Nevertheless, there is a 3-time one-sided 7/8-error PACA C for L. Checking that the input x = 0 k 1 l 2 m 3 n is such that (x is of the form 0 * 1 * 2 * 3 * and) l, m ≥ 2 can be done without need of randomness simply by looking at the infixes of length 5 of x: Every cell collects the infix m that corresponds to its position in the input and rejects if m is disallowed. (We refer to [10,15,18] for the general method.) This procedure is carried out in parallel to the one we describe next (and a cell turns accepting if and only if it both procedures dictate it to do so). Now we use randomness to check that one of m ≥ 3 and l ≥ 3 holds. In time step 1, every cell exposes its coin toss of step 0 so that its neighbors can read it and use it to choose their state in step 2. Let c σ denote the leftmost cell in which σ ∈ Σ appears, and let l σ and r σ be the coin tosses of the left and right neighbors of c σ , respectively. We have c 1 accept if and only if r 1 = 1, c 3 if and only if l 3 = 1, and c 2 if and only if l 2 + r 2 < 2. All other cells accept regardless of the coin tosses they see (as long as x satisfies the conditions we specified above).
For i ∈ {1, 2, 3}, let A i denote the event of cell c i accepting. The above results in the following behavior: If l = m = 2, we have r 1 = l 2 and r 2 = l 3 since the coin tosses belong to the same cells, in which case C never accepts. If l ≥ 3 and r 2 and l 3 belong to the same cell (i.e., r 2 = l 3 ), then r 1 and l 2 do not belong to the same cell and we have
Theorem 3 .
3The following hold: 1. If C is a one-sided error PACA with time complexity T , then there is an equivalent two-sided error PACA C with time complexity O(T ). 2. There is a language L recognizable by constant-time two-sided error PACA but not by any o( √ n)-time one-sided error PACA.
The element 0, 0
0is the center of the lightcone. The layers of L are indexed by i, where the i-th layer contains 2(m − i) + 1 elements. Hence, the top layer contains 2m + 1 elements and the bottom one 2(m − n) + 1; in particular, the bottom layer is a single element if and only if n = m. There are n i=0 (2(m − i) + 1) = (n + 1)(2m − n + 1) elements in a lightcone in total.
Figure 1
1Space-time diagram of a CA with 16 cells for an initial configuration s. (States have been omitted for simplicity.) The cells marked in red form the 2-neighborhood of cell 3, the ones in blue the 3-lightcone of cell number 11.
Definition 8 (
8Neighborhoods and lightcones). Let C be a CA and n ∈ N + . For i ∈ [n] and r ∈ N 0 , the interval [i − r, i + r] ∩ [n] forms the r-neighborhood of i. For t ∈ N 0 , the t-lightcone of i is the lightcone of radius and height t centered at i in the 0-th row (i.e., the initial configuration) of the space-time diagram of C.3
Every time the counter has looped over all possible values, the [M ] component of the cell is incremented (and the process begins anew with the counter set to all zeroes). When the value of the [M ] component is equal to M − 1 and the counter reaches its final value (i.e., i j = T − 1 for every j), then the cell conserves its current state indefinitely. We agree upon an enumeration M 0 , . . . , M M −1 of the subsets of [m] of size m/2 and identify a value of i in the [M ] component with M i . A cell whose [M ] component is equal to i is then accepting if and only if, for every j ∈ M i , its j-th component is an accepting state of C. Correctness. By construction, if C accepts in a time step where the T -ary counters have the value i 0 · · · i m−1 and the [M ] component the value j, then this is the case if and only if C k accepts in step i k for every k ∈ M j . Hence, C accepts if and only if at least m/2 of the simulated copies of C accept (which, by definition, must occur in a time step prior to T ); that is, C accepts if and only if a majority of the C 0 , . . . , C m−1 accept.
By the Chernoff bound (Theorem 7), this occurs with probability at most 2 −cmε 2 for some constant c > 0, so setting m such that m ≥ log(1/p )/cε 2 completes the proof. It remains open whether a similar result holds for general (i.e., non-constant-time) twosided error PACA. Generalizing our proof of Proposition 13 would require at the very least a construction for intersecting non-constant-time PACA languages. (Note we do show closure under intersection for the constant-time languages later in Proposition 20.) If such a construction were to be known, then extending the idea above one could use that the union of constantly many T -time PACA languages can be recognized in O(T ) time (as we prove later in Proposition 20) and represent the majority over m PACA languages L 0 , . . . , L m−1 as the union over all possible intersections of m/2 many L i . Note that closure under intersection is open in the deterministic setting (i.e., of DACA) as well[15].
Theorem 3 .
3The following hold: 1. If C is a one-sided error PACA with time complexity T , then there is an equivalent two-sided error PACA C with time complexity O(T ).
Lemma 15 .
15Let C be a T -time (one-or two-sided error) PACA for T ∈ N + , and let x ∈ L(C) ∩ Σ n . In addition, let t ∈ [T ] be a time step in which x is accepted by C with non-zero probability. Then there are 2 O(T 2 ) cells that are critical for x in step t. It follows there are T · 2 O(T 2 ) = 2 O(T 2 ) critical cells for x in total (i.e., all such time steps comprised).
PACA Theorem 5 .
5For any constant-time one-sided error PACA C, there is a constant-time DACA C such that L(C) = L(C ).
2 .
2The cell then determines whether it is critical itself in step t of C. If this is not the case, then it becomes accepting if and only if it is accepting in step t of C (regardless of the random input).3.Otherwise i is critical in step t. Let B i ⊆ [n] be the subset of cells that results from the following sequence of operations:a. Initialize B i to {i}. b. For every cell j ∈ B i , add to B i every k ∈ K such that |j − k| ≤ 2(T − 1). c. Repeatstep 2 until a fixpoint is reached. (This necessarily terminates since there are at most M critical cells in N and we checked the upper bound of M previously.) By choice of r, we have that |i − j| ≤ 2(M − 1)(T − 1)for every j ∈ B i . In particular, we have that the t-neighborhood of every j ∈ B i is completely contained in N , which means cell i is capable of determining B i .6 The cell i then accepts if and only if there is a setting of random bits in the lightcone L i of radius r and height t centered at i that causes every cell in B i to accept in step t of C.
Figure 3
3Placement of LLT in the subregular hierarchy. Arrows indicate inclusion relations; all are known to be strict except for the marked one (i.e., LLT∪ ⊆ LLT∪∩). Dashed lines between two classes denote they are incomparable.
Proposition 18 .
18The following hold (seeFigure 3):1. SLT LLT LLT ∪ ⊆ LLT ∪∩ LTT and SLT ∪ LLT ∪ . 2.LLT and LT as well as LLT ∪ and LT are incomparable.3. LTT equals the Boolean closure of LLT.Some relations between the classes are still open (seeFigure 3)and are left as a topic for future work.
Theorem 4 .
4Let d ≥ 1. The following hold: If there is ε > 0 such that every n ε -time (one-or two-sided error) PACA can be converted into an equivalent n d -time deterministic CA, then P = RP. If every polylog(n)-time PACA can be converted into an equivalent n d -time deterministic CA, then, for every ε > 0, RP ⊆ TIME[2 n ε ]. If there is b > 2 so that any (log n) b -time PACA can be converted into an equivalent n d -time deterministic CA, then, for every a ≥ 1 and c > a/(b − 1), RTIME[n a ] ⊆ TIME[2 O(n c ) ].
Pr[C(x, U T ×|x| ) = 1] = Pr[A 1 ] Pr[A 2 ∧ A 3 ] = Pr[r 1 = 1] Pr[l 2 = 0 ∧ r 2 = l 3 The case m ≥ 3 and r 1 and l 2 belonging to the same cell is similar. Finally, if l ≥ 3 and m ≥ 3, the values r 1 , l 2 , r 2 , and l 3 are all independent and we have Pr[C(x, U T ×|x| ) = 1] = 3 i=1 Pr[A i ] = Pr[r 1 = 1] Pr[l 2 + r 2 < 2] Pr[l 3 = 1] > 1 8 .
Theorem 4 .
4Let d ≥ 1. The following hold: If there is ε > 0 such that every n ε -time (one-or two-sided error) PACA can be converted into an equivalent n d -time deterministic CA, then P = RP. If every polylog(n)-time PACA can be converted into an equivalent n d -time deterministic CA, then, for every ε > 0, RP ⊆ TIME[2 n ε ]. If there is b > 2 so that any (log n) b -time PACA can be converted into an equivalent n d -time deterministic CA, then, for every a ≥ 1 and c > a/
Unfortunately, the literature uses the terms stochastic and probabilistic CA interchangeably. We deem "probabilistic" more suitable since it is intended as a CA version of a probabilistic Turing machine.
Some sources distinguish between future and past lightcones. Here we shall only need past lightcones.
If the lightcone's dimensions overstep the boundaries of the space-time diagram (i.e., i is too close to either of the borders of C (e.g., i < t)), then some cells in the t-lightcone will have undefined states. In this case, we set the undefined states to $, which ensures consistency with δ.
The number of rows of R is dependent on the choice of T . This is not an issue here since any superficial rows are ignored by C; that is, without restriction we may take T to be such that every value T (n) is minimal and set the number of rows of R to T (n). The motivation for letting R be larger is that, when simulating a PACA, it may be the case that it is more convenient (or even possible) to compute only an upper bound T (n) ≥ T (n) instead of the actual minimal value T (n).
In the same manner as we do for the indices of a word, we number the components starting with zero.
Note it is not necessary for i to be aware of the actual numbers of the cells in B i ; it suffices for it to compute their positions relative to itself. For example, if i = 5 and B i = {4, 5}, then it suffices for i to regard j = 4 as cell −1 (relative to itself). Hence, by "determining B i " here we mean that i computes only these relative positions (and not the absolute ones, which would be impossible to achieve in only constant time).
T (6h(n)p(n)) ≥ cp (n).
No DACA C accepts L in at most 3 steps. This can be shown using methods from[10,15,18]. Given a DACA C with time complexity 3, we can determine if C accepts a word x ∈ Σ * by looking only at the infixes of length 5 (and the prefix and suffix of length 4) of x. Consider the words x = 0 5 1 3 2 2 3 5 ∈ L, y = 0 5 1 2 2 3 3 5 ∈ L, and z = 0 5 1 2 2 2 3 5 / ∈ L(Figure 4). Then every infix of length 5 (and the prefix and suffix of length 4) of z appears in x except for the infixes 00112 and 01122, which both appear in y. It follows that, if x, y ∈ L(C), then C must also accept z, which proves there is no DACA for L with time complexity (at most) 3.
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Circuits and Local Computation. Andrew Chi-Chih Yao, 10.1145/73007.73025doi: 10.1145/ 73007.73025. REFERENCES XX:27Proceedings of the 21st Annual ACM Symposium on Theory of Computing. David S. Johnsonthe 21st Annual ACM Symposium on Theory of ComputingSeattle, Washington, USAACMAndrew Chi-Chih Yao. "Circuits and Local Computation." In: Proceedings of the 21st Annual ACM Symposium on Theory of Computing, May 14-17, 1989, Seattle, Washington, USA. Ed. by David S. Johnson. ACM, 1989, pp. 186-196. doi: 10.1145/ 73007.73025. REFERENCES XX:27
| [] |
[
"Control of protein-based pattern formation via guiding cues",
"Control of protein-based pattern formation via guiding cues"
] | [
"Tom Burkart \nDepartment of Physics\nArnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS)\nLudwig-Maximilians-Universität München\nTheresienstraße 37D-80333MunichGermany\n",
"Manon C Wigbers \nDepartment of Physics\nArnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS)\nLudwig-Maximilians-Universität München\nTheresienstraße 37D-80333MunichGermany\n",
"Laeschkir Würthner \nDepartment of Physics\nArnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS)\nLudwig-Maximilians-Universität München\nTheresienstraße 37D-80333MunichGermany\n",
"Erwin Frey \nDepartment of Physics\nArnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS)\nLudwig-Maximilians-Universität München\nTheresienstraße 37D-80333MunichGermany\n\nMax Planck School Matter to Life\nHofgartenstraße 8D-80539MunichGermany\n"
] | [
"Department of Physics\nArnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS)\nLudwig-Maximilians-Universität München\nTheresienstraße 37D-80333MunichGermany",
"Department of Physics\nArnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS)\nLudwig-Maximilians-Universität München\nTheresienstraße 37D-80333MunichGermany",
"Department of Physics\nArnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS)\nLudwig-Maximilians-Universität München\nTheresienstraße 37D-80333MunichGermany",
"Department of Physics\nArnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS)\nLudwig-Maximilians-Universität München\nTheresienstraße 37D-80333MunichGermany",
"Max Planck School Matter to Life\nHofgartenstraße 8D-80539MunichGermany"
] | [] | Proteins control many vital functions in living cells, such as cell growth and cell division. Reliable coordination of these functions requires the spatial and temporal organizaton of proteins inside cells, which encodes information about the cell's geometry and the cell-cycle stage. Such protein patterns arise from protein transport and reaction kinetics, and they can be controlled by various guiding cues within the cell. Here, we review how protein patterns are guided by cell size and shape, by other protein patterns that act as templates, and by the mechanical properties of the cell. The basic mechanisms of guided pattern formation are elucidated with reference to recent observations in various biological model organisms. We posit that understanding the controlled formation of protein patterns in cells will be an essential part of understanding information processing in living systems. | 10.1038/s42254-022-00461-3 | [
"https://arxiv.org/pdf/2202.10091v1.pdf"
] | 246,842,895 | 2202.10091 | dd1d6ce25fc1b23fe3f847eb6e8247cea9de0600 |
Control of protein-based pattern formation via guiding cues
(Dated: February 11, 2022)
Tom Burkart
Department of Physics
Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS)
Ludwig-Maximilians-Universität München
Theresienstraße 37D-80333MunichGermany
Manon C Wigbers
Department of Physics
Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS)
Ludwig-Maximilians-Universität München
Theresienstraße 37D-80333MunichGermany
Laeschkir Würthner
Department of Physics
Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS)
Ludwig-Maximilians-Universität München
Theresienstraße 37D-80333MunichGermany
Erwin Frey
Department of Physics
Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS)
Ludwig-Maximilians-Universität München
Theresienstraße 37D-80333MunichGermany
Max Planck School Matter to Life
Hofgartenstraße 8D-80539MunichGermany
Control of protein-based pattern formation via guiding cues
(Dated: February 11, 2022)
Proteins control many vital functions in living cells, such as cell growth and cell division. Reliable coordination of these functions requires the spatial and temporal organizaton of proteins inside cells, which encodes information about the cell's geometry and the cell-cycle stage. Such protein patterns arise from protein transport and reaction kinetics, and they can be controlled by various guiding cues within the cell. Here, we review how protein patterns are guided by cell size and shape, by other protein patterns that act as templates, and by the mechanical properties of the cell. The basic mechanisms of guided pattern formation are elucidated with reference to recent observations in various biological model organisms. We posit that understanding the controlled formation of protein patterns in cells will be an essential part of understanding information processing in living systems.
I. INTRODUCTION
To ensure their survival, cells must tightly regulate a wide range of cellular functions, such as cell migration, cell growth, DNA synthesis, and cell division. For example, in order to produce two viable daughter cells, a cell must precisely coordinate cell growth with the duplication and segregation of DNA, and with subsequent cell division. These cellular functions, in turn, are controlled and coordinated by proteins. Robust timing and reliable control of these functions requires cells to process spatiotemporal information, such as information about cell size and shape, cell cycle state, the cell's surroundings, and the current state of other cellular processes. Such spatiotemporal information is encoded in protein patterns -i.e., an inhomogeneous spatial distribution of proteinsthat regulate these cellular functions, whereby each type of protein may perform distinct tasks.
How then are proteins spatially and temporally organized in a cell? The idea that the collective organization of interacting chemicals (chemical reactions) in an initially homogeneous medium can give rise to spatial patterns dates back to Turing's seminal work on spontaneous pattern formation in reaction-diffusion systems [1]. While this work has greatly advanced the understanding of pattern formation in biological systems, many aspects of protein patterns such as their positioning, timing, reliability, and controllability -which are essential for the viability of living organisms -remain poorly understood. Since protein patterns in cells serve a timed and targeted functional purpose, they must form in response to certain signals and control mechanisms rather than spontaneously emerging from an initially homogeneous distribution. Indeed, an increasing number of theoretical and experimental studies find that protein distributions can * These authors contributed equally. † [email protected] respond and adapt to cell shape, size, and mechanics, as well as to signals encoded in previously established protein patterns [2][3][4][5][6][7][8]. This response is, in fact, bidirectional. Cells are not static objects but rather an active material whose size, shape, and mechanical properties can be altered dynamically through protein interactions in response to the cell's environment and the current state of the cell cycle [9][10][11][12]. These dynamic interactions between protein patterns and cell architecture are the subject of a rapidly developing field of study at the interface between cell biology and theoretical physics that benefits from constantly improving experimental techniques, as well as insights from physics that allow one to model and understand the guided organization of proteins into patterns.
In this review, we summarize recent advances in our understanding of how protein patterns are controlled by geometric, mechanical, and biochemical cues. The basics of pattern formation will only be summarized briefly, as recent reviews have provided a comprehensive introduction to this subject. The interested reader is referred to an elementary course on the mathematical tools that are required to study the physics of protein interactions and pattern formation, in particular ordinary differential equations (ODEs) and nonlinear dynamics [13]. For an introduction to the theory of pattern-forming systems, we direct the reader to pertinent textbooks [14,15], and to lecture notes for a review on quantitative modeling of pattern formation in mass-conserving systems [16]. Other recent reviews have focused on the theory of two specific aspects of pattern formation, namely the role of bistability for polarity [17] and the curvature-generating properties of proteins [18]. The relevance of protein patterns for cells has also been reviewed from a more biological perspective recently [19], in particular with respect to midcell localization [20], and current advances in understanding pattern formation at a molecular level [21] have been reviewed recently. We also want to highlight three recent reviews that emphasize the importance and role of modeling for understanding cell polarity [22,23] and biological phenomena in general [24].
Here, we discuss several theoretical models that have been developed with a view to reproducing and accounting for pattern guidance, together with examples of well-studied biological model organisms in which pattern guidance has been observed to play a critical role in cell viability. In particular, we discuss how biophysical theory has been instrumental in clarifying the underlying physical concepts of pattern guidance in living cells. We start by giving an overview of the predominant types of protein transport and chemical reactions that are predominately involved in the formation of patterns in cells. We then discuss how these factors can be affected by cell shape and size, pre-existing protein patterns, and cell mechanics, and how these cues guide and control protein pattern formation. We conclude with an outlook on the future research directions in this field.
II. BASIC PRINCIPLES OF PATTERN FORMATION
Protein patterns arise from the interplay of biochemical reaction kinetics with different types of transport mechanisms. While the amounts of locally available proteins are regulated by chemical reactions, their spatial distribution is altered by transport processes including diffusion, active transport and fluid flow (see Fig. 1). Some of the most important reaction and transport processes involved are presented in the following.
A. Protein reaction networks
Protein reaction networks differ in their degree of complexity, e.g., with respect to the number of different proteins and their conformations, as well as the number and type of reactions between them. Some of the most common types relevant to protein pattern formation are briefly discussed in the following.
Conformational state changes -The intracellular organization of proteins is largely controlled by protein reaction networks that contain nucleoside triphosphate a (NTP)-dependent regulatory modules. In prokaryotic cells, P-loop (phosphate binding loop) ATPases b such as ParA and MinD take on this role, and give rise to self-organized dynamic patterns at cellular interfaces -ParA on the nucleoid and MinD on the cell membrane [20,25,26]. Similarly, small GTPases like Cdc42 a Nucleoside tri-/diphosphate (NTP/NDP) -Nucleotide molecules with three (two) phosphate groups typically based on guanine (GTP), adenine (ATP) or cytosine (CTP), forming the main carriers of chemical energy in cells. b NTPase -Enzymes that bind to NTP and hydrolize it to NDP, thereby releasing energy.
and RhoA play an important role in establishing cell polarity in eukaryotic cells [27][28][29]. Basically, all these proteins serve as molecular switches that can cycle between an active and inactive state based on nucleotide binding and delayed hydrolysis, typically regulated by auxiliary proteins [30][31][32] (Fig. 1a). Similarly, proteins that are not NTPases can act as molecular switches if cycling between active and inactive states (phosphorylation c and dephosphorylation) is catalyzed by separate kinases and phosphatases, respectively [33,34]. These cycles have two key features. First, they are non-equilibrium processes driven by the supply of chemical energy, e.g. through ATP hydrolysis [35]. As such, they are the core element of most protein reaction networks, enabling them to drive selforganization processes. Secondly, the switch between active and inactive states is associated with changes in their affinity for targets such as the cell membrane and the nucleoid [35,36], as well as their specific binding affinity for other proteins or lipids. For example, MinD can only bind to the cell membrane in its ATP-bound, dimeric form and is released into the cytosol as an ADP-bound monomer upon ATP hydrolysis [37].
Binding and unbinding reactions -Many proteins can bind to different substrates in a cell, such as membranes. Typical residence times of proteins on membranes range from seconds to minutes [4,38,39]. In several biological model systems, the nonlinear binding kinetics of proteins to membranes plays a key role in the formation of spatiotemporal protein patterns.
One way to confer nonlinear binding kinetics is through limitation of binding sites on the membrane, which leads to saturated binding kinetics [40]. Another example is cooperative reactions that amplify or attenuate the attachment and detachment of other proteins to the membrane [41][42][43][44] (Fig. 1a). These feedback mechanisms were shown to be an integral part of the patterning mechanisms in the most important model organisms: In the MinDE system of E. coli, pole-to-pole oscillations of the Min proteins rely on recruitment of cytosolic MinD and MinE by membrane-bound, active MinD (positive feedback) and their release into the cytosol through MinE-induced hydrolysis and concomitant inactivation of MinD (negative feedback) [37,[45][46][47]. In budding yeast (S. cervisiae), the establishment of cell polarity via asymmetric distribution of Cdc42 involves multiple positive and negative feedback loops, which provide a high degree of robustness [32,40,48,49]. Finally, the PAR polarity system in the early C. elegans embryo exploits various antagonistic reactions that play a key role in specifying the correct orientation of the polarity axis [34,[50][51][52].
Complex formation -Proteins can also form oligomers d , in particular dimers (Fig. 1a). This can have Reaction and transport processes involved in pattern formation: (a) Protein reactions include binding to and detachment from the cell membrane or other intracellular structures, as well as conformational state changes due to (de-)phosphorylation or nucleotide exchange. Cooperative and antagonistic (nonlinear) reactions between multiple proteins can lead to assisted attachment (recruitment) or to detachment from the membrane. Multiple monomers can form oligomers with altered transport and reaction properties. (b) Proteins can be transported by diffusion (Dc, Dm, black arrows) and advection (vc, vm, pink) independently on surfaces -in particular cell cortex and membrane -and in the cytosol. In addition, directed protein transport can be established by subunit addition and disassembly of polymers, resulting in treadmilling of monomers, and by active transport along filamentous structures, mediated by energy-consuming motor proteins.
an impact on their ability to bind to cellular surfaces, as described above for active MinD dimers. The formation of higher-order protein aggregates leads to a change in Péclet number (see below), which in turn alters how they are affected by fluid flow as opposed to diffusion. Such an effect has been suggested to play a role in the transport of PAR-3 proteins in the C. elegans embryo. Here, diffusive transport may dominate for PAR-3 monomers (Pe < 1), whereas transport becomes dominated by flow (Pe > 1) upon cell-cycle-dependent aggregation of PAR-3 into complexes together with two other proteins -PAR-6 and aPKC [53]. Yet another process is the formation of higher-order oligomers, such as those observed for membrane-bound MinD [44,46]. Similar to the nonlinear attachment kinetics discussed above, cooperative reactions have also been suggested to participate in protein complex formation, potentially allowing for feedback loops [34].
Theory -Mathematically, the dynamics of well-mixed protein reaction networks are described by sets of coupled nonlinear differential equations for the concentraor a different type (homo-and hetero-oligomers, respectively).
tions u i (t) of each of the different protein types and conformations i ∈ {1, . . . , S},
∂ t u i (t) = f i ({u i }) .(1)
In such chemical rate equations, the nonlinear reaction terms f i (together with the reaction rates) must be inferred from the underlying reaction network using the law of mass action. An elaborate mathematical theory, called dynamic system theory, allows one to analyze systems of coupled nonlinear ordinary differential equations (ODEs). The basic idea of this theory, which goes back to the pioneering work of Poincaré [54], is to characterize the system dynamics in terms of certain geometric structures in the phase space spanned by the set of dynamical variables u i (t) [13,14].
Of particular interest are the asymptotic dynamics of the system over large time scales, which are characterized by the attractors in phase space within the framework of dynamic system theory. These include fixed points corresponding to reactive equilibria (see Supplementary Information), limit cycles corresponding to nonlinear oscillators, and more intricate geometric objects [13,14]. Importantly, the local properties of the fixed points (reactive equilibria), in particular their stability, can be determined using ODEs linearized around these fixed points [13,14].
B. Protein transport
Transport mechanisms play a crucial role in the control of spatial variations in protein concentration. In the following, we provide an overview of the most important modes of intracellular protein transport involved in pattern formation (Fig. 1b).
Diffusion -Perhaps the most basic means of protein transport is diffusion. It is a consequence of Brownian motion and is directed from regions of high to regions of low protein concentration u(x, t) with a diffusive current −D∇u(x, t) (Fick's law). For spherical particles of radius r, the diffusion constant is given by the Stokes-Einstein relation D = k B T /(6πηr), where η is the viscosity of the surrounding cytosol [55]; a qualitatively similar relation holds for transmembrane proteins [56,57]. This implies that the diffusive transport of proteins depends on their size and on the local properties of the surrounding medium.
Importantly, both the membrane and the cytoplasm e are highly heterogeneous environments crowded with macromolecular structures that interact with proteins, for example by temporarily binding or by taking up space [58]. For the purpose of studying pattern formation, however, one often disregards inhomogeneities and instead assumes an effective diffusion constant that takes into account such interactions that are not explicitly modeled. Hence, the diffusion constant is a mesoscopic quantity representing the mobility of proteins in a homogeneous, dilute fluid environment. In essence, the complex cytoplasmic environment is reduced to an effective cytosol for many applications in protein pattern formation, and similarly, the heterogeneous membrane is considered as an effective (dilute) fluid [59]. This simplification is justified since the length scale of patterns is typically larger than the length scale of heterogeneities in the cytoplasm or on the membrane, to which we will refer as substrates in the following. As a rough estimate, the diffusion coefficients of membrane-bound proteins are generically at least two orders of magnitude lower than those of their cytosolic counterparts: While characteristic values for membrane diffusion are D m ∼ 0.01 µm 2 /s, one observes D c ∼ 10 µm 2 /s in the cytosol [60]. Although the models discussed in this review suggest that the heterogeneous character of the cellular substrates are of minor importance for protein pattern formation, it would be interesting to explicitly probe the robustness of these models against more realistic substrates. For ex-ample, this could be incorporated into models via timeand space-dependent diffusion constants.
Active transport -Proteins can also be transported via active processes driven by the chemical energy of ATP, GTP or CTP at the molecular level. Of particular biological relevance are translational molecular motors f [61][62][63]. An important subclass of these motors is comprised of kinesins and dyneins that bind to, and 'walk' on microtubules g . In this way, cargo -such as other proteinscan be transported along the microtubules [61,62]. Depending on the type of motor and, in some cases, other factors such as external forces [64], this form of active transport is directed to either the plus or minus end of the microtubules [65]. Certain classes of myosin motors perform similar tasks by transporting cargo along actin filaments. Such active cargo transport is known to be involved in the polarization process of budding yeast. Here, the actin filaments are anchored to the polarity site, so that the myosin motors can deliver protein-coated vesicles towards the polarity site [66,67].
Another class of active transport processes is mediated by the directed polymerization of cytoskeletal filaments such as F-actin [68] and microtubules [69], which is driven by ATP and GTP hydrolysis, respectively. For instance, tubulin-like FtsZ filaments are particularly important active structures in bacterial cell division. These filaments exhibit treadmilling dynamics (see the segmented structure in Fig. 1b), as FtsZ monomers can only bind to the plus end and detach from the minus end [70,71]. By consuming GTP, this treadmilling allows FtsZ filaments to translocate directionally along the cell membrane, coordinating the activity of downstream cell division processes [72]. Similarly, treadmilling of actin filaments was shown to play a key role in cell migration, in particular for the extrusion of lamellipodia [73].
Both in vivo and in vitro experiments have shown how important these active transport processes are for the polarization of cells [74][75][76][77][78]. For example, during cell growth in fission yeast microtubules are aligned along the long axis of the cell, and direct the active transport of the tip factors Tea1 and Tea4 towards the cell poles in a two-fold manner [78][79][80][81]: The kinesin-like motor Tea2 mediates the transport of Tea1/Tea4 complexes along microtubules that emanate from the nucleus [82,83]. In addition, these complexes bind to microtubule tips assisted by Mal3, a tip-binding protein. Therefore, due to the directed microtubule polymerization along the long cell axis, the tip factors are transported to the cell poles [83]. At the poles, they then serve as a spatial cue for cell growth, and therefore facilitate the elongation of the cell along its long axis [84].
f Molecular motors -Enzymes that use energy released by NTP hydrolysis to perform mechanical work and that are generally associated with cytoskeletal filaments. g Microtubules and actin filaments -Protein filaments comprised of tubulin and actin proteins, respectively, which form an integral part of the cytoskeleton.
Advective transport -In the fluid environment of a cell, proteins can also be transported by cytoplasmic [85,86], cortical [51], and membrane flows [87,88], whose effect on protein transport through friction strongly depends -like diffusion -on the viscosity of the respective environment. An important force-generating active structure is the actin cortex h . In addition to actin filaments, it includes cross-linker proteins and myosin motors that cause cortical contractions which, in turn, can induce flows [89,90]. The cortical contractions that occur in the C. elegans zygote are a prominent example [4,91,92]. Here, local depletion of the concentration of the motor protein myosin at the cell cortex leads to a gradient of contractile stress, such that the cell cortex flows from the anterior to the posterior pole [93].
Cortical contractions can also lead to flows in the cytoplasm or membrane due to hydrodynamic coupling between membrane, cortex and cytoplasm [92]. In addition, they can also induce cell-shape changes that lead to flows in the cytoplasm. For example, surface contraction waves during the maturation of starfish oocytes have recently been shown to induce such flows [94,95]. Similarly, shape changes resulting from blebbing incidents coincide with intracellular flows [96].
The Péclet number -The relative impact of diffusion and flow on protein transport is quantified by the Péclet number Pe = ξ·v/D, where v is the typical protein advection velocity and ξ a characteristic length scale. Large values of the Péclet number correspond to protein transport that is dominated by flow rather than diffusion. Hence, small proteins with large diffusion constants are less affected by flow than large proteins or protein assemblies. In addition, the detailed chemical interactions of proteins with other biomolecules and cellular structures can affect the effective diffusivity and advection velocity [97]. As for diffusive transport, the advection velocity -and hence the Péclet number -is a mesoscopic quantity that disregards the heterogeneous structure of the environment. This approximation is justified since variations in the mobility coefficients within a given substrate are usually much smaller than the variations between different substrates, such as the cytoplasm and the membrane. In general, a protein that diffuses in the cytoplasm is less affected by flows than it is when bound to the more viscous membrane.
Theory -The spatiotemporal transport of, and reactions between proteins are mathematically described by nonlinear partial differential equations (PDEs) [16]. The protein dynamics in terms of their cytosolic (volume) concentrations c(r, t) and membrane (area) concentrations m(σ σ σ, t) generally take the form of general transport h Actin cortex -Thin and dynamic network that acts as a scaffold that determines the cell's shape and which is comprised of actin filaments, motor proteins, and other associated proteins.
equations with flux and source terms
∂ t c(r, t) = −∇ · J c + f cyt (c) ,(2)∂ t m(σ σ σ, t) = −∇ S · J m + f mem (m, c| S ) ,(3)
which represent a broad and general class of interesting dynamic systems far from thermodynamic equilibrium. The divergence of the cytosolic and membrane fluxes J c/m accounts for the (mass-conserving) spatial transport of proteins, and generally contains both diffusive and advective contributions. Here ∇ S denotes the covariant derivative for the curvilinear coordinates σ σ σ ∈ S on the membrane surface S. The membrane is often considered as a static object for simplicity, however models can in general be extended to dynamic surfaces. In particular, this requires to extend the dynamics by an explicit expression for the time evolution of the membrane geometry, S → S(t) [18,[98][99][100][101][102]. The source terms f cyt and f mem result from the chemical reactions of the underlying protein networks, as discussed above. Note that membrane-bound proteins not only react with each other, but membrane reactions also involve interactions with cytosolic proteins in close proximity to the membrane (c| S ). The set of nonlinear PDEs (Eqs. (2) and (3)) is closed by reactive boundary conditions at the membrane
J c ·n| S = g(m, c| S ) ,(4)
which ensures local mass conservation: cytosolic fluxes normal to the membrane (n denotes the outward normal vector) must be balanced by reactive fluxes g(m, c| S ) at the membrane [16]. An additional constraint for many models of protein pattern formation is the global conservation of protein mass, i.e., the assumption that no proteins are produced or degraded on the time scale of pattern formation. This assumption is violated on longer time scales, where protein production and degradation processes -in particular gene expression -need to be taken into account [23].
C. Lateral instabilities and trigger waves
This set of general transport equations provides the theoretical framework for studying the spatiotemporal dynamics of protein patterns. The interested reader may consult recent lecture notes [16] for an introduction to their analysis. Here, to conclude our introduction to the basic principles of pattern formation, we briefly introduce two particularly interesting phenomena: pattern-forming instabilities and trigger waves.
A pattern-forming instability arises when a spatially uniform steady state becomes unstable against spatially inhomogeneous perturbations (Fig. 2d). One example of such a pattern-forming instability is a massredistribution instability (see Supplementary Information), which amplifies spatial variations in protein number, thus leading to a protein concentration pattern [103].
The dynamics and length scale of these patterns on short time scales are determined by the growth rate and wavelength of the unstable modes, termed dispersion relation (see Supplementary Information). The growth rate of the unstable modes depends on the specific reaction kinetics and transport properties of the dynamics. The wavelength of the fastest growing unstable mode determines the characteristic length scale of the initially growing pattern. While the initial pattern is dominated by the dynamics of the unstable modes, the dynamics on longer timescales may be dominated by other processes, such as coarsening [104] and non-linear interactions of the unstable modes far away from the linear regime.
In addition, nonlinear protein reaction kinetics can give rise to several reactive equilibria at the same total protein concentration, which is a necessary requirement for trigger waves. This phenomenon is best exemplified by systems that show bistability (see Supplementary Information) [105]. In this case, the system can be at different reactive equilibria at different regions in the cell, giving rise to front-like protein activity patterns. Such front-like patterns propagate with a finite velocity, whose magnitude and sign depend on the details of the reaction kinetics [94,106]. This propagation is constrained by the limited abundance of proteins, which can result in localized wave fronts in cells [107][108][109]. Moreover, unstable reactive equilibria can give rise to spatially homogeneous oscillations and traveling spiral waves [103,110,111].
The spatiotemporal properties of these patterns, such as the orientation of static patterns or the direction of propagating wave fronts, need to be controlled tightly by the cell. This is achieved with the aid of guiding cues. In the following, we will discuss the most prominent types of guiding cues observed to play a role in pattern formation processes in cells.
III. GEOMETRIC GUIDING CUES
On the largest scales, cells are characterized by their size and shape, which together confine protein transport and protein reaction kinetics.
A. Cell size controls protein patterns
Experimental studies show that, in addition to reaction and transport properties of the cell, also the cell size affects protein patterns. Examples include the transition from pole-to-pole oscillatory patterns to stripe patterns of MinD in filamentous E. coli cells [112,113], and the observation that the PAR proteins in C. elegans fail to polarize in small cells [8].
Bulk-boundary-ratio.-On the time scale of pattern formation and dynamics, the total concentration of proteins remains constant. As a consequence of these resource limitations, protein concentrations on the membrane and in the cytosol will in general depend on the ratio of membrane area to cell volume. Moreover, the number and stability of reactive equilibria, as well as pattern-forming instabilities, are controlled by the total concentration of proteins (see Supplementary Information), and variations in cell size can therefore qualitatively affect protein patterns. To understand the underlying idea, we assume for simplicity that the concentrations of cytosolic proteins c and membrane-bound proteins m, respectively, are uniformly distributed. The total number of proteins N is then given by N = S · m + V · c, where S and V denote the membrane (surface) area and the cytosolic (bulk) volume, respectively (Fig. 2a). Rewriting this mass-conservation relation in terms of the total protein density ρ = N/V, one obtains ρ = S/V · m + c. Thus, the protein concentrations on the membrane and in the cytosol depend on the ratio of membrane to volume S/V; for example, for a spherical cell with radius R, one finds ρ = 3 m/R + c.
Cytosolic protein gradients.-Because the proteins of interest here are not permanently fixed to either the membrane or the cytosol, but circulate between these compartments due to various chemical processes such as membrane detachment, attachment, and recruitment, the cell membrane effectively acts both as a source and sink for cytosolic proteins. These chemical reactions need to be balanced by diffusive fluxes in the cytosol, otherwise local mass conservation would be violated. Hence, on these very general grounds, spatial gradients in the cytosolic protein density must be assumed [16,103]. Strikingly, these gradients generally do not equilibrate over time, but are maintained by an interplay between diffusion and non-equilibrium reaction kinetics (see Supplementary Information).
Indeed, a good example is the case where proteins in the cytosol can have two different conformations, an inactive and an active state. Only proteins in the active state are able to bind to the membrane, and they typically undergo a conformational change to the inactive state upon detachment from the membrane (Fig. 2b). In the cytosol, inactive proteins can switch back to the active state with a rate λ. This reactivation step requires the consumption of energy and is a generic feature in NT-Pase or phosphorylation/dephosphorylation cycles [30][31][32]. Since detached proteins cannot immediately bind to the membrane again, a protein concentration gradient may form in the cytosol [114,115]. The penetration depth of this gradient depends on the cytosolic diffusion constant D c and the reactivation rate λ, and is given by = D c /λ [2]. If the cell size is much smaller than this penetration depth, the cytosolic protein concentration is effectively nearly homogeneous throughout the cell. Conversely, if the cell is much larger than the penetration depth, protein gradients can be established in the cytosol (Fig. 2b). The presence of such cytosolic gradients can fundamentally affect the formation of patterns on the membrane [103,116,117]. This is well exemplified in the E.coli Min system, which shows standing wave patterns in vivo, but -strikingly -produces traveling and spiral wave patterns, among others, in reconstituted in vitro assays with large bulk volume [41,113,118].
Finite size effects.-In addition, cell size can affect pattern-forming instabilities. A pattern-forming instability arises when a spatially uniform steady state is unstable against spatially inhomogeneous perturbations (Fig. 2d). Due to the finite size of the cell, only particular unstable modes can grow, where the largest possible wavelength is constrained by the lateral length of the cell. Thus, while a reaction network can lead to a patternforming instability in large cells, it may result in a stable and spatially uniform steady state or a weak gradient in small cells (Fig. 2c,d). Indeed, this has been observed for the polarity pattern of PAR proteins in C. elegans (Fig. 2c) [8]. Similarly, cell size may not only limit the existence of a pattern, but also the type of protein pattern that can be established.
B. Cell shape and curvature sensing
For a wide range of cells, from bacteria [112,119,120] to migrating fibroblasts [121] to unicellular eukaryotes [122] and large zygotes [93], cell shape and local membrane curvature serve as important guiding cues for protein attachment to the membrane. The mechanisms underlying such curvature detection are based on the interaction of proteins with the membrane, in particular its membrane binding affinity (curvature-sensing proteins), and the probability that a protein will make contact with the membrane (collective curvature sensing). Both factors can be affected by cell shape (membrane curvature).
Curvature-sensing proteins
One prominent set of proteins that can individually sense membrane curvature are proteins containing a curved BAR domain i [123][124][125][126]. These proteins prefi BAR domain -A curved protein domain that binds to curved membranes, named after three proteins that contain this domain: erentially bind to membrane regions that have a curvature comparable to that of the BAR domain itself (Fig. 2e). For example, during persistent cell motion, the curvature-sensitive protein BAIAP2, which contains such a BAR domain, accumulates at curved membrane patches at the cell front, inducing the formation of lamellipodia [121]. Since BAR domains have a length of about 20 nm, the sensitivity of individual proteins to weakly curved surfaces is limited [124,127]. However, membrane curvature can facilitate the oligomerization of proteins into extended curved structures, which are capable of sensing membrane curvature on length scales larger than that of the individual protein [128]. Other important examples for such joint curvature sensing are dynamin, which forms helical collars around the thin neck during budding in yeast [129,130], and MreB, which assembles into filaments that orient along the highest membrane curvature [131,132]. Furthermore, some proteins recognize membrane curvature via defects in membrane structure. This mechanism is well exemplified by proteins with so-called ALPS motifs. ALPS motifs do not have a defined structure in solution, but insert into lipid bilayers by folding into an α-helix j . It has been shown that ALPS motifs bind preferably to regions with low lipid packing density [133]. Such low-density packing can arise from membrane curvature, where one sheet of the lipid bilayer is stretched compared to a flat membrane. In experiments, ALPS motifs were found to bind strongly to liposomes with sufficiently strong positive curvature (R < 50 nm), and to weakly curved liposomes with a high concentration of conically shaped lipids [133]. Thus, curvature-dependent binding affinity can lead to predominant accumulation of proteins at curved membrane regions.
It has been reported that proteins that sense curvature can also deform the membrane: The helical structure of dynamin oligomers induces membrane curvature during scission of the yeast bud [129,134,135]. Proteins with BAR domains play a curvature-sensing role at low concentrations, but stabilize membrane curvature at high protein concentrations [123,124]. Such a dual role can lead to a positive feedback loop, when a slightly curved membrane leads to the accumulation of curvature-sensitive proteins. These proteins, in turn, deform the membrane, leading to a further increase in the binding affinity. This has been proposed as a general mechanochemical mechanism for protein recruitment [7]. However, the formualation of a mechanistic theory for such curvature-regulating feedback loops remains an open and highly interesting challenge to this day.
Bin, Amphiphysin, and Rvs. j α-helix -Prevalent helical-like protein structure, which is highly stable due to hydrogen bonds.
Collective curvature sensing
It has recently been shown that the distribution of proteins on the membrane and in the cytosol can depend on the cell geometry, even when the binding affinity of proteins is independent of membrane curvature [2,3,136]. The underlying mechanism is based on the aforementioned cytosolic gradients of proteins that switch between an inactive and an active state in the cytosol. As the required reactivation step is a non-equilibrium process that consumes energy, these gradients are maintained by a constant cycling of such proteins between the membrane and the cytosol, and therefore do not equilibrate by cytosolic diffusion. Since cytosolic gradients from opposing membrane points overlap at curved regions, one generally expects accumulation of inactive proteins in regions of high curvature (e.g., near the cell poles of elongated cells, including the rod-shaped E. coli [2], the C. elegans zygote [3], and Bacillus subtilis [128]) and a corresponding depletion of active proteins (Fig. 2f). Moreover, the effect of such a cytosolic gradient on the protein distributions in curved geometries depends in particular on the characteristic length of the cytosolic gradient relative to the local membrane curvature [2,3].
While this explains where proteins are most likely to encounter the membrane, its effect on the ensuing protein pattern depends on the protein reaction kinetics. For proteins that exhibit a simple attachment-detachment dynamics with the membrane, the increased encounter probability leads directly to an increase in protein concentration at the poles, which is further enhanced if the protein autocatalytically promotes its own binding [2]. In contrast, if two proteins mutually inhibit each others binding, an increased encounter probability leads to the formation of an interface between two protein domains on the membrane [3].
IV. BIOCHEMICAL GUIDING CUES
For spatially homogeneous systems, several theoretical and experimental studies have identified biochemical circuits that are able to perform logic operations [138], generate pulses [139,140], act as noise-reduction filters [141], or process biochemical signals in other ways [142][143][144][145][146]. Here the information from an input signal -typically encoded in the concentration of a protein -is processed and an output signal is generated.
In general, however, protein concentrations tend to be spatially inhomogeneous, so that a locally varying input can lead to a locally varying output protein concentration in the cell. In this way, an input pattern can serve as a template or biochemical guiding cue for the formation of an output protein pattern. Such biochemical guidance has been observed in many biological processes and over widely varying scales, ranging from tissue development [147,148] to the positioning of the cell-division site [41,79,115,[149][150][151][152]. In all these cases, the input patterns encode positional information, as each concentration marks a specific location or region in space [153]. In fact, there are several known instances in which protein patterns (input) control the formation of other patterns (output) [115,[154][155][156][157]. However, the physical mechanisms responsible for the processing of the positional information encoded in patterns, and the generation of a qualitatively different output pattern (e.g., gradient vs. step profile) are still largely unclear.
Such input/output relations are found, for example, in the polarity mechanism of budding yeast. Here, several so-called landmark proteins mark specific locations in the cell, such as the previous bud site. These landmark proteins (input) alter the kinetics of nucleotide exchange in the polarity factor Cdc42 (output), and thus contribute to the control of cell polarity in a symmetry-breaking manner [158,159]. Another example is provided by the midcell localization machinery of Caulobacter crescentus. In these elongated cells, ParB-parS (input) complexes localized to the cell poles stimulate the ATP-dependent dimerization of MipZ (output), which results in the formation of a bipolar gradient of MipZ dimers with a minimum at midcell [114]. MipZ, in turn, inhibits the poly-merization of FtsZ, which is a central component of the cell-division machinery. Thus, the bipolar MipZ gradient also acts as an input for the control and positioning of FtsZ (output) to midcell [160]. Such a hierarchy of pattern control through multiple stages of protein interaction is a common feature of many biochemical guidance mechanisms [94,149,159,161].
In the following, we discuss some recent advances in this area, focusing on systems in which the concentration profile of an (input) protein is able to control the reaction kinetics of another (output) protein, such that one or more reaction rates become spatially inhomogeneous. This can result in an output protein pattern that is qualitatively different from the input pattern, which has been termed spatial network computations [162].
A. Spatially varying reaction kinetics
Since protein reaction kinetics can depend on the concentration of other proteins, a spatially varying input protein concentration can lead to locally varying reactive equilibria of the output protein. In particular, not only can the protein concentration at each local reactive equilibrium be altered; also, the number and the stability of these equilibria can change in response to a varying input concentration (see Supplementary Information). Heuristically, this means that space itself serves as a control parameter k for the protein reaction kinetics. Hence, the input protein pattern encodes positional information.
The dynamics of the output protein depend crucially on its explicit biochemical interactions with the input proteins. For example, for a particular interaction between proteins, this can lead to bistability of the output protein over a limited range of input concentrations, as observed in starfish oocytes [94]. Due to the correspondence between input protein concentration and space, such a bistable parameter range maps to a region in space where the output protein reactions are bistable, which we refer to as regional bistability. In a similar way, a protein pattern can cause a pattern-forming instability in a specific spatial region, which has been termed regional instability [163,164]. Thus, an input pattern can lead to a qualitatively different spatial concentration profile of the output protein, where the explicit output pattern strongly depends on the reaction kinetics (Fig. 3a). This fundamental property of protein interactions is likely to represent the mechanism that underlies many of the biochemically guided pattern-forming systems observed in experiments [156][157][158][159][160].
B. Wave localization by protein gradients
Biochemical trigger waves, consisting of a traveling front or pulse of biomolecule concentration, are a common means of long-ranged signal transmission in cells [105]. Prominent examples of such waves include calcium waves [165], the propagation of mitosis l [161] and apoptosis m [166] in Xenopus eggs, actin polymerization waves in Dictyostelium [167] and neutrophils [168], as well as intracellular signaling [169]. A key component of models for trigger waves, such as the FitzHugh-Nagumo model [170], are bistable reaction kinetics (see Supplementary Information). These bistable reaction kinetics, in addition to resulting in information transmission, allow trigger waves to serve as a readout for positional information encoded in other protein patterns.
To illustrate how spatially varying reactive equilibria allow proteins to read out this positional information, we now discuss how a protein gradient can lead to the localization of such a trigger wave, in particular a bistable k Control parameter -A parameter that alters the qualitative dynamics when it is changed, also referred to as a bifurcation parameter in nonlinear dynamics. l Mitosis -Stage of the cell cycle during which chromosomes are segregated into the two daughter cells. m Apoptosis -Cellular process leading to actively induced cell death.
front, to a specific position in the cell. We first consider a system with homogeneous bistable reaction kinetics forming a front pattern (see Supplementary Information). This front can propagate through the system at a speed and direction that depends on, among other factors, the concentration of the input protein [105,171].
In the presence of an input pattern, the reaction kinetics are no longer homogeneous, so that a regional bistability can emerge. Since the front only propagates in a bistable parameter range, propagation is constrained to this regional bistability. In particular, since the direction of propagation depends on the input concentration, the front is pinned at a threshold input concentration (Fig. 3a) [105]. Due to the correspondence between input concentration and space, this means that the front is localized to a specific position within the regional bistability. Thus, the position of the front interface marks the location of the input threshold concentration, allowing the positional information encoded in the input pattern to be read out. Such a threshold-sensing mechanism has been proposed to play a role in the propagation of surface contraction waves during meiosis n in starfish oocytes [94] and during chemotaxis o in eukaryotes [172].
C. Edge-sensing and ring formation
Proteins also have been found to localize at the edges of spatial domains that exhibit a high concentration of other proteins or macromolecules. For example, during cellular wound healing, the Rho-GTPase Cdc42 and an associated GTPase regulator, Abr, accumulate locally to form two concentric rings [173]. Experimental evidence suggests that this structure is hierarchically organized, with the outer Cdc42 ring being dependent on the presence of an inner Abr zone. While it is not particularly surprising that a given spatial protein profile serves as a template for creating another protein profile with a similar shape, it is quite interesting that the downstream profile assumes a qualitatively different shape, with a peak localized right at the edge of the upstream profile (inner Abr ring, see insets in Fig. 3b). To account for such edge-sensing, a regional instability has been suggested [97,164]. Here, the step-like Abr profile, acting as an input protein pattern, defines two spatial domains with qualitatively different reaction kinetics for Cdc42, which takes the role of the output protein. It was shown that the outer domain may effectively act as a stimulus that induces a lateral mass-redistribution instability in the inner domain, which leads to a concentration peak of the output protein at the template edge (Fig. 3b). Moreover, the formation of this output concentration ring can be controlled by both the magnitude of the input pattern step and the total amount of output protein. Thus, edge sensing is enabled by a regional mass-redistribution instability in a downstream protein pattern, which is itself triggered by an upstream protein pattern that acts as a step-like template.
Beyond the specific example discussed above, there are other biologically highly relevant processes that involve edge sensing. As in the case of wound healing, a ring of Rho forms around a patch of high Cdc42 concentration prior to polar body emission in Xenopus oocytes [174]. Another biological process in which protein templates appear to play an essential role is that of macropinocytosis, a form of endocytosis p associated with cell surface ruffling. Here, actin-recruiting proteins colocalize to highdensity patches of PIP3 (a charged phospholipid) and a Ras-GTPase, forming a ring around the edge of the PIP3 domain, which in turn leads to the assembly of a conctractile actomyosin ring [175]. This whole process is invariably linked to the presence of PIP3 and Ras patches, suggesting that these biomolecules serve as a biochemical guiding cue for the actin-recruiting proteins. The specific physical mechanisms responsible for each of these edgesensing processes have not yet been uncovered.
D. Tracking of moving patterns
In addition to varying in space, the input protein concentration can vary in time at a fixed location in the cell. Temporal changes of the input concentration can lead to sudden changes of the reactive equilibrium which, in turn, results in transient dynamics of the output concentration before the new reactive equilibrium is established -a phenomenon referred to as excitability in the field of nonlinear dynamics [14,171]. Such transient dynamics can mark the position of local changes in the input concentration. For example, in the case of a traveling front pattern, the input concentration changes in time at a fixed position as the front passes by. Due to the transient output dynamics, this can lead to a traveling output concentration peak that closely follows the moving front. This has been observed in starfish oocytes, where a traveling front pattern leads to a moving concentration peak which, is ultimately responsible for the surface contraction waves observed during meiosis [94,176,177]. Similar observations have been made in vitro for an artificial cortex based on frog egg extracts [178].
E. Phoretic transport
A more intricate mechanism by which spatiotemporal protein patterns could serve as cues for the developp Endocytosis -Cellular process that enables the uptake of biomolecules into the interior of the cell. ment of subsequent protein patterns are various types of phoretic transport processes. These are, in general, the result of an external field gradient acting on the protein [179,180]. Examples include concentration gradients of carrier particles (diffusiophoresis) [137,181], chemical potential gradients (chemophoresis) [182,183], electric potential gradients (electrophoresis) [184], or temperature gradients (thermophoresis) [185], along which cargo can be transported. Thus, cargo particles can form a pattern guided by such gradients [179]. Notably, in phoretic transport mechanisms, energy is consumed to maintain the gradient, resulting in a flux of cargo particles. This is substantially different from other transport mechanisms such as active transport, where energy is consumed to fuel molecular motors that move cargo particles.
In the field of phoretic transport, research has long been focused on colloidal particles [179][180][181]186]. Experimental evidence for phoretic transport in biological systems related to protein organisation and pattern formation has only recently been discovered [137,182]. For example, in-vitro experiments have shown that diffusiophoresis can result in the spatial organization of DNA origami nanostructures in a concentration gradient of MinD [137]. Here, the Min proteins self-organize into a stationary pattern [187], resulting in diffusive fluxes at the domain edges (c.f. Fig. 3c). These diffusive fluxes are transferred to the DNA nanostructures via friction, leading to diffusiophoretic transport of the latter along the Min gradients. Thus, the movement of the DNA nanostructures mimics the movement of the Min proteins, resulting in the formation of an anti-correlated pattern of the DNA nanostructures. Such diffusiophretic transport has been suggested to play an important role for the distribution of large particles in cells in general [188].
In the context of plasmid segregation, chemophoresis has been suggested to drive the movement of plasmids on the nucleoid [182]. Here, ParA proteins on the nucleoid surface are thought to bind to large cargo, such as plasmids. Upon unbinding, ParA proteins are released from the nucleoid, resulting in a local depletion of ParA at the position of the cargo. The ParA concentration gradient at the edge of this depletion zone creates a chemical potential gradient for the cargo, which tends to bind more strongly at regions of high ParA concentration. Thus, the cargo moves along the chemical potential gradient away from the depletion zone [182,183]. This chemophoretic movement is suggested to be sufficient to ensure a balanced distribution of plasmids on the nucleoid [182].
V. MECHANICAL GUIDING CUES
In addition to biochemical guiding and guidance by cell size and shape, also the mechanical properties of a cell can affect protein pattern formation by altering the transport and reaction kinetics of proteins.
Flows generally arise from stress gradients. In cells, such gradients can be generated via shape deformations ( Fig. 4a). For example, recent work has demonstrated the generation of flows in the cytoplasm due to shape deformations in starfish oocytes [95]. In these cells, a surface contraction wave travels across the membrane from the animal to the vegetal pole, which locally increases the pressure in the cytosol, and results in cytoplasmic flows along the oocyte's animal-vegetal axis q . Similar observations have been made for Drosophila embryos, where apical constrictions instead of surface contraction waves lead to cytoplasmic flows [189], and in Drosophila neuroblasts where cortical contractions induce flows in the cortex [190].
Next to deforming the cell shape, contractions of the actomyosin cortex can also lead to cortical flows, either as a consequence of spatially inhomogeneous actomyosin activity [4] or anisotropic cortical tension [191] (Fig. 4b).
For example, cortical flows in C. elegans zygotes prior to PAR polarization arise due to nonuniform actomyosin activity [4]. Through hydrodynamic coupling, such flows may also induce cytoplasmic flows [53,92].
How are protein patterns controlled by mechanical guiding cues? It has been suggested that a combination of pattern guidance by cortical flows and biochemical interactions may be ultimately responsible for the polarization mechanism in C. elegans zygotes [4]. Prior to polarization, a mechanical inhomogeneity in the cell cortex, induced by the symmetry-breaking introduction of a centrosome into the zygote, causes the cell cortex to contract asymmetrically. Here, the reduced actomyosin contractility at the posterior pole leads to anterior-directed cortical flow. Once symmetry is broken, the cortical flows and the associated anterior-directed cytoplasmic flows lead to a redistribution of PAR proteins, which in turn control and maintain the asymmetric actomyosin contractility of the q Animal-vegetal axis -Symmetry axis in oocytes, along which the developmental activity varies, separating the cell into two distinct poles.
cortex, thereby giving rise to a self-regulating polarization mechanism. These observations underline the key role of mechanical guiding cues in the process of protein pattern formation.
VI. UPCOMING CHALLENGES
In this review, we have focused on guidance mechanisms in model biological organisms that have been studied experimentally, and for which theoretical models exist. However, a much larger number of cellular processes rely on guiding cues and whose underlying biophysical mechanisms are still unknown. To conclude this review, we outline some promising recent developments in the field of protein pattern formation that build upon the recognition of the important role of guiding cues.
A. Robustness against guiding cues
Guiding cues can vary over time, as evidenced by cell size and shape, which change throughout the cell cycle. Moreover, these changes can affect the process of protein pattern formation in quite different ways: Protein patterns can either adapt to the changing guiding cues as discussed in this review, or they can be impervious to variations in geometric, mechanical, and biochemical factors. Pattern-forming mechanisms that are robust to changes in cell geometry or mechanics have recently been identified in various systems [94,144], but a general understanding of robustness in pattern formation is still lacking. Future research on pattern formation mechanisms in living cells will reveal whether there are more examples where the formation of protein patterns adapts to be robust to the effects of cell mechanics and geometry.
B. Mechanochemical feedback loops
We discussed above how protein patterns can flexibly adjust to changes in the physical properties of cells. However, proteins can also actively modify the mechanical properties of the cell, resulting in a feedback loop between cell mechanics and protein patterns. Various theoretical studies showed that the coupling to cell mechanics in such mechanochemical feedback loops can lead to the formation of protein patterns [100,[192][193][194][195][196][197]. For example, coupling of a contractility-regulating chemical agent to an active fluid surface can result in shape deformations of axisymmetric surfaces, accompanied by polarization of the chemical agent [101]. This phenomenon shows similarities to the aforementioned self-reinforcing polarity mechanism of C. elegans, where cortical flows are created by asymmetric actomyosin activity [191]. In addition, a recent experimental study showed that the spatiotemporal patterning of the Min protein system can induce substantial shape deformations in GUVs r [198,199]. This observation suggests a generic interplay between reaction-diffusion dynamics and membrane mechanics. We hypothesize that membrane properties, such as spontaneous curvature, may influence the kinetics of protein binding, and vice versa [7,98,102]. In combination with the hydrodynamic coupling of the cell membrane to the cortex and the cytosol, this can lead to a mutual feedback between the dynamics of protein patterns and cell shape.
A theoretical characterization of this two-way coupling between biochemical processes and cell mechanics is a promising avenue for future research [200]. Since such mechanochemical models need to account for protein reaction-diffusion dynamics as well as a dynamically varying three-dimensional cell shape, they are challenging to study both analytically and numerically [192,196,201,202]. In future research, it will be important to further develop methods and, in particular, biologically realistic three-dimensional models, such that they can be compared to quantitative experimental data and contribute to the interpretation of experimental results in mechanochemical model systems.
Mechanochemical feedback loops are a special case of a general phenomenon that can be observed in many pattern-forming systems: may patterns in cells are not the result of a single guiding cue, but are the products of multiple interacting cues and processes [53,75,79,[203][204][205]. However, it is often difficult to separate all the processes involved in the robust formation of functional protein patterns in living cells, as the example of C. elegans polarisation shows [3,8,53,93]. Recognizing and incorporating such interacting processes into the theoretical analysis of pattern-forming systems will therefore be a major task for future research on pattern formation. r GUV -Giant unilamellar vesicle, an artificial spherical chamber bounded by a lipid bilayer that mimics the membrane of cells.
C. Perspectives for pattern guidance
At the conceptual level, we currently face three main challenges in the context of understanding the biophysical basis of pattern guidance. These relate to (i) progress in the study of fundamental aspects of processes in living systems far from thermal equilibrium, (ii) finding the right level of simplification for a given complex biological system, and (iii) improving both computational and experimental tools. In the long term, meeting these challenges will be vital to advancing our knowledge of pattern guidance, pattern formation, and information processing in biology in general.
New frontiers in non-equilibrium physics
Several interesting physics questions arise from the biological model systems we have discussed in this review. A central issue concerns how the dynamics of patternforming systems are mechanistically controlled by spatial and temporal gradients. These gradients lead to a variety of fascinating phenomena including information processing [147], templating [164], and hierarchies of different patterns [94]. Since these gradients can form for different physical quantities they can influence the formation of patterns in many ways. Among others, we have discussed spatially varying reaction kinetics which can lead to the localization of trigger waves in bistable media. But any gradient in an intensive thermodynamic variable, such as a chemical potential, can give rise to corresponding particle currents, as described by the laws of non-equilibrium thermodynamics [206]. Transport properties are also strongly influenced by spatial variations in kinetic coefficients such as diffusion constants. These processes lead to additional advection currents which we have not addressed in this review. Moreover, due to dynamic feedback between these particle currents and protein patterns, the gradients themselves may become part of the dynamics rather than acting solely as external guiding cues. This greatly expands the possibilities for future theoretical and experimental research on this topic.
Levels of biological complexity
Another crucial and actually quite general challenge is how to deal with the different levels of complexity in biological systems. For example, the full extents of interaction networks of proteins are generally unknown, and it is often unclear whether integrating all possible interaction pathways into a theoretical model is actually necessary to explain a particular phenomenon [207,208]. Even in cases where networks are fully characterized, the information flow through the reaction network can be difficult to understand. Methods to analyse such information flows have been developed for well-mixed reaction systems, such as the modular response analysis [209]. For spatially extended systems, where information is stored and processed by patterns, such methods have yet to be developed.
In addition, temporal regulatory mechanisms, such as cell-cycle-induced gene regulation, are often excluded from models of pattern forming system, even though the relevance of such regulatory mechanisms for pattern formation is not fully understood yet [140,210]. Where such mechanisms are in place, global mass conservationwhich is a cornerstone in many models of protein pattern formation -does not apply anymore, opening an avenue to additional concepts for pattern formation [104].
Avoiding the overfitting of models, and separating important components of interaction networks from irrelevant interactions (on the time scale of interest), are both difficult to achieve, and this presents major difficulties for theory and mathematical modeling. Ultimately, theoretical frameworks need to be developed that allow for a systematic coarse-graining that shows how the manifold components of a biological system can be reduced to its core elements. Such reductionism, at least for someone trained in physics, is the silver bullet to determining fundamental principles and improving our understanding.
Finding the right level of geometric representation
Similarly, the question of how theory should deal with the dimensionality and geometric form of biological systems needs careful consideration. For example, reducing the dimension of a specific system, e.g., to simplified one-dimensional models, may help to obtain an analytically more accessible representation. While such a simplification can be useful for gaining insight into the underlying dynamics and for guiding experiments, it may also obscure important aspects of pattern guidance. As pointed out in this review, certain phenomena, such as curvature sensing, only occur in realistic geometries and would therefore be erased in simplified one-dimensional models [2,3]. In essence, the complexity of biological systems must be reduced in order to understand them better. However, the challenge for future models is to find the appropriate level of simplification without loss of crucial features.
How to face the challenge of multiphysics problems
In addition, many experimental results indicate that pattern formation, and pattern guidance in particular, are the result of a tight interplay between biochemical interactions, hydrodynamics of cellular substrates, and membrane mechanics [4,121,191,211]. While numerous theoretical advances have been made in each of these areas (e.g., reaction-diffusion dynamics and nonequilibrium physics), there is so far no unified theoretical and computational approach that would allow a thorough analysis of such multiphysics problems. Therefore, in order to gain a deeper understanding of pattern guidance in realistic biological systems, a comprehensive theoretical framework that allows the study of the interplay between these different fields of physics must be developed.
Improving experimental and computational methods
Another roadblock that impedes progress is the limited availability of experimental, analytical and computational tools. On the experimental side, the current challenges, to name just a few examples, are to improve the spatial and temporal resolution of the quantities of interest (e.g., proteins) and to access quantitative information such as local densities, reaction rates, transport properties, and forces. In addition, conducting experiments under well controlled conditions, where only one or a few parameters are adjusted at a time, is often difficult owing to the associated technical demands, as well as the inherent complexity of biological systems. Future progress in this area would greatly enhance our ability to make more detailed comparisons with theory.
Concerning computational approaches, the simulation of multiphysics problems presents a major obstacle. In particular, the numerical implementation of bulkboundary coupled reaction-diffusion systems in combination with hydrodynamics and deformable, time-evolving membranes, is an important task for future research. The primary difficulties here lie in the development of an efficient and stable numerical approach that allows one to solve multiphysics problems in which the numerical domain itself is part of the solution. In the case of reactiondiffusion dynamics on dynamic membranes without coupling to a bulk volume, this can be addressed by deriving the time-evolution of the surface from the (normal) variation of a free energy functional that describes the mechanical properties of the membrane [18,[98][99][100][101][102]. However, this does not account for dynamics in the bulk, such as intracellular flows and bulk-boundary coupling of protein reactions. Promising approaches that can cope with these problems in the future are the level-set and the phase-field methods [212,213]. These strategies allow one to segregate the computational domain into different regions (e.g. interior and exterior of a cell), where the interface between these regions corresponds to a (smooth) boundary (that could represent, e.g., the cell membrane). In this way, one can define and solve a coupled set of partial differential equations between different regions, including the interface, and at the same time allow these regions to evolve over time by solving the level-set or phase-field equation. Most notably, the phase-field method is being used in current research to model cell migration [214], with applications to reactiondiffusion systems arising only recently [215][216][217][218]. At the same time, new methods are being developed [219]. In the long run, it will be a challenge to not only model a deformable domain, but also incorporate the biochemical and mechanical details of membranes in computational approaches.
VII. SUMMARY
We have presented a summary of the recent progress in understanding the biophysical mechanisms underlying the guidance and control of protein patterns. In essence, one distinguishes between geometric, biochemical, and mechanical guidance cues.
First, geometric effects can control protein pattern formation, with the cell size affecting the bulk-boundary ratio and the relative penetration depth of cytosolic concentration gradients. In addition, pattern formation can be limited by finite-size effects. Geometric effects imposed by the cell shape -such as the local membrane curvature that controls the distribution of curvaturesensing proteins, and the overall cell shape, which affects the curvature-dependent probability that a protein will encounter the membrane -can also serve as guiding cues. Second, we reviewed how protein patterns can guide other protein patterns via biochemical interactions. Spatial information that is encoded in one protein pattern can be interpreted through protein-protein interactions, thereby transforming the spatial coordinate into a control parameter for downstream protein reactions. This gives rise to a wide range of different pattern guidance mechanisms, including threshold localization, edgesensing, and phoretic transport. Third, mechanical guiding cues, among which flow and stress gradients are of particular relevance, can affect protein pattern formation. Finally, we outlined open questions and the associated experimental, theoretical, and numerical challenges that need to be faced to improve our understanding of guided pattern formation.
We believe that the mechanisms presented in this review can be applied to a wide range of processes in which spatial information is processed, such as cell migration, cytokinesis, and morphogenesis. To advance our understanding of the physical basis and biological relevance of pattern formation, further research on the concepts of pattern guidance will be required, as well as more refined methods to explain experimental observations. Taken together, this could ultimately contribute to the characterization of general biophysical principles of spatial information processing in living cells.
SUPPLEMENTARY INFORMATION
A. Methods of analysing pattern formation
Reactive equilibrium
Chemical reactions convert reactants to products and vice versa, thus resulting in fluxes. An equilibrium state is reached if the sum of all fluxes equals zero, which determines the equilibrium concentrations of constituents. This equilibrium state is commonly referred to as a reactive equilibrium, and is generally distinct from a thermodynamic chemical equilibrium because fluxes can originate from non-equilibrium processes (broken detailed balance) [220]. One example are NTPase cycles, in which proteins detach from the membrane and must undergo a conformational change before they can re-attach. The reactive equilibrium in this case is given by a balance between reactive fluxes onto and off the membrane.
Mathematically, the reaction kinetics of a well-mixed system are expressed by ordinary differential equations (ODEs)
∂ t u(t) = f (u) ,(5)
where f (u) contains the (nonlinear) interactions between the components of u and therefore corresponds to the sum of individual reactive fluxes. Formally, a reactive equilibrium conforms to the steady state solution ∂ t u = 0 of Eq. (5) and is termed the fixed point of the ODE system, i.e. f (u * ) = 0 for steady state solutions u * . In general, the long-term dynamics are governed by attractors of the nonlinear system, whose properties are the subject of the field of dynamical systems theory [13].
Phase space analysis
To assess the qualitative dynamics of nonlinear dynamics systems, one must often resort to geometric phase space analysis (Fig. 5b). In phase space, each point corresponds to a specific state of the system, with the phase space flow tracing out the time evolution of the system. Next to the flow lines, fixed points (f (u * ) = 0) and nullclines (f i (u) = 0) are characteristic features which reflect the topology of phase space. In particular, this representation allows one to identify important features of the system, such as steady states or limit cycles.
As a characteristic example, consider the phase space diagram shown in Fig. 5b, for a two-component system whose dynamics are given by ∂ t u 1 = −u 1 + u 2 1 u 2 and ∂ t u 2 = u 1 − u 2 . Intersections of the nullclines correspond to fixed points, whose stability can be determined by visualizing the phase space flow. The system at hand possesses one stable fixed point and one saddle fixed point. Given a specific initial state, the time evolution of this state can be determined by following the flow line, which provides qualitative information about the system's dynamics.
Dispersion relation
In spatially extended systems, patterns typically form when a (spatially homogeneous) steady state is unstable against random spatial perturbations. The formal way to The reaction kinetics f (u) determine the speed v of the wave. Right: For noisy initial conditions interpolating between the two stable plateaus u±, the reaction kinetics first lead to a smoothening of the perturbation and then result in directed front propagation at velocity v. (d) Illustration of the massredistribution instability in phase-space for the biologically relevant limit Dc Dm. The homogeneous steady state (black open circle) is determined from the intersection between the local phase space of the total average massn (thick blue line) and the reactive nullcline f (m, c) = 0 (thick black line). A spatial perturbation δn around this homogeneous state causes spatial gradients of the local total density in real space (inset top right). In phase space, the perturbation is represented by local phase spaces (thin blue lines) that contain masses that differ from the homogeneous state, and therefore lead to reactive fluxes (red arrows) towards the reactive equilibria (orange filled circles). This leads to a growing inhomogeneous density distribution in real space, which is further amplified by diffusive fluxes (orange arrows). Note that, since cytosolic diffusion is much faster than membrane diffusion Dc Dm, diffusive fluxes must point along the vertical direction. The steady state density distribution in real space is represented by the flux-balance subspace in phase space (thick gray line) [163]. The constant η0 determines the vertical position of the flux-balance subspace in phase space and can be interpreted as the (spatial) average cytosolic density.
probe for instabilities is to perform a linear stability analysis: One first expands spatial perturbations in normal modes and then linearizes the dynamics around a spatially homogeneous steady state u * . From the linearized system, one can determine the dispersion relation σ(q n ), which relates the growth rate σ of perturbations to their respective mode number q n . A typical dispersion relation is shown in Fig. 2d. Positive values of the growth rate indicate that spatial perturbations are amplified and grow exponentially. Since the critical mode q c with the highest growth rate is expected to dominate near onset, this unstable mode sets the characteristic wavelength of the initial pattern. However, in general, the dispersion relation only informs about the characteristic length scale of the pattern in the vicinity of the homogeneous steady state [1]; the dominant length scale of the final pattern can be quite different.
B. Nonlinear feedback in protein pattern formation
Bistability and propagation of bistable fronts
Feedback loops are ubiquitous in biological systems and essential for many cellular processes [105,110,142]. For instance, the calcium waves that follow fertilization of an egg are the result of a positive feedback loop in which cytosolic calcium promotes the flow of additional calcium into the cytoplasm [105]. In general, feedback loops lead to nonlinear dynamics that exhibit multiple (linearly stable) reactive equilibria [221]. A common case is bistability, where the dynamics ∂ t u = f (u) has three reactive equilibria, two of which are (linearly) stable (u ± ) and one of which is (linearly) unstable (u 0 ). Consider a spatially extended bistable system with spatially uniform reaction kinetics f (u), described by the reaction-diffusion equation ∂ t u(x, t) = D∂ 2
x u(x, t) + f (u(x, t)) .
In such a system, a front-like profile, where an interface connects two plateaus at the two linearly stable fixed points u − and u + (Fig. 5c), will propagate [106]: one plateau invades the other with a constant velocity v ∼ − u+ u− du f (u). These fronts will come to a halt only for a certain choice of parameters, namely when the areas enclosed by f (u) in the intervals [u − , u 0 ] and [u 0 , u + ] are equal [16].
Mass-redistribution instability
A general design feature of biochemical networks underlying protein self-assembly is that their dynamics (approximately) preserve the mass of each protein species; i.e., on the time scale of pattern formation, both protein production and protein degradation can be neglected. Some key features of the patterning dynamics can already be seen with a two-component, mass-conserving system consisting of a cytosolic (c) and a membrane (m) species in one spatial dimension [16,163]:
∂ t m(x, t) = D m ∂ 2 x m + f (m, c),(7a)∂ t c(x, t) = D c ∂ 2 x c − f (m, c).(7b)
It is instructive to consider the system's dynamics in (m, c) phase space. The reactive nullcline (f (m, c) = 0) typically shows a N-shape. Since the reaction kinetics are mass-conserving, reactive flows tend to remain within the corresponding local phase spaces (n(x, t) = m + c), and point towards the reactive equilibria determined by the intersection points of these local phase spaces with the reactive nullcline [163]. Now consider a homogeneous steady staten in phase space that intersects the nullcline in a region of negative slope. Spatial perturbations δn around the homogeneous steady state lead to a shift of the local reactive equilibria. Due to the resulting reactive currents, an upward shift δn in total density leads to a decrease in cytosolic density and vice versa (Fig. 5d). This gives rise to cytosolic concentration gradients, which in turn lead to diffusive fluxes, creating a positive feedback loop. Eventually, a steady-state pattern is reached when the diffusion currents at the membrane and in the cytosol balance out. In phase space, the steady state is represented by a flux-balance subspace given by c(x) + D m /D c m(x) = η 0 , where η 0 is a constant. In summary, this pattern formation mechanism involves an intricate coupling between mass-redistribution and local reaction kinetics [16,163].
FIG. 1. Reaction and transport processes involved in pattern formation: (a) Protein reactions include binding to and detachment from the cell membrane or other intracellular structures, as well as conformational state changes due to (de-)phosphorylation or nucleotide exchange. Cooperative and antagonistic (nonlinear) reactions between multiple proteins can lead to assisted attachment (recruitment) or to detachment from the membrane. Multiple monomers can form oligomers with altered transport and reaction properties. (b) Proteins can be transported by diffusion (Dc, Dm, black arrows) and advection (vc, vm, pink) independently on surfaces -in particular cell cortex and membrane -and in the cytosol. In addition, directed protein transport can be established by subunit addition and disassembly of polymers, resulting in treadmilling of monomers, and by active transport along filamentous structures, mediated by energy-consuming motor proteins.
FIG. 2 .
2Size and shape as guiding cues: (a) Schematic illustration of protein distribution in the cytosol and on the membrane: the cell volume scales with cell size R as R 3 , whereas the cell surface scales as R 2 , implying that both membrane and cytosolic protein concentration change with cell size. (b) Left: cytosolic gradients can emerge when proteins undergo a 'reacivation' step after detaching from the membrane. Inactive proteins (red) diffuse over a characteristic length scale before being reactivated (purple). Right: cytosolic gradients are established when the cell size is much larger than this characteristic length scale. (c) Cell size controls pattern formation: protein patterns cannot be established in cells smaller than the characteristic length scale of a pattern. (d) Only certain unstable modes with a wavelength limited by the cell size L can be realised. In a cell of size L/2, no pattern-forming instability arises. (e) Proteins including BAR domains preferentially bind to similarly curved membranes. (f) Characteristic distribution of proteins with delayed reactivation in elongated cells. Inactive proteins are reactivated after diffusing over a characteristic length scale . At the cell poles, this leads to the accumulation of inactive proteins, while they are diluted at the center of the cell. A complementary distribution of active proteins is established.
FIG. 3 .
3Principles of biochemical pattern guidance: (a) Left: Characteristic bifurcation diagram for pattern-forming systems. For reaction kinetics where the concentration of the input protein is a control parameter, a spatially varying input protein concentration can serve as a map between space and varying reaction kinetics. Top right: an input protein concentration gradient corresponds to a cutline through the bifurcation diagram (gray line) laid out in space, which divides the cell into regions of distinct stability. Bottom right: for a system where the input concentration gradient connects two monostable regions via a bistable region, the resulting front pattern (red line) is pinned to a threshold concentration value of the input concentration. Fixed points of the protein reaction kinetics are indicated by filled (stable) and open (unstable) circles. (b) Edge detection: An input pattern (blue) spatially alters the reaction kinetics of the output protein, resulting in a regional instability of the output protein close to the input edge (gray filled area). This leads to a peak pattern of the output protein concentration (orange) that marks the position of the input edge. Insets show a possible realization of this edge-sensing process, leading to a ring around a template patch. The plots depict the concentration profiles along the black cutline. (c) Diffusiophoresis: Diffusive fluxes of pattern-forming proteins (carrier particles, shown in orange) are established at pattern interfaces. Carrier particles transport cargo particles (blue) via frictional interactions, resulting in a complementary pattern of cargo particles[137].
FIG. 4 .
4Principles of mechanical guidance by flow generation: Stress gradients result in flows. (a) Heterogeneous cell deformations, as indicated by the grayscale outline, lead to pressure gradients in the cytosol, which in turn induce cytosolic flows towards regions of low pressure. (b) Heterogeneous actomyosin activity (green gradient; actin filaments shown in red, myosin shown as green circles), as observed in C. elegans zygotes [53], leads to polarized contractions of the actomyosin cortex and a flow of the entire cortex towards regions of high actomyosin activity. Hydrodynamic coupling results in cytosolic flows.
FIG. 5 .
5Geometric analysis of nonlinear dynamics: (a) The number and stability of reactive equilibria (fixed points) depends, in general, on the reaction kinetics f (u; a), where a is a control parameter. (b) Characteristic phase space diagram showing the system's fixed points, which can be derived from the nullclines (blue, orange); the separatrices (black) divide the phase space into qualitatively distinct areas. The time evolution of a given initial state (red square) is represented by the flow line associated with this state (red line). (c) Front propagation. Left:
e Cytoplasm -Heterogeneous material making up most of the volume of a cell (excluding the nucleus), mainly consisting of the cytosol and macromolecular organelles.
n Meiosis -A type of cell division process that generates daughter cells that contain half as many chromosomes as the parent cell. o Chemotaxis -Directed locomotion of cells along chemical gradients.
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| [] |
[
"Discussion of cosmological acceleration and dark energy",
"Discussion of cosmological acceleration and dark energy"
] | [
"Felix M Lev [email protected] \nArtwork Conversion Software Inc\n509 N. Sepulveda Blvd Manhattan Beach90266CAUSA\n"
] | [
"Artwork Conversion Software Inc\n509 N. Sepulveda Blvd Manhattan Beach90266CAUSA"
] | [] | The title of this workshop is: "What comes beyond standard models?". Standard models are based on Poincare invariant quantum theory. However, as shown in the famous Dyson's paper "Missed Opportunities" and in my publications, such a theory is a special degenerate case of de Sitter invariant quantum theory. I argue that the phenomenon of cosmological acceleration has a natural explanation as a consequence of quantum de Sitter symmetry in semiclassical approximation. The explanation is based only on universally recognized results of physics and does not involve models and/or assumptions the validity of which has not been unambiguously proved yet (e.g., dark energy and quintessence). I also explain that the cosmological constant problem and the problem why the cosmological constant is as is do not arise. | null | [
"https://export.arxiv.org/pdf/2302.10794v1.pdf"
] | 246,940,957 | 2302.10794 | e4551802fe0c105922b8590de81d2fe4f413b805 |
Discussion of cosmological acceleration and dark energy
17 Feb 2023 February 22, 2023
Felix M Lev [email protected]
Artwork Conversion Software Inc
509 N. Sepulveda Blvd Manhattan Beach90266CAUSA
Discussion of cosmological acceleration and dark energy
17 Feb 2023 February 22, 2023quantum de Sitter symmetrycosmological accelerationirre- ducible representationsdark energy
The title of this workshop is: "What comes beyond standard models?". Standard models are based on Poincare invariant quantum theory. However, as shown in the famous Dyson's paper "Missed Opportunities" and in my publications, such a theory is a special degenerate case of de Sitter invariant quantum theory. I argue that the phenomenon of cosmological acceleration has a natural explanation as a consequence of quantum de Sitter symmetry in semiclassical approximation. The explanation is based only on universally recognized results of physics and does not involve models and/or assumptions the validity of which has not been unambiguously proved yet (e.g., dark energy and quintessence). I also explain that the cosmological constant problem and the problem why the cosmological constant is as is do not arise.
Introduction
The title of this workshop is: "What comes beyond standard models?". Standard models are based on Poincare invariant quantum theory. However, as shown in the famous Dyson's paper "Missed Opportunities" and in my publications, such a theory is a special degenerate case of de Sitter invariant quantum theory.
The problem of cosmological acceleration is an example where the approach based on de Sitter symmetry solves the problem proceeding only from universally recognized results of physics without involving models and/or assumptions the validity of which has not been unambiguously proved yet (e.g., dark energy and quintessence). This problem was considered in my papers published in known journals, and in the book recently published by Springer.
My publications are based on large calculations. To understand them, the readers must be experts not only in quantum theory, but also in the theory of representations of Lie algebras in Hilbert spaces. Therefore, understanding my results can be a challenge for many physicists. Since the problem of cosmological acceleration is very important and my approach considerably differs from approaches of other authors, in this presentation to the 25th Bled workshop I outline only the ideas of my approach without calculations.
History of dark energy
This history is well-known. First Einstein introduced the cosmological constant Λ because he believed that the universe was stationary and his equations can ensure this only if Λ = 0. But when Friedman found his solutions of equations of General Relativity (GR) with Λ = 0, and Hubble found that the universe was expanding, Einstein said (according to Gamow's memories) that introducing Λ = 0 was the biggest blunder of his life. After that, the statement that Λ must be zero was advocated even in textbooks.
The explanation was that, according to the philosophy of GR, matter creates a curvature of space-time, so when matter is absent, there should be no curvature, i.e., space-time should be the flat Minkowski space. That is why when in 1998 it was realized that the data on supernovae could be described only with Λ = 0, the impression was that it was a shock of something fundamental. However, the term with Λ in the Einstein equations has been moved from the left hand side to the right hand one, it was declared that in fact Λ = 0, but the impression that Λ = 0 was the manifestation of a hypothetical field which, depending on the model, was called dark energy or quintessence. In spite of the fact that, as noted in wide publications (see e.g., [1] and references therein), their physical nature remains a mystery, the most publications on the problem of cosmological acceleration involve those concepts.
Several authors criticized this approach from the following considerations. GR without the contribution of Λ has been confirmed with a good accuracy in experiments in the Solar System. If Λ is as small as it has been observed, then it can have a significant effect only at cosmological distances while for experiments in the Solar System the role of such a small value is negligible. The authors of [2] titled "Why All These Prejudices Against a Constant?" note that it is not clear why we should think that only a special case Λ = 0 is allowed. If we accept the theory containing the gravitational constant G, which cannot be calculated and is taken from outside, then why can't we accept a theory containing two independent constants?
Let us note that currently there is no physical theory which works under all conditions. For example, it is not correct to extrapolate nonrelativistic theory to the cases when speeds are comparable to c, and it is not correct to extrapolate classical physics for describing energy levels of the hydrogen atom. GR is a successful classical (i.e., non-quantum) theory for describing macroscopic phenomena where large masses are present, but extrapolation of GR to the case when matter disappears is not physical. One of the principles of physics is that a definition of a physical quantity is a description how this quantity should be measured. The concepts of space and its curvature are pure mathematical. Their aim is to describe the motion of real bodies. But the concepts of empty space and its curvature should not be used in physics because nothing can be measured in a space which exists only in our imagination. Indeed, in the limit of GR when matter disappears, space remains and has a curvature (zero curvature when Λ = 0, positive curvature when Λ > 0 and negative curvature when Λ < 0) while, since space is only a mathematical concept for describing matter, a reasonable approach should be such that in this limit space should disappear too.
A common principle of physics is that when a new phenomenon is discovered, physicists should try to first explain it proceeding from the existing science. Only if all such efforts fail, something exotic can be involved. But in the case of cosmological acceleration, an opposite approach was adopted: exotic explanations with dark energy or quintessence were accepted without serious efforts to explain the data in the framework of existing science.
Elementary particles in relativistic and de Sitterinvariant theories
In the problem of cosmological acceleration, only large macroscopic bodies are involved and that is why one might think that for considering this problem, there is no need to involve quantum theory. Most works on this problem proceed from GR with additional assumptions the validity of which has not been unambiguously proved yet (see e.g. [1] and references therein). However, ideally, the results for every classical (i.e., non-quantum) problem should be obtained from quantum theory in semiclassical approximation. We will see that considering the problem of cosmological acceleration from the point of view of quantum theory, sheds a new light on understanding this problem.
Standard particle theory and standard models are based on Poincare symmetry where elementary particles are described by irreducible representations (IRs) of the Poincare group or its Lie algebra. The representation generators of the Poincare algebra commute according to the commutation relations
[P µ , P ν ] = 0, [P µ , M νρ ] = −i(η µρ P ν − η µν P ρ ), [M µν , M ρσ ] = −i(η µρ M νσ + η νσ M µρ − η µσ M νρ − η νρ M µσ ) (1)
where µ, ν = 0, 1, 2, 3, P µ are the operators of the four-momentum, M µν are the operators of Lorentz angular momenta and η µν is such that η 00 = −η 11 = −η 22 = −η 33 = 1 and η µν = 0 if µ = ν.
Although the Poincare group is the group of motions of Minkowski space, the description in terms of relations (1) does not involve Minkowski space at all. It involves only representation operators of the Poincare algebra, and those relations can be treated as a definition of relativistic invariance on quantum level (see the discussion in [3,4]). In particular, the fact that η µν formally coincides with the metric tensor in Minkowski space does not imply that this space is involved.
In classical field theories, the background space (e.g., Minkowski space) is an auxiliary mathematical concept for describing real fields and bodies. In quantum theory, any physical quantity should be described by an operator, but there is no operator corresponding to the coordinate x of the background space. In quantum field theory, Minkowski space is an auxiliary mathematical concept for describing interacting fields. Here a local Lagrangian L(x) is used, and x is only an integration parameter. The goal of the theory is to construct the S-matrix in momentum space, and, when this construction has been accomplished, one can forget about space-time background. This is in the spirit of the Heisenberg S-matrix program according to which in quantum theory one can describe only transitions of states from the infinite past when t → −∞ to the distant future when t → +∞.
The fact that the S-matrix is the operator in momentum space does not exclude a possibility that, in semiclassical approximation, it is possible to have a space-time description with some accuracy but not with absolute accuracy (see e.g., [4] for a detailed discussion). For example, if p is the momentum operator of a particle then, in the nonrelativistic approximation, the position operator of this particle in momentum representation can be defined as r = i ∂/∂p. In this case, r is a physical quantity characterizing a given particle and is different for different particles.
In relativistic quantum mechanics, for considering a system of noninteracting particles, there is no need to involve Minkowski space. A description of a single particle is fully defined by its IR by the operators commuting according to Eq. (1) while the representation describing several particles is the tensor product of the corresponding single-particle IRs. This implies that the four-momentum and Lorenz angular momenta operators for a system are sums of the corresponding single-particle operators. In the general case, representations describing systems with interaction are not tensor products of single-particle IRs, but there is no law that the construction of such representations should necessarily involve a background space-time.
In his famous paper "Missed Opportunities" [5] Dyson notes that de Sitter (dS) and anti-de Sitter (AdS) theories are more general (fundamental) than Poincare one even from pure mathematical considerations because dS and AdS groups are more symmetric than Poincare one. The transition from the former to the latter is described by a procedure called contraction when a parameter R (see below) goes to infinity. At the same time, since dS and AdS groups are semisimple, they have a maximum possible symmetry and cannot be obtained from more symmetric groups by contraction.
The paper [5] appeared in 1972 (i.e., more than 50 years ago) and, in view of Dyson's results, a question arises why general theories of elementary particles (QED, electroweak theory and QCD) are still based on Poincare symmetry and not dS or AdS ones. Probably, physicists believe that, since the parameter R is much greater than even sizes of stars, dS and AdS symmetries can play an important role only in cosmology and there is no need to use them for describing elementary particles. We believe that this argument is not consistent because usually more general theories shed a new light on standard concepts. The discussion in our publications and, in particular, in this paper is a good illustration of this point.
By analogy with relativistic quantum theory, the definition of quantum dS symmetry should not involve dS space. If M ab (a, b = 0, 1, 2, 3, 4, M ab = −M ba ) are the operators describing the system under consideration, then, by definition of dS symmetry on quantum level, they should satisfy the commutation relations of the dS Lie algebra so (1,4), i.e.,
[M ab , M cd ] = −i(η ac M bd + η bd M ac − η ad M bc − η bc M ad )(2)
where η ab is such that η 00 = −η 11 = −η 22 = −η 33 = −η 44 = 1 and η ab = 0 if a = b. The definition of AdS symmetry on quantum level is given by the same equations but η 44 = 1. The procedure of contraction from dS and AdS symmetries to Poincare one is defined as follows. If we define the operators P µ as P µ = M 4µ /R where R is a parameter with the dimension length then in the formal limit when R → ∞, M 4µ → ∞ but the quantities P µ are finite, Eqs. (2) become Eqs. (1). This procedure is the same for the dS and AdS symmetries and it has nothing to do with the relation between the Minkowski and dS/AdS spaces.
In [4,6] it has been proposed the following Definition: Let theory A contain a finite nonzero parameter and theory B be obtained from theory A in the formal limit when the parameter goes to zero or infinity. Suppose that, with any desired accuracy, theory A can reproduce any result of theory B by choosing a value of the parameter. On the contrary, when the limit is already taken, one cannot return to theory A, and theory B cannot reproduce all results of theory A. Then theory A is more general than theory B and theory B is a special degenerate case of theory A.
As argued in [4,6], in contrast to Dyson's approach based on Lie groups, the approach to symmetry on quantum level should be based on Lie algebras. Then it has been proved that, on quantum level, dS and AdS symmetries are more general (fundamental) than Poincare symmetry, and this fact has nothing to do with the comparison of dS and AdS spaces with Minkowski space. It has been also proved that classical theory is a special degenerate case of quantum one in the formal limit → 0, and nonrelativistic theory is a special degenerate case of relativistic one in the formal limit c → ∞. In the literature the above facts are explained from physical considerations but, as shown in [4,6], they can be proved mathematically by using properties of Lie algebras.
Physicists usually understand that physics cannot (and should not) derive that c ≈ 3 · 10 8 m/s and ≈ 1.054 · 10 −34 kg·m 2 /s. At the same time, they usually believe that physics should derive the value of Λ, and that the solution of the dark energy problem depends on this value. However, background space in GR is only a classical concept, while on quantum level symmetry is defined by a Lie algebra of basic operators.
The parameters (c, , R) are on equal footing because each of them is the parameter of contraction from a more general Lie algebra to a less general one, and therefore those parameters must be finite. In particular, the formal case c = ∞ corresponds to the situation when the Poincare algebra does not exist because it becomes the Galilei algebra, and the formal case R = ∞ corresponds to the situation when the de Sitter algebras do not exist because they become the Poincare algebra.
Quantum de Sitter theories do not need the dimensionful parameters (c, , R) at all. They arise in less general theories, and the question why they are as are does not arise because the answer is:
is as is because people want to measure angular momenta in kg·m 2 /s, c is as is because people want to measure velocities in m/s, and R is as is because people want to measure distances in meters. The values of the parameters (c, , R) in (kg, m, s) have arisen from people's macroscopic experience, and there is no guaranty that those values will be the same during the whole history of the universe (see e.g., [4] for a more detailed discussion). The fact that particle theories do not need the quantities (c, ) is often explained such that the system of units c = = 1 is used. However, the concept of systems of units is purely classical and is not needed in quantum theory.
It is difficult to imagine standard particle theories without IRs of the Poincare algebra. Therefore, when Poincare symmetry is replaced by a more general dS one, dS particle theories should be based on IRs of the dS algebra. However, as a rule, physicists are not familiar with such IRs. The mathematical literature on such IRs is wide but there are only a few papers where such IRs are described for physicists. For example, an excellent Mensky's book [7] exists only in Russian.
Explanation of cosmological acceleration
In this section we explain that, as follows from quantum theory, the value of Λ in classical theory must be non-zero and the question why Λ is as is does not arise.
Consider a system of free macroscopic bodies, i.e., we do not consider gravitational, electromagnetic and other interactions between the bodies. Suppose that distances between the bodies are much greater than their sizes. Then the motion of each body as a whole can be formally described in the same way as the motion of an elementary particle with the same mass. In semiclassical approximation, the spin effects can be neglected, and we can consider our system in the framework of dS quantum mechanics of free particles.
The explicit expressions for the operators M ab in IRs of the dS Lie algebra have been derived in [8] (see also [4,6,9]). In contrast to standard quantum theory where the mass m of a particle is dimensionful, in dS quantum theory, the mass m dS of a particle is dimensionless. In the approximation when Poincare symmetry works with a high accuracy, these masses in units c = = 1 are related as m dS = Rm. Also, in dS quantum theory, the Hilbert space of functions in IRs is the space of functions depending not on momenta but on four-velocities
v = (v 0 , v) where v 0 = (1 + v 2 ) 1/2 .
Then in the spinless case, the explicit expressions for the operators M ab are (see e.g., Eq. (3.16) in [4]): The important observation is that, at this stage, we have no coordinates yet. For describing the motion of the particle in terms of coordinates, we must define the position operator. If Poincare symmetry works with a high accuracy, the momentum of the particle can be defined as p = mv and, as noted above, the position operator can be defined as r = i ∂/∂p. In semiclassical approximation, we can treat p and r as usual vectors. Then, if E = E/R, P = B/R and the classical nonrelativistic Hamiltonian is defined as H = E − mc 2 , it follows from Eq. (3) that
J = l(v), N = −iv 0 ∂ ∂v , E = m dS v 0 + iv 0 (v ∂ ∂v + 3 2 ) B = m dS v + i[ ∂ ∂v + v(v ∂ ∂v ) + 3 2 v](3)H(P, r) = P 2 2m − mc 2 r 2 2R 2(4)
Here the last term is the dS correction to the non-relativistic Hamiltonian. The representation describing a free N-body system is a tensor product of the corresponding single-particle IRs. This means that every N-body operator M ab is a sum of the corresponding single-particle operators.
Consider a system of two free particles described by the quantities P j and r j (j = 1, 2). Define standard nonrelativistic variables P = P 1 + P 2 , q = (m 2 P 1 − m 1 P 2 )/(m 1 + m 2 ) R = (m 1 r 1 + m 2 r 2 )/(m 1 + m 2 ), r = r 1 − r 2
Here P and R are the momentum and position of the system as a whole, and q and r are the relative momentum and relative radius-vector, respectively. Then as follows from Eqs. (4) and (5), the internal two-body Hamiltonian is
H nr (r, q) = q 2 2m 12 − m 12 c 2 r 2 2R 2(6)
where m 12 is the reduced two-particle mass. Then, as follows from the Hamilton equations, in semiclassical approximation the relative acceleration is given by a = rc 2 /R 2
where a and r are the relative acceleration and relative radius vector of the bodies, respectively. The fact that the relative acceleration of noninteracting bodies is not zero does not contradict the law of inertia, because this law is valid only in the case of Galilei and Poincare symmetries. At the same time, in the case of dS symmetry, noninteracting bodies necessarily repulse each other. In the formal limit R → ∞, the acceleration becomes zero as it should be.
Equations of relative motion derived from Eq. (6) are the same as those derived from GR if Λ = 0. In particular, the result (7) coincides with that in GR if the curvature of dS space equals Λ = 3/R 2 , where R is the radius of this space. Therefore the cosmological constant has a physical meaning only on classical level, the parameter of contraction from dS symmetry to Poincare one coincides with R and, as noted above, a question why R is as is does not arise.
In GR, the result (7) does not depend on how Λ is interpreted, as the curvature of empty space or as the manifestation of dark energy or quintessence. However, in quantum theory, there is no freedom of interpretation. Here R is the parameter of contraction from the dS Lie algebra to the Poincare one, it has nothing to do with dark energy or quintessence and it must be finite because dS symmetry is more general than Poincare one.
Every dimensionful parameter cannot have the same numerical values during the whole history of the universe. For example, at early stages of the universe such parameters do not have a physical meaning because semiclassical approximation does not work at those stages. In particular, the terms "cosmological constant" and "gravitational constant" can be misleading. General Relativity successfully describes many data in the approximation when Λ and G are constants but this does not mean that those quantities have the same numerical values during the whole history of the universe.
Discussion and conclusion
In view of the problem of cosmological acceleration, the cosmological constant problem is widely discussed in the literature. This problem arises as follows.
One starts from Poincare invariant quantum field theory (QFT) of gravity defined on Minkowski space. This theory contains only one phenomenological parameter -the gravitational constant G, and the cosmological constant Λ is defined by the vacuum expectation value of the energy-momentum tensor. The theory contains strong divergencies which cannot be eliminated because the theory is not renormalizable. Therefore, the results for divergent integrals can be made finite only with a choice of the cutoff parameter. Since G is the only parameter in the theory, a reasonable choice of the cutoff parameter in momentum space is the Planck momentum /l P where l P is the Plank length. In units = c = 1, G has the dimension 1/length 2 and Λ has the dimension length 2 . Therefore, the value of Λ obtained in this approach is of the order of 1/G. However, this value is more than 120 orders of magnitude greater than the experimental one.
In view of this situation, the following remarks can be made. As ex-plained in Sec. 3, Poincare symmetry is a special degenerate case of dS symmetry in the formal limit R → ∞. Here R is a parameter of contraction from dS algebra to Poincare one. This parameter has nothing to do with the relation between Poincare and dS spaces. The problem why R is as is does not arise by analogy with the problem why c and are as are.
As explained in Sec. 4, the cosmological constant Λ has a physical meaning only in semiclassical approximation and here it equals 3/R 2 . Therefore the cosmological constant problem and the problem why the cosmological constant is as is do not arise. As noted in Sec. 3, the background space-time is only a mathematical concept which has a physical meaning only in classical theory. This concept turned out to be successful in QED. In particular, the results for the electron and muon magnetic moments agree with experiments with the accuracy of eight decimal digits. However, QED works only in perturbation theory because the fine structure constant is small. There is no law that the ultimate quantum theory will necessarily involve the concept of background space-time. QFTs of gravity (for example, Loop Quantum Gravity) usually assume that in semiclassical approximation, the background space in those theories should become the background space in GR. However, in Sec. 4, the result for the cosmological acceleration in semiclassical approximation has been obtained without space-time background and this result is the same as that obtained in GR.
Although the physical nature of dark energy remains a mystery, there exists a wide literature where the authors propose QFT models of dark energy. These models are based on Poincare symmetry with the background Minkowski space. So, the authors do not take into account the fact that de Sitter symmetry is more general (fundamental) than Poincare symmetry and that the background space is only a classical concept. While in most publications, only proposals about future discovery of dark energy are considered, the authors of [1] argue that dark energy has been already discovered by the XENON1T collaboration. In June 2020, this collaboration reported an excess of electron recoils: 285 events, 53 more than the expected 232 with a statistical significance of 3.5σ. However, in July 2022, a new analysis by the XENONnT collaboration discarded the excess [10].
As shown in Sec. 4, the result (7) has been derived without using dS space and its geometry (metric and connection). It is simply a consequence of dS quantum mechanics in semiclassical approximation. We believe that this result is more important than the result of GR because any classical result should be a consequence of quantum theory in semiclassical approximation.
Therefore, the phenomenon of cosmological acceleration has nothing to do with dark energy or other artificial reasons. This phenomenon is purely a kinematical consequence of dS quantum mechanics in semiclassical approximation.
where J = {M 23 , M 31 , M 12 }, N = {M 01 , M 02 , M 03 }, B = {M 41 , M 42 , M 43 }, l(v) = −iv×∂/∂v and E = M 40 .
Direct detection of dark energy: the XENON1T excess and future prospects. S Vagnozzi, L Visinelli, P Brax, A -Ch, J Davis, Sakstein, Phys. Rev. 10463023S. Vagnozzi, L. Visinelli, P. Brax, A-Ch. Davis and J. Sakstein: Direct detection of dark energy: the XENON1T excess and future prospects, Phys. Rev. D104, 063023 (2021).
E Bianchi, C Rovelli, arXiv:1002.3966v3Why All These Prejudices Against a Constant. E. Bianchi and C. Rovelli: Why All These Prejudices Against a Con- stant, arXiv:1002.3966v3 (2010).
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Finiteness of Physics and its Possible Consequences. F M Lev, J. Math. Phys. 34F.M. Lev: Finiteness of Physics and its Possible Consequences, J. Math. Phys. 34, 490-527 (1993).
Could Only Fermions Be Elementary?. F M Lev, J. Phys. 37F.M. Lev: Could Only Fermions Be Elementary? J. Phys. A37, 3287- 3304 (2004).
E Aprile, K Abe, F Agostini, arXiv:2207.11330Search for New Physics in Electronic Recoil Data from XENONnT. E. Aprile, K. Abe, F. Agostini et. al.: Search for New Physics in Electronic Recoil Data from XENONnT, arXiv:2207.11330 (2022).
| [] |
[
"Decentralized Secondary Control Scheme for Frequency Restoration in Inverter-based Islanded Microgrids",
"Decentralized Secondary Control Scheme for Frequency Restoration in Inverter-based Islanded Microgrids"
] | [
"Siddharth Bhela ",
"Abhishek Banerjee ",
"Ulrich Muenz ",
"Joachim Bamberger "
] | [] | [] | Inverter-dominated microgrids are quickly becoming a key building block of future power systems. They rely on centralized controllers that can provide reliability and resiliency in extreme events. Nonetheless, communication failures due to cyber-physical attacks or natural disasters can make autonomous operation of islanded microgrids challenging. This paper examines a unified decentralized secondary control scheme that is robust to inverter clock synchronization errors and can be seamlessly applied to grid-following or grid-forming control architectures. The proposed scheme overcomes the wellknown stability problem that arises from parallel operation of local integral controllers. Theoretical guarantees for stability are provided along with criteria to appropriately tune the secondary control gains to achieve good frequency regulation performance while ensuring fair power sharing. The efficacy of our approach in eliminating the steady-state frequency deviation is demonstrated through simulations on a 5-bus microgrid with four grid-forming inverters. | 10.48550/arxiv.2211.03209 | [
"https://export.arxiv.org/pdf/2211.03209v1.pdf"
] | 253,383,754 | 2211.03209 | 4f4fa90968bd0978cff32c72b97fa4dedd7b98a0 |
Decentralized Secondary Control Scheme for Frequency Restoration in Inverter-based Islanded Microgrids
Siddharth Bhela
Abhishek Banerjee
Ulrich Muenz
Joachim Bamberger
Decentralized Secondary Control Scheme for Frequency Restoration in Inverter-based Islanded Microgrids
Inverter-dominated microgrids are quickly becoming a key building block of future power systems. They rely on centralized controllers that can provide reliability and resiliency in extreme events. Nonetheless, communication failures due to cyber-physical attacks or natural disasters can make autonomous operation of islanded microgrids challenging. This paper examines a unified decentralized secondary control scheme that is robust to inverter clock synchronization errors and can be seamlessly applied to grid-following or grid-forming control architectures. The proposed scheme overcomes the wellknown stability problem that arises from parallel operation of local integral controllers. Theoretical guarantees for stability are provided along with criteria to appropriately tune the secondary control gains to achieve good frequency regulation performance while ensuring fair power sharing. The efficacy of our approach in eliminating the steady-state frequency deviation is demonstrated through simulations on a 5-bus microgrid with four grid-forming inverters.
I. INTRODUCTION
Microgrids (MG) consist of a group of interconnected distributed energy resources and loads that act as a single controllable entity. Since MGs can operate in grid-connected or islanded modes they are touted as the key building blocks of future power systems [1], [2]. Inverter-based MGs are attracting attention in industry and academia as they can improve reliability and ensure support for critical services even during extreme events [3]. Operation of MGs in islanded mode is considered challenging as the dynamics of the MG are no longer dominated by the main grid. In such scenarios, advanced control mechanisms are needed to maintain the delicate demand-supply balance [4].
Hierarchical control schemes have been well-explored for operation of inverter-dominated microgrids [5], [6]. These schemes are classified into three levels of control that serve different functions: i) the primary control layer is the fastest and establishes power sharing; ii) the secondary control layer is responsible for providing frequency regulation and eliminating steady-state frequency deviations introduced by the primary control; and iii) the tertiary control layer is concerned with defining the the long-term set points based on economic dispatch [7].
The primary control layer is largely droop-based and relies purely on local measurements [8], [9], [10]. However, the secondary and tertiary control layers typically depend on communication. While centralized control architectures for secondary control provide good performance they are neither scalable nor robust to cyber-physical attacks [11]. For this reason, a variety of distributed and decentralized secondary control (DSC) schemes have been explored; see [12] and references therein. Despite their many benefits, communication-free control schemes in MGs can lead to poor performance and instability if the inverter digital processor clocks used to generate the time signals are not synchronized. The impact of clock synchronization on frequency regulation and power sharing has been briefly reviewed in literature [13], [11], [14], [15]. Nonetheless, there is no unified and robust DSC scheme that can be implemented in both Grid-forming (GFM) and Grid-following (GFL) inverters. Moreover, little thought is given on how to tune the secondary controller gains. In this paper, we address both these challenges.
Our contributions are as follows. First, in Section II we provide a unified modeling framework for investigating DSC schemes for both GFM and GFL inverters. We also show that an adhoc DSC approach based on local integral controllers is not robust to clock synchronization errors. Moreover, a novel DSC scheme with damping is proposed that overcomes these challenges. Second, we provide stability guarantees for our proposed DSC scheme with damping in Section III. We show that both the secondary controller dynamics and the MG frequency reach a steady state. Further, based on the desired objectives conditions for appropriately choosing the secondary controller gains are also provided. Section IV discusses simulation tests based on a 5-bus test case with four GFM inverters followed by conclusions and future research directions in Section V.
Notation: Sets are denoted by calligraphic symbols. Given a real-valued sequence {x a,1 , . . . , x a,N }, x a is the N × 1 vector obtained by stacking the entries x a,i , and dg(x a ) is the corresponding diagonal matrix. The operator (·) stands for transposition.
II. DECENTRALIZED SECONDARY CONTROL
An islanded microgrid having N nodes can be modeled as a connected graph G = (V, E), whose nodes V := {1, . . . , N } correspond to buses, and edges E to undirected lines. For simplicity of analysis we assume that there is an inverter at each bus i ∈ V. Each inverter is equipped with the standard droop control [9] which is further augmented by the decentralized secondary control schemes presented in this section.
A. Adhoc Decentralized Secondary Control
The adhoc DSC solution consists of local integral controllers at all inverters. This corresponds to the control architectures shown in Figures 1 and 2 where the damping gain k t,i and any saturation are neglected. Under this simplification, let us consider the power balance [16]
=P ref,i + k p,i (ω n − ω i ) + P s,i =P ref,i − k p,i ∆ω i + P s,i
where P gen,i , P ref,i , P prim,i , P s,i , and k p,i are the real power output, reference power, primary reserve power, secondary reserve power, and primary droop gain of the inverter at bus i, respectively. Note that k p,i is the inverse of the standard droop gain. The frequency at each bus i ∈ V is ω i and the nominal frequency is denoted by ω n . Observe that (1) without the corrective term P s,i is simply the primary droop equation [16], [9]. For notational convenience and to account for local loads let us define
P i :=P gen,i + P load,i (2a) P 0,i :=P ref,i + P load,i(2b)
where P load,i is the total load at bus i. Heed that the derived model will describe the secondary control dynamics which are much slower than the primary control dynamics. Therefore, the frequencies measured across the grid can be assumed to be identical, i.e., ω i ≈ ω * =⇒ ∆ω i = ∆ω = ω * − ω n . By rearranging the terms in (2a) and (2b) and substituting for P gen,i and P ref,i in (1), the power balance at each bus i ∈ V and collectively across the islanded microgrid can be expressed as
P i =P 0,i − k p,i ∆ω + P s,i = 0 (3a) 1 P =1 P 0 − 1 k p ∆ω + 1 P s = 0.(3b)
Here 1 is the N × 1 vector of ones, and P, P 0 , k p , P s are the N × 1 vectors obtained by respectively stacking the values {P i }, {P 0,i }, {k p,i }, {P s,i } at each bus. By rearranging the terms in (3b), the steady-state frequency can be inferred as
∆ω = 1 P 0 + 1 P s 1 k p .(4)
The adhoc solution can now be described by the following dynamical systeṁ
P s,i = − k s,i ∆ω = −k s,i 1 P 0 + 1 P s 1 k p (5a) P s = − 1 1 k p k s 1 P s − 1 P 0 1 k p k s (5b)
In practice, inverters operate with their own digital signal processors. The clocks used to generate the time signals differ from each other and without clock synchronization each inverter can have a different frequency offset γ i [13], [11]. Accounting for this offset in the original model (5) yieldṡ
P s,i = − k s,i 1 P 0 + 1 P s 1 k p − γ i (6a) P s = − 1 1 k p k s 1 A P s − 1 P 0 1 k p k s + dg(k s )γ (6b)
The stability and robustness of this system on γ depends on the system matrix A. Matrix A is rank one by construction, i.e., A has a zero eigenvalue with multiplicity N − 1 and a non-zero eigenvalue. The eigenmode with nonzero eigenvalue has the eigenvector k s and the corresponding eigenvalue − 1 ks 1 kp since
Ak s = − 1 1 k p k s 1 k s = − 1 k s 1 k p k s .(7)
Hence, the system has one stable eigenmode and all other eigenmodes are marginally stable [17]. The second term in (6b) is acting in the direction of the stable eigenmode of A. Therefore, sufficiently slow variations in P 0 are compensated by P s . With respect to this input, the system (6b) is stable and robust. The third term in (6b) representing the frequency offset usually acts in different directions than the stable eigenmode of A. In fact, only homogeneous frequency offsets γ = γ1, γ ∈ R, are damped by the system matrix A. All non-homogeneous frequency offsets are continuously integrated by the parallel integral controllers. This shows that the adhoc solution is not robust to frequency offsets introduced due to clock synchronization errors [15], [13].
B. Decentralized Secondary Control with Damping
To counteract the unstable frequency offset dynamics reported in subsection II-A we introduce an additional damping term −k t,i P s,i as shown in Figures 1 and 2
to obtaiṅ
P s,i = − k t,i P s,i − k s,i 1 P 0 + 1 P s 1 k p − γ i (8a) Ṗ s = − dg(k t ) + 1 1 k p k s 1 P s − 1 P 0 1 k p k s + dg(k s )γ. (8b)
The next section shows that this simple modification provides theoretical stability guarantees for an inverter dominated microgrid. Moreover, criteria are provided to appropriately tune the secondary control gains (k s , k t ) such that the steady-state frequency deviation is minimized. We also investigate how the ratio of aforementioned gains affects the secondary reserve unbalance and thereby the power sharing between inverters.
III. STABILITY ANALYSIS
To study the stability of the proposed DSC scheme with damping we first investigate the steady-state of the dynamical system in (8) by settingṖ s,i = 0 to obtain
P s,i = k s,i k t,i − 1 P 0 + 1 P s 1 k p + γ i .(9)
The total secondary reserve power at steady-state can then be expressed as
1 P s = −1 k st 1 P 0 + 1 P s 1 k p + k st γ (10a) = − 1 k st 1 k p + 1 k st 1 P 0 + 1 k p k st γ 1 k p + 1 k st (10b)
where k st refers to the vector with elements { ks,i /kt,i}. The equality in (10b) is derived by rearranging the terms in (10a). The steady-state frequency can now be computed as
∆ω = 1 P 0 + 1 P s 1 k p (11a) = 1 P 0 1 k p − 1 k st 1 k p + 1 k st 1 P 0 1 k p + k st γ 1 k p + 1 k st (11b) = 1 P 0 1 k p + 1 k st + k st γ 1 k p + 1 k st (11c)
and the steady-state secondary reserve power as
P s,i = k s,i k t,i − 1 P 0 1 k p − 1 P s 1 k p + γ i (12a) = k s,i k t,i − 1 P 0 1 k p + 1 k st 1 k p + 1 k st 1 P 0 1 k p − k st γ 1 k p + 1 k st + γ i (12b) = k s,i k t,i − 1 P 0 1 k p + 1 k st − k st γ 1 k p + 1 k st + γ i (12c)
The equalities in (11c) and (12c) are obtained by substituting the expression for 1 P s from (10b) in (11a) and (12a), correspondingly. We will revisit these equations later on. The subsequent analysis relies on the ensuing mild assumption that can removed with a more rigorous analysis. Assumption 1. All damping gains are identical, i.e., k t,i = k t for all i ∈ V.
To show that the damping termk t actually damps the frequency offsets γ i we first transform the dynamic system (8). We then separate the state space into a subspace that acts in the direction k s and an orthogonal subspace. Finally, we separate the dynamics in these two subspaces and show that the dynamics in both subspaces are stable. Using the transformationP
s,i = P s,i k s,i(13)
and the notationk s,i := k s,i for the dynamic system in (8) yieldṡ
P s = dg(k s )γ − dg(k t )P s − 1 1 k p k sk s ÃP s − 1 P 0 1 k pk s .(14)
As shown before in (6b), the matrixà is rank one by construction and has a single non-zero eigenvaluẽ
Ak s = − 1 k s 1 k pk s .(15)
Nevertheless, matrixà is symmetric after this transformation and therefore all eigenvectors are orthogonal to each other [17]. Especially, the eigenvectors corresponding to the zero eigenvalues are all orthogonal tok s , that is
Av =0 ∀v ∈ Z(16)
and Z := {v ∈ R N :k s v = 0}. Let us now separate the state space ofP s and the frequency offset γ as follows P s =αk s +P s (17a) γ =γ1 +γ.
(17b)
Here α is a scalar function,P s is orthogonal tok s , i.e., k sPs = 0 holds andγ is orthogonal to k s , i.e., it satisfies k sγ = 0. Note that the basis of the separation of γ is not orthogonal and therefore we may haveγ = 0 even when 1 γ = 0 because 1 γ = 0. Pre-multiplying (17b) with k s yields
k s γ =k s 1γ (18a) =⇒γ = k s γ k s 1 (18b) =⇒γ =γ − k s γ k s 1 1. (18c)
Therefore,γ is simply a weighted average of γ with weights ks,i 1 ks . With the separation (17) and under assumption 1
P s =αk s +Ṗ s (19a) = −k t αk s +P s − 1 k s 1 k p αk s − 1 P 0 1 k pk s +γk s + dg(k s )γ (19b) = −k t α − 1 k s 1 k p α − 1 P 0 1 k p +γ k s −k tPs + dg(k s )γ. (19c)
The equality in (19b) is obtained by substituting (17a) and (17b) in (14). Note that now the first term in (19c) is heading in the directionk s whereas the last two terms in the summand are orthogonal tok s becausek sPs = 0 andk s dg(k s )γ = k sγ = 0. Hence, we may separate the system of dynamic equations in (19c) as followṡ
α = − k t + 1 k s 1 k p α − 1 P 0 1 k p +γ (20a) P s = −k tPs + dg(k s )γ. (20b)
Notice that both dynamics are exponentially stable. As a last step, we recover our previously derived steady-states in (11c) and (12c). Recall from the transformation in (13) and the separation of state space in (17a) that
P s = dg(k s )P s (21a) = dg(k s )k s α + dg(k s )P s (21b) =k s α + dg(k s )P s . (21c)
The steady-states of α andP s can be derived from (20a) and (20b) as follows
α = − k t + 1 k s 1 k p −1 1 P 0 1 k p −γ (22a) P s = 1 k t dg(k s )γ. (22b)
We investigate first the total secondary reserve power
1 P s =1 k s α +k sPs = 1 k s α.(23)
where the equality is obtained by substituting for P s from (21). Sincek sPs = 0 by construction ofP s we can simplify the previous equation as follows
1 P s = 1 k s α = − k t 1 k s + 1 1 k p −1 1 P 0 1 k p −γ (24a) = − k t 1 k p + 1 k s 1 k s 1 k p −1 1 P 0 1 k p −γ (24b) = − 1 k s 1 k p k t 1 k p + 1 k s 1 P 0 1 k p −γ (24c)
Using the relation in (18b) it is not hard to show that (10) is equal to (24c) under Assumption 1. We naturally assume that 1 P0 1 kp γ because 1 P0 1 kp describes the frequency deviation after the reaction of the primary control which is certainly much larger than the weighted frequency offsetγ. Thus, we obtain
1 P s ≈ − 1 k s 1 k p k t 1 k p + 1 k s 1 P 0 1 k p (25a) = − 1 k s k t 1 k p + 1 k s 1 P 0 . (25b)
Without damping (k t = 0) we recover the original solution 1 P s = −1 P 0 , i.e., eventually the secondary reserve compensates the power imbalance in the microgrid. If we include damping, the solution should be close to the original solution. This requires that 1 ks kt1 kp+1 ks shall be close to one, which is achieved if
k t 1 k s 1 k p .(26)
Given the expression in (25b) the steady-state frequency deviation can now be determined as
∆ω = 1 P 0 + 1 P s 1 k p (27a) ≈ 1 1 k p 1 − 1 k s k t 1 k p + 1 k s 1 P 0 (27b) =k t 1 P 0 k t 1 k p + 1 k s (27c)
which corresponds to (11c) under Assumption 1. Note that (26) implies that the steady-state frequency deviation is close to zero, i.e. ∆ω ≈ 0. Finally, we compute the secondary reserve unbalance amongst the inverters in the microgrid. Recall from (21) that
P s =k s α + dg(k s )P s (28a) =k s α + 1 k t dg(k s )γ (28b)
The desired steady-state is P s = k s α because this implies that the secondary control reserve is split up as specified by the secondary control gains k s . Hence, the last term in (28b) should be as small as possible to achieve fair power sharing.
Remark 1. Notice from (26) and (28b) that there is an trade-off between the gains (k s , k t ). Larger ratios {k s,i /k t,i } minimize the frequency deviation but compromise on power sharing. Similarly, smaller ratios {k s,i /k t,i } improve power sharing, but provide poorer performance for frequency regulation. The gains can be tuned to achieve good frequency regulation performance while ensuring fair power sharing.
Finally, note that despite not considering a deadband in our secondary control architecture we were able to show that the dynamic system is stable and reaches a steady-state.
IV. SIMULATION RESULTS
The simulation test system comprised of a microgrid (MG) with four GFM-inverters, a fixed load, and a variable large load. Lines connecting the inverters are modelled with impedances Z Li . The single line diagram of the MG setup is shown in Figure 3 and nominal values of the modeled MG components are provided in Table I. An averaged model of GFM inverters was used [18], and each inverter had the same LCL filter at its output stage with (L f , C f ) defined in Table I. The inverters were programmed to operate in droop control mode with inner current and outer voltage control loops [19]. This droop-based GFM control architecture was augmented with our DSC scheme. Performance of the proposed DSC scheme was tested in our simulation test system using MATLAB/Simulink [20]. The nominal voltage of the the system was 480 V . For all test scenarios, the inverters were black started with a fixed load (FL) and a large load (LL) step was executed at t = 3 seconds. Each GFM inverter was programmed to have a different frequency offset between ±15 mHz. Unless noted otherwise, the same gain values were used for all GFMs. For For the second test scenario we varied the gains k t to observe the response of the MG test system; see Figures 6 and 7. Observe that for all positive pair of values (k s , k t ) the frequency and secondary reserved power reached a steadystate. For k t = 0 the secondary controller was continuously integrating the frequency offset, hence, the secondary reserve power was never able to reach a steady-state. For small values of k t ≈ 0 that satisfy (26) the steady-state frequency had smaller deviations from the nominal, however, this improvement in the frequency response came at the cost of increased settling time for the secondary reserve power dynamics. For the special case where no secondary control is involved, i.e., k s = 0 and k t = 0, the frequency settled at a much lower value.
For the third test scenario we varied the gains k s to observe Again, observe that for all pair of positive values (k s , k t ) both the frequency and the secondary reserve power reached a steady-state. For smaller values of k s that satisfy (26) the frequency response was poor, however, we observed that there was improved power sharing between the inverters. As observed through the simulations and from Remark 1, there is an inherent trade-off between the gains k s and k t . The ratio of the gains (k s , k t ) can be tuned to achieve the desired objectives.
V. CONCLUSIONS
This paper introduced a decentralized secondary control scheme that is robust to clock synchronization errors and is able to restore the frequency in inverter-based islanded microgrids. A unified control-theoretic approach was utilized to systematically show that the microgrid dynamics are stable and reach a steady-state. The proposed DSC scheme is agnostic to the inverter-type and can be used seamlessly with GFM or GFL inverter architectures. Criteria for designing the secondary control gains such that they provide good performance for frequency regulation while ensuring fair power sharing are also discussed. Current research efforts are focused on testing the novel DSC scheme under mixed setups of GFM and GFL inverters as well as validating the results in our microgrid hardware testbed comprised of several inverter-based resources.
Fig. 1 .
1Control architecture for Grid-following (GFL) inverters.
Fig. 2 .
2Control architecture for Grid-forming (GFM) inverters.
Fig. 4 .
4Frequency at the output of all GFMs for ks = 1000 and kt = 0.25
Fig. 5 . 0 Fig. 6 .
506Secondary reserve power (P s,i ) for ks = 1000 and kt = Frequency at GFM 1 for different values of kt the first test scenario we chose the gain parameters (k s , k t ) such that they satisfied criteria (26).Figures 4 and 5show that both the frequency and the secondary reserve power at all GFMs reached a steady-state.
Fig. 7 .Fig. 8 .
78Secondary reserve power at GFM 1 for different values of kt Frequency at GFM 1 for different values of ks the response of the MG test system; see Figures 8 and 9.
Fig. 9 .
9Secondary reserve power at GFM 1 for different values of ks
TABLE I NOMINAL
IVALUES OF THE MG COMPONENTSSymbol
Description
Nominal value
ωn
Nominal Frequency
377 rad/s
Z L1 , Z L3
Line Impedance
1.6965 + j0.9425 Ω
Z L2 , Z L4 , Z L5
Line Impedance
0.8482 + j0.4712 Ω
L f
All Inverter Filter L
1.125 mH
C f
All Inverter Filter C
11.5 µ F
F L
Fixed Load Power
5 kW
LL
Large Load Power
2.5 kW
P ref,i
Reference Power
2 kW
1/kp
Inverse Droop Gain
1.5%
Ts
Simulation Time Step
20µs
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| [] |
[
"Temporal evolution of the extreme excursions of multivariate kth order Markov processes with application to oceanographic data",
"Temporal evolution of the extreme excursions of multivariate kth order Markov processes with application to oceanographic data"
] | [
"Stan Tendijck \nDepartment of Mathematics and Statistics\nLancaster University\nLA1 4YWUnited Kingdom\n",
"Philip Jonathan \nDepartment of Mathematics and Statistics\nLancaster University\nLA1 4YWUnited Kingdom\n\nShell Research Limited\nSE1 7NALondonUnited Kingdom\n",
"David Randell \nShell Global Solutions International B.V\n1031 HWAmsterdamNetherlands\n",
"Jonathan Tawn \nDepartment of Mathematics and Statistics\nLancaster University\nLA1 4YWUnited Kingdom\n"
] | [
"Department of Mathematics and Statistics\nLancaster University\nLA1 4YWUnited Kingdom",
"Department of Mathematics and Statistics\nLancaster University\nLA1 4YWUnited Kingdom",
"Shell Research Limited\nSE1 7NALondonUnited Kingdom",
"Shell Global Solutions International B.V\n1031 HWAmsterdamNetherlands",
"Department of Mathematics and Statistics\nLancaster University\nLA1 4YWUnited Kingdom"
] | [] | We develop two models for the temporal evolution of extreme events of multivariate kth order Markov processes. The foundation of our methodology lies in the conditional extremes model ofHeffernan and Tawn (2004), and it naturally extends the work of Tawn (2016, 2017) and Tendijck et al.(2019)to include multivariate random variables. We use cross-validation-type techniques to develop a model order selection procedure, and we test our models on two-dimensional meteorological-oceanographic data with directional covariates for a location in the northern North Sea. We conclude that the newly-developed models perform better than the widely used historical matching methodology for these data. | null | [
"https://export.arxiv.org/pdf/2302.14501v1.pdf"
] | 257,232,552 | 2302.14501 | 9e69d9f36e741dfd157a87f23132a6edb8e1687c |
Temporal evolution of the extreme excursions of multivariate kth order Markov processes with application to oceanographic data
March 1, 2023
Stan Tendijck
Department of Mathematics and Statistics
Lancaster University
LA1 4YWUnited Kingdom
Philip Jonathan
Department of Mathematics and Statistics
Lancaster University
LA1 4YWUnited Kingdom
Shell Research Limited
SE1 7NALondonUnited Kingdom
David Randell
Shell Global Solutions International B.V
1031 HWAmsterdamNetherlands
Jonathan Tawn
Department of Mathematics and Statistics
Lancaster University
LA1 4YWUnited Kingdom
Temporal evolution of the extreme excursions of multivariate kth order Markov processes with application to oceanographic data
March 1, 2023extreme value theorytime-seriesMarkov processesoceanography
We develop two models for the temporal evolution of extreme events of multivariate kth order Markov processes. The foundation of our methodology lies in the conditional extremes model ofHeffernan and Tawn (2004), and it naturally extends the work of Tawn (2016, 2017) and Tendijck et al.(2019)to include multivariate random variables. We use cross-validation-type techniques to develop a model order selection procedure, and we test our models on two-dimensional meteorological-oceanographic data with directional covariates for a location in the northern North Sea. We conclude that the newly-developed models perform better than the widely used historical matching methodology for these data.
Introduction
Farmers, stock brokers and sailors have one thing in common: they or their businesses are most heavily affected by extreme events like droughts and rainfall, stock market crashes, or extreme winds and waves, respectively. Understanding the statistical behaviour of such events as a whole is crucial for risk analyses. To make this more precise, if we let (X t ) t∈Z be a stationary d-dimensional random process of interest, then we seek to model excursions of the process in and out of a set E ⊂ R d in time, i.e., the behaviour of
{X i : i = a, . . . , b; X i ∈ E; X a−1 , X b+1 ∈ E},(1)
where E is associated with extreme events of the random variable X which is identically distributed to any X j , j ∈ Z. Moreover, we assume that the random process consists of multiple components that can be extreme. To solve this task, we assume that the multivariate random process is a realisation of a kth order Markov chain. We use extreme value theory, a subfield of statistics, to characterise excursions. There is considerable attention to this area in the literature, but most of extreme value theory for stationary Markov chains dates back over 20 years. Rootzén (1988) and Perfekt (1997) develop limiting results for univariate Markov chains and multivariate Markov chains, respectively. Smith (1992) calculates the extremal index (Leadbetter et al., 1983) for a univariate Markov chain and Smith et al. (1997) use parametric bivariate transition distributions to model the extremes of a univariate first order Markov process. Finally, Yun (2000) develops asymptotic theory for functionals of univariate kth order Markov extreme events. All of these authors derive results under the assumption of asymptotic dependence (Joe, 1997), i.e., for a stationary process (X t ) t∈Z satisfying suitable long-range mixing conditions, under the assumption that for any lag l = 1, 2, . . . lim u→x * P(X t+l > u|X t > u) > 0, where x * is the right upper end point of the distribution of X t . This early work doesn't consider what happens when asymptotic independence is present, i.e., when this limiting probability converges to 0 for some l. The first paper which considers such processes is Bortot and Tawn (1998) who assume a first order Markov model, with Ledford and Tawn (2003) considering a general framework for the modelling of asymptotic independent processes, and key recent probabilistic developments given by Papastathopoulos et al. (2017) and Papastathopoulos et al. (2023). Randell et al. (2015) speculate that a statistical model for the evolution of (multivariate) trajectories would be a valuable enhancement of description of ocean storm events. The first statistical work the current authors are aware of, that defines a model for the distribution of all observations during an excursion is Winter and Tawn (2016), who assume a flexible univariate first order Markov process exhibiting either asymptotic independence or asymptotic dependence across lags. Winter and Tawn (2017) incorporate higher order dependence model to give kth order Markov processes with k > 1. Finally, Tendijck et al. (2019) extend that model to a kth order univariate Markov process with a directional covariate. We remark that their work cannot be considered to model the extremes of bivariate Markov processes since the associated directional covariate does not take on extreme values. Feld et al. (2015) use a sophisticated covariate model for the most extreme observation (the most extreme value of the dominant variable) in an excursion, combined with a historical matching approach for the intra-excursion trajectory; in Section 3.4 we adopt a version of this methodology as a benchmark for our case study. Finally, we mention well-established literature on multivariate time series, e.g., Tiao and Tsay (1989), which is not directly applicable to modelling environmental extremes because such models are only designed to model typical behaviours. Financial timeseries models, e.g., Bauwens et al. (2006), are also not applicable because these are specifically tailored to model data exhibiting volatility, with tail switching during extreme events (Bortot and Coles, 2003).
In this work, we present a natural extension to Tendijck et al. (2019) by defining two multivariate kth order Markov models that exhibit both asymptotic (in)dependence across variables and/or at some lags. The work is motivated by our case study in which we model excursions of meteorological-oceanographic (met-ocean) data: significant wave height, wind speed, and their associated directions, for a location in the northern North Sea.
We use the following set up. Assume that at each time t ∈ Z, the distribution of the d-dimensional random variable X t is stationary through time; that is, X t has the same distribution as some X = (X 1 , . . . , X d ) with distribution function F X . For 1 ≤ j ≤ d, write F Xj as the jth marginal distribution of F X . The distribution functions F Xj are unknown and must be estimated. For extreme arguments of F Xj , we use univariate extreme value theory to motivate a class of parametric tail forms. More precisely, we assume that for each 1 ≤ j ≤ d, the excesses tail above some high level u j ∈ R of the marginal distribution F Xj are approximated with a generalised Pareto distribution (Davison and Smith, 1990). For non-extreme arguments x < u j of the function F Xj , an empirical model usually suffices.
In multivariate extreme value theory, it is common to consider the marginals and the dependence of random variables separately, such that the usually-dominant marginal effect does not influence the modelling of a possibly complex dependence structure. So given the marginal models as discussed above, we transform the random process (X t ) t∈Z onto standard Laplace margins (Y t ) t∈Z using the transformation: X j → Y j := F −1 L (F Xj (X j )), where F −1 L is the inverse of the standard Laplace distribution function. Here the choice of Laplace margins is made to allow for the modelling of potential negative dependence at certain lags or across components (Keef et al., 2013).
For multivariate random processes, there are many ways of defining an extreme event. In our case study, we take the met-ocean variable significant wave height H S as the excursion-defining component. We follow Winter and Tawn (2017) and Tendijck et al. (2019) in adopting the conditional extremes model of Heffernan and Tawn (2004), see also Section 2.2, as the foundation of our approach. Without loss of generality, we first define the component X 1 of X as the defining variable for the extreme events. So, we set our excursion set E = E u := (F −1 X1 {F L (u)}, ∞) × R d−1 for some high threshold u ∈ R + and rewrite our definition of an excursion as
{Y i : i = a, . . . , b; Y i,1 > u; Y a−1,1 ≤ u, Y b+1,1 ≤ u} (2)
for a, b ∈ Z, indices for the start and the end time points of the excursion, respectively. In shorthand, the excursion is then Y a:b . We remark that in this definition, we accept that multiple excursions can occur close together in time, and thus these cannot be considered independent. The reason for this choice is that imposing a minimal separation of excursions would complicate the modelling significantly. We recognize that this is a feature of the current approach which can be improved upon in future work. The remaining part of this paper is organised as follows. In Section 2, we present our strategy for modelling excursions by defining time intervals corresponding to so-called "pre-peak", "peak" and "postpeak" periods, and we present our kth order Markov models for each of these time periods. In Section 3, we apply the two Markov model forms we propose to met-ocean data for a location in the northern North Sea. We compare the model performance with a baseline historical matching approach by assessing their respective performance in estimating the tails of the distributions of complex structure variables (Coles and Tawn, 1994), corresponding to approximations of the response of hypothetical offshore or coastal facilities to extreme met-ocean environments. We find that in general the new models are preferred. 2 The models
Modelling strategy
To model excursions as in definition (2), two types of approaches have been proposed in the literature of univariate extremes: a forward model (Rootzén, 1988) and a peak model (Smith et al., 1997). Both of these are two-step approaches by nature. The forward model first describes the distribution of a random exceedance Y t > u with a univariate extremes model and a conditional model for the distribution for any
j ≥ 1 of Y t+j |(Y t+j−i = y t+j−i i = 1, . . . , j)
where y t > u. Even though this approach does not directly model the univariate equivalent of excursions in formulation (2), estimates of some extremal properties of the process (Y t ) t≥1 , such as the extremal index (Leadbetter et al., 1983), can still be obtained by allowing the excursion threshold to be significantly lower than the cluster threshold used in extremal index estimators. Notably, Tawn (2016, 2017) use the forward approach in their work. The peak model, on the other hand, does model excursions as defined here. This method relies on a univariate extremes model for the largest observation of an excursion, e.g., Eastoe and Tawn (2012), and a conditional model for observations before and after the excursion maximum. Winter and Tawn (2016) use this approach for their first order model but not for their kth order model (Winter and Tawn, 2017). They avoid this method explicitly because of difficulties that arise in preserving model characteristics in forward and backward simulations near the excursion maximum (i.e., the time point at which the defining variate X 1 achieves its maximum value during the excursion). Tendijck et al. (2019) use the peak method, but they do not address the issues associated with forward and backward simulation under the method. Because the excursion maximum is usually the most important observation of an excursion for risk assessments, we also use the peak method in the current work, but with consideration of backward and forward models. We separate the modelling of excursions into three stages: the modelling of the period of the peak, and the modelling of the pre-peak and post-peak periods; see Figure 1 in which the three time periods are illustrated for k = 3. Without loss of generality, let t = 0 be the time point at which the first component Y t,1 takes its maximum value within an excursion such that Y 0,1 > u for the threshold u. The period of the peak P k 0 of an excursion of a kth order model is then defined as the set of 2k − 1 observations:
P k 0 := {Y t : −(k − 1) ≤ t ≤ k − 1} with Y 0,1 > u.
The pre-peak P pre and post-peak P post periods are defined as the sets of observations that include the excursion maximum and the observations before and after, respectively:
P pre := {Y t : t ≤ t ≤ 0, with t = min{s < 0 : min i=s,...,0 {Y i,1 } > u}} and P post := {Y t : 0 ≤ t ≤ t , with t = max{s > 0 : min i=0,...,s {Y i,1 } > u}},
so each of them intersects with P k 0 . The length of P k 0 can be longer or shorter than the length of an excursion if the excursion ends within the period of the peak. We choose to define the period P k 0 in this manner so that the pre-peak and post-peak parts of the excursion are both initialized with k observations.
We then model an excursion as follows: (i) we model the excursion maximum Y 0,1 using a generalised Pareto distribution; (ii) we model the period of the peak P k 0 conditional on the storm maximum Y 0,1 using the model described in Section 2.2; (iii-a) if min j=1,...,k−1 Y j,1 < u (min j=1,...,k−1 Y −j,1 < u), then the period P post (P pre ) of the excursion has ended; (iii-b) if min j=1,...,k−1 Y j,1 ≥ u (min j=1,...,k−1 Y −j,1 ≥ u), then the remaining part of the excursion is modelled with our time-series models from Sections 2.3-2.4 until there exist a j 1 , j 2 > 0 such that Y j1,1 < u and Y −j2,1 < u; (iv) if max −j2≤i≤j1 Y i,1 > Y 0,1 , then the model for the excursion contradicts the definition of the period of the peak of an excursion, and so we reject such occurrences.
In the next sections, we discuss forward models that are applicable to model the post-peak period P post . We model the pre-peak period P pre using the forward models applied to (Y −t ) t∈Z (with potentially different parameters, although these would be the same if the process was time reversible). Importantly, we do not impose consistency in the forward and backward models to yield a kth order Markov chain, e.g., in the case of asymptotic dependent Markov chains the precise dependence conditions between the forward and backward hidden tail chains are given by Janßen and Segers (2014). We make this choice for two reasons: (i) for environmental applications, such as in this work, the pre-peak and post-peak period have different distributions, see for example the asymmetry in Figure 5, which is due to different physics in the growth and decay of a storm; (ii) the assumption of a kth order Markov process is an approximation for the process that generates our data. Thus, imposing forward and backward consistency for a kth order Markov chain is likely to yield worse results for our application. So, we consider the violating of this assumption as a benefit more than a limitation as it can yield more flexible descriptions of excursions.
The conditional extremes model
We introduce the conditional extreme value model of Heffernan and Tawn (2004), henceforth denoted the HT model, with notation specific to modelling the period of the peak P k 0 . The HT model is widely studied and applied to extrapolate tails of multivariate distributions, e.g., in oceanography (Ross et al., 2020), finance (Hilal et al., 2011), spatio-temporal extremes (Simpson and Wadsworth, 2021), and multivariate spatial extremes (Shooter et al., 2022). The HT model is a limit model and its form was originally motivated by deriving possible limiting forms for numerous theoretical examples.
Let
Y −(k−1):(k−1) := Y −(k−1),1 · · · Y −(k−1),d . . . . . . Y k−1,1 · · · Y k−1,d
be a random matrix on R (2k−1)×d with standard Laplace margins (Keef et al., 2013), and define the irregular random matrix Y to be Y −(k−1):(k−1) without the (k, 1)th element Y 0,1 . That is, we define the irregular matrix x ∈ R (2k−1)d−1 as follows:
x = x −k+1,1 x −k+1,2 · · · x −k+1,d . . . . . . . . . x −1,1 x −1,2 · · · x −1,d x 0,2 · · · x 0,d x 1,1 x 1,2 · · · x 1,d . . . . . . . . . x k−1,1 x k−1,2 · · · x k−1,d ,
such that x does not contain the (k, 1)th element. Equivalently, we can write
x = x −(k,1) for x ∈ R (2k−1)×d .
Additionally, we assume that the joint density of Y −(k−1):(k−1) exists. The conditional extremes model for Y, conditional on Y 0,1 , assumes that irregular parameter matrices α ∈ [−1, 1] (2k−1)d−1 , β ∈ (−∞, 1) (2k−1)d−1 and a distribution function H with non-degenerate marginals on R (2k−1)d−1 (the space of irregular matrices) exist, such that for all irregular matrices z ∈ R (2k−1)d−1 the limit
lim u→∞ P Y − αY 0,1 Y β 0,1 ≤ z, Y 0,1 − u > y Y 0,1 > u
exists, assuming component-wise operations, and that
H(z) := lim y→∞ P Y − αY 0,1 Y β 0,1 ≤ z Y 0,1 = y (3)
exists, where α i,j , β i,j and z i,j are the (i, j)th elements of α, β and z, respectively. This then implies, according to l'Hopital's rule, that for
y > 0, z ∈ R (2k−1)d−1 lim u→∞ P Y − αY 0,1 Y β 0,1 ≤ z, Y 0,1 − u > y Y 0,1 > u = H(z) exp(−y).(4)
Limit (4) in turn has the interpretation that as u tends to infinity, (Y − αY 0,1 )Y −β 0,1 and (Y 0,1 − u) are independent conditional on Y 0,1 > u, and are distributed as H and a standard exponential, respectively.
In practice, we exploit these results by assuming they hold exactly above some high finite threshold u > 0. So, we approximate the conditional distribution of Y|Y 0,1 = y for y > u, y ∈ R (2k−1)d−1 as
P(Y ≤ y | Y 0,1 = y) = H y − αy y β ,(5)
and we assume independence of (Y − αY 0,1 )Y −β 0,1 and Y 0,1 . There is no finite-dimensional parametric form for H, so non-parametric methods are typically applied. However, we remark that there are applications of the conditional extreme value model where the copula H is assumed to be Gaussian (Towe et al., 2019) or a Bayesian semi-parametric model is used (Lugrin et al., 2016). For inference, see Section 2.5.
Multivariate Markov extremal model
For ease of presentation, we present the multivariate Markov extremal model (MMEM) of order k only for a two-dimensional time-series (Y t ) t∈Z such that Y t = (Y t,1 , Y t,2 ) in the notation of Section 1, i.e., Y t has standard Laplace margins. We only describe a forward model that is applicable to the post-peak period P post , since the backward model has a similar construction. As mentioned in Section 2.1, we apply a different forward MMEM model to (Y −t ) t∈Z to yield the backward model for the pre-peak period P pre . Concisely put, the MMEM exploits the HT model to estimate the distribution for Y t+k conditional on (Y t , . . . , Y t+k−1 ) when Y t,1 > u for a large threshold u > 0. As in Section 2.2, for each t ∈ Z, we definex t ∈ R k × R k+1 to be an irregular matrix with k + 1 rows and 2 columns without the element that is on the first row and first column:
x t = x t,2 x t+1,1 x t+1,2 . . . . . . x t+k,1 x t+k,2 .
Then, we assume that for a large threshold Winter and Tawn (2017), for t ∈ Z, j ≥ 1 when Y t+j,1 > u, we then get
u > 0, there exist parametersα 0 ∈ [−1, 1] k × [−1, 1] k+1 , β 0 ∈ (−∞, 1) k ×(−∞, 1) k+1 , and a residual random variableε t on R k ×R k+1 with non-degenerate marginals such that for t ∈ ZỸ t |(Y t,1 > u) =α 0 Y t,1 + Yβ 0 t,1ε t . Similar to[Y t+k+j,1 Y t+k+j,2 ]|(Y t+j:t+k+j−1 , Y t+j,1 > u) = [α k,1 , α k,2 ]Y t+j,1 + Y [β k,1 , β k,2 ] t+j,1 · ε C k,1:2 ,
where ε C k,1:2 is short-hand notation for [ε k,1 , ε k,2 ] conditional on (ε 1:k−1,1 , ε 0:k−1,2 ). For inference, we refer to Section 2.5.
Extremal vector autoregression
Here, we introduce extremal vector autoregression (EVAR) for extremes of the process (Y t ) t≥1 . This model combines the HT model with a vector autoregressive model for the joint evolution of the time-series at high levels. Here we focus on the post-peak period, but note that the pre-peak period is modelled analogously. We define an EVAR model of order k with parameters
Φ (i) ∈ R d × R d for i = 1, . . . , k and B ∈ (−∞, 1) d as Y t+k |(Y t , . . . , Y t+k−1 ) = k i=1 Φ (i) Y t+k−i + y B ε t ,(6)
with Y t,1 = y for y > u, where u > 0 is a large threshold and ε t is a d-dimensional multivariate random variable that has non-degenerate margins and is independent of (Y t , . . . , Y t+k−1 ). Usually for a vector autoregressive model, parameter constraints would be imposed so that the resulting process is stationary.
In the current extreme value context, stationarity is not of concern to us, since we reject trajectories that exceed the excursion maximum, and stop the process once the first component dips below threshold u. We define EVAR 0 as a special case of EVAR corresponding to B = 0. EVAR 0 therefore has clear similarities with a regular vector autoregressive model (Tiao and Box, 1981), yet we emphasise that there is considerable difference between the two, since the parameters of EVAR 0 do not need to yield a stationary process, and the parameters of EVAR 0 are estimated using only extreme observations. To estimate the EVAR model, we adopt the same approach as that used to estimate the HT model, see Section 2.5. As explained in Appendix A, the resulting parameter estimatorsΦ (i) are highly correlated. Hence a reparameterisation is introduced to reduce this correlation, and improve inference efficiency and computation. For practical applications, an advantage of EVAR over MMEM is that it provides a lower-dimensional residual distribution when k > 1 (with dimensions d and kd, respectively). As a consequence, the EVAR residual distribution is less affected by the curse of dimensionality. A drawback of EVAR is that it might be insufficiently flexible to describe complex dependence well.
Inference for conditional models
We discuss inference for each of the conditional extremes, MMEM and EVAR models with parameter vector θ. We discuss these together because they can be summarized in the same form. Specifically, let W = (W 1 , . . . , W d ) be a d-dimensional random variable and assume that for some high threshold u > 0,
W 2:d |(W 1 > u) = g 1 (W 1 ; θ) + g 2 (W 1 ; θ)ε(7)
for some parametric functions
g 1 ( · ; θ) : R → R d−1 and g 2 ( · ; θ) : R → R d−1 >0 , where g 1 (x, θ) := (g 1,2 (x, θ), . . . , g 1,d (x, θ)), and g 2 (x, θ) := (g 2,2 (x, θ), . . . , g 2,d (x, θ)), for x ∈ R where ε = (ε 2 , . . . , ε d ) is a (d − 1)-dimensional multivariate
random variable that is non-degenerate in each margin and independent of W 1 . As an example, for MMEM, g 1,j (x) = α j x for some α j and g 2,j (x) = x βj for some β j .
Next, assume that we have n observations D := {w 1 , . . . , w n } of the conditional random variable W|W 1 > u, where w i = (w i1 , . . . , w id ) with w i1 > u for i = 1, . . . , n. We then infer θ by calculating the likelihood of model (7) by temporarily assuming that the ε has a multivariate normal distribution with unknown mean µ = (µ 2 , . . . , µ d ) and unknown diagonal covariance matrix Σ = σ 2 I where σ 2 = (σ 2 2 , . . . , σ 2 d ). These assumptions imply that the mean and the variance of ε are estimated simultaneously with the model parameters. The likelihood is then evaluated as
L(θ, µ, σ 2 ; D) = n i=1 d j=2 1 √ 2πσ j g 2,j (w i1 ; θ) exp − 1 2σ 2 j w ij − g 1,j (w i1 ) − µ j g 2,j (w i1 ; θ) g 2,j (w i1 ; θ) 2 .
Finally, the parametric assumption on the distribution of ε is discarded and estimated conditional on the parametric estimateθ for θ, with a kernel densityĥ 2:d using the 'observations' {ε i : i = 1, . . . , n} where ε i = (ε i2 , . . . , ε id ) and
ε ij := w ij −ĝ 1,j (w i1 ;θ) g 2,j (w i1 ;θ)
for i = 1, . . . , n, j = 2, . . . , d. In case of MMEM, we additionally require estimates for the density of a conditional random variable ε l+1:d|2:l = (ε l+1 , . . . , ε d )|(ε 2 , . . . , ε l ) for some l ∈ {2, . . . , d − 1}. Given the same set of observations, we estimate its conditional density h l+1:d|2:l aŝ
h l+1:d|2:l (ε l+1 , . . . , ε d |ε 2 , . . . , ε l ) =ĥ 2:d (ε 2 , . . . , ε d ) h 2:l (ε 2 , . . . , ε l ) ,
where h 2:l is estimated as the (l − 1)-dimensional marginal ofĥ 2:d .
Case Study -Northern North Sea
Overview
We apply MMEM, EVAR and a historical matching procedure (introduced in Section 3.4, henceforth referred to as HM) to characterise excursions of significant wave height H S and wind speed W s with directional covariates for a location in the northern North Sea. Our goal is to estimate parsimonious predictive models for the joint evolution of H S and W s time-series conditional on H S being large. In Section 3.2, we describe the available met-ocean data. In Section 3.3, we outline a model for the evolution of storm direction that is needed for our time-series models. Section 3.4 then summarises the HM procedure, and in Section 3.5, we introduce structure variable responses that approximate fluid drag loading on a marine structure such as a wind turbine or coastal defence. Finally, in Section 3.6, we compare the predictive performance of MMEM and EVAR (over a set of model orders) with the HM method in estimating structure variables for withheld intervals of time-series.
Data
We have 53 years of hindcast data
D := {(H S,i , W s,i , θ H i , θ W i ) : i ∈ T } indexed with finite T ⊂ Z ≥1
consisting of time-series for four three-hourly met-ocean summary statistics at a location in the northern North Sea (Reistad et al., 2009): significant wave height (H S,i in metres), wind speed (W s,i in metres per second), wave direction (θ H i in degrees) and wind direction (θ W i in degrees) for each i ∈ T . To use MMEM and EVAR, we transform significant wave height and wind speed onto Laplace marginals: H S,i |θ H i → H L S,i and W s,i |θ W i → W L s,i , e.g., using directional marginal extreme value models for the tails (Chavez-Demoulin and Davison, 2005), but ignoring seasonality. This part of the analysis has been reported on numerous occasions, see for example Randell et al. (2015). Because the marginal transformation includes direction as a covariate and because direction is not constant during an excursion, we also establish a model for the directional evolution of excursions in order to transform them between standard and original margins, see Section 3.3.
Let D L be the collection of the transformed data
D L := {(H L S,i , W L s,i , θ H i , θ W i ) : i ∈ T }.
To define excursions in D L , we set the excursion threshold u equal to the 95% percentile of a standard Laplace distribution, i.e., u ≈ 2.3, yielding 1, 467 observations of extreme excursions E u . This choice of threshold is not unusual as similar conclusions are drawn for excursion thresholds that are slightly different from our original choice. Figure 2 shows four intervals of the time-series chosen to contain the observations corresponding to the 100%, 95%, 90% and 85% sample percentiles of the set of excursion maximum significant wave heights, on original and standard Laplace margins, with directional covariates. Excursions are centred around extreme events. There is a large dependence of H S and W s on both original and standard margins. Moreover, variables associated to significant wave height, i.e., H S , H L S and θ H , are much smoother than their wind speed counterparts. Additionally, the directional covariates θ H and θ W centre around each other with no large deviations during extreme events.
In Figure 3, we visualize the (across variable joint) dependence of key variables H L S and W L s on Laplace scale at time lags up to lag 4 using a series of scatterplots where a unit of lag corresponds to three hours of observation time. The figure illustrates the complex dependence of the bivariate time-series of significant wave height and wind speed on Laplace margins. As expected, we observe (slow) convergence to an independent variable model as lag increases. Most notably, we observe a similar level of dependence of (H L S,t , W L s,t+4 ) and (W L s,t , W L s,t+4 ) which suggests counter-intuitively that H L S,t would be a better predictor for W L s,t+4 than W L s,t . In Figure 4, we plot (cross) correlation functions for these variables, and also for the change in directional covariates at various lags. These show that the dependence of (H L S,t , H L S,t+τ ) decays relatively slowly as τ grows to 90 hours, and that indeed the cross dependence between (H L S,t , W L s,t+τ ) is larger than the dependence of (W L s,t , W L s,t+τ ) for large τ . Finally, the correlation plot of the change in directional covariates ∆θ H S,i := (θ H S,i+1 − θ H S,i , mod 360) and ∆θ W s,i := (θ W s,i+1 − θ W s,i , mod 360) on the right shows that a first order model for these covariates is appropriate since the correlations nearly vanish at lag 2 (for wind and wave) or 6 hours (for all other combinations).
Directional model
We model wave direction θ H i in a similar fashion as Tendijck et al. (2019), summarised as follows. Let I ⊂ T be the set of indices of the original data that correspond to all observations of any excursion. Next, let {d(θ H i+1 , θ H i ) : i ∈ I} be the set of changes in wave directions, where d(θ, θ ) = (θ − θ + 180; mod 360) − 180 ∈ [−180, 180) denotes the circular difference of θ and θ in degrees. In our application, the set of changes in wave directions during excursions do not contain values close to −180 or 180. In particular, all of the observed changes centre around 0.
For
i ∈ I, we transform observations d(θ H i+1 , θ H i ) → δ H i := Φ −1 (F (d(θ H i+1 , θ H i ))
) on Gaussian margins, whereF denotes the empirical distribution function of the set of changes in wave directions. Assume that {δ H i : i ∈ I} are realisations of the random variables {∆ H i : i ∈ I}. We estimate the following autoregressive model for ∆ H t of order p 1 = 1, 2, 3, . . . with parameters ϕ H j ∈ R for j = 1, . . . , p 1 as
∆ H t |(∆ H t−1 , . . . , ∆ H t−p1 ) = p1 j=1 ϕ H j ∆ H t−j + ζ(H S,t )ε t ,(8)
where ε t is a standard Gaussian random variable, and standard error ζ(h) is given by
ζ 2 (h) = λ 1 + λ 2 exp(−λ 3 h)
with λ j > 0 for j = 1, 2, 3, see Tendijck et al. (2019). In particular, the standard error ζ(h) decays as h grows due to the significantly larger amounts of energy needed to change the direction of more severe sea states. The parameters of this model are inferred with maximum likelihood, and in contrast to the inference discussed in Section 2.5, we do not reject the assumption that ε t is a standard Gaussian. In practice, we use p 1 = 1 in line with Tendijck et al. (2019). Given model (8), we propose the following model
θ W t = θ H t + γ t mod 360(9)
for wind direction θ W t conditional on wave direction θ H t , where γ t is a zero-mean stationary AR(p 2 ) process. That is, there exist parameters ϕ W j ∈ R, 1 ≤ j ≤ p 2 , and a non-degenerate residual distribution r t independent of γ t−j for j ≥ 1, such that
γ t |(γ t−1 , . . . , γ t−p2 ) = p2 j=1 ϕ W j γ t−j + r t ,
and such that the polynomial 1 − p2 j=1 ϕ W j z j has roots outside the unit circle. The model parameters and the distribution of r t are inferred as described in Section 2.5 conditional on the model order p 2 , which is selected by investigating the correlation function in Figure 4 and the partial autocorrelation function of γ t (not reported). In our application, we conclude that p 2 = 1 is sufficient.
Historical matching
An empirical method for simulating excursions is described in Feld et al. (2015) and termed historical matching (HM) in this work. They model trajectories of significant wave height, wave direction, season and wave period during extreme events. The key assumption they make is that storm trajectory (or excursion) profiles are not independent of storm maximum conditions. Specifically, the HM approach is a composition of four models: (i) a model for storm maximum wave direction; (ii) a model for storm maximum significant wave height conditional on storm maximum wave direction; (iii) a model that selects at random a historical storm trajectory with similar storm maximum characteristics to that simulated; (iv) a model that adjusts the historical storm trajectory by matching storm maximum characteristics of simulated and historical storms.
Specific details of the individual models are as follows, but this level of detail is not required for understanding the impact of the core methodology developments in Section 3. For model (i), we simply sample at random from the observed wave directions associated with storm maximum significant wave height (excursion maximum). In model (ii), storm maximum significant wave height are modelled using a generalised Pareto distribution conditional on the sampled storm maximum wave direction using a generalised additive model with the parameters as B-splines conditional on directional covariates (Chavez-Demoulin and Davison, 2005). In model (iii), we use a distance measure to calculate the dissimilarity between pairs of storm maximum significant wave heights and storm maximum wave directions for simulated and historical trajectories. Here, we use the heuristic recommended by Feld et al. (2015) ensuring that a difference of 5 degrees in storm maximum wave direction corresponds to the same dissimilarity as 0.5m of difference in storm maximum significant wave height; one of the closest 20 matching storms is then selected at random for associated with the simulated storm maximum. In model (iv), we match the variables of the chosen historical trajectory as follows: (a) the historical significant wave height series are multiplied by the ratio of the simulated maximum significant wave height to the maximum of the historical significant wave height; (b) the historical wave directions are shifted such that the storm maximum wave directions of simulated and historical trajectories coincide; (c) the associated historical wind directions are rotated in the exact same way as wave direction; (d) for the full set of historical storm maxima, storm maximum associated wind speed W M s (namely the value of wind speed at the time point corresponding to the storm maximum event) conditional on storm maximum significant wave height H M S is described using linear regression with parameters β 0 , β 1 ∈ R, σ > 0:
W M s |H M S = β 0 + β 1 H M S + σε
with ε a standard normal random variable; (e) wind speed for the selected historical trajectory is scaled linearly such that it agrees with the storm maximum associated wind speed from (d).
Perhaps the main deficiencies of the HM approach are (i) it does not provide a means for modelling the extremal temporal dependence characteristics of excursions, and the extremal dependence between different components of the time-series for excursions to levels beyond those observed in the historical sample, and (ii) it does not provide a model framework for the assessment of fit or uncertainty propagation.
Response variable
To measure the practical impact of extreme met-ocean excursions, we define structure response variables for a simple hypothetical marine offshore facility. A structure response variable is a function of the met-ocean variables, key to assessing the integrity of the design of a physical structure of interest. Specifically, we consider a structure in the form of a unit cube standing above the water, supported by thin rigid legs, with vertical cube faces aligned with cardinal directions. Only wave and wind impact on the cube itself is of interest to us, and we neglect the effects of other oceanic phenomena such as swell, surge, tide, and potential climate non-stationarity. For simplicity, we also assume that when H S < h, for some known value h > 0, the wave impact on the structure is negligible, and structural response is dominated by wind. When H S ≥ h, we assume that wave impact increases cubically with H S and quadratically with W s (see Morison et al. 1950 and Ma and Swan 2020 for supporting literature). Hence, the impact of an extreme excursion on the structure is defined by the instantaneous response variable R
R(H S , W s , θ H , θ W ; c, h) = c · I 2 W (W s , θ H − θ W ) for H S < h, c · I 2 W (W s , θ H − θ W ) + A(θ H ) · (H S − h) · H 2 S for H S ≥ h, where I W : R >0 × [−180, 180) → R is the inline wind-speed, defined below, A : [−180, 180) → [1,
√ 2] is the exposed cross-sectional area of the cube, see below, and the parameter c > 0 is specified such that both significant wave height and wind speed have an approximately equal contribution to the largest values of R.
Here both c and h are values that can be changed by altering structural features. The exposed cross-sectional area A(θ) ∈ [1,
√ 2] of the cube is given by
A(θE u := {(H S,i , W s,i , Θ H i , Θ W i ) : a ≤ i ≤ b},
for some a < b such that for a threshold u > 0 (on Laplace margins) H L S,i > u for a ≤ i ≤ b and H L S,a−1 , H L S,b+1 ≤ u. Next, let i * := i * (E u ) be the time of the excursion maximum, i.e., H S,i * is the maximum of H S,i over E u . We define two natural structure response variables representing the maximum impact of an excursion max {a≤i≤b} R i (c, h), and the cumulative impact of an excursion {a≤i≤b} R i (c, h), respectively. For our application, we consider slight alterations R max (c, h, E u ) and R sum (c, h, E u )
R max (c, h, E u ) := max {a≤i≤b, |i−i * |>2} R i (c, h), R sum (c, h, E u ) := {a≤i≤b, |i−i * |>2} R i (c, h).
That is, we consider responses that do not depend directly on the characteristics of the excursion near to the excursion maximum, to exaggerate the dependence of the structure variables on pre-peak and post-peak periods compared to the period of the peak, and hence the importance of estimating good models for the pre-peak and post-peak periods using MMEM or EVAR. Moreover, we define R max (c, h) and R sum (c, h) as the random structure responses related to a random excursion.
Model comparisons
Here, we use our time-series models to characterise extreme excursions for the met-ocean data D of Section 3.2 with structure responses R max and R sum . First, we investigate the model fits, then we describe our model comparison procedure, and finally we assess model performance using a visual diagnostic. We fit EVAR, EVAR 0 and MMEM with model orders k = 1, 2, . . . , 6 to data D L . The fitting of these 18 models is a two-stage procedure. In the first part, we fit (six) conditional extremes models for the period of the peak P k 0 for each k. In the second part, we fit 2 · 18 = 36 models to the pre-peak P pre and post-peak P post periods. In Table 1, we report parameter estimates of the period of the peak model, and in Tables 2-3, we report parameter estimates of MMEM on P post and P pre , respectively. Finally, we report parameter estimates of EVAR on P post and P pre in Tables 4-5, respectively. These indicate that all models agree on some level of asymptotic independence at each lag (coefficients ofα 0 are less than 1) with decreasing levels of dependence as lag increases, which can be seen by decreasing coefficients ofã 0 for entries further down the table. We remark that for EVAR(2) on P pre , the coefficient of H S at time t + 1 (0.96) is larger than the coefficient of W s at time t + 1 (0.50) for estimating W s at time t + 2. This has the interpretation that significant wave height might be a better predictor for wind speed than wind speed itself, also suggested by Figure 4.
For each of the 18 models and HM, we simulate 20, 000 excursions to estimate model properties. First, we illustrate model characteristics for EVAR(4) in Figure 5 by plotting simulated excursions such that the excursion maximum significant wave height takes on values between 11.5m and 12.5m (left). We visually compare these with observed excursions for the same interval of excursion maxima (middle). On the right, we summarize simulated and observed excursions in terms of the median, the 10% and 90% percentiles of the sampling distribution at each time period. Finally, in the bottom panel we plot P min{H L S,i : i = min(0, τ ), . . . , max(0, τ )} > u H S,0 ∈ [11.5, 12.5] ,
for τ ∈ Z, i.e., we plot the survival probability for an excursion relative to the time of the excursion maximum, conditional on the excursion maximum taking a value between 11.5m and 12.5m for both the Table 1: Estimates of model parameters α and β for the period of the peak P k 0 with model order k = 4. Also shown in parentheses are 90% bootstrap confidence intervals. The structure of the irregular matrix estimates of α and β is explained in Section 2.2. Table 2: Estimates of MMEM model parametersα 0 andβ 0 with model order k = 4 for P post . Also shown in parentheses are 90% bootstrap confidence intervals. The structure of the irregular matrix estimates ofα andβ is explained in Section 2.3. Table 3: Estimates of MMEM model parametersα 0 andβ 0 with model order k = 4 for P pre . Also shown in parentheses are 90% bootstrap confidence intervals. The structure of the irregular matrix estimates ofα andβ is explained in Section 2.3.
EVAR(1)
Φ ( -0.46 (-0.53, -0.41) 0.11 (0.09, 0.13) -0.67 (-0.76, -0.59) 0.10 (0.07, 0.14) B 0.53 (0.51, 0.64) 0.29 (0.26, 0.43) Table 5: Estimates of EVAR model parameters (Section 2.4) with model order k = 1 (left), 2 (right) for P pre . Also shown in parentheses are 90% bootstrap confidence intervals.
simulated excursions and the observed excursions. We observe good agreement in the distribution of the length of an excursion with respect to the excursion maximum as both estimates are close to each other.
In the supplementary material, we produce analogous plots for each of the 18 models considered and HM. We observe that EVAR(4) characterizes the period of the peak, and also the pre-peak and post-peak periods of the excursion well. Moreover, EVAR(4) also reproduces the observed excursion survival probability.
Next, in Figure 6, we plot estimates of conditional probabilities χ H (u, l) := P(H L S,t+l > u | H L S,t > u), χ HW (u, l) := P(W L s,t+l > u | H L S,t > u), and χ W (u, l) := P(W L S,t+l > u | W L S,t > u) using EVAR, MMEM and HM with model orders 1 and 4, and we compare these with empirical estimates. 1 We make the following observations: HM is significantly worse at characterizing each of χ H , χ W and χ HW compared to EVAR and MMEM. Moreover, estimates obtained from EVAR of large enough order, e.g., k ≥ 4, agree well with empirical estimates. MMEM, on the other hand, yields estimators that are slightly positively biased. In particular, larger model orders yield considerable improvements.
In Figure 6, we discuss goodness-of-fit of each of the models. To compare MMEM and EVAR with each other and with HM, we take a similar approach to Gandy et al. (2022), who adjust standard cross-validation techniques to extreme value applications by taking a small training set and a larger test set. We select at random 25% of the observed excursions for our training sample; the remaining 75% forms our test sample. Below, we calculate performance statistics for the response variables by averaging over 50 such random partitions of the sample.
For training, we fit EVAR, EVAR 0 and MMEM with model orders k = 1, 2, . . . , 6 as explained in the second paragraph of this section. For each of the 18 models and HM, we simulate 20, 000 excursions, calculate structure response variables R max and R sum , and compare distributions of simulated structure response variables with those corresponding to the withheld test data. This is achieved by defining a dissimilarity distance function D that measures the level of difference in tails of distribution functions. We select 20 equidistant percentiles p 1 , . . . , p 20 ranging from 97% to 99.9% corresponding to moderately extreme to very extreme levels with respect to the (smaller) training sample but not too extreme for the (larger) withheld data. We define the distance D of distribution functions F M (of model M ) and F E (an empirical distribution function) as the mean absolute relative error over these percentiles, i.e.,
D(F M , F E ; p 1 , . . . , p 20 ) = 1 20 20 j=1 F −1 E (p j ) − F −1 M (p j ) F −1 E (p j )
.
We remark that in the above definition, we never divide by zero because we only use D to measure the dissimilarity of distributions of positive random variables. In Figure 7, we show the results for the 50 random partitions of the original sample by plotting the average distance D (with 80% confidence intervals) for each model together with HM for four different structure response variables corresponding to two choices of c and h for each of R max and R sum . Note that similar studies for other values of c and h for R max and R sum were examined, and general findings are consistent with those illustrated in Figure 7. For legibility, we omit confidence bands for EVAR 0 since the , and data (middle; right) on original margins such that storm peak significant wave height is in [11.5, 12.5]; (right) summaries of the data (black) and EVAR(4) (red) excursions: median (solid), and the 10% and 90% quantiles (dashed). In the bottom panel, we plot survival probabilities for observed (black) and EVAR(4) (red) excursions relative to the time of the excursion maximum, see equation (10). difference with EVAR is minimal. Model selection now involves choosing the model that yields the smallest average dissimilarity D whilst keeping the model order as low as possible.
We make a number of observations. For the R max response, EVAR and MMEM clearly outperform HM regardless of model order. However, for the R sum response, high order (e.g., k = 4, 5, 6) EVAR and MMEM are necessary to be competitive with HM. We observe also that performance of EVAR and MMEM does not significantly improve or worsen for k > 4. This finding is further supported with an unpublished study with Markov model orders of k ≤ 10. We note that llustrations of excursions in the supplementary material demonstrate that MMEM(1) does not explain the variability of the pre-peak and post-peak periods well.
By looking at the average relative errors in R max and R sum of our proposed selection of methods, we conclude that a third or fourth order MMEM and a fourth order EVAR are competitive models within their class. Since these models have similar performance, we prefer EVAR(4) because of its simpler twodimensional residual distribution. Figure 7: Average mean relative errors of HM, EVAR, EVAR 0 and MMEM (dashed/dotted) and 80% confidence regions (shaded) for estimating the distribution of structure responses using 25% of data for training and 75% of data for testing. For details, see the text.
Conclusions and discussion
In this paper, we provide models for extreme excursions of multivariate time-series. Excursions are characterized by a three-stage modelling procedure for the period of the peak, the pre-peak and the post-peak periods. We model the period of the peak using the conditional extremes framework (Heffernan and Tawn, 2004), and for the pre-peak and post-peak periods, we define two classes of time-series models: MMEM, motivated by the Markov extremal model of Winter and Tawn (2017); and EVAR, an extreme-value extension of a vector autoregressive model. We compare these excursion models with a baseline historical matching method, motivated by Feld et al. (2015). We find that the excursion models -for a reasonably informed choice of k, the order of the Markov process -are at least competitive with historical matching and often outperform it in the estimation of the tail of a range of notional structure response variables for a met-ocean application in the northern North Sea.
Statistical modelling of extreme excursions of multivariate time-series is difficult as it requires the estimation of complex model forms. MMEM requires the estimation of the conditional distribution of highdimensional residual random variables and EVAR is highly parameterized. Nevertheless, for realistically sized directional samples of significant wave height and wind speed time-series, we found that MMEM(3), MMEM(4) and EVAR(4) perform well. Even when the empirical historical matching procedure is competitive, adoption of an excursion model is advantageous because it allows for rigorous uncertainty quantification. We expect that our excursion models are applicable more generally, e.g., for the modelling of higher-dimensional met-ocean time-series and spatial fields.
We model wind speed and significant wave height marginally conditional on directional covariates. However, we did not investigate the explicit effect of the directional components on the dependence models. Since, we remove the marginal effect of direction before modelling the dependence, we do not expect this covariate to have a significant impact on the dependence. However, it would be very interesting to adapt our models to be able to investigate this further in future research.
A Reparameterization of EVAR
As opposed to inference for vector autoregressive models, we cannot estimate the EVAR parameters by least squares due to the presence of the Y B t,1 term. Instead, we apply the inference methodology discussed in Section 2.5. Not surprisingly, the parameter estimatesΦ (i) for i = 1, . . . , k are highly intercorrelated because of the linear dependence between the components of Y t−1 , . . . , Y t−k . Reparameterization to reduce the correlation between parameter estimators is therefore attractive.
To reparameterize the model, we proceed as follows. First, we assume that the conditional extremes model is applicable to Y t−i,j conditional on Y t−k,1 for each i = 0, . . . , k and j = 1, . . . , d apart from (i, j) = (k, 1), i.e., there exist parameters α i,j ∈ [−1, 1] and β i,j < 1 such that
lim y→∞ P Y t−i,j − α i,j y y βi,j ≤ x Y t−k,1 = y = H i,j (x),
where H i,j is a non-degenerate distribution function. Following the EVAR model (6), we now must have
Y t+k,1 = Φ (1) 1,1 Y t+k−1,1 + · · · + Φ (1) d,1 Y t+k−1,d + · · · + Φ (k) 1,1 Y t,1 + · · · + Φ (k) d,1 Y t,d + Y B1 t,1 ε t,1 = Φ (1) 1,1 α k−1,1 + · · · + Φ (1) d,1 α k−1,d + · · · + Φ (k) 1,1 + · · · + Φ (k) d,1 α 0,d Y t,1 + o p (Y t,1 )
conditional on Y t,1 > v as v tends to infinity. On the other hand, we have Y t+k,1 |(Y t,1 > v) = α 0,1 Y t,1 + o p (Y t,1 ). So, α 0,1 = Φ
1,1 α k−1,1 + · · · + Φ (1) d,1 α k−1,d + · · · + Φ (k) 1,1 · 1 + · · · + Φ (k) d,1 α 0,d , which explains the collinearity of the estimators. We now propose the following reparameterization (B,Φ (1) , . . . ,Φ (k) ).
For each 1 ≤ l ≤ d, we acquireΦ (k−i) j,l , i.e., the (j, l)th element ofΦ (k−i) , inductively with 0 ≤ i ≤ k − 1, 1 ≤ j ≤ d. whereα i,j is the maximum likelihood estimate for α i,j . Under this reparametrization, estimators ofΦ (i) j,k are less correlated, which we demonstrated in unreported experiments comparing the dependence of the original and the reparameterized parameters using adaptive MCMC methodology (Roberts and Rosenthal, 2009).
Supplementary Information
The supplementary information consists of a series of figures following the format of Figure 5 of the main text, for different model choices.
Figure 1 :
1Illustration of the periods of the pre-peak, peak and post-peak periods for two excursions from a Markov model with order k = 3.
Figure 2 :
2Intervals of oceanographic time-series: (top) key variables: significant wave height H S,i and wind speed W s,i on original margins; (middle) on Laplace margins; (bottom) covariates: wave direction θ H i and wind direction θ W i . The four columns correspond to time periods that contain the 100%, 95%, 90% and 85% empirical percentiles of H S,i , respectively.
Figure 3 :Figure 4 :
34Matrix plot of observed H L S,i and W L s,i at various time lags up to lag 4 (corresponding to 12 hours in real time) including cross dependece. Estimated correlation and cross-correlation at various time lags of: (left) the key variables on Laplace margins: H L S,i and W L s,i ; (right) the covariates: change in wave direction ∆θ H i := (θ H i+1 − θ H i , mod 360), change in wind direction ∆θ W i := (θ W i+1 − θ W i , mod 360) and γ i , see definition (9).
Figure 5 :
5Excursions of H S and W s from EVAR(4) model (left; black)
Figure 6 :
6Estimates of measures of extremal dependence across time lags 1 and 4, and variables given by χ H , χ HW and χ W (left, middle, and right respectively) for each of the models: EVAR (red), MMEM (blue), HM (green), data (grey). For EVAR and MMEM, we plot these estimates for different model orders k = 1 and k = 4 with line types: one (solid), four (dotted). Moreover, the grey region depicts the confidence bounds for empirical estimates of these extremal dependence measures from the data.
,
for i = 0, . . . , k − 1, j = 2, . . . , d, conditional onΦ
Figure 9 :Figure 10 :Figure 11 :Figure 12 :Figure 13 :Figure 14 :Figure 15 :Figure 16 :Figure 17 :Figure 18 :Figure 19 :Figure 20 :
91011121314151617181920EVAREVAREVAREVAREVARMMEMMMEMMMEMMMEMMMEMMMEMHM
for a given wave direction θ H . The inline wind-speed I W is the component of the wind speed in the direction of the wave given by I W (W s , θ H − θ W ) = W s cos((θ H − θ W ) · π/180). for i ∈ T . To define a structure response for a complete excursion E u , we writeH ) := 1/ cos([(θ H + 45; mod90) − 45] · π/180)
To simplify notation, we write R i (c, h) := R(H S,i , W s,i , θ H
i , θ W
i ; c, h)
Table 4 :
4Estimates of EVAR model parameters (Section 2.4) with model order k = 1 (left), 2 (right) for
P post . Also shown in parentheses are 90% bootstrap confidence intervals.
We leave out EVAR 0 in this analysis for conciseness since its estimates are very similar to the estimates obtained using EVAR of the same model order.
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| [] |
[
"ON THE INTERSECTION FORM OF FILLINGS",
"ON THE INTERSECTION FORM OF FILLINGS"
] | [
"Zhengyi Zhou "
] | [] | [] | We prove, by an ad hoc method, that exact fillings with vanishing rational first Chern class of flexibly fillable contact manifolds have unique integral intersection forms. We appeal to the special Reeb dynamics (stronger than ADC in [25]) on the contact boundary, while a more systematic approach working for general ADC manifolds is developed independently by Eliashberg, Ganatra and Lazarev[18]. We also discuss cases where the vanishing rational first Chern class assumption can be removed. We derive the uniqueness of diffeomorphism types of exact fillings of certain flexibly fillable contact manifolds and obstructions to contact embeddings, which are not necessarily exact. | null | [
"https://export.arxiv.org/pdf/2208.07018v2.pdf"
] | 251,564,590 | 2208.07018 | 1ccb62f3c6e9ae790dc4a7fdba519b89d3e293b5 |
ON THE INTERSECTION FORM OF FILLINGS
24 Apr 2023
Zhengyi Zhou
ON THE INTERSECTION FORM OF FILLINGS
24 Apr 2023arXiv:2208.07018v2 [math.SG]
We prove, by an ad hoc method, that exact fillings with vanishing rational first Chern class of flexibly fillable contact manifolds have unique integral intersection forms. We appeal to the special Reeb dynamics (stronger than ADC in [25]) on the contact boundary, while a more systematic approach working for general ADC manifolds is developed independently by Eliashberg, Ganatra and Lazarev[18]. We also discuss cases where the vanishing rational first Chern class assumption can be removed. We derive the uniqueness of diffeomorphism types of exact fillings of certain flexibly fillable contact manifolds and obstructions to contact embeddings, which are not necessarily exact.
Introduction
In his seminal work of introducing pseudo-holomorphic curves into symplectic geometry, Gromov [22] proved the uniqueness of exact fillings of (S 3 , ξ std ). In higher dimensions, the celebrated Eliashberg-Floer-McDuff theorem [27] asserts the uniqueness of diffeomorphism type for exact fillings of (S 2n−1 , ξ std ) whenever n ≥ 3. Staring from those monumental results in late 1980s -early 1990s, understanding the uniqueness of exact fillings of some contact manifolds has been a fundamental and influential question. In dimension 3, the intersection theory of holomorphic curves can be used to construct foliations of symplectic fillings. A landmark result is Wendl's theorem on planar contact 3-folds [36], which translates the classification of symplectic fillings into factorizations in mapping class groups. In higher dimensions, only "homological" foliations by holomorphic curves can be obtained, just like the Eliashberg-Floer-McDuff theorem compared to Gromov's theorem. Based on "homological" foliations, various generalizations of the Eliashberg-Floer-McDuff theorem were obtained, e.g. Oancea-Viterbo [30] and Barth-Geiges-Zehmisch's [3] works on topological uniqueness of exact fillings of subcritically fillable contact manifolds, Bowden-Gironella-Moreno [9] and Geiges-Kwon-Zehmisch's [19] works on topological uniqueness of exact fillings of the cosphere bundle S * T n . On the other hand, we studied the filling question from the perspective of Floer theories and obtained various uniqueness results including uniqueness on cohomology groups [37,38] or rings [43], diffeomorphism types [42,39] as well as certain properties of the symplectic cohomology [38,41]. In this note, we show the uniqueness of the integral intersection form for exact fillings of some flexibly fillable contact manifolds, which shall yield uniqueness of diffeomorphism types in some cases. Theorem 1.1. Let (Y 2n−1 , ξ) be a flexibly fillable contact manifold with the rational first Chern class c Q 1 (ξ) = 0, then the integral intersection form of any exact filling W with c Q 1 (W ) = 0 is isomorphic to the integral intersection form of the flexible filling W 0 . Theorem 1.3 below lists some cases where the c Q 1 (W ) = 0 assumption can be dropped. The idea of the proof is as follows: take two closed cycles A, B in H n (W ; Z), under the transversality assumption, the intersection number A · B = #(A ⋔ B), where the later is a finite set of oriented points. By [38], we have SH * (W ; Z) = 0 for any topologically simple exact filling (i.e. c Q 1 (W ) = 0 and H 1 (Y ; Q) → H 1 (W ; Q) is injective). Geometrically, it means that there is one curve counted algebraically passing through a fixed point. If we choose the point constraint to be A ⋔ B, this allows us to view A · B as the counting of one boundary component of a 1-dimensional moduli space. Then we can establish the independence result by looking at the other boundaries and applying neck-stretching. It is important that Y is flexibly fillable rather than just being asymptotically dynamically convex (ADC) [25,Definition 3.6], as we need to use the special Reeb dynamics on the boundary of a flexibly fillable contact manifold constructed by Lazarev [25]. On the other hand, a more systematic approach is independently developed by Eliashberg, Ganatra and Lazarev [18] using the secondary coproduct defined by Ekholm and Oancea [17]. In particular, their results apply to general ADC contact manifolds. Our approach is ad hoc in the sense that we present the intersection number as counting of holomorphic curves contained in the cylindrical end of the filling. It is the combination of such curves carries the information, while some moduli spaces appear in the process do not bear meanings as structural maps in symplectic cohomology or symplectic field theory. We discuss some conjectural relations between two approaches at the end of §3.
1.1. Dropping the topologically simple assumption. The topologically simple condition (c Z 1 (W ) = 0,π 1 (Y ) → π 1 (W ) injective) in [37,38] was used to make sure the grading of symplectic cohomology of a filling is consistent with the grading induced from the contact boundary, so that we can do effective dimension computation after neck-stretching. Results in [37,38] hold under the condition c Q 1 (W ) = 0, H 1 (Y ; Q) → H 1 (W ; Q) injective by considering symplectic cohomology generated by orbits that are trivial in H 1 (W ; Q) instead of only those contractible in W as in [37,38]. We first note that the H 1 (Y ; Q) → H 1 (W ; Q) injective condition can be dropped for flexibly fillable contact manifolds (but not for general ADC manifolds).
Proposition 1.2. Let (Y 2n−1 , ξ) be a flexibly fillable contact manifold with c Q 1 (ξ) = 0. For any exact filling W of Y with c Q 1 (W ) = 0, we have that H * (W ; Z) → H * (Y ; Z) is isomorphic to H * (W 0 ; Z) → H * (Y ; Z) for the exact filling W 0 .
Following a different perspective which was used in [39], by exploiting the degeneracy of a spectral sequence from the "boundary" grading, we can drop the c Q 1 (W ) = 0 condition in some cases. Theorem 1.3. Let (Y 2n−1 , ξ) be a flexibly fillable contact manifold with c Q 1 (ξ) = 0.
(1) If H n (W 0 ; Q) → H n (Y ; Q)
is injective for the flexible filling W 0 , then for any exact filling W of Y ,
we have H * (W ; Z) → H * (Y ; Z) is isomorphic to H * (W 0 ; Z) → H * (Y ; Z).
In particular, c Q 1 (W ) = 0. (2) If n is even, then for any exact filling W of Y , we have SH * (W ; Z) = 0 and dim ⊕ 2n
* =0 H * (W ; Q) ≤ dim ⊕ 2n * =0 H * (W 0 ; Q). Moreover, (a) if H 1 (W 0 ; Q) = 0, then dim ⊕ 2n * =0 H * (W ; Q) = dim ⊕ 2n * =0 H * (W 0 ; Q); (b) if moreover dim ker(H n (W 0 ; Q) → H n (Y ; Q)) = 1, then H * (W ; Z) → H * (Y ; Z) is isomorphic to H * (W 0 ; Z) → H * (Y ; Z).
In particular, c Q 1 (W ) = 0. However, in the H n (W 0 ; Q) → H n (Y ; Q) injective case, the intersection form of W 0 is necessarily trivial.
1.2.
Uniqueness of diffeomorphism types. Using Theorem 1.3, we can get the following uniqueness result of diffeomorphism types for symplectic fillings following the same topological argument in [38,42]. Theorem 1.4. Let Q be a closed manifold such that π 1 (Q) is abelian and χ(Q) = 0, then the interior of any exact filling of the contact boundary of F lex(T * Q) (the flexible version of T * Q) is diffeomorphic to T * Q as an open manifold.
For highly connected manifolds, the diffeomorphism type is restricted to a large extend by its intersection form, c.f. [33]. In view of this, we have the following corollary of Theorem 1.1. Theorem 1.5. If n ≡ 6 mod 8 and W 2n 0 only has flexible n-handles except for the 0-handle such that the intersection form is even, then any exact filling W of ∂W 0 with c Q 1 (W ) = 0 is diffeomorphic to W 0 . If moreover the intersection form W 0 is of rank 1, then any exact filling W of ∂W 0 is diffeomorphic to W 0 .
It is worth-noting that the uniqueness of diffeomorphism type above follows from completely different topological argument compared to results in [3,9,19,27,42] as well as Theorem 1.4, where the uniqueness hinges on H * (W ; Z) → H * (Y ; Z) being injective and the h-cobordism theorem, which is rarely the case in Theorem 1.5. Moreover, Wall [35] proved that the diffeomorphism type of a smooth, closed, oriented, (n − 1)-connected 2n-manifold of dimension at least 6 is determined, up to connected sum with a homotopy sphere, by the middle homology group, the intersection pairing, and the so-called normal bundle data. It is an interesting question to study whether such normal bundle data can be approached symplectically in the flexibly domain case.
1.3. Obstructions to contact embeddings. Let V be a symplectic manifold, we call a closed hypersurface Y ⊂ V an contact hypersurface if there exists a local Liouville vector field near Y that is transverse to Y . When V is exact, and the Liouville vector field can be defined globally, we say Y is an exact contact hypersurface. When Y is separating, which is automatic if H 2n−1 (V ; Z) = 0, and is an exact contact hypersurface, the Liouville vector field must point out along Y w.r.t. the compact domain (inside the completion of V ) bounded by Y . Note that if we drop the exactness assumption, the local Liouville vector field can either point out or point in along Y w.r.t. the compact domain it bounds. Understanding whether a contact manifold can be embedded in a symplectic manifold, e.g. C n , is a fundamental question. In dimension 4, it is closely related to the Gompf conjecture [21], see [26] for recent advances on this conjecture. Obstructions to exact contact hypersurfaces were studied by Cieliebak and Frauenfelder [14] using the Rabinowitz-Floer homology. For example, they showed that there is no exact embedding of S * S n into C n . Using Theorem 1.1, we have the following obstructions to contact embeddings (not necessarily exact), where the obstructions from the Rabinowitz-Floer homology typically vanish. Theorem 1.6. Let V be a 2n-dimensional exact domain with c Q 1 (V ) = 0 and W a flexible Weinstein domain. Suppose the rank of the intersection form on H n (V ; Q) is smaller than that of W . Then ∂W can not be embedded into V as a separating contact hypersurface with the local Liouville vector field points out w.r.t. the bounded domain. If we assume moreover that V is P × D for an exact domain P , ∂W can not be embedded into V as a contact hypersurface.
For example, we can consider the flexible version W f lex of a Brieskorn variety x a 0 0 + . . . + x an n = 1 ⊂ C n+1 for a i ∈ N + . When a i ≥ 2 for all i, the intersection form of the Brieskorn variety/W f lex has positive rank if one of a i ≥ 3 or n is even. Then by Theorem 1.6, there is no contact embedding of the contact boundary ∂W f lex into C n . Note that for suitable choices of a i , the contact boundary ∂W f lex has the same almost contact structure as (S 2n−1 , ξ std ) [24], in particular, such embedding is not obstructed topologically. for some N ∈ N + , we can assign a rational Conley-Zehnder index to each non-degenerate Reeb orbit γ, i.e. 1/N of the Conley-Zehnder index of ⊕ N Φ(t) under a symplectic trivialization of ⊕ N γ * ξ inducing the fixed trivialization on det C ⊕ N ξ, where Φ(t) is the linearization of the Reeb flow along γ restricted to ξ, see [20,28] for details. For those orbits with torsion homology classes, the Conley-Zehnder index is independent of N and the trivialization [20,Proposition 3.8].
A flexible domain, introduced by Cieliebak and Eliashberg [13], is a Weinstein domain obtained from subcritical handles and flexible handles, i.e. those with attaching spheres being loose Legendrians. They enjoy the h-principle [13] and have vanishing symplectic cohomology [5,29], hence should be considered as the simplest Weinstein domains. In [25], Lazarev studied the Reeb dynamics on contact manifolds admitting flexible fillings and introduced the notion of asymptotically dynamical convexity (ADC). In fact, Lazarev proved that contact manifolds with flexible fillings have stronger properties than ADC as follow:
Proposition 2.1. Let (Y 2n−1 , ξ) be a flexibly fillable contact manifold with c Q 1 (ξ) = 0 and a fixed contact form α 0 . Then for any D ≫ 0, there exists a contact form α < α 0 such that Reeb orbits of α with period smaller than D are non-degenerate and have Conley-Zehnder index ≥ 1 (for any fixed trivialization of det C ⊕ N ξ ). And those orbits with Conley-Zehnder index 1 are simple.
Proof. This follows from the proof of [25,Theorem 3.15,3.17,3.18]. For D ≫ 0 and a suitable α < α 0 , if we consider Reeb orbits of period smaller than D, then they fall into the following two classes: (1) Each subcritical handle of index k creates a simple contractible Reeb orbit with Conley-Zehnder index n + 1 − k, and all multiple covers of it have higher Conley-Zehnder indices; (2) Every loose handle attachment creates (several) contractible simple Reeb orbits of Conley-Zehnder index 1 and many other orbits with Conley-Zehnder index strictly greater than 1.
Remark 2.2 (Clarification on asymptotically dynamical convexity and dynamical convexity). Dynamical convexity was introduced by Hofer, Wysocki, and Zehnder [23] as a dynamical consequence of convexity, namely a contact form α on (S 2n−1 , ξ std ) is dynamical convex if all Reeb orbits have Conley-Zehnder indices at least n + 1. It was shown recently by Chaidez and Edtmair [11,12] that dynamical convexity does not imply convexity. In terms of cylindrical contact homology, dynamical convexity means that there are no Reeb orbits of small Conley-Zehnder indices that are not visible in the cylindrical contact homology. This is the perspective used by Abreu and Macarini in [2] for contact manifolds other than spheres. On the other hand, Bourgeois, Cieliebak, Oancea, and Uebele [6,15,34] introduced index positivity of a contact form, namely if the SFT degree µ CZ + n − 3 of any contractible Reeb orbit is positive. For (S 2n−1 , ξ std ), this means that the Conley-Zehnder indices are at least 2 − n = n + 1. From this perspective, the ADC condition is "asymtotical" index positivity by considering a sequence of contact forms instead of just one.
Symplectic cohomology.
We will only recall basics of symplectic cohomology to set up notations and relevant structures for our main results. We refer readers to [15,31,32] for a more complete treatment of the subject.
Given an exact filling (W, λ), roughly speaking, symplectic cohomology is the "Morse cohomology" of the free loop space w.r.t. the symplectic action functional
A H (x) := − x * λ + S 1 (x * H)dt (2.1) where H : S 1 × W → R is a Hamiltonian on the completion ( W , λ) := (W ∪ ∂W × (1, ∞) r , λ ∪ rλ| ∂W ) such
that "the slope d dr H" goes to infinity as r goes to infinity. Let R be any commutative ring, the cochain complex C * (H) is a free R-module generated by 1-periodic orbits of X H and the differential is defined by counting rigid solutions to the Floer equation. Let x, y be two generators in C * (H) represented by periodic orbits, we use M x,y to denote the compactified moduli space of Floer cylinders from x to y, whose specific meaning depends on the construction of the symplectic cohomology as follows.
(1) If H is a non-degenerate Hamiltonian, i.e. all 1-periodic orbits are non-degenerate, then we have
M x,y = u : R s × S 1 t → W ∂ s u + J(∂ u t − X H ) = 0, lim s→∞ u = x, lim s→−∞ u = y /R.
(2) If H is Morse-Bott non-degenerate, including the case of H = 0 on W considered in [16,38], then x, y are critical points of the auxiliary Morse functions on the Morse-Bott families of periodic orbits (which can be viewed as a submanifold in W using the starting point of the orbit). In this case, M x,y is the compactified moduli space of cascades from x to y, which can be pictorially described as In both cases, for a generic compatible (which is also cylindrically convex to guarantee the integrated maximum principle [1,15]) almost complex structure J : S 1 → End(T * W ), M x,y is cut out transversely as a manifold with boundary for those with expected dimension virdim M x,y ≤ 1. Moreover, M x,y can be oriented in a coherent way such that
δ(x) = y,virdim Mx,y=0
(#M x,y )y defines a differential on C * (H). The symplectic cochain complex is graded by n − µ CZ (x). In general, since µ CZ is only well-defined in Z/2, symplectic cohomology always has a Z/2 grading. If c Z 1 (W ) = 0, upon fixing a trivialization of det C T W , µ CZ is well-defined in Z, which is independent of the trivialization for any periodic orbit with finite order homology class. Moreover, if c Q 1 (W ) = 0, then µ CZ is well-defined in Q (as in §2.1) if we fix a trivialization of det C ⊕ N T W and symplectic cohomology can be graded by Q. We say an exact filling W of Y is a topologically simple if H 1 (Y ; Q) → H 1 (W ; Q) is injective and c Q 1 (W ) = 0. This condition makes sure that for some N ∈ N + , any trivialization of det C ⊕ N ξ can be extended to a trivialization of det C ⊕ N T W . In particular, the Q-grading in symplectic cohomology is given by the boundary Conley-Zehnder index, which is moreover independent of trivialization if the orbit represents a torsion homology class in W .
Symplectic cohomology SH * (W ; R) has the following properties:
(1) If we choose H to be C 2 small on W and to be h(r) on ∂W × (1, ∞) r with h ′′ (r) > 0, then the periodic orbits of X H are either constant orbits on W or non-constant orbits on ∂W × (1, ∞), which, in pairs after a small perturbation, correspond to Reeb orbits on (∂W, λ| ∂W ). Those constant orbits generate a subcomplex corresponding to the cohomology of W , and those non-constant orbits generate a quotient complex C * + (H), whose cohomology is called the positive symplectic cohomology SH * + (W ; R). Then we have a tautological long exact sequence, (1), assume there is no Reeb orbit with period D, the Hamiltonian-Floer cohomology defines filtered symplectic cohomology SH * ,<D (W ; R) and SH * ,<D + (W ; R) with a similar tautological long exact sequence. And we have with all the structures, e.g. the tautological long exact sequence, the ring structure and so on. For the filtered version, from the symplectic field theory perspective, we should have SH * ,<D (W ) → SH * ,<D (V ). However, due to the Hamiltonian setup of symplectic cohomology, we have the following form of the filtered Viterbo transfer:
. . . → H * (W ; R) → SH * (W ; R) → SH * + (W ; R) → H * +1 (W ; R) → . . . (2.2) (2) SH * (W ; R) is a unital ring and H * (W ; R) → SH * (W ; R) is a unital ring map. (3) If we consider h(r) with h ′ (r) = D for r ∈ (1 + ǫ, +∞) in the setup inSH * (W ; R) = lim − → D→∞ SH * ,<D (W ; R), SH * + (W ; R) = lim − → D→∞ SH * ,<D + (W ; R). (4) We consider the map δ ∂ : SH * + (W ; R) → H * +1 (W ) → H * +1 (∂W ),SH * ,<D (W ) → SH * ,<D ′ (V ) (2.3) where D ′ ǫ ≥ D if we have V ∪ ∂V ∂V × [1, 1 + ǫ] ⊂ V embeds exactly into W , see [15, §5] or [40, §2.2].
2.3. Neck-stretching. We first recall some basics of the neck-stretching procedure in [4]. We also recommend [15, §2.3, 9.5] for applications of neck-stretching in Floer theories. We recall the setup of neck-stretching for general case following [41, §3.2]. Let (W, λ) be an exact domain and (Y, α := λ| Y ) be a contact type hypersurface inside W . 1 The hypersurface divides W into a cobordism X union with a domain W ′ . Then we can find a small slice
(Y × [1 − η, 1 + η] r , d(rα)) symplectomorphic to a neighborhood of Y in W . Assume J| Y ×[1−η,1+η]r = J 0 , where J 0 is independent of S 1 and r and J 0 (r∂ r ) = R α , J 0 ξ = ξ for ξ := ker α. Then we pick a family of diffeomorphism φ R : [(1 − η)e 1− 1 R , (1 + η)e 1 R −1 ] → [1 − η, 1 + η] for R ∈ (0, 1] such that φ 1 = id and φ R near the boundary is linear with slope 1. Then the stretched almost complex structure N S R (J) is defined to be J outside Y × [1 − η, 1 + η] and is (φ R × id) * J 0 on Y 1 × [1 − η, 1 + η].
Then N S 1 (J) = J and N S 0 (J) gives almost complex structures on the completions X, W ′ and Y × R + , which we will refer as the fully stretched almost complex structure. 1 The process works for strong filling W as long as Y is contact hypersurface.
We will consider the degeneration of curves solving the Floer equation with one positive cylindrical end asymptotic to a non-constant Hamiltonian orbit of X H . Here we require that H = 0 near the contact hypersurface Y . Since either the orbit is simple or J depends on the S 1 coordinate near non-simple orbits, the topmost curve in the SFT building, i.e. the curve in X, has the somewhere injectivity property. In particular, we can find regular J on X such that all relevant moduli spaces, i.e. those with point constraint from X (used in §3), or with negative cylindrical ends asymptotic to non-constant Hamiltonian orbits of X H , possibly with negative punctures asymptotic to Reeb orbits of Y and multiple cascades levels, are cut out transversely.
u 1 u 2 u 1 u 2 u ∞ 1 u ∞ 2 X Y × R + W ′ Figure 2. Neck-stretching
In the figure, we use to indicate the puncture that is asymptotic to a Reeb orbit. For the compactification of curves in the topmost SFT level, in addition to the usual SFT building in the symplectization Y × R + stacked from below [4], we also need to include Hamiltonian-Floer breakings near the cylindrical ends. If we use autonomous Hamiltonians and cascades, we need to include curves with multiple cascades levels and their degeneration, e.g. l i = 0, ∞ in the cascades for some horizontal level i. A generic configuration is described in the top-right of the figure above, but we could also have more cascades levels with the connecting Morse trajectories degenerate to 0 length or broken Morse trajectories.
A useful fact from the non-negativity of energy is the following action constraint. Let u be a Floer cylinder in X with negative punctures asymptotic to a multiset Γ of Reeb orbits (i.e. a set of Reeb orbits with possible duplications). Assume lim s→∞ u = x and lim s→−∞ u = y, then we have
A H (y) − A H (x) − γ∈Γ γ * α ≥ 0 (2.4)
If we apply neck-stretching to the contact boundary Y = ∂W in the completion W , assume H = h(r) on ∂W × [0, +∞) and we use the cascades model and a cylindrical convex almost complex structure, for a top level curve as above, we get
γ * x α − γ * y α − γ∈Γ γ * α ≥ 0 (2.5)
where γ x , γ y are the corresponding Reeb orbits for x, y. The follows from the non-negativity of u * π * α, where π :
X = Y × (0, +∞) → Y .
3. Proof of Theorem 1.1
Let Y be a flexibly fillable contact manifold with c Q 1 (ξ) = 0 and α a contact form as in Proposition 2.1. Let γ 1 , . . . , γ N denote all the Reeb orbits of action smaller than D with Conley-Zehnder index 1. We assume they are ordered increasingly with respect to their periods. In the definition of filtered positive symplectic cohomology SH * ,<D + (W ) with slope D, we will use the following special Hamiltonian H. Here W is a topologically simple exact filling of (Y, α).
(1) H = 0 on W and H ′ (r) = D for r > 1 + w for w > 0.
(2) H on Y × [1, 1 + w] is a small perturbation of H = f (r) with f ′′ (r) > 0 such that the periodic orbits of X H are non-degenerate and in a two-to-one correspondence with Reeb orbits of period smaller than D. More precisely, every non-degenerate Reeb orbits γ will split into two Hamiltonian orbitsγ andγ with µ CZ (γ) = µ CZ (γ) + 1 and µ CZ (γ) = µ CZ (γ) following [7]. [25,38].
In the following, we will use α, β, γ to stand for Reeb orbits andα,α, α to stand for Hamiltonian orbits, where α means that we do not specify whether it is a check or a hat orbit. In the following, for the simplicity of notation, we will assume a i α i is represented by a single Hamiltonian orbit α. The argument below works for linear combinations as long as they represent a closed class in the positive cochain complex. In the following, we will suppress D in W D , α D , i.e. W means W D in the previous two propositions.
Fixing any two closed chains A, B representing classes in H n (W ; Z) with transverse intersections and a periodic orbit α, we consider the compactified moduli space of the following
M α,A,B := u : C → W (du − v) 0,1 = 0, u(∞) = α, u(0) ∈ A, u(1) ∈ B
where v = X H ⊗ β with H a Hamiltonian as before and β a one form, such that β = dt near the ends w.r.t. fixed cylindrical coordinates (i.e. we fix biholomorphisms from (−∞, 0) × S 1 t to neighborhoods of 0, 1, and from (0, +∞) × S 1 t to a neighborhood of ∞) and dβ ≤ 0. Here u(∞) = α is a short hand for lim s→∞ u = α for the cylindrical coordinate (0, +∞) s × S 1 t → C, (s, t) → e 2π(s+it) . Since H = 0 near A and B, the removal of singularity implies that u can be viewed as a map on C. Similarly for another orbit β, we can define M α,β,B and M α,A,β . We also define M α,A to be the compactification of the following.
2, then the intersection number is
A · B = N i=1 (# (M α,γ i ,B × Mγ i ,A ) + # (M α,A,γ i × Mγ i ,B )) .
Proof. It follows from the boundary configuration of M α,A,B whose dimension is 1. Since dim M β,A = µ CZ (β) − 1, and all periodic orbits have Conley-Zehnder indices greater than 1 unless they are one ofγ i . By degree reason, the Floer type breakings near 0, 1 give rise to the right hand side. Since α is closed in the positive symplectic cohomology, the Floer type breakings near ∞ at a non-constant orbit sum up to 0. If we consider the Floer type breakings near ∞ at interior of W . Then by the integrated maximal principle the curve is contained in W , where the equation is the Cauchy-Riemann equation. By the exactness of W , such curve must be constant. Therefore such degeneration can be identified with curves u : C → W solves (du − X H dt) 0,1 = 0 with u(∞) = α and u(0) ∈ A ∩ B modulo the R translation. Since α is mapped to 1 ∈ H 0 (W ; Z), i.e. the count of curves u with a point constraint at u(0) and u(∞) = α modulo R is 1 when transversality holds. Therefore, this type of degeneration is counted as A · B. Proof. Given a Morse function f on W such that ∂ r f > 0 on ∂W , then we can represent a cochain complex of H * (W ; Z) by critical points of f , then the paring of critical point x with a closed cycle B is the intersection number of the ascending manifold of x with B. Following [38, §2], the map δ can be represented counting the moduli space of (u, l) with u solves the Floer equation and u(0) ∈ W and l is a gradient trajectory from u(0) to a critical point x. Therefore δ([γ j ]), B counts the moduli space of (u, l 1 , l 2 ) with l 1 , l 2 be two half infinite gradient trajectories connected at an index n critical point of f . Then by shrinking the time of the gradient flow lines from ∞ to 0 as in [38, §3.1], and note that [γ i ] is closed in positive symplectic cohomology and B is closed, the count equals to length 0 count, which is #Mγ i ,B .
To compute M α,γ j ,B , we perform a full neck-stretching on the boundary. Let Y denote the symplectization Y × (0, ∞), which is equipped with a Hamiltonian H such that H = 0 on Y × (0, 1) and after that it is the same as H on W . Then we define N α,γ i ,γ j to be the compactification of the following moduli space.
u : CP 1 \{∞, 0, 1} → Y (du − X H ⊗ β) 0,1 = 0, u(∞) = α, u(0) =γ i , u(1) = (0, γ j )
i.e. u(∞), u(0) are asymptotic to Hamiltonian orbits and u(1) is asymptotic to a Reeb orbit at a negative puncture. We define N γ j ,B to be the compactification of the following moduli space.
u : C → W (du) 0,1 = 0, u(∞) = (+∞, γ j ), u(0) ∈ B /R × S 1
Proposition 3.5. For a sufficiently stretched almost complex structure, we have
#M α,γ j ,B = N k=1 # N α,γ j ,γ k × N γ k ,B .
Proof. We perform a full neck-stretching along the boundary, then any curve in M α,γ j ,B will converge to a SFT building type curve, since B ⊂ W . The top level curve is necessarily connected by [15,Proposition 9.17], with one fixed negative puncture at 1 which will connect to the component that eventually intersects B. But there might be other free moving punctures that will eventually be closed off by holomorphic planes. Let γ denote the Reeb orbit on the puncture 1, and β i be those Reeb orbits on those free punctures. Then the virtual dimension of this moduli space is
µ CZ (α) − µ CZ (γ j ) − (µ CZ (γ) + n − 1) − (µ CZ (β i ) + n − 3).
We have all Reeb orbits that can potentially appear must have µ CZ ≥ 1. Since we can assume transversely for the upper level curve. Therefore the only possibility is γ is one of γ i and there is no β i , and we have the expected dimension is 0, for otherwise, the expected dimension is negative. After the top level, we might have several levels of curves in the symplectization, with the topmost curve with only one positive puncture asymptotic to γ i . Since γ i is simple, the topmost curve is necessarily somewhere injective, hence we can assume transversality for this curve. Since the curve must connect to some component that eventually intersects B, therefore the curve must have at least one negative end γ ′ , then the expected dimension of the moduli space of this curve is
µ CZ (γ i ) − µ CZ (γ ′ ) − j (µ CZ (β j ) + n − 3) − 1. Since µ CZ (γ i )
is the lowest and all SFT grading µ CZ (β i ) + n − 3 are positive, we have the dimension is negative. As a result, there is no curve in the symplectization. The last part is the curve in the completion W , which is exactly N γ i ,B with expected dimension 0. Since γ i is simple, transversality is not a issue. Therefore the right hand side is the count from the fully stretched almost complex structure. If we assume we start with an almost complex structure that is stretched enough, we may assume in the process of stretching there is no curve in M α,β and M β,γ i with expected dimension −1. Moreover, we also assume there is no curve in M β,γ,B with expected dimension −1 in the process of stretching, for otherwise, we have a curve in a moduli space of negative dimension after the full stretch. Since B is closed, in the process of neck-stretching, we only have M α,γ j ,B and N k=1 #N α,γ j ,γ k × N γ k ,B as boundary corresponding to the two ends of the neck-stretching parameter.
We define Nγ i ,γ j to be the compactified moduli space of
u : R × S 1 → Y (du − X H dt) 0,1 = 0, u(∞) =γ i , u(−∞) = (0, γ j ) /R Proposition 3.6.
For a sufficiently stretched almost complex structure and H sufficient close to the autonomous one which only depends on r, we have
j i=1 #Nγ j ,γ i × N γ i ,B = δ([γ j ]), B ,
and #Nγ j ,γ j = 1.
Proof. The proof is similar to Proposition 3.5 by fully stretching the moduli space Mγ j ,B . By a similar dimension argument, the moduli space must break into Nγ j ,γ i × N γ i ,B . Therefore it suffices to prove i ≤ j. When H is autonomous and only depends on r, X H is parallel to the Reeb vector. Therefore for any solution u ∈ Nγ j ,γ i , we have the α-energy u * π * α ≥ 0 (as in (2.5)), which implies that the period of γ j must be greater than γ i unless γ i = γ j . Then for H sufficient close to the autonomous one, we have Nγ j ,γ i = ∅ implies that i ≤ j. Moreover, for the autonomous Hamiltonian, curves in Nγ j ,γ j is necessarily reparametrization of the trivial cylinder since the α-energy is 0. The moduli space is diffeomorphic to S 1 and is cut out transversely in the Morse-Bott sense. Then by the same analysis in [7] and γ j is simple, we have #Nγ j ,γ j = 1. One can avoid such perturbation if uses a cascades setup with a autonomous Hamiltonian as in [6,7].
Proof of Theorem 1.1. In view of Proposition 1.2, c Q 1 (W ) = 0 implies that W is topologically simple. If we are given δ([γ j ]), B and #Nγ i ,γ j , we can solve uniquely #N γ i ,B by Proposition 3.6, since the coefficient matrix is triangular with ones on the diagonal. Then by Proposition 3.3, 3.4 and 3.5, we can represent the intersection A · B by M α,γ i ,γ j , M α,γ i ,γ j , Nγ i ,γ j and δ([γ j ]), A/B . The first three moduli spaces are independent of the filling, as they are contained in the symplectization. Note that H n (W ; Z) is independent of filling, and a basis can be represented by combinations of [γ i ] by Proposition 3.1. By the universal coefficient theorem, H n (W ; Z) is isomorphic to the free part of H n (W ; Z) since H * (W ; Z) is supported in degree ≤ n. Fixing a basis of a fixed free part of H n (W ; Z) induces a dual basis on H n (W ; Z). We use this dual basis to identify the homology of two fillings. Such identification means that we identify δ([γ j ]), A(B) for both fillings, hence the intersection form can be identified.
Remark 3.7.
A natural question is whether some of the above counts bear homological meaning, i.e. can they be phrased as structural maps in (S 1 -equivariant) symplectic cohomology or symplectic field theory.
(1) Under the identification of linearized contact homology and positive S 1 -equivariant symplectic cohomology proved by Bourgeois and Oancea [6], the counting of N γ i ,B should be contained in the map
SH * +,S 1 (W ; R) → H * +1 (W ; R) ⊗ R R[u, u −1 ]/u → H * +1 (W ). (2)
The counting of Nγ i ,γ j should be contained in the map ι : SH * + (W ) → SH * +,S 1 (W ) from the Gysin exact sequence [8].
(3) The counting of M α,γ j ,γ k does not give rise to structural maps. This is because that the moduli space counts solutions to (du − X H ⊗ β) 0,1 = 0, where we can have Floer type breaking as well as SFT type breaking at 1. It is important in Proposition 3.5 that we ask the almost contact structure to be sufficiently stretched, for otherwise the relation could fail. However if we change β to be dt on C * = (−∞, +∞) × S 1 , and count anchored version (as in [5]) of the following curves,
u : CP 1 \{∞, 0, 1} → Y (du − X H ⊗ β) 0,1 = 0, u(∞) = α, u(0) = β, u(1) = (0, γ)
It should give rise to a map SH * + (W ) → SH * + (W )⊗SH * +,S 1 (W ) of degree 2n−1, which we conjecture to be isomorphic to the secondary coproduct SH * + (W ) → SH * + (W ) ⊗ SH * +,S 1 (W ) (also of degree 2n − 1) in [18,17] composed with id ⊗ι.
Removing the topologically simple assumption
Exploiting the independence of augmentations using grading constraints was initiated by Bourgeois and Oancea [6], also see the work of Cieliebak and Oancea [15], and Uebele [34], where they introduced the notion of index-positive contact manifolds (Remark 2.2). This notion was generalized by Lazarev [25] to the notion of asymptotically dynamically convex (ADC) manifolds to contain examples like flexibly fillable contact manifolds with vanishing (integral) first Chern class. Several structural maps on (S 1 -equivariant) symplectic cohomology of exact fillings of ADC manifolds are independent of topologically simple fillings [38,41]. Those topological conditions are used to get a Z grading for the symplectic cohomology generated by contractible orbits, as the ADC condition only requires that µ CZ (γ) + n − 3 > 0 for a contractible Reeb orbit γ (which has a canonical Z-valued Conley-Zehnder index, as c 1 (ξ) = 0). Because of this, (S 2n−1 /G, ξ std ) is ADC for any finite G ⊂ U (n) acting freely on S 2n−1 and n ≥ 2, as contractible orbits of (S 2n−1 /G, ξ std ) are the same as those on (S 2n−1 , ξ std ). However, those non-contractible orbits on (S 2n−1 /G, ξ std ) play an important role in [39], and by [39,Theorem A], there are no topological simple fillings (even strong fillings) of (S 2n−1 /G, ξ std ). Therefore, it is natural to generalize the notion of ADC manifolds as follows to impose conditions on non-contractible orbits.
Definition 4.1. Let (Y, ξ) be a contact manifold such that c Q 1 (ξ) = 0. Let Ψ be a trivialization of det C ⊕ N ξ for some N ∈ N + . We say (Y, ξ, Ψ) is generalized ADC if there exist contact forms α 1 > α 2 > . . ., positive real numbers D 1 < D 2 < . . . converging to infinity, such that all Reeb orbits of α i of period up to D i are non-degenerate and have rational SFT grading µ CZ (γ) + n − 3 > 0. We say (Y, ξ, Ψ) is generalized TADC, if in addition, there is a contact form α such that all α i > α. (1) Let G ⊂ U (n) such that the quotient C m /G has an isolated singularity at 0, the contact link (S 2n−1 /G, ξ std ) is generalized ADC if and only if C n /G is a terminal singularity by the work of McLean [28]. (2) The contact boundary of a flexible Weinstein domain with vanishing rational first Chern class for any trivialization by the arguments in [25]. More precisely, the contact boundary of a subcritical Weinstein domain is generalized ADC as all of the relevant orbits can be assumed to be contractible (as they wind around cores of handles). When we attach a flexible handle, non-contractible orbits could appear, however the argument of lifting the Conley-Zehnder indices by adding zig-zags in [25,Theorem 3.18] works for non-contractible orbits and any fixed trivialization. (3) For a closed manifold Q, we have that det C ⊕ 2 T T * Q is trivialized using the trivial real bundle det R ⊕ 2 T Q. We use Ψ to denote the trivialization. Then (S * Q, Ψ) is generalized ADC if dim Q ≥ 4, as the Conley-Zehnder index using such trivialization is the Morse index when the contact form is induced from a metric. This is an example where the notion of generalized ADC depends on Ψ, as changing Ψ will increase the Conley-Zehnder indices of some orbits with nontrivial homotopy classes and decrease the same amount for orbits with the opposite homotopy classes, e.g. T * T n . The same holds for any closed orbifold Q with only isolated singularities (then S * Q is a contact manifold). Proof of Proposition 1.2. If c Q 1 (W ) = 0, then we can trivialize det ⊕ N T W for some N ∈ N + . Note that
H 1 (W 0 ; Z) → H 1 (Y ; Z)
is an isomorphism, hence there is a trivialization of det C ⊕ N T W 0 whose restriction to ∂W is the same as the restriction of the trivialization of det ⊕ N T W . Since Y is generalized ADC for any trivialization, we then run the argument of [38, Corollary B] using such trivializations and conclude the claim.
Proposition 4.3. Assume (Y, ξ, Ψ) is generalized ADC and there is an exact filling W , such that Ψ extends to a trivialization Ψ of det C ⊕ N T W . Then for any exact filling V of Y , there is a spectral sequence converging to SH * + (V ; R) (not graded), such that (1) The (N + 1)th page of the spectral sequence is isomorphic to SH * + (W ; R; Ψ) (filtered by the Q-grading using Ψ) for any coefficient ring R.
(2) The cochain map δ ∂ from the positive cochain complex to the Morse cochain complex of Y is compatible with spectral sequence. On the (N +1)th page, the induced map is isomorphic to SH * + (W ; R; Ψ) → H * +1 (Y ; R). If (Y, ξ, Ψ) is generalized TADC, the same holds for (semi-positive) strong fillings V, W and R the Novikov field.
Proof. First note that µ CZ (x) computed using the trivialization Ψ is always a multiple of 1 N . Hence the generalized ADC property implies that µ CZ + n − 3 ≥ 1 N . The proof follows from applying arguments in [39, §3] to the spectral sequence associated to the filtration
F k C + (H) := x |x| ∂ ≥ k N , k ∈ Z,
where |x| ∂ = n − µ CZ (x). By the same argument of [39,Proposition 3.3], the differential is compatible with filtration by neck-stretching, and moreover, there is no differential before the N th page, and on N th page, there are differentials from x to y with |y| ∂ − |x| ∂ = 1, whose underlying curve is contained in the cylindrical end of the completion for a sufficiently stretched almost complex structure. This differential computes SH * + (W ; R; Ψ), yielding the first claim. The second claim follows from [39,Proposition 3.4]. Strictly speaking, we need to apply arguments in [39, §3] to the infinite telescope construction of filtered (by D i ) positive cochain complexes of α i in the definition of generalized ADC, but this imposes no essential change to the argument. Proof. By Example 4.2, there is a trivialization Ψ, such that (Y, ξ, Ψ) is generalized ADC. Since the trivialization Ψ is the restriction of a trivialization of det C ⊕ N T W 0 , by [5,29], we have SH * + (W 0 ; Q) = H * +1 (W 0 , Q). Then by Proposition 4.3, we have dim ⊕SH * + (W ; Q) ≤ dim ⊕H * (W 0 ; Q) for any exact filling W .
In general, a morphism between two spectral sequences in (2) of Proposition 4.3 only captures the morphism on the associated grade of the limits, i.e. leading terms. But we can exploit the degeneracy of the spectral sequence in some special cases, which improves some of the results in [38].
Proof of Theorem 1.3. For (1), by assumption, we have SH * −1
+ (W 0 ; Q) ≃ H * (W 0 ; Q) → H * (Y ; Q)
is injective. Then by Proposition 4.3, the spectral sequence map from the spectral sequence of SH * + (W ; Q) to that of H * (Y ; Q), on the (N + 1)th page, is isomorphic to the injective map SH * + (W 0 ; Q) → H * +1 (Y ; Q). Since the spectral sequence on H * (Y ; Q) of index gap 1/N degenerates from the (N + 1) page, the injectivity implies that the spectral sequence on SH * + (W ; Q) also degenerates at the (N + 1) page. And the map SH * (W ; Q) → H * +1 (Y ; Q) on the associated graded is the same as SH * + (W 0 ; Q) → H * +1 (Y ; Q). Since 1 is in the image of SH * + (W 0 ; Q) → H * +1 (Y ; Q), we know that 1+a is in the image of SH * + (W ; Q) → H * +1 (Y ; Q) for deg(a) > 0. Therefore, we have SH * (W ; Q) = 0 and SH * + (W ; Q) ≃ H * +1 (W ; Q). The injectivity of
SH * + (W 0 ; Q) → H * +1 (Y ; Q) implies that SH * + (W ; Q) ≃ H * +1 (W ; Q) → H * +1 (Y ; Q)
is also injective. As a consequence, we have c Q 1 (W ) = 0. Then claim follows from Proposition 1.2. For (2), when n is even, we have H * (W 0 ; Z) → H * (Y ; Z) is injective on odd degrees. Note that symplectic cohomology is canonically graded by Z/2, and the differentials on the spectral sequence is compatible with the Z/2 grading. By looking at the (N + 1)th page of spectral sequence map as before, which is injective on even (Z/2) degrees of SH * + (W ; Z), the differential of the (N + 1)th page of the spectral sequence for SH * + (W ; Z) must be zero on odd degrees. As a consequence, by induction, the (N + k)th page of spectral sequence map is injective on even degrees and the differential of the (N + k)th page of the spectral sequence for SH * + (W ; Z) must be zero on odd degrees for k > 0. As a consequence, we also have that 1 is in the image of the morphism of spectral sequences in (2) of Proposition 4.3 on the ∞th page, hence SH * (W ; Z) = 0 and SH * + (W ; Q) ≃ H * +1 (W ; Q). Then Corollary 4.4 implies that dim ⊕ 2n * =1 H * (W ; Q) ≤ dim ⊕ 2n * =1 H * (W 0 ; Q). For (2a), we claim that the spectral sequence of SH * + (W ; Q) degenerates at the (N + 1)th page. For otherwise, the non-trivial differential must be from even degrees of SH * + (W ; Q) to odd degrees by the argument in (2). However, this implies that the total dimension of even degrees of SH * + (W ; Q) is smaller than that of SH *
+ (W 0 ; Q), which is the total dimension of ⊕ n i=1 H 2i+1 (W 0 ; Q) ≃ ⊕ n i=1 im(H 2i+1 (W 0 ; Q) → H 2i+1 (Y ; Q)). By [10, Theorem 4.4], we have H * (W ; Q) → H * (Y ; Q) is surjective onto the image of H * (W 0 ; Q) → H * (Y ; Q) for 2 ≤ * ≤ n. As H 1 (W 0 ; Q) = 0, we know that dim ⊕ n i=1 H 2i+1 (W ; Q) ≥ dim ⊕ n i=1 H 2i+1 (W 0 ; Q)
. Then this contradicts with that SH * + (W ; Q) → H * +1 (W ; Q) is surjective (for * even). Therefore the spectral sequence degenerates at the (N + 1)th page and we have SH * + (W ; Q) ≃ SH * + (W 0 ; Q), and the claim follows from that SH * (W ; Q) = 0 in (2). For (2b), we already know that the spectral sequence of SH * + (W ; Q) degenerates at the (N + 1)th page. We claim that H 2 (W ; Q) → H 2 (Y ; Q) is injective, for otherwise, we have dim H 2 (W, Y ; Q) ≥ 1. Then by Lefschetz duality and the universal coefficient theorem we have dim H 2n−2 (W ; Q) ≥ 1. As a consequence, we have dim ⊕ 2n * =0 H * (W ; Q) ≥ 2 + dim ⊕ n * =0 im(H * (W 0 ; Q) → H * (Y ; Q)). When dim ker(H n (W 0 ; Q) → H n (Y ; Q)) = 1, we have dim ⊕ 2n * =0 H * (W 0 ) = 1 + ⊕ n * =0 im(H * (W 0 ; Q) → H * (Y ; Q)). Hence we arrive at a contradiction with (2). Now since H 2 (W ; Q) → H 2 (Y ; Q) is injective, we have c Q 1 (W ) = 0 and we can apply Proposition 1.2.
The main difference between Theorem 1.3 and results in [38] is that we can prove topological simplicity instead of assuming it in Theorem 1.3, but we need addition information, e.g. Y being flexibly fillable, to get the degeneracy of the spectral sequence. When n is odd, a prior, there could be differentials in the spectral sequence acting non-trivially on the element in SH * + (W ) that is supposed to kill the unit. The H n (W 0 ; Q) → H n (Y ; Q) injective condition prevents such differentials by exploiting the tautological degeneracy on the spectral sequence on H * (Y ). It is possible to strengthen (2b) to the case where the intersection form on the cokernel of H n (Y ; Q) → H n (W 0 ; Q) is positive/negative definite.
Applications
Proof of Theorem 1.4. We use D * Q, S * Q to denote the unit disk bundle and the sphere bundle in T * Q. When χ(Q) = 0, we have H n (D * Q; Q) → H n (S * Q; Q) is injective for n = dim Q. Then by Theorem 1.3, we have that H * (W ; Z) → H * (S * Q; Z) is isomorphic to H * (D * Q; Z) → H * (S * Q; Z), which is injective, for any exact filling W of ∂(F lex(T * Q)). Then by [38,Proposition 3.24], we have W is D * Q glued with a homology cobordism between S * Q. By the same argument in [42, §4], we can improve the homology cobordism to an h-cobordism when π 1 (Q) is abelian. Then the interior of W is diffeomorphic to T * Q as an open manifold by the Mazur trick as in [42,Theorem 1.2].
Proof of Theorem 1.5. By Proposition 1.2 and [38, Theorem E], any exact filling W with c Q 1 (W ) = 0 is simply connected. Since H * (W ; Z) = H * (W 0 ; Z) by Proposition 1.2 and is freely generated and supported in degree 0 and n and dim W ≥ 6, we have W has a handle decomposition of one 0-handle and several n-handles. By [33, the remark after Corollary 4.6], the diffeomorphism type of such manifold is uniquely determined by the intersection form under the conditions listed. Therefore W is diffeomorphic to W 0 by Theorem 1.1. When the rank of the intersection form of W 0 is 1, then Theorem 1.3 implies that c Q 1 (W ) = 0 automatically for any exact filling W .
Proof of Theorem 1.6. In the first case, if ∂W embeds into V as a separating contact hypersurface with the local Liouville vector field points out w.r.t. the bounded domain U . Then U is a symplectically aspherical filling of ∂W with an exact symplectic form. Although we might not be able to find all the contact hypersurface realizing the ADC property in U (see [38, §8]), hence Proposition 1.2 does not apply directly. However, by comparing δ ∂ : SH * ,<D + (U ; Q) → H * +1 (∂W ; Q) to that of W for a suitable D, we can prove that H 1 (U ; Q) → H 1 (∂W ; Q) is surjective onto the image of H 1 (W ; Q) → H 1 (∂W ; Q), i.e. all of H 1 (∂W ; Q). The surjectivity of H 1 (U ; Q) → H 1 (∂W ; Q) implies that U becomes a Liouville filling of ∂W after a modification of the Liouville form on V with a closed 1-form on U . Then Proposition 1.2 and Theorem 1.1 implies that the rank of the intersection form on H n (U ; Q) is the same as that of H n (W ; Q), which is larger than that of H n (V ; Q). Hence it is impossible to embed U to V topologically.
In the second case, if ∂W embeds in P × D as a contact surface, as H 2n−1 (P × D; Z) = 0, it must be separating. Now if the local Liouville vector field points into the compact domain U bounded by ∂W , by deleting U , we get a symplectically aspherical filling of ∂(P × D) ⊔ ∂W . Combining the result in [42] and [39, Proposition B], we know that ∂(P × D) is not co-fillable, i.e. such a filling with two boundary components can not exist. Hence we reduce the situation back to the first case.
Figure 1 .
12 level cascades Here the horizontal lines are negative gradient flow of the auxiliary Morse function used in dealing the Morse-Bott family of non-constant orbits (so that critical points with larger critical value corresponding to longer periodic orbits using the perturbation in [7]) except for the bottom line of the right of the figure above, which is the gradient flow of an admissible Morse function [38, Definition 2.1] on W when H = 0 on W . For the more formal description of the moduli spaces see [38, Definition 4.7] for details.
this can be defined by counting rigid configurations in the right of Figure 2 with the bottom flow line replaced by a gradient flow line in ∂W (using an auxiliary Morse function on ∂W ) viewed as fiber product over W , see [38, §3.1] for details. (5) For an exact subdomain V ⊂ W , we have a Viterbo transfer map SH * (W ) → SH * (V ) compatible
Then by degree reason, [γ 1 ], . . . , [γ N ] represent classes in SH n−1,<D + (W ; Z).In the following, we fix a contact form α 0 . For every D > 0, there exists a contact form α D < 1 2 α 0 such that Proposition 2.1 holds for the period threshold D. Let M D denote the cobordism from α D to α 0 in the symplectization of (Y, α 0 ), W D the strict exact filling of (Y, α D ). Then W D ∪ M D is a strict exact filling of (Y, α 0 ). Then by (2.3) (here we can choose ǫ = 1), we have a transfer mapSH * ,<D + (W D ∪ M D ; Z) → SH * ,<D + (W D ; Z)which is compatible with the connecting map to H * (W (≃ W D ≃ W D ∪ M D ); Z). We will stretch on the contact boundary of W D , the following propositions hold if stretch the almost complex structure sufficiently.
Proposition 3. 1 .
1For D ≫ 0 and a sufficiently stretched almost complex structure, we have δ : γ 1 , . . . ,γ N → SH n−1,<D+ (W D ; Z) → H n (W ; Z)is surjective and the kernel is independent of the topologically simple exact filling W .Proof. By[38], SH n−1+ (W D ∪M D ; Z) → H n (W D ∪M D ; Z) = H n (W ; Z)is an isomorphism for any topological simple filling. In particular, for D big enough, the map SH n−1,<D+ (W D ∪ M D ; Z) → H n (W ; Z) is a surjection (W Dvaries w.r.t. to D, W D ∪M D does not). We can assume threshold of D for it to hold works for the flexible filling as well. We have that SH n−1,<D + (W D ; Z) → H n (W ; Z) is surjective by the Viterbo transfer. Moreover, SH n−1,<D + (W D ; Z) must be spanned by [γ i ] by degree reason. The remaining part of the proposition follows from thatγ i is matched in the identification of SH n−1,<D + (W D ; Z) → SH n−1 + (W D ; Z) ≃ H n (W ; Z) with that of the flexible filling for a sufficiently stretched almost complex structure
Proposition 3. 2 .
2For D ≫ 0 and a sufficiently stretched almost complex structure, there is a linear combination of Hamiltonian orbitsa i α i of Conley-Zehnder index n + 1, such that [ a i α i ] ∈ SH −1,<D + (W D ; Z) is sent to 1 in H 0 (W ; Z)and it is independent of the topological simple filling.Proof. This element represents the element hitting 1 under the map SH −1 * (W ; Z) → H 0 (W ; Z) → H 0 (Y ; Z). The proposition follows from that the map above is independent of the filling by[38, Corollary B].
u
: C → W (du − X H dt) 0,1 = 0, u(∞) = α, u(0) ∈ A /RProposition 3.3. Let α be the class in Proposition 3.
Proposition 3 . 4 .
34For a sufficiently stretched almost complex structure, we have#Mγ i ,B = δ([γ j ]), B ,where the last pairing is the natural map H n (W ; Z) ⊗ H n (W ; Z) → Z.
Since the Conley-Zehnder index of a contractible orbit is an integer and independent of the trivialization Ψ, it is clear that generalized ADC implies ADC. Moreover, if c Q 1 (ξ) = 0 and H 1 (Y ; Q) = 0, then rational Conley-Zehnder indices are independent of Ψ. But in general, if H 1 (Y ; Q) = 0, then the notion of generalized ADC depends on the trivialization Ψ.
Example 4. 2 .
2We have the following examples of generalized ADC contact manifolds.
( 4 )
4Let V be a Liouville domain, such that c Q 1 (V ) = 0, then ∂(V × D) is generalized ADC for any trivialization by (the proof of ) [38, Theorem K].
Corollary 4 . 4 .
44Let (Y 2n−1 , ξ) be the contact boundary of a flexible Weinstein domain W 0 with c Q 1 (W 0 ) = 0. Then for any exact filling W of Y , we have dim ⊕SH * + (W ; Q) ≤ dim ⊕H * (W 0 ; Q).
Acknowledgments. The author is grateful to Sheel Ganatra for sharing the draft of[18]and helpful conversions, to Jonathan Bowden and Oleg Lazarev for enlightening discussions. The author is supported by National Natural Science Foundation of China under Grant No. 12288201 and 12231010.
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| [] |
[
"Perfect optomechanically induced transparency in two-cavity optomechanics",
"Perfect optomechanically induced transparency in two-cavity optomechanics"
] | [
"Lai-Bin Qian \nSchool of Physics and Electronic Engineering\nNortheast Petroleum University\n163318DaqingChina\n",
"Xiao-Bo Yan \nSchool of Physics and Electronic Engineering\nNortheast Petroleum University\n163318DaqingChina\n"
] | [
"School of Physics and Electronic Engineering\nNortheast Petroleum University\n163318DaqingChina",
"School of Physics and Electronic Engineering\nNortheast Petroleum University\n163318DaqingChina"
] | [] | Here, we study the controllable optical responses in a two-cavity optomechanical system, especially on the perfect optomechanically induced transparency (OMIT) in the model which has never been studied before. The results show that the perfect OMIT can still occur even with a large mechanical damping rate, and at the perfect transparency window the long-lived slow light can be achieved. In addition, we find that the conversion between the perfect OMIT and optomechanically induced absorption can be easily achieved just by adjusting the driving field strength of the second cavity. We believe that the results can be used to control optical transmission in modern optical networks. | 10.1007/s11467-023-1279-1 | [
"https://export.arxiv.org/pdf/2303.16069v1.pdf"
] | 257,771,336 | 2303.16069 | e017cfac2bd6dd6acbacb21c164b20e45bfe6178 |
Perfect optomechanically induced transparency in two-cavity optomechanics
25 Mar 2023
Lai-Bin Qian
School of Physics and Electronic Engineering
Northeast Petroleum University
163318DaqingChina
Xiao-Bo Yan
School of Physics and Electronic Engineering
Northeast Petroleum University
163318DaqingChina
Perfect optomechanically induced transparency in two-cavity optomechanics
25 Mar 2023(Dated: March 29, 2023)
Here, we study the controllable optical responses in a two-cavity optomechanical system, especially on the perfect optomechanically induced transparency (OMIT) in the model which has never been studied before. The results show that the perfect OMIT can still occur even with a large mechanical damping rate, and at the perfect transparency window the long-lived slow light can be achieved. In addition, we find that the conversion between the perfect OMIT and optomechanically induced absorption can be easily achieved just by adjusting the driving field strength of the second cavity. We believe that the results can be used to control optical transmission in modern optical networks.
I. INTRODUCTION
The interaction between light and matter is an interesting and important research subject in quantum optics. It is an important means to understand the microstructure of matter. Cavity optomechanics [1] can provide such a research platform where the macroscopic mechanical resonators and light fields interact with each other. With the development of nanotechnology, various physical systems which can exhibit such interaction have been proposed and investigated, such as Fabry-Perot cavities [2,3], whispering-gallery microcavities [4][5][6], superconducting circuits [7,8] and membranes [9][10][11][12]. The optomechanical interaction can strongly affect the motion of mechanical oscillator and the optical properties in these systems, and then various interesting quantum phenomena can be generated, such as ground-state cooling of mechanical modes [13][14][15][16][17][18][19][20], quantum entanglement [21][22][23][24][25][26][27][28], mechanical squeezing [29], unconventional photon blockade [30], and optomechanically induced transmission and absorption [31][32][33][34][35][36][37][38][39][40][41][42].
Recently, the researches on optomechanically induced transparency (OMIT) [43][44][45][46][47][48][49][50][51][52][53][54][55][56][57] and the associated slow light in optomechanical systems have attracted much attentions. A remarkable feature of OMIT is that there is a deep dip in the absorption curve accompanied with a steep dispersion behavior at the transparency window. The steep dispersion behavior can generate a drastic reduction in the group velocity of light passing through the system [56]. According to this effect, many schemes have been proposed to slow or stop light [56][57][58][59][60][61][62][63][64][65], which is very meaningful in the construction of quantum information networks. However, the ideal depth of the transparency window cannot be achieved due to the nonzero mechanical damping rate in the usual OMIT theory [48,54,56], which will cause a very limited slow light effect (usually on the order of milliseconds) [58][59][60]. Until recently, the perfect OMIT can be easily achieved using the mechanism of non-rotating wave approximation [47,57], and the slow light effect at the perfect transparency window * Electronic address: [email protected] can be greatly improved [57]. The perfect OMIT theory above is studied in standard single-cavity optomechanics. In fact, there are more abundant and interesting quantum phenomena in multi-mode optomechanical systems, such as quantum entanglement [23][24][25], optical nonreciprocity [66,67] and quantum nonlinearity [68,69]. Therefore, it is necessary to generalize the perfect OMIT theory to multi-mode optomechanical systems.
Here, we theoretically study the optical responses including perfect OMIT, long-lived slow light and optomechanically induced absorption in a multi-mode optomechanical system (comprising two optical modes and one mechanical mode), see Fig. 1 [25,[68][69][70]. First, we give the conditions under which the perfect OMIT will occur, and the window width expression of the perfect OMIT. As long as the conditions are satisfied, the perfect OMIT can be achieved even with a large mechanical damping rate. Secondly, the dispersion curve becomes very steep at the perfect transparency window where it can be proofed that the negative dispersion curve slope is exactly equal to the value of time delay (slow light) in the model. It means that the long-lived slow light can be achieved at the perfect transparency window. Thirdly, the driving strength and dissipation of the second cavity have a great effect on the optical response of the system. Especially, the conversion between perfect OMIT and optomechanically induced absorption in the model can be easily achieved just by adjusting the driving field strength of the second cavity, which can be used as an optical switch in modern optical networks.
II. MODEL AND EQUATIONS
We consider an optomechanical system consisting of a mechanical membrane and two cavities (see Fig. 1). The mechanical membrane with frequency ω m and mass m is coupled to the two cavities via radiation pressure effects. The frequencies of the two cavities are described as ω 1 and ω 2 , and the annihilation operators of the two cavities are denoted byâ 1 andâ 2 , respectively. The position and momentum operators of the mechanical membrane are represented byq andp, respectively. The radiation pressure effect can be described by q(g 2â † 2â 2 − g 1â † Sketch of a two-cavity optomechanical system consists of a mechanical membrane with frequency ωm interacted with cavityâ1 andâ2 via radiation pressure. The cavityâ1 is driven by a coupling field εc with frequency ωc and a weak probe field εp with frequency ωp. The cavityâ2 is driven by a driving field ε d with frequency ω d .
with the optomechanical coupling rate g i = ω i /L i where L i is the geometric length of the cavity i. The cavitŷ a 1 /â 2 is driven by a strong coupling/driving field with frequency ω c /ω d and amplitude ε c /ε d . In addition, a weak probe field with frequency ω p and amplitude ε p is injected into cavityâ 1 . For simplicity and without loss of generality, we can set ω 1 = ω 2 = ω 0 and L 1 = L 2 = L. Then, the Hamiltonian of the system H s is
H s = ω 0â † 1â 1 + ω 0â † 2â 2 +p 2 2m + 1 2 mω 2 mq 2(1)+ qg 0 (â † 2â 2 −â † 1â 1 ) + i (â † 1 ε p e −iωpt −â 1 ε * p e iωpt ) +i ε c (â † 1 e −iωct −â 1 e iωct ) + i ε d (â † 2 e −iω d t −â 2 e iω d t )
with g 0 = ω 0 /L. Since in quantum mechanics the unitary transformation will not change the physics properties of the system, we can take the unitary transformationÛ = e i(ωcâ †
1â 1 +ω dâ † 2â 2)t using the formula H =Û H sÛ † − i Û ∂Û † ∂t .
After this transformation, the Hamiltonian H s will turn to H which can be given by
H = ∆ câ † 1â 1 + ∆ dâ † 2â 2 +p 2 2m + 1 2 mω 2 mq 2 + qg 0 (â † 2â 2 −â † 1â 1 ) + i (â † 1 ε p e −iδt −â 1 ε * p e iδt ) + i ε c (â † 1 −â 1 ) + i ε d (â † 2 −â 2 ).(2)
Here, ∆ c/d = ω 0 − ω c/d is the detuning between cavity fieldâ 1 /â 2 and coupling/driving field, and δ = ω p − ω c is the detuning between the probe field and coupling field.
In this paper, we deal with the mean response of the system to the probe field in the presence of the coupling and driving field, hence we do not include quantum fluctuations. According to the Heisenberg-Langevin equation and from Eq. (2), we can obtain the motion equations of mean values of the system operators as follow
ȧ 1 = −[κ 1 + i(∆ c − g 0 q )] â 1 + ε c + ε p e −iδt ,ȧ 2 = −[κ 2 + i(∆ d + g 0 q )] â 2 + ε d , ṗ = −γ m p − mω 2 m q + g 0 ( â † 1 â 1 − â † 2 â 2 ), q = p m .
(3) Here, we have used the usual factorization assumption, i.e. XŶ = X Ŷ (which holds in the case of singlephoton weak coupling, i.e., g 0 ≪ ω m [71]), and γ m , κ 1 and κ 2 are the damping rates of the mechanical membrane, cavityâ 1 andâ 2 , respectively.
It is very difficult to obtain the exact solution of Eq.
(3) because it is a nonlinear equation. However, in general the strength of the probe field is much smaller than that of the coupling field in cavity optomechanics. Hence, we can solve Eq. (3) by perturbation method. To this end, we assume that the solution of the mean values of the operators in Eq. (3) has the following form
ŝ = s 0 + ε p e −iδt s + + ε * p e iδt s −(4)
where s = {q, p, a 1 , a 2 }. It means that we can safely ignore the higher order terms of the probe field ε p .
We are particularly interested in the properties of the field with frequency ω p in the output field of cavityâ 1 , which can be determined by the term a 1+ in Eq. (4). The term a 1− denotes the anti-Stokes effect and we don't care here. Substituting Eq. (4) in Eq. (3) and ignoring the higher order terms of ε p , we can obtain the important expression for the term a 1+ (See Appendix A for detailed calculations) as
a 1+ = 1 κ 1 − i(δ − ∆ 1 ) + β1 δ 2 −ω 2 m +iδγm 2iωm − β 1 κ 1 −i(δ+∆ 1 ) + β 2 κ 2 −i(δ−∆ 2 ) − β 2 κ 2 −i(δ+∆ 2 ) ,(5)
here,
β 1 = g 2 0 |a 10 | 2 2mω m , β 2 = g 2 0 |a 20 | 2 2mω m ,(6)
and ∆ 1 = ∆ c −g 0 q 0 , ∆ 2 = ∆ d +g 0 q 0 , and the expressions of q 0 , a 10 and a 20 can be found in Appendix A.
Because it is known that the coupling between the cavity and the resonator is strong at the near-resonant frequency, in this paper, we consider ∆ 1 ∼ ∆ 2 ∼ ω m and δ ∼ ω m . Then we have δ 2 − ω 2 m ∼ 2ω m (δ − ω m ) and
δ + ∆ 1 ∼ δ + ∆ 2 ∼ 2ω m . If we set x = δ − ω m , Eq. (5)
can be simplified as
a 1+ = 1 κ 1 − ix + β1 γm 2 −ix− β 1 κ 1 −2iωm + β 2 κ 2 −ix − β 2 κ 2 −2iωm . (7)
Next, based on Eq. (7), we will study the optical response of the optomechanical system to the probe field, including the perfect optomechanically induced transparency, slow light and optomechanically induced absorption, respectively.
III. PERFECT OPTOMECHANICALLY INDUCED TRANSPARENCY
According to input-output relation [48], the quadrature of the optical component with frequency ω p in the output field can be defined as ε T = 2κ 1 a 1+ [48]. The real part Re[ε T ] and imaginary part Im[ε T ] represent the absorptive and dispersive behavior of the optomechanical system to the probe field, respectively. We first give the conditions of perfect OMIT in the model. According to the conclusions in Ref. [47], the conditions of perfect OMIT are determined by the pole location of the subfraction in Eq. (7), i.e.,
γ m 2 − ix − β 1 κ 1 − 2iω m + β 2 κ 2 − ix − β 2 κ 2 − 2iω m = 0. (8)
It can be known from Eq. (8) that the perfect OMIT cannot occur at the resonant frequency x = 0 because in this case β 2 = − γmκ2(κ 2 2 +4ω 2 m ) 2(κ1κ2+4ω 2 m ) < 0 which is in contradiction with the definition in Eq. (6).
We first study the case of large κ 2 , i.e., the cavity damping rate κ 2 is much larger than the detuning x where the perfect OMIT appears (κ 2 ≫ |x|), and from Eq. (8) we can obtain the conditions of perfect OMIT as
x = 2κ 2 ω m [γ m κ 2 (κ 2 1 + 4ω 2 m ) − 2β 1 (κ 1 κ 2 + 4ω 2 m )] (κ 2 1 + 4ω 2 m )ξ − 2β 1 κ 1 (κ 2 2 + 4ω 2 m ) , β 2 = κ 2 (κ 2 2 + 4ω 2 m )(2β 1 κ 1 − γ m (κ 2 1 + 4ω 2 m )) 8ω 2 m (κ 2 1 + 4ω 2 m ) ,(9)
with ξ = 8κ 2 ω 2 m + γ m (κ 2 2 + 4ω 2 m ). The perfect OMIT can be achieved if the conditions in Eq. (9) are satisfied. In Fig. (2), we plot the real part Re[ε T ] (red-solid) vs. frequency detuning x with ω m = κ 1 = κ 2 = 10 4 , γ m = 1, β 1 = 3×10 4 and β 2 = 1250 according to Eq. (9). With these parameters, according to Eq. (9), the transparency window will appear at x ≃ −1.25 which is consistent with the numerical result in Fig. (2). The inset in Fig. (2) shows the OMIT profile in a large scale, from which it can be seen that the width of the transparency window is very narrow. From Eq. (7), we can obtain the expression of the width Γ OMIT (full width at half maximum) of transparent window, but it is too lengthy to be reported here. However, if we take the case of equal cavity damping rate (κ 1 = κ 2 = κ), the width Γ OMIT can be given as
Γ OMIT = κ[32β 1 ω 2 m η + κ(η + ω m (4β 1 − 2κγ m )) 2 ] 2η + κ[32β 1 ω 2 m η + κ(η − ω m (4β 1 − 2κγ m )) 2 ] 2η − κ,(10)
with η = 8κω 2 m − 2β 1 κ + γ m (κ 2 + 4ω 2 m ). With the parameters above, the width Γ OMIT ≃ 5.998 which shows an excellent agreement with the numerical result (see the blue-dashed line) in Fig. 2.
One of the advantages of the perfect OMIT theory is the perfect transparency window can still appear even with a large mechanical damping rate γ m . In Fig. 3, we plot the real part Re[ε T ] vs. frequency detuning x for large mechanical damping rate γ m = 10 (red-solid) with β 1 = 3 × 10 5 , and β 2 = 1.25 × 10 4 according to Eq. (9), and for γ m = 100 (blue-dashed) with β 1 = 3 × 10 6 , and β 2 = 1.25 × 10 5 according to Eq. (9). The other parameters are ω m = κ 1 = κ 2 = 10 4 . It can be clearly seen from Fig. 3 that the perfect OMIT can do occur with large mechanical damping rate γ m as long as the conditions in Eq. (9) are satisfied. In addition, according to Eq. (10), the width Γ OMIT = 59.83 (red-solid) and Γ OMIT = 583.79 (blue-dashed), which are consistent with the results in Fig. 3. In Fig. 4, we plot the imaginary part Im[ε T ] vs. frequency detuning x with the same parameters as Fig. 3. It can be seen from Fig. 4 that the dispersion curve becomes very steep at the perfect transparency window and the slope is negative there.
The above discussion is based on the condition that the cavity damping rate κ 2 is much larger than the transparent window position x. While if the second cavity is the microwave cavity, the cavity damping rate κ 2 can be very small [72,73]. If κ 2 is very small, the above calculation would be a little more complicated. However, we can always obtain the window position x and driving strength β 2 through numerical methods according to Eq. (7). For example, for parameters κ 2 = 10, ω m = 10 4 , κ 1 = 4×10 3 , γ m = 1 and β 1 = 10 5 , we numerically obtain the window position x ≃ −5.55 and β 2 = 5.91. With these parameters, in Fig. 5 we plot the real part Re[ε T ] vs. frequency detuning x, and the inset in Fig. 5 shows a zoom-in of the transparency window at x ≃ −5.55. It can be clearly seen from Fig. 5 that the perfect OMIT can still occur with small cavity damping rate κ 2 .
IV. ULTRASLOW LIGHT
The time delay (slow light effect) of the probe field with frequency ω p in the output field can be determined by [54,56,59] The positive (negative) value of the time delays represents slow (fast) light [74] in the system. According to Eqs. (7), (9) and (11), we can obtain the analytic expressions of time delays τ , but it is too tedious to be reported here. However, the time delay at the transparent window can be obtained as
τ = ∂arg[ε T − 1] ∂ω p .(11)τ = κ 1 [8κ 2 ω 2 m + γ m (κ 2 2 + 4ω 2 m )] 4β 1 κ 2 ω 2 m − κ 2 1 (κ 2 2 + 4ω 2 m ) 2κ 2 ω 2 m (κ 2 1 + 4ω 2 m )
.
In Fig. 6, we plot the time delay τ vs. frequency detuning x for γ m = 1 (red-solid) with β 1 = 3 × 10 4 and β 2 = 1250 (same as Fig. 2), and for γ m = 10 (bluedashed) with β 1 = 3 × 10 5 and β 2 = 1.25 × 10 4 (same as Fig. 3). The other parameters are ω m = κ 1 = κ 2 = 10 4 (same as Fig. 2). It can be seen from Fig. 6 that the time delay τ exactly takes the maximum at the transparency window where the steepest dispersion appears. In fact, it can be proofed that the time delay at the transparency window is exactly equal to the negative dispersion curve slope there. It means that the steeper the slope of dispersion curve is, the larger the slow light effect becomes. From Fig. 6, the maximum delay τ max ≃ 0.67 for redsolid line and τ max ≃ 0.067 for blue-dashed line, which are very consistent with the results according to Eq. (12).
We also study the effect of cavity damping rate (κ 1 , κ 2 ) on the time delay τ . For simplicity, we also take κ 1 = κ 2 here. In Fig. 7, we plot the time delay τ vs. frequency detuning x for κ 1 = κ 2 = 2 × 10 4 (red-solid) with β 1 = 3 × 10 4 and β 2 = 10 4 according to Eq. (9), and for κ 1 = κ 2 = 8 × 10 3 (blue-dashed) with β 1 = 3 × 10 4 and β 2 = 160 according to Eq. (9). The other parameters are ω m = 10 4 and γ m = 1. It can be seen from Fig. 7 that the maximum time delay (at the transparency window) in the unresolved sideband regime (κ 1 > ω m ) will larger than that in the resolved sideband regime (κ 1 < ω m ). With the parameters, the maximum time delay τ max ≃ 1.33 in the unresolved sideband regime (see red-solid). It means that the τ max ≃ 1.33 sec (if the units of physical quantities above are Hertz), which is actually a long-lived slow light. This long-lived slow light may be used for OMIT-based memories in the future.
V. OPTOMECHANICALLY INDUCED ABSORPTION
Compared with Re[ε T ] = 0 at the window of the perfect OMIT, the phenomenon of optomechanically induced absorption will appear at the near resonance position x = 0 (Re[ε T ] shows a noticeable increase at position x = 0), if some conditions are satisfied. We first do some qualitative analyses of these conditions. At the position x = 0, we have
ε T (x = 0) = 2κ 1 κ 1 + β1 γm 2 − β 1 κ 1 −2iωm + β 2 κ 2 − β 2 κ 2 −2iωm ,(13)
which means that Re[ε T ] will display a noticeable increase at x = 0 if the ratio β 2 /κ 2 is large enough. In other words, if κ 2 is small enough or the driving strength β 2 is large enough, the phenomenon of optomechanically induced absorption at x = 0 can occur.
In Fig. 8, we plot the Re[ε T ] vs. the frequency detuning x for κ 2 = 10 (blue-dashed) and κ 2 = 1 (redsolid) with parameters κ 1 = 4 × 10 3 , ω m = 10 4 , γ m = 1, β 1 = 10 5 and β 2 = 100. It can be seen from Fig. 8 that the phenomenon of optomechanically induced absorption at x = 0 becomes very significant with the decrease of κ 2 . In addition, according to Eq. (13), we have ε T (x = 0) ≃ 0.58 for κ 2 = 10 and ε T (x = 0) ≃ 1.60 for κ 2 = 1, which are clearly consistent with the numerical results in Fig. 8.
Generally, once a quantum device has been manufactured, its characteristic parameters are fixed. Therefore, it is not easy to achieve optomechanically induced absorption by changing the dissipation rate κ 2 . While it is very convenient to adjust the driving strength β 2 in experiments. In Fig. 9, we plot the Re[ε T ] vs. the frequency detuning x for β 2 = 10 5 (red-solid) with parameters κ 2 = 10, κ 1 = 4 × 10 3 , ω m = 10 4 , γ m = 1, and β 1 = 10 5 . With the parameters, Re[ε T ] = 1.9995 at x = 0 according to Eq. (13), which is very much in agreement with the results in Fig. 9. Hence the optomechanically induced absorption can be achieved at x = 0 with the driving strength β 2 = 10 5 . In addition, the absorption curve is approximately a horizontal line over a very wide frequency range, see the red-solid line in the inset in Fig. 9. For comparison, we also plot the curve of perfect OMIT with the same parameters but β 2 = 5.91 (bluedashed), the transparency window x ≃ −5.55 (see the blue-dashed line in the inset of Fig. 9). It means that the Fig. 9 shows a zoom-in of the transparency window at x ≃ −5.55.
conversion between perfect OMIT and optomechanically induced absorption of the probe field can be achieved simply by adjusting the driving strength β 2 . These results can be used to achieve the optical switch in modern optical networks.
VI. CONCLUSION
In summary, we have theoretically studied the controllable optical responses in a two-cavity optomechanical system, especially on the perfect optomechanically induced transparency (OMIT), long-lived slow light and optomechanically induced absorption in the model. From the theoretical results, we can draw some conclusions. First, the perfect OMIT can be still achieved even with a large mechanical damping rate γ m , which is difficult to be realized in the usual OMIT theory. Secondly, at the transparency window of the perfect OMIT, the longlived slow light can be achieved, which can be used for OMIT-based memories in the future. Thirdly, an optical switch taking advantage of the conversion between perfect OMIT and optomechanically induced absorption can be achieved just by adjusting the driving field strength of the second cavity. We believe that the results can be used to control optical transmission in quantum information processing.
with ∆ 1 = ∆ c − g 0 q 0 and ∆ 2 = ∆ d + g 0 q 0 .
In order to obtain the expression of a 1+ , we need to give the expression of q + in Eq. (A2). To this end, according to Eqs. (A3)-(A9) and using the fact q * − = q + , we have Aq + = a * 10 a 1+ + a 10 a * 1− − a * 20 a 2+ − a 20 a * 2− , (A10) a * 10 a 1+ = B(ig 0 |a 10 | 2 q + + a * 10 ), (A11) a 10 a * 1− = −Cig 0 |a 10 | 2 q + , (A12) a * 20 a 2+ = −Dig 0 |a 20 | 2 q + , (A13) a 20 a * 2− = Eig 0 |a 20 | 2 q + , (A14)
with A = m(ω 2 m − δ 2 − iγ m δ) g 0 ,(A15)B = 1 κ 1 − i(δ − ∆ 1 ) ,(A16)C = 1 κ 1 − i(δ + ∆ 1 ) ,(A17)D = 1 κ 2 − i(δ − ∆ 2 ) ,(A18)E = 1 κ 2 − i(δ + ∆ 2 )
.
From Eqs. (A11)-(A14), we can obtain the expression of a * 10 a 1+ + a 10 a * 1− − a * 20 a 2+ − a 20 a * 2− , and then combining Eq. (A10), the expression of q + can be obtained as
FIG. 1: Sketch of a two-cavity optomechanical system consists of a mechanical membrane with frequency ωm interacted with cavityâ1 andâ2 via radiation pressure. The cavityâ1 is driven by a coupling field εc with frequency ωc and a weak probe field εp with frequency ωp. The cavityâ2 is driven by a driving field ε d with frequency ω d .
FIG. 2 :
2The real part Re[εT ] (red-solid) vs. frequency detuning x with parameters ωm = κ1 = κ2 = 10 4 , γm = 1, β1 = 3 × 10 4 and β2 = 1250 according to Eq. (9). The bluedashed line indicates width ΓOMIT. The inset in Fig. 2 shows the OMIT profile in a large scale. FIG. 3: The real part Re[εT ] vs. frequency detuning x for large mechanical damping rate γm = 10 (red-solid) with β1 = 3 × 10 5 and β2 = 1.25 × 10 4 according to Eq. (9), and for γm = 100 (blue-dashed) with β1 = 3×10 6 and β2 = 1.25×10 5 according to Eq. (9). The other parameters are same as Fig. 2.
FIG. 4 :
4The imaginary part Im[εT ] vs. frequency detuning x with the same parameters as those inFig. 3.
FIG. 5 :
5The real part Re[εT ] vs. frequency detuning x for κ2 = 10 with parameters ωm = 10 4 , κ1 = 4 × 10 3 , γm = 1, β1 = 10 5 and β2 = 5.91. The inset inFig. 5shows a zoom-in of the transparency window at x ≃ −5.55.FIG. 6:The time delay τ vs. frequency detuning x for γm = 1 (red-solid) with β1 = 3 × 10 4 and β2 = 1250 (same asFig. 2), and for γm = 10 (blue-dashed) with β1 = 3 × 10 5 and β2 = 1.25 × 10 4 (same asFig. 3). The other parameters are ωm = κ1 = κ2 = 10 4 .
FIG. 7 :
7The time delay τ vs. frequency detuning x for κ1 = κ2 = 2 × 10 4 (red-solid) with β1 = 3 × 10 4 and β2 = 10 4 according to Eq. (9), and for κ1 = κ2 = 8 × 10 3 (blue-dashed) with β1 = 3 × 10 4 and β2 = 160 according to Eq.(9). The other parameters are ωm = 10 4 and γm = 1.
FIG. 8 :
8The real part Re[εT ] vs. frequency detuning x for κ2 = 10 (blue-dashed) and κ2 = 1 (red-solid) with parameters κ1 = 4 × 10 3 , ωm = 10 4 , γm = 1, β1 = 10 5 and β2 = 100.
FIG. 9 :
9The real part Re[εT ] vs. frequency detuning x for β2 = 5.91 (blue-dashed) and β2 = 10 5 (red-solid) with parameters ωm = 10 4 , κ1 = 4 × 10 3 , γm = 1, β1 = 10 5 and κ2 = 10. The inset in
A
− ig 0 [(B − C)|a 10 | 2 + (D − E)|a 20 | 2 ] . (A20)Substituting Eq. (A20) into Eq. (A5), we obtain a 1+ asa 1+ = ig 0 B 2 |a 10 | 2 A − ig 0 [(B − C)|a 10 | 2 + (D − E)|a 20 | 2 ] + B = [A − ig 0 (−C|a 10 | 2 + (D − E)|a 20 | 2 )]B A − ig 0 [(B − C)|a 10 | 2 + (D − E)|a 20 | text for a 1+ .
Appendix A: Derivation of a1+Substituting Eq. (4) in Eq.(3), just keeping the constant term and the first order term of ε p , and then comparing the coefficients of the terms e iδt and e −iδt on both sides of the equation, we can obtain, (A8), (A9)
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| [] |
[
"Supplementary Figure 1",
"Supplementary Figure 1"
] | [] | [] | [] | Supplementary Figure 1. Simulation of the broadening of an ultrafast pulse when it transmits through a high-Q micro-resonator. We simulate a ~100 fs pulse transmitting through a 5μm-radius, over-coupled silicon microring add-drop filter using a 3D finitedifference time-domain (FDTD) method based numerical solver MEEP 1 . The microring is the same design as that of a replica generation (recombination) ring in the shaper shown inFig. 1ain the main text, where the ring-waveguide power coupling coefficient is κ 2 ≈ 0.1. (a) The optical spectra of the input (in red) and output (in blue) pulses. The input pulse has a center wavelength of ~1550 nm, and its ultrawide spectrum covers multiple free spectral ranges (FSRs) of the microring. (b) The temporal profiles of the input pulse (in red) and output pulse train (in blue). The peak value of each pulse in the output pulse train decays exponentially with a time constant of Q/ω 0 , where ω 0 is the resonance frequency close to 1550 nm and Q ≈ 2600 is the corresponding quality factor. In the RF AWG experiment, the photo-detector responds only to the envelope of the output optical pulse train, whose 3dB temporal width is given by ln2×Q/ω 0 . The quality factor Q of the over-coupled microring is related to the power coupling coefficient κ 2 through ω 0 /Q ≈ 2κ 2 v g /L, where v g is the group velocity of light in the waveguide and L is the circumference of the microring. The input and output pulses are offset in time for ease of viewing. | 10.1038/ncomms6957 | null | 15,121,574 | 1404.3227 | 48ecdf0a1a4df9e52cdc26d5fd80352d3e73f5fe |
Supplementary Figure 1
Supplementary Figure 1
Supplementary Figure 1. Simulation of the broadening of an ultrafast pulse when it transmits through a high-Q micro-resonator. We simulate a ~100 fs pulse transmitting through a 5μm-radius, over-coupled silicon microring add-drop filter using a 3D finitedifference time-domain (FDTD) method based numerical solver MEEP 1 . The microring is the same design as that of a replica generation (recombination) ring in the shaper shown inFig. 1ain the main text, where the ring-waveguide power coupling coefficient is κ 2 ≈ 0.1. (a) The optical spectra of the input (in red) and output (in blue) pulses. The input pulse has a center wavelength of ~1550 nm, and its ultrawide spectrum covers multiple free spectral ranges (FSRs) of the microring. (b) The temporal profiles of the input pulse (in red) and output pulse train (in blue). The peak value of each pulse in the output pulse train decays exponentially with a time constant of Q/ω 0 , where ω 0 is the resonance frequency close to 1550 nm and Q ≈ 2600 is the corresponding quality factor. In the RF AWG experiment, the photo-detector responds only to the envelope of the output optical pulse train, whose 3dB temporal width is given by ln2×Q/ω 0 . The quality factor Q of the over-coupled microring is related to the power coupling coefficient κ 2 through ω 0 /Q ≈ 2κ 2 v g /L, where v g is the group velocity of light in the waveguide and L is the circumference of the microring. The input and output pulses are offset in time for ease of viewing.
Fig. 1
. The transmission spectra are taken at the common through and common drop ports of the replica generation rings and at the common through port of the replica recombination rings, respectively. (a) The through-port transmission spectra are taken before thermal resonance matching. (b) The through-port and (c) drop-port transmission spectra are taken after thermal resonance matching. The fibre-to-fibre loss of ~22 dB measured at the through ports is attributed to waveguide facet coupling loss.
Supplementary Figure 3
Supplementary Figure 3. Simulated delay and transmission spectra of two tuneable optical delay line designs using coupled-mode theory 2 . In both designs, all the microrings side couple to the bus waveguide at the same coupling gap of 100 nm (with a power coupling coefficient of κ 2 ≈ 0.6) to satisfy the over-coupling condition. The propagation loss in the microrings is set as 10 dB/cm in the simulations. (a) The delay (blue) and transmission (green) spectra of an optical delay line that consists of 41 microrings with a radius increment step of 1 nm and a median radius of 5 μm. The simulated delay ranges from 4.5 ps to 41.5 ps, with the insertion loss increasing from 0.32 dB to 3 dB. To test the robustness of the design, we next assumed a random variation of ±2 nm in the radii of each of the 41 rings, which corresponds to the estimated precision of E-beam lithography in fabricating the delay line. 1000 simulations with random radius variations were performed and overlaid with that of the original design, showing a small variation of ~4 ps in delay and of 0.3 dB in transmission (shaded areas). (b) The delay and transmission spectra of another optical delay line design where the 41 microrings have a radius increment step of 1.5 nm and a median radius of 5μm. Both spectra are overlaid with the simulation results where random radius variations of ±2 nm in the radii of each of the 41 rings are incorporated.
Supplementary Figure 4
Supplementary Figure 4. Characterization of the silicon intensity modulator (embedded in the pulse shaper) with a continuous-wave (CW) laser input at 1549.7 nm. The modulator was tested at a modulation speed of (a) 250 Mbps, (b) 1.06 Gbps, (c) 2.5 Gbps, and (d) 5 Gbps, respectively. The modulator is the same design as those reported in Ref.
[3] except that some modification is made to the traveling-wave electrode design to meet the testing requirement but at a price of a compromised modulation efficiency. In the experiment of rapidly reconfigurable AWG, we applied external electrical controls to the silicon chip using RF/DC probes, which required much more setup space than the wire bonding method. In the setup, we coupled light into and out of the chip through fibre butt coupling and the remaining setup space could only accommodate one 16-pin DC probe to reconfigure the pulse shaper and one RF probe to drive the modulator. However, driving a traveling-wave modulator typically requires using a pair of 'GSG' RF probes, one feeding the RF signal at one end of the electrodes and one terminating the RF wave at the other end. Thus using only one RF probe not only limits the number of modulators that can be incorporated into the shaper but also requires modification to the electrode design of the modulator. In our design, the electrodes were wrapped to one side of the modulator (See Fig. 4a in the main text) so that a single 'GSGSG' RF probe can feed and terminate the RF driving signal simultaneously. In addition, the thermal tuning element was also dropped in this modulator design due to lack of space for additional DC probing. As a consequence of the compromised electrode design and lack of thermal tuning element, the extinction ratio of the modulator was limited to ~3:1 and the modulation speed was limited to 5 Gbps.
Supplementary Figure 5
Supplementary Figure 5. Illustration of RF frequency synthesis by adjusting delays of some channels and disabling selected channels. The fixed delay is 25 ps between consecutive channels, corresponding to a 40 GHz fundamental RF tone. One can achieve a 33.3 ps channel spacing (30 GHz fundamental RF tone) by adjusting the 2nd and 3rd channel delays and disabling the 4th channel. Our architecture can easily disable any channel by shifting the recombination ring resonance away from that of the replica generation resonator. Experimental demonstrations are shown in Fig. 2e and Fig. 3b. Mathematically, since each channel can tune the delay (27 ps) more than the fixed channel spacing (25 ps), one can synthesize any RF waveforms as long as the highest frequency component is less than 1/(25 ps) = 40 GHz. To increase the highest frequency, one can reduce the fixed delay and increase the number of channels.
Supplementary Figure 2 Supplementary Figure 2 .
22Measured transmission spectra of the Si pulse shaper described in
Yariv, A. Universal relations for coupling of optical power between microresonators and dielectric waveguides. Electron. Lett. 36, 321-322 (2000). 3 Wang, J., et al. Optimization and demonstration of a large-bandwidth carrierdepletion silicon optical modulator. J. Lightwave Technol. 31,4119-4125 (2013).
Supplementary Table 1Supplementary Table 1. Measured and simulated characteristics of racetrack resonators in the foundry chip. The racetrack resonator has a bending portion of ~5 μm in radius and a straight portion of ~7.5 μm in length that side couples to the bus waveguide at a coupling gap of 200 nm. The experimental values of the free spectral range, Q-factor and power coupling coefficient κ 2 were retrieved from the transmission spectrum taken at the common through port of the replica generation rings, and they were averaged over 8 channels. The numerical values were determined using a 3D FDTD solver MEEP 1 for both the transverse-magnetic (TM) and transverse-electric (TE) modes of the racetrack resonators. By comparing the three groups of values, we confirmed that the racetrack resonators in the shaper worked in TE polarization as designed.Measurement | [] |
[
"Positive Vacuum Energy and the N -bound",
"Positive Vacuum Energy and the N -bound"
] | [
"Raphael Bousso [email protected] \nInstitute for Theoretical Physics\nUniversity of California\n93106-4030Santa BarbaraCalifornia\n"
] | [
"Institute for Theoretical Physics\nUniversity of California\n93106-4030Santa BarbaraCalifornia"
] | [] | We argue that the total observable entropy is bounded by the inverse of the cosmological constant. This holds for all space-times with a positive cosmological constant, including cosmologies dominated by ordinary matter, and recollapsing universes. The argument involves intermediate steps which may be of interest in their own right. We note that entropy cannot be observed unless it lies both in the past and in the future of the observer's history. This truncates space-time to a diamond-shaped subset well-suited to the application of the covariant entropy bound. We further require, and derive, a novel Bekenstein-like bound on matter entropy in asymptotically de Sitter spaces. Our main result lends support to the proposal that universes with positive cosmological constant are described by a fundamental theory with only a finite number of degrees of freedom. | 10.1088/1126-6708/2000/11/038 | [
"https://arxiv.org/pdf/hep-th/0010252v2.pdf"
] | 1,439,541 | hep-th/0010252 | 50f028089d752c60387d3c86c1f5031867c7853a |
Positive Vacuum Energy and the N -bound
12 Dec 2000
Raphael Bousso [email protected]
Institute for Theoretical Physics
University of California
93106-4030Santa BarbaraCalifornia
Positive Vacuum Energy and the N -bound
12 Dec 2000Preprint typeset in JHEP style. -HYPER VERSION hep-th/0010252
We argue that the total observable entropy is bounded by the inverse of the cosmological constant. This holds for all space-times with a positive cosmological constant, including cosmologies dominated by ordinary matter, and recollapsing universes. The argument involves intermediate steps which may be of interest in their own right. We note that entropy cannot be observed unless it lies both in the past and in the future of the observer's history. This truncates space-time to a diamond-shaped subset well-suited to the application of the covariant entropy bound. We further require, and derive, a novel Bekenstein-like bound on matter entropy in asymptotically de Sitter spaces. Our main result lends support to the proposal that universes with positive cosmological constant are described by a fundamental theory with only a finite number of degrees of freedom.
Introduction
Banks's proposal
Banks [1] has proposed that the cosmological constant should not be viewed as an effective parameter to be derived in a theoretical framework like QFT or string theory. Instead, it is determined as the inverse of the number of degrees of freedom, N, in the fundamental theory. 1 It should thus be considered an input parameter at the most fundamental level of physics.
The proposal can be motivated as follows. In the presence of a positive cosmological constant, Λ, the universe tends to evolve to empty de Sitter space. de Sitter space has a finite entropy S = 3π/Λ, given by the area of the cosmological horizon. Thus the universe is most economically described by a theory with the corresponding number of degrees of freedom, N = 3π/Λ. Conversely, a quantum gravity theory with a finite number of degrees of freedom, N, requires for consistency a cosmological constant Λ = 3π/N to provide a geometric entropy cutoff.
The Λ-N correspondence does not solve the cosmological constant problem except by fiat. It is not clear why the fundamental theory should happen to possess the enormous but finite number of degrees of freedom N ∼ 10 122 that corresponds to the observationally favoured value of the cosmological constant. But the proposal offers a radical, and potentially fruitful, change of perspective.
Its most profound implication is the following: A quantum gravity theory with an infinite number of degrees of freedom, such as M theory, cannot describe space-times with a positive cosmological constant. 2 This is consistent with the fact that no stable de Sitter vacua are known in M theory. If the proposal is correct, this gap would not be due to our limited understanding of the theory, but must be ascribed to an obstruction in principle.
The correspondence thus suggests that one should look for a theory with finite N that is self-consistent and complete; i.e., it will not do to impose a naive cut-off on an N = ∞ theory. If such theories exist for arbitrarily large values of N, one might expect them to limit to M theory. However, finite N theories will contain certain qualitative features, such as positive vacuum energy and perhaps supersymmetry breaking, 3 which would be entirely absent in the infinite N limit and could not have been studied there.
How can the proposal be tested? It asserts that a universe with Λ > 0 is a system with N = 3π/Λ degrees of freedom. Unfortunately, this cannot be verified at the semi-classical level, as we have no understanding what the true degrees of freedom are. However, a system with N degrees of freedom certainly cannot have entropy greater than N. Thus, the Λ-N correspondence predicts that a universe with Λ > 0 cannot have entropy greater than N = 3π/Λ. We call this prediction the N-bound. It can be tested.
It is not difficult to see that the N-bound is true for vacuum solutions like de Sitter space (a trivial case). Moreover, one can argue that it is satisfied for all space-times which are asymptotically de Sitter at late times, by the generalized second law of thermodynamics. Indeed, in Ref. [1] the Λ-N correspondence was conjectured to apply only within this class of space-times. This includes, for example, the Λ > 0 flat Friedman-Robertson-Walker (FRW) solution which appears to describe our universe: it starts out with a big bang and is initially radiation-or matter-dominated; then the matter is diluted by the cosmological expansion; and at some time (as it happens, roughly now) the vacuum energy-which is not diluted-starts to dominate and leads the universe to evolve towards empty de Sitter space in the far future.
However, solutions with Λ > 0 need not necessarily become de Sitter at late times. Consider, for example, the time reversal of the cosmological solution just described: it starts out as empty de Sitter; then more and more matter condenses, which eventually causes the space-time to collapse in a big crunch. This illustrates, in particular, that one can never be sure to have reached the safety of asymptotic de Sitter space; there is always the possibility of a huge collapsing shell of matter that cannot be seen yet but will cause an apocalypse in the future. Another example is a Λ > 0 closed FRW universe. Given a sufficiently large matter density, the cosmological constant will not be strong enough to prevent recollapse. Indeed, Λ might be a negligible contribution to the total energy density at all times.
These Λ > 0 solutions are perfectly valid from the perspective of semi-classical gravity. Many of them are physically quite reasonable, and we would find it unconvincing to exclude them a priori. In some cases, a small perturbation can make all the difference between collapse and expansion to asymptotically de Sitter space. These arguments lead us to advocate a stronger version of Banks's proposal. We conjecture that the Λ-N correspondence holds for all Λ > 0 universes, including those that do not evolve to de Sitter in the future. But if de Sitter is not the 'final state', the second law will be of no help, and it is no longer obvious that the N-bound holds. The N-bound is thus a non-trivial prediction of the Λ-N correspondence.
Indeed, at first sight, some solutions may appear to have entropy greater than N, in contradiction with the correspondence. Nevertheless, it will be argued in this paper that the N-bound is valid for all universes with Λ > 0. This statement is far from obvious, and its proof will be seen to require a number of non-trivial intermediate results. Therefore, our conclusion may be viewed as evidence in favour of the proposed correspondence.
Outline
Our goal is to prove the following conjecture:
N-bound In any universe with a positive cosmological constant Λ (as well as arbitrary additional matter that may well dominate at all times) the observable entropy S is bounded by
N = 3π/Λ. (1.1)
Here S includes both matter and horizon entropy, but excludes entropy that cannot be observed in a causal experiment. Note that N is the Bekenstein-Hawking entropy of empty de Sitter space. The bound becomes trivial in the limit of vanishing cosmological constant. As we have argued above, an independent proof of the N-bound provides strong support to the proposed Λ-N correspondence [1]; hence, the correspondence will not be used anywhere in the paper. In Sec. 2 we ask what constitutes observable entropy. We argue that one should restrict attention to the causal diamond of an observer: the space-time region that can be both influenced and seen by an observer. Thus, the observable entropy lies in a region bounded by the past and future light cones from the endpoints of the observer's world line.
The covariant entropy bound [8][9][10][11], reviewed in Sec. 3, can be applied to the cones bounding the observable region. This turns out to imply only S ≤ 2N, however, which does not quite suffice. In Sec. 4, we derive a novel bound on the entropy of matter systems in de Sitter space, the 'D-bound', which can be tighter than the covariant bound. In Sec. 5 we argue that the two bounds can be combined to imply the Nbound. The results are discussed in Sec. 6.
Banks's discussion [1] of the consistency of his proposal involved many of the considerations that enter our derivation of the N-bound. The arguments presented here are strongly influenced by Susskind's emphasis on the operational meaning of physical quantities. The covariant entropy bound [8], which plays a central role in the present work, generalizes a proposal by Fischler and Susskind [12] and is thought to have its origin in the holographic principle, first formulated by 't Hooft [13] and Susskind [14]. The application of the covariant entropy bound to the past light-cone of an observer was proposed by Banks in Ref. [15]. The D-bound is related to Bekenstein's bound on the entropy of finite systems in flat space [16]. Its derivation adapts the original arguments of Geroch and Bekenstein, and extends to cosmologically large systems Schiffer's use of the cosmological horizon to obtain Bekenstein's bound [17]. Bekenstein's generalized second law of thermodynamics [18][19][20] underlies most of the work in this paper. The semi-classical description of asymptotically de Sitter space-times was laid out by the work of Gibbons and Hawking [21]; see also Ref. [22]. Other recent work exploring connections between the holographic principle and the cosmological constant includes Refs. [23][24][25].
Causal diamonds
We first address the question of which entropy (or information) is actually accessible to a given observer. We will argue that certain space-time regions can be eliminated from consideration, and that the N-bound need only hold for the remaining region, the 'causal diamond' associated with an observer. It will also be shown that these restrictions are necessary, in the sense that the inclusion of unobservable entropy would easily allow the violation of the bound.
Implicit in this approach is the principle, long advocated by Susskind, that a fundamental theory need only answer questions that are operationally meaningful. For example, it need not (and, from an aesthetic standpoint, should not) simultanously describe the experiments made by two separate observers who, for reasons of causal structure, will never be able to compare results. Of course, it must be able to describe each experiment separately. This principle has previously been used to resolve certain apparent paradoxes in the evaporation of black holes [26][27][28].
We will consider an experiment that begins at point p and ends at a later point q on the observer's world line. It will be seen that causality limits the space-time region whose entropy can play a role in the experiment. It may be sufficient to consider only 'the longest experiment possible', i.e., the limit in which p is taken to be in the far past, and q in the far future, on the world line. However, it will be simpler and more instructive to carry out the discussion for arbitrary p and q. As experiments often have finite duration, this is the most general case; and all results will continue to hold in the limit of early p and late q.
The past light-cone
There are two independent restrictions. The first is:
(R1) Consider only the observer's causal past, J − (q). Ignore everything else. This is a sensible restriction. At the point q, the endpoint of the experiment, the observer can only have received signals from the past of q. The rest of space-time has not yet been seen. For the purposes of the experiment in question, its entropy is operationally meaningless and can be ignored.
For the later application of entropy bounds, note that the observer's past is bounded by the past light-cone from the point q, and that all matter within the observer's past must pass through this cone. 4 Thus, if one wishes to bound the observable entropy, it will be sufficient to bound the entropy on the past light-cone of the endpoint, q. The restriction R1 is necessary for the N-bound. Consider a Λ > 0 flat FRW universe starting with a big bang-possibly a good approximation to the universe we inhabit. The entropy density on any homogeneous spacelike slice is constant; thus, the total entropy on the slice is formally infinite, in apparent violation of the N-bound. The restriction R1 resolves this problem. Because the cosmological constant dominates at late times, any observer has a future event horizon (Fig. 1). The entropy in its interior is finite. Because the event horizon contains the observer's past for any endpoint q, the observed entropy is also finite. (We do not show quantitatively that it satisfies the N-bound as this will follow from the general arguments given in Sec. 5.) In this example, space-time is asymptotically de Sitter in the future, with entropy N. Thus, R1 is not only sensible and necessary for the N-bound, but indeed necessary for the validity of the generalized second law of thermodynamics.
It is instructive to contrast the above example with the case of a Λ = 0 flat FRW universe (Fig. 1). The latter has a different infinity structure. Arbitrarily large portions of any flat hypersurface lie within the past light-cone at sufficiently late times.
Even with restriction R1, the observed entropy is unbounded. Of course, this is not a problem, because N = ∞ in this case.
The future light-cone
The second restriction is:
(R2) Consider only the observer's causal future, J + (p). Ignore everything else.
Note that the observer's future is bounded by the future light-cone of the point p, and that all matter within the observer's future must have entered through this cone. 5 This restriction may seem less obvious than the previous one. But it is just as sensible. It is not enough for entropy, or information, to lie in the observer's past. To be observed, it actually has to get to the observer, or at least to a region that can be probed by the observer. But an experiment that commences at p can only probe what is in the causal future of p.
Put differently, all information that reaches the observer, or at least is accessible to the observer, must have passed through the future light cone of p. For the purpose of describing the experiment in question, one can ignore the space-time region outside the cone; instead, one may think of the initial conditions as residing on the cone. Entropy that fails to enter through the cone is operationally meaningless: though it may well be present in the observer's causal past, an experiment that starts at p will not know about it, because it cannot probe the region where such entropy resides. How is this consistent with cosmological observations of distant galaxies? By measuring the cosmic microwave background radiation, are we not collecting information about the early universe? These regions are indeed outside the future of our entire world-line, let alone the future of the point when the experiment began. However, all the information we gathered was in photons that interacted with some local apparatus. They had to enter through the future cone to get here. So the entropy we actually observe is quite local. It is certainly insightful to interpret this information in terms of models that involve inaccessible regions. For example, one might say that the early universe contained certain density perturbations. But the information used to obtain this conclusion is here, now. Thus, it is subject to entropy bounds associated with a much smaller region than the one it is interpreted to be an imprint of.
Without the additional restriction R2, the N-bound would fail. Fig. 2 shows a Λ > 0 space-time in which the observer's causal past contains an arbitrarily large entropy. Consider a universe that approaches empty de Sitter space asymptotically in the past. The geometry will resemble the lower half of the de Sitter hyperboloid at early times (see Appendix). It contains exponentially large three-spheres, on which one can place dilute matter with arbitrary entropy. If the total entropy exceeds N, the universe will necessarily be dominated by this matter at a later time. It will collapse in a big crunch, and there will be no future de Sitter region. One can arrange for the energy and entropy density to be constant on the observer's past light-cone (by giving it an increasing profile on the early S 3 , in the radial direction away from the observer's world line). The past light-cone keeps going forever, and so the total entropy on it will be infinite.-Note that the area of surfaces on the past light-cone diverges, so this example does not contradict the covariant entropy bound discussed in Sec. 3.
The causal diamond
Recall that p and q are two points on an observer's world line, with q later than p. One can think of p as the beginning and q as the end of some experiment. The restrictions R1 and R2 define the space-time region that can come into play in such an experiment. According to R2, one can ignore what is outside the causal future of p, and R1 states that regions outside the causal past of q are operationally meaningless as well. Combining both conditions, one can restrict to the points which are both in the future of p and in the past of q. This set,
C(p, q) = J + (p) ∩ J − (q), (2.1)
will be called the causal diamond associated with an experiment beginning at p and ending at q. Thus, one obtains the condition (R1+R2) Consider only the entropy in causal diamonds, i.e., in regions of the form C(p, q).
(The notion of an observer's world line was a crutch that can be dropped now. If q is in the future of p, there will be world lines connecting them; if not, then C(p, q) will be empty or degenerate.)
Of a fundamental theory, one may demand that it describe any experiment, but no more than that. Hence, it should describe the physics in any causal diamond, that is, in any region of the form C(p, q) for some pair of points (p, q), but only one causal diamond at a time. One should not demand that the theory simultaneously describe two separate causal diamonds, unless they are both contained in a single larger causal diamond.
For example, the theory should be able to describe an experiment inside a black hole, as well as an experiment outside a black hole. But it should not describe correlation functions between a point inside and a point outside a black hole if those points do not lie in any causal diamond. This example is just a reformulation of some of the arguments that established the concept of 'black hole complementarity' [26][27][28]. (In this case only the restriction R1 really matters, since R2 can easily be satisfied.) An analogous argument can be made for pairs of points near a big bang singularity. If they are sufficiently far, they cannot lie in a single causal diamond. Then no experiment can be set up that will involve both points. (In this case, R2 is the crucial restriction.)
In space-times that are asymptotically de Sitter in the past and future, any causal diamond lies within both the past and future event horizon. (Both R1 and R2 are used here.) The exponentially large regions beyond those horizons are operationally meaningless. This result has long been advocated by Susskind.
A causal diamond is bounded by a top cone (a portion of the past light-cone of q), and a bottom cone (a portion of the future light-cone of p); see Fig. 3. The cones usually, though not necessarily, intersect at a two-dimensional spatial surface, the edge of the causal diamond. In any case, the entropy in the causal diamond must pass through the top cone (and all matter must have entered through the bottom cone). 6 It will be seen below that the nature of the boundaries allows for a straightforward application of the covariant entropy bound. For this reason, the entropy within a causal diamond is under good theoretical control.
The covariant entropy bound
The covariant entropy bound [8] bounds the entropy on certain null hypersurfaces or 'light-sheets'. It was developed in order to formulate the holographic principle [13,14] for general space-times [9], and can be viewed as a generalization of the approach of Fischler and Susskind [12]. The use of null hypersurfaces to relate entropy and area was originally suggested by Susskind [14]. Several concepts crucial to a general formulation were first recognized by Corley and Jacobson [30]. The bound is conjectured to hold for any spacial surface in any space-time with reasonable energy conditions. It will be useful here because it applies even to regions, such as recollapsing universes or black hole interiors, where the second law is of no help. The conjecture has passed a number of non-trivial tests [8]. It has been proven in space-time regions where a fluid approximation to entropy can be made with plausible relations between entropy and energy density [11].
Consider some 2-dimensional spa-0000 0000 1111 1111
A Figure 4: The four light-like hypersurfaces orthogonal to a spatial surface (in this example, two cones going in and two 'skirts' going out). In a Penrose diagram the four null directions are indicated by an 'X' (right).
tial surface of area A. (We will mostly be interested in closed surfaces, but this is not a necessary restriction.) Any surface has four orthogonal light-like directions. Namely, the surface has two sides, and on each side there is a family of orthogonal light-rays arriving from the past (past-directed light-rays), and a family of future-directed light-rays. In Fig. 4 this is illustrated for the example of a spherical surface. In a Penrose diagram, where light travels at 45 degrees, the four orthogonal light-like directions are indicated by the legs of an 'X' centered on the point that represents the sphere. The orthogonal light-rays generate four 2+1 dimensional null hypersurfaces. On some of them, the light-sheets of the surface A, the cross-sectional area spanned by the light-rays will be decreasing or constant in the direction away from the original surface. (In the example in Fig. 4, the two cones.) The entropy on any light-sheet is less than A/4:
S(light-sheet of A) ≤ 1 4 A. (3.1)
Any surface has at least two light-sheets, since two of the four families of light-rays are just continuations of the opposite two. E.g., if the area is increasing in the future direction to one side, it will necessarily decrease in the past direction to the other side.
If it is constant in some direction, both opposing legs will be allowed. sectional area is a local condition and it must hold everywhere on the lightsheet. This means that the light-sheet must be terminated at or before one reaches a caustic, i.e., before neighbouring light-rays intersect (Fig. 5). In Fig. 4, the tips of the cones are caustics, and the light-sheets end there. The focussing theorem guarantees that contracting light-rays will not become expanding without going through a caustic. 7 If one chooses to terminate the light-sheet before each light-ray reaches a caustic, the end-points will span a non-zero area A ′ . Then the covariant bound can be strengthened [11]:
S ≤ 1 4 (A − A ′ ) . (3.2)
The light-sheet directions associated with a surface can be indicated, in a causal diagram, by the corresponding legs of the 'X' (Fig. 6). The two allowed directions form a wedge. One may classify closed surfaces as follows. For normal surfaces, both legs of the wedge point to one side, which is called the inside by definition. If both lightsheets are future-directed, the surface is trapped ; if the area is contracting in both past directions, it is called anti-trapped . Marginal cases arise for surfaces on the interface between a normal and a trapped or anti-trapped region. Then the expansion vanishes along at least one opposing pair of legs, and three or four legs must be drawn. The covariant entropy bound is particularly powerful when applied to such surfaces, and we will focus on them in Sec. 5.
The D-bound on matter entropy in de Sitter space
By studying the second law of thermodynamics in asymptotically flat space, Bekenstein 7 The null convergence condition [31] is assumed to hold: T ab k a k b ≥ 0 for all null vectors k a . -It has been suggested [32] that a light-sheet be terminated also at points where non-neighbouring light-rays intersect. As this can only make the light-sheet smaller, it gives a weaker bound, but the smaller light-sheet may be easier to compute practically. The light-sheets in Sec. 5 below are of this simple type, because they are a portion of the boundary of the causal past of a point. found that the total entropy is given by the sum of ordinary matter entropy, S m , plus the semiclassical Bekenstein-Hawking entropy, S h = 1 4 A h , associated with the horizons of black holes [18][19][20]33]. Similarly, in asymptotically de Sitter space, the cosmological horizon contributes with
S c = 1 4 A c (4.1)
to the total entropy [21]. Empty de Sitter space has a cosmological horizon of area
A 0 = 12π Λ = 4N (4.2)
(see Appendix). Therefore, empty de Sitter space has horizon entropy S c = N. One might think that even a tiny amount of matter entropy would already increase the total entropy, S c + S m , above N. However, the cosmological horizon surrounding a matter system in asymptotically de Sitter space is smaller than A 0 : the more matter, the smaller the cosmological horizon. Thus it is possible that the total entropy remains bounded by N. (It is instructive to verify this explicitly for the simple case of Schwarzschild-de Sitter black holes; see also [34].) It will now be shown that the N-bound is in fact implied by the second law if space-time contains an asymptotically de Sitter region in the future. This will allow us, by subtracting the horizon entropy from N, to derive a bound on the matter entropy in de Sitter space. Despite the restrictive assumption of an asymptotic de Sitter region, this bound will be useful to our purpose; we will argue later that it may also be applied to certain portions of more general space-times. Thus it will join the covariant bound, and the concept of causal diamonds, as a third ingredient in the argument constructed in Sec. 5 to show that the N-bound is valid for all Λ > 0 space-times.
Consider the following process. The initial configuration is a matter system in asymptotically de Sitter space. The matter system may contain black holes, whose entropy is taken to be included in the matter entropy, S m . The system is surrounded by a cosmological horizon of area A c . The final state is empty de Sitter space. The transition is achieved by taking the observer to move into the asymptotic region. (To the observer, the matter system appears to fall into the cosmological horizon.) In this process, the matter entropy S m is lost, while the entropy of the cosmological horizon increases by an amount
∆S c = 1 4 (A 0 − A c ) . (4.3)
The generalized second law of thermodynamics [18][19][20] implies that the total entropy must not decrease:
∆S c ≥ S m . (4.4)
With A 0 = 4N, one obtains a bound on the matter entropy:
S m ≤ N − 1 4 A c . (4.5)
To distinguish this bound from the covariant entropy bound and the N-bound, it will be called the D-bound ('D' as in Difference between N and the horizon entropy). The D-bound is less general than the covariant bound of Sec. 3, because it only applies to matter systems within a de Sitter horizon. For a dilute system, one has A c ≈ A 0 , and therefore, N − 1 4 A c ≪ 1 4 A c . So the D-bound can be tighter than the covariant bound applied to a surface enclosing the system. In the next section it will be seen that causal diamonds can contain portions to which the D-bound applies.
The N -bound
In Sec. 1 the N-bound was presented as a conjecture: The observable entropy in any Λ > 0 universe cannot exceed N = 3π/Λ. In Sec. 2 it was shown that only the entropy within space-time regions of a particular form, causal diamonds, is observable. Hence, to prove the N-bound, it suffices to show that the entropy of an arbitrary causal diamond does not exceed N. By applying the covariant entropy bound (Sec. 3) and the D-bound (Sec. 4), we will now give a proof for spherically symmetric causal diamonds. Spherical symmetry allows us to keep the discussion fairly non-technical and focus on the key idea, the interplay between the D-bound and the covariant bound. We expect that the assumption of spherical symmetry can be eliminated in a more refined treatment; this will be discussed briefly at the end of the section.
Consider an experiment beginning at a point p and ending at q, in a universe with Λ > 0. We must show that the matter entropy, S m , within the causal diamond, C(p, q), plus the Bekenstein-Hawking entropy of any black hole or cosmological horizons identified by the experiment, will not exceed N. To limit S m , it suffices to consider the matter entropy passing through the top cone bounding the diamond, by the second law. It will be seen that the horizon entropy is bounded by the area of the diamond's edge.
Neither the top nor the bottom cone contain any caustics, except at endpoints, because each is a portion of the boundary of the future or past of a point. Then, by the focussing theorem, each cone has exactly one maximal cross-sectional area. The maximum may be local, or it may lie on the intersection of the two cones, the edge, where they terminate. Depending on the location of the maxima, we distinguish three cases.
Case 1: No local maximum on either cone. Then the maximum area of each cone lies on the edge. Thus, the edge will be a normal surface, with the observer on the inside (Fig. 7). This case applies, for example, to regions within the horizon in an asymptotically de Sitter universe, and to sufficiently small causal diamonds in arbitrary spacetimes.
Consider a space-like hypersurface containing the edge. One can consistently assume, for the sake of argument, that the exterior of the edge is a vacuum solution. With the assumption of spherical symmetry, Birkhoff's theorem implies that the exterior will be a portion of a Schwarzschildde Sitter (or a Reissner-Nordström-de Sitter) solution. The space-time thus constructed will be called the auxiliary space-time. It is asymptotically de Sitter in the future and past; hence, it invites the application of the D-bound. The causal diamond lies within the cosmological horizon of the auxiliary space-time, because the edge is normal and the cosmological horizon is the outermost normal surface. With spherical symmetry it follows that c ≥ A edge . The hat indicates that the cosmological horizon is a surface in the auxiliary space-time. Because the auxiliary spacetime is asymptotically de Sitter, the D-bound can be applied to the interior of the cosmological horizon, yielding
S m ≤ N − 1 4Â c ≤ N − 1 4 A edge . (5.1)
Recall from Sec. 4 that the entropy of black holes is already subsumed in S m , 8
but not the entropy of the cosmological horizon (supposing that one exists in the actual space-time under consideration). However, the observer cannot assign more cosmological horizon entropy than a quarter of the area of the outermost surface that has been probed:
S c ≤ 1 4 A edge . (5.2)
The two inequalities imply the N-bound,
S = S m + S c ≤ N. (5.3)
Case 2: Local maximum on the top cone. Now assume that the top cone contains a locally maximal area A max , an apparent horizon.
No assumption is made about the bottom cone. Cases of this type include large expanding or collapsing cosmological regions. They correspond to highly dynamical situations without a quasi-static cosmological horizon, so S c = 0. Then we need to show only that the matter and black hole entropy on the top cone does not exceed N. The maximal area divides the top cone into two parts. We will show that the D-bound can be applied to one part and the covariant bound to the other. Recall the wedge formalism summarized in Fig. 6. The wedge of the surface A max is constructed by drawing a leg for each light-like direction with decreasing cross-sectional area. Because A max is the largest surface on the top cone, the area obviously decreases in the two null directions that generate the cone. Of the other two null directions orthogonal to A max , at least one must have decreasing area, because they oppose each other. Hence, the wedge associated with A max has at least three legs. Necessarily, two of them will be pointing to the same spatial side (Fig. 8). Therefore, A max is a marginally normal surface. The side with two legs is, in the wedge sense, the inside of A max . The corresponding portion of the top cone will be called T 1 . Note that T 1 need not be the portion that includes the tip; it may be on the 'far side' of A max (Fig. 8, right).
Consider the inside portion, T 1 , in isolation. To this hypersurface one can apply the D-bound, using an argument similar to that of Case 1. One can take T 1 to be embedded in an otherwise vacuous auxiliary space-time. Because T 1 is the interior of a normal surface, in the auxiliary space-time it will be surrounded by a cosmological horizon. The area of the cosmological horizon will be no less than A max . Because the auxiliary space-time is asymptotically de Sitter, the D-bound applies to the interior of the cosmological horizon. Hence, the entropy on T 1 satisfies
S 1 ≤ N − 1 4Â c ≤ N − 1 4 A max . (5.4)
The other part, T 2 , of the top cone, is a light-sheet of the surface A max . The covariant entropy bound yields
S 2 ≤ 1 4 A max . (5.5)
It follows that the entropy on the top cone is bounded by N:
S = S 1 + S 2 ≤ N. (5.6)
In this result, S already includes black hole entropy. A horizon is probed by an experiment only if the edge of the causal diamond contains a portion in the vicinity of the horizon. The edge lies on the far side of the top cone. If this is T 1 , the side to which the D-bound applies, then the black hole entropy is already subsumed in S 1 , as discussed in Sec. 4. If the far side is T 2 , let us split S 2 into black hole horizon entropy, S h , and ordinary matter entropy, S m :
S 2 ≡ S h + S m . (5.7)
If A edge > 0, the covariant bound on T 2 can be strengthened [11]:
S m ≤ 1 4 (A max − A edge ) . (5.8)
The horizon cannot be larger than the area of the edge:
S h ≤ 1 4 A edge . (5.9)
So Eq. (5.5) holds, and S in Eq. (5.6) is indeed the total observable entropy.
Case 3: Local maximum on the bottom cone but not on the top cone.
Finally, consider the case where the top cone has no local maximum, but the bottom cone does (Fig. 9). Examples include large regions in collapsing universes, or black hole interiors. The edge of the cone is a trapped surface in this case. This implies a dynamical situation without Bekenstein-Hawking entropy. It will suffice to show that the matter entropy on the top cone does not exceed N.
In the absence of a local maximum, the largest surface p q A max A edge = Figure 9: Case 3. The edge is trapped. We apply only the covariant entropy bound.
on the top cone is the edge. The entire top cone is a lightsheet of the edge. By the covariant entropy bound,
S ≤ 1 4 A edge . (5.10)
By the arguments used in Case 2, the maximal area on the bottom cone, A max , is a normal surface. Hence, it can be embedded in an asymptotically de Sitter auxiliary space time, where it is surrounded by a cosmological horizon of area c . By the second law, c cannot exceed the horizon area of empty de Sitter space, A 0 . Moreover, by construction, A max is larger than the edge. In summary, one finds
A edge ≤ A max ≤ c ≤ A 0 = 4N. (5.11)
Therefore the N-bound is satisfied:
S ≤ N (5.12)
In all three cases, we have used an auxiliary construction by which (portions of) the causal diamond were embedded in an asymptotically de Sitter auxiliary space-time. This method is rigorous only for spherically symmetric situations. The assumption was used in applying Birkhoff's theorem to establish the auxiliary space-time, and in taking the cosmological horizon as an upper bound on the area of normal surfaces on the lightcone. Spherical symmetry has also simplified the case distinction, since it implies that the maximum is either local or entirely on the edge; in general, the maximal area on the top cone may have locally maximal components as well as portions that lie on the edge.
Our assumption of spherical symmetry notwithstanding, we expect that the above arguments represent the core of a general proof. Causal diamonds, light-sheets, and the entropy bounds are all defined without reference to spherical symmetry. The task of combining them to derive the N-bound in the non-spherical case is left to future work.
Outlook
Non-perturbative definitions of quantum gravity have been given for certain spacetimes that are asymptotically flat or AdS [2,4]. No such description has been found for space-times with a positive cosmological constant. As no de Sitter solutions of M-theory are known, one does not even have a microscopic framework. Banks [1] has opened a new perspective on this problem by suggesting that an asymptotically de Sitter universe is described by a microscopic theory with finite-dimensional Hilbert space. Quantitatively, the Λ-N correspondence relates the cosmological constant of a stable vacuum, Λ, to a theory with N = 3π/Λ degrees of freedom (i.e., with a Hilbert space of dimension e N ).
If this is correct, M-theory (as it is currently understood) will arise only in the limit of vanishing Λ and infinite N. The cosmological constant problem becomes a problem of understanding the particular dimension of Hilbert space chosen for the theory.
In Sec. 1.1, considerations of consistency with semi-classical gravity led us to propose the stronger conjecture that the Λ-N correspondence applies to all universes with Λ > 0, whether they are de Sitter in the future or not. This conjecture makes the nontrivial, testable prediction that the observable entropy in all such universes is bounded by N. We then argued that this statement, the 'N-bound', is correct. This required the combination of the covariant entropy bound with two intermediate results derived in Secs. 2 and 4: the D-bound, and the restriction to causal diamonds. It is hard to see what, other than the Λ-N correspondence, would offer a compelling explanation why such disparate elements appear to join seamlessly to imply a simple and general result.
The D-bound has a number of properties that merit further investigation. In particular, one can show that it is closely related to Bekenstein's bound [16]. Bekenstein's bound, valid for systems in flat space, can be written as S m ≤ πr g R, where r g = 2m is the 'gravitational radius' of the system and R is the circumscribing radius. For dilute, spherically symmetric systems in de Sitter space, the D-bound takes precisely this form as well, despite the significant deviation from flat space. A full discussion is given elsewhere [35].
The restriction to causal diamonds arose in Sec. 2 from the requirement to include only operationally meaningful parts of a space-time in a microscopic description. This principle is independent of the present context of positive Λ, and one may expect that causal diamonds will be of wider use. Banks [15] has sketched a framework for the combination of quantum mechanics and cosmology, in which the variable size of the quantum Hilbert space is related to the maximal area of the observer's past lightcone. The arguments of Sec. 2.2 suggest a possible modification of this approach that may lead to a more time-symmetric treatment based on the Hilbert space of causal diamonds.
In de Sitter space, an observer will be immersed in quantum radiation coming from the cosmological horizon. At the semi-classical level, this radiation is thermal [21]. One would expect that the radiation will occasionally contain large fluctuations that appear to an observer as classical objects. Taking a global view of the de Sitter space, one would say that quantum fluctuations originate behind the future horizon, while classical objects enter through the past horizon. When one restricts to causal diamonds, however, both of these outside regions are eliminated. Then it is no longer clear how an observer can distinguish between a genuine classical object and a fluctuation in the quantum radiation. (This view has previously been advocated by Susskind.) Indeed, all of standard cosmology may be a rare fluctuation in a long-lived de Sitter space [1].
Most of high energy physics is based on the S-matrix, with the implicit assumptions that an observer of infinite size is located at the infinity of an asymptotically flat spacetime-the observer is 'outside looking in'. This point of view will have to be transcended in order to describe experiments in cosmology, where the observer is always of finite size, and is 'inside looking out'. Indeed, a Minkowski infinity typically does not exist in cosmology; but even if it did, real observers would not live there. On the other hand, the approximation of an observer as a point in the space-time bulk is also unsatisfactory, because an experiment involves the collection and analysis of information. According to the holographic principle, a non-vanishing amount of information can be obtained only by an observer of non-zero size. The maximal information involved in an experiment is related not to the size of the universe, but to the size of the experiment. One may be motivated by these considerations to abandon the distinction between observer and experiment, and also to claim that a general experiment is a causal diamond. The bottom cone is best thought of as arising from the limitation of preparing the apparatus in a causal way; the future cone reflects the limitation of analysing the data causally.
The correspondence between finite N and positive Λ would impose a surprisingly strong restriction on the fundamental theory, if indeed we live in a universe with positive (and true) vacuum energy. We believe that its implications deserve to be further explored.
A. de Sitter space
This Appendix summarizes a number of properties of de Sitter space that are used in the text. An excellent discussion of the classical geometry is found in Ref. [31]. The semi-classical properties are laid out in Ref. [21].
de Sitter space is the maximally Globally, de Sitter space can be written as a closed FRW universe:
ds 2 = −dT 2 + r 2 0 cosh T r 0 2 dΩ 2 3 (A.
2) The spacelike slices are three-spheres. The space-time can be visualized as a hyperboloid [31] (Fig. 10). The smallest S 3 is at the throat of the hyperboloid, at T = 0. For T > 0, the three-spheres expand exponentially without bound. The time evolution is symmetric about T = 0, so three-spheres in the past are arbitrarily large and contracting.
The Penrose diagram of de Sitter space is a square (Fig. 10). The spatial threespheres are horizontal lines. As usual, every point represents a two-sphere, except the points on the left and right edge of the square, which represent the poles of the threesphere. The top and bottom edge are the future and past infinity, where all spheres become arbitrarily large.
In de Sitter space, an observer is surrounded by a cosmological horizon at r = r 0 . This is best seen in the static coordinate system:
ds 2 = −V (r) dt 2 + 1 V (r) dr 2 + r 2 dΩ 2 2 , (A.3) where V (r) = 1 − r 2 r 2 0 . (A.4)
This system covers only part of the space-time, namely the interior of a cavity bounded by r = r 0 . By the arguments given in Sec. 2, this is precisely the operationally meaningful portion of de Sitter space, because it is the largest causal diamond possible. It corresponds to a quarter of the Penrose diagram (e.g., for an observer at the left pole, the 'left triangle' shown in Fig. 11). The upper and lower triangles contain exponentially large regions that cannot be observed. The spheres with r = r 0 , the past and future event horizons, are the entire diagonals of the square. However, the spheres with r = r 0 − ǫ (the stretched horizon [26]) lie within the left (or right) triangle and represent the cosmological horizon of an observer at the corresponding pole.
An object held at a fixed distance from the observer is redshifted; the red-shift diverges near the horizon. If released, the object will accelerate towards the horizon. If it crosses the horizon, it cannot be retrieved. Thus, the cosmological horizon acts like a black hole 'surrounding' the observer. Note that the symmetry of the space-time implies that the location of the cosmological horizon is observer-dependent.
The black hole analogy carries over to the semiclassical level [21]. Because matter entropy can be lost when it crosses, the cosmological horizon must be assigned a Bekenstein-Hawking entropy equal to a quarter of its area, in order for the generalized second law of thermodynamics [18][19][20] to remain valid:
S 0 = A 0 4 , (A.5) where A 0 = 4πr 2 0 = 12π Λ (A.6)
The horizon emits Hawking radiation with temperature (2πr 0 ) −1 .
de Sitter space can also be written as a flat expanding FRW universe:
ds 2 = −dτ 2 + exp 2τ r 0 dx 2 + dy 2 + dz 2 . (A.7)
This metric covers half of the Penrose diagram (Fig. 12). If matter is present, it gives rise to a singularity on a space-like slice at finite time τ 0 , which one can take to be 0. One thus obtains a space-time which starts with a big bang and becomes asymptotically de Sitter in the future. Its Penrose diagram is given by a portion of the flat slicing, between some finite τ and asymptotic infinity.
The remaining half of de Sitter space is covered by Figure 12: The flat slicing covers half of de Sitter space. The dark shaded region is the Penrose diagram of a flat big bang-de Sitter cosmology (Fig. 1). the contracting flat FRW universe obtained by timereversal of Eq. (A.7). By analogy with the previous paragraph, the introduction of matter leads to a flat FRW universe that is asymptotically de Sitter in the past and collapses in a big crunch, with a time-reversed Penrose diagram. (These space-times are used in Sec. 2 to illustrate the restriction to causal diamonds.)
Figure 1 :
1Flat FRW universe with Λ > 0 (left). The entropy on any constant-time slice is infinite, but only a finite portion (heavy line) can be seen by the observer at q. Because of the future de Sitter horizon (dotted line), this portion will not diverge. Right: flat FRW universe with Λ = 0. The entropy within the observer's past light-cone diverges at late times.
Figure 2 :
2A collapsing universe approaching de Sitter in the past. The past light-cone of q may contain infinite entropy (arrows), but only a finite amount will enter the future of p (shaded region).
Figure 3 :
3A causal diamond, with top cone T , bottom cone B, and edge E.
Figure 5 :
5A light-sheet is a null hypersurface with shrinking cross-sectional area. It terminates at caustics (if not earlier).
Figure 6 :
6Wedge symbols for different types of surfaces. A leg is drawn for each direction in which light-rays are non-expanding.
Figure 7 :
7Causal diamond in Case 1 (to be rotated about the p-q axis). The edge is normal. The Dbound applies to the top cone.
Figure 8 :
8Case 2. One side of the maximal area must be normal. We apply the D-bound to this side, T 1 , and the covariant bound to the other side, T 2 . Both possibilities are shown.
Figure 10 :
10de Sitter space as a hyperboloid. Right: Penrose diagram. Horizontal lines represent three-spheres. symmetric solution of the vacuum Einstein equations with a positive cosmological constant, Λ. It is positively curved with characteristic length
Figure 11 :
11Past and future event horizon (diagonal lines). The static slicing covers the interior of the cosmological horizon (shaded).
In this paper, N always denotes the number of degrees of freedom; it should not be confused with the size of a gauge group, or the level of supersymmetry. The space-time dimension is taken to be 4 in order to keep equations simple, but generalization to arbitrary dimensions is trivial. Planck units are used throughout.
For a quantum theory, N is defined to be the logarithm of the dimension of Hilbert space; thus, a theory with finite N has a finite-dimensional Hilbert space. Superstring theories, and current nonperturbative proposals for M-theory[2][3][4], have an infinite-dimensional Hilbert space. (Indeed, even a single harmonic oscillator has an infinite-dimensional Hilbert space.)3 Ref.[1] also explores the possibility of a connection between finite N and supersymmetry (SUSY) breaking, noting that no stable SUSY-violating vacuum states have been firmly identified in M theory (see, however, Refs.[5][6][7]). One may therefore speculate that SUSY breaking can only occur in theories with finite N , and that both the SUSY and the vacuum energy scales arise from finite N . One then faces the challenge of explaining why the SUSY scale is much larger than N −1 . We shall not pursue the connection with supersymmetry in the present paper.
This is intuitively obvious but can be made precise as follows. The causal past of q, J − (q), is defined as the set of points that can be reached from q via a smooth curve that is everywhere pastdirected timelike or null. Define the past light-cone, L − (q), as the hypersurface generated by the past-directed null geodesics that start at q and are terminated if and only if they run into a point conjugate to q (a 'caustic'). Assuming global hyperbolicity one can show[29] that the boundary of the past of q,J − (q), is a portion of L − (q). For any point r ∈ J − (q), we claim that all future inextendible causal curves through r must intersect L − (q). In fact, the stronger statement holds that they must intersectJ − (q); in the notation of Wald[29], D − [J − (q)] = J − (q). This follows from the compactness of J + (r) ∩ J − (q) (Theorem 8.3.10) and Lemma 8.2.1 in Wald[29].
This follows by exchanging 'past' and 'future', − and +, and q and p, in the previous footnote.
This follows from the previous two footnotes, withĊ(p,q) = [J + (p) ∩J − (q)] ∪ [J + (p) ∩ J − (q)].The first term in square brackets is the top cone, T (p, q); the second is the bottom cone, B(p, q); their intersection is the edge, E(p, q). Clearly, T (p, q) ⊂J − (q) ⊂ L − (q), and similarly, B(p, q) ⊂ L + (p).
This may play a role when spherical symmetry is abandoned. In the presence of black holes, the edge can contain additional disconnected components, namely spherical surfaces surrounding the black hole horizons, within the cosmological horizon.
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| [] |
[
"Kinks versus fermions, or the 2D sine-Gordon versus massive Thirring models, at T > 0 and µ = 0 *",
"Kinks versus fermions, or the 2D sine-Gordon versus massive Thirring models, at T > 0 and µ = 0 *",
"Kinks versus fermions, or the 2D sine-Gordon versus massive Thirring models, at T > 0 and µ = 0 *",
"Kinks versus fermions, or the 2D sine-Gordon versus massive Thirring models, at T > 0 and µ = 0 *"
] | [
"A Gómez Nicola \nDepartamento de Física Teórica\nUniversidad Complutense\n28040MadridSpain\n",
"R J Rivers \nTheoretical Physics\nBlackett Laboratory\nImperial College\nPrince Consort Road, Silver StreetSW7 2BZ, CB3 9EWLondon, CambridgeU.K. c) D.A.M.T.P., U.K. (February\n",
"D A Steer ",
"\n2022)\n",
"A Gómez Nicola \nDepartamento de Física Teórica\nUniversidad Complutense\n28040MadridSpain\n",
"R J Rivers \nTheoretical Physics\nBlackett Laboratory\nImperial College\nPrince Consort Road, Silver StreetSW7 2BZ, CB3 9EWLondon, CambridgeU.K. c) D.A.M.T.P., U.K. (February\n",
"D A Steer ",
"\n2022)\n"
] | [
"Departamento de Física Teórica\nUniversidad Complutense\n28040MadridSpain",
"Theoretical Physics\nBlackett Laboratory\nImperial College\nPrince Consort Road, Silver StreetSW7 2BZ, CB3 9EWLondon, CambridgeU.K. c) D.A.M.T.P., U.K. (February",
"2022)",
"Departamento de Física Teórica\nUniversidad Complutense\n28040MadridSpain",
"Theoretical Physics\nBlackett Laboratory\nImperial College\nPrince Consort Road, Silver StreetSW7 2BZ, CB3 9EWLondon, CambridgeU.K. c) D.A.M.T.P., U.K. (February",
"2022)"
] | [] | The two dimensional (1+1) sine-Gordon model finds many applications in condensed matter physics. These in turn provide an experimental means for the study of topological defects, some of which may have had a huge impact on the early universe. As a first step in trying to exploit this analogy and also others which exist with low-energy QCD, we study bosonisation in the massive Thirring and sine-Gordon models at finite temperature T and nonzero fermion chemical potential µ. Both canonical operator and path integral approaches are used to prove the equality of the partition functions of the two models at T > 0 and µ = 0, as was recently shown. This enables the relationship between thermal normal ordering and path-integral renormalisation to be specified. Furthermore, we prove that thermal averages of zero-charge operators can be identified as long as one uses the usual T = 0 identification between coupling constants. Analysis of the point-split regularised fermion current then leads to the thermal equivalence between sine-Gordon kinks and Thirring fermions. At µ = 0 and T > 0, we show, in perturbation theory around the massless Thirring model, that the bosonised theory is the sine-Gordon model plus an additional topological term which accounts for the existence of net fermion charge excitations (the fermions or the kinks) in the thermal bath. This result generalises that recently obtained for the massless case, and it is the two-dimensional version of the low-energy QCD chiral Lagrangian at finite baryon density. * Proceedings of the talk based on [1] and given by D. | null | [
"https://export.arxiv.org/pdf/hep-th/9809080v3.pdf"
] | 118,924,722 | hep-th/9809080 | 3d9527d40ac115fddbefaf5b50012448ef98d00a |
Kinks versus fermions, or the 2D sine-Gordon versus massive Thirring models, at T > 0 and µ = 0 *
arXiv:hep-th/9809080v3 30 Oct 1998
A Gómez Nicola
Departamento de Física Teórica
Universidad Complutense
28040MadridSpain
R J Rivers
Theoretical Physics
Blackett Laboratory
Imperial College
Prince Consort Road, Silver StreetSW7 2BZ, CB3 9EWLondon, CambridgeU.K. c) D.A.M.T.P., U.K. (February
D A Steer
2022)
Kinks versus fermions, or the 2D sine-Gordon versus massive Thirring models, at T > 0 and µ = 0 *
arXiv:hep-th/9809080v3 30 Oct 1998
The two dimensional (1+1) sine-Gordon model finds many applications in condensed matter physics. These in turn provide an experimental means for the study of topological defects, some of which may have had a huge impact on the early universe. As a first step in trying to exploit this analogy and also others which exist with low-energy QCD, we study bosonisation in the massive Thirring and sine-Gordon models at finite temperature T and nonzero fermion chemical potential µ. Both canonical operator and path integral approaches are used to prove the equality of the partition functions of the two models at T > 0 and µ = 0, as was recently shown. This enables the relationship between thermal normal ordering and path-integral renormalisation to be specified. Furthermore, we prove that thermal averages of zero-charge operators can be identified as long as one uses the usual T = 0 identification between coupling constants. Analysis of the point-split regularised fermion current then leads to the thermal equivalence between sine-Gordon kinks and Thirring fermions. At µ = 0 and T > 0, we show, in perturbation theory around the massless Thirring model, that the bosonised theory is the sine-Gordon model plus an additional topological term which accounts for the existence of net fermion charge excitations (the fermions or the kinks) in the thermal bath. This result generalises that recently obtained for the massless case, and it is the two-dimensional version of the low-energy QCD chiral Lagrangian at finite baryon density. * Proceedings of the talk based on [1] and given by D.
I. INTRODUCTION
The sine-Gordon (SG) theory in two dimensional Minkowski space-time with metric (+, −) is described by the Lagrangian density
L SG [φ] = 1 2 ∂ µ φ∂ µ φ + α 0 λ 2 cos λφ − γ 0 ,(1)
where φ is a real scalar field, γ 0 , α 0 and λ are bare parameters to be renormalised later, and x µ = (t, x), µ = 0, 1. Notice that apart from the usual kinetic term, the potential term is periodic so that there are an infinite number of degenerate vacua whose value, φ v , depends on the coupling constant λ: φ v = 2nπ/λ where n ∈ Z Z.
We comment that the Lagrangian (1) is invariant under φ → φ + φ v and φ → −φ meaning that the sign of α 0 is unimportant, and the name sine-Gordon is related to the fact that the equations of motion contain a sin φ term (see [2,3]). The massive Thirring (MT) model, on the other hand, is a model with a fermionic field ψ = (ψ 1 , ψ 2 ) and a four Fermi interaction with coupling g:
L T h [ψ, ψ] = iψ( ∂ − m 0 )ψ + 1 2 gj µ (x)j µ (x),(2)
where j µ (x) =ψ(x)γ µ ψ(x) and m 0 is a bare mass. For positive coupling constant, g > 0, the interaction term is attractive and there are fermion-antifermion bound states.
Though it maybe somewhat surprising since one is a model with a bosonic field and the other one with a fermionic field, it is very well known that these two theories are linked. In particular, they provide one of the earliest examples of duality in which the weak limit of one theory describes the same physics as the strong limit of the other and conversely since
λ 2 4π = 1 1 + g/π .(3)
Hence perturbative calculations in one theory tell us about non-perturbative effects in the other. Identity (3) was proved in perturbation theory about α 0 = 0 = m 0 (see below) at T = µ = 0 by Coleman using canonical operator methods [4] and later on using path integrals [5,6]. Note the special value g = 0 in which the Thirring model is a free theory of massive fermions, corresponding to λ 2 = 4π in the SG model. Also the MT bound states exist for g > 0 ⇒ λ 2 < 4π. The duality (3) is directly related to the fact that the models exhibit bosonisation, in which a theory of fermions is equivalent to a theory of bosons. In general the bosonic (fermionic) theory may also have fermionic (bosonic) excitations; this indeed occurs in the SG and MT models and we will see another example below in low-energy QCD. In the case of the SG and MT models, bosonisation schematically works in the following way (see [3] for a summary as well as a list of the relevant articles). Consider first the SG model. Perturbations in λ about one of the minima of the degenerate potential give rise to the usual bosonic simple harmonic oscillator spectrum. The model also has other excitations. Recall that there are topologically non-trivial solitonic solutions to the classical equations of motion, and that these arise from the vacuum degeneracy: there is no reason why at x → ∞ the system should be in the same vacuum as at x → −∞. As a result one can have finite energy solutions with non-zero charge
Q = ∞ −∞ dx λ 2π ∂ ∂x 1 φ(x) = λ 2π [φ(∞) − φ(−∞)] = ∆n.(4)
The simplest 'kink' solution has Q = 1 and an energy proportional to α 0 /λ 2 , whilst the anti-kink (Q = −1) is obtained from the kink by taking φ → −φ. Furthermore, since the system is integrable, exact expressions for multikink and anti-kink states are known and one can construct moving kinks by Lorentz transformation. These may then be quantised using semi-classical methods [3,7]. There are also classical solutions with Q = 0 corresponding to 'breather' solutions in which kinks and antikinks oscillate periodically about each other [3]. These can be quantised by WKB methods [3,7]. The relationship between these SG excitations and the fundamental fermionic excitations and bound state excitations of the MT model is summarised schematically in table I. MT MODEL SG MODEL fermionic fields bosonic fields
g λ ∼ 1/ḡ f f bound states (g > 0)
bosons, kink -anti-kink breather solutions (λ 2 < 4π) f kinks + µj 0 m 0 = 0 → see later m 0 = 0 → topological term [8] What is the motivation for studying these models at T > 0 and µ = 0? One first reason is that these models could be used to develop techniques which may then be applicable to more realistic 4D theories, in particular QCD. Recall that at low temperatures, where the quarks and gluons are strongly confined into hadrons, one can describe the system by an effective chiral bosonic lagrangian (CBL) for the lightest mesons (pions, kaons and eta) which are the Nambu Goldstone bosons (NGB) of the chiral symmetry breaking. The relationship between this chiral bosonic theory and the original fermionic QCD has many similarities with the relationship between the SG and MT models (see table II). QCD CBL quarks and gluons bosonic fields α s (large at low energies) small expansion parameter mπ Λχ , Λ χ ≃ 4πf π ≃ 1 GeV qq bound states π, K, η (NGB of chiral symmetry) baryon skyrmion (topological defect) + µ for baryons topological term [9] For example, the bound states of QCD are the mesons of the chiral theory whilst the baryons of QCD correspond to skyrmions-topological defects [10,11]-in the chiral theory which to lowest order is the non-linear sigma model [12] (c.f. the relationship between the SG and MT models in table I). Note also that the SG Lagrangian in (1) corresponds to a non-linear sigma model in 2D for a single Nambu-Goldstone-like field φ. Although there is no spontaneous symmetry breaking in 2D [13], the potential term in (1) breaks explicitly the symmetry φ → φ+ a with a ∈ IR (which we will call the chiral symmetry for reasons to become clear below) still preserving the symmetry φ → φ + φ v . These two symmetries are, respectively, the counterparts of the chiral and isospin symmetries for QCD, α 0 and λ playing the rôle of the pion mass squared and the inverse of the pion decay constant f π respectively. On the other hand, the chiral symmetry transformations in terms of the Thirring fermion are ψ → exp(iaγ 5 )ψ. The massless Thirring model is chiral invariant, the fermion mass term breaking that symmetry in the same way as the α term does in the SG Lagrangian. A second motivation for studying these models at T > 0 and µ = 0 comes from cosmology. The SG kinks are the 2D analogues of cosmic strings, line-like defects formed in 4D when a system goes through a symmetry breaking phase transition (say of some group G to a subgroup H) for which the first homotopy group π 1 (G/H) = 0 [14,15]. The strings trap regions of the unbroken high energy phase and so have energies per unit length which can be very large, depending on the critical temperature. In the context of the early universe and cosmology, cosmic strings may have played an important rôle because as the universe cooled and expanded after the big bang, it went through a number of phase transitions some of which may have led to the formation of strings. In particular, strings formed at the GUT phase transition have huge energies per unit length (µ ∼ 10 32 GeV 2 ) and hence significant gravitational effects, and so it is thought that they may have been responsible for the temperature fluctuations in the cosmic microwave background radiation and for seeding the perturbations which led to the formation of structures such as galaxies [15].
However, in order to make detailed predictions as to their effects, it is important to know the initial distribution of strings and whether or not it contains infinite strings. This is a very difficult task to undertake analytically [16], but recently progress has been made by using the analogy between experimentally observable systems such as 3 He and 4 He and the early universe [17]. It would seem, however, that there is an even more simple experimentally accessible system with which one could try to test ideas of defect formation, and that is a Josephson junction [18]. This device consists of two layers of superconductors separated by a thin dielectric barrier, typically of the order of 5nm. Denoting the macroscopic wave function of each of the superconductors by Ψ a = |Ψ a | exp{iθ a } where a = 1, 2 labels each of the two layers, then Josephson tunnelling of the Cooper pairs across the dielectric layer results in ∆θ = θ 1 − θ 2 satisfying a SG equation [18]. As the two superconductors are taken through the phase transition, kinks are formed in the junction and these are observed experimentally [19]. Experiments are also done to see what is, for example, the effect of the geometry of the set up on the kinks and their dynamics [20] and these devices are used as sources of some of the highest energy micro-waves finding applications in satellites [18,19]. One idea might therefore be to see whether one can indeed use such experiments with Josephson junctions to test ideas of defect formation. The hope is in particular that the situation can be much simplified in this case because of the duality between the SG and MT models. First, however, one has to check what is the relation between these two models at T > 0 and µ = 0. In particular, could it be possible that the relationship between coupling constants (3) is a function of temperature? As we will comment in the conclusions, these models also provide examples of other phenomena which may too be more easy to study in 2D rather than 4D.
Here we report on our first steps in these directions. In section II we briefly summarise some of the particularities and basic results of 2D which will be useful to bear in mind for the rest of the work. Results are then given in section III. First, in subsection III A, we outline the main steps which must be taken to extend Coleman's work [4] on the SG model to T > 0 using an operator approach. The results of [21] showing the equivalence of the partition functions are reproduced though the approach is entirely different. Path integral methods are used in the rest of the work. We show that not only the partition functions are equivalent but also that thermal averages of correlators of zero-charge operators evaluated at different space-time points coincide. The relationship between normal ordering and regularisation is specified and we analyse the T > 0 situation in which there is a net number of fermions, µ = 0. The partition function is calculated in that case so extending [8] to the massive case and [21] to µ = 0. The analogy between these models and a classical gas of particles is then noted in the conclusions. Details of the calculations and results presented here may be found in references [1,22].
II. PECULIARITIES OF 2D
The following three basic, though important, results hold for free fields in 2D at T > 0. They will be useful in the following sections.
1) Equality of free massless boson and fermion partition functions
Consider a free bosonic field of massμ and a massless free fermionic field. The partition functions can be obtained by writing the Hamiltonians in terms of a normal ordered part plus an infinite vacuum energy; they are respectively
ln Zμ ,B 0 (T ) = −L +∞ −∞ dk 2π βω k,μ 2 + + ln 1 − e −βω k,μ ,(5)ln Z F 0 (T ) = 2L +∞ −∞ dk 2π β|k| 2 + ln 1 + e −β|k| .(6)
Here ω k,μ = k 2 +μ 2 , β is the inverse temperature β = 1/T , and L is the spatial dimension which we are taking to infinity. As usual, all the thermodynamic observables are obtained from the logarithm of the partition function and its derivatives by dividing by βL.
Observe that whenμ = 0,
+∞ −∞ dk 2π ln 1 − e −β|k| = −2 +∞ −∞ dk 2π ln 1 + e −β|k| = − πT 6 ,
and hence it follows that ignoring vacuum terms, the two partition functions (5)-(6) are equivalent forμ = 0:
Z F 0 (T ) = Z 0,B 0 (T ) = exp πLT 6 .(7)
Thus one cannot tell the difference between the bulk quantities of these two systems: equality (7) is perhaps the simplest example of bosonisation. As will be seen below, the equality of the SG and MT partition functions at T > 0 rests on (7) since in those models we work in perturbation theory, expanding about α 0 = 0 in the SG model and about m 0 = 0 in the MT model (and hence about massless bosonic and fermionic theories).
2) Free thermal propagators
We will need the free propagators for boson and fermion fields at T > 0. Again working with a boson field of massμ and a massless fermion field, and calculating in the imaginary time formalism witht = it, these are found to be respectively [21] ∆
T (x) = − 1 4π lnμ 2 β 2 Q 2 (x) + K + O(μβ),(8)S αβ (x) = − 1 2β Q αβ (x) Q 2 (x) ,(9)
where we have expanded the boson propagator aboutμ = 0, and K is a constant. The Q variable is given by
Q 2 = Q 2 0 + Q 2 1 where Q 0 (x,t) = − cosh( πx β ) sin( πt β ), Q 1 (x,t) = − sinh( πx β ) cos( πt β ),(10)
so that Q(x) is a Lorentz scalar. The indices α, β are Dirac indices and we have worked with the 2D Euclidean γ matrices
γ 0 = 0 1 1 0 γ 1 = 0 −i i 0 γ 5 = −iγ 0 γ 1 = 1 0 0 −1 so that {γ µ , γ ν } = 2δ µν , γ µ γ 5 = −iǫ µν γ ν with ǫ 01 = 1. Observe that (8) is both ultra-violet diver- gent (x → 0) as well as infra-red divergent (x → ∞ or µ → 0) since Q 2 (x,t) |x|→∞ −→ 1 4 e 2π|x|/β ∀t (x,t)→(0,0) −→ (πT ) 2 (x 2 +t 2 ).(11)
An interesting property of the propagators (8)-(9) is that they can be directly obtained from the corresponding T = 0 propagators through the substitution x µ → βQ µ . This is in fact true for all contours in the complex time plane, and can be viewed either as a result of the conformal invariance of the free theories, or as arising from solving the Green function equation for the Coulomb potential on a cylinder of radius β [22].
3) Ultra-violet divergences and normal ordering
Finally, recall that the UV divergence structure of 2D bosonic theories is much simpler than that of 4D ones. The reason follows from the fact that in d dimensions with an interaction of the form φ r , a diagram with n vertices and E external lines has a UV degree of divergence D of [23]
D = d − d 2 − 1 E + n r 2 (d − 2) − d . So with d = 2, D = 2 − 2n
and the only divergent diagrams ∀r are tadpole diagrams. Since the SG lagrangian contains a term cos λφ = (−1) 2 φ 2n /(2n!) then in the operator formalism it follows that all UV divergences of the theory should be removed through normal ordering (as this removes tadpole diagrams). In path integral methods there is no operator normal ordering, and the divergences are removed using different methods (see below).
III. RESULTS
A. Bosonisation in the canonical operator approach at T > 0 and µ = 0
We have used operator methods to extend the paper of Coleman [4] to T > 0 and µ = 0. This enabled us to prove [1] that the partition functions of the SG and MT models are identical
Z SG (T, µ = 0) = Z MT (T, µ = 0)(12)
provided the coupling constants of the theories satisfy (3) which is temperature independent. In fact, (12) was already proved using very different path integral methods in [21] (see also section III B). In the SG model the main steps in the calculation Z SG (T, µ = 0) are the following:
1. Remove all UV divergences by normal ordering the SG Hamiltonian.
2. Deal with the IR divergence of the propagator (8).
3. Calculate the partition function Z SG , a sum of thermal expectation values of free (interaction picture) operators each of which have been normal ordered as a result of step 1.
We now outline the main features of each of these steps, and also comment on the differences between this T > 0 case and the T = 0 one discussed in [4].
Step 1: Removal of all UV divergences by normal ordering As stated in point 3 above, the partition function contains thermal expectation values (TEV's) of normal ordered operators. Therefore to simplify its calculation, we remove UV divergences by using thermal normal ordering (TNO) introduced in [24] rather than standard zero temperature normal ordering (which places annihilation operators to the left of creation operators). By construction TNO, which is denoted by N ES , guarantees that for any 1 operator (other than the identity) • in the interaction picture,
≪ N ES [•] ≫ 0 = 0,(13)
whereas for usual normal ordered products
≪: • :≫ 0 = 0.
The operation N ES [•] was defined in [24] to place the "positive" parts of the operator, • + (a combination both of annihilation and creation operators), to the right of the "negative" part, • − . See [24] for the exact definitions. In the above equations the angular brackets denote a thermal expectation value, and the subscript zero indicates that we work with free (interaction picture) fields, and hence in perturbation theory. Identity (13) will greatly simplify calculations below. The SG Hamiltonian iŝ
H SG = L 0 dx π 2 2 + 1 2 ∂φ ∂x 2 − α 0 λ 2 cos λφ − γ 0 =:Ĥ 0 − L 0 dx α 0 λ 2 cos λφ + γ 0(14)
which must be divided into a free and interacting part so as to apply TNO. Although the term cos λφ itself contains a mass term on expansion in powers of λ, we want to keep λ of arbitrary size. Consider thereforê
H SG = Ĥ 0 + L 0 dx 1 2 ρ 2φ2 − L 0 dx α 0 λ 2 cos λφ + 1 2 ρ 2φ2 + γ 0 ,
so that perturbations are about a scalar field of mass ρ.
To take account of this fact, TNO is now denoted by
N ES ρ [•]
and similarly we add an extra mass label to the propagators (8); ∆ T (x) → ∆ T (x;μ). Hence in the case ofĤ SG above we will be dealing with ∆ T (x; ρ). The link between ρ and the regularisation scale in path-integral methods is discussed in section III B.
TNO of (14) may be carried out by using the identity
where T c means contour ordering 2 . Had we used T = 0 normal ordering, the zero-temperature propagator would have appeared in (15) rather than the finite temperature one: TNO means that the Q variables of (10) are built in from the start of the calculation. Finally the UV divergence of ∆ T (x; ρ) is regulated by cutting off the theory replacing
∆ T (x; ρ) −→ ∆ T (x; ρ; Λ) := ∆ T (x; ρ) − ∆ T (x; Λ)(16)
where Λ is a large mass. Hence the constant K in the propagator (8) cancels, and ∆ T (x; ρ; Λ) is now both non-singular as well as β independent for x → 0; ∆ T (0; ρ; Λ) = − 1 4π ln ρ 2 /Λ 2 . Combining (16) with (15) for j(x) = λδ(x − y) gives, for example
e iλφ(y) = ρ 2 Λ 2 λ 2 8π N ES ρ e iλφ(y) .(17)
After similar manipulations, (14) can be written as [1] H
SG = N ES ρ Ĥ 0 − α λ 2 L 0 dx cos λφ − Lγ(18)
where α 0 has been multiplicatively renormalised,
α = α 0 ρ 2 Λ 2 λ 2 8π(19)
and γ 0 has been renormalised according to
γ = γ 0 − E T (ρ)(20)
where
E T (ρ) = 1 2 π + ,π − + (∂ 0φ ) + , (∂ 0φ ) − = E 0 (ρ) + L +∞ −∞ dk 2π 1 2 N k,ρ 2k 2 + ρ 2 ω k,ρ .
Here E 0 (ρ) is an infinite temperature independent contribution;
E 0 (ρ) = L +∞ −∞ dk 2π 1 4 2k 2 +ρ 2 ω k,ρ .
The temperature dependent part proportional to the Bose-Einstein distribution N k,ρ = (e βω k,ρ − 1) −1 is finite. The coupling λ is unchanged.
Thermal normal ordering the SG Hamiltonian has therefore absorbed all UV infinities just as zero temperature normal ordering does [4], but it has also introduced some extra T -dependent finite terms.
Steps 2 and 3: IR divergences and calculation of the partition functions
The IR divergence of the boson propagator (8) is removed by introducing a massμ into the SG Hamiltonian (18). At the end of the calculation we letμ → 0, and hence are free to add the extra mass term within the normal ordering giving the Hamiltonian
H = N ES ρ Ĥμ 0 − γL − α λ 2 L 0 dx cos λφ = N ES ρ Ĥμ 0 − α λ 2 L 0 dx cos λφ =:Â 0 +Â I .(21)
HereĤμ 0 =Ĥ 0 + L 0 dx 1 2μ 2φ2 , and 0 and I denote the free (α = 0) and interacting Hamiltonians respectively. In perturbation theory the SG partition function is therefore given by
≪ T c n j=1 N ES ρ e iλjφ(xj) ≫ 0 = μ 2 ρ 2 j λ 2 j 8π n j=1 ≪ N ES µ e i j (λjφ(xj )) ≫ 0 × × n j>k β 2μ2 |Q(x j − x k )| 2 λ j λ k 4π 3
In fact one has to be extremely careful when considering the precise form of the thermal weight exp{−βÂ0}. The reason is that it contains two different mass scales ρ andμ, meaning thatÂ0 is not obviously diagonal and so not obviously of the form dk ω(k)â † (k)â(k) as is usually the case and as was assumed in [24]. See [1] for details about this point.
= μ 2 ρ 2 1 8π j λ 2 j n j=1 n j>k β 2μ2 |Q(x j − x k )| 2 λ j λ k 4π ,(23)
where we have expanded the exponential inside the normal ordered term and noted that all terms which contain φ n vanish (by (13)) apart from that with n = 0. Then observe that the terms proportional toμ in (23) have a contributionμ ( λj ) 2 /4π . Thus in the limitμ → 0, only configurations with λ j = 0 contribute (a condition which will become analogous to the fermion chiral selection rule of section III B). In (23) n must therefore be even. If we let m = n/2, y j = x j for j = n/2+1, . . . , n and defineÂ
± = N ES ρ e ±iλφ ,(24)
then from (23)
≪ T c m j=1Â + (x j )Â − (y j ) ≫ 0 = m j=1 m j>k β 4 ρ 4 |Q(x j − x k )| 2 |Q(y j − y k )| 2 λ 2 4π m k=1 [β 2 ρ 2 |Q(x j − y k )| 2 ] λ 2 4π .(25)
Hence the partition function (22) is [1] Z SG (T, µ = 0) =
Z B 0 (T ) n 1 (n!) 2 α 2λ 2 T ρ λ 2 4π 2n n j=1 T d 2 x j d 2 y j × n j>k |Q(x j − x k )| 2 |Q(y j − y k )| 2 λ 2 4π n k=1 [|Q(x j − y k )| 2 ] λ 2 4π(26)
where T d 2 x ≡ β 0 dt +∞ −∞ dx and Z B 0 (T ) = Tr e −βÂ0 = e βγ0L Z 0,B 0 (T ) which is finite for γ 0 satisfying (20) and λ 2 < 4π. 4 For the MT model, calculation of Z MT (T, µ = 0) in the operator approach is rather more complicated (it is based on a paper by Klaiber [25]) and is discussed in [1]. In perturbation theory one can show that the partition function Z MT (T, µ = 0) is given by [1]
Z MT (T, µ = 0) = Z F 0 (T ) ∞ n=0 1 n! 2 m 2β ρ T κ 2 /π 2n n j=1 T d 2 x j d 2 y j × k<j Q 2 (x j − x k )Q 2 (y j − y k ) 1−κ 2 /π n k=1 [Q 2 (x j − y k )] 1−κ 2 /π(27)
where we have chosen to renormalise the MT model at the scale ρ. Here Z F 0 is given in (6), (28) and m is the renormalised mass (see [1] and also section III B)
κ 2 = g 1 + g/πm = m 0 (Λ/ρ) κ 2 /π .(29)
Thus term by term (26) and (27) are identical provided that a) the parameters of the two theories are identified as in (3), b) that α and m are related by
ρm = α λ 2(30)
and c) that γ 0 = 0 as then Z B 0 (T ) = Z F 0 (T ) by (7). We have therefore extended the work of Coleman [4] to T > 0, µ = 0 using operator methods. This was rendered more simple through the use of TNO. The results obtained are in fact the same as those of [21] though the approach has been entirely different. We now turn to path integral methods.
B. Path Integral bosonisation at T > 0 and µ = 0
We have used path integral methods not only to prove the equality of the two partition functions as in (12) [1,21], but also to prove the equivalence of certain sets of correlators of operators evaluated at different space time points. These correlators cannot be obtained from the partition function which only contains information about global thermodynamic observables like the pressure or the condensates, but not about correlators which physically yield, for instance, thermal correlation lengths. We outline the main points of such PI calculations for the SG model and then the MT model. This will enable the link between normal ordering and PI regularisation to be made. Our results for the correlators will also be stated more precisely.
SG model
As always in path integral methods, one works with the generating functional. Once again it is useful to start with free boson fields and in particular to calculate the correlator (25) which may be obtained from the free boson Euclidean generating functional
Z B 0 [J; T ] = N β periodic dφ exp − T d 2 x 1 2 × (∂ α φ) 2 +μ 2 φ 2 + J(x)φ(x) = Z B 0 [0; T ] exp 1 2 T d 2 x T d 2 y× J(x)∆ T (x − y)J(y)} .(31)
Here N β is an infinite T -dependent constant arising in the path integral description [26], the propagator is given in (8) for smallμβ and the free boson partition function is
Z B 0 (T ) = Z B 0 [0;
T ] as in (5). Note that we have removed theμ labels on propagator and partition function as there is no longer any possible confusion with other mass scales. Now define
A ± = e ±[iλφ]
(remember that we do not normal order the operators in path integral and so this differs from the definition (24)). Then
≪ T c n j=1 A + (x j )A − (y j ) ≫ 0 = Z B 0 [J; T ] Z B 0 (T ) withJ (z) = −iλ n j=1 δ (2) (z − x j ) − δ (2) (z − y j ) .
As opposed to section III A, here the UV divergence of the propagator (8) is regulated by replacing
Q 2 (0, 0) → Q 2 (ε 0 , ε 1 ) = T 2 ε 2 + O(ε 3 )(32)
where ε α → 0 + and ε 2 = π 2 (ε 2 0 + ε 2 1 ). From (31), (8) and (32) it follows that
≪ T c n j=1 A + (x j )A − (y j ) ≫ 0 = (T ε) nλ 2 /2π n j=1 j>k Q 2 (x j − x k )Q 2 (y j − y k ) λ 2 /4π n k=1 [Q 2 (x j − y k )] λ 2 /4π . (33)
The correlator is divergent due to the short-distance (UV) divergent behaviour of the composite operator exp[iφ(x)], which needs to be renormalised. We do this in the usual way through the replacement of exp[iaφ(x)], with a ∈ IR arbitrary, by
[exp [iaφ(x)]] bare = (ερ) a 2 /4π [exp [iaφ(x)]] R .(34)
Here ρ is an arbitrary renormalisation scale and the superscript R will denote renormalised operators. Note that (34) is analogous to (17) with the identification
ε = 1 Λ (35)
though in the operator formalism the renormalisation was carried out through TNO. Also observe that the ρ's appear in the same way though they have different origins-in the operator approach ρ corresponded to an arbitrary mass at which normal ordering was performed whereas here it is the arbitrary renormalisation scale. From equations (35) and (34) observe that (33) reduces to (25) as required.
In the full SG model one again works with the generating functional which is expanded formally in powers of α 0 /λ 2 [1]. The partition function Z SG (T, µ = 0) is just obtained by setting the external sources to zero. We find [1] that with the regularisation (32) of the propagator, α 0 is renormalised just as in (19) for all the divergences to be eliminated. Once again the partition function is given by (26).
MT model
In calculating the MT partition function in perturbation theory about m 0 = 0, the correlator analogous to (25) in the SG model is just the TEV of insertions of the operators σ ± (x) =ψ(x)P ± ψ(x). Note, however, that the massless fermion theory is invariant under chiral transformations ψ → exp(iαγ 5 )ψ. Under such transformation σ ± (x) → exp(±2iα)σ ± (x) and therefore the thermal average of a product of σ ± (x) operators will vanish in the massless case unless the number of σ + and σ − is the same. This is the chiral selection rule, which only holds for m 0 = 0. Following [27], the required correlator is obtained by shiftingψ →ψγ 0 so that, naming ψ a with a = 1, 2 the two components of the bispinor ψ, the free massless theory decouples into two free theories for the spinors ψ a , and we have σ + →ψ 2 ψ 1 and σ − →ψ 1 ψ 2 . One obtains [1,27]
≪ T c n j=1 σ + (x j )σ − (y j ) ≫ 0 = (2β) −2n n j=1 j>k Q 2 (x j − x k )Q 2 (y j − y k ) n k=1 [Q 2 (x j − y k )]
.
(36)
Notice that the above correlator has exactly the same structure as the boson correlator (33)-this is another peculiarity of 2D. However, unlike (33), (36) is finite since it contains no product of fields at the same space-time point and there are no mixing terms between ψ 1 and ψ 2 in the Lagrangian. For the MT model one does have to worry about renormalisation (section III A). The reason is that whilst the chiral symmetry is still unbroken so that σ ± (x) correlators still appear in the same number if g = 0 and m 0 = 0, now (ψγ µ ψ) 2 → 4ψ 1 ψ 1ψ2 ψ 2 whenψ →ψγ 0 and therefore there is mixing between ψ 1 and ψ 2 . Thus products of fields at the same point appear and the σ ± (x) correlator becomes divergent: in the same way as the boson operator exp[iaφ(x)], the σ ± (x) composite operators need renormalisation. Also as in the SG model, those are the only infinities we have to worry about and they are absorbed in the renormalised mass m as in (29) whilst
[σ ± (x)] R = (ερ) κ 2 /π [σ ± (x)] bare .(37)
Given these renormalisations, the MT model partition function is obtained from the generating functional when the sources are set to zero. Its calculation requires some standard manipulations [5,6,27,21] (writing the quartic Thirring interaction as a 'gauge-like' interaction as well as calculating the axial anomaly [28]) and one again obtains the result (27).
Zero-charge operators equivalences
So far we have shown the equivalence of the SG and MT partition functions with µ = 0 using both operator and PI techniques. Further equalities between correlators of different operators in each of the two models may also be shown to hold at T > 0 and µ = 0-we simply state the results here. Further calculational details may be found in [1].
The equivalence of (33) and (36) in the free massless bosonic and fermionic theories (after renormalisation) may be extended to the SG and MT models where it becomes
≪ T c N j=1 σ R + (x j )σ R − (y j ) ≫ MT = ρ 2 2N ≪ T c N j=1 A R + (x j )A R − (y j ) ≫ SG .
We have also considered more complicated cases 5 . There the calculation goes through in a similar way and leads to the expected result in which
σ R ± (x j ) → ρ 2 A R ± (x j ).(38)
When dealing with correlators including insertions of the current operator j µ (x) in the MT model one is again faced with a product of field operators at the same point, and hence with additional divergences. Using pointsplitting regularisation [29] and taking particular care to ensure that the Ward identities are satisfied, we have proved that there are no extra divergences to renormalise and
≪ T c j R µ (x)σ R + (z 1 )σ R − (z 2 ) ≫ MT = ρ 2 2 λ 2π ǫ µν ≪ T c ∂ x ν φ(x)A R + (z 1 )A R − (z 2 ) ≫ SG where j R µ (x)
is the regularised current [1]. More generally we find that the identities which hold between different correlators seem to be simply obtainable through the replacements (38) and
j R µ (x) → λ 2π ǫ µν ∂ x ν φ(x)(39)
which are usually called operator bosonisation identities. We stress, however, that firstly we do not believe these identities to be strong-that is, we only expect them to hold between thermal expectation values-and secondly it is not immediately obvious that they hold inside all expectation values, though it does certainly seem to be the case for the zero fermion charge ones we have considered.
C. Bosonisation of the massive Thirring model at T > 0, µ = 0
Finally we have studied the Thirring model at nonzero chemical potential µ for the conserved charge Q F = dxj 0 (t, x) which is the net number of fermions minus anti-fermions;
L T h [ψ, ψ; µ] = −ψ( ∂ + m 0 )ψ + 1 2 g 2 j α (x)j α (x) + µj 0 (x),
and we have calculated the grand-canonical ensemble partition function [1]. Recall that now the averaged net fermion number density ρ(µ) = (βL) −1 ≪ Q F ≫ is no longer necessarily zero, and so a natural question to ask is what is the bosonised version of this theory. The answer was obtained in [8] for the massless case where the free boson partition function acquires an extra µ-dependent term. In the massive case with partition function
Z T h (T, µ) = N F β antiper dψdψ exp − T d 2 xL T h [ψ, ψ; µ]
we find in perturbation theory around the massless case, after using some of the results in [8] and performing some functional integral manipulations, that
Z T h (T, µ) = Z SGµ (T, µ) where Z SGµ (T, µ) = N β periodic dφ exp − T d 2 x (L SG − µλ 2π ∂ ∂x 1 φ(x) .
(See [1] for a discussion of the boundary conditions.) That is, the bosonised action is the SG model plus an extra term which is topological in that it only depends on the value of the field at the spatial boundary (x = ±∞). From (4) this term is interpreted as the result of excitations with net kink (fermion) charge being present in the thermal bath and having associated chemical potential µ in the grand-canonical ensemble. Recall that an analogous contribution was found in the chiral Lagrangian for low-energy QCD [9], where the rôle of kinks (Thirring fermions) is played by the skyrmions (QCD baryons): the chiral Lagrangian at finite baryon density acquires, amongst other things, a new factor µQ SK with Q SK the skyrmion topological charge.
IV. CONCLUSIONS AND OUTLOOK
After motivating the study of the SG and MT models at T > 0 and µ = 0, we have summarised some of the results obtained in [1,22]. Firstly, with zero chemical potential we were able to show using both operator methods and path integral methods that the partition functions were equivalent. This also enabled a link to be made between the arbitrary scale ρ at which we carried out thermal normal ordering in the operator approach and the arbitrary renormalisation scale in the path integral-these are identical. We then studied correlators of operators at different space time points in each of the models. Such results will be of crucial importance for the application of this work to the estimation of the number of topological (kink) defects formed in a phase transition, an extension of this work which we motivated in the introduction [22]. Also of relevance to this future work is the relationship between the two models in the presence of a net number of excitations; we stated our results in sections III B 3 and III C.
Finally, we are studying the link between the sine-Gordon model and a 1D classical gas of positive and negative charges with non-zero fugacity [22]. As a result of this (highly studied system) and of the relationship between the sine-Gordon model, Josephson junctions and the massive Thirring model, we would tentatively suggest that such models should perhaps not be overlooked as ones in which to develop or test calculational methods. For example, one could try to investigate such difficult quantities as the pressure or the fermion number density, which could then give some insight into real physical problems. Similarly we hope to investigate the precise nature of the transition at λ 2 = 8π. ACKNOWLEDGMENTS D.A.S. thanks Peter Landshoff both for useful arguments and for originally drawing her attention in this direction. Thanks also to Tim Evans for numerous helpful discussions, and to participants of TFT '98 for noting the conformal origin of the Q variables. M.Ogilvie has also pointed out some useful references. Our warmest thanks go to the organisers of TFT '98 for an enjoyable and stimulating conference and also a very useful pre-conference summer school. D.A.S. is supported by P.P.A.R.C. of the UK through a research fellowship and is a member of Girton College, Cambridge. A.G.N has received support through CICYT, Spain, project AEN96-1634 and through a fellowship of MEC, Spain, and wants to thank the Imperial College Theory Group for their hospitality during the completion of this work. This work was supported in part by the E.S.F.
1
This could be the field operatorφ, the momentum operator π or any other composite operator.
e −βÂ0 T c  I (t 1 ) . . . I (t n ) ,(22) so that one only needs calculate free expectation values ≪ • ≫ 0 = Tr exp{−β 0 •}/Tr exp{−β 0 }.3 Indeed, the correlator appearing in(22) can be obtained by showing first that[1]
TABLE I .
ITable showing schematically some of the links between the SG and MT models at T = 0We see that the kinks themselves are fermion-like excita-
tions corresponding to the fundamental fermions of the
Thirring model. For λ 2 < 4π, the bosonic (Q = 0) exci-
tations of the SG model correspond to the bound states
of the MT model. This is what is meant by bosonisation.
TABLE II .
IITable showing schematically some of the links between QCD and the low energy chiral lagrangian.
These equalities may at first sight seem surprising, but they might be clarified by noting that here the advanced and retarded thermal propagators are equal so that ∆ > T (x − y, ρ) = ∆T (x−y, ρ) = [φ + (x), φ − (y)] where the positive and negative parts refer to those of[24].
From the behaviour of the Q variables in(11), one can see that for λ 2 > 4π there are extra divergences in(26). The treatment of these is commented on in[1].
For example there could be unequal numbers of σ+ and σ−'s, since the chiral symmetry is broken in the MT model with m0 = 0.
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| [] |
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"Correlation-driven insulator-metal transition in near-ideal vanadium dioxide films",
"Correlation-driven insulator-metal transition in near-ideal vanadium dioxide films"
] | [
"A X Gray [email protected] \nStanford Institute for Materials and Energy Sciences\nSLAC National Accelerator Laboratory\n2575 Sand Hill Road94025Menlo ParkCaliforniaUSA\n\nDepartment of Physics\nTemple University\n1925 N. 12 th St19130PhiladelphiaPennsylvaniaUSA\n",
"J Jeong \nIBM Almaden Research Center\n650 Harry Road95120San JoseCaliforniaUSA\n",
"N P Aetukuri \nIBM Almaden Research Center\n650 Harry Road95120San JoseCaliforniaUSA\n",
"P Granitzka \nStanford Institute for Materials and Energy Sciences\nSLAC National Accelerator Laboratory\n2575 Sand Hill Road94025Menlo ParkCaliforniaUSA\n\nWaals-Zeeman Institute\nUniversity of Amsterdam\n1018XEAmsterdamThe Netherlands\n",
"Z Chen \nStanford Institute for Materials and Energy Sciences\nSLAC National Accelerator Laboratory\n2575 Sand Hill Road94025Menlo ParkCaliforniaUSA\n\nDepartment of Physics\nStanford University\n94305StanfordCaliforniaUSA\n",
"R Kukreja \nStanford Institute for Materials and Energy Sciences\nSLAC National Accelerator Laboratory\n2575 Sand Hill Road94025Menlo ParkCaliforniaUSA\n\nDepartment of Materials Science and Engineering\nStanford University\n94305StanfordCaliforniaUSA\n",
"D Higley \nStanford Institute for Materials and Energy Sciences\nSLAC National Accelerator Laboratory\n2575 Sand Hill Road94025Menlo ParkCaliforniaUSA\n\nDepartment of Applied Physics\nStanford University\n94305StanfordCalifornia\n",
"T Chase \nStanford Institute for Materials and Energy Sciences\nSLAC National Accelerator Laboratory\n2575 Sand Hill Road94025Menlo ParkCaliforniaUSA\n\nDepartment of Applied Physics\nStanford University\n94305StanfordCalifornia\n",
"A H Reid \nStanford Institute for Materials and Energy Sciences\nSLAC National Accelerator Laboratory\n2575 Sand Hill Road94025Menlo ParkCaliforniaUSA\n",
"H Ohldag \nStanford Synchrotron Radiation Lightsource\nSLAC National Accelerator Laboratory\n2575 Sand Hill Road94025Menlo ParkCaliforniaUSA\n",
"M A Marcus \nAdvanced Light Source\nLawrence Berkeley National Laboratory\nOne Cyclotron Road94720BerkeleyCaliforniaUSA \n",
"A Scholl \nAdvanced Light Source\nLawrence Berkeley National Laboratory\nOne Cyclotron Road94720BerkeleyCaliforniaUSA \n",
"A T Young \nAdvanced Light Source\nLawrence Berkeley National Laboratory\nOne Cyclotron Road94720BerkeleyCaliforniaUSA \n",
"A Doran \nAdvanced Light Source\nLawrence Berkeley National Laboratory\nOne Cyclotron Road94720BerkeleyCaliforniaUSA \n",
"C A Jenkins \nAdvanced Light Source\nLawrence Berkeley National Laboratory\nOne Cyclotron Road94720BerkeleyCaliforniaUSA \n",
"P Shafer \nAdvanced Light Source\nLawrence Berkeley National Laboratory\nOne Cyclotron Road94720BerkeleyCaliforniaUSA \n",
"E Arenholz \nAdvanced Light Source\nLawrence Berkeley National Laboratory\nOne Cyclotron Road94720BerkeleyCaliforniaUSA \n",
"M G Samant \nIBM Almaden Research Center\n650 Harry Road95120San JoseCaliforniaUSA\n",
"S S P Parkin \nIBM Almaden Research Center\n650 Harry Road95120San JoseCaliforniaUSA\n",
"H A Dürr \nStanford Institute for Materials and Energy Sciences\nSLAC National Accelerator Laboratory\n2575 Sand Hill Road94025Menlo ParkCaliforniaUSA\n"
] | [
"Stanford Institute for Materials and Energy Sciences\nSLAC National Accelerator Laboratory\n2575 Sand Hill Road94025Menlo ParkCaliforniaUSA",
"Department of Physics\nTemple University\n1925 N. 12 th St19130PhiladelphiaPennsylvaniaUSA",
"IBM Almaden Research Center\n650 Harry Road95120San JoseCaliforniaUSA",
"IBM Almaden Research Center\n650 Harry Road95120San JoseCaliforniaUSA",
"Stanford Institute for Materials and Energy Sciences\nSLAC National Accelerator Laboratory\n2575 Sand Hill Road94025Menlo ParkCaliforniaUSA",
"Waals-Zeeman Institute\nUniversity of Amsterdam\n1018XEAmsterdamThe Netherlands",
"Stanford Institute for Materials and Energy Sciences\nSLAC National Accelerator Laboratory\n2575 Sand Hill Road94025Menlo ParkCaliforniaUSA",
"Department of Physics\nStanford University\n94305StanfordCaliforniaUSA",
"Stanford Institute for Materials and Energy Sciences\nSLAC National Accelerator Laboratory\n2575 Sand Hill Road94025Menlo ParkCaliforniaUSA",
"Department of Materials Science and Engineering\nStanford University\n94305StanfordCaliforniaUSA",
"Stanford Institute for Materials and Energy Sciences\nSLAC National Accelerator Laboratory\n2575 Sand Hill Road94025Menlo ParkCaliforniaUSA",
"Department of Applied Physics\nStanford University\n94305StanfordCalifornia",
"Stanford Institute for Materials and Energy Sciences\nSLAC National Accelerator Laboratory\n2575 Sand Hill Road94025Menlo ParkCaliforniaUSA",
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"Stanford Institute for Materials and Energy Sciences\nSLAC National Accelerator Laboratory\n2575 Sand Hill Road94025Menlo ParkCaliforniaUSA",
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"Advanced Light Source\nLawrence Berkeley National Laboratory\nOne Cyclotron Road94720BerkeleyCaliforniaUSA ",
"Advanced Light Source\nLawrence Berkeley National Laboratory\nOne Cyclotron Road94720BerkeleyCaliforniaUSA ",
"Advanced Light Source\nLawrence Berkeley National Laboratory\nOne Cyclotron Road94720BerkeleyCaliforniaUSA ",
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] | [] | We use polarization-and temperature-dependent x-ray absorption spectroscopy, in combination with photoelectron microscopy, x-ray diffraction and electronic transport measurements, to study the driving force behind the insulator-metal transition in VO 2 . We show that both the collapse of the insulating gap and the concomitant change in crystal symmetry in homogeneously strained single-crystalline VO 2 films are preceded by the purely-electronic softening of Coulomb correlations within V-V singlet dimers. This process starts 7 K (±0.3 K) below the transition temperature, as conventionally defined by electronic transport and x-ray diffraction measurements, and sets the energy scale for driving the near-roomtemperature insulator-metal transition in this technologically-promising material. | 10.1103/physrevlett.116.116403 | [
"https://arxiv.org/pdf/1503.07892v1.pdf"
] | 10,148,846 | 1503.07892 | fa1c8d71a48366f5b4b83a34c440e9e0ad2a2f13 |
Correlation-driven insulator-metal transition in near-ideal vanadium dioxide films
A X Gray [email protected]
Stanford Institute for Materials and Energy Sciences
SLAC National Accelerator Laboratory
2575 Sand Hill Road94025Menlo ParkCaliforniaUSA
Department of Physics
Temple University
1925 N. 12 th St19130PhiladelphiaPennsylvaniaUSA
J Jeong
IBM Almaden Research Center
650 Harry Road95120San JoseCaliforniaUSA
N P Aetukuri
IBM Almaden Research Center
650 Harry Road95120San JoseCaliforniaUSA
P Granitzka
Stanford Institute for Materials and Energy Sciences
SLAC National Accelerator Laboratory
2575 Sand Hill Road94025Menlo ParkCaliforniaUSA
Waals-Zeeman Institute
University of Amsterdam
1018XEAmsterdamThe Netherlands
Z Chen
Stanford Institute for Materials and Energy Sciences
SLAC National Accelerator Laboratory
2575 Sand Hill Road94025Menlo ParkCaliforniaUSA
Department of Physics
Stanford University
94305StanfordCaliforniaUSA
R Kukreja
Stanford Institute for Materials and Energy Sciences
SLAC National Accelerator Laboratory
2575 Sand Hill Road94025Menlo ParkCaliforniaUSA
Department of Materials Science and Engineering
Stanford University
94305StanfordCaliforniaUSA
D Higley
Stanford Institute for Materials and Energy Sciences
SLAC National Accelerator Laboratory
2575 Sand Hill Road94025Menlo ParkCaliforniaUSA
Department of Applied Physics
Stanford University
94305StanfordCalifornia
T Chase
Stanford Institute for Materials and Energy Sciences
SLAC National Accelerator Laboratory
2575 Sand Hill Road94025Menlo ParkCaliforniaUSA
Department of Applied Physics
Stanford University
94305StanfordCalifornia
A H Reid
Stanford Institute for Materials and Energy Sciences
SLAC National Accelerator Laboratory
2575 Sand Hill Road94025Menlo ParkCaliforniaUSA
H Ohldag
Stanford Synchrotron Radiation Lightsource
SLAC National Accelerator Laboratory
2575 Sand Hill Road94025Menlo ParkCaliforniaUSA
M A Marcus
Advanced Light Source
Lawrence Berkeley National Laboratory
One Cyclotron Road94720BerkeleyCaliforniaUSA
A Scholl
Advanced Light Source
Lawrence Berkeley National Laboratory
One Cyclotron Road94720BerkeleyCaliforniaUSA
A T Young
Advanced Light Source
Lawrence Berkeley National Laboratory
One Cyclotron Road94720BerkeleyCaliforniaUSA
A Doran
Advanced Light Source
Lawrence Berkeley National Laboratory
One Cyclotron Road94720BerkeleyCaliforniaUSA
C A Jenkins
Advanced Light Source
Lawrence Berkeley National Laboratory
One Cyclotron Road94720BerkeleyCaliforniaUSA
P Shafer
Advanced Light Source
Lawrence Berkeley National Laboratory
One Cyclotron Road94720BerkeleyCaliforniaUSA
E Arenholz
Advanced Light Source
Lawrence Berkeley National Laboratory
One Cyclotron Road94720BerkeleyCaliforniaUSA
M G Samant
IBM Almaden Research Center
650 Harry Road95120San JoseCaliforniaUSA
S S P Parkin
IBM Almaden Research Center
650 Harry Road95120San JoseCaliforniaUSA
H A Dürr
Stanford Institute for Materials and Energy Sciences
SLAC National Accelerator Laboratory
2575 Sand Hill Road94025Menlo ParkCaliforniaUSA
Correlation-driven insulator-metal transition in near-ideal vanadium dioxide films
1
We use polarization-and temperature-dependent x-ray absorption spectroscopy, in combination with photoelectron microscopy, x-ray diffraction and electronic transport measurements, to study the driving force behind the insulator-metal transition in VO 2 . We show that both the collapse of the insulating gap and the concomitant change in crystal symmetry in homogeneously strained single-crystalline VO 2 films are preceded by the purely-electronic softening of Coulomb correlations within V-V singlet dimers. This process starts 7 K (±0.3 K) below the transition temperature, as conventionally defined by electronic transport and x-ray diffraction measurements, and sets the energy scale for driving the near-roomtemperature insulator-metal transition in this technologically-promising material.
A clear understanding of how new electronic and structural phases of matter arise and evolve is not only important for basic science but is also becoming crucial for information technology. Our ability to navigate and ultimately control the complex dynamical pathways in the multidimensional landscape of the electronic, spin, and lattice degrees of freedom is starting to play a critical role in achieving technical feasibility and efficient performance of future electronic devices [1].
One of the prime material candidates for such devices, vanadium dioxide (VO 2 ), undergoes an insulator-metal transition with a four-orders-of-magnitude increase in conductivity and a concomitant monoclinic-rutile structural transformation accompanied by the dimerization of neighboring vanadium atoms just above room temperature [2]. This makes VO 2 suitable for technological applications, from solid-state sensors and optical detectors to field-effect transistors and memristors [1], and motivates investigations aimed at controlling the insulator-metal transition by external stimuli [3][4][5] and epitaxial strain [6].
The nature of the driving force behind the insulator-metal transition in this prototypical stronglycorrelated electron system is one of the longest-standing problems in condensed matter physics [2]. Both electron-electron correlations and electron-lattice interactions are believed to be relevant. However, the question of which drives the other and the interplay between Mott-Hubbard [7,8] and Peierls [9,10] mechanisms remain under debate [11][12][13][14][15].
The structural and electronic dichotomy of the insulator-metal transition in VO 2 , depicted schematically in Fig. 1, opens the door for two possible explanations as to what could be the driving force and the physical mechanism behind this phenomenon. On one hand, a structural transformation with strong dimerization (Figs. 1(a),(c)) presents seemingly convincing evidence for the Peierls-like picture, wherein opening of the insulating gap shown schematically in Fig. 1(d) is caused by the lattice distortion [9,10]. On the other hand, the evidence of strong electron-electron correlation effects suggests the Mott-Hubbard scenario in which strong Coulomb interaction between electrons plays the key role in triggering the insulator-metal transition by splitting the near-Fermi-level electronic states ( Fig. 1(b)) into the bonding and anti-bonding bands, thus opening-up an insulating gap ( Fig. 1(d)) [7,8]. To this day, however, the underlying physics of the transition remains elusive due to the lack of success in consolidating all of its experimentally observed structural and electronic aspects in a singular selfconsistent quantitative theoretical picture [7][8][9][10][11][12][13][14]. A further complication arises due to phase separation scenarios often accompanying insulator-metal transitions [16][17][18]. In bulk-like VO 2 films the coexistence of insulating and metallic-like patches was observed in various experiments probing electronic and structural aspects [16,17]. Similarly, in thick epitaxial films the formation of unidirectional metallic-like stripes [18] and a monoclinic-like metallic phase [19] was observed in the vicinity of the transition.
Here we show that near-ideal, high-quality ultrathin epitaxial VO 2 films grown on TiO 2 (001) substrates display a spatially homogeneous insulator-metal transition without phase segregation. We track the VO 2 insulator-metal transition by monitoring with polarization-dependent x-ray absorption spectroscopy (XAS) spectroscopic features of the dimer V-V electronic correlations (Fig. 2d), the electronic band gap (Fig. 2c) and the dimer V-V Peierls lattice distortion (Fig. 2d), respectively. This enables the unambiguous assignment of the insulator-metal transition to two consecutive processes occurring with increasing temperature. Initially, the insulating phase is modified by a weakening of electronic correlations in dimer V-V singlet states [15]. This process sets in as much as 7 K (±0.3 K) below the proper insulator-metal transition temperature (T IMT = 295±0.3 K) [20] probed by the electronic band gap collapse. The gap collapse starts as soon as the V-V singlet correlations have completely disappeared at T IMT . Only at T IMT do we detect structural changes in V-V dimerization [20] as implied by a Peierls scenario, as well as electronic changes consistent with the disappearance of a dimer V-V Peierls lattice distortion.
For our experiment, high-quality single-crystalline VO 2 thin films were grown epitaxially on TiO 2 (001) substrates by pulsed laser deposition, following the procedures described in-depth in Refs. 5 and 6 (our prior studies). This results in coherently-strained 10 nm-thick VO 2 (001) films with a c R /a R lattice-constant ratio of 0.617 [20,6]. The films undergo an abrupt insulator-metal transition at 295 K upon heating that is accompanied by a change in crystal symmetry. This transition is sharper than that observed for comparable films [19] and indicative of the absence of any detectable electronic phase separation (see below).
Temperature-and x-ray polarization-dependent XAS measurements were carried out at the elliptically polarized undulator beamline 4.0.2 of the Advanced Light Source, using the Vector Magnet endstation equipped with a 4-axis sample manipulator with cryogenic cooling as well as conductive heating capabilities [21]. The nearly 100% linearly-polarized x-ray beam was focused down to 100 μm diameter spot at the sample surface. Average probing depth in the total electron yield XAS detection mode was estimated to be about 5 nm, providing bulk-sensitive information with minimal contribution from surface adsorbates. The temperature was varied using a closed-loop feedback-activated temperature controller, providing temperature stability of ±0.1 K. Measurements were carried out at several locations on the sample, to exclude the possibility of x-ray sample damage. Figure 2 shows high-resolution polarization-dependent XAS measurements at the O K edge. O K edge XAS probes the O 2p-projected unoccupied density of states resulting from dipole-allowed x-ray transitions from the 1s core shell. It is well known [24][25][26] that this provides access to all relevant states in the unoccupied VO 2 conduction band via O 2p -V 3d orbital hybridization. We use the x-ray polarization dependence of XAS to determine the orbital symmetry. In the grazing incidence experimental geometry shown in Fig. 2(a), linearly polarized x-rays with the electric-field orientation parallel to the c R axis (E ∥ c R ) preferentially probe the unoccupied d ∥ states of a 1g symmetry. d ∥ orbitals are preferentially oriented along the c R direction (see Fig. 2(b)). X-rays with their electric-field orientation perpendicular to the c R axis (E c R ) are more sensitive in probing the states with symmetry, comprised of d xz and d yx orbitals (π* states) (see Fig. 2(b)) [6,24]. as measured for the photon energy range between 528 and 531 eV. The spectra have been normalized to equal edge jump accounting for the incomplete alignment of x-ray polarization with the d ∥ orbital orientation and the varying orbital multiplicity for the two polarizations [24,27]. The leading edge found at 528.7 eV (in the insulating phase) is known to sensitively probe the closing of the electronic band gap with increasing temperature [24]. This is observed as the edge shift to lower photon energy with increasing temperature which is one of the major effects seen in Fig. 2(c) especially between spectra taken with fixed x-ray polarization. Overlaid with this change in band gap are polarization effects seen in the insulating phase but not for metallic VO 2 . Two most prominent polarization-dependent differences are observed at the photon energies of 528.7 eV and 530.2 eV (see also Fig. 2(d)) and are related to the dimer V-V electronic correlations and Peierls lattice distortion, respectively. We will first describe these three spectroscopic features in Figs. 2(c) and 2(d) before proceeding to the temperature dependence.
The XAS spectra measured at the temperature of 310 K, well above the insulator-metal transition (T IMT =295 K), are shown in Fig. 2(c) as open and solid red symbols. These spectra are good representations of the metallic unoccupied density of states immediately above the Fermi level [6,28].
Consistent with prior experimental results [24] and theoretical calculations [15] no significant difference between the spectra collected using two mutually-orthogonal x-ray linear polarizations is observed due to an almost isotropic distribution of the t 2g orbitals near the Fermi level [15,27].
The insulating-state spectra (shown in blue) collected at 280 K (15 K below the insulator-metal transition) are shifted by about 80-100 meV to higher photon energy due to the opening of the insulating gap and, in stark contrast with the metallic-state spectra, are significantly different. Their intensity difference plotted in Fig. 2(d) features two major polarization-dependent peaks favoring E ∥ c R polarization. The more prominent peak centered at the photon energy of 530.2 eV is a well-known d ∥ state arising from the Peierls distortion of the lattice [11,15,24]. This d ∥ Peierls peak is depicted schematically (red outline) in Fig. 1(d). The second feature is observed at the photon energy of 528.7 eV. Such additional XAS intensity for E ∥ c R presents direct evidence of the additional d ∥ orbital character at the onset of the conduction band (depicted schematically in Fig. 1(d)) which has been identified as the fingerprint of the strong electronic correlations within the V-V dimers [15,24]. In addition to being present in the spectra for the coherently-strained thin VO 2 (001) films, such a d ∥ V-V singlet state has also been observed for the bulk VO 2 single-crystals [24]. It represents a unique feature of the stronglycorrelated insulating phase of crystalline VO 2 .
The temperature dependence of the electronic band gap (red solid symbols in Fig. 3(a)) displays the insulator metal transition with T IMT = 295±0.3 K. This insulator-metal transition is closely tracked by the monoclinic-rutile structural transition [20] monitored by the dimer V-V Peierls peak (yellow solid symbols in Fig. 3(a)). Extended datasets with individual spectra for each temperature are presented in Fig. S2 of [20]. Interestingly, the temperature dependence of the dimer V-V singlet state is very different (blue/white symbols in Fig. 3(a)). We observe that the V-V state intensity (as defined in Fig. 2(d)) begins to decay at 288 K which is 7 K below T IMT . The dimer V-V singlet state disappears completely just before the onset of the electronic band gap collapse at 294 K. Thus, our results strongly suggest that the insulator-metal transition in VO 2 follows a three-stage pathway depicted schematically in Figs. 3(b)-(d).
(1) At temperatures that are up to 7 K below the insulator-metal transition (denoted T≪T IMT in Fig. 3(a)), VO 2 is in the insulating monoclinic phase, with two 3d 1 electrons of two adjacent dimer V-V atoms forming a strongly-correlated singlet state of d ∥ symmetry [15] (Fig. 3(b)). (2) Upon heating above T IMT -7K, VO 2 remains in the insulating monoclinic state, i.e. the electronic band gap remains unchanged as demonstrated by the upper band gap edge shown in Fig. 3(a). However, the electronic correlations start to soften, as evidenced by the decay of the dimer V-V singlet state intensity (Fig. 3(c)). (3) Finally, once the electronic correlations are sufficiently diminished, the band gap collapse is initiated. This latter process is accompanied by the change in crystal symmetry from insulating monoclinic to metallic rutile as evidenced by the dimer V-V Peierls peak change (Fig. 3(d)). Above 297 K VO 2 is in a homogeneous metallic state. Process (3) seems to correspond to the conventional Peierls mechanism where lattice distortions keep the band gap intact. However, our results show that it is the preceding decay of electron correlations giving rise to dimer V-V singlet state that set the high-temperature energy scale in VO 2 . As an important consequence of this transition pathway, one can define a second distinct critical temperature (T corr =290 K ± 0.3 K) at which VO 2 undergoes a purely-electronic transition between a stronglycorrelated and a conventional monoclinic Peierls insulator. This transition temperature T corr is 5 K (± 0.3 K) below the insulator-metal transition.
We finally need to address the question of spatial homogeneity of the observed phenomena since phase segregation scenarios have been observed for bulk-like and thin-film VO 2 [16][17][18][19]. A temperatureand polarization-dependent spectro-microscopic investigation of the sample using photoemission microscopy (PEEM) was carried out at the EPU beamline 11.0.1.1 of the Advanced Light Source, using the PEEM-3 microscope routinely facilitating sub-50 nm spatial resolution. Measurements at the O K absorption edge (528.7 eV) reveal that our thin coherently-strained VO 2 film grown in (001) crystallographic orientation on a single-crystalline TiO 2 substrate does not undergo any detectable insulator/metal phase segregation across the insulator-metal transition (see Fig. 3(e)). We note that the allover PEEM contrast in the images of Fig. 3(e) follows the electronic band gap collapse in Fig. 3(a) when normalized to defects (dark dots in Fig. 3(e)). Indirectly, such quasi-instantaneous single-domain switching is evidenced by the sharpness of the transition (2-4 K width) observed in the electronic structure via XAS and electronic transport measurements, as well as structurally via x-ray diffraction spectroscopy [20]. This is in stark contrast to bulk-like films where epitaxial strain is fully relaxed. We find that such films do display a separation into metallic and insulating regions during the insulator-metal transition as shown in Fig S3 of [20].
In summary, our results indicate that the temperature-driven insulator-metal transition in a prototypical strongly-correlated oxide VO 2 is preceded and, possibly, driven by the purely-electronic phase transition occurring at a lower temperature (T corr <T IMT ). We have observed the emergence and the and [110] directions. Each dimerized V-V atomic pair shares a singlet electronic state composed of two strongly-correlated V 3d 1 electrons (shown in blue) [15]. (d) V-V dimerization splits the highlydirectional d ∥ orbitals into the bonding and anti-bonding bands, and the tilting of the dimers shifts the π* band to higher energies due to the increase of the p-d orbital overlap [11,15], together producing an insulating gap of 0.6 eVthe key aspect of the monoclinic phase observed via a wide variety of experimental techniques [22,23]. Additional d ∥ feature in DOS, which arises at the onset of the conduction band (shown in blue) is a unique fingerprint of the strong electron-electron correlations within the dimers [15], accessible experimentally via polarization-dependent XAS at the O K edge (see Ref. 24 and this work).
T<T IMT
Monoclinic Insulator
c. character at the bottom of the unoccupied conduction band, which is predicted by theory [15] and is shown schematically in Fig. 1(d).
d ǁ π* E F c R a R b R T>T IMT Rutile Metal a. V atoms b. d. d ǁ π* 0.6 eV d ǁ d ǁ V-VE ∥ c R E c R ( ∥ ) ( ) b. c R 15°E ∥ c R E c R hν A TEY c R (x) b R a RIns. E ∥ c R Ins. E c R Met. E ∥ c R Met. E c R Ins. I ∥ -I c.
d.
T IMT
In this supplemental section we would like to present some additional materials related to characterization of the samples mentioned in the letter, as well as some expanded temperature-dependent XAS and PEEM datasets detailing the results shown in the main text. These provide additional insights into the validity and interpretation of our experimental data. The supplementary materials include temperature-dependent x-ray diffraction measurements (θ -2θ) of the TiO 2 substrate (002) and the VO 2 (002) R film peaks in the metallic and insulating phases, and expanded dataset showing individual spectra of the temperature-dependent evolution of the d ∥ V-V singlet peak, d ∥ Peierls peak, as well as the collapse of the upper band gap.
Temperature-dependent x-ray diffraction (XRD) measurements
High-angular-resolution (<0.01°) temperature-dependent XRD measurements of the VO 2 /TiO 2 (001) films were carried out in a vacuum of ~1 mTorr, using a Bruker D8 Discover system equipped with variable-temperature stage enclosed in an x-ray transparent beryllium dome. Figure S1 shows two typical θ-2θ scans across the TiO 2 substrate (002) and the VO 2 film peaks, which are indexed as (002) R for the high-temperature rutile phase, and (4 ̅ 02) M for the low-temperature monoclinic phase. The scans were collected immediately below and above the insulator-metal transition, and show an abrupt change of the out-of-plane inter-planar spacing in the VO 2 film at the transition temperature T IMT =295K (green circles in the inset). Concomitant in-situ electronic transport measurements (purple circles in the inset) confirm that the insulator-metal transition and the structural monoclinic-rutile transition occur at the same temperature.
Figure S1 | Monoclinic/Rutile structural transition in near-ideal coherently-strained VO2/TiO2(001). Temperaturedependent x-ray diffraction θ-2θ scans across the TiO 2 substrate (002) and the VO 2 film peaks show a clear structural transition in VO 2 manifested by an abrupt change in the inter-planar atomic spacing along the direction normal to the film surface at 295K (inset). Concomitant XRD and electronic transport measurements were collected for the heating cycle, with 1 hour temperature equilibration time between scans. Inset: Temperature-dependent resistance (purple circles) and c R lattice parameter (green circles) obtained concomitantly via electronic transport and x-ray-diffraction measurement respectively. In this supplementary section we expand on the data shown in Fig. 3a of the main text by showing individual spectra of the temperature-dependent evolution of the d ∥ V-V singlet peak, d ∥ Peierls peak, as well as the collapse of the upper band gap. These provide additional insight into the validity of our experimental data. Since all three datasets are extracted from the same XAS spectra, no temperature calibration is required to compare these two plots. The plots clearly illustrate that the two electronic-structure transitions happen at two distinct temperatures, with T corr =290K and T IMT =295K. Temperature-dependent PEEM measurements of bulk-like VO2 film -Phase Separation
In this supplementary section we show results of the temperature-dependent photoelectron microscopy (PEEM) measurements of bulk-like VO 2 film on Al 2 O 3 (1010) substrate showing clear separation into metallic and insulating regions during the insulator-metal transition (at 340K), which is in stark contrast with the high-quality ultrathin epitaxial VO 2 films grown on TiO 2 (001) substrate (see Fig. 3 of the main text). Measurements were carried out at the leading slope of the O K absorption edge (528.8 eV) which is most sensitive to the collapse of the insulating gap. As expected from the XAS spectroscopic characterization of the same sample (plot on the right), metallic phase appears in the form of lighter patches at 528.8 eV.
. Figure S3| Phase separation in bulk-like VO2 film. Temperature-dependent PEEM images of bulk-like VO 2 film on Al 2 O 3 (1010) substrate measured in the insulating state (left panel), during the insulator transition (middle panel) and in the high-temperature metallic state (right panel). Images show clear separation into metallic and insulating regions during the insulator-metal transition (at 340K), in contrast with the high-quality ultrathin epitaxial VO 2 films grown on TiO 2 (001) substrate described in the main text (see Fig. 3).
Figure 2 (
2c) shows typical temperature and polarization-dependent O XAS spectra for the VO 2 film,
temperature-dependent evolution of the new intermediate monoclinic/insulating phase in VO 2 which is characterized by softening and disappearance of the strong correlations within the V-V dimers. These findings have a far-reaching impact on our understanding of the complex physics of the insulator-metal transition in strongly-correlated oxides, since they suggest that the fundamental electronic and structural transformations in these materials arise from precursory changes in the electronic correlations. Thus, for all the future practical applications of VO 2 as a building-block for next-generation electronic devices, as well as for a wide range of time-resolved pump-probe experiments, this underpinning transition and the characteristic T corr should be carefully characterized and understood. Research at Stanford was supported through the Stanford Institute for Materials and Energy Sciences (SIMES) under contract DE-AC02-76SF00515 and the LCLS by the US Department of Energy, Office of Basic Energy Sciences. The Advanced Light Source is supported by the Director, Office of Science, Office of Basic Energy Sciences, US Department of Energy under Contract No. DE-AC02-05CH11231. Authors would like to thank C.-C Chen, B. Moritz, T. P. Devereaux and M. van Veenendaal for helpful discussions.
FIG. 1 .
1(a) In the high-temperature (T>T IMT ) metallic phase VO 2 forms rutile lattice structure with P4 2 /mnm space symmetry where V atoms (yellow) occupy centers of six-fold oxygen-coordinated sites (O atoms not shown). (b) The near-Fermi-level t 2g states are separated in energy by the orthorhombic component of the crystal field into the twofold-degenerate (π*) states and a single a 1g orbital, which is aligned parallel to the rutile c axis (c R ) and is thus commonly denoted d ∥ . The two bands overlap in energy, and the resulting non-zero density of states at the Fermi level accounts for the metallic behavior of the rutile phase [11,15]. (c) In the low-temperature (T<T IMT ) insulating phase the lattice undergoes a structural transition to a lower-symmetry (P2 1 /c) monoclinic crystal system via dimerization of the neighboring V atoms along the c R direction and tilting of the resultant V-V dimers along the rutile [110]
FIG. 2 .
2(a) Schematic diagram of the polarization-dependent XAS measurement geometry used in this study. Linearly-polarized x-rays are incident at 15° to the surface of the coherently-strained epitaxial 10 nm-thick VO 2 /TiO 2 (001) sample, with the photon polarization set to either parallel (E ∥ c R ) or perpendicular (E c R ) to the c R axis (along the sample normal) of the film enabling preferential probing of the strongly-directional d ∥ ( 2 − 2 ) and π* (d xz and d yx ) orbitals depicted in (b), respectively. (c) Polarization-dependent O K edge XAS measurements of VO 2 in the high-temperature (T=T IMT +15K) metallic state (shown as open and solid red symbols) and in the low-temperature (T=T IMT -15K) insulating state (shown as open and solid blue symbols). Shift of the leading slope of the edge towards lower energy between the insulating and the metallic phases of VO 2 corresponds to the collapse of the insulating band gap on the unoccupied side of the energy band diagram (upper gap collapse). Distinct dichroic signal at the onset of the absorption edge in the insulating state presents direct evidence of the additional d ∥ orbital
(d) d ∥ V-V singlet peak at the onset of the conduction band (528.7 eV) and the d ∥ Peierls peak at 530.2 eV, both obtained by calculating the I ∥ -I XAS intensity taken directly from (c).
FIG. 3 .
3(a) Temperature-dependent evolution of the d ∥ V-V singlet peak intensity (blue/white symbols) showing distinctly different transition temperature (T corr =290 K) as compared to the T IMT (295 K), the critical temperature at which the collapse of the upper band gap (red solid symbols) as well as the structural transition (yellow solid symbols) is observed in VO 2 . (b)-(d) Schematic representation of the three phases of VO 2 implied by the experimental data in the plots on the left. (b) The low-temperature phase (T≪T IMT ) is a monoclinic insulator with a strongly-correlated singlet electronic state on each V-V dimer. At T=T corr the strong Coulomb correlations within the dimers soften, giving rise to a new monoclinic insulating phase shown in (c) via a purely-electronic phase transition. Finally, only after the ee correlations are sufficiently diminished (T=T corr +3 K), the system undergoes a transition to a rutile metallic phase shown in (d). (e) Temperature-and polarization-dependent PEEM images measured at the photon energy of the leading slope of the O K absorption edge (528.7 eV) which is most sensitive to the collapse of the insulating gap in VO 2 , and provides a clear contrast mechanism (up to 20%) between the insulating and metallic phases (see Fig. S3 in the Supplemental Material[20]). In contrast to the bulk-like VO 2 films, our high-quality (near-ideal) ultrathin epitaxial VO 2 film grown on TiO 2 (001) substrate exhibits a homogeneous insulator-metal transition without phase segregation. Darker defects (likely specs of dust) in the left part of the images were used for fiducial alignment and focusing. The overall intensity of the images increases with temperature, which is consistent with the changes in XAS intensity at 528.7 eV as VO 2 undergoes the insulator-metal transition.
Figure S2|
S2|Two electronic transitions in VO2 -expanded datasets. Temperature-dependent evolution of the d∥ V-V singlet peak (left panel) showing distinctly different transition temperature (T corr =290 K) as compared to the T IMT (295 K), the critical temperature at which the decay of the d∥ Peierls peak (middle panel) as well as the collapse of the upper band gap (right panel) is observed in VO 2 .
Ins. E ∥ c R Met. E ∥ c R528.4
528.6
528.8
529.0
0.0
0.5
1.0
Photon Energy (eV)
289 K
291 K
293 K
295 K
297 K
299 K
301 K
XAS
Intensity
(a.u.)
upper gap collapse
d ǁ singlet peak decay
528.2 528.4 528.6 528.8 529.0
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287 K
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301K
d ǁ Peierls peak decay
528.5
529.0
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XAS
Intensity
(a.
u.)
Photon Energy (eV)
PEEM
contrast
320K
(insulator)
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(transition)
360K
(metal)
5 μm
5 μm
(320K)
(360K)
upper gap
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See Supplemental Material below for details related to characterization of the samples mentioned in the letter, as well as some expanded temperature-dependent XAS and PEEM datasets detailing the results shown in the main text. See Supplemental Material below for details related to characterization of the samples mentioned in the letter, as well as some expanded temperature-dependent XAS and PEEM datasets detailing the results shown in the main text.
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| [] |
[] | [
"Johannes Buchner [email protected] \nMillenium Institute of Astrophysics\nVicuña Mackenna 48607820436Macul, SantiagoChile\n\nInstituto de Astrofísica\nPontificia Universidad Católica de Chile\nVicuña Mackenna 48607820436Macul, SantiagoChile\n",
"Steve Schulze \nMillenium Institute of Astrophysics\nVicuña Mackenna 48607820436Macul, SantiagoChile\n\nInstituto de Astrofísica\nPontificia Universidad Católica de Chile\nVicuña Mackenna 48607820436Macul, SantiagoChile\n",
"Franz E Bauer \nMillenium Institute of Astrophysics\nVicuña Mackenna 48607820436Macul, SantiagoChile\n\nInstituto de Astrofísica\nPontificia Universidad Católica de Chile\nVicuña Mackenna 48607820436Macul, SantiagoChile\n\nSpace Science Institute\n4750 Walnut Street, Suite 20580301BoulderColorado\n"
] | [
"Millenium Institute of Astrophysics\nVicuña Mackenna 48607820436Macul, SantiagoChile",
"Instituto de Astrofísica\nPontificia Universidad Católica de Chile\nVicuña Mackenna 48607820436Macul, SantiagoChile",
"Millenium Institute of Astrophysics\nVicuña Mackenna 48607820436Macul, SantiagoChile",
"Instituto de Astrofísica\nPontificia Universidad Católica de Chile\nVicuña Mackenna 48607820436Macul, SantiagoChile",
"Millenium Institute of Astrophysics\nVicuña Mackenna 48607820436Macul, SantiagoChile",
"Instituto de Astrofísica\nPontificia Universidad Católica de Chile\nVicuña Mackenna 48607820436Macul, SantiagoChile",
"Space Science Institute\n4750 Walnut Street, Suite 20580301BoulderColorado"
] | [
"MNRAS"
] | An important constraint for galaxy evolution models is how much gas resides in galaxies, in particular at the peak of star formation z = 1 − 3. We attempt a novel approach by letting longduration Gamma Ray Bursts (LGRBs) x-ray their host galaxies and deliver column densities to us. This requires a good understanding of the obscurer and biases introduced by incomplete follow-up observations. We analyse the X-ray afterglow of all 844 Swift LGRBs to date for their column density N H . To derive the population properties we propagate all uncertainties in a consistent Bayesian methodology. The N H distribution covers the 10 20−23 cm −2 range and shows no evolutionary effect. Higher obscurations, e.g. Compton-thick columns, could have been detected but are not observed. The N H distribution is consistent with sources randomly populating a ellipsoidal gas cloud of major axis N major H = 10 23 cm −2 with 0.22 dex intrinsic scatter between objects. The unbiased SHOALS survey of afterglows and hosts allows us to constrain the relation between Spitzer-derived stellar masses and X-ray derived column densities N H . We find a well-constrained powerlaw relation of N H = 10 21.7 cm −2 × M /10 9.5 M 1/3 , with 0.5 dex intrinsic scatter between objects. The Milky Way and the Magellanic clouds also follow this relation. From the geometry of the obscurer, its stellar mass dependence and comparison with local galaxies we conclude that LGRBs are primarily obscured by galaxyscale gas. Ray tracing of simulated Illustris galaxies reveals a relation of the same normalisation, but a steeper stellar-mass dependence and mild redshift evolution. Our new approach provides valuable insight into the gas residing in high-redshift galaxies. | 10.1093/mnras/stw2423 | [
"https://arxiv.org/pdf/1610.09379v1.pdf"
] | 119,098,236 | 1610.09379 | 1962feea278eec9850fcdb36f87c026b5432468d |
2016
Johannes Buchner [email protected]
Millenium Institute of Astrophysics
Vicuña Mackenna 48607820436Macul, SantiagoChile
Instituto de Astrofísica
Pontificia Universidad Católica de Chile
Vicuña Mackenna 48607820436Macul, SantiagoChile
Steve Schulze
Millenium Institute of Astrophysics
Vicuña Mackenna 48607820436Macul, SantiagoChile
Instituto de Astrofísica
Pontificia Universidad Católica de Chile
Vicuña Mackenna 48607820436Macul, SantiagoChile
Franz E Bauer
Millenium Institute of Astrophysics
Vicuña Mackenna 48607820436Macul, SantiagoChile
Instituto de Astrofísica
Pontificia Universidad Católica de Chile
Vicuña Mackenna 48607820436Macul, SantiagoChile
Space Science Institute
4750 Walnut Street, Suite 20580301BoulderColorado
MNRAS
0002016Accepted XXX. Received YYY; in original form ZZZPreprint 1st November 2016 Compiled using MNRAS L A T E X style file v3.0 Galaxy gas as obscurer: I. GRBs x-ray galaxies and find a N 3 H ∝ M relationgamma-ray burst: general -galaxies: general -X-rays: galaxies - galaxies: structure -galaxies: ISM -X-rays: ISM -Magellanic Clouds -dust, extinction
An important constraint for galaxy evolution models is how much gas resides in galaxies, in particular at the peak of star formation z = 1 − 3. We attempt a novel approach by letting longduration Gamma Ray Bursts (LGRBs) x-ray their host galaxies and deliver column densities to us. This requires a good understanding of the obscurer and biases introduced by incomplete follow-up observations. We analyse the X-ray afterglow of all 844 Swift LGRBs to date for their column density N H . To derive the population properties we propagate all uncertainties in a consistent Bayesian methodology. The N H distribution covers the 10 20−23 cm −2 range and shows no evolutionary effect. Higher obscurations, e.g. Compton-thick columns, could have been detected but are not observed. The N H distribution is consistent with sources randomly populating a ellipsoidal gas cloud of major axis N major H = 10 23 cm −2 with 0.22 dex intrinsic scatter between objects. The unbiased SHOALS survey of afterglows and hosts allows us to constrain the relation between Spitzer-derived stellar masses and X-ray derived column densities N H . We find a well-constrained powerlaw relation of N H = 10 21.7 cm −2 × M /10 9.5 M 1/3 , with 0.5 dex intrinsic scatter between objects. The Milky Way and the Magellanic clouds also follow this relation. From the geometry of the obscurer, its stellar mass dependence and comparison with local galaxies we conclude that LGRBs are primarily obscured by galaxyscale gas. Ray tracing of simulated Illustris galaxies reveals a relation of the same normalisation, but a steeper stellar-mass dependence and mild redshift evolution. Our new approach provides valuable insight into the gas residing in high-redshift galaxies.
INTRODUCTION
Galaxies radiate because of gas condensed into stars and accretion onto compact objects. Reconciling that emission with the gas actually present in galaxies is important to understand processes of galaxy evolution, including the efficiency of star formation and accretion processes and the in-fall of cosmological gas into galaxy halos. Most interesting are constraints at the peak of star formation (z = 1 − 3, Madau & Dickinson 2014) and at the peak of the accretion history (z = 0.5 − 3, Aird et al. 2010;Ueda et al. 2014;Buchner et al. 2015;Aird et al. 2015). At that time, the gas content in galaxies was probably higher, as indicated by molecular gas measurements (e.g. Tacconi et al. 2013), the current primary tracer of galaxy gas at high redshifts.
The gas in galaxies in turn can attenuate the radiation by absorption along the line of sight (LOS). This allows one to con-
LGRBs emits X-rays which can be used to assess the column density through photo-electric absorption, primarily from electrons in O and Fe atoms along the LOS 1 . This work uses the distribution of LOS column densities to reconstruct the galactic gas of GRB host galaxies at high redshift.
Three major methodological contributions lead up to this work: 1) The launch of the Swift satellite (Gehrels 2004) resulted in a wealth of GRB data with accurate positioning and timely afterglow spectroscopy. A series of papers by Campana and collaborators already exploited these data using best-fit column densities (Campana et al. 2006(Campana et al. , 2010a(Campana et al. , 2012. 2) By 2012 it was appreciated that current redshift determination practices introduce systematic sample biases that tend to exclude dusty, massive host galaxies, potentially under-estimating the entire column density distribution of the GRB population (see e.g. Fynbo et al. 2009;Krühler et al. 2011;Perley et al. 2013). This work thus also investigates the nature of that bias (see also Campana et al. 2012, for a bright sample analysis). 3) Reichart (2001) and Reichart & Price (2002) developed a hierarchical Bayesian inference framework for GRB population analysis which incorporates the uncertainties of each individual Xray spectral analysis. We use this framework to also consistently include GRBs without redshift information, and provide a more advanced statistical analysis for potential redshift evolution of the GRB population.
This work begins with describing the statistical framework in Section 2. We describe the LGRB samples used in this work, the data reduction and spectral analysis procedures in Section 3. Section 4 presents the results of the spectral analysis and population properties. There we begin with empirical models and in turn investigate more physical models, redshift evolution, host mass dependence and the effect of redshift-incomplete samples. We discuss our results in Section 5, putting our results in the context of previous investigation of LGRB obscurers (5.2) and comparing our results to local galaxies. Finally we look inside simulated galaxies to investigate their gas columns (Section 5.5) before summarising our conclusions in Section 6.
METHODOLOGY
To analyse the GRB population we consider several parameterised models. Each model M predicts the distribution of column densities, p(N H |M, θ), between the GRB and the observer. The parameters θ of each model are constrained with X-ray spectral data of a GRB sample, described in the next section, which is a (Poisson) draw from the general GRB population. We define the likelihood of observing data D for n objects as
L M (θ) = n i=1 p(D i |N H , φ i ) · p(N H , φ i |M, θ) d log N H dφ i ,(1)
where p(D i |·) is the likelihood of object i to have column density N H and other properties φ i relevant for the X-ray spectrum, such as redshift and the photon index Γ (see below in Section 3.3). That likelihood p(D i |·) is discussed in more detail Section 3.3, but briefly speaking it is a Poisson likelihood due to the count nature of X-ray photon detections. The sample and its X-ray data are discussed in the section below.
The distribution of properties within the population is described by the p(N H , φ|M, θ), which is the product of the column density distribution p(log N H |M, θ), the photon index distribution and further distributions assumed for the remaining spectral parameters (defined below). The photon index distribution is assumed to be Gaussian, with the mean and standard deviation free parameters in θ and determined simultaneously. Models for the column density distribution, which are focus of this paper, p(N H |M, θ) are presented in Section 4.1.
To illustrate the meaning of Equation 1 consider the case of objects with no information, i.e., a flat p(D i |N H ). Any population model p(log N H ) is then equally probable, including delta functions that predict a constant N H for all objects. This follows from p(log N H ) being normalised in log N H space. In contrast, if an object has a well-constrained likelihood p(D i |N H ), the integral will be maximised by population models with some probability there. If two objects have well-constrained, but mutually exclusive N H constraints, then the population model p(log N H ) has to distribute its probability weight, thereby representing scatter in the population. Objects without data constraints are effectively "wildcards": their integral mass is concentrated wherever the specific population model p(log N H ) is concentrated, creating degenerate solutions of equal likelihood L.
Equation 1 defines the likelihood which can be explored with maximum likelihood methods or a Bayesian approach by varying the population model parameters θ. This likelihood is well-known in luminosity function works (e.g., Marshall et al. 1983;Loredo 2004;Kelly et al. 2008 2 ) and has also been used in previous studies of the column density of GRBs (Reichart 2001;Reichart & Price 2002). To compare various models, the Akaike information criterion (AIC; Akaike 1974) is adopted. The AIC prefers the model with the highest maximum likelihood L M (θ max ), but punishes by the number of parameters m according to AIC := 2 · L M (θ max ) − 2 · m (higher AIC is better).
DATA
Sample Selection
The Swift satellite (Gehrels 2004) is a dedicated GRB mission which features on-board detection of GRBs with a wide field γray detector and automatic followup using X-ray and optical/UV telescopes for determining the source position to arcsec accuracy. We analysed the X-ray spectrum of all GRBs in the Swift Burst Analyser ) archive 3 up to June 24th, 2015. Our parent sample is 920 GRBs detected by Swift.
This work ultimately aims to constrain the intrinsic column density distribution of GRB host galaxies. We select all detected long-duration GRB, which are associated with the death of massive stars, and therefore can be expected to trace the galactic gas content through star formation. To illustrate our sample selection, Figure 1 shows the distribution of all Swift-detected GRBs on the sky. The positions of GRBs are constrained on-board by combining the gamma-ray, X-ray and optical/UV telescopes. Bursts of short duration (green star symbols, T 90 < 2s) were excluded. These have been identified through mentions of short duration in associated GRB Coordinates Network (GCN) Circulars. This leaves 844 LGRBs,
60°S 40°S 20°S 0°2 0°N 40°N 60°N Redshift known --, excluded Redshift unknown --, excluded Short N H >5 ·10 20 cm −2 N H >10 21 cm −2 N H >10 22 cm −2 Figure 1.
Distribution of the GRB sample in Galactic coordinates (points). To avoid Galactic absorption the Galactic latitude |b| < 20°and regions with N H > 10 21 cm −2 are excluded from the sample (circles). A smoothened Galactic column density map is shown in the background (Kalberla et al. 2005). indicated as squares and circles in Figure 1. Regions where the Milky Way contributes substantial column densities (over-plotted shades) have to be excluded. We exclude positions with Galactic column densities N MW H > 10 21 cm −2 and Galactic latitudes |b| < 20°. We call this sample with 512 objects the complete sample (see Table 1) as it is unbiased against obscuration.
The availability of redshift information is important to constrain the obscuring column density from the X-ray spectrum. For 208 LGRBs, redshifts have been determined previously and are indicated by red squares in Figure 1. Those 163 objects with redshifts in the range z = 0.3 − 3.2 are called the redshift subsample (see Table 1). This criterion excludes very high-redshift afterglows for which the imprint of absorption is not observable, and lowluminosity LGRBs at low redshifts which may have different progenitors or emission mechanisms (see Dereli et al. 2015, and references therein).
Whether a redshift has been successfully determined for a particular LGRB depends on many factors and thus the LGRBs with available redshifts constitute a biased subsample, showing higher fluxes, lower absorption than carefully constructed samples with dedicated follow-up ). Working with unbiased samples is thus important to determine the underlying distribution (e.g. Campana et al. 2012). Such samples are pre-selected by GRB position relative to the Sun, Moon, galactic plane and available observatories, but importantly not by afterglow detection or magnitude. The sample is then followed up with deep groundbased observations to determine redshifts and host properties (e.g. Jakobsson et al. 2006a;Fynbo et al. 2009;Greiner et al. 2011;Krühler et al. 2012;Jakobsson et al. 2012;Schulze et al. 2015;Perley et al. 2016b). The largest unbiased sample to date is the Swift Gamma-Ray Burst Host Galaxy Legacy Survey (SHOALS, Perley et al. 2016b). Their redshift distribution is depicted in Figure 2 as a red dotted line. In comparison, the redshift distribution of the complete sample, where redshifts are available, peaks at lower redshifts than the redshift subsample (black histogram).
To overcome the biases of redshift selection, two approaches are followed: (a) The entire sample is used, including objects without determined redshift. This sample does then not have a redshift selection bias, but has low redshift completeness (∼ 40%). The distribution in the sky of this sample is shown in Figure 1 with square symbols. (b) The SHOALS sample is adopted with the same Milky Way absorption criteria. The resulting 105 objects form an unbiased sample with ∼ 90% redshift completeness. For objects with known spectroscopic redshift we fix the redshift during spectral analysis, for objects without redshifts or photometric redshifts we adopt as a redshift prior the unbiased distribution of SHOALS. Perley et al. (2016c) investigated the masses of GRB host galaxies. They obtained Spitzer follow-up observations of the SHOALS sample in the 3.6µm band, where −22 mag corresponds approximately to a stellar mass of 10 10 M . We split our unbiased sample further into low-mass and high-mass subsamples using this criterion (see Table 1). These subsamples have 94 and 25 objects, respectively.
Data reduction
X-ray observations were taken using the XRT instrument on-board Swift (Burrows et al. 2005) which is sensitive in the 0.2−10 keV energy range. To minimise pile-up, XRT is operated in Window Timing (WT) mode for high-flux GRB afterglows. Otherwise, Photon Counting mode (PC) is used. GRBs show strong evolution in their light curve and spectral hardness. This work focuses on the emission of the X-ray afterglow, which is assumed here to be intrinsically a powerlaw, and to have a time-invariant spectral shape. However, the early evolution of an afterglow (t 1000 s) can be affected by prompt emission. This phase can be easily identified by its rapid decay (t −α with α > 2) and spectral softening. For this we turn to the Swift Burst Analyser, which analyses the light curve as described in Evans et al. (2009). Briefly speaking, the light curve is approximated by piece-wise powerlaw evolutions. We discard the initial two time intervals if either show a powerlaw decline with a slope steeper than −2, and any immediately following intervals that also abide by this criterion. Flare intervals are also discarded. The remaining time segmentation was checked and corrected by individually inspecting each light curve and hardness ratio. The final time segmentation is listed for each source in the catalogue released with this paper (see below).
The spectra from the chosen time segmentation was then extracted using the Swift Burst Analyser, which automatically screens event files, selects appropriate energy ranges, applies grade filtering, and chooses spectral extraction regions to avoid pile-up while maximising S/N. Background spectra were extracted from appropriate surrounding areas. Ancillary Response Files (ARFs) and Response Matrix Files (RMFs) were computed. The 0.5 − 5keV data were used in the spectral analysis below, as XRT is most sensitive there with the background well-behaved and sub-dominant.
Because XRT automatically observes bright sources in WT mode, but faint sources and late-time observations in PC mode, two data sets may be available for any GRB. For the majority of GRBs, WT mode is not or only very briefly used. Brief WT observations were not analysed here as it is difficult to constrain the background and in practice they do not improve constraints over the PC mode observations. The criterion for including WT spectra was that the background spectrum must contain more than 150 counts. The PC mode data are always used, if available.
In some cases, no time interval can be safely used. This can occur if only the prompt emission is bright enough to be detected. Furthermore there are sources with no detected afterglow 4 . These sources lacking X-ray data were included (if Swift-triggered) in our analysis nevertheless, as they could be heavily obscured, and comprise part of the unbiased sample.
X-ray spectral analysis
The intrinsic afterglow spectrum is thought to be due to synchrotron radiation (Piran 2005). We model the relevant portion for the 0.5 − 5 keV energy range as a power law φ(E) = A × E −Γ , which is then photo-electrically absorbed: once by an intrinsic absorption within the host galaxy N H (see below), and once with a Galactic absorption N MW H using the TBABS ISM absorption model (Wilms et al. 2000, with cross-sections from Verner et al. 1996). The source model parameters are thus the normalisation A, the photon index Γ, and the absorbing column densities N H and N MW H . The intrinsic power law is assumed to remain constant with time: hardness ratio variations where not found (see above) and luminosity variations do not affect the Poisson fit when we are only interested in the spectral shape. The background spectrum is empirically modelled as described in Appendix B.
Towards Compton-thick densities, here N H > 10 24 cm −2 for simplicity, effects beyond photo-electric absorption become important. Such columns in a dense GRB environment or host galaxy gas could block even the X-ray afterglow emission. Therefore we search for evidence of high column densities. We adopt the SPHERE model of Brightman & Nandra (2011), which describes photoelectric absorption, Compton-scattering and line fluorescence computed self-consistently in a spherical, constant-density obscurer geometry with a powerlaw source in the centre. The SPHERE model supports column densities up to N H = 10 26 cm −2 . Solar metallicities (Anders & Ebihara 1982) are assumed when deriving the neutral hydrogen-equivalent column densities N H . However, LGRBs appear to be often found in low-metallicity environments (Graham & Fruchter 2013). Derived column densities should thus be primarily considered as metal column densities as relevant for photo-electric absorption of X-rays. We also repeated our entire analysis using the spectral analysis the TBABS ISM absorption model (Wilms et al. 2000, with cross-sections from Verner et al. 1996) and find consistent column densities within the uncertainties. However, in highobscuration sources (N H 10 22.5 cm −2 ) TBABS (and other photoabsorption models) sometimes produces a secondary, Comptonthick solution of lower probability. This solution is not physical as it is not present when analysing with the more appropriate SPHERE model. We therefore use the SPHERE model throughout.
To obtain probability distributions for the column density N H , fitted absorbed powerlaw unabsorbed powerlaw + without galactic absorption data Figure 3. XRT convolved X-ray spectrum of GRB 081221. Left: PC mode spectrum. Right: WT mode spectrum. The absorbed powerlaw model (black line, with 3σ uncertainties in grey) fits the data (black error bars, binned for plotting) well. For this source, the column density is N H ≈ 10 22.5 cm −2 . The dashed green line indicates the fit of an unabsorbed powerlaw, which is clearly ruled out by the low number of counts at 0.5 − 1 keV. The dotted line shows the effect if the Galactic absorption, 2 × 10 21 cm −2 for this source, is also removed. a Bayesian methodology is adopted for analysing the X-ray spectrum (van Dyk et al. 2001;Buchner et al. 2014). The Bayesian X-ray Analysis (BXA) software, which connects the Sherpa X-ray spectral analysis tool (Freeman et al. 2001) to the MultiNest algorithm (Feroz et al. 2009(Feroz et al. , 2013, is used with a Poisson likelihood (Cash 1979). This methodology has the benefit of exploring the full parameter space and propagating correlated uncertainties, e.g., between redshift, obscuring column and the powerlaw slope. The Bayesian approach requires the explicit specification of priors. They are listed for each parameter separately in Table 2. For the Galactic column density N MW H , a informative normal prior is adopted around the value measured by the Leiden/Argentine/Bonn (LAB) Survey of Galactic HI (Kalberla et al. 2005) at the source position. This estimate may be slightly off in unfortunate conditions, if the Galactic gas is very structured and/or the position is not precisely known. This uncertainty is allowed by putting a Gaussian prior around log N MW H with a standard deviation of σ = 1 3 log 3, i.e., allowing a three sigma deviation of a factor of 3 in N MW H . This only broadens our uncertainties in N H , especially when N MW H ≈ N H . The uninformative priors adopted for N H and Γ are later replaced by the populations' column density distribution and thus do not influence the results. For completeness, we include GRBs without redshift information. For these we adopt as a redshift prior the distribution of the SHOALS sample (see Figure 2, red dotted line), which encodes the assumption that these GRBs stem from the same underlying distribution. Perley et al. (2016b) tested the influence of their fluence cut on the redshift distribution and found it to be negligible. Figure 3 shows an example of a fitted X-ray spectrum in both PC and WT mode of a highly obscured GRB. None of the objects show contradictory constraints between the two modes, which could occur due to poor fits of the source or background spectrum. As we have analysed the WT and PC mode spectra separately, no assumptions about the two spectra having the same luminosity or photon index are made. Finally, the constraints for column density N H from the WT and PC mode spectra are merged (if both available) by multiplying the PC mode probability distributions (in N H and Γ) by the WT N H probability distribution, thus tightening the constraints on N H . The photon index Γ can show degeneracies with N H in low-count spectra, so its probability distribution is carried along as described in Section 2. The population distributions of Γ, N H and z are constrained simultaneously. The photon index distribution is reported in Appendix A.
RESULTS
Before inferring the population properties of GRB obscurers, we briefly describe the spectral analysis results of the sample. Their distribution in column density and redshift is shown in Figure 4 for the redshift subsample at z = 0.3 − 3.2. Out of the 163 objects, 28 can be securely identified as intrinsically obscured (N H > 10 22 cm −2 with 90% probability), and four can be securely identified as intrinsically unobscured (N H < 10 21 cm −2 with 90% probability). GRB 080207 shows the highest obscuration with N H ≈ 10 23 cm −2 . In the complete sample, which comprises all LGRB detections of Swift, all sources with X-ray data are constrained to N H < 2 × 10 23 cm −2 . A large portion of the sample only has upper limits for the column density distribution. The lowest upper limit is N H < 10 20.54 cm −2 (90% quantile) in GRB 061021. A catalogue of the sample analysis results of all GRBs is released with this paper. Its columns are described in Table 3.
To test whether we might have missed any heavily obscured (dark) LGRB, we simulate PC mode spectra for all sources with z = 1 − 3, focusing on the bulk of the sample redshift distribution. We use the same spectral parameters (normalisation, photon index) but set the obscuring column to N H = 10 24 cm −2 . This reduces the median number of detected counts in the 0.5 − 5keV range from (7) Redshift (0 indicates unknown). (8) Time selection (in seconds since trigger) from which the afterglow emission was analysed in X-ray. If "n/a", no interval could be used. If not Swift triggered, this is also noted in this column. (9-12) Derived column density distribution from the X-ray spectrum analysis, listing (9) the mean in logarithmic units of cm −2 , (10) the 10% quantile and (11) the 90% quantile. This catalogue excerpt contains only the first and last GRB of our sample, as well as the lowest and highest obscuration GRBs found. 10266 (of which 2 are expected to be background counts) to 66 counts. Therefore there would still be enough contrast to detect and characterise the X-ray emission of such heavily obscured sources via their extremely hard spectra, which at these redshifts exposes the FeKα feature and the low-energy end of the Compton-hump.
Empirical Population Models
To analyse the population properties, specifically the obscurer column density distribution (CDD), models are adopted which predict the CDD. To start, we simply want to visualise the data constraints, which contain large uncertainties and upper limits (shown before in Figure 4). Figure 5 shows the CDD fitted with several models. The points with error bars are derived by adopting 11 bins (the last bin extends up to 10 26 cm −2 ). We find that the column densities are confined to the 10 20.5−23 cm −2 range. Two models have been adopted in the literature to describe the CDD empirically, and are shown in
p(N H |M BKNPL , a, b, c) = ln(10) · b · c c − b · (N H /a) b if N H ≤ a (N H /a) c if N H > a .(2)
The parameters b and c give the powerlaw slopes at the low and high-obscuration ends respectively, separated at the break a. Alternatively, a Gaussian distribution of log N H (solid line fit in Figure 5) has been used (e.g. Campana et al. 2010aCampana et al. , 2012
p(N H |M GAUSS , µ, σ) = 1 √ 2πσ · exp − log N H − µ 2 2σ 2 .(3)
The parameters are the centre of the distribution µ and its width σ.
The constrained parameter values for the two models are listed in Table 4. The parameters are constrained by sampling the posterior distribution using MultiNest (Feroz et al. 2009(Feroz et al. , 2013 through PyMultiNest (Buchner et al. 2014). Flat priors have been
Low N H slope High N H slope z < 0.3 log a 1 = 20.67 ± 0.25 b 1 = 2.44 ± 1.13 c 1 = −2.35 ± 1.14 z = 0.3 − 1 log a 2 = 21.81 ± 0.18 b 2 = 0.74 ± 0.18 c 2 = −1.85 ± 0.70 z = 1 − 2 log a 3 = 21.83 ± 0.09 b 3 = 1.02 ± 0.22 c 3 = −1.24 ± 0.18 z = 2 − 4 log a 4 = 22.03 ± 0.12 b 4 = 0.68 ± 0.12 c 4 = −1.67 ± 0.44 z > 4 log a 5 = 21.05 ± 0.61 b 5 = 2.19 ± 1.15 c 5 = −1.48 ± 1.16
Gaussians, 5 redshift bins 12 -13.9 Mean Standard deviation assumed on µ, log σ and ν, log a, b, and c. In general we find the distribution to be centred at N H ≈ 10 21.8 cm −2 and effectively spreading two orders of magnitude (see Figure 5). The Gaussian model yields a higher likelihood and since it also has one parameter fewer, is preferred through a lower AIC value. One possibly important difference is that the Gaussian has lighter tails (declining square-exponential) than the broken powerlaw model.
z < 0.3 µ 1 = 20.55 ± 0.16 σ 1 = 0.14 ± 0.26 z = 0.3 − 1 µ 2 = 21.42 ± 0.09 σ 2 = 0.54 ± 0.08 z = 1 − 2 µ 3 = 21.77 ± 0.05 σ 3 = 0.50 ± 0.04 z = 2 − 4 µ 4 = 21.63 ± 0.10 σ 4 = 0.61 ± 0.08 z > 4 µ 5 = 21.20 ± 0.40 σ 5 = 0.55 ± 0.28
SingleEllipsoid: A simplistic physically motivated model
While we can model the CDD empirically, we ultimately would like to understand the gas clouds which give rise to the observed column densities. To this end, we present a simple model of a cloud population, which could represent the star-forming region the GRB originated in or the host galaxy. The considered models are gross over-simplifications of the real scenario, which may include multiple absorbers with sub-structure and density gradients sampled in a biased fashion by GRBs. However, as we will show, the simple models are useful to understand the width and shape of the arising distribution.
As an initial toy model, consider a sphere of constant density (radius 1, density 1). For sources distributed uniformly like the gas, the emerging CDD 6 is plotted in Figure 6 (black solid line). The largest possible LOS column density is 2 in these units, corresponding to a full crossing. The most probable column to be observed under random orientations is around 1, with a long tail down to one order of magnitude lower. This scenario is illustrated in the top right corner of Figure 6, with blue crosses indicating the randomly placed sources. Now consider a flatter geometry, an ellipsoid of relative height z/R = 0.1 in cylindrical coordinates, illustrated in red in Figure 6. The red curve in the plot shows the corresponding CDD. Here, the distribution is centred at much lower values, around 0.1 (i.e., close to the vertical extent), and it is also very broad, spanning almost three orders of magnitude.
Such a ellipsoid, representing gas that simultaneously obscures GRBs and hosts their progenitors, forms the baseline model of our approach (inspired by, but a generalisation of Reichart & Price 2002, see also Vergani et al. 2004). But this model cannot 10 −3 10 −2 10 −1 1 2 Column density Frequency z/R=1 (sphere) z/R=0.4 z/R=0.1 z/R=0.04 Figure 6. Normalised column density distributions of ellipsoids. A spherical gas distribution, illustrated in the top right corner, is sampled with uniformly distributed sources (blue crosses). The distribution of their column density in random directions is shown by the thick black line in the plot. In other lines we show the cases for cylindrical symmetric ellipsoids with flatter height z to radius R ratios. For instance, the red solid curve shows the case of a z/R = 0.1 ellipsoid, also illustrated in red. N H along major axis 10 20 − 10 26 cm −2 log-uniform σ scatter of log N H major 10 −3 − 1 log-uniform z/R height/radius ratio 10 −3 − 1 log-uniform match the data: the derived CDD in Figure 5 is broader and less peaked than those of Figure 6). Actually, it would be very surprising if it did match, because that would imply that all gas clouds in which GRBs reside have the same mass and geometry. We thus define the first physical model to be a population of ellipsoids with the same height/radius ratio, but with a Gaussian distribution of total gas densities. The variance of the population in their column density along the major axis is defined through the para-
meter σ = var(log N major H )
. This is mathematically equivalent to convolving the distributions of Figure 6 with a Gaussian of width σ. Subsequently, this model is referred to as the SingleEllipsoid model. The parameters are summarised in Table 5. We constrain them from the redshift subsample in the same fashion as in the previous section.
The parameters exhibit strong degeneracies between the z/R ratio and N H along the major axis, illustrated in the left panel of Figure 7. The black and red dots indicate the sphere and disk geometries discussed before in Figure 6. However, all the possibilities in this degeneracy yield relatively similar CDDs, shown in to the data than the empirical Gaussian mixture or Broken Powerlaw models, and is also preferred by the AIC.
More importantly, however, the SingleEllipsoid model allows us to derive the CDD as it would be seen from the centre of the cloud. This is shown in Figure 9. The central CDD spans the N H = 10 21 −10 23 cm −2 range, with no Compton-thick lines of sight. Under this SingleEllipsoid geometry, up to 50% of the sky would appear obscured with N H > 10 22 cm −2 .
We also investigated possible density gradients by using several co-centred ellipses, each having free shape and density parameters. Such a model can, given enough components, reproduce any monotonically declining density profile. It should also be noted that by embedding a small, dense component arbitrarily large cent- ral column density distribution can be produced. However, we find that such complications are not justified by AIC model comparison.
Redshift evolution
We investigate evolutionary trends with redshift by adopting independent column density distributions in each of five redshift bins,
p(N H , z|M X,z , θ) = p(N H |M X , φ 1 ) z < 0.3 p(N H |M X , φ 2 ) 0.3 < z < 1 p(N H |M X , φ 3 ) 1 < z < 2 p(N H |M X , φ 4 ) 2 < z < 4 p(N H |M X , φ 5 ) z > 4 .(4)
In each redshift bin we adopt the Gaussian or Broken Powerlaw empirical models. This is chosen despite the above finding that the SingleEllipsoid model is a better fit; the empirical models provide a sufficient characterisation of the CDD and are faster to evaluate. Their parameters are also easier to understand and to compare with other works. The constrained parameter values, using the complete sample, are listed in Table 4 on page 7. The last column lists the AIC model comparison values relative to the single broken powerlaw model (lower is better). The redshift-independent broken powerlaw is preferred over the redshift-dependent variant. For the Gaussian model, the redshift-dependent variant is preferred. This is caused by the significantly lower average column density in the lowest redshift bin z < 0.3. Such very local LGRBs are dominated by low-luminosity afterglows which may be a distinct population (Dereli et al. 2015). Uncertainties however remain substantial as few LGRBs exist at low redshifts. At higher redshifts we find no evidence of any redshift evolution, with the CDD always centred at N H ≈ 10 21.4−21.8 cm −2 with small uncertainties. If any redshift evolution exists there, its effect on the mean column is less than a factor of 3. At very high redshifts (z > 4) the evolution LGRBs originating in low-mass host galaxies (cyan, M < 10 10 M ) show lower column densities than those in high-mass galaxies (red, M > 10 10 M ). For the SHOALS sample Spitzer observations were used to derive stellar masses (Perley et al. 2016c). In grey, the SingleEllipse model for all LGRBs is shown. is again uncertain because the imprint of absorption is redshifted below the X-ray regime. The uncertainties in Table 4 indeed highlight that the strongest constraints come from the z = 0.3 − 4 redshift range. If we replace the model in that range with a SingleEllipsoid model, the uncertainties of Figure 7 shrink and the spherical obscurer is ruled out with 99% probability. We also note that the redshift-independent models are always preferred, i.e., no significant redshift evolution is found, when adopting the SHOALS sample. LGRBs had dusty/obscured afterglows. Their SHOALS sample was constructed only using observability criteria, i.e., quantities that are unrelated to the host galaxy mass and LGRB obscuration (see Perley et al. 2016b, for details). This subsample selection was then targeted with very deep follow-up observations to derive redshifts and host galaxy masses. We adopt their highly redshift-complete sample and split it at M = 10 10 M into a low-mass and high-mass subsample (data are described in more detail in Section 3). We analysed each subsample with the Gaussian model. We find that the column densities are drastically different between the low-mass and high-mass samples, as illustrated in Figure 10: The high-mass subsample shows a five times higher mean column density. In other words, high-mass host galaxies are preferentially associated with obscured LGRBs, while a sizeable fraction of low-mass host galaxies have unobscured
Host mass dependence of the N H distribution
LGRBs. The corresponding parameter values are listed in Table 6.
A mass -column density relationship
We have now established that the primary driver of the diversity of LGRB column densities is host galaxy mass, not evolution with redshift. Consequently we fit a model for the distribution that is stellar mass dependent. We convert the Spitzer 3.4µm magnitudes given in Perley et al. (2016c) to masses according to their model. Adopting instead the relation of Meidt et al. 2014 does not change our results significantly. Figure 11 shows our data in stellar mass -N H space. A clear increase in the column density with mass can be observed, with no LGRBs in host galaxies of M < 10 9 M exhibiting N H > 10 22 cm −2 , while such sources exist for more massive hosts. However, there is substantial scatter in the diagram. We first fit a powerlaw model including a systematic Gaussian scatter for log N H . Our fitting method is as before and takes into account the upper limits, but the CDD model is now mass-dependent. Our relationship can be written as:
log N H = 21.7 + 0.38 · log M /M − 9.5 (5) Figure 11 plots the relation (red dashed line) with its data uncertainties (grey shading). The data uncertainties are for the intercept N H = 21.67 ± 0.06 and the slope 0.38 ± 0.06. The determined exponent of approximately 1 3 is noteworthy. The powerlaw relation of equation 5 connects the line integral N H on the left with the volume integral M on the right hand side. If both simply scale geometrically with galaxy size (N H ∼ r, M ∼ r 3 ), and the obscuration is primarily due to the host galaxy, a exponent of 1 3 is expected. We have simultaneously constrained the remaining intrinsic scatter as a normal distribution. Its standard deviation is σ = 0.49 ± 0.05 around the relation, shown as a blue error bar on the right of Figure 11. If instead of a normal distribution we adopt the SingleEllipse model, the scatter is consistent with zero. In other words, the observed scatter can be fully explained by the mass distribution and geometric effects.
Redshift incompleteness bias
Many previous works have only considered
LGRBs with determined redshifts, which is liable to introduce a bias against faint, dust-extincted hosts. To investigate the nature of this redshift incompleteness bias, we analyse the redshift subsample, i.e., limit ourselves to LGRBs with determined redshifts in the z = 0.3 − 3.2 range. The derived Gaussian model parameters are listed in Table 6. Compared with the complete sample, the N H distribution is centred at lower column densities. This indicates a bias against obscured
LGRBs. In fact, the derived parameter values are most similar to the low-mass subsample, with the break and low-N H slope having the exact same values, and the high-N H slope falling within 1σ of the uncertainty. This finding reproduces early works which used only LGRBs with determined redshifts and found that these only occur in low-mass host galaxies: The bias on the log N H distribution appears as if only low-mass host galaxies had been selected. Red squares indicate constraints in mass and column density, green downward pointing triangles are upper limits in column density, grey triangles indicate upper limits in both. Columns thicker than N H = 10 22 cm −2 are only observed for galaxies more massive than a billion suns. A powerlaw fit is shown as a red dashed line. Grey indicates the uncertainty around the slope, while the blue error bar indicates the systematic scattering around the powerlaw for individual objects.
DISCUSSION
The column densities of the GRB population
We have analysed the column density distribution of GRBs as probes of the gas distribution in their host galaxies. We used stateof-the-art statistical methods to propagate uncertainties in the spectral analysis and redshift into the population analysis, while remaining careful of biases from incomplete redshift information.
In the complete sample, most of the 512 GRBs have column densities below 10 21.5 cm −2 , with the most extreme spectrum showing 10 23 cm −2 (see Figure 4). The population can be empirically fitted using a broken powerlaw distribution which shows a steep decline towards high obscurations (slope of ∼ −1.2) and a long tail towards low obscurations (slope of 0.75), spanning the 10 20−23 cm −2 range. Thus, heavily obscured (e.g. Compton-thick) LGRBs, if they exist at all, must be extremely rare. They have not been seen although XRT is sensitive enough to detect 7 and characterise them (see Section 4). Using model selection we concluded that a better empirical description is provided by a normal distribution centred at log N H = 21.6 with intrinsic scatter of σ = 0.6.
In and c = −0.78 +0.42 −0.26 . This is approximately the same peak as found in this work, but they find a steeper decline towards low-N H (b) and a shallower decline towards high-N H (c) in the population. This difference is probably due to the handling of the uncertainties in X-ray spectra and the population analysis. Many sources in their as well as our analysis show large uncertainties in N H as derived from spectral analysis. Errors such as in 2.4 +1.7 −1.5 × 10 21 cm −2 are common, and essentially include the possibility of negligible intrinsic obscuration. Notably the lower error estimate often includes values one or two orders of magnitude lower, while the upper error only doubles the value. In log-space, the best-fit N H estimator is thus biased towards the upper limit. This work adopts a Bayesian methodology to propagate the uncertainties into the population analysis, which also allows us to treat upper limits consistently.
L o w -M a s s H o s t s H ig h -M a s s H o s t s
The obscuration of
LGRBs depends strongly on their host galaxies.
LGRBs in high-mass galaxies show higher absorbing columns by ∼ 0.7 dex versus those originating in low-mass galaxies, as shown in Figure 10. This agrees with the findings at optical wavelengths of Perley et al. (2016c), where massive host galaxies are virtually always associated with absorbed/dusty afterglows. This suggests that the obscurer may be primarily the host galaxy itself, with high-mass galaxies being capable of attracting and holding larger quantities of gas (see more discussion in the next section). Importantly, this biases the results when incomplete samples are used: Campana et al. (2012) noted the bias of Campana et al. (2010a), which appears when considering only LGRBs where the redshift is determined, as dust-extincted afterglows are fainter and often are harder to obtain spectra of. They use a unbiased sample of 58 bright LGRBs and find similar results to Campana et al. (2010a), when comparing a Gaussian fit. This work adopted the SHOALS sample, which is similar in spirit but larger in size (112 objects, Perley et al. 2016b). Using newer spectral models which incorporate effects relevant at high obscuring columns and improved spectral analysis methodology we are able to make stronger inferences in the derived column density distribution (Table 4). We find that the bias of considering only LGRBs with determined redshifts is severe, and that it approximates the exclusion of all massive host galaxies (see Table 6). The aforementioned effects can be seen in the left panel of Figure 12, where we compare the redshift sample to the complete sample. Campana et al. (2010a) and Campana et al. (2012) also investigated a possible redshift evolution of the obscuration. This is interesting because star forming regions at high redshift, particularly at the peak of star formation at z = 1 − 3, may be more compact. They claimed that high-redshift GRBs (z > 4) are more obscured than low-redshift GRBs. This is based on a KS-test which yields a p-value of 0.08 that the best-fit column densities are drawn from the same distribution. P-values are uniform random variates, such that the frequency of yielding such a result or a more extreme one is high (10%, but increasing with the number of tests performed), indicating a substantial probability of a false positive. Campana et al. (2012) makes more cautious claims due to the smaller sample size of their unbiased sample. Even if significant, the best-fit N H values cannot be drawn from the same distribution in principle, because the spectral window probed is different. Furthermore, splitting the sample is problematic because a high percentage of GRBs have uncertain redshifts. To overcome the limitations of the KS test, in this work we simultaneously fitted independent distributions in 5 redshift intervals (z < 0.3, z = 0.3−1, 1−2, 2−4, z > 4) and compared their parameters. We find consistent parameters (see Table 4) in the relevant redshift bins, indicating no redshift evolution around the peak of star formation. If any redshift evolution of the obscurer is present, it is limited to modifying the obscuring columns by a factor of 3, and thus less important than the host galaxy mass. An exception is the z < 0.3 redshift bin, which shows lower obscuration on average. This may be explained by a dominant low-luminosity GRB population in that redshift range, which form a distinct population (Dereli et al. 2015). Alternatively, it could be a side-effect of galaxy-mass downsizing, which is more pronounced in GRBs (Schulze et al. 2015;Perley et al. 2016c) than in the general galaxy population (e.g. Fontanot et al. 2009).
LGRB obscurer models
Reichart & Price (2002, RP02 hereafter) developed a obscurer model based on the distribution of molecular clouds in the Milky Way. Their mean radial column densities are N H major ≈ 10 22 cm −2 with a scatter of 0.2 dex in their population. Such a cloud distribution, when including random placement and orientation in such clouds, was found to be consistent with observations in the analysis of RP02 with 15 GRBs, and also in the analysis of Campana et al. (2006) and Campana et al. (2010a) which included Swift observations. Vergani et al. (2004) developed a multi-component gas model of the Milky Way and simulated the LGRB column density distribution with ray-tracing. They however assumed that a large portion of LGRBs may occur in diffuse gas. This includes the disk, leading to a high percentage of LGRBs with low column densities. Campana et al. (2006) ruled out that model based on their derived column density distribution, and concluded that LGRBs likely originate in molecular clouds (the remaining model). A limitation of the RP02 molecular cloud model is that it is based on the Milky Way, which is atypical in mass and metallicity for LGRB host galaxies. In this work we developed a more general approach by deriving the properties of the LGRB obscurer population from the data.
We find that the column density distribution of LGRBs can be well-described by a simple model: a single gas component of uniform density, in which LGRBs are randomly located. The geometry and major axis column density of the cloud population were tentatively constrained (see Figure 7) to a flat disk with a height-toradius ratio of 1 : 20 and a major axis column density of N major H ≈ 10 23 cm −2 . To explain the broadness of the column density distribution, the best-fit model has a scatter in log N major H of σ ≈ 0.22 ± 0.14 (the 10% quantile is at 0.03). For comparison, the RP02 model used an essentially flat distribution of scatter σ = 0.2 dex between molecular clouds in the Milky Way. The clouds in the RP02 model have a density gradient, are spherical and therefore match the data with a lower maximal column density. The right panel of Figure 12 shows the RP02 Milky Way model in comparison to our results using the high-mass and low-mass subsamples, which probe stellar masses comparable to, and below that of the Milky Way. The RP02 model falls in between the constraints from those samples, implying that LGRBs in Milky Way-size galaxies are more obscured than what the RP02 Milky Way model predicts. Figure 12 also compares the multi-component model of the Milky Way gas components by Vergani et al. (2004). That model assumes that GRBs can also originate in the atomic hydrogen of the thin disk, which leads to many GRBs with low column densities. This model is clearly ruled out, leaving the origin of GRBs in galactic molecular clouds as a plausible scenario. In that case however the column density of the molecular clouds would have to increase with galaxy host mass, which is not the case in nearby galaxies (Larson 1981;Bolatto et al. 2008;Lombardi et al. 2010). The geometry constraints suggest another . Column densities of local galaxies. We derived the densest column densities seen in the LMC and SMC, by taking the 99% highest values from HI radio maps and correcting for metal abundances to derive a N H as would be seen by X-ray observations. For the Milky Way (MW) we use the map of Dickey & Lockman (1990). The obscuration of galaxy gas of each of these local galaxies falls exactly on the relationship we derive from GRB obscuring column densities (black line). possibility however: giant molecular clouds could be arranged in a relatively flat disk in which the GRBs are produced. The number of clouds (and thus the major axis density) should then scale with the size of the galaxy, as more massive galaxies can hold larger quantities of gas. This scenario of the galaxy acting as the primary obscurer appears more likely due to the N H 3 ∝ M relation found (Equation 5 in Section 4.5).
Local Galaxies as Obscurers
We verify the mass dependence of the obscurer by determining how well local galaxies act as obscurers. The Milky Way, the Large Magellanic Cloud (LMC) and the Small Magellanic Cloud (SMC) cover a large stellar mass range and their column density of N H is well mapped. Our relationship depicted in Figure 11 predicts that at Milky Way stellar masses, M = 6 × 10 10 M (McMillan 2011), the average galaxy should have some LOS column densities above N H > 10 22 cm −2 . Galactic column density maps of the Milky Way, as depicted in Figure 1, show that a small fraction of the sky (∼ 1%) is indeed obscured with N H > 10 22 cm −2 as seen from our vantage point. The fraction may be larger from more central regions of the Galaxy, or from the vicinity of blue, star forming regions where LGRBs are typically found (Bloom et al. 2002;Fruchter et al. 2006). The Milky Way however lies at the massive extreme when compared with the host galaxies of LGRBs. The LMC is perhaps a more appropriate galaxy to consider, with a stellar mass of M ∼ 3 × 10 9 M . Its observed LOS column density reaches values up to N H ≈ 3·10 21 cm −2 , with the star-forming region 30 Dor reaching N H ≈ 9×10 21 (Brüns et al. 2005). Using the ellipticity ( ≈ 0.3) and inclination (i ≈ 30°) of the LMC (see van der Marel 2006, for a review of various measurements) the major axis column density should be lower than the observed column density by a factor of 60%. Therefore, from the centre of the LMC the entire sky has LOS column densities below N H < 10 22 cm −2 . That is again consistent with our N H 3 ∝ M relationship, which predicts that such sight lines should be very rare for LMC-mass galaxies. LGRBs where the He and metal abundances could be constrained. The 3-sigma contours (orange thick lines) of six spectra exclude metal-poor and He-abundant solutions. For GRB 090618 (red thick line), metal-rich abundances are excluded. All shown LOS's are consistent with the abundances of the local ISM (cross, Wilms et al. 2000). A metal-free, Helium-dominated absorber can be rejected at high significance.
LGRBs marked with asterisks show super-galactic N H /A V ratios.
N HI is higher than in the LMC, reaching N H = 10 22 cm −2 (Brüns et al. 2005). One should keep in mind that for the SMC and LMC we relied on HI columns derived from radio observations, whereas everywhere else in this paper we consistently use X-ray determined metal columns converted into N H assuming solar abundances. Therefore the N H values we should compare with for the SMC and LMC -their metal columns -actually should be lower given their low gas metallicity of 0.2 and 0.5 solar, respectively (Tchernyshyov et al. 2015). Taking the metal abundance into account, we contrast these three local galaxies (MW, LMC and SMC) in Figure 13 against our relationship and find excellent agreement.
A Helium-dominated, circum-burst absorber?
An open issue in the understanding of LGRB afterglow is the inconsistency between optically and X-ray-derived absorption. The restframe UV extinction, A V , and the X-ray derived column density, N H , show a tendency toward lower-than-galactic ratios and broad scatter (e.g. Schady et al. 2011;Watson et al. 2013). This can be caused by deviations in abundances, dust-to-gas ratios and/or ionisation states compared to the galactic ISM. To resolve this inconsistency goes beyond the scope of this work. However, we study one recently proposed explanation in detail. Watson et al. (2013, W13 hereafter) postulated that gas in the vicinity (< 30pc) of the burst provides the bulk of the X-ray absorption. The powerful burst emission can destroy dust, fully ionise all hydrogen atoms (which have a low ionisation energy) and O and Fe atoms (which are few in number) along the LOS. These atoms would therefore not absorb X-rays. However, for certain luminosities, not all He atoms would become ionised, because of their large number. W13 showed that a He-dominated X-ray absorption spectrum is observationally indis- LGRBs where the X-ray afterglow spectrum exceeds 1000 counts. The circles indicate the same objects shown in Figure 14, where there is evidence of metals in the X-ray absorption. The red arrow indicates a reduction in the effective column density by a factor of seven if dense circum-burst material were ionised by highly energetic bursts. However, no such systematic shift is apparent when comparing to lower energy bursts.
tinguishable from one with local ISM abundances in typical X-ray spectra.
In a few cases where photon statistics are robust, the effective abundance of metals and helium can be constrained directly. We set the helium and metal abundances each as free parameters in our fit (TBVARABS model, Wilms et al. 2000), and analysed all LGRB XRT spectra with more than 5000 counts (42 sources). The normalisation, photon index and column density were also free parameters. In seven cases, shown in Figure 14, the abundances could be constrained. These have more than 30000 photon counts each. We comment on some individual sources in Figure 14 in detail: We note that Giuliani & Mereghetti (2014) found a metal-free solution (Z/Z < 0.05) when analysing a XMM-Newton spectrum of GRB 120711A. The difference may be because we use a slightly higher galactic column density of 1.06 × 10 21 cm −2 following Evans et al. (2009) and more recent absorption model and cross-sections in our fit. Our abundance contours for GRB 090618 are consistent with those from the XMM-Newton spectral analysis of Campana et al. (2011). They could also place a lower limit of Z/Z > 0.2. For GRB 130925A and GRB 130907A we additionally excluded the (early-time) WT mode spectra that may still be, despite our efforts of Section 3.2, slightly contaminated by prompt emission. Nevertheless we obtain constraints with the (late-time) PC mode data alone. For GRB 130925A, we note that Schady et al. (2015) derived super-solar metallicity from optical spectroscopy. Particular noteworthy is GRB 130702A. This LGRB is associated with a supernova hosted by a dwarf galaxy with M ≈ 10 8 M (Kelly et al. 2013), and hence represents a typical LGRB. In all cases, the constraints exclude a He-dominant absorber but are consistent with the abundances of the local ISM.
Additionally, if the LGRB is responsible for ionising substantial fractions of the absorber, the effective absorbing column
Illustris-1 simulation
LGRBs intrinsic relation LGRBs are overlaid as red error bars (same as in Figure 11). While in the left panel redshift snapshots have been combined according to the SHOALS redshift distribution, the right panel shows the median as a function of redshift (thick lines). Dotted lines indicate the effect of only using simulated galaxies with sub-solar stellar metallicity. In red, our N H − M relation is shown. The ellipse indicates (for z = 0) a sub-population in the simulation showing elevated obscuring columns. These are predominantly star-burst galaxies.
should be reduced for more energetic bursts. Campana et al. (2012) find no significant difference in the column distribution of bright bursts. In Figure 15 we show the distribution of sources in isotropic energy and effective column density N H for sources with redshift information. The isotropic energies were computed using the method of Bloom et al. (2003) 8 . Effective column densities were derived using a local ISM abundance, neutral absorber model. The SPHERE model is used, which is valid also for the highest absorbing columns, as discussed in Section 3.3. There does not appear to be a burst energy-dependence in the absorber properties. If the burst energy ionised metals and hydrogen, reducing the effective column density by a factor of seven (example calculation in W13, red arrow), we would expect a deficit of sources in the upper right quadrant of the plot. However, no deficit of high-obscuration sources at the luminous end is apparent in Figure 15, and the column density distribution appears independent of energy. In Figure 15, only GRBs for which redshifts have been determined are shown, which reduces the number of faint, obscured bursts (lower right quadrant). To avoid such biases, black crosses show LGRB from the SHOALS survey. Additionally, metals have been detected in highly-absorbed, energetic bursts: Orange circles in Figure 15 show the same sources as in Figure 14.
The correlation of column density and stellar mass dependence is a strong argument that the X-ray column density is predominately due to the host galaxy-scale gas. The lack of energydependence of the absorber support the dominance of a distant obscurer. Abundance measurements suggest that the X-ray obscurer can be modelled similar to the local ISM. Under local ISM abundances, the dominant absorbers are Fe and O. Partial ionisation of metals may still be present and account for deviations in A V , but its effect on the X-ray spectrum appears negligible. Furthermore, relatively low galaxy-scale column densities can occur if the LOS does not pass through the galaxy (left tail in Figure 6, particularly objects below the M − N H relation in Figure 11). In these cases, the dominant obscurer could be the local environment, where hard burst radiation destroying dust may reduce the A V /N H ratio. However, fully resolving the discrepancy between N H and A V measurements is beyond the scope of this work.
Obscuration in simulated galaxies
At higher redshifts the gas content of galaxies is not easily accessibly through observations. Instead we turn to simulated galaxies from hydrodynamic cosmological simulations. This exercise is potentially predictive because the amount of gas inside galaxies is constrained by the simulation's requirement to start with the Big Bang's density and to reproduce today's stellar mass function. In galaxy evolution models, the massive end of the existing stellar population expels metals into the galaxy. The metal gas produced per stellar mass is determined by the chosen IMF and the metal yield, with the latter tuned to reproduce the stellar mass function (Lu et al. 2015). The total metal gas mass residing in galaxies further depends on the chosen feedback models, which can expel gas out of the galaxy. Typically the metal gas mass inside galaxies follows a M Z /M = 1 : 30 − 1 : 100 relation in semi-analytic models at z = 0 − 3 (e.g. Croton et al. 2006Croton et al. , 2016. The crucial remaining question surrounds the arrangement of that gas inside galaxies, as the concentration of gas defines its column density -this requires hydrodynamic simulations.
The Illustris simulation (Vogelsberger et al. 2014b,a) is a cosmological hydrodynamic simulation which attempts to reproduce the galaxy population using state-of-the-art star formation, supernovae and AGN feedback mechanisms inside dark matter haloes. Illustris reproduces many observed quantities; most relevant for this work it reproduces roughly the stellar mass distribution of galaxies, their morphology, and gas content from CO observations (Vogelsberger et al. 2014b;Genel et al. 2014). The gas particle resolution in Illustris is adaptive, with some cells being as small as 48pc in the highest resolution simulation (Illustris-1) used here, indicating that today's cosmological simulations indeed resolve galaxies into small substructures.
We apply ray-tracing, treating each simulated galaxy (subhalo) separately. The starting point is the densest region, presumably representing a region of star formation. From that position, we radiate along random sight-lines all metal gas bound to the subhalo. In Illustris, gas is represented by Voronoi cells, therefore we Voronoi-tessellate the ray and assign each part the corresponding cells density and finally sum to a total metal column density. We then compute a equivalent hydrogen column density distribution by converting under Wilms et al. (2000) solar abundances to N H . This mimics how N H is derived in X-ray observations. We adopt h = 0.7 and work in physical units at redshift slices z = 0, 1, 2 and 3. We investigate all galaxy subhaloes with M > 10 9 M (as smaller galaxies are difficult to resolve). The first question to address is whether the gas in simulated galaxies reproduces the same N H values as observed. The left panel of Figure 16 shows several individual LGRBs from the SHOALS sample, specifically those with host mass and N H measurements. Overlaid are the results of the simulation snapshots, redshift-weighted according to the SHOALS redshift distribution. Grey shading represents the distribution of the median N H of individual simulated galaxies. We find that the observations overlap well with the simulations under the assumption that LGRBs originate in dense regions of galaxies. We also find that the simulations predict a diversity of galaxies with a scatter of ∼ 0.5 dex in N H , in agreement with our observations. We test the importance of the immediate vicinity to the N H by excluding the inner 100 pc radius, which reduces N H by a factor of 2 on average. This indicates that distant, i.e., galaxy-scale obscuration is important.
In both the observations and simulations, some rare objects occupy the upper left quadrant of Figure 16, showing high obscurations N H > 10 22 cm −2 despite low masses M < 10 10 M . In the simulations, these galaxies have high star formation rates, with the majority being starburst galaxies and many having recently experienced mergers. Figure 16 indicates the relevant range with an ellipse for z = 0 (other redshifts are slightly higher).
The right panel of Figure 16 depicts the redshift evolution of simulated galaxies. In Illustris, z = 1 − 3 galaxies have a slightly higher specific gas content, which affects the median column densities. At low redshifts (z = 0 − 1), massive galaxies in particular loose gas by strong feedback from active galactic nuclei to avoid over-production of massive galaxies. From our observations we ruled out a redshift-dependent trend of LGRBs in Section 4.4 9 . Furthermore, at each redshift the slope of the N H − M relation derived from Illustris is substantially steeper than 1 3 . These two issues indicate that the Illustris simulation may not represent the gas in galaxies correctly, because it predicts substantially more obscured LGRBs and higher columns for LGRBs in massive galaxies. If remaining concerns e.g., regarding absorber geometry and substructure can be addressed, our X-ray tomography of galaxy gas could be used in the future to distinguish (the strength of) feedback models and star formation efficiencies (see also Paper II). Alternatively, the lower observed obscuration may be due to a environmental preferences of LGRBs inside galaxies, such as a metal/dust aversion. 9 The same result occurs when adding a redshift-dependence to the N H − M relation; the individual constrained SHOALS LGRBs also do not appear to follow that redshift evolution.
SUMMARY
We analysed a large sample of Swift-detected long-duration Gamma-Ray Bursts using modern statistical techniques, incorporating the uncertainties from spectral analysis and investigating the effect of redshift incompleteness from dust-extinct/dark LGRBs. Our findings can be summarised as follows:
(i) The column density of the LGRB population lies in the 10 20−23 cm −2 range and can be described by a normal distribution.
(ii) A well-suited model for the column density distribution is a axisymmetric ellipsoid of gas with randomly placed GRBs within. This set-up generalises previous models based on the giant molecular clouds of the Milky Way. Those in fact have lower column densities than observed from GRBs in host galaxies of similar mass. Permitted solutions for the obscuring clouds include a degenerate range of densities and flatness (∼1:20). Additionally it is necessary that the gas ellipsoid population has a distribution in its total gas density of about 0.22 dex.
(iii) We systematically search the Swift archive for evidence of heavily-obscured LGRBs. We note that such LGRB could have been detected and characterised by Swift/ XRT given their intrinsic X-ray luminosities, but are not observed.
LGRBs therefore do not reach heavily obscured column densities of N H > 10 23 cm −2 .
(iv) The column density of LGRBs shows no significant evolution with redshift. If present its effect is at most a factor of 3.
(v) LGRBs in galaxies of high stellar mass show substantially more obscuration. We find a novel relation: N H = 10 21.7 cm −2 × M /10 9.5 M 1/3 (vi) The scatter in column densities can be fully explained by the mass-dependence (v) and geometric effects (ii).
(vii) We argue based on the mass-dependence of the obscuration and the derived geometry of the obscurer as well as analysis of well-mapped local galaxies, that the obscurer is predominantly the GRB host galaxy itself.
(viii) This conclusion is corroborated by investigating the metal gas mass in simulated galaxies. These predict the same magnitude of obscuring X-ray column densities, similar scatter as well as a mass-dependence, although of a steeper slope. tion package CIAO and Sherpa. Additionally, the BXA 10 , PyMul-tiNest 11 , Astropy (Astropy Collaboration et al. 2013) and Cosmo-loPy 12 software packages were used. Tanvir N. R., Wiersema K., Levan A. J., 2010, GRB Coordinates Network, 11230 Tanvir N. R., Levan A. J., Cucchiara A., Fox D. B., 2012a, GRB Coordinates Network, 13251 Tanvir N. R., et al., 2012b, GRB Coordinates Network, 13348 Tanvir N. R., Levan A. J., Krogsrud D., 2012c Tanvir For completeness, we report the intrinsic photon index distribution of GRB afterglows, which is a side-products of our analysis. The X-ray spectrum of the complete sample was analysed to obtain constraints on the slope of the intrinsic powerlaw. The constraints for the complete sample is shown in Figure A1. We approximate the distribution by a Gaussian distribution, plotted in red. The overall distribution is centred at Γ = 1.94 with a width of σ = 0.20. Table A1 shows the mean and standard deviations obtained from the various samples. The Gaussian distribution provides a good approximation in the bulk of the population. However the data show heavier tails, including Γ < 1.5 and Γ > 2.4. The individual spectra show no obvious signs of bad fits in either extreme. The high-mass subsample and redshift subsample show steeper slopes with a narrower distribution, but the difference is smaller than the uncertainties. Curran et al. (2010) investigate the photon index distribution based on the automatic fitting results of Evans et al. (2009) of 301 GRBs. They find a peak at Γ ≈ 2.1, and the distribution spreads the full range of 1 − 3. In contrast, we find that the standard deviation of the distribution is much narrower (σ = 0.2) than their analysis suggests (σ ≈ 0.5). This is probably because of their use of best-fit values, which introduce additional scatter, and the fact that the completely automated analysis of Evans et al. (2009) sometimes chooses time windows affected by prompt emission while we manually verified each time window. Wang et al. (2015) performed temporally resolved fitting of X-ray and optical data and found Γ = 1.98 with standard deviation σ = 0.15, which is consistent with our results.
1.0 1.5 2.0 2.5 3.0 Photon Index Gaussian fit Spectral fit Figure A1. Caterpillar plot of the photon index. Constraints on the photon index of the intrinsic powerlaw are shown as error bars for each object, sorted by their mean. During the population analysis we assumed a Gaussian for the distribution of photon indices (red). The spectral index β is related to the photon index as β = Γ − 1. Curran et al. (2010) then considers two regimes in which synchrotron radiation is produced: (a) The electron energy distribution index p relates to the spectral index (β = Γ − 1) as p = 2 · β when the cooling frequency is below the X-ray frequency (ν c < ν X ), and (b) as p = 2 · β + 1 otherwise (β = (p[−1])/2 + 1. Adopting a Gaussian distribution for p centred at, e.g., 2.2 then yields a doublepeaked distribution for the photon index centred at Γ = 2.1, with a secondary, small component (case b) offset by 0.5. Here, we find additional contributions at both lower and higher photon indices which necessitate a different model than two Gaussian peaks, and we can therefore not unambiguously conclude that ν c > ν X applies in most cases. Nevertheless, if that is assumed, we find p = 2 for the electron density distribution index distribution with a standard deviation of 0.4.
APPENDIX B: X-RAY BACKGROUND MODEL
The shape of the XRT background has been analysed by Pagani et al. (2007) (see Figure 2 there). It shows a steep increase below 0.5 keV and several bumps. The background spectra analysed in this work show the same shape. We fit the background spectrum with a broken powerlaw model and four Gaussian components at ∼ 0.7, 2.2, 1.2 and 0.4 keV, by order of importance. The parameters of this model are optimised according to the Poisson likelihood. In further analysis, the background model parameters are held fixed, and the background model is added to the source spectral fit, scaled by the area ratio of the spectral extraction regions. Our code and model for fitting the Swift/ XRT background is available as part of the BXA software.
Table 3 .
3Catalogue (excerpt). Columns: (1) Name. (2) Duration (-1 indicates unknown). (3) Right ascension (J2000) in degrees. (4). Declination (J2000) in degrees. (5) Galactic latitude in degrees. (6) Galactic absorption in 10 20 cm −2 .
Figure 4 .
4Redshift z and column density N H distributions of the redshift subsample. Blue circles with error bars show objects with upper and lower limits; the circle indicates the posterior median, while the error bars show the 90% posterior probability quantiles. The black crosses show the Galactic column density corresponding to each object. Grey circles and crosses show the same, but for unobscured GRBs which only have upper limits. The red squares at the top indicate GRBs where no X-ray information is available, either because no afterglow was detected or no time intervals are free of prompt emission and flares. They are placed here at an arbitrarily for visualisation, but in the population analysis all N H values are considered possible for these sources.
Figure 5 .
5Reichart & Price (2002) and Campana et al. (2010a) define a broken powerlaw model (dashed line fit) as 5
Figure 5 .
5Column density distributions of Swift-detected LGRBs. Error bars show 2σ equivalent quantiles.
Figure 7 .
7. The population scatter σ is constrained to 0.22 ± 0.14. This makes the distribution as broad as the empirical model shown inFigure 5. This simple model (SingleEllipsoid) is already a Degeneracy between scale height parameter z/R and the column density along the major axis N major H in the SingleEllipsoid model. The black and red circle indicate the sphere and disk geometries illustrated inFigure 6. The contours encapsulate 50%, 84% and 99% of the probability.
Figure 8 .
8Column density distribution of the SingleEllipsoid model. The dashed line shows the median distribution from the posterior probability distribution. In dark grey shading, the 1σ uncertainty is shown, while light grey shading show the quantiles encapsulating 90% of the probability distribution.
Figure 9 .
9Cumulative column density distribution of the SingleEllipsoid model as seen from the centre. The solid line shows the median distribution from the posterior probability distribution. In dark shading, the 1σ uncertainty is shown, while light shading represent the quantiles encapsulating 90% of the probability distribution. In this model, about ∼ 40% of the sky as seen from the centre is obscured with N H > 10 22 cm −2 , while there are no Compton-thick lines of sight.
Figure 10 .
10Mass dependence of the column density distribution.
Perley et al. (2016c) investigated the galaxy mass of LGRB hosts and found that above M = 10 10 M virtually all of their
Figure 11 .
11Host mass -column density relation. Points are individual GRBs in the SHOALS survey.
a series of papers, Campana and collaborators investigated the column density distribution of LGRBs as a population. Campana et al. (2010a) updated the results ofCampana et al. (2006)
Figure 12 .
12Comparison to literature results. Thick solid lines show the models ofReichart & Price (2002) andVergani et al. (2004) where GRBs are placed randomly in gas components of the Milky Way;Reichart & Price (2002) is restricted to molecular clouds. The cumulative column density distribution from the four samples used in this work is plotted when adopting the Gaussian model (solid: median, shading: 1σ quantiles). Right panel: Selection of low-mass (cyan) and high-mass (red) subsamples show the importance of the host galaxy. Left panel: The complete sample distribution (grey) is very close to theVergani et al. (2004) model. The (biased) redshift subsample (green) lies systematically at lower columns by about 0.3dex, similar to the low-mass subsample in the right panel.and analysed a sample of 93 Swift-detected LGRBs with redshift measurements. They performed a broken powerlaw fit and find the break of the distribution at a = 21.71 +0.14 −0.15 , with slopes b = 1.59 +1.81 −0.57
Figure 13
13Figure 13. Column densities of local galaxies. We derived the densest column densities seen in the LMC and SMC, by taking the 99% highest values from HI radio maps and correcting for metal abundances to derive a N H as would be seen by X-ray observations. For the Milky Way (MW) we use the map of Dickey & Lockman (1990). The obscuration of galaxy gas of each of these local galaxies falls exactly on the relationship we derive from GRB obscuring column densities (black line).
Figure 14 .
14Finally we consider the SMC, with a stellar mass M ∼ 3 × 10 8 M (van der Marel et al. 2009). While that galaxy has a smaller HI massLOS metal abundances for seven
Figure 15 .
15The relation between obscurer and LGRB energy. We plot the effective column density (neutral absorber with local ISM abundances) against the isotropic energy for sources where the redshift is known. Black crosses are from the SHOALS sample, grey crosses denote other
Figure 16 .
16Obscuration in simulated galaxies. Left panel: The grey shaded regions show the 1 and 3 sigma range of the distributions in median N H as seen from the densest region. The black line marks the median. Individual observed
N. R., Fynbo J. P. U.,Melandri A., Levan A. J., Xu D., D'Elia V., 2012d, GRB Coordinates Network, 13890 Tanvir N. R., Wiersema K., Levan A. J., Starling R. L. C., Margheim S., Hayward T., 2012e, GRB Coordinates Network, 13929 Tanvir N. R., Wiersema K., Xu D., Fynbo J. P. U., 2013, GRB Coordinates Network, 14882 Tanvir N. R., Levan A. J., Wiersema K.,Cucchiara A., 2014a, GRB Coordinates Network, 15961 Tanvir N. R., Levan A. J., Coulson D., 2014b
Table 1 .
1Sample selection.Figure 2. Redshift distribution. The histogram of redshifts is shown in black. The grey dotted line indicates a fitted Beta distribution. The red dotted line shows the redshift distribution of the SHOALS unbiased sample, which peaks at slightly higher redshifts (filled circles indicate the respective medians). Most LGRBs are found in the redshift interval z = 0.5 − 3.Sample name
Parent sample
Size
Criteria
Swift
http://www.swift.ac.uk/,
including non-detected afterglows
920
Swift-detected GRBs
Long
Swift
844
T 90 > 2
Complete sample
Long
512
N MW
H
< 10 21 cm −2 and |b > 20°.
Redshift subsample
Complete sample
163
redshift known and z = 0.3 − 3.2
SHOALS
-
119
Perley et al. (2016b)
unbiased sample
SHOALS
119
N MW
H
< 10 21 cm −2 and |b| < 20°.
High-mass subsample
unbiased sample
25
Spitzer 3.6µm band < −22 mag
Low-mass subsample
unbiased sample
94
Spitzer 3.6µm band > −22 mag, or uncertain
0 1 2 3 4 5 6 7 8 9 10
redshift
0
20
40
number
GRB redshifts
SHOALS distribution
Table 2 .
2Parameters of the spectral model and their priors.Parameter
Symbol Prior
Normalisation
A
log-uniform
10 −10 − 10 2 keV −1 cm −2 s −1
Powerlaw Slope
Γ
uniform 1 − 3
Column density (intrinsic)
N H
log-uniform 10 19 − 10 26 cm −2
Galactic column density
N MW
H
log-normal around LAB value;
standard deviation 1
3 ln 3
Redshift
z
fixed if spectroscopy available;
otherwise SHOALS distribution
(Figure 2)
Table 4 .
4Empirical models for the column density distribution for the complete sample.Model
Component
Parameters
N params
∆AIC
Broken powerlaw
Break N H
Low N H slope
High N H slope
5
0.0
log a = 21.87 ± 0.06
b = 0.78 ± 0.09
c = −1.39 ± 0.16
Gaussians
Mean
Standard deviation
4
-2.3
µ = 21.64 ± 0.03
σ = 0.55 ± 0.03
Broken Powerlaws, 5 redshift bins
17
7.2
Break N H
Table 5 .
5Parameters of SingleEllipsoid modelName
Description
Range
Prior
N
major
H
Table 6 .
6Gaussian model parameters for different samples.Parameter
Mean log N H
Std. deviation σ
Complete sample
21.64 ± 0.03
0.55 ± 0.03
Low-mass sub-sample
21.44 ± 0.09
0.65 ± 0.07
High-mass sub-sample
22.13 ± 0.10
0.43 ± 0.09
Redshift sub-sample
21.49 ± 0.06
0.58 ± 0.05
Table A1 .
A1Photon index Gaussian fits. mass sub-sample 1.98 ± 0.04 0.18 ± 0.04 de Ugarte Postigo A.,Thoene C., Lombardi G., Perez A., 2015c, GRB Coordinates Network, 18274 de Ugarte Postigo A., Malesani D., Xu D., 2015d, GRB Coordinates Network, 18426 de Ugarte Postigo A., Pugliese G., Xu D., Malesani D., 2016a, GRB Coordinates Network, 18886 de Ugarte Postigo A., Thoene C. C., Sanchez-Ramirez R., 2016b, GRB Coordinates Network, 18966 van Dyk D. A., Connors A., Kashyap V. L., Siemiginowska A., 2001, ApJ, 548, 224 van der Marel R. P., 2006, in Livio M., Brown T. M., eds, The Local Group as an Astrophysical Laboratory. pp 47-71 (arXiv:astro-ph/0404192), http://adsabs.harvard.edu/ cgi-bin/nph-data_query?bibcode=2009IAUS..256... 81V&link_type=ARTICLE&db_key=AST&high= van der Marel R. P., Kallivayalil N., Besla G., 2009, in Van Loon J. T., Oliveira J. M., eds, IAU Symposium Vol. 256, The Magellanic System: Stars, Gas, and Galaxies. pp 81-92 (arXiv:0809.4268), doi:10.1017/S1743921308028299 von Kienlin A., 2010, GRB Coordinates Network, 11015 von Kienlin A., 2014a, GRB Coordinates Network, 15790 von Kienlin A., 2014b, GRB Coordinates Network, 16152 von Kienlin A., et al., 2014, ApJS, 211, 13 APPENDIX A: PHOTON INDEX DISTRIBUTIONParameter
Mean
Standard deviation
Complete sample
1.95 ± 0.01
0.19 ± 0.01
Redshift sub-sample
1.93 ± 0.02
0.19 ± 0.01
SHOALS
1.95 ± 0.02
0.23 ± 0.02
low-mass sub-sample
1.95 ± 0.03
0.25 ± 0.02
high-
The alternative hypothesis of a He-dominated, ionised absorber(Watson et al. 2013) is discussed extensively in Section 5.4.
See the Appendix of Buchner et al. 2015 for a derivation and how intermediate priors are handled. 3 http://www.swift.ac.uk/
MNRAS 000, 1-22(2016)
Listed separately on the http://www.swift.ac.uk/ website
In previous works the normalising factor ln(10) is erroneously divided, which is important for model comparison.MNRAS 000, 1-22(2016)
Numerical details of the computation can be found in Appendix C. MNRAS 000, 1-22 (2016) 8 Buchner, Schulze & Bauer
In principle, a second class of GRBs could exist behind even more extreme columns (e.g., N H 10 25 cm −2 ) so as to render the GRBs undetectable even in the BAT energy band. This would require a conspicuously bimodal column density distribution which we do not consider probable apriori. Such a high column density would almost certainly have to be local to the GRB.MNRAS 000, 1-22(2016)
In a few cases where the peak energy could not be constrained, the values were taken fromButler et al. (2010).
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This paper has been typeset from a T E X/L A T E X file prepared by the author.MNRAS 000, 1-22(2016)
We thank the builders and operators of Swift. This work made extensive use of data supplied by the UK Swift Science Data Centre at the University of Leicester. This research has made use of software provided by the Chandra X-ray Center (CXC) in the applica-APPENDIX C: NUMERICAL DETAILS ON FITTING THE ELLIPSOIDS MODELSThis section describes how the SingleEllipsoid model is computed. Monte Carlo ray-tracing simulations are used to compute the column density distribution p(N H |M, θ).First, random points inside the ellipsoid are generated. The ellipsoids equation,describes whether a point p = (p x , p y , p z ) is inside. The radii are r x = r y = R and r z = z under cylindrical symmetry. For constant density sampling, (p x , p y , p z ) are first drawn uniformly in p x ∼ U(−R, R), p y ∼ U(−R, R), p z ∼ U(−z, z) and the vector p is rejected if outside the ellipsoid. Second, a random unit direction vector is generated. Three unit normal variates are combined to a vector d ∼ (N(0, 1), N(0, 1), N(0, 1)) which is then normalised to unit length n = d/|d|.Third, the length l of the ray inside the ellipsoid needs to be computed, which is the distance between p and the point where the ray exits the ellipsoid, q. The coordinates of q are governed by Equation C1 and the line equation,which yield the quadratic equationwith l unknown. Equation C3 has two solutions for l (one for the positive, one for the negative direction). Only the positive one iswe can finally write the column probed by the ray as N H = N H major ·l, where N H major is the column density of the ellipsoid in a unit length. The problem is fundamentally degenerate (N H major and size), so R = 1 is assumed for the SingleEllipse model.To compute the column density distribution, p(N H |M, θ), 400000 random rays are generated. A histogram of their N H between 10 19 − 10 26 cm −2 with 100 logarithmically spaced bins provides a well-sampled approximation. A problem may occur with less obscured column densities in the simulation, which can never be measured due to Milky Way absorption. For simplicity, the column density of each ray is modified as N H = N H + 10 u with the random number u ∼ U(19,20)to ensure all rays have N H > 10 19 cm −2 . This redistributes unobscured rays to the range 10 19 − 10 20 cm −2 , uniformly, and, although done primarily for numerical reasons, may be interpreted as placing the ellipsoid in a low-density gas with N H < 10 20 cm −2 .The arising distribution (seeFigure 6) cannot be approximated by simple analytic formulas. Towards low N H values, the distribution rises exponentially. For very low z/R ratios, the distribution declines exponentially toward high N H values, but the peak is too wide/narrow to be fitted by a broken/bending powerlaw. Additionally, for moderate z/R ratios, there is a steep truncation at N H = 2 (seeFigure 6) which declines faster than a exponential cut-off.To emulate a dispersion in the population of log N H major of standard deviation σ, the histogram is convolved with a Gaussian. Finally, linear interpolation of the histogram is used to evaluate p(N H |M, θ) at arbitrary N H values.The MultiEllipsoid model generates points proportional to the mass in each ellipsoid, which isThe ellipsoid to draw from is chosen randomly in proportion to its masses M. Rays now may probe multiple ellipsoids, and the final N H is the sum of all ellipsoids encountered. The definition of l has to be modified because some rays may not originate inside the ellipsoid at hand (checked with Equation C1), but cross it. In that case the distance l is between the two quadratic solutions, giving l cross = 2 · √ d/a.Otherwise, the same procedure as in the SingleEllipsoid model is applied.To simulate the column density from the centre, p = (0, 0, 0) is set fixed, and the procedure of generating random rays is applied in the same fashion.
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| [] |
[
"Investigation of two-color magneto-optical trap with cesium 6S 1/2 -6P 3/2 -7S 1/2 ladder-type system",
"Investigation of two-color magneto-optical trap with cesium 6S 1/2 -6P 3/2 -7S 1/2 ladder-type system"
] | [
"Jie Wang \nState Key Laboratory of Quantum Optics and Quantum Optics Devices (Shanxi University)\n\n\nInstitute of Opto-Electronics\nShanxi University\n\n",
"Guang Yang \nState Key Laboratory of Quantum Optics and Quantum Optics Devices (Shanxi University)\n\n\nInstitute of Opto-Electronics\nShanxi University\n\n",
"Baodong Yang ",
"Jun He \nState Key Laboratory of Quantum Optics and Quantum Optics Devices (Shanxi University)\n\n\nInstitute of Opto-Electronics\nShanxi University\n\n\nCollaborative Innovation Center of Extreme Optics (Shanxi University)\nNo.92 Wu Cheng Road, Shan Xi Province030006Tai YuanPeople's Republic of China\n",
"Junmin Wang \nState Key Laboratory of Quantum Optics and Quantum Optics Devices (Shanxi University)\n\n\nInstitute of Opto-Electronics\nShanxi University\n\n\nCollaborative Innovation Center of Extreme Optics (Shanxi University)\nNo.92 Wu Cheng Road, Shan Xi Province030006Tai YuanPeople's Republic of China\n",
"Junmin Wang \nState Key Laboratory of Quantum Optics and Quantum Optics Devices (Shanxi University)\n\n\nInstitute of Opto-Electronics\nShanxi University\n\n",
") "
] | [
"State Key Laboratory of Quantum Optics and Quantum Optics Devices (Shanxi University)\n",
"Institute of Opto-Electronics\nShanxi University\n",
"State Key Laboratory of Quantum Optics and Quantum Optics Devices (Shanxi University)\n",
"Institute of Opto-Electronics\nShanxi University\n",
"State Key Laboratory of Quantum Optics and Quantum Optics Devices (Shanxi University)\n",
"Institute of Opto-Electronics\nShanxi University\n",
"Collaborative Innovation Center of Extreme Optics (Shanxi University)\nNo.92 Wu Cheng Road, Shan Xi Province030006Tai YuanPeople's Republic of China",
"State Key Laboratory of Quantum Optics and Quantum Optics Devices (Shanxi University)\n",
"Institute of Opto-Electronics\nShanxi University\n",
"Collaborative Innovation Center of Extreme Optics (Shanxi University)\nNo.92 Wu Cheng Road, Shan Xi Province030006Tai YuanPeople's Republic of China",
"State Key Laboratory of Quantum Optics and Quantum Optics Devices (Shanxi University)\n",
"Institute of Opto-Electronics\nShanxi University\n"
] | [] | A novel cesium (Cs) two-color magneto-optical trap (TC-MOT), which partially employs the optical radiation forces due to photon scattering of the 6P 3/2 (F'=5) -7S 1/2 (F"=4) excited-state transition in the Cs 6S 1/2 -6P 3/2 -7S 1/2 (852 + 1470 nm) ladder-type system, has been proposed and experimentally investigated. One of the three pairs of 852 nm cooling/trapping beams (CTBs) in a conventional Cs MOT is replaced with a pair of the 1470 nm CTBs (type-I) or with one 852 nm CTB plus another counter-propagating 1470 nm CTB (type-II). Both the type-I and type-II Cs TC-MOTs can cool and trap atoms on both the red-and blue-detuning sides of the two-photon resonance. The Cs TC-MOT demonstrated in this work may have applications in the background-free detection of cooled and trapped atoms, and the photon-pair sources compatible with the ensemble-based quantum memory and the long-distance quantum communication via optical fiber. | null | [
"https://export.arxiv.org/pdf/1601.02746v1.pdf"
] | 119,236,219 | 1601.02746 | eb3640cde58b0fd2429eb9feb1098577c00e6ce1 |
Investigation of two-color magneto-optical trap with cesium 6S 1/2 -6P 3/2 -7S 1/2 ladder-type system
Jie Wang
State Key Laboratory of Quantum Optics and Quantum Optics Devices (Shanxi University)
Institute of Opto-Electronics
Shanxi University
Guang Yang
State Key Laboratory of Quantum Optics and Quantum Optics Devices (Shanxi University)
Institute of Opto-Electronics
Shanxi University
Baodong Yang
Jun He
State Key Laboratory of Quantum Optics and Quantum Optics Devices (Shanxi University)
Institute of Opto-Electronics
Shanxi University
Collaborative Innovation Center of Extreme Optics (Shanxi University)
No.92 Wu Cheng Road, Shan Xi Province030006Tai YuanPeople's Republic of China
Junmin Wang
State Key Laboratory of Quantum Optics and Quantum Optics Devices (Shanxi University)
Institute of Opto-Electronics
Shanxi University
Collaborative Innovation Center of Extreme Optics (Shanxi University)
No.92 Wu Cheng Road, Shan Xi Province030006Tai YuanPeople's Republic of China
Junmin Wang
State Key Laboratory of Quantum Optics and Quantum Optics Devices (Shanxi University)
Institute of Opto-Electronics
Shanxi University
)
Investigation of two-color magneto-optical trap with cesium 6S 1/2 -6P 3/2 -7S 1/2 ladder-type system
1cold atomstwo-color magneto-optical trapladder-type systemtwo-color polarization spectroscopyfrequency offset locking
A novel cesium (Cs) two-color magneto-optical trap (TC-MOT), which partially employs the optical radiation forces due to photon scattering of the 6P 3/2 (F'=5) -7S 1/2 (F"=4) excited-state transition in the Cs 6S 1/2 -6P 3/2 -7S 1/2 (852 + 1470 nm) ladder-type system, has been proposed and experimentally investigated. One of the three pairs of 852 nm cooling/trapping beams (CTBs) in a conventional Cs MOT is replaced with a pair of the 1470 nm CTBs (type-I) or with one 852 nm CTB plus another counter-propagating 1470 nm CTB (type-II). Both the type-I and type-II Cs TC-MOTs can cool and trap atoms on both the red-and blue-detuning sides of the two-photon resonance. The Cs TC-MOT demonstrated in this work may have applications in the background-free detection of cooled and trapped atoms, and the photon-pair sources compatible with the ensemble-based quantum memory and the long-distance quantum communication via optical fiber.
Introduction
Laser cooling and trapping of neutral atoms played an important role, and caused a profound impact in many fields such as precision measurements, optical atomic clock, quantum degenerate gases, quantum information processing and so on [1][2]. Up to now, most laser cooling schemes used the optical radiation forces due to photon scattering from the single-photon transition between atomic ground states and excited states. This approach has been extremely successful, leading to a range of techniques including the Doppler cooling [3], the polarization gradient cooling [4], and the velocity-selective coherent population trapping [5]. However, there are a few theoretical and experimental investigations of two-photon laser cooling in a ladder-type atomic system. Furthermore, these studies mostly focused on the cooling of alkaline-earth-metal atoms as a second stage using the narrow 1 S 0 -3 P 1 inter-combination transition after initial precooling with the strong 1 S 0 -1 P 1 dipole transition [6][7][8]. Most recently, a two-color magneto-optical trap (TC-MOT) based on cesium (Cs) 6S 1/2 -6P 3/2 -8S 1/2 (852 + 795 nm) ladder-type system, which partially uses the optical radiation forces due to photon scattering between two excited states (6P 3/2 and 8S 1/2 ), has been demonstrated experimentally [9,10]. The TC-MOT can cool and trap atoms on both the red-and blue-detuning sides of the two-photon resonance. This approach has been applied to a background-free detection of trapped atoms from the related transitions driven by no laser beam with a help of narrow-bandwidth high-contrast interference filters in our previous work [10]. Also, this approach has application in assisting cooling of certain atomic or molecular species which require lasers at inconvenient wavelengths. For instance, a laser cooling technique to cool hydrogen or anti-hydrogen atoms using cooling transition between excited states was proposed [11].
More potential and distinctive applications of the TC-MOT may be the photon-pair source compatible with the ensemble-based quantum memory and the long-distance quantum communication via optical fiber [12,13]. For many years, the spontaneous parametric down conversion in a nonlinear crystal has become the standard method for generating entangled photon pairs [14,15]. This kind of photon-pair source has limitations for certain applications due to the broad line-width (~ GHz) and the short coherence time (sub-picosecond). In 2005, Balić et al [16] reported a cold rubidium (Rb) ensemble based four-wave mixing to generate paired photons with coherence time of ~ 50 ns and line-width of ~ 9 MHz. They used a four-level system with two hyperfine ground states to generate the Stokes and anti-Stokes photons at 780 and 795 nm, respectively. Then they improve the results by using a two-dimensional MOT with a higher optical depth [17,18]. However, previously described photon sources do not offer much flexibility with the photon wavelength. In particular, it is not possible to directly create photons at 1.5 μm. In 2006, Chaneliere et al [13] demonstrated 1529 and 780 nm entangled photon pairs by using of a cascade transition of a ladder-type system in a cold Rb ensemble. 1529 nm photons are ideal for the long-distance quantum communication due to low loss in silica fiber, while 780 nm photons are naturally suitable for storing quamtun information into and retrieving it from a long-lived Rb quantum memory. Inspired by this idea, the Cs TC-MOT based on 6S 1/2 -6P 3/2 -7S 1/2 (852 + 1470 nm) ladder-type system may be an alternative approach of the photon-pair source compatible with the long-lived Cs quantum memory and the long-distance quantum communication via optical fiber. The 852 and 1470 nm laser beams in the Cs TC-MOT will not only serve as the cooling/trapping beams (CTBs), but also serve as the pump beams of four-wave mixing for the paired-photon generation. In detail, there are two possible ways: (1) One possible way is that the CW pair-production experiment is directly operated in type-II TC-MOT. The 852 and 1470 nm CTBs counter-propagate with a small angle (for example, ~1°, it almost does not affect the TC-MOT operation), serve as both the CTBs in the cooling and trapping process and the pump beams in the four-wave mixing, while the 852 nm CTBs in x-y plane with low power take part in the cooling and trapping process. In previous paired-photon source with a cold ensemble [16][17][18], the photon generation window of 500 μs is followed by a cooling and trapping period of 4.5 ms. The duty cycle of 10% surely reduces the pair generation rate and limits application. Maybe our paired-photon source can overcome this limitation. However, it is not clear that pair-production experiments could easily be performed in the complicated polarization configuration and the multiple beams of a MOT in any case;
(2) Another possible way is that we firstly cool and trap atoms in type-II TC-MOT, and then turn off the CTBs in x-y plane, while the CTBs along z axis remain for four-wave mixing. At least this arrangement can save physical space (do not need additional pump beams any more) and laser powers.
The primary motivation of this work is not only for the 852 and 1470 nm photon-pair source compatible with the Cs ensemble-based quantum memory and the long-distance quantum communication via optical fiber, but also for the understanding of cooling and trapping mechanism from the multi-photon transitions. Fig. 1(a) shows the decay channels from Cs 8S 1/2 and 7S 1/2 states, besides the different decay rate (γ = 2π × 3.30 MHz for 7S 1/2 , and γ = 2π × 1.52 MHz for 8S 1/2 ), the significant difference is that: compared with the Cs 6S 1/2 -6P 3/2 -8S 1/2 TC-MOT [9, 10], the Cs 6S 1/2 -6P 3/2 -7S 1/2 TC-MOT is significantly simpler due to less decay channels. Consequently, it is profitable for the understanding of cooling and trapping mechanism in this simple and different ladder-type system. To be specific, the behaviors are a bit different between them in type-I TC-MOT, see Fig. 2(a) and the discussion.
In addition, the main virtue of less decay channels is that the trapped atom's number is much more linearly dependent on the fluorescence from the TC-MOT to a certain degree. More strictly, the relationship between the atom number and the fluorescence is not perfectly linear due to different optical power, different detuning, non-cyclic transition, and so on. However, the fluorescence indicates a large number of atoms are trapped in the MOT. Consequently, we directly measure the fluorescence to diagnose whether the TC-MOT operates and determine the range of two-photon detuning for the TC-MOT operation.
In this article, a novel Cs TC-MOT based on Cs 6S 1/2 -6P 3/2 -7S 1/2 (852 + 1470 nm) ladder-type system is proposed and experimentally investigated, which partially employs the optical radiation forces due to photon scattering of the 6P 3/2 (F'=5) -7S 1/2 (F"=4) excited-state transition. One of the three pairs of 852 nm CTBs in a conventional Cs MOT is replaced with a pair of the 1470 nm CTBs (type-I) or with one 852 nm CTB plus another counter-propagating 1470 nm CTB (type-II). Both the type-I and type-II Cs TC-MOTs can cool and trap atoms on both the red-and blue-detuning sides of the two-photon resonance. We measured and analyzed qualitatively the dependence of peak fluorescence (cold atom number) on the two-photon detuning, the intensity of CTBs and the different combination of CTBs along the z direction (the axis of the anti-Helmholtz coils of the TC-MOT). These results provide optimized experimental parameters to trap atoms and pave the first step towards application of type-II Cs TC-MOT for 852 and 1470 nm entangled photon pair generation.
The rest of this paper is organized as follows. In Sec. 2 we briefly introduce the experimental setup of type-I and type-II Cs TC-MOTs and discuss the cooling and trapping effects. In Sec. 3 we develop a frequency offset locking system to control the detuning of two lasers. In Sec. 4 we measured and analyzed qualitatively the dependence of peak fluorescence on the two-photon detuning, the intensity of CTBs in type-I and type-II TC-MOT. Lastly, we draw our concluding remarks in Sec. 5.
Experimental apparatus and principle
In this section, we firstly introduce the relevant energy-level transitions and the experimental setup of type-I and type-II Cs TC-MOTs, and then we discuss the cooling and trapping effects. beam configuration for the TC-MOT. Fig. 1(b) shows an energy-level diagram of relevant transitions. The 852 nm CTBs interacted with |g>−|e> transition (ω 1 ) has a detuning of ∆ 1 . The 1470 nm CTBs interacted with |e>−|e'> excited-state transition (ω 2 ) have a detuning of ∆ 2 and the two-photon detuning is δ 2 . The 852 nm repumping beams resonant with 6S 1/2 (F=3) -6P 3/2 (F'=4) transition. The excited state line-widths for |e> and |e'> states are Γ/2π = 5.2 MHz and γ/2π = 3.3MHz [19], respectively. We consider two types of CTB configuration for the Cs TC-MOT: type-I as shown in Fig. 1(c), the CTBs in the x-y plane comprises the two pairs of conter-propagating 852 nm beams (~ 14 mm, 1/e 2 diameter), the CTBs along the z axis (the axis of the anti-Helmholtz coils of the TC-MOT) is a pair of conter-propagating 1470 nm beams (~ 10 mm, 1/e 2 diameter); type-II as shown in Fig. 1(d), the CTBs in the x-y plane is the same as type-I, the CTBs along the z axis comprises one 852 nm CTB (~ 14 mm, 1/e 2 diameter) and one counter-propagating 1470 nm CTB (~ 10 mm, 1/e 2 diameter). The 852 nm repumping beams (not shown in Fig. 1) with ~ 16 mm 1/e 2 diameter are sent along the ±y axis. In our experiment, the gradient of the quadrupole magnetic field generated by a pair of anti-Helmholtz coils with current I is 1.0 mT/cm (10 Gauss/cm) along the z direction. The pressure of Cs atomic vapor inside the stainless steel vacuum chamber is 1.06×10 -6 Pa (8×10 -9 Torr).
Experimental apparatus
(c)
6P 3/2 F'=4 7S 1/2 F''=4 (e') 6P 3/2 F'=5 (e) 6S 1/2 F=4 (g) △ 1 2 ω 3 (1359nm) 6P 1/2 F'=3,4 ω 2 (1470nm) Ω ee' ω 1 (852.3nm) Ω ge ω 4 (894.6nm) (b) 6S 1/2 F=3 Repumping 6P 3/2 6S 1/2 8S 1/2 7S 1/2 6P 1/2 7P 3/2 5D 5/2 5D 3/2 (a)
Principle
The cooling and trapping process in type-I TC-MOT arises from two effects [9]: one is the velocity-dependent scattering force, associated with 2-photon or 3-photon scattering process; another is the position-dependent restoring force, which is essential for trapping. Here the restoring force was found when the helicities of the |e>−|e'> CTBs are opposite to those for the conventional MOT. The restoring force has the correct sign for both positive and negative δ 2 when ∆ 1 < 0 [9]. We mainly introduce the velocity-dependent scattering force as following.
Two-photon scattering process
In low intensity regime, the dominant radiation pressure along the z axis is due to 2-photon scattering, where the first photon is absorbed from the in the x-y plane laser beams and the second photon is absorbed from the CTBs along the z axis. The scattering forces along the z axis can be written as (2) (2) ,ẑ ee velocity v , the 2-photon scattering rate in the low intensity limit can be written as:
k R ′ = ∑ i j i, j f j , where { }ˆˆˆ∈ i x, -x,2 (2) , 2 1 2 =ˆˆ1 6 ( )(δ ) ge ee ge ge ee R k k k γ ′ ′ Ω Ω ∆ − ⋅ − ⋅ − ⋅ i j i v i v j v .
(1)
Here ge Ω and ge Ω are the Rabi frequencies of the laser induced couplings per beam; ge k and ee k ′ are the wave numbers of the |g>−|e> and |e>−|e'> CTBs; 1 1 i 2 ∆ =∆ + Γ
Three-photon scattering process
At moderate and high intensity of the |g>−|e> CTBs, the cooling also works, which is opposed to the 2-photon scattering cooling. This should be attributed to 3-photon and higher order scattering process. In the 3-photon process, the 2-photon absorption is followed by a stimulated |e'>−|e> emission. These multi-photon processes can lead to efficient cooling along the z axis in a manner similar to the "Doppleron" cooling [20]. In the same way as for the 2-photon force calculations, the 3-photon scattering force can be written as (3)
(3) ,ẑ ee k R ′ = ∑ i j i, j f j
, where, for atoms moving at velocity v , the 3-photon scattering rate (3) ,
R i j is 6 2 (3) (2) , , 2 1 =ˆ4 2 ee ge ee R R k k γ ′ ′ Ω Γ ∆ − ⋅ − ⋅ i j i j i v j v .
(2)
We Taylor-expand
(3) z f near z v 0 = to find the 3-photon damping coefficient (3) α . 1 1 i 2 ∆ =∆ + Γ and 2 2 δ δ iγ 2 = + ; For 1 0 ∆ < and 2 2 2 1 γ =Γ +4Δ , we find (3) 0 α > for either 2 δ <0 or 1 2 δ > 2 − ∆ .
Here the (3) α involves only the 3-photon process and ignores the 2-photon process, light shift, and higher order process. The 3-photon cooling effect can be understood qualitatively from the |g>−|e>−|e'> Raman process. At large |δ 2 |, the Doppler sensitivity along the z axis becomes independent of δ 2 , but remains dependent on ∆ 1 . The fact that (3) α is positive is determined by the negative ∆ 1 . In addition, compare to the cascade transitions in the 6S 1/2 -6P 3/2 -8S 1/2 TC-MOT, the cascade transitions in the 6S 1/2 -6P 3/2 -7S 1/2 TC-MOT is significantly simpler due to less decay channels, as shown in Fig. 1(a). It is not only profitable for the understanding of cooling and trapping mechanism (see Fig. 2(a) and the discussion) but also for the fluorescence detection. Usually, the absorption detection or laser-induced fluorescence after turning off the MOT can be used to estimate the atom number. For simplicity here we use the in-situ fluorescence measurement via a CCD camera to estimate cold cloud's size and atom numbers. In 6S 1/2 -6P 3/2 -8S 1/2 TC-MOT, atoms can decay from 8S 1/2 to 6S 1/2 state through the cascaded 8S 1/2 -7P 3/2 (7P 1/2 ) -6S 1/2 and 8S 1/2 -6P 3/2 (6P 1/2 ) -6S 1/2 two-photon transitions, and the cascaded 8S 1/2 -7P 3/2 (7P 1/2 ) -7S 1/2 -6P 3/2 (6P 1/2 ) -6S 1/2 and 8S 1/2 -7P 3/2 (7P 1/2 ) -5D 5/2 (5D 3/2 ) -6P 3/2 (6P 1/2 ) -6S 1/2 four-photon transitions. Even though interference filters can be employed to choose the desired wavelength, still it is a little bit difficult to estimate cold atom number from the fluorescence signal because too many branches have to be considered. Now the situation is much simpler for 6S 1/2 -6P 3/2 -7S 1/2 TC-MOT, in which atoms can decay from 7S 1/2 to 6S 1/2 state through the cascaded 7S 1/2 -6P 3/2 (6P 1/2 ) -6S 1/2 two-photon transitions. The cold atom number is much more linearly dependent on the fluorescence to a certain degree. Strictly this is inaccurate, but it is a simple method to diagnose whether the TC-MOT operates and determinate the range of two-photon detuning for TC-MOT operation. For simplicity, the peak fluorescence of trapped atoms is recorded in our experiment.
Frequency offset locking system
Doppler-free spectroscopy between atomic ground and excited states, such as saturated absorption spectroscopy, polarization spectroscopy, and modulation transfer spectroscopy, provides a reference to control the frequency of 852 nm laser. The spectroscopy between excited states, such as optical-optical double resonance [21,22], double-resonance optical pumping [23-25], ladder-type electromagnetically induced transparency [26,27], and two-color polarization spectroscopy [28,29], can be used to stabilize 1470 nm laser frequency. The frequency control system of cooling lasers is shown in Fig. 2. The frequency detuning ∆ 1 of 852 nm cooling laser is controlled by two acousto-optic modulators (AOM3 and AOM2). For example, in order to set ∆ 1 to be -12.5 MHz, the laser interacted with Cs vapor cell is locked to Cs 6S 1/2 (F=4) -6P 3/2 (F'=4, 5) crossover line by using of polarization spectroscopic locking scheme, and the radio frequency (RF) signals applied on AOM3 and AOM2 should be 96.52 and 80 MHz, respectively. The frequency detuning ∆ 2 of 1470 nm cooling laser is controlled by AOM3 and a fiber-pigtailed phase-type wave-guide electro-optic modulator (EOM). For example, the RF signals applied on AOM3 and EOM are 96.52 MHz and Ω RF (Ω RF : 90 ~ 410 MHz), respectively, and these yield that the 1470 nm laser frequency is [ω 2 + (Ω RF -227.5 MHz) ×852/1470], thus δ 2 =∆ 1 +∆ 2 ranges from -92.2 to +93.3 MHz. Here the 1470 nm laser frequency is offset locked by the velocity-selective off-resonant two-color polarization spectroscopy [28]. The 852 nm laser detuned +227.5 MHz to ω 1 pass through EOM and generate two sidebands (See Fig. 2), the -1 order sideband with frequency of (ω 1 +227.5 MHz-Ω RF ) serves as pump laser for the velocity-selective off-resonance two-color polarization spectroscopy, which is used to lock 1470 nm laser. By comparison, spectroscopy arising from the +1 order sideband with frequency of (ω 1 +227.5 MHz+Ω RF ) is weak because the large detuning laser only interacts with a fraction of atoms moving with large velocity which obey the Maxwell-Boltzmann distribution. Fig. 3 shows the velocity-selective off-resonance two-color polarization spectroscopy. Once the 1470 nm laser frequency is locked to the spectroscopy generated by the -1 order modulation sideband, the laser frequency should be [ω 2 + (Ω RF -227.5 MHz) ×852/1470]. This setup is the modification of our previous offset locking system in [10] and [28] where the RF modulation is added to 1470 nm laser. The main advantage of this locking scheme is that all frequency-shift elements are placed in the 852 nm optical path, thus to save the 1470 nm laser power. The 852 nm repumping laser which is sent along the ±y axis (not shown in Fig. 2) is locked to the 6S 1/2 (F=3) -6P 3/2 (F'=4) transition by saturated absorption spectroscopic locking scheme, to avoid atoms accumulating at the 6S 1/2 (F=3) ground state which does not interact with the cooling lasers. . 3. (Color online) The velocity-selective off-resonance two-color polarization spectra versus the 1470 nm laser frequency detuning with different Ω RF . The 852 nm laser with frequency of (ω 1 +227.5 MHz) is coupled into EOM (Seen in Fig.2) to generate the modulation sidebands. The velocity-selective off-resonance two-color polarization spectroscopy can be obtained by scanning the 1470 nm laser frequency.
Experimental results and discussions
As motioned above, the fluorescence is approximately proportional to cold atom number. Here the peak fluorescence directly measured by a CCD camera in order to diagnose whether the TC-MOT operates and determine the range of two-photon detuning for the TC-MOT operation. The dependence of the peak fluorescence upon the two-photon detuning δ 2 , and the intensity of CTBs for type-I and type-II TC-MOTs is measured qualitatively. And some relevant discussions are also made.
Type-I Cs TC-MOT
Based on conventional Cs MOT, two 852 nm CTBs along the ±z directions are replaced with 1470 nm CTBs, as shown in Fig. 1(c), we named this case as the type-I Cs TC-MOT. The peak fluorescence as function of the single-photon detuning ∆ 1 , the two-photon detuning δ 2 , the 852 and 1470 nm CTBs' power, and the repumping beams' power are shown in Fig. 4.
The significant characteristic of the TC-MOT is that it can cool and trap atoms on both the redand blue-detuning sides of the two-photon resonance. As motioned above, the two-photon detuning δ 2 is controlled (from -90 to +90 MHz) by EOM. Two typical false-color fluorescence images of cold cloud are shown as the insets of Fig. 4(d), corresponding to the data points for δ 2 = +28.4 and -36. 6 MHz with a maximum repumping power, respectively. The sizes of the two clouds are about 0.8 mm (z) × 0.3 mm (x, y) and 0.9 mm (z) × 0.2 mm (x, y), respectively. Typical atom number is estimated to be ~ 5×10 6 , and the corresponding atomic density is 6.9×10 10~1 .3×10 11 /cm 3 . Fig. 4(a) shows the peak fluorescence of atoms trapped in the type-I Cs TC-MOT as a function of δ 2 with different 1470 nm CTBs' power, while the 852 nm CTBs' total power is 4×6.10 mW, and the single-photon detuning is ∆ 1 = -12.5 MHz. On the red-detuning side of the two-photon resonance, as the 1470 nm CTBs' power increases, the range of δ 2 for TC-MOT operation broadens and is red-shifted. On the blue-detuning side, TC-MOT works with δ 2 > 12 MHz, and the range broadens when the 1470 nm CTBs' power increases. The required CTBs' power for TC-MOT operation on the blue-detuning side is less than that on the red-detuning side. One point should be addressed here, this is inversed to that in 852+795 nm Cs TC-MOT in [9], in which the required CTBs' power for TC-MOT operation on the blue-detuning side is larger. The different behaviors between 852+1470 nm and 852+795 nm TC-MOTs are probably due to following reasons: (1) The photon momentum of 1470 nm is less than that of 795 nm, thus the scattering force in former TC-MOT is weaker than that in latter one, so it is more difficult to cool and trap atoms in δ 2 <0 region. In other words, more optical power is needed at the blue-detuning side for TC-MOT operation; (2) The decay channels in former TC-MOT are much less than that in latter, in detail, the decay branching ratio for 7S 1/2 -6P 3/2 channel is ~ 65% and that for 8S 1/2 -6P 3/2 channel is ~ 37%, hence the Raman process in former TC-MOT is more pure and the 3-photon scattering rate is higher in δ 2 > 12 MHz region; (3) In Eq. (2), the 3-photon scattering rate is inversely proportional to 2 1ˆ2 ge ee
k k ′ ∆ − ⋅ − ⋅ i v j v
, consequently, it is large due to small ee k ′ along the z direction. In addition, the former velocity capture range is larger than that of the latter. Fig. 4(b) shows the peak fluorescence of atoms trapped in type-I Cs TC-MOT as a function of δ 2 with different 852 nm CTBs' power, while the 1470 nm CTBs' total power is 2×20.0 mW, and the single-photon detuning is ∆ 1 = -12.5 MHz. As the 852 nm CTBs' power increases, the range of δ 2 for TC-MOT operation on both the red-and blue-detuning sides does not change much (neither shift nor broaden much) due to force balance along the z direction. The 852 nm CTBs are orthogonal to the 1470 nm CTBs, as a result, when the 852 nm CTBs' power increase, the atom number increase, but the force balance along the z direction does not break. Fig. 4(c) shows the peak fluorescence of atoms trapped in type-I Cs TC-MOT as a function of δ 2 for atoms in type-I TC-MOT with different single-photon detuning ∆ 1 , while the 1470 nm CTBs' total power is 2×20.0 mW, and the 852 nm CTBs' total power is 4×6.10 mW. As the change of single-photon detuning ∆ 1 , the peak fluorescence has an optimized intensity at ∆ 1 = -12.5 MHz. Larger or smaller than this value, the peak fluorescence reduces. Note that the single-photon detuning ∆ 1 seems not to shift the range of δ 2 for TC-MOT operation due to perpendicular between 852 and 1470 nm CTBs. The gray lines representing ∆ 2 =0 (δ 2 =∆ 1 ) in each curve provide a most direct impression why we consider the two-photon detuning δ 2 instead of the detuning ∆ 2 . Fig. 4(d) shows the peak fluorescence of atoms trapped in type-I Cs TC-MOT as a function of the 852 nm repumping beams' power. The single-photon detuning is ∆ 1 = -12.5 MHz. The two photon detunings are δ 2 = 28.4 MHz for upper curve, and δ 2 = -36.6 MHz for lower curve. The CTBs' powers are 4×6.10 mW for 852 nm and 2×20.0 mW for 1470 nm, respectively. The insets are fluorescence images of cold cloud for the two different δ 2 with a maximum repumping beams' power (2×2.0 mW). As the repumping beams' power increases, the peak fluorescence increases and tends to be saturated on both red-and blue-detuning sides.
Type-II Cs TC-MOT
Based on the type-I Cs TC-MOT, one of the 1470 nm CTBs along the ±z directions is replaced with one 852 nm CTB, as shown in Fig. 1(d), we named this case as the type-II Cs TC-MOT. Fig. 5 shows the peak fluorescence of atoms trapped in type-II TC-MOT as a function of δ 2 with different 852 nm CTB's power along the +z direction, while the 1470 nm CTB's power is 9.5 mW. The 852 nm CTBs' total power in the x-y plane is 4×1.99 mW (a) and 4×6.10 mW (b). The single-photon detuning is ∆ 1 = -12.5 MHz. The type-II TC-MOT can also cool and trap atoms on both the red-and blue-detuning sides of the two-photon resonance. As the 852 nm CTB's power along the +z direction increases, on the red-detuning side, the range of for TC-MOT operation shifts close to resonance until the power is larger than a threshold value 2.66 mW in Fig. 5(a) and 5.80 mW in Fig. 5(b), respectively. On the blue-detuning side, the TC-MOT works with δ 2 > 8 MHz, the range of δ 2 for TC-MOT operation narrows, and the fluorescence disappears when the 852 nm CTB's power is larger than a threshold value 1.30 mW in Fig. 5(a) and 4.09 mW in Fig. 5(b), respectively. There are two distinguished characteristics: (1) In each of the figures, as the 852 nm CTB's power along the +z direction increases, the range of δ 2 for TC-MOT operation noticeably shifts. This is because the scattering force arising from 852 and 1470 nm CTBs should be equal along the z direction. For the 852 nm CTB with ∆ 1 =-12.5 MHz, as its power increases, the scattering force increases. For the 1470 nm CTB with fixed power, as 852 nm CTB's scattering force increases, the detuning δ 2 has to shift close to resonance to keep the balance between the scattering forces. (2) For TC-MOT operation, the 852 nm CTB's upper limit power along the +z direction in Fig. 5 (b) is higher than the power in Fig. 5 (a). One reason is that the scattering force of 1470 nm CTB is higher in Fig. 5 (b) due to larger Rabi frequency Ω ge according to Eq. 1 and 2 for higher 852 nm CTBs' power in the x-y plane. Hence the 852 nm CTB's upper limit power along the +z direction in Fig. 5 (b) should be higher to keep the force balance. Another possible reason is the misalignment of the two beams (<5 mrad) along the z axis, producing a resultant force in the x-y plane, which push the atoms away from the center of TC-MOT. Although the scattering forces between 852 nm and 1470 nm CTBs keep balance along the z axis, to keep the force balance in the x-y plane, the binding force from 852 nm CTBs in the x-y plane should be large enough to against this resultant force. In other words, the 852 nm CTB's upper limit power along the +z direction in Fig. 5 (b) is higher due to the stronger binding force in the x-y plane. Fig. 6 shows the peak fluorescence of Cs atoms trapped in the type-II Cs TC-MOT as a function of δ 2 with other different parameters. In Fig. 6 (a), with higher 1470 nm CTB's power, to keep the force balance between scattering forces of 852 and 1470 nm CTB, the two-photon detuning δ 2 for TC-MOT operation has to shift away from resonance. In Fig. 6 (b), with higher 852 nm CTBs' power in the x-y plane, the scattering force of 1470 nm CTB is higher due to larger Rabi frequency Ω ge according to Eq. 1 and 2. So the two-photon detuning δ 2 for TC-MOT operation has to shift away from resonance to keep the force balance. Moreover, this feature is different from that in type-I TC-MOT illustrated in Fig. 5 (b). In that case, the range of δ 2 for TC-MOT operation on both the red-and blue-detuning sides does not change much due to force balance along the z direction. In Fig. 6 (c), larger single-photon detuning ∆ 1 will weaken the scattering force arising from 852 nm CTBs both in the x-y plane and along the z axis, hence the δ 2 for TC-MOT operation should slightly shift away from resonance to keep the scattering force balance. In sum, one needs to carefully adjust all the parameters, such as the 1470 nm CTB's power, the 852 nm CTB's power both in the x-y plane and along the z direction, the single-photon detuning ∆ 1 , and the two-photon detuning δ 2 , to keep the scattering force balance and trap more atoms.
Conclusion
In conclusion, a novel Cs TC-MOT, in which the optical forces due to photon scattering of the Cs 6P 3/2 (F'=5) -7S 1/2 (F"=4) excited-state transition in the Cs 6S 1/2 -6P 3/2 -7S 1/2 (852 + 1470 nm) ladder-type system are partially employed, has been proposed and experimentally investigated. One of the three pairs of 852 nm CTBs in a conventional Cs MOT is replaced with a pair of the 1470 nm CTBs (type-I) or with one 852 nm beam plus another counter-propagating 1470 nm beam (type-II). Both the type-I and type-II Cs TC-MOTs can cool and trap atoms on both the red-detuning and blue-detuning sides of the two-photon resonance. We measured and analyzed qualitatively the dependence of peak fluorescence on the two-photon detuning, the intensity of CTBs in type-I and type-II TC-MOT.
These results not only provided optimized experimental parameters to trap atoms, but also provided helpful evidence to deeply investigate the mechanism of cooling and trapping atoms in TC-MOT. The experiment demonstrated in this work may have wide applications, such as the background-free detection of trapped atoms, the laser cooling of Rydberg atoms in a cascade atomic configuration, and the assisting laser cooling of certain atomic species which require cooling lasers at inconvenient wavelengths. Especially, type-II TC-MOT is an alternative scheme to cool and trap atoms and pump the atomic ensemble for the practical application in the photon-pair sources compatible with the ensemble-based quantum memory and the long-distance quantum communication via optical fiber. Note that high optical depth is profitable to this kind of photon-pair generation based on four-wave mixing. Higher power and large spot size can effectively enlarge the optical depth. In our experiment, the 1470 nm beam size is smaller than 852 nm beam size in order to enhance the optical intensity, but this may reduce the velocity capture range. Moreover, the technique which used to enlarge optical depth in 2D MOT [30], like a cylindrical quadrupole field, large atom number, a temporally dark and compressed MOT, and Zeeman-state optical pumping, will also be used to achieve a high optical depth.
Fig. 1
1shows the relevant energy-level transitions and the schematic diagram of two type of laser4
FIG. 1 .
1(Color online) Relevant energy-level transitions and schematic diagram of two type of laser beam configuration for the TC-MOT. (a) Relevant energy-level transitions of Cs atoms. There are very less decay channels from 7S 1/2 state than from 8S 1/2 state, which is profitable to the analysis of cooling mechanism. (b) The transitions interacting with the 852 and 1470 nm CTBs and repumping beams. (c) type-I TC-MOT, two 1470 nm CTBs counter-propagate along the z axis, (d) type-II TC-MOT, one 1470 nm CTB and one 852 nm CTB counter-propagate along the z axis, σ ± are specified with respect to the positive x, y and z axis, and I is the electric current of the anti-Helmholtz coils.
y, -y is one of the four directions of the |g>−|e> CTBs, and { }ˆ∈ j z, -z is one of the two directions of the |e>−|e'> CTBs. For a Cs atom moving with a
. This is similar to the Doppler cooling process in the conventional MOT, where the Doppler effect enhances the absorption cross section for the |e>−|e'> CTBs opposing the velocity v .
The discussion above has shown the cooling and trapping in type-I TC-MOT. With respect to type-II TC-MOT, thez directions of the |e>−|e'> CTBs, and the analyses should be similar as the type-I TC-MOT.
FIG. 2 .
2(Color online) Schematic diagram of frequency offset locking of 852 and 1470 nm cooling lasers for the TC-MOT. Keys to figure: OI, optical isolator; PBS, polarization beam splitter cube; λ/2, half-wave plate; λ/4, quarter-wave plate; EOM, fibre-pigtailed phase-type wave-guide electro-optic modulator; PD, photodiode; DPD, differential photodiode; The frequency labelled in the figure shows the operated frequency of the lasers and the AOMs while the detuning ∆ 1 is set to be -12.5 MHz: ω 1 , Cs 6S 1/2 (F=4) − 6P 3/2 (F'=5) transition frequency; ω 2 , Cs 6P 3/2 (F'=5) −7S 1/2 (F''=4) transition frequency; Ω RF , radio frequency applied on EOM.
-
FIG. 4 .
4(Color online) The peak fluorescence of Cs atoms trapped in the type-I Cs TC-MOT as a function of the two-photon detuning δ 2 (a) with different 1470 nm CTBs' power, while the 852 nm CTBs' total power is 4×6.10 mW, and the single-photon detuning is ∆ 1 = -12.5 MHz; (b) with different 852 nm CTBs' power, while the 1470 nm CTBs' total power is 2×20.0 mW, and the single-photon detuning is ∆ 1 = -12.5 MHz; (c) with different single-photon detuning ∆ 1 , while the 1470 nm CTBs' total power is 2×20.0 mW, and the 852 nm CTBs' total power is 4×6.10 mW. The repumping beams' total power is 2×2.0 mW for (a), (b) and (c). The vertical grey line represents ∆ 2 = 0, the vertical red line represents δ 2 = 0, and the vertical blue line represents δ 2 ≈ +12 MHz. (d) The peak fluorescence of Cs atoms trapped in type-I TC-MOT as a function of the 852 nm repumping beams' power. The CTBs' powers are 4×6.10 mW for 852 nm and 2×20.0 mW for 1470 nm, respectively. The insets are fluorescence images of cold cloud for the two different δ 2 with a maximum repumping beams' power (2×2.0 mW).
FIG. 5 .
5(Color online) The peak fluorescence of Cs atoms trapped in the type-II Cs TC-MOT as a function of the two-photon detuning δ 2 with different 852 nm CTB power along the +z direction, while the 1470 nm CTB power is 9.5 mW. The 852 nm CTBs' total power in the x-y plane is 4×1.99 mW (a) and 4×6.10 mW (b). The single-photon detuning is ∆ 1 = -12.5 MHz. The vertical grey line represents ∆ 2 = 0, the vertical red line represents δ 2 = 0, and the blue vertical line represents δ 2 ≈ +8 MHz.
FIG. 6 .
6(Color online) The peak fluorescence of Cs atoms trapped in the type-II Cs TC-MOT as a function of the two-photon detuning δ 2 with other different parameters. (a) With different 1470 nm CTB; (b) With different 852 nm CTBs' power in the x-y plane; (c) With different single-photon detuning ∆ 1 . The details are: (i) black line, ∆ 1 = -12.5MHz, P CTB xy =4×6.10 mW, P CTB z+ =0.33 mW, P CTB z-=19.5 mW; (ii) red line, ∆ 1 = -12.5 MHz, P CTBxy =4×6.10 mW, P CTB z+ =0.33 mW, P CTB z-=9.5 mW; (iii) blue line, ∆ 1 = -12.5 MHz, P CTB xy =4×1.99 mW, P CTB z+ =0.33 mW, P CTB z-=9.5 mW; (ix) magenta line, ∆ 1 = -17.5 MHz, P CTB xy =4×1.99 mW, P CTB z+ =0.33 mW, P CTB z-=9.5 mW.
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| [] |
[
"Dirty black holes: Entropy as a surface term",
"Dirty black holes: Entropy as a surface term"
] | [
"Matt Visser \nPhysics Department\nWashington University St. Louis Missouri\n63130-4899\n"
] | [
"Physics Department\nWashington University St. Louis Missouri\n63130-4899"
] | [] | It is by now clear that the naive rule for the entropy of a black hole, (entropy) = 1/4 (area of event horizon), is violated in many interesting cases. Indeed, several authors have recently conjectured that in general the entropy of a dirty black hole might be given purely in terms of some surface integral over the event horizon of that black hole. A formal proof of this conjecture, using Lorentzian signature techniques, has recently been provided by Wald. This note performs two functions. Firstly, by extending a previous analysis due to the present author [Physical Review D48, ???? (15 July 1993)] it is possible to provide a rather different proof of this result -a proof based on Euclidean signature techniques. The proof applies both to arbitrary static [aspheric] black holes, and also to arbitrary stationary axisymmetric black holes. The total entropy is note proceeds via Euclidean signature techniques the result can be checked against certain special cases previously obtained by other techniques, e.g. (Ricci) n gravity, R n gravity, and Lovelock gravity. | 10.1103/physrevd.48.5697 | [
"https://export.arxiv.org/pdf/hep-th/9307194v1.pdf"
] | 15,737,861 | hep-th/9307194 | 8d03c42774dffd15b9b01257b6ba23a48615b4c1 |
Dirty black holes: Entropy as a surface term
arXiv:hep-th/9307194v1 30 Jul 1993 30 July 1993
Matt Visser
Physics Department
Washington University St. Louis Missouri
63130-4899
Dirty black holes: Entropy as a surface term
arXiv:hep-th/9307194v1 30 Jul 1993 30 July 1993
It is by now clear that the naive rule for the entropy of a black hole, (entropy) = 1/4 (area of event horizon), is violated in many interesting cases. Indeed, several authors have recently conjectured that in general the entropy of a dirty black hole might be given purely in terms of some surface integral over the event horizon of that black hole. A formal proof of this conjecture, using Lorentzian signature techniques, has recently been provided by Wald. This note performs two functions. Firstly, by extending a previous analysis due to the present author [Physical Review D48, ???? (15 July 1993)] it is possible to provide a rather different proof of this result -a proof based on Euclidean signature techniques. The proof applies both to arbitrary static [aspheric] black holes, and also to arbitrary stationary axisymmetric black holes. The total entropy is note proceeds via Euclidean signature techniques the result can be checked against certain special cases previously obtained by other techniques, e.g. (Ricci) n gravity, R n gravity, and Lovelock gravity.
S = kAH 4ℓ 2 P + H S √ 2g d 2 x.
The integration runs over a spacelike cross-section of the event horizon H. The surface entropy density, S, is related to the behaviour of the matter Lagrangian under time dilations. Secondly, I shall consider the specific case of Einstein-Hilbert gravity coupled to an effective Lagrangian that is an arbitrary function of the Riemann tensor (though not of its derivatives).
In this case a more explicit result is obtained
S = kAH 4ℓ 2 P + 4π k h H ∂L ∂R µνλρ g ⊥ µλ g ⊥ νρ √ 2gd 2 x.
The symbol g ⊥ µν denotes the projection onto the two-dimensional subspace orthogonal to the event horizon. Though the derivation exhibited in this
INTRODUCTION
The entropy versus area relationship for generic "dirty" black holes has recently engendered considerable interest [1,2,3,4,5,6]. Generically a dirty black hole [8] is a black hole distorted by either: (1) various classical matter fields, (2) higher curvature terms in the gravity Lagrangian [e.g. (Riemann) n ], or (3) infestation with some version of quantum hair.
The present note addresses two main points: Firstly, it has recently been conjectured that the entropy of a dirty black hole can always be cast into the form of an integral of some quantity over the event horizon [3,9,10]. A formal proof of this result, based on Lorentzian signature Lagrangian techniques, has recently been announced [5]. Details and applications may be found in [6,7]. In this note I present an alternative proof of this result. The present proof is obtained by utilizing Euclidean space techniques in the manner of [1], and is ultimately an extension of the original Gibbons-Hawking Euclidean signature technology [11]. The proof applies both to arbitrary static [aspheric] black holes, and also (with additional technical complications) to arbitrary stationary axisymmetric black holes. The total entropy is
S = kA H 4ℓ 2 P + H S √ 2 g d 2 x.(1)
The integration runs over a spacelike cross-section of the event horizon H. The surface entropy density, S, is related (in a particular manner involving time dilations) to the surface term arising in the integration by parts that connects the stress-energy tensor with the variation of the action under a variation of the spacetime metric. For definiteness, the calculations are carried out in fourdimensional spacetime, but the generalization to arbitrary dimensionality is immediate. Secondly, as a specific example, this note will focus on the case of black holes in Einstein-Hilbert gravity coupled to an effective Lagrangian that is any arbitrary function of the Riemann tensor (though not of its derivatives). Interest in such a toy model is justified by noting that whatever the underlying quantum theory of gravity is, one would expect on general grounds that the low energy theory should be describable by an effective Lagrangian that contains at least the class of terms indicated above. Applying the general formalism developed in this note to this particular case yields
S = kA H 4ℓ 2 P + 4π k h H J µνλρ g ⊥ µλ g ⊥ νρ √ 2 gd 2 x.(2)
The tensor J µνλρ ≡ ∂L/∂R µνλρ has the same symmetries at the Riemann tensor. The symbol g ⊥ µν denotes the projection onto the two-dimensional subspace orthogonal to the event horizon.
It is instructive to check this formula against several special cases that have been derived by rather different methods. For instance, Jacobson and Myers [2] have recently evaluated the entropy for black holes in Lovelock gravity using Hamiltonian methods. The present analysis reproduces their result with no difficulty. More recently, Jacobson, Kang, and Myers [3,4] have extended their analysis to the case where the Lagrangian is an arbitrary function of the Ricci scalar. The entropy for black holes of this type was extracted by using a combination of field redefinition and Hamiltonian techniques. [Consider the behaviour of the black hole under conformal deformations.] Again, this result can be shown to be a special case of the general formula given above. Furthermore, Jacobson, Kang, and Myers [3,4] have also considered the case where the Lagrangian is the Einstein-Hilbert Lagrangian augmented by the square of the Ricci tensor. The present techniques allow a simple extension of that result to the case of an arbitrary function of the Ricci tensor. The fact that different techniques give the same answer where they overlap is encouraging.
Notation: Adopt units where c ≡ 1, but all other quantities retain their usual dimensionalities, so that in particular G ≡ ℓ P /m P ≡h/m 2 P ≡ ℓ 2 P /h. The metric signature is (+, +, +, +). The symbol T will always denote a temperature. The stress-energy tensor will be denoted by t µν , and its trace by t.
GENERAL THEOREM
Reprise
In a previous paper [1], I have derived a general formula for the entropy of a dirty black hole in terms of: (1) the area of the event horizon, A H , (2) the energy density in the classical fields surrounding the black hole, ̺, (3) the Euclideanized Lagrangian describing those fields, L, (4) the Hawking temperature, T H , (5) the entropy density, s, associated with the fluctuations [quantum hair, statistical hair], and finally (6) the metric. The total entropy is:
S = kA H 4ℓ 2 P + 1 T H Σ {̺ − L}K µ dΣ µ + Σ sV µ dΣ µ .(3)
This formula applies to all static black holes [not necessarily spherically symmetric], and to stationary non-static [axisymmetric] black holes. K µ is the timelike Killing vector. V µ is the four-velocity of a co-moving observer. For a static black hole, this is just the four-velocity of a FIDO [fiducial observer]. For a rotating black hole this is the four-velocity of a co-rotating observer. Σ denotes a constant time hypersurface. The first term in this formula agrees with Bekenstein's original suggestion [13], with the normalization constant fixed by Hawking's calculation [14]. For the time being fluctuations are ignored, (s = 0, no quantum hair, no statistical mechanics effects). The issue of interest is the evaluation of the term:
1 T H Σ {̺ − L}K µ dΣ µ = k h Ω {̺ − L} √ gd 4 x.(4)
Here Ω denotes the entire Euclidean four-manifold. As is usual in the Euclidean formulation the time direction is compact with periodhβ =h/(kT H ). The Hawking temperature, T H , is related to the surface gravity, κ, by kT H =hκ/2π. By their very construction, Euclidean signature techniques are capable of addressing only the equilibrium thermodynamics of that class of black holes whose surface gravity is constant over the event horizon. Consequently, the "zeroth law" of black hole thermodynamics will be adopted by fiat. By judicious use of several integrations by parts this integral will be transformed into a surface integral over the two-dimensional event horizon. To show this one must first introduce some extra technical machinery.
Metric
Static geometry
In the case of a static, possibly aspheric, black hole the Euclidean signature metric can be cast into the form
g = +N 2 dt ⊗ dt + g ij dx i ⊗ dx j .(5)
The quantity N is known as the lapse function. The event horizon occurs at N = 0. The timelike Killing Vector is given by K ≡ ∂/∂t. In coordinates K µ = (1, 0, 0, 0); K µ = (N 2 , 0, 0, 0). FIDOS [fiducial observers] follow integral curves of the Killing vector, thus the four-velocity of a FIDO is V ≡ K/||K||. In coordinates V µ = (1/N, 0, 0, 0), V µ = (N, 0, 0, 0). Consider the one-form dt. Note that ||dt|| = 1/N . In coordinates (dt) µ = (1, 0, 0, 0); (dt) µ = (1/N 2 , 0, 0, 0). Consequently the one-form dt and Killing vector K are parallel, indeed dt = K/N 2 .
The four-acceleration of a FIDO is given by a ≡ (V · ∇)V . In coordinates a µ = V ν ∇ ν V µ = −(1/N )g µν ∇ ν N . Define the unit normal to the constant lapse hypersurface by n µ , then by construction a µ = −||a||n µ . Using the fact that the Killing vector is hypersurface orthogonal, a brief computation shows
∇ µ K ν = − 1 N (K µ ∇ ν N − K ν ∇ µ N ) = (K µ a ν − a µ K ν ) = {||a||N } (n µ V ν − n ν V µ ) .(6)
Furthermore
∇ µ V ν = V µ a ν = −||a||V µ n ν .
Stationary geometry
In the case of a stationary non-static black hole the Euclidean signature metric can be put into the form (see e.g. [12])
g = +N 2 dt ⊗ dt + g ij (dx i − β i dt) ⊗ (dx j − β j dt).(7)
In this more complicated situation it is possible to distinguish at least four interesting classes of fiducial observers -STATORS, ZEVOS, ZAMOS, and ROTORS. As previously, N is known as the lapse function. The timelike Killing Vector is still K ≡ ∂/∂t. In coordinates K µ = (1, 0, 0, 0). A STATOR [stationary observer at rest] is one who follows the integral curves of the Killing vector.
V µ S = K µ /||K||. Note that ||K|| 2 = N 2 + g ij β i β j .
In Lorentzian signature the vanishing of ||K|| defines the ergosphere, a concept that has no analogue in Euclidean signature. The notion of a STATOR will not prove particularly useful in what follows.
Consider the one-form dt. Even in a stationary [as opposed to static] geometry it is still true that ||dt|| = 1/N . In either Lorentzian or Euclidean signature the vanishing of N = 1/||dt|| defines the event horizon. This one-form may be used to introduce the notion of a "minimally dragged " observer -a ZEVO [zero vorticity observer]. A ZEVO is an observer whose (covariant) four-velocity is defined to be V Z = dt/||dt|| = N dt. The appelation is justified by calculating the vorticity of such a system of observers:
̟ = * (V ∧ dV ) = 0.
In coordinates V µ Z = (1; β i )/N . Define the relevant four-acceleration to be a Z ≡ (V · ∇)V . A brief computation shows that the ZEVOS inherit much but not all of the structure of the FIDOS of a static geometry. For instance
a µ Z = −(1/N )(g µν − V µ V ν )∇ ν N . The projection operator is needed because (V · ∇N ) = (β i ∂ i N )/N = 0, unless further assumptions are made. The surface gravity is defined by κ = lim H {N ||a||} = lim H {||∇ ⊥ N ||}.
Note that if the stationary geometry is in fact static that the system of ZEVOS coincides with the system of STATORS and one recovers the system of FIDOS.
It is believed that every black hole that is stationary but not static must be axially symmetric. Physically, the reason for this is that a rotating (i.e. nonstatic) black hole induces tidal dissipation in any system that is not axially symmetric. The final equilibrium state should thus be either static or axially symmetric. While some rigorous theorems along these lines can be proved for Einstein-Hilbert gravity the situation regarding more general theories is far from clear.
Nevertheless, if one adopts these physical arguments above to justify specializing to axial symmetry the metric may be further reduced to the form (see e.g. [12])
g = +N 2 dt ⊗ dt + g φφ (dφ − ωdt) ⊗ (dφ − ωdt) + g AB dx A ⊗ dx B . (8)
There are now two Killing vectors, the timelike Killing vector K ≡ ∂/∂t, and the axial Killing vectorK ≡ ∂/∂φ. [K µ = (1, 0, 0, 0);K µ = (0, 0, 0, 1).] Because of the axial symmetry it is now possible to define the notion of angular momentum. The notion of the "minimally dragged" ZEVO system discussed above now particularizes to the notion of the ZAMO [zero angular momentum observer]. For a ZAMO:
[V ω ] µ = (1, 0, 0, ω)/N . This implies V ω ∝ (K + ωK). Note that ||K + ωK|| = N . Thus V ω ≡ dt/||dt|| = (K + ωK)/||K + ωK||. Rearranging yields the useful result dt = (K + ωK)/N 2 .(9)
Because V · ∇N ∝ (K + ωK) · ∇N = 0, the formulae for locally measured acceleration and surface gravity simplify from those appropriate to the "minimally dragged" ZEVOS, and one has results more closely related to those of the static FIDOS. For instance, one recovers
a µ = V ν ∇ ν V µ = −(1/N )g µν ∇ ν N , while for the surface gravity κ = lim H {N ||a||} = lim H {||∇N ||}.
Next, define the angular velocity of the event horizon by Ω H ≡ lim H ω. For Kerr and Kerr-Newman black holes it is possible to show, as a mathematical theorem, that Ω H is a constant everywhere on the event horizon. Indeed, for Kerr and Kerr-Newman black holes, as one approaches the horizon ω = Ω H + O(N 2 ). For arbitrary theories with arbitrary stress-energy tensors the truth or falsity of such results is far from clear. To obtain such results would require, at a minimum, the use of the field equations together with some form of the energy conditions. [This parallels the question of the constancy of the surface gravity over the event horizon.] As with the question of the surface gravity, Euclidean signature techniques cannot even be set up unless Ω H is a constant. Physically, this is due to the fact that Euclidean signature techniques are intrinsically limited to the analysis of equilibrium thermodynamics. If Ω H is not a constant then the implied differential rotation leads to shearing and dissipation so that the situation is decidedly not in equilibrium. Consequently the constancy of Ω H will be adopted by fiat. [The assumed constancy of Ω H is equivalent to assuming that the horizon of a stationary axisymmetric black hole is a Killing horizon, cf [6].]
Having done this, it is now possible to introduce a fourth class of fiducial observers -the ROTORS [co-rotating observers]. Consider the Killing vector K Ω = K + Ω HK . In coordinates K µ Ω = (1, 0, 0, Ω H ). Consequently ||K Ω || 2 = N 2 + g φφ (Ω H − ω) 2 . Thus K Ω is that unique Killing vector that is null on the event horizon. The co-rotating observers are defined by V Ω ≡ K Ω /||K Ω ||. Note that the ROTOR system of co-rotating observers and the ZAMO system have the same limit as one approaches the horizon. For convenience I shall sometimes write N Ω for ||K Ω ||. Note that both N and N Ω vanish on the event horizon.
In Lorentzian signature the system of co-rotating fiducial observers breaks down at sufficiently large distances. [K Ω becomes spacelike for Ω H r ≥ c.] There is no analogue of this behaviour in Euclidean signature, and it can be safely ignored.
One
has a Ω ≡ (V Ω · ∇)V Ω = (K Ω · ∇)K Ω /||K Ω || 2 = −∇N Ω /N Ω . The surface gravity is given by κ = lim H {N Ω ||a Ω ||} = lim H {||∇(N Ω )||}.
That this definition in terms of ROTORS coincides with the definition in terms of ZAMOS is yet another manifestation of the fact that these two systems tend to the same limit at the event horizon.
The necessity for this extended discussion of fiducial observers arises from the fact these distinctions are both useful and necessary for the following discussion. For static black holes it suffices to use the simple system of FIDOS. For rotating black holes it is the ROTOR system of co-rotating observers that plays a primary role, first in defining the entropy, and secondly in performing the manipulations to be discussed below. The ZAMO system is also used, but is of secondary importance. It is to be emphasised that whatever stress-energy is surrounding the black hole it must, by the assumed internal equilibrium, be co-rotating with the hole. That is, the four-velocity of the ROTOR system of co-rotating observers must be an eigenvector of the stress-energy tensor.
Action
Take the Euclidean action to be
I tot (g) = I EH (g) + I m (g).(10)
The Einstein-Hilbert action is
I EH (g) = − 1 16πG Ω R √ g d 4 x − 1 8πG ∂Ω K √ 3 g d 3 x,(11)
and consists of: (1) the original Einstein-Hilbert Lagrangian, to be integrated over the entire Euclidean manifold, and (2) the Gibbons-Hawking surface term [11]. Here 3 g denotes the induced three-metric on the three-dimensional hypersurface ∂Ω, while K denotes the trace of the second fundamental form. By the assumed asymptotic flatness of the black hole spacetime this term is to be integrated only over the three surface at spatial infinity [11,15].
For an arbitrary variation of the metric
δI EH (g) = 1 16πG Ω G µν δ(g µν ) √ g d 4 x − ∂Ω Θ EH (δg) √ 3 g d 3 x.(12)
The surface term, Θ EH depends in a linear fashion on δg and its first derivative. For the augmented Einstein-Hilbert action, it is a special case result that these surface terms Θ EH vanish provided that δg, though not necessarily its normal derivative, vanishes on the boundary. The "matter" action is of the form
I m (g) = Ω L √ g d 4 x.(13)
Here L denotes the Euclideanized "matter" Lagrangian (All higher order geometrical terms [e.g. (Riemann) n ] are lumped into this "matter" Lagrangian.) For an arbitrary variation of the metric
δI m (g) = 1 2 Ω t µν δ(g µν ) √ g d 4 x − 1 2 ∂Ω Θ(δg) √ 3 g d 3 x.(14)
The surface term, Θ depends in a linear fashion on δg and its first n − 1 derivatives, where n denotes the highest order of the metric derivatives appearing in L. In general, there is no particular reason to expect Θ to vanish unless δg and its first n − 1 normal derivatives vanish on the boundary. The Einstein-Hilbert Lagrangian is special in this regard, as is the Lovelock Lagrangian [16].
Lemma: volume term versus surface term
Static geometries
For clarity, I shall first discuss the case of a static, possibly aspheric, geometry. Consider the object Ω ̺ √ gd 4 x. For the time being, let Ω denote a four-volume that is bounded by hypersurfaces of constant lapse N . Let ∂Ω denote its threeboundary, whose normal is by construction orthogonal to the Killing flow. Note that by definition
̺ = t µν V µ V ν = t µν ∇ µ t K ν . Thus by considering δ(g µν ) ≡ ǫV µ V ν , one has Ω ̺ √ gd 4 x = Ω t µν V µ V ν √ gd 4 x (15) = d dǫ 2I m (g + ǫV V ) + ∂Ω Θ(δg = ǫV ⊗ V ) √ 3 g d 3 x .(16)
The derivative in the above equation is to be evaluated at ǫ = 0. Introduce the notation g ǫ ≡ g + ǫV V . Differentiation yields
d dǫ [I m (g + ǫV V )] = d dǫ Ω L(g ǫ ) √ g ǫ d 4 x = 1 2 Ω L √ g d 4 x + Ω dL dǫ √ g d 4 x.(17)
Again, everything in the above equation is evaluated at ǫ = 0. Now note that the substitution g → g ǫ ≡ g + ǫV ⊗ V merely corresponds to a coordinate change, a rescaling of the time direction by an amount √ 1 + ǫ. [One might also profitably think of this as a time dilation.] To be more explicit
δg µν = ǫV µ V ν = ǫ∇ (µ t K ν) = ∇ (µ [ǫt K] ν) .(18)
Under a coordinate change x µ → x µ + ξ µ (x), any arbitrary scalar transforms as δL = ξ µ (x)∂ µ L. Because the particular coordinate change under consideration is parallel to the Killing vector, the value of the Lagrangian is unaltered. That is: dL/dǫ = 0. Introduce the notation f µν = V µ V ν = ∇ (µ t K ν) . Consequently, for any four-volume Ω, bounded by constant lapse hypersurfaces, one has:
Ω {̺ − L} √ g d 4 x = ∂Ω Θ(δg = f ) √ 3 g d 3 x.(19)
Stationary geometries
One must now repeat a minor variant of the above analysis, with additional technical complications to take care of the black hole's rotation. Recall that by the assumed internal equilibrium of the distribution one can show [1] that the stress energy tensor has as one of its eigenvectors the four-velocity of the ROTOR system of co-rotating observers, V Ω , with the associated eigenvalue being the energy density, ̺. Indeed
t µ ν (V Ω ) ν = ̺(V Ω ) µ .(20)
Now it is certainly true, but not useful, to observe that in the stationary case
̺ = t µν (V Ω ) µ (V Ω ) ν .
The reason that it is not useful is that explicit computation shows that it is not possible to interpret δg = (V Ω ) ⊗ (V Ω ) in terms of the effects of a coordinate transformation. This is, fortunately, only a technical difficulty and not a fundamental problem. Introduce the notation V ⊥ to denote some arbitrary four-vector that is constrained only by the fact that it is assumed to be perpendicular to V Ω . That is V ⊥ · V Ω ≡ 0. Then, because V Ω is an eigenvector of the stress-energy tensor, for any such V ⊥ one has
̺ = t µν (V Ω ) µ [(V Ω ) ν + (V ⊥ ) ν ].
The trick is to pick V ⊥ in some appropriate manner. Without further ado, consider
V ⊥ = ||K Ω ||dt − V Ω .(21)
Note that dt · V Ω = dt · K Ω /||K Ω || = 1/||K Ω ||, so that the perpendicularity requirement is indeed satisfied. Furthermore, by explicit construction,
V Ω ⊗ [V Ω + V ⊥ ] = K Ω ⊗ dt. Consequently ̺ = t µν (K Ω ) µ (dt) ν .(22)
Now repeat the analysis used for the static case, this time considering
δg µν = ǫ∇ (µ t [K Ω ] ν) = ∇ (µ {ǫt [K Ω ]} ν) .(23)
This is nothing more than the effect of the coordinate change x µ → x µ +ǫt[K Ω ] µ . Consequently the logic of the preceding case continues to hold, and the lemma is not disturbed by the black hole's rotation. For this stationary case introduce the notation f µν = ∇ (µ t [K Ω ] ν) . Again, for any four-volume Ω, bounded by constant lapse hypersurfaces, one has:
Ω {̺ − L} √ g d 4 x = ∂Ω Θ(δg = f ) √ 3 g d 3 x.(24)
Entropy
The preceding lemma essentially solves the problem. Apply the lemma to the volume term in the entropy formula. As one pushes Ω outward to cover the whole Euclidean four-manifold two potential sources of surface term should be considered: surface terms arising at spatial infinity, and surface terms arising at the horizon. The surface terms arising at spatial infinity should be quietly discarded by the assumed asymptotic flatness of spacetime. The only remaining piece is the boundary term at the horizon (cf [17]). A suitably careful definition of the entropy is in terms of the limit
S = kA H 4ℓ 2 P + k h lim H ∂Ω Θ(f ) √ 3 gd 3 x.(25)S = kA H 4ℓ 2 P + k H β lim H [N Θ(f )] √ 2 g d 2 x.(26)
This can be interpreted in terms of a surface entropy density defined on the event horizon:
S = kβ lim H [N Θ(f )] . (27) Whence S = kA H 4ℓ 2 P + H S √ 2 g d 2 x.(28)
An interesting non-zero result is obtained only if Θ(f ) blows up as N → 0. To see why and when this occurs requires a deeper understanding of the surface term. This is as far as I have currently been able to push the program in the general case. Further advances seem to require the choice of some specific class of Lagrangian as template.
Finally, let us reinstate the (volume) entropy density term associated with the statistical and quantum fluctuations occurring outside the black hole event horizon. Then
S = kA H 4ℓ 2 P + H S √ 2 g d 2 x + Σ s √ 3 g d 3 x.(29)
Insofar as the quantum fluctuations can be described by some effective Lagrangian L ef f , they may be extracted from the volume density s, and pushed into the surface density term S. This trade-off between volume and surface effects parallels the trade-off between integrating out fast modes (and describing them by an effective Lagrangian), and keeping the slow modes available for explicit computation.
SPECIFIC EXAMPLES
L = L(Riemann)
Consider now the case where one takes L to be some arbitrary function of the Riemann tensor, (though not of its derivatives). Interest in this class of Lagrangians is justified on the grounds that any quantum theory of gravity will induce terms of this type in the low-energy effective theory. Many of the examples considered in the literature are special cases of this reasonably large class. By the preceding general analysis, evaluation of the entropy is equivalent to the determination of the value of the surface term Θ at the horizon. This surface term is best evaluated by indirection. Define the object:
J µν λρ ≡ ∂L ∂R µν λρ .(30)
So that in particular δ(L) = J µν λρ δ(R µν λρ ) .
Without loss of generality one may take J µν λρ to inherit the symmetry structure of the Riemann tensor itself. Specifically J µν λρ = J [µν]
[λρ] = J λρ µν . Consider a general variation of the metric. Define δg µ ν = g µσ δg σν . One has
δ(R µν λρ ) = −2∇ [µ ∇ [λ (δg) ν] ρ] + R µν σ[λ δg σ ρ] .(32)
This allows us to write
Ω δ(L) √ g d 4 x = Ω J µν λρ −2∇ µ ∇ λ (δg) ν ρ + R µν σλ (δg) σ ρ √ g d 4 x. (33)
Here one has been able to drop the explicit antisymmetrization in view of the symmetry properties of J µν λρ itself. From the above, one reads off
t µν = −2∇ α ∇ β J αµβν − 2∇ α ∇ β J ανβµ −J αβγµ R αβγ ν − J αβγν R αβγ µ + Lg µν .(34)
Recall the notation
f µν ≡ [K Ω ] (µ ∇ ν) t.(35)
For the static case interpret Ω H = 0, and discard the axial symmetry. Thus this definition is seen to make sense for both the static [aspheric] and stationary axisymmetric cases. In either case we have by construction ̺ = t µν f µν . Construct the integral
X ≡ Ω {̺ − L} √ gd 4 x = Ω {t µν f µν − L} √ g d 4 x.(36)
Then
X = Ω {−4∇ α ∇ β J αµβν − 2J αβγµ R αβγ ν }f µν √ gd 4 x.(37)
Integrate by parts once
X = Ω +4∇ β J αµβν ∇ α (f µν ) + −2J αβγµ R αβγ ν f µν √ g d 4 x − 4 ∂Ω n α (∇ β J αµβν )f µν √ 3 g d 3 x.(38)
Integrate by parts a second time
X = Ω −4J αµβν ∇ β ∇ α (f µν ) + −2J αβγµ R αβγ ν f µν √ g d 4 x − 4 ∂Ω n α (∇ β J αµβν )f µν − n β J αµβν (∇ α f µν ) √ 3 g d 3 x.(39)
Rearrange
X = −2 Ω J µνλρ {2∇ µ ∇ λ (f νρ ) + R µνλ σ f ρσ } √ g d 4 x − 4 ∂Ω n α (∇ β J αµβν )(f µν ) − J αµβν (∇ α f µν )n β √ 3 g d 3 x(40)
The volume integral above vanishes identically. To see this, note that after appropriate explicit antisymmetrization the volume term is just
Ω J µνλρ δ(R µνλρ ) √ g d 4 x.(41)
Where δ(R µνλρ ) is just that due to taking δ(g µν ) = f µν . But, as we have already seen f µν = ∇ (µ [tK Ω ] ν) , which corresponds to just the effect of a coordinate transformation. The evaluation of the surface terms proceeds as follows. First note that the surface term at spatial infinity is automatically suppressed by the assumption of asymptotic flatness. Second, near the horizon
√ 3 g d 3 x → N √ 2 g d 2 xdt.
Since the Riemann tensor and its derivatives are well behaved at the horizon, as are the limits of n α and f µν , it is easy to see that the first surface term vanishes, being suppressed by the factor of N in the metric determinant.
The only remaining term is
X = 4 lim H ∂Ω J αµβν ∇ α (f µν )n β √ 3 g d 3 x.(42)
This formula is, of course, nothing more nor less than the special case explicit evaluation of the surface term Θ(f ) previously encountered in the general argument. At this stage it proves useful to treat the stationary and static cases separately. Static geometry:
For a static geometry f µν = V µ V ν . The gradient term includes pieces such as
∇ α (V µ V ν ) = (∇ α V µ )V ν + V µ (∇ α V ν) = −||a|| (V α n µ V ν + V µ V α n ν ) .(43)
Now, take the limit as one approaches the horizon. The only surviving term in the surface integral comes from the cancellation between the N arising from the metric determinant and ||a||. Note that
lim H hβ 0 dtN ||a|| = lim Hh βN ||a|| = 2π.(44)
This yields
Ω {̺ − L} √ g d 4 x = 8π H J αµβν V α V β n µ n ν √ 2 g d 2 x.(45)
Stationary geometry:
The limiting procedure is now a little more delicate, and requires some tedious technical steps. Recall that f µν = [K Ω ] (µ [dt] ν) . The gradient term includes pieces such as
∇ α ([K Ω ] µ [dt] ν ) = (∇ α [K Ω ] µ )[dt] ν + [K Ω ] µ (∇ α ∇ ν t).(46)
Because the Killing vector K Ω is not hypersurface orthogonal, in general, the best we can say is that the covariant derivative of the Killing vector satisfies
∇ µ [K Ω ] ν = 2(a Ω ) [µ (K Ω ) ν] + π µν .(47)
Here π µν is an antisymmetric tensor orthogonal to K Ω . On the other hand, it is known that K Ω is hypersurface orthogonal on the event horizon, so that π µν vanishes in that limit. Consequently
∇ µ [K Ω ] ν = −2N Ω ||a Ω ||n [µ (V Ω ) ν] + O(N Ω ).(48)
Now consider
(∇ µ ∇ ν t) = ∇ µ [(K ν + ωK ν )/N 2 ] = −2N −3 ∇ (µ N [K + ωK] ν) + N −2 ∇ (µ ωK ν) = −2N −1 [a ω ] (µ [V ω ] ν) + N −2 ∇ (µ ωK ν) = +2 ||a ω || N [n ω ] (µ [V ω ] ν) + N −2 ∇ (µ ωK ν) .(49)Now because ω = Ω H + O(N 2 ) one has ∇ω = O(N ∇N ) = O(N 2 a). So finally (∇ µ ∇ ν t) = +2 ||a ω || N [n ω ] (µ [V ω ] ν) + O(N ) .(50)
Now, take the limit as one approaches the horizon. The ZAMOS, V ω , and the ROTORS, V Ω , approach the same limit. Ditto for the relevant normals. Again, the only surviving term in the surface integral comes from the cancellation between the N arising from the metric determinant ||a ω ||, and ||a Ω ||. Various subdominant pieces vanish in the limit. As before, this yields
Ω {̺ − L} √ g d 4 x = 8π H J αµβν V α V β n µ n ν √ 2 g d 2 x.(51)
Returning to the general case (static or stationary), and backtracking to the general entropy formula, one now obtains
S = kA H 4ℓ 2 P + 8π k h H J µνλρ V µ n ν V λ n ρ √ 2 g d 2 x.(52)
A further refinement is to define g ⊥ µν ≡ V µ V ν + n µ n ν . This is essentially the metric in the two directions perpendicular to the event horizon, in terms of which the symmetries of J µνλρ imply
S = kA H 4ℓ 2 P + 4π k h H J µνλρ g ⊥ µλ g ⊥ νρ √ 2 g d 2 x.(53)
This is our final form for the entropy. Note that it exhibits all of the properties expected from the general analysis.
L = L(Ricci)
A further specialization of the above, is to consider the case where the Lagrangian is an arbitrary function of the Ricci tensor, rather than the full Riemann tensor. The analysis is straightforward.
DefineJ µν ≡ ∂L ∂R µν .(54)
Then
J µν λρ ≡ ∂L ∂R µν λρ =J [µ [λ g ν] ρ] = 1 4 J µ λ g ν ρ −J µ ρ g ν λ +J ν ρ g µ λ −J ν λ g µ ρ .
(55) Insert into the previous formula, one extracts
S = kA H 4ℓ 2 P + 2π k h H ∂L ∂R µν g ⊥ µν √ 2 gd 2 x.(56)
This formula is instructively similar to that obtained by Jacobson, Kang, and Myers [3,4]. Using field redefinition techniques under conformal rescalings they were limited to the case L = R µν R µν + L matter . (The extra matter was required to enforce a nontrivial solution to the fields equations, it was assumed that the extra matter was sufficiently well behaved not to contribute to the entropy in its own right.) Under these assumptions, Jacobson, Kang, and Myers showed that
S = kA H 4ℓ 2 P + 4π k h H R µν g ⊥ µν √ 2 gd 2 x.(57)
L = L(T r[Ricci])
A completely analogous analysis can be applied in the case that the Lagrangian is an arbitrary function of the scalar Ricci curvature.
Consider J µν λρ ≡ ∂L ∂R µν λρ = ∂L ∂R g [µ [λ g ν] ρ] .(58)
Insert into the general result, one obtains
S = kA H 4ℓ 2 P + 4π k h H ∂L ∂R √ 2 gd 2 x.(59)
This is exactly the result enunciated by Jacobson, Kang, and Myers [3,4]. A simple consistency check is to lump the Einstein-Hilbert action in with L. Taking L = 1 16πG R =h 16πℓ 2 P R reproduces the ordinary area term.
Lovelock gravity
As a final example, I shall discuss Lovelock gravity. While the analysis presented so far has, for definiteness, been presented in four dimensions there is nothing essentially four-dimensional about these techniques. In D dimensions the Lovelock Lagrangian is given by (see e.g. [2]).
L = [D/2] m=0 c m L m .(60)
In this sum, [D/2] indicates the integer part of D/2. The individual terms are given by
L m (g) = 1 2 m δ µ1ν1···µmνm λ1ρ1···λmρm R µ1ν1 λ1ρ1 · · · R µmνm λmρm .(61)
The δ symbol is a totally antisymmetric product of 2m Kronecker deltas, suitably normalized to take values 0 and ±1. It is convenient to define L 0 = 1, this term corresponding to a cosmological constant. Furthermore L 1 = R is the Einstein-Hilbert Lagrangian. In general, L m is the Euler density for a 2mdimensional manifold. Because of the antisymmetrization, no derivative appears at higher than second order in the equations of motion. For the purposes currently at hand, consider the object
[J m ] µν λρ ≡ ∂L m ∂R µν λρ = m 2 m δ µ1ν1···µm−1νm−1µν λ1ρ1···λm−1ρm−1λρ R µ1ν1 λ1ρ1 · · · R µm−1νm−1 λm−1ρm−1 .(62)
Applying the general formula, the contractions with g ⊥ , together with the total antisymmetrization of the indices, imply that the only components of Riemann tensor that contribute to the entropy density are those that are tangential to the D − 2 dimensional event horizon. Specifically, we note that
δ λ1ρ1···µm−1νm−1µν λ1ρ1···λm−1ρm−1λρ (g ⊥ ) [λ [µ (g ⊥ ) ρ] ν] =δ µ1ν1···µm−1νm−1 λ1ρ1···λm−1ρm−1 .(63)
Hereδ is the totally antisymmetric product of 2(m − 1) Kronecker deltas, restricted to the subspace orthogonal to g ⊥ . The rest of the derivation now parallels that due to Jacobson and Myers [2]. The entropy is
S = 4π k h H [D/2] m=1 m c m L m−1 (h) √ h d D−2 x.(64)
In this particular case the entropy is given solely in terms of the intrinsic geometry (h) of the event horizon -this result is special to this particular type of Lagrangian and does not generalize.
DISCUSSION
The computation of black hole entropies in various model theories is an issue of great current interest. Based on the work of a several authors the situation has by now become significantly clarified. Several points are worth making. First: The naive area law for black hole entropies is in general false.
S = kA H 4ℓ 2 P .(65)
The naive law certainly holds for Einstein-Hilbert gravity coupled to matter whose kinetic energy is quadratic [1]. Once one moves beyond quadratic kinetic energies the naive law fails in general. Second: For a general higher derivative Lagrangian the entropy of a black hole is given by an integral of some suitable density over (a fixed time spacelike cross-section of) the event horizon
S = kA H 4ℓ 2 P + H S √ 2 g d 2 x.(66)
The entropy surface density is a simple function of the surface term that connects the stress-energy tensor with the variation of the action under a variation of the spacetime metric. Third: In the specific case of a Lagrangian that is solely a function of the Riemann tensor
S = kA H 4ℓ 2 P + 4π k h H ∂L ∂R µνλρ g ⊥ µλ g ⊥ νρ √ 2 g d 2 x.(67)
This relatively general formula can be checked against a number of more specific examples where the entropy is known by other means. The fact that different types of calculation give the same answer where they overlap is certainly encouraging. Fourth: The present paper has obtained its results via extensive use of Euclidean signature techniques. The underlying physics is perhaps somewhat obscured by this formalism. It is encouraging to note that similar results have by now been presented using a number of different techniques [3,4,5,6]. The overall agreement between these various different techniques is a further useful consistency check.
The surface gravity is defined byκ = lim H {N ||a||} = lim H {||∇N ||} = lim H {∂N/∂η}.
Acknowledgements This research was supported by the U.S. Department of Energy. I wish to thank Ted Jacobson, Robert Myers, and Robert Wald for stimulating discussions, and for a reading of the manuscript.
Electronic mail: visser@kiwi. wustl.eduElectronic mail: [email protected]
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Black hole entropy and higher curvature interactions, talk delivered by T. Jacobson at the conference on Quantum aspects of black holes. T Jacobson, G Kang, R Myers, Santa BarbaraT. Jacobson, G. Kang, and R. Myers, Black hole entropy and higher curvature interactions, talk delivered by T. Jacobson at the conference on Quantum aspects of black holes, Santa Barbara, June 1993.
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| [] |
[
"HODGELETS: LOCALIZED SPECTRAL REPRESENTATIONS OF FLOWS ON SIMPLICIAL COMPLEXES",
"HODGELETS: LOCALIZED SPECTRAL REPRESENTATIONS OF FLOWS ON SIMPLICIAL COMPLEXES",
"HODGELETS: LOCALIZED SPECTRAL REPRESENTATIONS OF FLOWS ON SIMPLICIAL COMPLEXES",
"HODGELETS: LOCALIZED SPECTRAL REPRESENTATIONS OF FLOWS ON SIMPLICIAL COMPLEXES"
] | [
"T Mitchell Roddenberry ",
"Florian Frantzen \nDept. of Computer Science\nRWTH Aachen University\nAachenGermany\n",
"Michael T Schaub \nDept. of Computer Science\nRWTH Aachen University\nAachenGermany\n",
"Santiago Segarra ",
"\nDept. of Electrical and Computer Engineering\nRice University\nHoustonTXUSA\n",
"T Mitchell Roddenberry ",
"Florian Frantzen \nDept. of Computer Science\nRWTH Aachen University\nAachenGermany\n",
"Michael T Schaub \nDept. of Computer Science\nRWTH Aachen University\nAachenGermany\n",
"Santiago Segarra ",
"\nDept. of Electrical and Computer Engineering\nRice University\nHoustonTXUSA\n"
] | [
"Dept. of Computer Science\nRWTH Aachen University\nAachenGermany",
"Dept. of Computer Science\nRWTH Aachen University\nAachenGermany",
"Dept. of Electrical and Computer Engineering\nRice University\nHoustonTXUSA",
"Dept. of Computer Science\nRWTH Aachen University\nAachenGermany",
"Dept. of Computer Science\nRWTH Aachen University\nAachenGermany",
"Dept. of Electrical and Computer Engineering\nRice University\nHoustonTXUSA"
] | [] | We develop wavelet representations for edge-flows on simplicial complexes, using ideas rooted in combinatorial Hodge theory and spectral graph wavelets. We first show that the Hodge Laplacian can be used in lieu of the graph Laplacian to construct a family of wavelets for higher-order signals on simplicial complexes. Then, we refine this idea to construct wavelets that respect the Hodge-Helmholtz decomposition. For these Hodgelets, familiar notions of curl-free and divergence-free flows from vector calculus are preserved. We characterize the representational quality of our Hodgelets for edge flows in terms of frame bounds and demonstrate the use of these spectral wavelets for sparse representation of edge flows on real and synthetic data. | 10.1109/icassp43922.2022.9747203 | [
"https://export.arxiv.org/pdf/2109.08728v1.pdf"
] | 237,572,315 | 2109.08728 | d2d7dca8c481f03571bfc71e69f10e6b8a61191a |
HODGELETS: LOCALIZED SPECTRAL REPRESENTATIONS OF FLOWS ON SIMPLICIAL COMPLEXES
T Mitchell Roddenberry
Florian Frantzen
Dept. of Computer Science
RWTH Aachen University
AachenGermany
Michael T Schaub
Dept. of Computer Science
RWTH Aachen University
AachenGermany
Santiago Segarra
Dept. of Electrical and Computer Engineering
Rice University
HoustonTXUSA
HODGELETS: LOCALIZED SPECTRAL REPRESENTATIONS OF FLOWS ON SIMPLICIAL COMPLEXES
Index Terms-Graph signal processingHodge LaplacianSimplicial complexWaveletDiscrete calculus
We develop wavelet representations for edge-flows on simplicial complexes, using ideas rooted in combinatorial Hodge theory and spectral graph wavelets. We first show that the Hodge Laplacian can be used in lieu of the graph Laplacian to construct a family of wavelets for higher-order signals on simplicial complexes. Then, we refine this idea to construct wavelets that respect the Hodge-Helmholtz decomposition. For these Hodgelets, familiar notions of curl-free and divergence-free flows from vector calculus are preserved. We characterize the representational quality of our Hodgelets for edge flows in terms of frame bounds and demonstrate the use of these spectral wavelets for sparse representation of edge flows on real and synthetic data.
INTRODUCTION
There has been substantial interest in graph-based techniques to understand data with a complex relational structure [1][2][3], with applications ranging from biology [4] to system robustness [5]. In this context, graph signal processing (GSP) has proven to be a useful way to understand the processing of signals defined on graphs, leveraging ideas from both signal processing and graph theory [6]. The primary focus of GSP has been on signals supported on the nodes of a graph. For such signals, the graph Laplacian and adjacency matrix are natural shift operators, from which we can define notions of filtering and Fourier transformations [6].
However, there has been a recent flurry of interest in studying flows on the edges of graphs and simplicial complexes [7][8][9][10][11][12], which can be used to model the motion of mass, energy, or information. Since flows carry a natural orientation that does not arise when studying signals on the nodes of a graph, recent works have leveraged tools from algebraic topology [13] and discrete exterior calculus [14] to form appropriate Laplace operators that respect the orientation of edge flows. This approach has allowed for the study of edge flows through the lens of the celebrated Hodge-Helmholtz decomposition [7][8][9][10]. This viewpoint has even been leveraged to define neural network architectures for edge flows [15][16][17][18][19].
In the literature thus far, the primary focus has been on understanding filtering and signal representation in the spatial and Fourier domains, where we take the Fourier modes to be the eigenvectors of a suitably defined Laplacian. However, just as in classical signal This work was partially supported by USA NSF under award CCF-2008555. FF and MTS acknowledge partial support from the Excellence Strategy of the Federal Government and the Länder in Germany, and the NRW Rückkehrprogramm. processing, the Fourier modes are highly delocalized. That is to say, the support of a Fourier mode is typically not restricted to one small region of the graph. In GSP, this has motivated the development of spectral graph wavelets [20,21], which proposes to use a dictionary of atoms for signal representation that is localized in both the spatial and frequency domains. Here, we introduce a family of wavelets for edge flows, seeking to balance localization in the spatial and frequency domains, while also respecting the Hodge decomposition. Contributions and outline. We consider the design of spectral wavelets for edge flows on simplicial complexes. In particular, we discuss how the orthogonal decomposition of the space of edge flows in terms of the Hodge Laplacian can be leveraged to design interpretable wavelets that yield high-quality sparse and localized representations of edge flows.
We begin by discussing preliminaries in graph signal processing for edge flows in Section 2. Then, we propose a simple construction for spectral graph wavelets based on previous literature in Section 3, as well as a modification that respects the Hodge decomposition. Theoretical properties of both models are considered in Section 4. In particular, we state frame bounds for both models, in terms of the family of spectral kernels used in their definition. Finally, we demonstrate the utility of our constructions for sparse representation and flow clustering on real and synthetic data in Section 5.
NOTATION AND BACKGROUND
For a positive integer N , we denote the set of integers {1, 2, . . . , N } by [N ]. We use ∼ = to denote isomorphism between vector spaces, and ⊕ to denote the orthogonal direct sum of vector spaces. For a linear operator A between two vector spaces, we denote the set of eigenvalues of A by s(A). Simplicial complexes and the Hodge Laplacian. We consider data supported on (abstract) simplicial complexes, which generalize graphs to allow for higher-order connectivity. An (abstract) simplicial complex X is a finite collection of finite sets that is closed under restriction: that is to say, for any σ in X, all nonempty subsets of σ are also contained in X. We call the elements of X simplices and denote by X k the set of all simplices in X with cardinality k + 1, also referred to as k-simplices. In particular, X0 is the set of all singleton sets in X, X1 is the set of all simplices with cardinality 2, and so on. Grounded in our intuition for graphs, we call X0 the set of nodes in X, X1 the set of edges in X, and X2 the set of triangles.
We identify the set X k with the integers [N k ], and denote the cardinality of X k by N k . By convention, we label the nodes with 1, 2, . . . , N0. Further, we assign to each k-simplex an orientation 1 , or a canonical ordering, following the ordering induced by the node labeling, e.g., a triangle {i, j, k} is given the orientation [i, j, k], where i < j < k. Given this reference orientation, we define the space of k-cochains, denoted by C k (X), as the vector space of functions from the oriented simplices in X k to R. One can check that for each k, C k (X) is naturally isomorphic to R N k . For a given C k (X), we take as an orthonormal basis for that space the set of functions {eσ}σ∈X k taking unit value on each oriented k-simplex σ, and zero elsewhere. Of particular interest is the space C 1 (X), which models flows on the edges of a graph or simplicial complex [7][8][9][10].
The spaces C k (X) are related by the set of incidence matrices. The 2nd incidence matrix ∂2 : C 2 (X) → C 1 (X) is a linear map defined over the standard orthonormal basis for C 2 (X) as follows: k] . Similarly, the 1st incidence matrix ∂1 : C 1 (X) → C 0 (X) is the familiar node-edge incidence matrix, defined according to ∂1e [i,j] = e [j] − e [i] . We also call the matrices ∂2, ∂1 the boundary maps, since they map each k-simplex to the k − 1-simplices on its boundary, as well as obeying the important property ∂1∂2 = 0. The following well-known result characterizes the space C 1 (X) in terms of these maps.
∂2e [i,j,k] = e [i,j] +e [j,k] −e [i,
Theorem 1 (Hodge Decomposition). Let X be a finite simplicial complex. Define the first Hodge Laplacian as ∆1 = ∂2∂ 2 + ∂ 1 ∂1.
The vector space C 1 (X) can be written as the direct sum of orthogonal subspaces:
C 1 (X) ∼ = Im(∂2) ⊕ Im(∂ 1 ) ⊕ ker(∆1).(1)
See [23] for the proof. Viewing the standard graph Laplacian ∆0 = ∂1∂ 1 as an operator that measures smoothness for node signals based on their edgewise incidence, the Hodge Laplacian ∆1 similarly measures "smoothness" for 1-cochains based on nodewise and trianglewise incidence. For convenience, we define the upper Hodge Laplacian ∆ U 1 = ∂2∂ 2 and the lower Hodge Laplacian
∆ L 1 = ∂ 1 ∂1, so that ∆1 = ∆ U 1 + ∆ L 1 .
Spectral graph wavelets.
A key component of many modern signal processing and machine learning tasks is the choice of a proper representation for the data. In graph signal processing, this often amounts to using a spatial representation or a frequency domain representation. In time-domain signal processing, one can construct a dictionary of wavelets that interpolates between these two extremes [24]. Similarly, in [20,21], methods to construct spectral graph wavelets are proposed, in which a family of kernel functions {gm : R → R}m∈Γ for some index set Γ is applied to the standard basis for C 0 (X) on a graph X to yield a dictionary for localized, bandlimited representations of graph signals as follows:
D = {ψj,m = gm(∆0)ej : m ∈ Γ, j ∈ [N0]} ,(2)
where ∆0 indicates the graph Laplacian.
AN INTERPRETABLE SPECTRAL WAVELET MODEL
Much like in graph signal processing, we seek useful representations of edge flows that balance spatial localization and bandlimitedness in the frequency domain. As done by [11], we can define a simplicial Fourier transform by projecting an edge flow onto each eigenvector of ∆1. To illustrate this, we construct a simplicial complex by picking 40 points randomly distributed in the unit square, then taking their Delaunay triangulation [25]. We then create one hole in this simplicial complex, and construct the Hodge Laplacian ∆1 for the remaining structure. We illustrate this complex as well as one of the eigenvectors in its nullspace in Fig. 1 (a). We see that the flow "wraps around" the hole of the complex, an inherently global phenomenon, but does not have localized support on the complex. In the ensuing discussion, we aim to construct dictionaries for edge flows that find a balance in this local-global tradeoff. Jointly designed wavelets. As the Hodge Laplacian generalizes the graph Laplacian for higher-order signals, it is natural to apply the methods of [20,21] for constructing graph wavelets by substituting the Hodge Laplacian for the graph Laplacian. In that direction, let {ej} N 1 j=1 be the standard orthonormal basis for C 1 (X), and let {gm} M m=1 be a set of continuous, non-negative functions on the real line. For each j ∈ [N1], m ∈ [M ], we define the atom ψj,m by applying the polynomial gm(∆1) to ej, where we have implicitly labeled the oriented 1-simplices with the integers [N1]. That is, if the Hodge Laplacian ∆1 admits an eigendecomposition
{(λi, vi)} N 1 i=1 , ψj,m = gm(∆1)ej = N k i=1 gm(λi)viv i ej.(3)
Since this construction uses the sum of the upper and lower Hodge Laplacians and thus both components of the Laplacian, we refer to such atoms as joint Hodgelets.
We illustrate this approach in Fig. 1 (b), using the same simplicial complex as before. We construct wavelets using the log-scaled Hann kernels of [21], and illustrate a single wavelet atom. Note that this atom is spatially localized, due to the spectral kernel being wellapproximated by a low-order polynomial of the Hodge Laplacian. Separately designed wavelets. The approach of directly applying the wavelet construction of [20,21] using the Hodge Laplacian presents some shortcomings. Importantly, it fails to differentiate between the spectral features of the Hodge Laplacian due to the upper and lower components being present in each atom. To address this issue, we treat each subspace of the Hodge decomposition [cf. Theorem 1] separately, by defining upper and lower wavelets for C 1 (X).
A similar approach was taken in the design of filters for signals on simplicial complexes by [11].
As before, let {ej} N 1 j=1 be the standard orthonormal basis for
C 1 (X), and let {g U m } M U m=1 , {g L m } M L m=1 be sequences of continuous functions on the real line. For each j ∈ [N k ], m ∈ [MU ], m ∈ [ML], define ψ U j,m = g U m (∆ U 1 )ej ψ L j,m = g L m (∆ L 1 )ej,(4)
where g U m (∆ U 1 ), g L m (∆ L 1 ) are defined via the functional calculus as before. The set {ψ U j,m } forms what we call the upper atoms, and similarly {ψ L j,m } forms the set of lower atoms for C 1 (X). Since this construction separates the upper and lower components of the Hodge Laplacian, we refer to such atoms as separate Hodgelets.
By separately treating the upper and lower Hodge Laplacian, we can construct atoms with greater interpretability than those designed jointly. In particular, we can show the following result: Proposition 1. Suppose g U m and g L m are kernels that take value 0 at 0. Then, for all j, j ∈ [N1], ψ U j,m ∈ Im(∂2) and ψ L j ,m ∈ Im(∂ 1 ).
Thus, the upper wavelets are dictated by the boundaries of 2simplices (triangles), and the lower wavelets are dictated by the coboundaries of 0-simplices (nodes). We leave the proof to Appendix A. Indeed, Proposition 1 reflects the properties of [26, Theorem 3], in which curl and divergence wavelets are constructed for differential forms in Euclidean space.
We illustrate this in Fig. 1 (c1,c2), by plotting wavelet atoms with the same kernels as the joint wavelet in Fig. 1 (b), except with a separated construction. One can see that there is a clear distinction between the upper wavelet (c1) which corresponds to a curl around triangles, and the lower wavelet (c2) which corresponds to the gradient of a node signal.
FRAME BOUNDS ON DICTIONARIES
In signal processing on graphs, the graph Fourier transform has the appealing property of being an orthogonal transform, thus preserving the norm of the signal it acts upon. Since wavelet dictionaries are typically overcomplete in their construction, we do not have orthogonality, but rather have frame bounds for the dictionary. For a Hilbert space V , a dictionary of vectors D with at most countably many elements is said to be an
(A, B)-frame with 0 ≤ A ≤ B < ∞ if for all v ∈ V , we have A v 2 ≤ ψ∈D | ψ, v | 2 ≤ B v 2 .(6)
If A = B, we say that D forms a tight frame. We allow for the case where A = 0, in which case D is a degenerate frame. The frame bounds of a dictionary dictate its representational quality, as well as the performance of reconstruction algorithms [24]. Moreover, if A = B = 1, the coefficients | ψ, v | 2 are analogous to the spectrogram representation of a signal [21]. Here, in the same vein as [20,21], we characterize the frame bounds for both Hodgelet constructions in terms of the spectral properties of the kernels gm. Joint wavelets. Given that the jointly designed wavelets are a direct adaptation of those proposed in [20,21], we can show a similar result in our context:
Theorem 2.
(based on [20,21]) Let {ej} N 1 j=1 be the standard orthonormal basis of C 1 (X), and let {gm} M m=1 be continuous nonnegative functions on the real line. Let D be the dictionary of atoms defined by (3), and define
G(λ) = M m=1 |gm(λ)| 2 .(7)
Then, D forms an (A, B)-frame for C 1 (X), where
A = min λ∈s(∆ 1 ) G(λ) B = max λ∈s(∆ 1 ) G(λ).(8)
We omit the proof, as it directly mirrors that of [21]. In particular, if G is constant on s(∆0), then D is a tight frame. Separate wavelets. We now state frame bounds for the separate Hodgelet construction, keeping the mutual orthogonality of the upper and lower Hodge Laplacians in mind.
G(µ, ν) = M U m=1 |g U m (µ)| 2 + M L m=1 |g L m (ν)| 2 (9)
Then, D forms an (A, B)-frame for C 1 (X), where
A = min min µ∈s(∆ U 1 ) G(µ, 0), min ν∈s(∆ L 1 ) G(0, ν) B = max max µ∈s(∆ U 1 ) G(µ, 0), max ν∈s(∆ L 1 ) G(0, ν) .(10)
We leave the proof to Appendix B. By Theorem 3, we see that the frame bounds are determined by the quality of the kernels for the upper and lower parts of the spectrum independently. In particular, if G(·, 0) and G(0, ·) are constant on s(∆ U 1 ) and s(∆ L 1 ), respectively, then the dictionary forms a tight frame.
In the context of Proposition 1, Theorem 3 indicates that we can construct norm-preserving representations of edge flows that are also interpretable in terms of harmonic flows in ker(∆1), curl flows in Im(∂2), and divergence flows in Im(∂ 1 ). This is aligned with the development in [11], where edge flow filters were designed using ∆ L 1 and ∆ U 1 separately, rather than the total Hodge Laplacian ∆1.
EXPERIMENTS
We demonstrate the utility of applying spectral wavelets based on the Hodge Laplacian for sparse, localized representations of flow data. For all experiments, 2 we take the spectral kernels {gm} M m=1 to be the log-scaled Hann kernels proposed by [21] with R = 3, where R dictates the degree of overlap between each kernel, and M is chosen based on the particular task. Sparse representations. To illustrate the advantage of using wavelets based on the upper and lower Hodge Laplacians separately, we consider a vector field on [−2, 2] 2 , given by where we define B1, B2 to be closed balls of radius 0.7 centered at (±π/4, ±π/4), respectively, as illustrated in Fig. 2 (a). We discretize [−2, 2] 2 with a hexagonal grid, then construct a simplicial complex X (N0 = 225, N1 = 629, N2 = 405) by treating each hexagon as a node, with edges for each pair of hexagons that share a side, and triangles for each set of three hexagons that share a corner. This vector field is converted to a vector f ∈ C 1 (X) by taking the flow on each edge to be the total flow perpendicular to the corresponding side between hexagons. Then, dictionaries of joint and separated Hodgelets are constructed with M = MU = ML = 4.
F (x, y) = [cos(x + y), sin(x − y)] (x, y) ∈ B1 ∪ B2, [0, 0] otherwise,(11)
The joint dictionary has 2516 atoms and forms a tight frame, while the separated dictionary has 5032 atoms and also forms a tight frame. We now consider the sparsity of the representation of f in each dictionary. For each dictionary D of atoms in C 1 (X), we construct the sparsest linear combination f of atoms in D via orthogonal matching pursuit [27,28] such that f − f ≤ , for ∈ (0, f ] sampled on a logarithmic scale. As a baseline, we repeat this task using the Fourier basis vectors, i.e., the eigenvectors of the Hodge Laplacian, as well as with Fourier basis vectors using the linegraph Laplacian [7], and wavelets constructed from the linegraph Laplacian. The results of this are plotted in Fig. 2 (b), where it is apparent that the separately designed dictionary outperforms the jointly designed one, with both performing better than the Fourier bases and the linegraph dictionary. Clustering buoy trajectories. One setting in which 1-cochains find particular utility is in modeling flows and trajectories [7,18,22]. We consider a dataset from the Global Drifter Program dataset, restricted to the region around Madagascar. 3 In this dataset, a set of buoys floating in the ocean have their location logged every 12 hours, which we use to construct trajectories along the edges of a triangulation (N0 = 133, N1 = 320, N2 = 186) of the region. Since these trajectories consist of a combination of oriented edges, they are naturally modeled as vectors in C 1 (X), following the approach taken by [18,22]. We picture the sum of all P = 334 such trajectories in Fig. 2 (c), where the hole in the simplicial complex corresponds to the landmass of Madagascar.
We aim to find a good set of representative trajectories that captures both the local and global structure of the simplicial complex. For a given dictionary, we first transform each trajectory by taking the inner product with each element of the dictionary. Then, we perform sparse k-means [29,30] with K = 2 clusters on Ptr = 0.75P trajectories in the dataset, which yields a set of centroids as well as feature selection weights.
To evaluate the quality of these centroids, we use the remaining Pts = 0.25P trajectories as a test set, and compute the average normalized inner product between each test trajectory and the nearest centroid flow. That is, for a set of centroids {c k } K k=1 in C 1 (X), we compute for the test set {fj} M ts j=1 the value
L = 1 Pts P ts j=1 max 1≤k≤K fj, c k fj 2 c k 2 ,(12)
where the norms and inner products are taken with respect to the feature weighting kernel obtained from the sparse k-means procedure. If a set of centroids yields a large value of L, indicating that the set of nearest flows aligns to each centroid well, we interpret it to be a realistic model for the trajectories. We gather the results of this experiment when using the standard basis, the Fourier basis, the joint Hodgelet dictionary, and the separate Hodgelet dictionary in Fig. 2 (d). The wavelet dictionaries were constructed with M = M U = M L = 16 filter banks. We observe that the standard basis is not suitable for this task, since two trajectories that are close in space could have disjoint support, so the highly localized basis of edges does not perform well. The Fourier basis and joint dictionary perform similarly, while the separate dictionary significantly outperforms the other representations. This is due to the ability of the sparse clustering algorithm to both pick the proper scale for representation and discern between different qualitative features of the signal in this representation, as revealed by Proposition 1.
CONCLUSION
Signals supported on the edges of simplicial complexes have been of great interest lately, with applications in computer graphics, mobility analysis, and modeling of physical flow phenomena. We have considered the extension of spectral graph wavelets to such signals by replacing the graph Laplacian with the analogous Hodge Laplacian. In doing so, we open up the possibility of considering the upper and lower components of the Hodge Laplacian separately, in order to yield a dictionary of wavelet atoms that respects the Hodge decomposition. Based on these constructions, we state frame bounds for each type of wavelet dictionary, and then illustrate their utility for sparse representation on synthetic flow data. Leveraging the ability of these wavelet dictionaries to sparsely represent flow signals, we demonstrate how they can be used to find high-quality representative cluster centroids on real-world buoy trajectory data via a sparse k-means procedure.
A. PROOF OF PROPOSITION 1
Let g U m , g L m : R >0 → R be given, such that g U m (0) = g L m (0) = 0. For any j ∈ [N1], we have
ψ U j,m = g U m (∆ U 1 )ej = g U m (∂2∂ 2 )ej.(13)
Let
{(u k , s k , v k )} rank ∂ 2 k=1
be the (low-rank) singular values and vectors of ∂2, so that
∂2 = rank ∂ 2 k=1 s k u k v k .(14)
Then, due to the fact that g U m (0) = 0, applying the functional calculus yields
g U m (∂2∂ 2 )ej = rank ∂ 2 k=1 g U m (s 2 k )u k u k , ej .(15)
That is, ψ U j,m can be written as a linear combination of the left (lowrank) singular vectors of ∂2, hence ψ U j,m ∈ Im(∂2). A similar argument holds for the lower wavelets ψ L j ,m , completing the proof.
B. PROOF OF THEOREM 3
Let f ∈ C 1 (X) be given, and put
S = ψ∈D | ψ, f | 2 .(16)
This can be expanded to
S = M U m=1 N 1 j=1 | g U m (∆ U 1 )ej, f | 2 + M L m=1 N 1 j=1 | g L m (∆ L 1 )ej, f | 2 .
(17) Since ∆ U 1 , ∆ L 1 are Hermitian, this can be rewritten as
S = M U m=1 N 1 j=1 | g U m (∆ U 1 )f , ej | 2 + M L m=1 N 1 j=1 | g L m (∆ L 1 )f , ej | 2 .
(18) We now lower bound S. Since the set {ej} N 1 j=1 is an orthogonal basis for C 1 (X), Parseval's identity holds. This yields
S ≥ M U m=1 g U m (∆ U 1 )f 2 + M L m=1 g L m (∆ L 1 )f 2 .(19)
Following the simplicial Fourier transform of [11], let (µj, vj) be the nonzero eigenpairs of ∆ U 1 and (νj, uj) be the nonnull eigenpairs of ∆ L 1 . Then, there exists f H ∈ ker ∆1 and real coefficients αj, βj such that
f = f H + rank ∆ U 1 j=1 αjvj + rank ∆ L 1 j=1 βjuj(20)
and
f 2 = f H 2 + rank ∆ U 1 j=1 α 2 j + rank ∆ L 1 j=1 β 2 j .(21)
With this in mind, coupled with the fact that f H ∈ ker(∆1) = ker(∆ U 1 ) ∩ ker(∆ L 1 ), (19) can be written as
S ≥ M U m=1 |g U m (0)| 2 + M L m=1 |g L 1 (0)| 2 f H 2 + rank ∆ U 1 j=1 M U m=1 |g U m (µj)| 2 + M L m=1 |g L m (0)| 2 α 2 j + rank ∆ L 1 j=1 M U m=1 |g U 1 (0)| 2 + M L m=1 |g L m (νj)| 2 β 2 j .(22)
Considering our definition of the function G, (22) can be more concisely written as
S ≥ G(0, 0) f H 2 + rank ∆ U 1 j=1 G(µj, 0)α 2 j + rank ∆ L 1 j=1 G(0, νj)β 2 j .(23)
One can then see that
S ≥ A f 2 ,(24)
where A is as defined in (10). A similar argument using the upper frame bounds yields
S ≤ B f 2 ,(25)
as desired.
Fig. 1 .
1Dictionary atoms for representing flows on simplicial complexes. All signals are scaled to have unit ∞-norm. Edge orientations are indicated by the direction of the arrowheads. (a) An eigenvector in the nullspace of the Hodge Laplacian ∆1. (b) Jointly designed wavelet based on ∆1. (c1) Upper wavelet based on ∆ U 1 . Triangles are colored according to the 2-cochain c such that the wavelet atom is equal to ∂2c, with orientation of the flow induced by each triangle indicated by arrows (space permitting). (c2) Lower wavelet based on ∆ L 1 . Nodes are colored according to the 0-cochain x such that the wavelet atom is equal to ∂ 1 x.
Theorem 3 .
3Let {ej} N 1 j=1 be the standard orthonormal basis forC 1 (X), and let {g U m } M U m=1 , {g L m } M L m=1be collections of continuous non-negative functions on the real line. Let D be the dictionary of separate Hodgelets defined by (4), and define
Fig. 2 .
2Representation of flows on simplicial complexes with spectral wavelets. (a) Vector field F (x, y). (b) Sparsity of representing a discretization of the vector field using different dictionaries as error tolerance increases. 'LG' indicates the use of the linegraph Laplacian. (c) Sum of all buoy trajectories in the Global Drifter dataset around Madagascar. Signal is scaled to have unit ∞-norm. (d)Average maximum inner product L between test set and centroids obtained via sparse k-means clustering using standard basis, Fourier basis, joint Hodgelets, and separate Hodgelets.
Code is available at https://www.git.roddenberry.xyz/ hodgelets/
Data available from NOAA/AOML at http://www.aoml.noaa. gov/envids/gld/
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| [] |
[
"Energy levels, radiative rates and electron impact excitation rates for transitions in He-like Ga XXX, Ge XXXI, As XXXII, Se XXXIII and Br XXXIV",
"Energy levels, radiative rates and electron impact excitation rates for transitions in He-like Ga XXX, Ge XXXI, As XXXII, Se XXXIII and Br XXXIV"
] | [
"Kanti M Aggarwal [email protected] \nAstrophysics Research Centre\nSchool of Mathematics and Physics\nQueen's University Belfast\nBT7 1NNBelfastNorthern IrelandUK\n",
"Francis P Keenan \nAstrophysics Research Centre\nSchool of Mathematics and Physics\nQueen's University Belfast\nBT7 1NNBelfastNorthern IrelandUK\n"
] | [
"Astrophysics Research Centre\nSchool of Mathematics and Physics\nQueen's University Belfast\nBT7 1NNBelfastNorthern IrelandUK",
"Astrophysics Research Centre\nSchool of Mathematics and Physics\nQueen's University Belfast\nBT7 1NNBelfastNorthern IrelandUK"
] | [] | We report calculations of energy levels, radiative rates and electron impact excitation cross sections and rates for transitions in He-like Ga XXX, Ge XXXI, As XXXII, Se XXXIII and Br XXXIV. The grasp (general-purpose relativistic atomic structure package) is adopted for calculating energy levels and radiative rates. For determining the collision strengths, and subsequently the excitation rates, the Dirac Atomic Rmatrix Code (darc) is used. Oscillator strengths, radiative rates and line strengths are reported for all E1, E2, M1 and M2 transitions among the lowest 49 levels of each ion. Additionally, theoretical lifetimes are provided for all 49 levels of the above five ions. Collision strengths are averaged over a Maxwellian velocity distribution and the effective collision strengths obtained listed over a wide temperature range up to 10 8 K. Comparisons are made with similar data obtained using the Flexible Atomic Code (fac) to highlight the importance of resonances, included in calculations with darc, in the determination of effective collision strengths. Discrepancies between the collision strengths from darc and fac, particularly for some forbidden transitions, are also discussed. Finally, discrepancies between the present results for effective collision strengths with the darc code and earlier semi-relativistic R-matrix data are noted over a wide range of electron temperatures for many transitions in all ions.2The 1s 2 , 1s2ℓ, 1s3ℓ, 1s4ℓ and 1s5ℓ configurations of He-like ions give rise to the lowest 49 levels listed inTable 1(a-e), in which we compare our level energies with grasp (obtained without and with the inclusion of Breit and QED effects) with the critically evaluated data compiled by NIST. Also included in these tables are our results obtained from the fac code (FAC1), including the same CI (configuration interaction) as in grasp. Our level energies obtained without the Breit and QED effects (GRASP1) are consistently higher than the NIST values by up to ∼ 1.7 Ryd, similar to the effect observed for other He-like ions [11]-[20]. However, the orderings are nearly the same as those of NIST. The inclusion of Breit and QED effects (GRASP2) lowers the energies by a maximum of ∼ 1.9 Ryd, depending on the ion. As expected, the contribution of Breit and QED effects increases with increasing Z, but our GRASP2 energies are lower than the NIST listings by up to ∼ 0.4 Ryd, depending on the ion. In addition, the orderings have slightly altered in a few instances, see for example the 4f and 5f levels. However, the energy differences between these swapped levels are very small. It may be noted that NIST compilations are mostly based on a variety of theoretical works resulting in small differences with our listed energies. Our FAC1 level energies are consistently higher by up to 0.2 Ryd than the GRASP2 results, for all ions, and hence are comparatively in better agreement with the NIST listings. The level orderings from FAC1 are also in general agreement with the calculations from grasp, but differ in some instances, particularly for the n = 5 levels. This is mainly because the non-degeneracy among the levels of the n = 5 configurations is very small. A further inclusion of the 1s6ℓ configurations, labelled FAC2 calculations inTable 1(a-e), makes no appreciable difference either in the magnitude or orderings of the levels.Finally, inTable 1(a-e) we also list the energies of Whiteford et al[28], which are obtained with the AutoStructure (as) code of Badnell[29], and are available at the apap (Atomic Processes for Astrophysical Plasmas) website: http://amdpp.phys.strath.ac.uk/UK_APAP/. The as energies are higher by up to ∼ 2 Ryd, depending on the ion, than the corresponding reference values from NIST and our results from the fac and grasp codes. Differences between the NIST and our theoretical energy levels with the latter calculations increase with increasing Z, and are due to the higher-order relativistic effects being neglected in the as calculations. More importantly, the level orderings are slightly different, particularly for levels 42 and higher, and the as energies for some of the levels of the 5g configuration are non-degenerate. However, the energy differences among the levels of a configuration are very small, as noted above. To conclude, we may state that overall there is general agreement between our calculations with the fac and grasp codes for the energy levels of the He-like ions considered here, and there is no major discrepancy with the NIST listings. However, the NIST energies are not available for some levels, particularly of the n = 5 configurations, for which our energy levels either from the GRASP2 or FAC1 calculations should be adopted in modelling applications. For the remaining levels, the critically compiled listings of NIST may be preferred. | 10.1088/0031-8949/87/04/045304 | [
"https://arxiv.org/pdf/1303.3220v1.pdf"
] | 119,174,580 | 1303.3220 | 8681672e52d9f07a7dc25238514bd64703b39635 |
Energy levels, radiative rates and electron impact excitation rates for transitions in He-like Ga XXX, Ge XXXI, As XXXII, Se XXXIII and Br XXXIV
13 Mar 2013 Published xx Month 2013
Kanti M Aggarwal [email protected]
Astrophysics Research Centre
School of Mathematics and Physics
Queen's University Belfast
BT7 1NNBelfastNorthern IrelandUK
Francis P Keenan
Astrophysics Research Centre
School of Mathematics and Physics
Queen's University Belfast
BT7 1NNBelfastNorthern IrelandUK
Energy levels, radiative rates and electron impact excitation rates for transitions in He-like Ga XXX, Ge XXXI, As XXXII, Se XXXIII and Br XXXIV
13 Mar 2013 Published xx Month 2013Received 10 Sptember 2012 Accepted for publication 06 March 2013arXiv:1303.3220v1 [physics.atom-ph] Online at stacks.iop.org/PhysScr/vol/number PACS Ref: 32.70 Cs, 34.80 Dp, 95.30 Ky S This article has associated online supplementary data files Tables 2 and 4 are available only in the electronic version at stacks.iop.org/PhysScr/vol/number 1
We report calculations of energy levels, radiative rates and electron impact excitation cross sections and rates for transitions in He-like Ga XXX, Ge XXXI, As XXXII, Se XXXIII and Br XXXIV. The grasp (general-purpose relativistic atomic structure package) is adopted for calculating energy levels and radiative rates. For determining the collision strengths, and subsequently the excitation rates, the Dirac Atomic Rmatrix Code (darc) is used. Oscillator strengths, radiative rates and line strengths are reported for all E1, E2, M1 and M2 transitions among the lowest 49 levels of each ion. Additionally, theoretical lifetimes are provided for all 49 levels of the above five ions. Collision strengths are averaged over a Maxwellian velocity distribution and the effective collision strengths obtained listed over a wide temperature range up to 10 8 K. Comparisons are made with similar data obtained using the Flexible Atomic Code (fac) to highlight the importance of resonances, included in calculations with darc, in the determination of effective collision strengths. Discrepancies between the collision strengths from darc and fac, particularly for some forbidden transitions, are also discussed. Finally, discrepancies between the present results for effective collision strengths with the darc code and earlier semi-relativistic R-matrix data are noted over a wide range of electron temperatures for many transitions in all ions.2The 1s 2 , 1s2ℓ, 1s3ℓ, 1s4ℓ and 1s5ℓ configurations of He-like ions give rise to the lowest 49 levels listed inTable 1(a-e), in which we compare our level energies with grasp (obtained without and with the inclusion of Breit and QED effects) with the critically evaluated data compiled by NIST. Also included in these tables are our results obtained from the fac code (FAC1), including the same CI (configuration interaction) as in grasp. Our level energies obtained without the Breit and QED effects (GRASP1) are consistently higher than the NIST values by up to ∼ 1.7 Ryd, similar to the effect observed for other He-like ions [11]-[20]. However, the orderings are nearly the same as those of NIST. The inclusion of Breit and QED effects (GRASP2) lowers the energies by a maximum of ∼ 1.9 Ryd, depending on the ion. As expected, the contribution of Breit and QED effects increases with increasing Z, but our GRASP2 energies are lower than the NIST listings by up to ∼ 0.4 Ryd, depending on the ion. In addition, the orderings have slightly altered in a few instances, see for example the 4f and 5f levels. However, the energy differences between these swapped levels are very small. It may be noted that NIST compilations are mostly based on a variety of theoretical works resulting in small differences with our listed energies. Our FAC1 level energies are consistently higher by up to 0.2 Ryd than the GRASP2 results, for all ions, and hence are comparatively in better agreement with the NIST listings. The level orderings from FAC1 are also in general agreement with the calculations from grasp, but differ in some instances, particularly for the n = 5 levels. This is mainly because the non-degeneracy among the levels of the n = 5 configurations is very small. A further inclusion of the 1s6ℓ configurations, labelled FAC2 calculations inTable 1(a-e), makes no appreciable difference either in the magnitude or orderings of the levels.Finally, inTable 1(a-e) we also list the energies of Whiteford et al[28], which are obtained with the AutoStructure (as) code of Badnell[29], and are available at the apap (Atomic Processes for Astrophysical Plasmas) website: http://amdpp.phys.strath.ac.uk/UK_APAP/. The as energies are higher by up to ∼ 2 Ryd, depending on the ion, than the corresponding reference values from NIST and our results from the fac and grasp codes. Differences between the NIST and our theoretical energy levels with the latter calculations increase with increasing Z, and are due to the higher-order relativistic effects being neglected in the as calculations. More importantly, the level orderings are slightly different, particularly for levels 42 and higher, and the as energies for some of the levels of the 5g configuration are non-degenerate. However, the energy differences among the levels of a configuration are very small, as noted above. To conclude, we may state that overall there is general agreement between our calculations with the fac and grasp codes for the energy levels of the He-like ions considered here, and there is no major discrepancy with the NIST listings. However, the NIST energies are not available for some levels, particularly of the n = 5 configurations, for which our energy levels either from the GRASP2 or FAC1 calculations should be adopted in modelling applications. For the remaining levels, the critically compiled listings of NIST may be preferred.
Introduction
Emission lines of He-like ions have been widely observed from a variety of astrophysical and laboratory plasmas. For example, lines of many He-like ions detected in solar plasmas at x-ray wavelengths (1-50Å) have been listed by Dere et al [1], and in the regions 170-211Å and 245-291Å by Feldman et al [2]. Similarly, transitions from these ions have been observed in laboratory plasmas [3] - [5]. Of particular interest are the resonance (w: 1s 2 1 S 0 -1s2p 1 P o 1 ), intercombination (x and y: 1s 2 1 S 0 -1s2p 3 P o 2,1 ), and forbidden (z: 1s 2 1 S 0 -1s2s 3 S 1 ) lines, which are highly useful for plasma diagnostics -see, for example, [6] - [9] and references therein.
However, to analyse observations, atomic data are required for a variety of parameters, such as energy levels, radiative rates (A-values), and excitation rates or equivalently the effective collision strengths (Υ), which are obtained from the electron impact collision strengths (Ω). These data are also required for the modelling of fusion plasmas [10]. Therefore, in a series of papers we have already reported atomic data for He-like ions up to Z=30 [11] - [19] and for Z=36, i.e. Kr XXXV [20]. In this work we report similar data for the remaining five ions with 31 ≤ Z ≤ 35, i.e. Ga XXX, Ge XXXI, As XXXII, Se XXXIII and Br XXXIV.
The National Institute of Standards and Technology (NIST) have compiled and critically evaluated energy levels for many He-like ions, and have posted data on their website http://www.nist.gov/pml/data/asd.cfm. Additionally, the A-value for the 1s 2 1 S 0 -1s2s 3 S 1 (1-2) magnetic dipole transition of Ga XXX is available on the NIST website. However, no collisional data are available in the literature for the ions under consideration. Therefore, in this paper we report a complete set of results (namely energy levels, radiative rates, lifetimes, and effective collision strengths) for all transitions among the lowest 49 fine-structure levels of these He-like ions, which belong to the 1s 2 , 1s2ℓ, 1s3ℓ, 1s4ℓ and 1s5ℓ configurations. Finally, we also provide the A-values for four types of transitions, namely electric dipole (E1), electric quadrupole (E2), magnetic dipole (M1) and magnetic quadrupole (M2), as these are required in a complete plasma model.
For our calculations we employ the fully relativistic grasp (general-purpose relativistic atomic structure package) code for the determination of wavefunctions, originally developed by Grant et al [21] and subsequently revised by several workers, under the names grasp [22], grasp92 [23], and grasp2k [24]. However, the version adopted here is grasp0, which is based on [21] and is revised by Dr P H Norrington. This version contains most of the modifications undertaken in the other revised codes and is available on the website http://web.am.qub.ac.uk/DARC/. grasp is a fully relativistic code, and is based on the jj coupling scheme. Further relativistic corrections arising from the Breit interaction and QED effects (vacuum polarization and Lamb shift) have also been included in the same way as in the original version described in [21] and [25]. Additionally, we have used the option of extended average level (EAL), in which a weighted (proportional to 2j+1) trace of the Hamiltonian matrix is minimized. This produces a compromise set of orbitals describing closely lying states with moderate accuracy. For our calculations of Ω, we have adopted the Dirac Atomic R-matrix Code (darc) of P H Norrington and I P Grant (http://web.am.qub.ac.uk/DARC/). However, the code does not include the Breit and QED corrections, and hence the target energies obtained are slightly different (and comparatively less accurate) than from grasp. Finally, for comparison purposes we have performed parallel calculations with the Flexible Atomic Code (fac) of Gu [26], available from the website
Radiative rates
The absorption oscillator strength (f ij ) and radiative rate A ji (in s −1 ) for a transition i → j are related by the following expression:
f ij = mc 8π 2 e 2 λ 2 ji ω j ω i A ji = 1.49 × 10 −16 λ 2 ji (ω j /ω i )A ji(1)
where m and e are the electron mass and charge, respectively, c is the velocity of light, λ ji is the transition energy/wavelength inÅ, and ω i and ω j are the statistical weights of the lower (i) and upper (j) levels, respectively. Similarly, the oscillator strength f ij (dimensionless) and the line strength S (in atomic unit, 1 a.u. = 6.460×10 −36 cm 2 esu 2 ) are related by the standard equations given below.
For the electric dipole (E1) transitions
A ji = 2.0261 × 10 18 ω j λ 3 ji S E1 and f ij = 303.75 λ ji ω i S E1 ,(2)λ ji ω i S M1 ,(3)
for the electric quadrupole (E2) transitions
A ji = 1.1199 × 10 18 ω j λ 5 ji S E2 and f ij = 167.89 λ 3 ji ω i S E2 ,(4)
and for the magnetic quadrupole (M2) transitions
A ji = 1.4910 × 10 13 ω j λ 5 ji S M2 and f ij = 2.236 × 10 −3 λ 3 ji ω i S M2 .(5)
In Table 2 (a-e) we present transition energies/wavelengths (λ, inÅ), radiative rates (A ji , in s −1 ), oscillator strengths (f ij , dimensionless), and line strengths (S, in a.u.), in length form (Babushkin gauge) only, for all 336 electric dipole (E1) transitions among the 49 levels of the He-like ions considered here. The indices used to represent the lower and upper levels of a transition have already been defined in Table 1 (a-e). Similarly, there are 391 electric quadrupole (E2), 316 magnetic dipole (M1), and 410 magnetic quadrupole (M2) transitions among the 49 levels. However, for these transitions only the A-values are listed in Table 2, and the corresponding results for the f -or S-values can be easily obtained using Eqs. (1)(2)(3)(4)(5).
As stated earlier, no other A-values are available in the literature with which to compare our results. Therefore, we have performed another calculation with the fac code of Gu [26]. For all strong transitions (f ≥ 0.01), the A-values from grasp and fac, in the Babushkin gauge, agree to better than 10% for the five ions. Furthermore, for a majority of the strong E1 transitions (f ≥ 0.01) the length and velocity (Coulomb gauge) forms in our grasp calculations agree within 10%. However, the differences are larger for a few transitions, which are among the degenerate levels of a configuration, such as 10-11 (f ∼ 1.4×10 −4 ), 24-25 (f ∼ 8.5×10 −6 ) and 27-29 (f ∼ 1.9×10 −7 ) in Ga XXX. This is because their transition energy (∆E) is very small and hence a slight variation in ∆E has a considerable effect on the A-values. For a few such weaker transitions (f < 10 −3 ) the two forms of the f -value differ by orders of magnitude, for all ions. Finally, as for the energy levels the effect of additional CI is negligible on the A-values, as results for all transitions agree within 10% with those obtained with the inclusion of the n = 6 configurations. Therefore, for almost all strong E1 transitions our radiative rates should be accurate to better than 10%. However, for a few weaker transitions the accuracy is comparatively lower.
Lifetimes
The lifetime τ for a level j is defined as follows:
τ j = 1 i A ji .(6)
Since this is a measurable parameter, it provides a check on the accuracy of the calculations. Therefore, in Table 1 (a-e) we have also listed our calculated lifetimes, which include the contributions from four types of transitions, i.e. E1, E2, M1 and M2. Unfortunately to our knowledge no similar theoretical or experimental data are available with which to compare our results. However, based on the accuracy assessment of our Avalues, we expect our lifetimes to have the same level of accuracy.
Collision strengths
Collision strengths (Ω) are related to the more commonly known collision cross section (σ ij , πa 0 2 ) by the following relationship:
Ω ij (E) = k 2 i ω i σ ij (E)(7)
where k 2 i is the incident energy of the electron and ω i is the statistical weight of the initial state. Results for collisional data are preferred in the form of Ω because it is a symmetric and dimensionless quantity.
For the computation of collision strengths Ω, we have employed the Dirac atomic R-matrix code (darc), which includes the relativistic effects in a systematic way, in both the target description and the scattering model. It is based on the jj coupling scheme, and uses the Dirac-Coulomb Hamiltonian in the R-matrix approach. The R-matrix radii adopted for Ga XXX, Ge XXXI, As XXXII, Se XXXIII and Br XXXIV are 2.88, 2.72, 2.56, 2.40 and 2.24 au, respectively. For all five ions, 60 continuum orbitals have been included for each channel angular momentum in the expansion of the wavefunction, allowing us to compute Ω up to an energy of 2150, 2350, 2650, 3000 and 3400 Ryd for Ga XXX, Ge XXXI, As XXXII, Se XXXIII and Br XXXIV, respectively. These energy ranges are sufficient to calculate values of effective collision strengths Υ (see section 6) up to T e = 10 8 K, appropriate for applications to the modelling of high temperature laboratory plasmas. The maximum number of channels for a partial wave is 217, and the corresponding size of the Hamiltonian matrix is 13 076. To obtain convergence of Ω for all transitions and at all energies, we have included all partial waves with angular momentum J ≤ 40.5, although a larger number would have been preferable for the convergence of some allowed transitions, especially at higher energies. However, to account for higher neglected partial waves, we have included a top-up, based on the Coulomb-Bethe approximation [30] for allowed transitions and geometric series for others.
For illustration, in Figs. 1-3 we show the variation of Ω with angular momentum J for three transitions of Ga XXX, namely 2-5 (1s2s 3 S 1 -1s2p 3 P o 1 ), 2-11 (1s2s 3 S 1 -1s3p 3 P o 1 ), and 9-12 (1s3p 3 P o 0 -1s3p 3 P o 2 ), and at three energies of 1000, 1400 and 1800 Ryd. Values of Ω have fully converged for all resonance transitions (including the allowed ones), plus a majority of the allowed transitions among the higher excited levels, as shown in Fig. 2 for the 2-11 transition. It is also clear from Fig. 2 that the need to include a larger range of partial waves increases with increasing energy. However, values of Ω have not converged for those allowed transitions whose ∆E is very small (mainly within the same n complex), as shown for the 2-5 transition in Fig. 1. Similarly, values of Ω have (almost) converged for all forbidden transitions, including those whose ∆E is very small, such as the 9-12 transition shown in Fig. 3. Therefore, for the allowed transitions within the same n complex, our wide range of partial waves is not sufficient for the convergence of Ω, for which a top-up has been included as mentioned above, and has been found to be appreciable.
In Table 3 (a-e) we list our values of Ω for resonance transitions of Ga XXX, Ge XXXI, As XXXII, Se XXXIII and Br XXXIV at energies above thresholds. The indices used to represent the levels of a transition have already been defined in Table 1 (a-e). Unfortunately, no similar data are available for comparison purposes as already stated in section 1. Therefore, to assess the accuracy of Ω, we have performed another calculation using the fac code of Gu [26]. This code is also fully relativistic, and is based on the well-known and widely-used distorted-wave (DW) method. Furthermore, the same CI is included in fac as in the calculations from darc. Therefore, also included in Table 3 (a-e) for comparison purposes are the Ω values from fac at a single excited energy E j , which corresponds to an incident energy of ∼ 2000, 2100, 2300, 2400 and 2600 Ryd for Ga XXX, Ge XXXI, As XXXII, Se XXXIII and Br XXXIV, respectively. For a majority of transitions the two sets of Ω generally agree well (within ∼ 20%). However, the differences are larger for a few (particularly weaker) transitions. For example, for 70% of the Ga XXX transitions, the values of Ω agree to within 20% at an energy of 2000 Ryd, and the discrepancies for others are mostly within a factor of two, although for some transitions (such as: 19-49, 33-38/41/43/47/49), the differences are up to an order of magnitude. However, most of these transitions are weak (Ω ∼ 10 −6 ) and forbidden, i.e. the values of Ω have fully converged at all energies within our adopted range of partial waves in the calculations from the darc code. For such weak transitions, values of Ω from the fac code are not assessed to be accurate. The values of Ω are higher from darc for some transitions, whereas for others are higher from the fac code. Additionally, for a few transitions, such as 18-28/29/30/31, values of Ω from the fac code show an anomalous behaviour towards the higher end of the energy range. This problem is common for all ions and examples can be seen in Fig. 6 of Aggarwal and Keenan [16] - [17]. The sudden anomalous behaviour in values of Ω from the fac code is also responsible for the differences noted above for some of the transitions. Such anomalies for some transitions (both allowed and forbidden) from the fac calculations arise primarily because of the interpolation and extrapolation techniques employed in the code, which is designed to generate a large amount of atomic data in a comparatively very short period of time, and without too large loss of accuracy. Similarly, some differences in Ω are expected because the DW method generally overestimates the results due to the exclusion of channel coupling. Such discrepancies, for a similar number of transitions, have also been noted for Ge XXXI, As XXXII, Se XXXIII and Br XXXIV.
As a further comparison between the darc and fac values of Ω, in Fig. 4 we show the variation of Ω with energy for three allowed transitions among the excited levels of Ga XXX, namely 5-14 (1s2p 3
P o 1 -1s3d 3 D 2 ), 12-26 (1s3p 3 P o 2 -1s4d 3 D 3 ), and 15-27 (1s3p 1 P o 1 -1s4d 1 D 2 )
. Also included in this figure are the corresponding results obtained with the fac code. For many transitions there are no discrepancies between the f -values obtained with the two different codes (grasp and fac), and therefore the values of Ω also agree to better than 20%. However, the values of Ω obtained with fac are underestimated in comparison to our calculations with darc, particularly towards the lower end of the energy range. Similar comparisons between the two calculations with darc and fac are shown in Fig. 5 for three forbidden transitions of Ga XXX, namely 2-8 (1s2s 3 S 1 -1s3s 3 S 1 ), 2-16 (1s2s 3 S 1 -1s3d 3 D 3 ), and 6-12 (1s2p 3 P o 2 -1s3p 3 P o 2 ). As in the case of allowed transitions, for these forbidden ones the agreement between the two calculations improves considerably with increasing energy, but the differences are significant towards the lower end of the energy range, as Ω from fac are underestimated. These anomalies are due to the interpolation and extrapolation techniques employed in the fac code, as stated above. Therefore, on the basis of these and other comparisons discussed above, collision strengths from the fac code are not assessed to be very accurate, over the entire energy range, for a majority of transitions for the above named five He-like ions. However, we do not see any apparent deficiency in our darc calculations for Ω, and estimate our results to be accurate to better than 20% for a majority of the (strong) transitions.
Effective collision strengths
Excitation rates, in addition to energy levels and radiative rates, are required for plasma modelling, and are determined from the collision strengths (Ω). Since the threshold energy region is dominated by numerous closed-channel (Feshbach) resonances, values of Ω need to be calculated in a fine energy mesh to accurately account for their contribution. Furthermore, in a plasma electrons have a wide distribution of velocities, and therefore values of Ω are generally averaged over a Maxwellian distribution as follows:
Υ(T e ) = ∞ 0 Ω(E)exp(−E j /kT e )d(E j /kT e ),(8)
where k is Boltzmann constant, T e is electron temperature in K, and E j is the electron energy with respect to the final (excited) state. Once the value of Υ is known the corresponding results for the excitation q(i, j) and de-excitation q(j, i) rates can be easily obtained from the following equations:
q(i, j) = 8.63 × 10 −6 ω i T 1/2 e Υexp(−E ij /kT e ) cm 3 s −1(9)
and q(j, i) = 8.63 × 10 −6
ω j T 1/2 e Υ cm 3 s −1 ,(10)
where ω i and ω j are the statistical weights of the initial (i) and final (j) states, respectively, and E ij is the transition energy. The contribution of resonances may enhance the values of Υ over those of the background collision strengths (Ω B ), especially for the forbidden transitions, by up to an order of magnitude (or even more) depending on the transition, and particularly at low temperatures. Similarly, values of Ω need to be calculated over a wide energy range (above thresholds) in order to obtain convergence of the integral in Eq. (8), as demonstrated in Fig. 7 of Aggarwal and Keenan [31].
To delineate resonances, we have performed our calculations of Ω at up to ∼ 127 500 energies in the thresholds region, depending on the ion. Close to thresholds (∼0.1 Ryd above a threshold) the energy mesh is 0.001 Ryd, and away from thresholds is 0.002 Ryd. Thus care has been taken to include as many resonances as possible, and with as fine a resolution as is computationally feasible. The density and importance of resonances can be appreciated from Figs. 6-9, where we plot Ω as a function of energy in the thresholds region for the four most important transitions of He-like ions, namely 1-2 (z: 1s 2 1 S 0 -1s2s 3 S 1 ), 1-5 (y:
1s 2 1 S 0 -1s2p 3 P o 1 ), 1-6 (x: 1s 2 1 S 0 -1s2p 3 P o 2 )
, and 1-7 (w: 1s 2 1 S 0 -1s2p 1 P o 1 ). Resonances shown in these figures are for transitions in Ga XXX, but similar dense resonances have been noted for all He-like ions. For some transitions, such as 1-2, 1-5 and 1-6, the resonances are dense, particularly at energies just above the thresholds, and are spread over a wide energy range of ∼ 150 Ryd. These near-threshold resonances affect the values of Υ particularly towards the lower end of the temperature range.
Our calculated values of Υ are listed in Table 4 (a-e) over a wide temperature range up to 10 8 K, suitable for applications in a variety of plasmas. As stated in section 1, there are no other results available with which to compare. Therefore, we have also calculated values of Υ from our non-resonant Ω data obtained with the fac code. These calculations are particularly helpful in providing an estimate of the importance of resonances in the determination of excitation rates. In Table 4 (a-e) we have included these results from fac at the lowest and the highest calculated temperatures for Ga XXX, Ge XXXI, As XXXII, Se XXXIII and Br XXXIV. However, for discussion we focus solely on transitions in Ga XXX. At T e = 10 6.4 K, our resonance-resolved values of Υ are higher by over 20% for about 37% of the transitions. Generally, the differences for a majority of the transitions are within a factor of two, but are up to a factor of five for a few. Furthermore, for a majority of transitions the Υ values from the darc code are higher, partly due to the inclusion of resonances in the calculations with darc but also because of the underestimation of Ω at lower energies in the fac calculations, as demonstrated in Figs. 4 and 5 and discussed in section 5. In the case of the most important w, x, y, and z lines, resonances have significantly enhanced the values of Υ, by about a factor of 2.5, for the z (1-2: 1s 2 1 S 0 -1s2s 3 S 1 ) transition. A similar comparison at the highest temperature of our calculations, i.e. 10 7.8 K,
indicates that about 39% of the transitions in Ga XXX show differences of over 20% between the darc and fac values for Υ. These differences are generally within a factor of two, but are higher (up to an order of magnitude) for a few, such as: 18-28 (allowed transition) and 18-29/30/31 (forbidden). For all such transitions the fac results are higher, mainly because of the sudden anomalies in the values of Ω, as discussed in section 5. Similar differences in Υ are noted for a comparable number of transitions in the other He-like ions, as shown in Table 4 (a-e).
Whiteford et al [28] have undertaken semi-relativistic calculations for Ar XVII and Fe XXV, using the standard R-matrix code of Berrington et al [32]. Recently they have placed their results for Υ for He-like ions up to Z = 36 on the apap website: http://amdpp.phys.strath.ac.uk/UK_APAP/. In their calculations electron exchange was included for partial waves up to J = 10.5, but was neglected for higher J values. Furthermore, the calculations were performed in the LS coupling scheme and the corresponding results for Ω and subsequently Υ for fine-structure transitions obtained using their intermediate coupling frame transformation (icft) method. However, the data obtained by such procedures are generally comparable with our fully relativistic results from the darc code, as has already been demonstrated in several papers -see for example, Liang and Badnell [33]. Furthermore, Whiteford et al included the effect of radiation damping. This can reduce the contribution of resonances to the determination of Υ for some transitions, such as 1s2p 1 P o 1 -1s3s 1 S 0 , particularly towards the lower end of the temperature range (see their Fig. 4). However, at temperatures relevant to plasma modelling the effect is negligible. Radiation damping may be significant for highly charged ions, because radiative decay rates are large and compete with autoionisation rates [34]- [35]. However, similar R-matrix calculations performed by Delahaye et al [35] for several He-like ions up to Z = 20, with the inclusion of radiation damping, show that their contribution (i.e. reduction in values of Υ) is appreciable only at lower temperatures (below 10 6 K), which should not be important in plasma modelling applications for such highly ionised species. This has also been confirmed in calculations by Whiteford et al [28], who have shown that the resultant uncertainties in the excited level populations for the important w, x, y and z lines of Ar XVII and Fe XXV, at appropriate temperatures and densities, are under 10%. Finally, in a comparatively more recent calculations for Fe XXV and Kr XXXV using the the darc code, Griffin and Ballance [36] have concluded that the damping effect on excitation to the vast majority of levels for both ions is small. Specifically, the differences between the two sets of calculations (with and without damping) when averaged over six temperatures ranging from 1.25×10 6 to 3.12×10 8 K for the lowest 30 resonance transitions are only 4% for Fe XXV and 4.4% for Kr XXXV. Similarly, the corresponding differences for all 1176 transitions over a wider range of nine temperatures are only 1.1% and 1.3%, for the respective ions. These differences are much smaller than the errors introduced by other parameters, such as inadequacies of partial waves, configuration interaction, energy mesh, and inclusion of a limited energy range. We can therefore confidently conclude that radiation damping is not important for the He-like ions considered here.
We now compare our results of Υ with those of Whiteford et al [28], and for illustration focus only on transitions in Ga XXX. Differences between the two sets of results are quite significant (over 20%) for many transitions, and throughout the temperature range of the calculations. To be specific, at the lowest common temperature of 1.8×10 5 K, the two sets of Υ differ by over 20% for ∼35% of the transitions. For a majority of transitions, these differences are within a factor of two, and for some our results are higher whereas for most the reverse is true. However, for a few transitions the differences are up to a factor of five, and for the 'elastic' transitions, such as: 14-16/17, 16-17, 23-24/26/27, and 24-26/27, the discrepancies are up to two orders of magnitude, with the Υ values of Whiteford et al being invariably higher. Most of these belong to the degenerate levels of a state/configuration, and hence have very small transition energies. To demonstrate the differences between the two calculations, in Fig. 10 we compare values of Υ for three transitions of Ga XXX, namely 14-16 (1s3d 3 D 2 -1s3d 3 D 3 ), 16-17 (1s3d 3 D 3 -1s3d 1 D 2 ), and 24-26 (1s4d 3 D 2 -1s4d 3 D 3 ). The differences in the Υ values are not due to resonances, but arise from the limitation of the approach adopted by Whiteford et al [28], as recently discussed and demonstrated by Bautista et al [37]. Similar large discrepancies are also observed with the calculations of Whiteford et al [38] for transitions in Li-like ions, as discussed and demonstrated by Aggarwal and Keenan [39], [40]. The problem in the R-matrix code adopted by Whiteford et al [28], [38] has been identified and rectified by Liang and Badnell [33]. However, since the calculations of Whiteford et al [28], [38] were performed more than a decade ago, limitations in their data for He-like ions remain.
Differences between our data for Υ from darc and those of Whiteford et al [28] are not confined to lower temperatures, but cover the entire range of temperatures. For example, at the highest common temperature of 4.5×10 7 K, the two sets of Υ differ by over 20% for ∼13% of the transitions of Ga XXX. Hence, there is comparatively a better convergence of the results at higher temperatures, but the Υ values of Whiteford et al are invariably higher. The differences for most transitions are within a factor of two, but for some are up to an order of magnitude, such as: 19-43/45, 29-31 and 33-43/45, and all transitions with the lower levels I ≥ 46. Most of these transitions are forbidden and Ω for these have converged within our adopted partial waves range, as discussed in section 5. To demonstrate the differences, in Fig. 11 we compare our results of Υ from darc with those of Whiteford et al [28] for three transitions of Ga XXX, namely 46-48 (1s5g 3 G 3 -1s5g 3 G 5 ), 47-48 (1s5g 3 G 4 -1s5g 3 G 5 ) and 48-49 (1s5g 3 G 5 -1s5g 1 G 4 ). Since the collision strengths of Whiteford et al are overestimated for many transitions, particularly at the lower energies, the corresponding results for Υ are affected throughout the entire temperature range. To conclude, we may state that the Υ results of Whiteford et al [28] for transitions in Ga XXX and other He-like ions should not be as accurate as those presented here.
Conclusions
In this paper we have presented results for energy levels and radiative rates for four types of transitions (E1, E2, M1 and M2) among the lowest 49 levels of Ga XXX, Ge XXXI, As XXXII, Se XXXIII and Br XXXIV belonging to the n ≤ 5 configurations. Additionally, lifetimes for all the calculated levels have been reported, although unfortunately no other theoretical or experimental results are available with which to compare. Furthermore, based on a variety of comparisons among various calculations with the grasp and fac codes, our data for radiative rates, oscillator strengths, line strengths, and lifetimes are judged to be accurate to better than 10% for a majority of the strong transitions (levels). Similarly, the accuracy of our results for collision strengths and effective collision strengths is estimated to be better than 20% for most transitions. We have considered a large range of partial waves to achieve convergence of Ω at all energies, included a wide energy range to accurately calculate values of Υ up to T e = 10 8 K, and resolved resonances in a fine energy mesh to account for their contributions. Hence we see no apparent deficiency in our reported results. However, the present data for effective collision strengths for transitions involving the levels of the n = 5 configurations may be improved slightly by the inclusion of the levels of the n = 6 configurations. We believe the present set of complete results for radiative and excitation rates for transitions in five He-like ions should be useful for the modelling of a variety of plasmas.
Figure 1 :
1Partial collision strengths for the 1s2s 3 S 1 -1s2p 3 P o 1 (2-5) transition of Ga XXX, at three energies of: 1000 Ryd (circles), 1400 Ryd (stars) and 1800 Ryd (diamonds).
Figure 2 : 13 Figure 3 :
2133Partial collision strengths for the 1s2s 3 S 1 -1s3p 3 P o 1 (2-11) transition of Ga XXX, at three energies of: 1000 Ryd (circles), 1400 Ryd (stars) and 1800 Ryd (diamonds). Partial collision strengths for the 1s3p 3 P o 0 -1s3p 3 P o 2 (9-12) transition of Ga XXX, at three energies of: 1000 Ryd (circles), 1400 Ryd (stars) and 1800 Ryd (diamonds).
Figure 4 :
4Comparison of collision strengths from our calculations from darc (continuous curves) and fac (broken curves) for the 5-14 (circles: 1s2p 3 P o 1 -1s3d 3 D 2 ), 12-26 (triangles : 1s3p 3 P o 2 -1s4d 3 D 3 ), and 15-27 (stars: 1s3p 1 P o 1 -1s4d 1 D 2 ) allowed transitions of Ga XXX.
Figure 5 : 16 Figure 6 :
5166Comparison of collision strengths from our calculations from darc (continuous curves) and fac (broken curves) for the 2-8 (circles: 1s2s 3 S 1 -1s3s 3 S 1 ), 2-16 (triangles: 1s2s 3 S 1 -1s3d 3 D 3 ), and 6-12 (stars: 1s2p 3 P o 2 -1s3p 3 P o 2 ) forbidden transitions of Ga XXX. Collision strengths for the 1s 2 1 S 0 -1s2s 3 S 1 (1-2) transition of Ga XXX.
Figure 7 :
7Collision strengths for the 1s 2 1 S 0 -1s2p 3 P o 1 (1-5) transition of Ga XXX.
Figure 8 :
8Collision strengths for the 1s 2 1 S 0 -1s2p 3 P o 2 (1-6) transition of Ga XXX.
Figure 9 :
9Collision strengths for the 1s 2 1 S 0 -1s2p 1 P o 1 (1-7) transition of Ga XXX.
Figure 10 :
10Comparison of effective collision strengths for the 14-16 (circles: 1s3d 3 D 2 -1s3d 3 D 3 ), 16-17 (triangles: 1s3d 3 D 3 -1s3d 1 D 2 ), and 24-26 (stars: 1s4d 3 D 2 -1s4d 3 D 3 ) transitions of Ga XXX. Continuous and dotted curves are from the present darc and earlier R-matrix codes[28], respectively.
Figure 11 :
11Comparison of effective collision strengths for the 46-48 (circles: 1s5g 3 G 3 -1s5g 3 G 5 ), 47-48 (triangles: 1s5g 3 G 4 -1s5g 3 G 5 ), and 48-49 (stars: 1s5g 3 G 5 -1s5g 1 G 4 ) transitions of Ga XXX. Continuous and dotted curves are from the present darc and earlier R-matrix codes[28], respectively.
Table 1 :
1a. Energy levels (in Ryd) of Ga XXX and their lifetimes (τ , s). a±b ≡ a×10 ±b .Index Configuration/Level
NIST GRASP1 GRASP2
FAC1
FAC2
AS
τ (s)
1 1s 2
1 S 0
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
........
2 1s2s 3 S 1
700.86597 701.83875 700.59790 700.73132 700.73102 702.11621 8.137-10
3 1s2p 3 P o
0
703.52441 704.29773 703.26678 703.43005 703.42999 704.47528 2.070-09
4 1s2s 1 S 0
703.73382 704.60309 703.51764 703.66174 703.66138 704.94012 2.190-03
5 1s2p 3 P o
1
703.71860 704.72906 703.46155 703.63263 703.63251 704.91608 6.376-15
6 1s2p 3 P o
2
706.14166 707.17432 705.88123 706.04395 706.04382 707.36456 3.196-11
7 1s2p 1 P o
1
707.66900 708.70074 707.42346 707.61078 707.61066 708.94946 1.123-15
8 1s3s 3 S 1
830.87033 831.90155 830.60962 830.76263 830.76239 832.19769 1.617-13
9 1s3p 3 P o
0
831.60281 832.58154 831.34540 831.50067 831.50061 832.83984 5.362-14
10 1s3s 1 S 0
831.63510 832.63214 831.38556 831.53259 831.53241 832.90070 1.610-13
11 1s3p 3 P o
1
831.65690 832.70105 831.40076 831.55780 831.55774 832.95496 1.565-14
12 1s3p 3 P o
2
832.38076 833.43121 832.12354 832.27997 832.27979 833.66095 5.612-14
13 1s3d 3 D 1
832.76987 833.80634 832.51001 832.65704 832.65704 834.03058 1.859-14
14 1s3d 3 D 2
832.75629 833.81580 832.49683 832.64435 832.64435 834.04120 1.856-14
15 1s3p 1 P o
1
832.79611 833.84375 832.54443 832.70276 832.70270 834.07385 3.797-15
16 1s3d 3 D 3
833.01846 834.07513 832.75812 832.90497 832.90497 834.30786 1.892-14
17 1s3d 1 D 2
833.04750 834.09094 832.78772 832.93549 832.93549 834.32520 1.891-14
18 1s4s 3 S 1
875.91260 876.96307 875.65991 875.80420 875.80396 877.24005 2.391-13
19 1s4p 3 P o
0
876.21880 877.24286 875.96198 876.09937 876.09924 877.50201 9.218-14
20 1s4s 1 S 0
876.22970 877.26007 875.97601 876.11591 876.11560 877.51215 2.235-13
21 1s4p 3 P o
1
876.24250 877.29211 875.98511 876.12311 876.12305 877.54706 3.362-14
22 1s4p 3 P o
2
876.54690 877.60040 876.29034 876.42865 876.42859 877.83502 9.582-14
23 1s4d 3 D 1
876.71090 877.75427 876.44928 876.59924 876.59924 877.98401 4.323-14
24 1s4d 3 D 2
876.70540 877.75928 876.44476 876.59497 876.59503 877.98950 4.314-14
25 1s4p 1 P o
1
876.72270 877.76935 876.46326 876.60138 876.60126 877.99951 8.917-15
26 1s4d 3 D 3
876.81570 877.86780 876.55408 876.70355 876.70355 878.09973 4.408-14
27 1s4d 1 D 2
876.82840 877.87640 876.56842 876.71838 876.71838 878.10712 4.397-14
28 1s4f 3 F o
2
877.87708 876.56818 876.70862 876.70874 878.10901 8.863-14
29 1s4f 3 F o
3
877.87720 876.56390 876.70441 876.70441 878.10699 8.861-14
30 1s4f 3 F o
4
877.93347 876.62061 876.76111 876.76111 878.16522 8.912-14
31 1s4f 1 F o
3
877.93359 876.62390 876.76440 876.76440 878.16541 8.917-14
32 1s5s 3 S 1
896.64490 897.69702 896.38977 896.52985 896.52966 897.96375 3.879-13
33 1s5p 3 P o
0
897.83850 896.54236 896.67773 896.67767 898.09497 1.594-13
34 1s5s 1 S 0
896.80620 897.84711 896.54980 896.68646 896.68610 898.09277 3.163-13
35 1s5p 3 P o
1
896.81250 897.86346 896.55420 896.68994 896.68976 898.11639 6.572-14
36 1s5p 3 P o
2
896.96840 898.02118 896.71039 896.84631 896.84613 898.25702 1.655-13
37 1s5d 3 D 1
898.09888 896.79077 896.93866 896.93866 898.33087 8.334-14
38 1s5d 3 D 2
898.10168 896.78870 896.93671 896.93671 898.33380 8.250-14
39 1s5p 1 P o
1
897.05860 898.10681 896.79816 896.93341 896.93323 898.33759 1.838-14
40 1s5d 3 D 3
898.15704 896.84442 896.99194 896.99194 898.38898 8.523-14
41 1s5d 1 D 2
898.16187 896.85217 897.00000 897.00000 898.39313 8.331-14
42 1s5f 3 F o
2
898.16223 896.85205 896.99219 896.99219 898.39404 1.710-13
43 1s5f 3 F o
3
898.16229 896.84991 896.99005 896.99005 898.39307 1.711-13
44 1s5f 3 F o
4
898.19110 896.87891 897.01904 897.01904 898.42267 1.721-13
45 1s5f 1 F o
3
898.19122 896.88068 897.02075 897.02075 898.42273 1.722-13
46 1s5g 3 G 3
898.19122 896.88049 897.02026 897.02026 898.42194 2.886-13
47 1s5g 3 G 4
898.19122 896.87921 897.01904 897.01904 898.42194 2.886-13
48 1s5g 3 G 5
898.20850 896.89661 897.03638 897.03638 898.43976 2.893-13
Table 1 :
1b. Energy levels (in Ryd) of Ge XXXI and their lifetimes (τ , s). a±b ≡ a×10 ±b .Index Configuration/Level
NIST GRASP1 GRASP2
FAC1
FAC2
AS
τ (s)
1 1s 2
1 S 0
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
........
2 1s2s 3 S 1
748.31400 749.35278 747.98206 748.11768 748.11737 749.63153 5.872-10
3 1s2p 3 P o
0
751.01363 751.90405 750.76447 750.93030 750.93018 752.07312 1.956-09
4 1s2s 1 S 0
751.22441 752.21918 751.01996 751.16595 751.16565 752.57642 1.711-03
5 1s2p 3 P o
1
751.28783 752.36719 750.96320 751.13751 751.13733 752.54663 5.186-15
6 1s2p 3 P o
2
754.00687 755.18738 753.75824 753.92365 753.92346 755.36475 2.482-11
7 1s2p 1 P o
1
755.57598 756.74915 755.34058 755.52960 755.52948 756.98779 1.001-15
8 1s3s 3 S 1
887.14927 888.31311 886.88550 887.04102 887.04077 888.60968 1.413-13
9 1s3p 3 P o
0
887.91273 889.01868 887.65234 887.81000 887.81000 889.27332 4.692-14
10 1s3s 1 S 0
887.94618 889.07037 887.69275 887.84222 887.84198 889.34064 1.408-13
11 1s3p 3 P o
1
887.96780 889.14655 887.70868 887.86829 887.86823 889.39624 1.307-14
12 1s3p 3 P o
2
888.80359 889.98871 888.54340 888.70233 888.70221 890.20734 4.926-14
13 1s3d 3 D 1
889.20682 890.37683 888.94409 889.09344 889.09344 890.58875 1.629-14
14 1s3d 3 D 2
889.19124 890.38666 888.92889 889.07880 889.07880 890.59973 1.626-14
15 1s3p 1 P o
1
889.23052 890.41095 888.97571 889.13641 889.13635 890.62976 3.384-15
16 1s3d 3 D 3
889.49123 890.68347 889.22791 889.37708 889.37708 890.90466 1.660-14
17 1s3d 1 D 2
889.52231 890.69977 889.25946 889.40955 889.40955 890.92255 1.658-14
18 1s4s 3 S 1
936.45941 935.01917 935.16589 935.16571 936.73291 2.088-13
19 1s4p 3 P o
0
936.74963 935.33398 935.47363 935.47357 937.00336 8.067-14
20 1s4s 1 S 0
936.76715 935.34802 935.49023 935.48993 937.01495 1.958-13
21 1s4p 3 P o
1
936.80231 935.35748 935.49786 935.49774 937.05133 2.829-14
22 1s4p 3 P o
2
937.15784 935.70990 935.85071 935.85052 937.38153 8.407-14
23 1s4d 3 D 1
937.31708 935.87476 936.02710 936.02710 937.53510 3.788-14
24 1s4d 3 D 2
937.32227 935.86945 936.02203 936.02203 937.54077 3.778-14
25 1s4p 1 P o
1
937.33075 935.88757 936.02808 936.02795 937.54968 7.964-15
26 1s4d 3 D 3
937.44659 935.99463 936.14648 936.14648 937.66681 3.868-14
27 1s4d 1 D 2
937.45551 936.00989 936.16217 936.16217 937.67432 3.854-14
28 1s4f 3 F o
2
937.45624 936.00952 936.15247 936.15247 937.67645 7.768-14
29 1s4f 3 F o
3
937.45630 936.00482 936.14771 936.14771 937.67419 7.767-14
30 1s4f 3 F o
4
937.52051 936.06946 936.21234 936.21234 937.74054 7.814-14
31 1s4f 1 F o
3
937.52063 936.07312 936.21600 936.21600 937.74066 7.818-14
32 1s5s 3 S 1
958.61139 957.16663 957.30908 957.30884 958.87256 3.383-13
33 1s5p 3 P o
0
958.75806 957.32556 957.46332 957.46326 959.00800 1.394-13
34 1s5s 1 S 0
958.76678 957.33301 957.47198 957.47162 959.00586 2.783-13
35 1s5p 3 P o
1
958.78479 957.33759 957.47565 957.47552 959.03070 5.538-14
36 1s5p 3 P o
2
958.96661 957.51794 957.65625 957.65607 959.19153 1.451-13
37 1s5d 3 D 1
959.04706 957.60132 957.75153 957.75153 959.26764 7.301-14
38 1s5d 3 D 2
959.04999 957.59882 957.74921 957.74921 959.27075 7.220-14
39 1s5p 1 P o
1
959.05426 957.60809 957.74567 957.74554 959.27380 1.637-14
40 1s5d 3 D 3
959.11340 957.66266 957.81256 957.81256 959.33368 7.477-14
41 1s5d 1 D 2
959.11841 957.67090 957.82111 957.82111 959.33795 7.299-14
42 1s5f 3 F o
2
959.11877 957.67078 957.81323 957.81323 959.33893 1.499-13
43 1s5f 3 F o
3
959.11884 957.66833 957.81085 957.81085 959.33777 1.500-13
44 1s5f 3 F o
4
959.15173 957.70148 957.84387 957.84387 959.37146 1.509-13
45 1s5f 1 F o
3
959.15179 957.70337 957.84583 957.84583 959.37152 1.510-13
46 1s5g 3 G 3
959.15186 957.70312 957.84528 957.84528 959.37073 2.531-13
47 1s5g 3 G 4
959.15186 957.70172 957.84393 957.84393 959.37073 2.530-13
48 1s5g 3 G 5
959.17157 957.72156 957.86377 957.86377 959.39099 2.537-13
Table 1 :
1c. Energy levels (in Ryd) of As XXXII and their lifetimes (τ , s). a±b ≡ a×10 ±b .Index Configuration/Level
NIST
GRASP1
GRASP2
FAC1
FAC2
AS
τ (s)
1 1s 2
1 S 0
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
........
2 1s2s 3 S 1
797.34582
798.48352
796.97406
797.11176
797.11145
798.76208 4.281-10
3 1s2p 3 P o
0
800.26160
801.12836
799.87250
800.04065
800.04053
801.28674 1.848-09
4 1s2s 1 S 0
800.43502
801.45300
800.13220
800.27991
800.27954
801.83307 1.345-03
5 1s2p 3 P o
1
800.46973
801.62323
800.07312
800.25043
800.25031
801.79297 4.269-15
6 1s2p 3 P o
2
803.56011
804.86084
803.28619
803.45416
803.45398
805.02167 1.941-11
7 1s2p 1 P o
1
805.17105
806.45795
804.90912
805.09967
805.09955
806.68280 8.956-16
8 1s3s 3 S 1
945.39582
946.65387
945.08112
945.23901
945.23883
946.94891 1.239-13
9 1s3p 3 P o
0
946.19956
947.38519
945.87970
946.03967
946.03955
947.63403 4.122-14
10 1s3s 1 S 0
946.21232
947.43805
945.92023
946.07190
946.07166
947.70905 1.236-13
11 1s3p 3 P o
1
946.25059
947.52148
945.93640
946.09845
946.09827
947.76471 1.100-14
12 1s3p 3 P o
2
947.19649
948.48816
946.89569
947.05701
947.05695
948.69220 4.343-14
13 1s3d 3 D 1
948.88940
947.31073
947.46228
947.46228
949.08521 1.434-14
14 1s3d 3 D 2
948.89960
947.29333
947.44543
947.44543
949.09668 1.430-14
15 1s3p 1 P o
1
947.62570
948.92029
947.33972
947.50256
947.50244
949.12433 3.026-15
16 1s3d 3 D 3
948.02483
949.23773
947.63397
947.78534
947.78534
949.44379 1.463-14
17 1s3d 1 D 2
948.05855
949.25452
947.66754
947.81982
947.81982
949.46222 1.460-14
18 1s4s 3 S 1
996.69743
997.99231
996.40552
996.55450
996.55432
998.25989 1.830-13
19 1s4p 3 P o
0
997.93277
998.29315
996.73328
996.87506
996.87506
998.53888 7.088-14
20 1s4s 1 S 0
997.02822
998.31085
996.74725
996.89166
996.89142
998.55237 1.722-13
21 1s4p 3 P o
1
997.05373
998.34918
996.75690
996.89948
996.89941
998.58966 2.400-14
22 1s4p 3 P o
2
997.44649
998.75720
997.16187
997.30493
997.30481
998.96649 7.407-14
23 1s4d 3 D 1
998.92188
997.33270
997.48724
997.48724
999.12457 3.333-14
24 1s4d 3 D 2
998.92731
997.32654
997.48126
997.48138
999.13049 3.322-14
25 1s4p 1 P o
1
997.62874
998.93420
997.34442
997.48712
997.48694
999.13831 7.137-15
26 1s4d 3 D 3
999.06903
997.46918
997.62323
997.62323
999.27405 3.408-14
27 1s4d 1 D 2
999.07819
997.48535
997.63983
997.63983
999.28149 3.391-14
28 1s4f 3 F o
2
999.07898
997.48499
997.63007
997.63007
999.28387 6.838-14
29 1s4f 3 F o
3
999.07904
997.47980
997.62482
997.62482
999.28143 6.836-14
30 1s4f 3 F o
Table 1 :
1d. Energy levels (in Ryd) of Se XXXIII and their lifetimes (τ , s). a±b ≡ a×10 ±b .Index Configuration/Level
NIST
GRASP1
GRASP2
FAC1
FAC2
AS
τ (s)
1 1s 2
1 S 0
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
........
2 1s2s 3 S 1
847.96060
849.23895
847.58148
847.72168
847.72144
849.51550 3.150-10
3 1s2p 3 P o
0
850.99949
851.97845
850.59851
850.76965
850.76959
852.12384 1.747-09
4 1s2s 1 S 0
851.16899
852.31256
850.86206
851.01196
851.01160
852.71826 1.064-03
5 1s2p 3 P o
1
851.20918
852.50494
850.79877
850.97968
850.97961
852.66272 3.554-15
6 1s2p 3 P o
Table 1 :
1e. Energy levels (in Ryd) of Br XXXIV and their lifetimes (τ , s). a±b ≡ a×10 ±b .Index Configuration/Level
NIST
GRASP1
GRASP2
FAC1
FAC2
AS
τ (s)
1 1s 2
1 S 0
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
........
2 1s2s 3 S 1
900.19625
901.62781
899.81268
899.95477
899.95453
901.89984 2.338-10
3 1s2p 3 P o
0
903.35862
904.46326
902.95123
903.12457
903.12451
904.59241 1.652-09
4 1s2s 1 S 0
903.52584
904.80664
903.21796
903.36945
903.36914
905.24097 8.463-04
5 1s2p 3 P o
1
903.57058
905.02130
903.14862
903.33252
903.33234
905.16406 2.988-15
6 1s2p 3 P o
Table 3 :
3a. Collision strengths for transitions in Ga XXX. (a±b ≡ a×10 ±b ).Transition
Energy (Ryd)
i
j 900
1200
1500
1800
2100
FAC a
1
2 2.038−04 1.462−04 1.094−04 8.475−05 6.927−05 7.760−05
1
3 1.182−04 7.456−05 5.071−05 3.585−05 2.683−05 3.192−05
1
4 6.395−04 7.418−04 8.145−04 8.656−04 9.152−04 7.652−04
1
5 6.197−04 6.550−04 7.202−04 7.941−04 8.706−04 9.110−04
1
6 5.386−04 3.352−04 2.256−04 1.580−04 1.173−04 1.506−04
1
7 2.084−03 3.016−03 3.821−03 4.535−03 5.174−03 4.637−03
1
8 6.533−05 4.557−05 3.321−05 2.534−05 2.039−05 2.073−05
1
9 3.945−05 2.432−05 1.621−05 1.130−05 8.370−06 8.770−06
1
10 1.091−04 1.353−04 1.526−04 1.650−04 1.766−04 1.489−04
1
11 1.474−04 1.387−04 1.433−04 1.528−04 1.648−04 1.779−04
1
12 1.828−04 1.111−04 7.319−05 5.052−05 3.712−05 4.176−05
1
13 1.558−05 7.880−06 4.534−06 2.884−06 1.957−06 2.191−06
1
14 2.396−05 2.099−05 2.252−05 2.507−05 2.771−05 2.753−05
1
15 3.077−04 4.836−04 6.347−04 7.669−04 8.841−04 8.307−04
1
16 3.368−05 1.677−05 9.531−06 6.006−06 4.044−06 4.958−06
1
17 2.192−05 2.724−05 3.467−05 4.162−05 4.773−05 4.435−05
1
18 2.927−05 1.951−05 1.406−05 1.084−05 8.699−06 8.385−06
1
19 1.735−05 1.058−05 6.983−06 4.882−06 3.594−06 3.589−06
1
20 3.851−05 4.858−05 5.527−05 6.107−05 6.553−05 5.526−05
1
21 5.969−05 5.373−05 5.413−05 5.700−05 6.095−05 6.590−05
1
22 8.097−05 4.862−05 3.173−05 2.195−05 1.602−05 1.713−05
1
23 9.235−06 4.567−06 2.603−06 1.654−06 1.115−06 1.186−06
1
24 1.337−05 1.032−05 1.047−05 1.144−05 1.258−05 1.276−05
1
25 1.032−04 1.682−04 2.239−04 2.724−04 3.152−04 2.989−04
1
26 2.005−05 9.758−06 5.490−06 3.453−06 2.310−06 2.688−06
1
27 1.134−05 1.274−05 1.608−05 1.943−05 2.246−05 2.135−05
1
28 6.393−07 2.494−07 1.227−07 6.916−08 4.287−08 4.771−08
1
29 7.822−07 4.054−07 3.355−07 3.273−07 3.337−07 3.244−07
1
30 1.073−06 4.116−07 2.004−07 1.121−07 6.910−08 8.406−08
1
31 7.281−07 4.202−07 3.926−07 4.100−07 4.331−07 3.999−07
1
32 1.651−05 1.005−05 7.175−06 5.503−06 4.410−06 4.171−06
1
33 9.155−06 5.482−06 3.603−06 2.501−06 1.847−06 1.802−06
1
34 1.911−05 2.358−05 2.688−05 2.969−05 3.208−05 2.697−05
1
35 3.060−05 2.670−05 2.659−05 2.779−05 2.965−05 3.193−05
1
36 4.291−05 2.528−05 1.643−05 1.128−05 8.260−06 8.596−06
1
37 5.541−06 2.637−06 1.498−06 9.478−07 6.393−07 6.644−07
1
38 8.011−06 5.600−06 5.518−06 5.959−06 6.526−06 6.675−06
1
39 4.896−05 8.006−05 1.072−04 1.306−04 1.514−04 1.434−04
1
40 1.208−05 5.643−06 3.165−06 1.982−06 1.326−06 1.506−06
1
41 6.743−06 6.724−06 8.415−06 1.018−05 1.180−05 1.135−05
1
42 5.849−07 2.078−07 1.019−07 5.733−08 3.552−08 3.871−08
1
43 7.338−07 3.271−07 2.602−07 2.501−07 2.556−07 2.496−07
1
44 9.845−07 3.434−07 1.667−07 9.302−08 5.729−08 6.823−08
1
45 6.939−07 3.359−07 3.016−07 3.121−07 3.317−07 3.081−07
1
46 1.956−08 4.033−09 1.614−09 7.745−10 4.241−10 4.864−10
1
47 2.461−08 5.978−09 3.393−09 2.663−09 2.598−09 2.091−09
1
48 2.893−08 5.825−09 2.307−09 1.098−09 5.984−10 7.523−10
28
Table 3 :
3b. Collision strengths for transitions in Ge XXXI. (a±b ≡ a×10 ±b ).Transition
Energy (Ryd)
i
j 1000
1250
1500
1750
2000
2250
FAC a
1
2 1.831−04 1.414−04 1.124−04 9.138−05 7.501−05 6.333−05 7.219−05
1
3 1.047−04 7.323−05 5.374−05 4.027−05 3.147−05 2.527−05 2.965−05
1
4 6.151−04 6.927−04 7.514−04 7.937−04 8.305−04 8.672−04 7.202−04
1
5 6.075−04 6.444−04 6.990−04 7.585−04 8.224−04 8.855−04 9.258−04
1
6 4.735−04 3.272−04 2.380−04 1.770−04 1.371−04 1.094−04 1.393−04
1
7 2.049−03 2.719−03 3.320−03 3.862−03 4.363−03 4.822−03 4.331−03
1
8 5.860−05 4.390−05 3.418−05 2.766−05 2.256−05 1.905−05 1.930−05
1
9 3.499−05 2.390−05 1.723−05 1.278−05 9.928−06 7.927−06 8.151−06
1
10 1.063−04 1.247−04 1.392−04 1.501−04 1.591−04 1.680−04 1.400−04
1
11 1.396−04 1.351−04 1.390−04 1.462−04 1.557−04 1.658−04 1.789−04
1
12 1.611−04 1.087−04 7.755−05 5.706−05 4.395−05 3.485−05 3.868−05
1
13 1.347−05 7.866−06 5.021−06 3.424−06 2.461−06 1.833−06 2.023−06
1
14 2.175−05 1.977−05 2.062−05 2.240−05 2.439−05 2.637−05 2.629−05
1
15 3.088−04 4.345−04 5.470−04 6.474−04 7.396−04 8.235−04 7.757−04
1
16 2.889−05 1.667−05 1.052−05 7.110−06 5.075−06 3.755−06 4.569−06
1
17 2.082−05 2.481−05 3.016−05 3.549−05 4.037−05 4.475−05 4.174−05
1
18 2.584−05 1.901−05 1.459−05 1.178−05 9.532−06 7.895−06 7.806−06
1
19 1.538−05 1.045−05 7.468−06 5.527−06 4.241−06 3.369−06 3.336−06
1
20 3.757−05 4.509−05 5.034−05 5.473−05 5.836−05 6.156−05 5.194−05
1
21 5.589−05 5.229−05 5.262−05 5.467−05 5.760−05 6.109−05 6.604−05
1
22 7.133−05 4.784−05 3.385−05 2.484−05 1.889−05 1.491−05 1.587−05
1
23 7.928−06 4.581−06 2.897−06 1.969−06 1.404−06 1.044−06 1.096−06
1
24 1.183−05 9.820−06 9.727−06 1.032−05 1.113−05 1.199−05 1.221−05
1
25 1.045−04 1.507−04 1.924−04 2.296−04 2.629−04 2.939−04 2.790−04
1
26 1.709−05 9.743−06 6.092−06 4.100−06 2.904−06 2.144−06 2.479−06
1
27 1.044−05 1.166−05 1.399−05 1.648−05 1.886−05 2.102−05 2.008−05
1
28 5.201−07 2.547−07 1.423−07 8.748−08 5.750−08 3.988−08 4.383−08
1
29 6.404−07 3.943−07 3.244−07 3.072−07 3.074−07 3.131−07 3.063−07
1
30 8.663−07 4.186−07 2.318−07 1.414−07 9.238−08 6.380−08 7.712−08
1
31 5.965−07 4.021−07 3.668−07 3.727−07 3.893−07 4.067−07 3.776−07
1
32 1.387−05 9.845−06 7.449−06 6.018−06 4.923−06 4.044−06 3.884−06
1
33 8.123−06 5.404−06 3.844−06 2.834−06 2.193−06 1.734−06 1.676−06
1
34 1.844−05 2.175−05 2.443−05 2.657−05 2.858−05 2.998−05 2.534−05
1
35 2.852−05 2.591−05 2.580−05 2.665−05 2.805−05 2.968−05 3.194−05
1
36 3.780−05 2.483−05 1.748−05 1.278−05 9.797−06 7.695−06 7.971−06
1
37 4.664−06 2.640−06 1.664−06 1.128−06 8.066−07 5.974−07 6.140−07
1
38 6.825−06 5.342−06 5.157−06 5.400−06 5.796−06 6.226−06 6.393−06
1
39 4.961−05 7.157−05 9.171−05 1.098−04 1.260−04 1.412−04 1.339−04
1
40 1.007−05 5.625−06 3.504−06 2.354−06 1.670−06 1.229−06 1.389−06
1
41 5.856−06 6.161−06 7.328−06 8.630−06 9.893−06 1.105−05 1.066−05
1
42 4.407−07 2.119−07 1.182−07 7.255−08 4.768−08 3.301−08 3.558−08
1
43 5.537−07 3.193−07 2.537−07 2.358−07 2.352−07 2.400−07 2.356−07
1
44 7.352−07 3.489−07 1.927−07 1.174−07 7.666−08 5.285−08 6.261−08
1
45 5.198−07 3.228−07 2.837−07 2.842−07 2.970−07 3.117−07 2.909−07
1
46 1.154−08 4.197−09 1.985−09 1.060−09 6.229−10 3.943−10 4.442−10
1
47 1.515−08 6.024−09 3.629−09 2.724−09 2.414−09 2.385−09 1.970−09
1
48 1.683−08 6.037−09 2.828−09 1.499−09 8.759−10 5.524−10 6.862−10
29
Table 3 :
3c. Collision strengths for transitions in As XXXII. (a±b ≡ a×10 ±b ).Transition
Energy (Ryd)
i
j 1100
1400
1700
2000
2300
2600
FAC a
1
2 1.681−04 1.254−04 9.695−05 7.778−05 6.367−05 5.333−05 6.734−05
1
3 9.376−05 6.353−05 4.526−05 3.359−05 2.589−05 2.048−05 2.763−05
1
4 5.950−04 6.689−04 7.265−04 7.704−04 8.122−04 8.460−04 6.795−04
1
5 5.975−04 6.516−04 7.211−04 7.947−04 8.697−04 9.419−04 9.364−04
1
6 4.205−04 2.816−04 1.981−04 1.458−04 1.113−04 8.736−05 1.293−04
1
7 1.992−03 2.675−03 3.286−03 3.833−03 4.335−03 4.794−03 4.052−03
1
8 5.347−05 3.897−05 2.955−05 2.325−05 1.921−05 1.582−05 1.801−05
1
9 3.115−05 2.072−05 1.448−05 1.060−05 8.135−06 6.376−06 7.599−06
1
10 1.035−04 1.222−04 1.356−04 1.459−04 1.566−04 1.647−04 1.320−04
1
11 1.325−04 1.325−04 1.396−04 1.499−04 1.618−04 1.735−04 1.792−04
1
12 1.423−04 9.339−05 6.449−05 4.675−05 3.557−05 2.764−05 3.594−05
1
13 1.182−05 6.580−06 4.057−06 2.710−06 1.921−06 1.408−06 1.874−06
1
14 1.995−05 1.874−05 2.009−05 2.218−05 2.435−05 2.642−05 2.512−05
1
15 3.034−04 4.325−04 5.459−04 6.471−04 7.389−04 8.214−04 7.259−04
1
16 2.520−05 1.381−05 8.411−06 5.565−06 3.909−06 2.848−06 4.224−06
1
17 1.979−05 2.447−05 3.015−05 3.561−05 4.046−05 4.480−05 3.937−05
1
18 2.360−05 1.668−05 1.268−05 9.956−06 7.970−06 6.608−06 7.287−06
1
19 1.378−05 9.002−06 6.294−06 4.591−06 3.461−06 2.722−06 3.111−06
1
20 3.696−05 4.382−05 4.974−05 5.388−05 5.730−05 6.055−05 4.893−05
1
21 5.272−05 5.058−05 5.231−05 5.557−05 5.949−05 6.373−05 6.594−05
1
22 6.341−05 4.086−05 2.822−05 2.038−05 1.524−05 1.188−05 1.476−05
1
23 6.987−06 3.809−06 2.344−06 1.557−06 1.095−06 8.022−07 1.016−06
1
24 1.074−05 9.140−06 9.388−06 1.018−05 1.110−05 1.204−05 1.169−05
1
25 1.035−04 1.507−04 1.923−04 2.296−04 2.634−04 2.940−04 2.611−04
1
26 1.495−05 8.026−06 4.879−06 3.208−06 2.237−06 1.628−06 2.292−06
1
27 9.809−06 1.142−05 1.398−05 1.658−05 1.895−05 2.111−05 1.892−05
1
28 4.418−07 2.046−07 1.105−07 6.629−08 4.277−08 2.937−08 4.040−08
1
29 5.510−07 3.440−07 2.951−07 2.876−07 2.917−07 3.004−07 2.897−07
1
30 7.303−07 3.334−07 1.781−07 1.060−07 6.802−08 4.649−08 7.097−08
1
31 5.148−07 3.601−07 3.438−07 3.575−07 3.761−07 3.953−07 3.571−07
1
32 1.241−05 8.669−06 6.412−06 5.071−06 4.084−06 3.346−06 3.627−06
1
33 7.165−06 4.656−06 3.229−06 2.350−06 1.782−06 1.392−06 1.563−06
1
34 1.775−05 2.118−05 2.406−05 2.610−05 2.788−05 2.955−05 2.386−05
1
35 2.643−05 2.493−05 2.552−05 2.699−05 2.887−05 3.083−05 3.185−05
1
36 3.312−05 2.121−05 1.453−05 1.047−05 7.868−06 6.091−06 7.412−06
1
37 4.045−06 2.196−06 1.343−06 8.921−07 6.269−07 4.580−07 5.690−07
1
38 6.060−06 4.933−06 4.947−06 5.315−06 5.775−06 6.252−06 6.125−06
1
39 4.879−05 7.153−05 9.179−05 1.099−04 1.264−04 1.411−04 1.252−04
1
40 8.676−06 4.634−06 2.802−06 1.841−06 1.282−06 9.309−07 1.285−06
1
41 5.366−06 6.015−06 7.316−06 8.686−06 9.950−06 1.111−05 1.005−05
1
42 3.673−07 1.702−07 9.168−08 5.498−08 3.543−08 2.432−08 3.280−08
1
43 4.638−07 2.758−07 2.287−07 2.199−07 2.234−07 2.309−07 2.229−07
1
44 6.081−07 2.778−07 1.480−07 8.803−08 5.639−08 3.854−08 5.764−08
1
45 4.351−07 2.856−07 2.639−07 2.720−07 2.876−07 3.041−07 2.752−07
1
46 8.920−09 3.219−09 1.456−09 7.558−10 4.362−10 2.742−10 4.070−10
1
47 1.173−08 4.819−09 2.966−09 2.342−09 2.195−09 2.240−09 1.853−09
1
48 1.291−08 4.591−09 2.054−09 1.058−09 6.073−10 3.803−10 6.281−10
30
Table 3 :
3d. Collision strengths for transitions in Se XXXIII. (a±b ≡ a×10 ±b ).Transition
Energy (Ryd)
i
j 1100
1500
2000
2500
3000
FAC a
1
2 1.688−04 1.178−04 8.198−05 5.882−05 4.510−05 6.298−05
1
3 9.769−05 5.972−05 3.578−05 2.360−05 1.654−05 2.582−05
1
4 5.430−04 6.415−04 7.215−04 7.767−04 8.292−04 6.421−04
1
5 5.752−04 6.483−04 7.653−04 8.856−04 1.001−03 9.430−04
1
6 4.367−04 2.620−04 1.545−04 1.005−04 6.952−05 1.203−04
1
7 1.705−03 2.516−03 3.378−03 4.119−03 4.775−03 3.799−03
1
8 5.414−05 3.627−05 2.472−05 1.736−05 1.347−05 1.685−05
1
9 3.260−05 1.932−05 1.131−05 7.350−06 5.127−06 7.104−06
1
10 9.269−05 1.162−04 1.357−04 1.496−04 1.619−04 1.246−04
1
11 1.287−04 1.292−04 1.439−04 1.628−04 1.824−04 1.789−04
1
12 1.487−04 8.641−05 4.973−05 3.183−05 2.191−05 3.348−05
1
13 1.276−05 6.077−06 2.994−06 1.718−06 1.094−06 1.741−06
1
14 1.964−05 1.769−05 2.023−05 2.348−05 2.655−05 2.403−05
1
15 2.544−04 4.068−04 5.670−04 7.028−04 8.221−04 6.807−04
1
16 2.713−05 1.266−05 6.120−06 3.465−06 2.183−06 3.915−06
1
17 1.806−05 2.311−05 3.142−05 3.874−05 4.490−05 3.722−05
1
18 2.407−05 1.561−05 1.039−05 7.375−06 5.598−06 6.820−06
1
19 1.434−05 8.413−06 4.860−06 3.161−06 2.189−06 2.909−06
1
20 3.266−05 4.186−05 4.954−05 5.489−05 5.991−05 4.616−05
1
21 5.122−05 4.912−05 5.323−05 5.976−05 6.661−05 6.563−05
1
22 6.594−05 3.792−05 2.152−05 1.378−05 9.420−06 1.375−05
1
23 7.552−06 3.529−06 1.721−06 9.803−07 6.227−07 9.439−07
1
24 1.084−05 8.624−06 9.335−06 1.073−05 1.214−05 1.120−05
1
25 8.576−05 1.418−04 2.008−04 2.508−04 2.943−04 2.449−04
1
26 1.615−05 7.384−06 3.531−06 1.983−06 1.246−06 2.126−06
1
27 9.241−06 1.078−05 1.459−05 1.816−05 2.123−05 1.787−05
1
28 5.086−07 1.882−07 7.589−08 3.788−08 2.180−08 3.734−08
1
29 6.117−07 3.194−07 2.700−07 2.762−07 2.896−07 2.744−07
1
30 8.312−07 3.047−07 1.209−07 5.971−08 3.415−08 6.550−08
1
31 5.676−07 3.352−07 3.284−07 3.577−07 3.851−07 3.383−07
1
32 1.323−05 8.043−06 5.343−06 3.761−06 2.820−06 3.395−06
1
33 7.637−06 4.383−06 2.504−06 1.622−06 1.116−06 1.462−06
1
34 1.613−05 2.040−05 2.400−05 2.667−05 2.912−05 2.249−05
1
35 2.628−05 2.423−05 2.587−05 2.895−05 3.216−05 3.166−05
1
36 3.528−05 1.983−05 1.113−05 7.098−06 4.822−06 6.911−06
1
37 4.516−06 2.043−06 9.869−07 5.620−07 3.553−07 5.289−07
1
38 6.378−06 4.659−06 4.893−06 5.582−06 6.310−06 5.871−06
1
39 4.088−05 6.737−05 9.596−05 1.204−04 1.413−04 1.174−04
1
40 9.691−06 4.283−06 2.028−06 1.139−06 7.126−07 1.192−06
1
41 5.344−06 5.680−06 7.638−06 9.531−06 1.118−05 9.488−06
1
42 4.472−07 1.572−07 6.291−08 3.141−08 1.806−08 3.033−08
1
43 5.451−07 2.570−07 2.075−07 2.114−07 2.235−07 2.112−07
1
44 7.171−07 2.549−07 1.003−07 4.956−08 2.830−08 5.321−08
1
45 5.211−07 2.667−07 2.506−07 2.736−07 2.977−07 2.607−07
1
46 1.292−08 2.950−09 9.123−10 3.792−10 1.922−10 3.741−10
1
47 1.662−08 4.415−09 2.344−09 2.031−09 2.138−09 1.746−09
1
48 1.848−08 4.179−09 1.272−09 5.236−10 2.640−10 5.766−10
31
Table 3 :
3e. Collision strengths for transitions in Br XXXIV. (a±b ≡ a×10 ±b ). .722−05 2.182−05 2.774−05 3.327−05 3.812−05 4.247−05 3.525−05 1 18 2.226−05 1.483−05 1.082−05 8.152−06 6.502−06 5.091−06 6.398−06 1 19 1.322−05 7.955−06 5.176−06 3.631−06 2.688−06 2.058−06 2.727−06 1 20 3.237−05 4.003−05 4.581−05 5.006−05 5.393−05 5.643−05 4.363−05 1 21 4.906−05 4.780−05 5.084−05 5.571−05 6.090−05 6.635−05 6.Transition
Energy (Ryd)
i
j 1200
1600
2000
2400
2800
3200
FAC a
1
2 1.557−04 1.105−04 8.404−05 6.450−05 5.230−05 4.178−05 5.905−05
1
3 8.866−05 5.595−05 3.776−05 2.697−05 2.022−05 1.563−05 2.419−05
1
4 5.262−04 6.075−04 6.740−04 7.179−04 7.597−04 7.883−04 6.077−04
1
5 5.638−04 6.411−04 7.325−04 8.267−04 9.178−04 1.007−03 9.459−04
1
6 3.930−04 2.438−04 1.620−04 1.144−04 8.491−05 6.501−05 1.122−04
1
7 1.651−03 2.360−03 2.978−03 3.529−03 4.024−03 4.487−03 3.569−03
1
8 5.012−05 3.424−05 2.571−05 1.959−05 1.547−05 1.228−05 1.580−05
1
9 2.977−05 1.820−05 1.208−05 8.517−06 6.305−06 4.861−06 6.659−06
1
10 9.115−05 1.105−04 1.266−04 1.371−04 1.472−04 1.536−04 1.178−04
1
11 1.235−04 1.263−04 1.377−04 1.520−04 1.670−04 1.823−04 1.780−04
1
12 1.346−04 8.084−05 5.287−05 3.681−05 2.694−05 2.057−05 3.127−05
1
13 1.138−05 5.695−06 3.304−06 2.102−06 1.428−06 1.028−06 1.622−06
1
14 1.819−05 1.677−05 1.849−05 2.083−05 2.315−05 2.538−05 2.300−05
1
15 2.505−04 3.830−04 4.977−04 5.990−04 6.894−04 7.736−04 6.398−04
1
16 2.403−05 1.178−05 6.729−06 4.226−06 2.846−06 2.032−06 3.638−06
1
17 1513−05
1
22 6.031−05 3.562−05 2.283−05 1.582−05 1.157−05 8.773−06 1.285−05
1
23 6.766−06 3.317−06 1.893−06 1.202−06 8.122−07 5.840−07 8.796−07
1
24 9.954−06 8.182−06 8.598−06 9.551−06 1.060−05 1.163−05 1.073−05
1
25 8.523−05 1.335−04 1.755−04 2.130−04 2.460−04 2.768−04 2.301−04
1
26 1.436−05 6.893−06 3.870−06 2.426−06 1.623−06 1.159−06 1.977−06
1
27 8.711−06 1.019−05 1.286−05 1.551−05 1.791−05 2.008−05 1.692−05
1
28 4.461−07 1.763−07 8.636−08 4.892−08 3.038−08 2.035−08 3.462−08
1
29 5.424−07 3.001−07 2.565−07 2.553−07 2.637−07 2.741−07 2.604−07
1
30 7.311−07 2.835−07 1.370−07 7.686−08 4.742−08 3.159−08 6.062−08
1
31 4.996−07 3.150−07 3.042−07 3.231−07 3.450−07 3.650−07 3.210−07
1
32 1.181−05 7.695−06 5.519−06 4.201−06 3.295−06 2.644−06 3.185−06
1
33 6.818−06 4.103−06 2.680−06 1.868−06 1.385−06 1.056−06 1.371−06
1
34 1.538−05 1.924−05 2.230−05 2.429−05 2.633−05 2.750−05 2.125−05
1
35 2.438−05 2.344−05 2.477−05 2.696−05 2.947−05 3.200−05 3.137−05
1
36 3.125−05 1.845−05 1.187−05 8.168−06 5.986−06 4.513−06 6.460−06
1
37 3.891−06 1.911−06 1.091−06 6.881−07 4.658−07 3.333−07 4.929−07
1
38 5.620−06 4.406−06 4.526−06 4.986−06 5.515−06 6.052−06 5.631−06
1
39 3.987−05 6.344−05 8.390−05 1.020−04 1.181−04 1.329−04 1.103−04
1
40 8.284−06 3.979−06 2.236−06 1.391−06 9.325−07 6.624−07 1.108−06
1
41 4.800−06 5.361−06 6.727−06 8.131−06 9.410−06 1.058−05 8.976−06
1
42 3.697−07 1.467−07 7.187−08 4.053−08 2.519−08 1.686−08 2.812−08
1
43 4.582−07 2.399−07 1.986−07 1.953−07 2.023−07 2.119−07 2.004−07
1
44 6.073−07 2.364−07 1.142−07 6.377−08 3.934−08 2.620−08 4.926−08
1
45 4.249−07 2.488−07 2.333−07 2.464−07 2.647−07 2.825−07 2.474−07
1
46 9.556−09 2.739−09 1.099−09 5.303−10 2.930−10 1.792−10 3.448−10
1
47 1.232−08 4.089−09 2.408−09 1.943−09 1.898−09 1.981−09 1.642−09
1
48 1.374−08 3.855−09 1.526−09 7.293−10 4.005−10 2.440−10 5.309−10
32
http://sprg.ssl.berkeley.edu/~mfgu/fac/. This is also a fully relativistic code which provides a variety of atomic parameters, and (generally) yields results for energy levels and radiative rates comparable to grasp -see, for example,[18] and[27], and references therein. However, the differences in collision strengths and subsequently in effective collision strengths between fac and darc can be large, particularly for forbidden transitions, as demonstrated in our earlier papers[11]-[20], and also discussed below in sections 5 and 6. Hence results from fac will be helpful in assessing the accuracy of our energy levels and radiative rates, and in estimating the contribution of resonances to effective collision strengths, included in calculations with darc but not in fac.3
AcknowledgmentKMA is grateful to AWE Aldermaston for financial support.
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| [] |
[
"The fine-tuning price of the early LHC",
"The fine-tuning price of the early LHC"
] | [
"Alessandro Strumia \nDipartimento di Fisica dell'Università di Pisa and INFN\nNational Institute of Chemical Physics and Biophysics\nRavala 10TallinItalia, Estonia\n"
] | [
"Dipartimento di Fisica dell'Università di Pisa and INFN\nNational Institute of Chemical Physics and Biophysics\nRavala 10TallinItalia, Estonia"
] | [] | LHC already probed and excluded most of the parameter space of the Constrained Minimal Supersymmetric Standard Model allowed by previous experiments. Only about 0.3% of the CMSSM parameter space survives. This fraction rises to about 0.9% if the bound on the Higgs mass can be circumvented. | 10.1007/jhep04(2011)073 | [
"https://arxiv.org/pdf/1101.2195v3.pdf"
] | 119,205,939 | 1101.2195 | ef0e9e69c2056d984a3f92ef9c935b19bbb19b13 |
The fine-tuning price of the early LHC
Alessandro Strumia
Dipartimento di Fisica dell'Università di Pisa and INFN
National Institute of Chemical Physics and Biophysics
Ravala 10TallinItalia, Estonia
The fine-tuning price of the early LHC
LHC already probed and excluded most of the parameter space of the Constrained Minimal Supersymmetric Standard Model allowed by previous experiments. Only about 0.3% of the CMSSM parameter space survives. This fraction rises to about 0.9% if the bound on the Higgs mass can be circumvented.
Introduction
When LEP started 20 years ago, the main topic of high-energy physics was finding the right supersymmetric unified model, its embedding in string theory and understanding how it predicts a zero cosmological constant.
But supersymmetry was not found and LEP opened a "little hierarchy problem" [1,2]: experimental bounds on sparticle masses made difficult to fully solve the higgs mass hierarchy problem. Furthermore, cosmological observations strongly suggested a small but non-zero cosmological constant [3], opening a new hierarchy problem with no known solution. Now LHC is starting and its most fundamental goal is telling why the the weak scale is much below the Planck scale: is it small due to some natural reason (as many theorists expect) or for some other "unnatural" reason (such as anthropic selection in a multiverse)?
Supersymmetry remains the main candidate natural solution to the weak scale hierarchy problem. The CMS and ATLAS collaborations published the first results of searches for supersymmetric particles looking at events with jets and missing energy, without and with a lepton, first in data at √ s = 7 TeV with 35/pb of integrated luminosity [4], later increased to about 1.1/fb [5].
Supersymmetry was not found in such LHC data, and the LHC collaborations produced bounds in the CMSSM model: degenerate squarks and gluinos must be typically heavier than up to 1.1 TeV (see fig. 2 below). This is significantly stronger than previous bounds.
Again, the main implications of such negative experimental searches concerns naturalness, which is the heart of the main question: is the weak scale naturally small?
Naturalness
To illustrate the naturalness problem of the CMSSM model we recall that it predicts the Z mass to be
M 2 Z ≈ 0.7M 2 3 + 0.2m 2 0 − 2µ 2 = (91 GeV) 2 × 100( M 3 1.1 TeV ) 2 + · · ·(1)
where M 3 ≈ 2.6M 1/2 is the gluino mass, M 1/2 and m 0 are the unified gaugino and scalar masses at the unification scale; the µ term is renormalized at the weak scale, and · · · denotes the m 2 0 and µ 2 terms. We here assumed tan β = 3 and A 0 = 0, such that the top Yukawa coupling renormalized at the unification scale is λ t (M GUT ) ≈ 0.5. Eq. (1) means that the natural sparticle scale is M 1/2 ∼ m 0 ∼ µ ∼ M Z and that an accidental cancellation by a part in ≈ 100 is needed if M 3 > 1.1 TeV. Eq. (1) can be used to fix the overall SUSY mass scale, such that the CMSSM model has two free adimensional parameters: the ratios M 1/2 /µ and m 0 /µ (tan β = 3 and A 0 = 0 are for the moment kept fixed). Such parameter space is plotted in fig. 1:
• The light-gray regions are theoretically excluded because the minimum of the potential is not the physical one: in the left region one would have M 2 Z < 0 which means that the true minimum is at v = 0; in the bottom-right region the potential is unstable when the two higgses have equal vev.
• The red region in the middle is theoretically allowed, but has now been experimentally excluded. The darker red shows the new region probed and excluded by LHC with respect to the previous LEP bounds, approximated to be M 2 > 100 GeV.
• The green region is allowed. Indeed it is close to the boundary where M Z = 0 and thereby has M Z m 0 , M 1/2 , µ. Figure 2: Left: naturalness scan of the CMSSM. Red points are excluded by LHC, black points have been excluded earlier, green points are allowed. The darker pink region was excluded by LEP and the pink region by early LHC (the red lines show the various bounds from ATLAS and CMS). Right: "naturalness probability distribution" for the gluino mass in the CMSSM. Only its tail was allowed after LEP, and the tail of the tail remains allowed after first LHC data.
The smallness of the allowed region is a manifestation of the "little hierarchy problem".
We now relax the restriction on A 0 and tan β (or equivalently B 0 ) and study naturalness proceeding along the lines of [1], as briefly summarized below.
We randomly scan the full theoretically allowed adimensional parameters of the model (the adimensional ratios between m 0 , M 1/2 , µ, A 0 , B 0 as well as the top Yukawa coupling λ t , all renormalized at the unification scale) determining the overall SUSY mass scale and tan β from the potential minimization condition. Thanks to the last step, we sample the full CMSSM parameter space according to its natural density (rare accidental cancellations that make sparticles heavy happen rarely). We compute how rare are the still allowed sparticle spectra, as in [1] that claimed that only 5% of the CMSSM parameter space survived to LEP.
More precisely we perform the following scan
m 0 = ( 1 3 2 ÷ 3) log m SUSY , |µ 0 |, M 1/2 = ( 1 3 ÷ 3) log m SUSY , A 0 , B 0 = (−3 ÷ 3) lin M 1/2 (2)
and verify that it gives results similar to other possibilities such as
m 0 , |µ 0 |, M 1/2 = ( 1 3 ÷ 3) log m SUSY , A 0 , B 0 = (−3 ÷ 3) lin m 0(3)
or as m 0 , |µ 0 |,
M 1/2 , |B 0 |, |A 0 | = (0 ÷ 1) lin m SUSY .(4)
where the pedices 'lin' and 'log' respectively denote a flat probability distribution in linear or logarithmic scale within the given range. More formally, this is a Monte Carlo Bayesian technique that starts with an arbitrary non-informative prior probability density function (implicitly defined by the 'random scans' in eq.s (2) to (4)) and gives a set of points in parameter space with probability density roughly equal to the inverse of the various fine-tuning measures proposed to approximate the naturalness issue [6]. The above procedure makes no use of any fine-tuning parameter, and automatically takes into account all fine-tunings: not only the one needed to have M Z m 0 , M 1/2 , µ, but also the one needed to have tan β 1, or the fine-tuning on λ t that can give a small or even negative m 2 0 coefficient in eq. (1), such that the M 2 3 term can be cancelled by m 2 0 rather than by µ 2 . The scanning is restricted to top quark masses within 3 standard deviations of the present measured value, m t = (173.1 ± 1.1) GeV [7].
A technical detail. The MSSM minimization equations generalize eq. (1) taking into account one loop corrections to the potential. To understand their relevance, we recall that at tree level the higgs mass is predicted to be m tree h ≤ M Z cos 2β, while at loop level it can be above the experimental limit m h > 114 GeV. The effect of minimizing the one loop potential (rather than the tree level potential) is essentially equivalent to rescaling the overall SUSY mass scale by a factor m h /m tree h , which helps naturalness. We consider the three main bounds on sparticles, that can be roughly summarized as follows:
1) The LHC bound on (mainly) the gluino and squark masses is plotted in fig. 2a. 1 We find that this bound alone excludes about 99% of the CMSSM parameter space.
2) LEP tells that all charged sparticles (charginos, sleptons, stops...) are heavier than about 100 GeV, unless they are quasi-degenerate with the lightest supersymmetric particle. Such bounds alone excluded about 90% of the CMSSM parameter space [1].
3) The LEP bound on the Higgs mass (m h > 114 GeV in the SM) is potentially even stronger but it is not robust and deserves a dedicated discussion.
As well known, the bound on the Higgs mass can exclude the whole MSSM, because the MSSM predicts at tree level a higgs lighter than M Z and at loop level a higgs lighter than about 125 GeV. The precise value logarithmically depends on the sparticle mass scale; we compute it using our own code and using the more precise SoftSusy code [8]. However, one can modify the MSSM to increase the predicted higgs mass (e.g. adding a singlet as in the NMSSM), avoiding the fine-tuning price of the higgs mass bound (and alleviating the whole fine tuning [9]). Furthermore, in regions of the MSSM parameter space with µ ∼ M Z and large tan β the higgs coupling to the Z is reduced and a weaker bound m h > ∼ 100 GeV applies. More generically, the bound on the higgs mass can be weakened (down to about 100 GeV) modifying the theory such that the higgs has new dominant decay modes which are more difficult to CMSSM dynamical m SUSY Figure 3: As in the previous figure 2a, but assuming that the sparticle mass scale is dynamically determined by minimizing the MSSM potential [13], such that it is naturally heavier than M Z . see experimentally, or which have not been experimentally studied [10]. Summarizing, many models can alleviate the fine-tuning problem related to the higgs mass.
The motivation for such models gets now washed out by the LHC bound, which has nothing to do with the higgs mass. Table 1 shows that naturalness remains a problem, even if one can circumvent the bound on the higgs mass.
In its last column we stick to the CMSSM and approximate the LEP bound m h > 114 GeV as m th h > 110 GeV, where m th h is the higgs mass as computed by state-of-the-art codes [8], that have a theoretical uncertainty estimated to be about ±3 GeV. With such a bound the allowed fraction of the CMSSM parameter space decreases down to about 0.3%. The other scans (eq. (3) or eq. (4)) would give lower comparable fractions of allowed parameter space.
Conclusions
LHC data sharpen the "little hierarchy problem" of supersymmetry. Various tentative solutions were proposed. The simplest solution, a gluino lighter than what predicted by unification [11,1], seems now excluded by the new LHC bound.
It is maybe useful to see the problem in this way: in SUSY models renormalization effects from the Planck scale down to the weak scale can radiatively break the weak symmetry, making the determinant of the squared higgs mass term negative below some scale Q 0 which in principle is anyway between the weak and Planck scales: the little hierarchy problem means that breaking of the weak gauge group happens at the last moment, Q 0 ∼ v [2]. Therefore, one possible interpretation is that the Higgs is the pseudo-Goldstone boson of some symmetry broken at some scale Q 0 around the weak scale; however concrete models often look less plausible than the fine-tuning they avoid [12].
Another possibility considered in [13] is assuming that the overall scale of soft supersymmetry breaking is dynamically fixed by minimizing the weak part only of the potential (this assumption looks implausible, as recognized in [13]), such that the SUSY scale must be just below Q 0 ; the precise computation gives a neat prediction of the form m SUSY ≈ 4πM Z /12, where 4π is a loop factor and the factor 12 counts the spins and colors in the most relevant diagram [13]. Applied to the CMSSM model such prediction is plotted as dashed line in fig. 1: it lies around the present allowed/excluded border. Fig. 3 shows the naturalness scan in the full parameter space: we see that even in this case LHC probed most of the parameter space. LEP tested SUSY masses around the Z mass, and now LHC reached the next milestone, testing SUSY masses a loop factor above the Z mass.
Maybe the weak scale is small due to anthropic selection, and attempts of keeping it technically small are like attempts of dragging the aether. The scenario of [13] was reconsidered in [14] with a different motivation: the authors imagine a supersymmetric multiverse where for some unknown reason the weak scale is "almost never" broken (Q 0 m SUSY ), such that Q 0 ∼ m SUSY is "more likely". The qualitative expectation is similar to the prediction of [13], but m SUSY can be made heavier by order one factors making the "almost never" stronger, at the price of making MSSM vacua more rare than the SM with an unnaturally light higgs.
All these beautiful ideas and the history of the Michelson-Morley experiment teach us that a negative experimental search can have deep theoretical implications.
"History repeats itself, first as tragedy, second as farce". Karl Marx
Figure 1 :
1A typical example of the parameter space of the CMSSM model. The green region is allowed (see it in the enlarged box). The dashed line around the boundary of the allowed region is the prediction of the model considered in[13].
Table 1 :
1Fraction of the CMSSM parameter space that survives to the various bounds.experimental fraction of surviving CMSSM parameter space
bound
any m h m h > 100 GeV
m h > 110 GeV
LEP
10%
4%
1%
LHC
0.9%
0.5%
0.3%
The CMS and ATLAS collaborations computed bounds for tan β = 3 or 10 and A 0 = 0[4,5]. The dominant bound from events with jets and missing energy is essentially a bound on the gluino and squark masses, so that the dependence on A 0 and tan β can be neglected. The subleading bound from events with one lepton has only a moderate dependence on tan β, that we also ignore.
Acknowledgements The author thanks Ben Allanach, Riccardo Rattazzi and Andrea Romanino for useful communications. This work was supported by the ESF grant MTT8 and by SF0690030s09 project.
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| [] |
[
"Transition to turbulence when the Tollmien-Schlichting and bypass routes coexist",
"Transition to turbulence when the Tollmien-Schlichting and bypass routes coexist",
"Transition to turbulence when the Tollmien-Schlichting and bypass routes coexist",
"Transition to turbulence when the Tollmien-Schlichting and bypass routes coexist"
] | [
"Stefan Zammert \nLaboratory for Aero and Hydrodynamics\nDelft University of Technology\n2628 CDDelftThe Netherlands\n\nFachbereich Physik\nPhilipps-Universität Marburg\nD-35032MarburgGermany\n",
"Bruno Eckhardt \nFachbereich Physik\nPhilipps-Universität Marburg\nD-35032MarburgGermany\n\nJ.M. Burgerscentrum\nDelft University of Technology\n2628 CDDelftThe Netherlands\n",
"Stefan Zammert \nLaboratory for Aero and Hydrodynamics\nDelft University of Technology\n2628 CDDelftThe Netherlands\n\nFachbereich Physik\nPhilipps-Universität Marburg\nD-35032MarburgGermany\n",
"Bruno Eckhardt \nFachbereich Physik\nPhilipps-Universität Marburg\nD-35032MarburgGermany\n\nJ.M. Burgerscentrum\nDelft University of Technology\n2628 CDDelftThe Netherlands\n"
] | [
"Laboratory for Aero and Hydrodynamics\nDelft University of Technology\n2628 CDDelftThe Netherlands",
"Fachbereich Physik\nPhilipps-Universität Marburg\nD-35032MarburgGermany",
"Fachbereich Physik\nPhilipps-Universität Marburg\nD-35032MarburgGermany",
"J.M. Burgerscentrum\nDelft University of Technology\n2628 CDDelftThe Netherlands",
"Laboratory for Aero and Hydrodynamics\nDelft University of Technology\n2628 CDDelftThe Netherlands",
"Fachbereich Physik\nPhilipps-Universität Marburg\nD-35032MarburgGermany",
"Fachbereich Physik\nPhilipps-Universität Marburg\nD-35032MarburgGermany",
"J.M. Burgerscentrum\nDelft University of Technology\n2628 CDDelftThe Netherlands"
] | [
"J. Fluid Mech",
"J. Fluid Mech"
] | Plane Poiseuille flow, the pressure-driven flow between parallel plates, shows a route to turbulence connected with a linear instability to Tollmien-Schlichting (TS) waves, and another route, the bypass transition, that can be triggered with finite-amplitude perturbation. We use direct numerical simulations to explore the arrangement of the different routes to turbulence among the set of initial conditions. For plates that are a distance 2H apart, and in a domain of width 2πH and length 2πH, the subcritical instability to TS waves sets in at Re c = 5815 and extends down to Re TS ≈ 4884. The bypass route becomes available above Re E = 459 with the appearance of three-dimensional, finite-amplitude travelling waves. Below Re c , TS transition appears for a tiny region of initial conditions that grows with increasing Reynolds number. Above Re c , the previously stable region becomes unstable via TS waves, but a sharp transition to the bypass route can still be identified. Both routes lead to the same turbulent state in the final stage of the transition, but on different time scales. Similar phenomena can be expected in other flows where two or more routes to turbulence compete. | 10.1017/jfm.2019.724 | null | 119,075,897 | 1702.08416 | 0af739775f8bde6b0f511e622e8d67def4e7bf37 |
Transition to turbulence when the Tollmien-Schlichting and bypass routes coexist
2019
Stefan Zammert
Laboratory for Aero and Hydrodynamics
Delft University of Technology
2628 CDDelftThe Netherlands
Fachbereich Physik
Philipps-Universität Marburg
D-35032MarburgGermany
Bruno Eckhardt
Fachbereich Physik
Philipps-Universität Marburg
D-35032MarburgGermany
J.M. Burgerscentrum
Delft University of Technology
2628 CDDelftThe Netherlands
Transition to turbulence when the Tollmien-Schlichting and bypass routes coexist
J. Fluid Mech
8802201910.1017/jfm.2019.724(Received 27 February 2017; revised 22 August 2019; accepted 30 August 2019; first published online 9 October 2019)transition to turbulencenonlinear instabilitybifurcation
Plane Poiseuille flow, the pressure-driven flow between parallel plates, shows a route to turbulence connected with a linear instability to Tollmien-Schlichting (TS) waves, and another route, the bypass transition, that can be triggered with finite-amplitude perturbation. We use direct numerical simulations to explore the arrangement of the different routes to turbulence among the set of initial conditions. For plates that are a distance 2H apart, and in a domain of width 2πH and length 2πH, the subcritical instability to TS waves sets in at Re c = 5815 and extends down to Re TS ≈ 4884. The bypass route becomes available above Re E = 459 with the appearance of three-dimensional, finite-amplitude travelling waves. Below Re c , TS transition appears for a tiny region of initial conditions that grows with increasing Reynolds number. Above Re c , the previously stable region becomes unstable via TS waves, but a sharp transition to the bypass route can still be identified. Both routes lead to the same turbulent state in the final stage of the transition, but on different time scales. Similar phenomena can be expected in other flows where two or more routes to turbulence compete.
Introduction
The application of ideas from dynamical systems theory to the turbulence transition in flows without linear instability of the laminar profile, such as pipe flow or plane Couette flow, have provided a framework in which many of the observed phenomena can be rationalized. This includes the sensitive dependence on initial conditions (Darbyshire & Mullin 1995;Schmiegel & Eckhardt 1997), the appearance of exact coherent states around which the turbulent state can form (Nagata 1990;Clever & Busse 1997;Waleffe 1998;Faisst & Eckhardt 2003;Wedin & Kerswell 2004;Gibson, Halcrow & Cvitanović 2009), the transience of the turbulent state (Brosa 1989;Bottin et al. 1998;Hof et al. 2006;Kreilos & Eckhardt 2012), or the complex spatio-temporal dynamics in large systems (Bottin et al. 1998; Barkley & Tuckerman 2005;Manneville 2009; Moxey & Barkley 2010;Avila et al. 2011). Extensions to open external flows, such as asymptotic suction boundary layers Khapko et al. 2013Khapko et al. , 2014Khapko et al. , 2016 and developing boundary layers (Cherubini et al. 2011a;Duguet et al. 2012;Wedin et al. 2014), have been proposed.
Plane Poiseuille flow (PPF), the pressure-driven flow between parallel plates, shows a transition to turbulence near a Reynolds number of approximately 1000 (Carlson, Widnall & Peeters 1982;Lemoult, Aider & Wesfreid 2012;Tuckerman et al. 2014). In the subcritical range the flow shows much of the transition phenomenology observed in other subcritical flows, such as plane Couette flow or pipe flow, but it also has a linear instability of the laminar profile at a Reynolds number of 5772 (Orszag 1971). This raises a question about the relation between the transition via an instability to the formation of Tollmien-Schlichting (TS) waves and the transition triggered by large-amplitude perturbations that do not need the linear instability (henceforth referred to as the 'bypass' transition) (Schmid & Henningson 2001). For instance, one could imagine that the exact coherent structures related to the bypass transition are connected to the TS waves in some kind of subcritical bifurcation. That, however, would require a connection between the two very different flow structures: the exact coherent structures that are dominated by downstream vortices (Zammert & Eckhardt 2014, 2015, and the TS waves that are dominated by spanwise vortices.
Many studies of the transition to turbulence have focused on the identification of optimal perturbations that are distinguished by being the smallest ones that can trigger the transition (Farrell 1988;Butler & Farrell 1992;Duguet, Brandt & Larsson 2010;Cherubini et al. 2011b;Monokrousos et al. 2011;Pringle, Willis & Kerswell 2012;Duguet et al. 2013). With a suitable choice of norm (for example, energy density or dissipation) this leads to an optimization problem that can be solved with suitable adjoint techniques (see Schmid (2007) and Kerswell, Pringle & Willis (2014) for reviews of the method and the connection to the dynamics in the state space of the system). In the presence of two possible routes to turbulence, via the Tollmien-Schlichting states and the bypass transition, one can expect that there are two different kinds of optimal perturbation, one for each route, but they have not been determined.
As an alternative to the determination of the optimal perturbations, we here use direct numerical simulations to map out the regions of initial conditions that follow one or the other path. Given the high dimensionality of the space, we can explore a subset of initial conditions only, and choose a plane that contains both transition states. Such explorations of the state space of a flow have been useful in the identification of the sensitive dependence on initial conditions for the transition (Schmiegel & Eckhardt 1997;Faisst & Eckhardt 2004), and in the exploration of the bifurcations (Kreilos & Eckhardt 2012;Kreilos, Eckhardt & Schneider 2014).
We start with a description of the system and the bifurcations of the relevant coherent states in § 2. Afterwards, in § 3, we describe the exploration of the state space of the system, and continue with an description of the influence of the operating conditions in § 4. Conclusions are summarized in § 5. The red and blue surfaces correspond to u = ±0.5u max , respectively. (b) Visualization of the edge state TW E for the bypass transition. As before, the yellow surface indicates values of 0.3Q max for the Q-vortex criterion. The levels for the red and blue surfaces are now u = 0.008 and u = −0.014, respectively.
Plane Poiseuille flow and its coherent structures
To fix the geometry, let x, y and z be the downstream, normal and spanwise directions, and let the flow be bounded by parallel plates at y = ±H. Dimensionless units are formed with the height H and the centreline velocity U 0 of the laminar parabolic profile so that the unit of time is H/U 0 and the Reynolds number becomes Re = U 0 H/ν, with ν the fluid viscosity. In these units the laminar profile becomes u 0 = (1 − y 2 )e x . The equations of motion, the incompressible Navier-Stokes equations, are solved using Channelflow (Gibson 2012), with a spatial resolution of N x = N z = 32 and N y = 65 for a domain of length 2π and width 2π and at fixed mass flux. The chosen resolution is sufficient to resolve the exact solutions and the transition process but under-resolved in the turbulent case. In the studied domain, the linear instability occurs at Re c = 5815, slightly higher than the value found by Orszag (1971), on account of the slightly different domain size.
The full velocity field U = u 0 + u can be written as the sum of the laminar flow u 0 and deviations u = (u, v, w). In the following we always mean u when we refer to the velocity field. To characterize the size of perturbations we will use the root mean square (r.m.s.) velocity in u, 1) or in one of the components, for example,
a(u) = u = 1 L x L y L z u 2 dx dy dz,(2.u rms = 1 L x L y L z u 2 dx dy dz, (2.2)
for the downstream component. Tollmien-Schlichting (TS) waves are travelling waves formed by spanwise vortices. They appear in a subcritical bifurcation that extends down to Re ≈ 2610 for a streamwise wavenumber of 1.36. The TS wave is independent of spanwise position z and consists of two spanwise vortices, as shown in figure 1(a). The Reynolds number range over which the transition to TS waves is subcritical depends on the domain size. For our domain (streamwise wavenumber of 1.0) the turning point is at Re ≈ 4685. A bifurcation diagram of this exact solution, referred to as TW TS in the remainder of the paper, is shown in figure 2(a). The ordinate in the bifurcation diagram is the amplitude of the deviation from laminar flow (2.1). A study of the stability of the state in the full three-dimensional space shows that this lower branch state has only one unstable direction in the used computational domain for 5727 < Re < 5815 = Re c . Thus, for these Reynolds numbers the state is an edge state whose stable manifold divides the state space in two parts (Skufca, Yorke & Eckhardt 2006). For lower Re, there are secondary bifurcations that add more unstable directions to the state. Specifically, near the turning point at Re = 4690, the lower branch has acquired approximately 350 unstable directions. Because of the high critical Reynolds numbers this state cannot explain the transition to turbulence observed in experiments at Reynolds numbers around 1000 (Carlson et al. 1982;Nishioka & Asai 1985;Lemoult et al. 2012;Lemoult, Aider & Wesfreid 2013) or even lower (Sano & Tamai 2016).
The states that are relevant to the bypass transition can be found using the method of edge tracking (Toh & Itano 2003;Schneider, Eckhardt & Yorke 2007;. Starting from an arbitrary turbulent initial condition at Re = 1400, trajectories in the laminar-turbulent boundary that are followed with the edge-tracking algorithm converge to a travelling wave (Zammert & Eckhardt 2014), which we referred to as TW E in the following. After its initial identification, the state can be tracked easily to higher Reynolds numbers. The visualization in figure 1(b) shows that this state has a strong narrow upstream streak, a weaker but more extended downstream streak, and streamwise vortices. Moreover, TW E has a wall-normal reflection symmetry
s y : [u, v, w](x, y, z) = [u, −v, w](x, −y, z), (2.3)
a shift-and-reflect symmetry
s z τ x : [u, v, w](x, y, z) = [u, v, −w](x + 0.5 · L x , y, −z), (2.4)
and exists for a wide range in Reynolds numbers. It is created in a saddle-node bifurcation near Re ≈ 459 (see the bifurcation diagram in figure 2a); for other combinations of spanwise and streamwise wavelengths the state appears at an even lower Reynolds numbers of 319 (Zammert & Eckhardt 2017). The corresponding lower branch state can be continued to Reynolds numbers far above 3 × 10 5 , and its amplitude decreases with increasing Reynolds number, as shown in figure 2(b). A fit to the amplitude for large Reynolds numbers gives a scaling like Re −0.52 , similar to that of the solution embedded in the edge of plane Couette flow (Itano et al. 2013). A stability analysis of the lower branch of TW E shows that the travelling wave has one unstable eigenvalue for 510 < Re < 5850. Therefore, TW E is a second travelling wave with a stable manifold that can divide the state space into two disconnected parts. How the two edge states interact and divide up the state space will be discussed in § 3.
At Re = 510 the lower branch undergoes a supercritical pitchfork bifurcation that breaks the s y symmetry and adds a second unstable eigenvalue for Re < 510. The upper branch of the travelling wave has three unstable eigenvalues for Re < 1000. Investigation of different systems which show subcritical turbulence revealed that bifurcations of exact solutions connected to the edge state of the system lead to the formation of a chaotic saddle that shows transient turbulence with exponential distributed lifetimes (Kreilos & Eckhardt 2012;Ritter, Mellibovsky & Avila 2016). In the present systems the formation of chaotic saddles cannot be studied in detail since it takes place in an unstable subspace. However, previous investigations in a symmetry-restricted system did show that the states follow such a sequence of bifurcations to the formation of a chaotic saddle , 2017, so that we expect that also the states in the unstable subspace follow this phenomenology.
The edge state TW E discussed here is not identical to the travelling waves identified by Waleffe (2001) or Nagata & Deguchi (2013), as they appear at considerably higher Reynolds numbers (even for their optimal wavenumbers) and are not edge states of the system. The two travelling waves described above are clearly related to the two different transition mechanisms that exist in the flow. For Reynolds numbers below the linear instability (here, Re c = 5815), initial conditions that start close to TW E in the state space will either decay or become turbulent without showing any approach to a TS wave: they will follow the bypass transition to turbulence. Initial conditions that start close to TW TS can also either decay or swing up to turbulence, but they will first form TS waves. Above Re c all initial conditions will show a transition to turbulence, but it will still be possible to distinguish whether they follow the bypass or TS route to turbulence, as we will see.
State space structure
In order to explore the arrangement of the different routes to turbulence in the space of initial conditions we pick initial conditions and integrate them until the flow either becomes turbulent or until it returns to the laminar profile. The initial conditions are taken in a two-dimensional slice of the high-dimensional space, spanned by two flow fields u 1 and u 2 . The choice of the flow fields allows one to explore different crosssections of state space. For the most part, we will use u 1 and u 2 to be the travelling waves TW E and TW TS , so that both states are part of the cross-section. The initial conditions are then parametrized by a mixing parameter α and an amplitude A, i.e.,
x x x x (a) (b) (c) (d)u(α, A) = A (1 − α)u 1 + αu 2 (1 − α)u 1 + αu 2 .
(3.1)
For α = 0 one explores the state space along velocity field u 1 and for α = 1 along velocity field u 2 . If the upper and lower branch of TW E are used to span such a slice, one finds that the regions in initial conditions that become turbulent have shapes that are similar to the ones in plane Couette flow (Kreilos & Eckhardt 2012;. Lower branch states are on the boundary between laminar flow and turbulence, so we begin by exploring the slice spanned by the lower branches of TW E and TW TS . In figure 3, visualization for different values of α are shown. The plots illustrate how the topology of the initial flow state changes from a state dominated by streamwise streaks and vortices to a state where streaks and vortices are aligned perpendicular to the flow direction. As quantitative measures of the states, we use the r.m.s. velocity (2.2) of the individual velocity components, and the turbulent intensity components, due to the strong low-speed streak contained in TW E . With increasing α the relative contribution of the wall-normal component becomes larger until for α = 1, where we have pure spanwise rolls, it is nearly as strong as the streamwise component.
Tu = 1 3 ( u 2 + v 2 + w 2 ).
In figure 4(b), streamwise velocity profiles for different α are shown. For low α there is a pronounced minimum in the centre of the channel, which is related to the lowspeed streak of TW E , and there are maxima close to the wall. When α is increased the profile becomes nearly inverted. For large values of α there is a maximum in the centre of the channel, and minima close to the walls. We assign to each initial condition the time it takes to become turbulent, with an upper cutoff for initial conditions that either take longer or that never become turbulent because they return to the laminar profile. Colour-coded transition-time plots are shown in figure 5(a-e) for different Reynolds numbers below Re c . The boundary between initial conditions that relaminarize and those that become turbulent stands out clearly. They are formed by the stable manifold of the states and their crossings with the cross-section. The part of the laminar-turbulent boundary connected with TW E can be distinguished from that connected with TW TS by the huge differences in transition times: for TW TS , transition times are significantly longer and even exceed 2 × 10 4 time units. The slow growth rates of TS waves has been known for a long time. An stability analysis of the base profile shows that there is an optimal Reynolds number around Re = 40.000 for which the largest growth rate is obtained. But even for this Reynolds number a growth by only a factor of 10 within 300 viscous time units is achieved (Bayly, Orszag & Herbert 1988).
The interaction between the two domains is rather intricate. For Reynolds number 5780, shown in figure 5(d), it seems that the borders do not cross, but rather wind around each other in a spiral shape down to very small scales. Although the wave TW TS has still only one unstable eigenvalue, the size of the structure that is directly connected to TW TS shrinks with decreasing Re and is not visible in this kind of projection for Re < 5727, where TW TS has more than one unstable eigenvalue. In Re > Re c the plane is spanned by the flow field of TW E and by the unstable TS mode of the laminar state. In all panels the colour indicates the time it takes to reach the turbulent states, up to a maximum integration time of 70 000 time units. Accordingly, initial conditions that do not become turbulent or return to the laminar state are indicated by dark blue. In ( f ) the dashed white line indicates the stable manifold of TW E . The coloured triangles and dots in (d) mark initial conditions whose time evolution is shown in figure 6. particular, a zoom of the area around TW TS for Re = 5720, which is included as an inset in figure 5(a), shows that initial conditions which pass near TW TS appear for α = 1 only. Thus, the number of initial conditions in this projection that undergoes TS transition is of measure zero. Some of the initial conditions that miss the bypass transition become turbulent nevertheless, because they are captured by the TS instability (for example, the full blue, and both green lines).
In figure 6 the evolution of the amplitude for different initial conditions marked in figure 5(d) is shown. The green and blue lines are typical representatives of the slow TS transition. Starting with a three-dimensional initial condition, their amplitude decays and the two-dimensional TS wave TW TS , whose amplitude is marked by the black line, is approached.
Afterwards, they depart from TW TS again, which is a slow process because of the small growth rate. As commonly known, the transition in the case of the TS transition is eventually caused by secondary instabilities of the TS waves (Herbert 1988).
The solid yellow line in figure 6 is an initial condition that undergoes bypass transition. It quickly swings up to higher amplitudes and does not approach the TS wave on its way to turbulence. The dashed yellow line is of an intermediate type.
It takes a long time to become turbulent but it does not come very close to the TS wave. The relation between time evolution, transient amplification and final state is complicated and non-intuitive. For instance, the dashed red and green trajectories share a transient increase near t ≈ 4000, but differ in their final state: the red curve, with the higher maximum, eventually returns to the laminar profile, but the green curve, with the smaller maximum, approaches the TS level and eventually becomes turbulent following the TS route. Similarly, the red, blue and green continuous lines start with high amplitude slightly below the threshold for the bypass route. They all decay, but while the red initial conditions ends up on the decaying side of the TS wave, the green and blue cases eventually become turbulent via the TS route.
For plane Couette flow it was found that a small chaotic saddle can appear inside of existing larger ones . There, trajectories that escape from the inner saddle are still captured by the outer saddle and thus they cannot decay directly. There is evidence that the appearance of TS transition in PPF follows a comparable but slightly different mechanism. At Reynolds number lower than 1000, the chaotic saddle related to bypass transition is created and subcritical turbulence exists. With increasing Reynolds number the chaotic saddle gets partly enclosed by the stable manifold of the TS wave, which above Re = 5727 can separate two parts of the state space and therefore prevent trajectories in the interior from becoming laminar.
With increasing Reynolds number the number of initial conditions becoming turbulent increases. Finally, for Re > Re c , initial conditions that return to the laminar state no longer exist. Nevertheless, also in this supercritical regime a sudden change in the type of transition can be identified: when the amplitude increases and crosses the stable manifold of TW E , the transition time drops dramatically and turbulence is reached via the bypass route. In the state space visualization for Re = 5855, which is shown in figure 5( f ), the change of the transition type presents itself in the rapid drop of the transition time with increasing A for α values between 0 and 0.6.
In the supercritical range the stable manifold of the bypass edge state TW E separates initial conditions undergoing the quick bypass transition from initial conditions that become turbulent by TS transition. The state space picture at a higher Reynolds number of 6000 looks qualitatively similar to the one shown in figure 5(c), including the switch from TS to bypass transition when the stable manifold of TW E is crossed.
State space for constant pressure gradient
Up to this point all results were obtained by simulations which keep the bulk velocity constant. Since previous studies showed differences between the system operating at constant bulk velocity and at constant pressure gradient (Barkley 1990;Soibelman & Meron 1991), we here analyse the changes in the state space structure when forces driving the flow are changed.
If the pressure gradient is fixed the Reynolds number Re P can be defined by using the centreline velocity of the laminar flow corresponding to the pressure gradient, together with the half-channel width and the viscosity. In dimensionless form the pressure gradient then is given by dP/dx = −2/Re P . This definition ensures that for the laminar base flow, the pressure-and bulk-based Reynolds numbers coincide.
We chose a Reynolds number of 5780 for a comparison of two constraints. For this Reynolds number the state space structure shows the highest complexity for the case of constant bulk velocity. Using the same parametrization of initial conditions, we obtain the transition-time plot shown in figure 7. Although there are small differences from the case of constant bulk velocity -for example, the tongue-like structure connected to TW TS appears to be slightly thinner -the main features are unchanged.
The relative stability to a change of the operating conditions can be rationalized by the large distance to the turning points of the solution branches. Lower branch states commonly differ only slightly between pressure and bulk Reynolds number if the distance to the turning point is large. However, these differences can become huge close to the turning point of the solution branch (e.g. Zammert 2015, chap. 3.2).
These results show that organization of the state space depends only weakly on driving with constant pressure difference or constant bulk flow, and that the slices are similar. Furthermore, since the flow states used to generate the state space slice are also solutions for the case of constant bulk velocity, but for slightly different values of the bulk Reynolds number, the calculations suggest that the obtained state space slices are to some degree robust to small disturbances of the flow fields used to generate the slices.
Conclusions
We have explored the coexistence of two types of transition in subcritical plane Poiseuille flow connected with the existence of states dominated by streamwise and spanwise vortices (bypass and TS transition). Probing the state space by scanning initial conditions in two-dimensional cross-sections gave information on the sets of initial conditions that follow one or the other route to turbulence. The results show that the transition via TS waves initially occupies a tiny region of state space. As this region expands, it approaches the bypass-dominated regions, but a boundary between the two remains visible because of the very different times needed to reach turbulence. This extends to the parameter range where the laminar profile is unstable to the formation of TS waves. The investigated state TW E lies on the border between the two transition types also for Reynolds number far above the critical one. Since its amplitude scales approximately as Re −0.5 for high Reynolds numbers, the minimal threshold required to trigger bypass transition must decrease with this power or even faster.
The results shown here are obtained for small domains, where the extensive numerical computations for very many initial conditions are feasible. For larger domains, the corresponding exact coherent structures are localized, as shown by Jiménez (1990) and Mellibovsky & Meseguer (2015) for TS waves and by Zammert & Eckhardt (2014) for the bypass transition. Since the bifurcation diagrams for the localized states are similar to that of the extended states, we anticipate a similar phenomenology also for localized perturbations in spatially extended states.
The methods presented here can also be used to explore the relation between bypass transition and TS waves in boundary layers (Duguet et al. 2012;Kreilos et al. 2016). More generally, they can be applied to any kind of transition where two different paths compete: examples include shear-driven or convection-driven instabilities in thermal convection (Clever & Busse 1992;Zammert, Fischer & Eckhardt 2016), the interaction between transitions driven by different symmetries (Faisst & Eckhardt 2003;Wedin & Kerswell 2004;, or the interaction between the established subcritical scenario and the recently discovered linear instability in Taylor-Couette flow with a rotating outer cylinder (Deguchi 2017).
FIGURE 1 .
1The exact coherent states for the transition to turbulence in plane Poiseuille flow. (a) Visualization of the Tollmien-Schlichting wave TW TS . The yellow surface indicates values of 0.3Q max for the Q-vortex criterion.
FIGURE 2 .
2The bifurcation diagrams for TW E (red) and TW TS (blue) are shown in (a). A solid line is used if the travelling wave has just one unstable eigenvalue, while a dashed line is used when the wave has further unstable eigenvalues. The inset zooms in on the region where both waves have only one unstable eigenvalue. The bifurcation points of the waves are marked with black dots. In (b) the amplitude a(u) of TW E is shown in a double-logarithmic plot, together with a power-law decay like Re −0.52 for large Re.
FIGURE 3 .
3Visualizations of the velocity fields for perturbations with parameter α = 0.2, 0.4, 0.6 and 0.8 (a-d, respectively) at Reynolds number 5720. The states are scaled to have an amplitude A of 0.0105. The visualizations use iso-contours of the Q-vortex criterion for Q = 2 × 10 −5 , shown in yellow. In addition, iso-surfaces for the streamwise velocity u = 0.002 and u = −0.01 are shown in red and blue, respectively.
Figure 4 FIGURE 4 .
44(a) shows the dependence of these quantities on α. For small values of α the contribution of the streamwise component is much stronger than that of the other (a) Turbulence intensities Tu and r.m.s.-velocities u , v and w and (b) mean profiles for the initial fields shown in figure 3 for different parameters α at Reynolds number 5720, and turbulence intensity Tu (red) in dependence on α. The streamwise, wall-normal and spanwise components are shown in blue, green and yellow, respectively. For α = 0 the contribution of the wall-normal component does not vanish but is only of order 10 −5 . (b) Mean streamwise velocity profiles (deviation from laminar) for different values of α.
FIGURE 5 .
5Two-dimensional slices of the state space for various Reynolds numbers. In (a-e), where Re < Re c = 5815, the parameter α interpolates between the flow fields of both travelling waves, and both are indicated by white dots in the figures. In ( f ), where
FIGURE 6 .
6Time evolution of the velocity amplitude a(u) for the initial conditions marked in figure 5(d) for Re = 5780. The initial conditions for the trajectories drawn with solid and dashed lines are marked in figure 5(b) with triangles and circles, respectively. The black line indicates the amplitude of the lower branch of the TS state TW TS at Re = 5780.
FIGURE 7 .
7Two-dimensional slice of the state space for a pressure-based Reynolds number (Re P ) of 5780.
AcknowledgementB.E. passed away on 7 August 2019 before finalization of the revision of this manuscript. He was a constant source of inspiring new ideas, an astute mind, a wonderful teacher and a good friend. He will be dearly missed.
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| [] |
[
"Confounder-Aware Visualization of ConvNets",
"Confounder-Aware Visualization of ConvNets"
] | [
"Qingyu Zhao \nSchool of Medicine\nStanford University\nStanfordUSA\n",
"Ehsan Adeli \nSchool of Medicine\nStanford University\nStanfordUSA\n",
"Adolf Pfefferbaum \nSchool of Medicine\nStanford University\nStanfordUSA\n\nCenter of Health Sciences\nSRI International\nMenlo ParkUSA\n",
"Edith V Sullivan \nSchool of Medicine\nStanford University\nStanfordUSA\n",
"Kilian M Pohl \nSchool of Medicine\nStanford University\nStanfordUSA\n\nCenter of Health Sciences\nSRI International\nMenlo ParkUSA\n"
] | [
"School of Medicine\nStanford University\nStanfordUSA",
"School of Medicine\nStanford University\nStanfordUSA",
"School of Medicine\nStanford University\nStanfordUSA",
"Center of Health Sciences\nSRI International\nMenlo ParkUSA",
"School of Medicine\nStanford University\nStanfordUSA",
"School of Medicine\nStanford University\nStanfordUSA",
"Center of Health Sciences\nSRI International\nMenlo ParkUSA"
] | [] | With recent advances in deep learning, neuroimaging studies increasingly rely on convolutional networks (ConvNets) to predict diagnosis based on MR images. To gain a better understanding of how a disease impacts the brain, the studies visualize the salience maps of the ConvNet highlighting voxels within the brain majorly contributing to the prediction. However, these salience maps are generally confounded, i.e., some salient regions are more predictive of confounding variables (such as age) than the diagnosis. To avoid such misinterpretation, we propose in this paper an approach that aims to visualize confounder-free saliency maps that only highlight voxels predictive of the diagnosis. The approach incorporates univariate statistical tests to identify confounding effects within the intermediate features learned by ConvNet. The influence from the subset of confounded features is then removed by a novel partial back-propagation procedure. We use this two-step approach to visualize confounder-free saliency maps extracted from synthetic and two real datasets. These experiments reveal the potential of our visualization in producing unbiased model-interpretation. | 10.1007/978-3-030-32692-0_38 | [
"https://arxiv.org/pdf/1907.12727v2.pdf"
] | 198,986,148 | 1907.12727 | 228d7b24621c6ff91a5ccb18d49a7d0980071337 |
Confounder-Aware Visualization of ConvNets
Qingyu Zhao
School of Medicine
Stanford University
StanfordUSA
Ehsan Adeli
School of Medicine
Stanford University
StanfordUSA
Adolf Pfefferbaum
School of Medicine
Stanford University
StanfordUSA
Center of Health Sciences
SRI International
Menlo ParkUSA
Edith V Sullivan
School of Medicine
Stanford University
StanfordUSA
Kilian M Pohl
School of Medicine
Stanford University
StanfordUSA
Center of Health Sciences
SRI International
Menlo ParkUSA
Confounder-Aware Visualization of ConvNets
With recent advances in deep learning, neuroimaging studies increasingly rely on convolutional networks (ConvNets) to predict diagnosis based on MR images. To gain a better understanding of how a disease impacts the brain, the studies visualize the salience maps of the ConvNet highlighting voxels within the brain majorly contributing to the prediction. However, these salience maps are generally confounded, i.e., some salient regions are more predictive of confounding variables (such as age) than the diagnosis. To avoid such misinterpretation, we propose in this paper an approach that aims to visualize confounder-free saliency maps that only highlight voxels predictive of the diagnosis. The approach incorporates univariate statistical tests to identify confounding effects within the intermediate features learned by ConvNet. The influence from the subset of confounded features is then removed by a novel partial back-propagation procedure. We use this two-step approach to visualize confounder-free saliency maps extracted from synthetic and two real datasets. These experiments reveal the potential of our visualization in producing unbiased model-interpretation.
Introduction
The development of deep-learning technologies in medicine is advancing rapidly [1]. Leveraging labeled big data and enhanced computational power, deep convolutional neural networks have been applied in many neuroscience studies to accurately classify patients with brain diseases from normal controls based on their MR images [1,2]. State-of-the-art saliency visualization techniques are used to interpret the trained model and to visualize specific brain regions that significantly contribute to the classification [2]. The resulting saliency map therefore provides fine-grained insights into how the disease may impact the human brain.
Despite the promises of deep learning, there are formidable obstacles and pitfalls [1,3]. One of the most critical challenges is the algorithmic bias introduced by the model towards confounding factors in the study [4]. A confounding factor (or confounder) correlates with both the dependent variable (group label) and independent variable (MR image) causing spurious association. For instance, if the age distribution of the disease group is different from that of the normal controls, age might become a potential confounder because one cannot differentiate whether the trained model characterizes neurodegeneration caused by the disease or by normal aging.
Since the end-to-end training scheme disfavors any additional intervention, controlling for confounding effects in deep learning is inherently difficult. This often leads to misinterpretation of the trained model during visualization: while some salient regions correspond to true impact of the disease, others are potentially linked to the confounders. In this paper, we present an approach that identifies confounding effects within a trained ConvNet and removes them to produce confounder-free visualization of the model. The central idea is first to detect confounding effects in each intermediate feature via univariate statistical testing. Then, the influence of confounded features is removed from the saliency map by a novel "partial back-propagation" operation, which can be intuitively explained by a chain-rule derivation on voxelwise saliency scores. This operation is efficiently implemented with a model refactorization trick. We apply our visualization procedure to interpret ConvNet classifiers trained on a synthetic dataset with known confounding effects and on two real datasets, i.e., MRIs of 345 adults for analyzing Human Immunodeficiency Virus (HIV) effects on the brain and MRIs of 674 adolescents for analyzing sexual dimporphsim. In all three experiments, our visualization shows the potential in producing unbiased saliency maps compared to traditional visualization techniques.
Confounder-Aware Saliency Visualization
We base our approach on the saliency visualization proposed in [5]. Given an MR image I and a trained ConvNet model, saliency visualization produces a voxelwise saliency map specific to I indicating important regions that strongly impact the classification decision. Without loss of generality, we assume a ConvNet model is trained for a binary classification task (pipeline generalizable to multi-group classification and regression), where the prediction output is a continuous score s ∈ [0, 1]. Then, the saliency value at voxel v is computed as the partial derivative |∂s/∂I v |. Intuitively, it quantifies how the prediction changes with respect to a small change in the intensity value at voxel v. Computationally, this quantity can be computed efficiently using back-propagation.
As discussed, when the ConvNet is confounded, some salient regions may actually contribute to the prediction of confounding variables rather than the group label. To address this issue, we propose a two-step approach to remove confounding effects from the saliency map enabling an unbiased interpretation of a trained ConvNet. To do this, we assume that a typical ConvNet architecture is composed of an encoder and a predictor. The encoder contains convolutional layers (including their variants and related operations such as pooling, batch normalization and ReLU) that extract a fixed-length feature vector
f i ∈ R M = [f 1 i , .
.., f M i ] from the i th training image. The predictor, usually a fully connected network, takes the M features as input and produces a prediction score s i for image i. To disentangle confounding effects from the saliency map, we propose in Section 2.1 to first test each of the M features separately for confounding effects using a general linear model (GLM). Next, the influence from the subset of features with significant confounding effects can be removed from the saliency map by performing a novel partial back-propagation procedure based on an intuitive chain-rule derivation (Section 2.2).
Univariate Test for Identifying Confounding Effects
This section introduces a way to test for the presence of confounding effect within a specific feature. Let f j = [f j 1 , ..., f j N ] denote the j th feature derived from all N training images. Likewise, denote s = [s 1 , ..., s N ] as the N prediction scores and z = [z 1 , ..., z N ] as a confounding variable (e.g., age of the N subjects). In this work, we use GLM [6] to perform a group-level statistical test for detecting whether the relationship between s and f j is confounded by z. Specifically, GLM decomposes the variance in f j into variance explained by s and variance explained by z. The model reads
f j = β 0 + β 1 s + β 2 z.(1)
We claim feature f j is confounded by z if the null hypothesis that linear coefficient β 2 is zero can be rejected (e.g., p < 0.05 by t-test). In other words, when the variance in f j is partially explained by z, f j potentially contributes to the prediction of the confounder rather than the key variable of interest. This analysis can be extended to handle multiple confounding variables, where all confounders are included in the GLM as independent covariates. Then, f j is confounded when the p-value for at least one confounder is significant. Note, this model is a specific instance of the mediation model [7], a popular model for confounding analysis. However, our model makes fewer assumptions so that it is more sensitive in detecting confounding effects than the mediation model. We also emphasize that such confounding analysis can only be performed on the feature-level instead of voxel-level. Unlike features encoding geometric patterns that are commensurate within a group, voxel intensities are only meaningful within a neighborhood but variant across MRIs. As such, removing confounding effects based on feature-analysis is prevalent in traditional feature-based models (non-deep-learning models) [8,9].
Repeating the above analysis for all M features, we generate a binary mask b
∈ [0, 1] M = [b 1 , ..., b M ], where b j = 0
indicates the presence of confounding effect in the j th feature and b j = 1 otherwise.
Visualization via Partial Back-Propagation
To generate a saliency map unbiased towards the subset of confounded features, we further investigate the voxelwise partial derivative. Based on the chain-rule,
∂s i ∂I v = ∂s i (f 1 i , ..., f M i ) ∂I v = M j=1 ∂s i ∂f j i ∂f j i ∂I v .(2)
Eq.
(2) factorizes the voxelwise partial derivative with respect to the M features, where each ∂s i /∂f j i quantifies the impact of the j th feature on the prediction. Therefore, to derive a confounder-free saliency map, we set this impact to zero for the confounded features. In doing so, the saliency score can be computed as
M j=1 b j ∂s i ∂f j i ∂f j i ∂I v .(3)
Computationally, this corresponds to a partial back-propagation procedure, where the gradient is only back-propagated through the un-confounded features.
The Refactorization Trick. We show that performing the partial back-propagation for a training image I can be implemented by refactorizing the trained Con-vNet model and then applying the original visualization pipeline of full backpropagation. As enforcing a zero ∂f j i /∂I v is equivalent to fixing f j i to a constant value independent of the input image, we design a dummy layer L between the encoder and the predictor that performs L(
x) = x ⊗ b i ⊕ ((1 − b i ) ⊗ y i ),
where ⊗ and ⊕ denote element-wise operators, and y i is a constant feature vector for image i pre-computed by the trained ConvNet. As shown in Fig. 1b, the dummy layer fixes the value of confounded features while keeping un-confounded features dependent on the input image. As such, the partial back-propagation of Eq. (3) can by simply computed by running the full back-propagation on the refactorized model. Note, model refactorization is performed for each MR image independently to yield subject-specific saliency maps.
Experiments
We first performed synthetic experiments, in which image data were imputed by known confounding effects so that we could test whether the proposed approach can successfully remove those effects during visualization. Next, we applied the approach to two real datasets to visualize (1) the impact of HIV on brain structures while controlling for aging effects; (2) sexual dimorphism during adolescence while controlling for 'puberty stage'.
Synthetic Data
We first generated a synthetic dataset containing two groups. Each group consisted of 512 2D images (dimension: 32 × 32 pixels). Each image was generated by 4 Gaussians (Fig. 2a), the width of which was controlled by the standard deviation σ. For each image of Group 1, we sampled σ from the uniform distribution U(2, 6). Images of Group 2 generally had wider distributions as we sampled from U(4, 8) instead. To predict group labels from the synthetic images, we constructed a simple ConvNet with the encoder consisting of 3 stacks of 2*2 convolution/ReLu/max-pooling layers and producing 32 intermediate features.
The fully-connected predictor had one hidden layer of dimension 16 with tanh as the non-linear activation function. We trained the network for binary classification on the entire synthetic dataset as the focus here was to interpret the trained model as opposed to measuring classification accuracy. With the trained ConvNet, we first applied the original visualization pipeline to each image and averaged the resulting subject-specific saliency maps. The average saliency map shown in Fig. 2b indicates that all 4 Gaussians contributed to the classification.
Next, we viewed the width of the two off-diagonal Gaussians, i.e., the standard deviations σ B of Block B and σ C of Block C as confounders. Based on Eq. 1, we then tested the presence of confounding effects in each of the 32 intermediate features with the following GLM:
f j = β 0 + β 1 s + β B σ B + β C σ C .
The results revealed that all features extracted from Blocks B and C were detected as confounded (p < 0.05 for either β B or β C ), while only features from Blocks A and D were identified as unconfounded (p ≥ 0.05 for both β B and β C ). We can see that our conservative test was sensitive in detecting confounding effects (no false negative but several false positives), thereby potentially removing some features representing true group difference. Such trade-off can be controlled by the p-value threshold used in the GLM tests. Finally, using the binary mask (yellow mask in Fig. 2d) for partial back-propagation, we produced a confounder-free average saliency map (Fig. 2e) that successfully removed the confounding effects.
Visualizing HIV Effects
The second experiment examined the impact of HIV on the human brain. The classification was performed on the T1-weighted MRI data of 223 control subjects (CTRL) and 122 HIV patients [8]. Participants ranged in age between 18-86 years, and there was a significant age difference between CTRL and HIV subjects (CTRL: 45 ± 17, HIV: 51 ± 8.3, p<0.001 by two-sample t-test). As HIV has been frequently suggested to accelerate brain aging [10], age is therefore a confounder that needs to be controlled for when interpreting the saliency map associated with the trained classifier.
Preprocessing and Classification. The MR images were first preprocessed by denoising, bias field correction, skull striping, affine registration to the SRI24 template (which accounts for differences in head size), and re-scaling to a 64 × 64 × 64 volume [8]. Even though the present study focused on the visualization technique, we measured the classification accuracy as a sanity check via 5-fold cross validation. To ensure the classifier can reasonably learn the group difference between HIV and CTRL subjects, the training dataset was augmented by random shifting (within one-voxel distance), rotation (within one degree) in all 3 directions, and left-right flipping. Note, the flipping was based on the assumption that HIV infection affects the brain bilaterally [8]. The data augmentation resulted in a balanced training set of 1024 CTRLs and 1024 HIVs.
As the flipping removed left-right orientation, the ConvNet was built on half of the 3D volume containing one hemisphere. The encoder contained 4 stacks of 2*2*2 3D convolution/ReLu/batch-normalization/max-pooling layers yielding 4096 intermediate features. The fully-connected predictor had 2 hidden layers of dimension (64, 32) with tanh as the non-linear activation function. An L2regularization (λ = 0.1) was applied to all fully-connected layers. Based on this ConvNet architecture, we achieved 73% normalized accuracy for HIV/CTRL classification, which was comparable to other recent studies on this dataset [8].
Model Visualization. To visualize the HIV effect, we re-trained the ConvNet on a dataset of 1024 CTRLs and 1024 HIVs augmented from the entire dataset of 345 MRIs. We first visualized the average saliency map produced by the original visualization pipeline. Since the ConvNet operated on only one hemisphere, we mirrored the resulting average saliency map to the other hemisphere to create bilaterally symmetric display and overlaid it on the SRI24 T1 atlas (Fig. 3a). For comparison, we then visualized the confounder-free saliency map produced by our approach. Specifically, we tested each of the 4096 features with f j = β 0 + β 1 s + β 2 age, and 804 were identified to be confounded by age. Fig. 3b shows the saliency map after removing aging effects, and shows that saliency at the posterior ventricle (red regions) was attenuated by our approach indicating those regions contained aging effects instead of HIV effects. This finding is consistent with current concept that the ventricular volume significantly increases with age [11].
Visualizing Sexual Dimorphism
The third experiment aimed to improve understanding of sexual dimorphism in brain development that emerges during adolescence. The classification was performed on the baseline T1 MR images of 334 boys and 340 girls (age 12-21) from the National Consortium on Alocohol and NeuroDevelopment in Adolescence (NCANDA) [12]. All subjects met the no-to-low alcohol drinking criteria of the study, and there was no significant age-difference between boys and girls (p>0.5 two-sample t-test). As puberty stage [12] of girls was significantly higher than boys during adolescence, the pubertal development score (PDS: boys 2.86±0.7, girls 3.41±0.6, p<0.001 by two-sample t-test) was a potential confounder of the study.
All experimental setups complied with the previous HIV study. As the first attempt of predicting sex on the NCANDA data, we achieved 89.5% normalized accuracy based on a 5-fold cross-validation. The original saliency map produced for the ConvNet trained on the entire augmented dataset is shown in Fig. 3d. After testing and removing PDS effects, the confounder-free saliency map is shown in Fig. 3e. Consistent with existing adolescence literature, sex difference was mainly found in the temporal lobe [13]. Fig. 3f indicates PDS effects mainly existed in the frontal and inferior parietal region. Another interesting observation is in the caudate, which has been frequently reported as proportionately larger in female participants across different ages [7]. As shown in our results, the saliency at the caudate region attenuated after removing confounding effects, suggesting a potential compounding effect of PDS in that region.
Discussion and Conclusion
In this paper, we introduced a novel approach for confounder-free visualization and interpretation of a trained ConvNet. By performing partial back-propagation with respect to a set of unconfounded intermediate features, the approach disentangled true group difference from confounding effects and produced unbiased saliency maps. We successfully illustrated its usage on a synthetic dataset with groundtruth confounding effects and two real neuroimaging datasets. Because our approach is a type of post-hoc analyses with respect to a trained model, further extension could potentially integrate similar confounder-control procedures during model-training time to fully explore unbiased group differences within a dataset.
Fig. 1 :
1Our confounder-aware visualization is composed two steps: (a) A GLM test is performed on each individual feature collected over all training images to detect confounding effects. (b) For each image, the model is refactorized to fix the value of confounded features, thereby enabling a partial back-propagation to derive a confounder-free saliency map.
Fig. 2 :
2Synthetic experiments: (a) Each synthetic image contains 4 Gaussians that are created differently between the two groups; (b) Average saliency map produced by the original visualization pipeline; (c) Widths of the two off-diagonal Gaussians are considered as confounders; (d) GLM identifies selective features, mainly in Blocks B and C, as confounded; (e) Removing the confounded features in the visualization leads to a confounder-free saliency map.
Fig. 3c
Fig. 3 :
3Visualization of ConvNets trained for HIV/CTRL classification (top row) and sexual dimorphism (bottom row).
Acknowledgements. This research was supported in part by NIH grants AA017347, AA005965, AA010723, AA021697, AA013521, AA026762 and MH113406.
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| [] |
[
"Frugal and Truthful Auctions for Vertex Covers, Flows, and Cuts *",
"Frugal and Truthful Auctions for Vertex Covers, Flows, and Cuts *"
] | [
"David Kempe [email protected] ",
"Mahyar Salek [email protected] ",
"Cristopher Moore [email protected] ",
"\nDepartment of Computer Science\nDepartment of Computer Science\nUniversity of Southern California\n90089-0781CAUSA\n",
"\nComputer Science Department and Department of Physics and Astronomy\nUniversity of Southern California\n90089-0781CAUSA\n",
"\nUniversity of New Mexico\n87131-0001AlbuquerqueNMUSA\n",
"\nand Santa Fe Institute\n87501Santa FeNMUSA\n"
] | [
"Department of Computer Science\nDepartment of Computer Science\nUniversity of Southern California\n90089-0781CAUSA",
"Computer Science Department and Department of Physics and Astronomy\nUniversity of Southern California\n90089-0781CAUSA",
"University of New Mexico\n87131-0001AlbuquerqueNMUSA",
"and Santa Fe Institute\n87501Santa FeNMUSA"
] | [] | We study truthful mechanisms for hiring a team of agents in three classes of set systems: Vertex Cover auctions, k-flow auctions, and cut auctions. For Vertex Cover auctions, the vertices are owned by selfish and rational agents, and the auctioneer wants to purchase a vertex cover from them. For k-flow auctions, the edges are owned by the agents, and the auctioneer wants to purchase k edge-disjoint s-t paths, for given s and t. In the same setting, for cut auctions, the auctioneer wants to purchase an s-t cut. Only the agents know their costs, and the auctioneer needs to select a feasible set and payments based on bids made by the agents.We present constant-competitive truthful mechanisms for all three set systems. That is, the maximum overpayment of the mechanism is within a constant factor of the maximum overpayment of any truthful mechanism, for every set system in the * A preliminary version of this article appeared in the Proceedings of FOCS 2010[19]. 1 class. The mechanism for Vertex Cover is based on scaling each bid by a multiplier derived from the dominant eigenvector of a certain matrix. The mechanism for k-flows prunes the graph to be minimally (k + 1)-connected, and then applies the Vertex Cover mechanism. Similarly, the mechanism for cuts contracts the graph until all s-t paths have length exactly 2, and then applies the Vertex Cover mechanism.1. If e is a saturated forward edge, then ℓ e = 1. | 10.1109/focs.2010.76 | [
"https://arxiv.org/pdf/0912.3310v2.pdf"
] | 682,674 | 0912.3310 | 47bd21e3f30822734c815c69f6782ebaa7c44313 |
Frugal and Truthful Auctions for Vertex Covers, Flows, and Cuts *
13 Jun 2011 June 14, 2011
David Kempe [email protected]
Mahyar Salek [email protected]
Cristopher Moore [email protected]
Department of Computer Science
Department of Computer Science
University of Southern California
90089-0781CAUSA
Computer Science Department and Department of Physics and Astronomy
University of Southern California
90089-0781CAUSA
University of New Mexico
87131-0001AlbuquerqueNMUSA
and Santa Fe Institute
87501Santa FeNMUSA
Frugal and Truthful Auctions for Vertex Covers, Flows, and Cuts *
13 Jun 2011 June 14, 2011
We study truthful mechanisms for hiring a team of agents in three classes of set systems: Vertex Cover auctions, k-flow auctions, and cut auctions. For Vertex Cover auctions, the vertices are owned by selfish and rational agents, and the auctioneer wants to purchase a vertex cover from them. For k-flow auctions, the edges are owned by the agents, and the auctioneer wants to purchase k edge-disjoint s-t paths, for given s and t. In the same setting, for cut auctions, the auctioneer wants to purchase an s-t cut. Only the agents know their costs, and the auctioneer needs to select a feasible set and payments based on bids made by the agents.We present constant-competitive truthful mechanisms for all three set systems. That is, the maximum overpayment of the mechanism is within a constant factor of the maximum overpayment of any truthful mechanism, for every set system in the * A preliminary version of this article appeared in the Proceedings of FOCS 2010[19]. 1 class. The mechanism for Vertex Cover is based on scaling each bid by a multiplier derived from the dominant eigenvector of a certain matrix. The mechanism for k-flows prunes the graph to be minimally (k + 1)-connected, and then applies the Vertex Cover mechanism. Similarly, the mechanism for cuts contracts the graph until all s-t paths have length exactly 2, and then applies the Vertex Cover mechanism.1. If e is a saturated forward edge, then ℓ e = 1.
Introduction
Many tasks require the joint allocation of multiple resources belonging to different bidders. For instance, consider the task of routing a packet through a network whose edges are owned by different agents. In this setting, it is necessary to obtain usage rights for multiple edges simultaneously from the agents. Similarly, if the agents own the vertices of a graph, and we want to monitor all edges, we need the right to install monitoring devices on nodes, and again obtain these rights from distinct agents.
Providing access to edges or nodes in such settings makes the agents incur a cost c e , which the agents should be paid for. A convenient way to determine "appropriate" prices to pay the agents is by way of auctions, wherein the agents e submit bids b e to an auctioneer, who selects a feasible subset S of agents to use, and determines prices p e to pay the agents. The most basic case is a single-item auction. The auctioneer requires the service of any one of the agents, and their services are interchangeable. Single-item auctions have a long history of study, and are fairly well understood [20,21]. Motivated by applications in computer networks and electronic commerce, several recent papers have considered the extension to a setup termed hiring a team of agents [3,11,12,18,28]. In this setting, there is a collection of feasible sets, each consisting of one or more agent. The auctioneer, based on the agents' bids b e , selects one feasible set S, and pays each agent e ∈ S a price p e .
Some of the well-studied special cases of set systems are path auctions [3,12,18,24,30], in which the feasible sets are paths from a given source s to a given sink t, and spanning tree auctions [4,14,18,28], in which the feasible sets are spanning trees of a connected graph. In both cases, the agents are the edges of the graph. In this paper, we extend the study to more complex examples of set systems, namely:
1. Vertex Covers: The agents are the vertices of the graph G, and the auctioneer needs to select a vertex cover [5,11,28]. Not only are vertex covers of interest in their own right, but they give a key primitive for many other set systems as well, an approach we explore in depth in this paper.
2. Flows: The agents are the edges of G, and the auctioneer wants to select k edge-disjoint paths from s to t. Thus, this scenario generalizes path auctions; the generalization turns out to require significant new techniques in the design and analysis of mechanisms.
3. Cuts: In the same setting as for flows, the auctioneer wants to purchase an s-t cut.
In choosing an auction mechanism for a set system, the auctioneer needs to take into account that the agents are selfish. Ideally, the auctioneer would like to know the agents' true costs c e . However, the costs are private information, and a rational and selfish agent will submit a bid b e = c e if doing so leads to a higher profit. The area of mechanism design [23,24,25] studies the design of auctions for selfish and rational agents.
We are interested in designing truthful (or incentive-compatible) auction mechanisms: auctions under which it is always optimal for selfish agents to reveal their private costs c e to the auctioneer. Such mechanisms are societally desirable, because they make the computation of strategies a trivial task for the agents, and obviate the need for gathering information about the costs or strategies of competitors. They are also desirable from the point of view of analysis, as they allow us to identify bids with costs, and let us dispense with any kinds of assumptions about the distribution of agents' costs. Thus, the outcomes of truthful mechanisms are stable in a stronger sense than Nash equilibria, and may give bidders more confidence that the right outcome will be reached. For this reason, truthful mechanism design has been a mainstay of game theory for a long time.
It is well known that any truthful mechanism will have to pay agents more than their costs at times; in this paper, we study mechanisms approximately minimizing the "overpayment." The ratio between the payments of the "best" truthful mechanism and natural lower bounds has been termed the "Price of Truth" by Talwar [28], and studied in a number of recent papers [3,4,11,12,14,18,28,30]. In particular, [18] and [11] define and analyze different natural measures of lower bounds on payments, and define the notions of frugality ratio and competitiveness. The frugality ratio of a mechanism is the worst-case ratio of payments to a natural lower bound (formally defined in Section 2), maximized over all cost vectors of the agents. A mechanism is competitive for a class of set systems if its frugality ratio is within a constant factor of the frugality ratio of the best truthful mechanism, for all set systems in the class.
Our Contributions
In this paper, we present novel frugal mechanisms for three general classes of set systems: Vertex Covers, k-Flows, and Cuts. Vertex Cover auctions can be considered a very natural primitive for more complicated set systems. Under the natural assumption that there are no isolated vertices, they capture set systems with "minimal competition": if the auction mechanism decides to exclude an agent v from the selected set, this immediately forces the mechanism to include all of v's neighbors, thus giving these neighbors a monopoly. Thus, a different interpretation of Vertex Cover auctions is that they capture any set system whose feasible sets can be characterized by positive 2SAT formulas: each edge (i, j) corresponds to a clause (x i ∨ x j ), stating that any feasible set must include at least one of agents i and j.
Our mechanism for Vertex Cover works as follows: based solely on the structure of the graph G, we define an appropriate matrix K and compute its dominant eigenvector q. After agents submit their bids b v , the mechanism first scales each bid to c ′ v = b v /q v , and then simply runs the VCG mechanism [29,9,15] with these modified bids. We prove that this mechanism has a frugality ratio equal to the largest eigenvalue α of K, and that this is within a factor of 2 of the frugality ratio of any mechanism. The lower bound is based on pairwise competition between adjacent bidders for any truthful mechanism, and in a sense can be considered the natural culmination of the lower bound techniques of [12,18]. The upper bound is based on carefully balancing all possible worst cases of a single non-zero cost against each other, and showing that the worst case is indeed one of these cost vectors. We stress here that the mechanism does not in general run in polynomial time: the entries of K are derived from fractional clique sizes in G, which are known to be hard to compute, even approximately. We discuss the issue of polynomial time briefly in Section 6.
Based on our Vertex Cover mechanism, we present a general methodology for designing frugal truthful mechanisms. The idea is to take the original set system, and prune agents from it until it has "minimal competition" in the above sense; subsequently, the Vertex Cover auction can be invoked. So long as the pruning is "composable" in the sense of [1] (see Section 3), the resulting auction is truthful. The crux is then to prove that the pruning step (which removes a significant amount of competition) does not increase the lower bound on payments too much. We illustrate the power of this approach with two examples.
1. For the k-flow problem, we show that pruning the graph to a minimum-cost (k + 1) s-t-connected graph H is composable, and increases the lower bound at most by a factor of k + 1. Hence, we obtain a 2(k + 1)-competitive mechanism. Establishing the bound of k + 1 requires significant technical effort.
2. For the cut problem, we show that pruning the graph to a minimum-cost set of edges such that each s-t path is cut at least twice gives a composable selection rule. Furthermore, it increases the lower bound by at most a factor of 2, leading to a 4-competitive mechanism. For the pruning step, we develop a primal-dual algorithm generalizing the Ford-Fulkerson Minimum-Cut algorithm.
We note that while the Vertex Cover mechanism is in general not polynomial, for both special cases derived here, the running time is in fact polynomial.
Relationship to Past and Parallel Work
As discussed above, a line of recent papers [3,4,11,12,14,18,28,30] analyze frugality of auctions in the "hiring a team" setting, where the auctioneer wants to obtain a feasible set of agents, while paying not much more than necessary. In this context, the papers by Karlin, Kempe, and Tamir [18] and Elkind, Goldberg, and Goldberg [11] are particularly related to our work.
Karlin et al. [18] introduce the definitions of frugality and competitiveness which we use here. They also give competitive mechanisms for path auctions, and for so-called r-out-of-k systems, in which the auctioneer can select any r out of k disjoint sets of agents. At the heart of both mechanisms is a mechanism for r-out-of-(r + 1) systems. Our mechanism for Vertex Covers can be considered a natural generalization of this mechanism. Furthermore, both r-out-of-k systems and path auctions are special cases of r-flows, since choosing an r-flow in a graph consisting of k vertex-disjoint s-t paths is equivalent to an r-out-of-k system. Our approach of pruning the graph is similar in spirit to the approach in [18], where graphs were also first pruned to be minimally 2-connected, and set systems were reduced to r-out-of-(r + 1) systems. However, the combinatorial structure of k-flows makes this pruning (and its analysis) much more involved in our case.
Elkind et al. [11] study truthful mechanisms for Vertex Cover. They present a polynomialtime mechanism with frugality ratio bounded by 2∆, where ∆ is the maximum degree of the graph, and also show that there exist graphs where the best truthful mechanism must have frugality ratio at least ∆/2. Notice, however, that this does not guarantee that the mechanism is competitive. Indeed, there are graphs where the best truthful mechanism has frugality ratio significantly smaller than ∆/2, and our goal is to have a mechanism which is within a constant factor of best possible for every graph.
Several recent papers have extended the problem of hiring a team of agents in various directions. Cary, Flaxman, Hartline, and Karlin [6] combine truthful auctions for hiring a team with revenue-maximizing auctions for selling items. Du, Sami, and Shi [10] and Iwasaki, Kempe, Saito, Salek, and Yokoo [17] study path auctions under the additional requirement that not only should they be truthful, but false-name proof: agents owning multiple edges have no incentive to claim that these edges belong to different agents. Du et al. show that there are no false-name proof mechanisms that are also Pareto-efficient, and Iwasaki et al. analyze the frugality ratio of false-name proof mechanisms, showing exponential lower bounds.
Results very similar to ours have been derived independently and simultaneously by Chen, Elkind, Gravin, and Petrov [7]. Both papers first derive mechanisms for Vertex Cover auctions. Our mechanism is based on scaling the agents' bids by the entries of the dominant eigenvector of a scaled adjacency matrix. It has constant competitive ratio for all graphs, but may not run in polynomial time. The mechanism of Chen et al., on the other hand, uses eigenvectors of the unscaled adjacency matrix. It may not be constant competitive on some inputs, but it always runs in polynomial time.
Chen et al. also propose the approach of reducing other set systems to Vertex Cover instances, called "Pruning-Lifting Mechanisms" there. In particular, they derive the same mechanism as the present paper for k-flows, with similar key lemmas in the proof. While their Vertex Cover mechanism is different from ours in general, on inputs derived from flow and cut problems, the scaling factor in our matrix is the same for all entries, and the mechanisms therefore coincide. In particular, the mechanisms in both papers are thus competitive and run in polynomial time.
While the mechanism of Chen et al. [7] may not always be competitive due to the lack of scaling factors in the matrix, their proof of a lower bound involves a clever application of Young's Inequality, and thus avoids losing the factor of 2 in our lower bound. Thus, whenever their mechanism coincides with ours, both mechanisms are optimal. In particular, this also implies that the k-flow mechanism of the present paper is (k + 1)-competitive and our mechanism for s-t cuts is 2-competitive. Moreover, they prove stronger bounds on the k-flow mechanism: when compared against the lower bound from [11] (used in this paper, and defined formally in Section 2), the mechanism is in fact optimal.
Finally, in collaboration with the authors of [7], we recently showed that Young's In-equality can be applied to the analysis of our Vertex Cover mechanism, removing the factor of 2 from the lower bound. In other words, we show that our Vertex Cover mechanism is indeed optimal for all Vertex Cover instances. This result will be included in a joint full version of both papers.
Preliminaries
A set system (E, F ) has n agents (or elements), and a collection F ⊆ 2 E of feasible sets. We call a set system monopoly-free if no element is in all feasible sets, i.e., if S∈F S = ∅. The three classes of set systems studied in this paper are:
1. Vertex Covers: here, the agents are the vertices of a graph G, and F is the collection of all vertex covers of G. To avoid confusion, we will denote the agents by u, v instead of e in this case. Notice that every Vertex Cover set system is monopoly-free.
2. k-flows: here, we are given a graph G with source s and sink t. The agents are the edges of G. A set of edges is feasible if it contains at least k edge-disjoint s-t paths.
A k-flow set system is monopoly-free if and only if the minimum s-t cut cuts at least k + 1 edges.
3. Cuts: With the same setup as for k-flows, a set of edges is feasible if it contains an s-t cut. Thus, the set system is monopoly-free if and only if G contains no edge from s to t.
The set system (E, F ) is common knowledge to the auctioneer and all agents. Each agent e ∈ E has a cost c e , which is private, i.e., known only to e. We write c(S) = e∈S c e for the total cost of a set S of agents, and also extend this notation to other quantities (such as bids or payments). A mechanism for a set system proceeds as follows:
1. Each agent submits a sealed bid b e .
2. Based on the bids b e , the auctioneer selects a feasible set S ∈ F as the winner, and computes a payment p e ≥ b e for each agent e ∈ S. The agents e ∈ S are said to win, while all other agents lose.
Each agent, knowing the algorithm for computing the winning set and the payments, will choose a bid b e maximizing her own profit, which is p e − c e if the agent wins, and 0 otherwise. We are interested in mechanisms where self-interested agents will bid b e = c e . More precisely, a mechanism is truthful if, for any fixed vector b −e of bids by all other agents, e maximizes her profit by bidding b e = c e . If a mechanism is known to be truthful, we can use b e and c e interchangeably. It is well-known [3,21] that a mechanism is truthful only if its selection rule is monotone in the following sense: if all other agents' bids stay the same, then a losing agent cannot become a winner by raising her bid. Once the selection rule is fixed, there is a unique payment scheme to make the mechanism truthful. Namely, each agent is paid her threshold bid: the supremum of all winning bids she could have made given the bids of all other agents.
Nash Equilibria and Frugality Ratios
To measure how much a truthful mechanism "overpays," we need a natural bound to compare the payments to. Karlin et al. [18] proposed using as a bound the solution of a natural minimization problem. Let S be the cheapest feasible set with respect to the true costs c e ; ties are broken lexicographically.
Minimize
ν − (c) := e∈S x e subject to x e ≥ c e for all e ∈ S x e = c e for all e / ∈ S e∈S x e ≤ e∈T x e for all T ∈ F For every e ∈ S, there is a T e ∈ F , e /
∈ T e such that
e ′ ∈S x e ′ = e ′ ∈Te x e ′(1)
The intuition for this optimization problem is that it captures the bids of agents in the cheapest "Nash Equilibrium" of a first-price auction with full information, under the assumption that the actual cheapest set S wins, and the losing agents all bid their costs. That is, the mechanism selects the cheapest set with respect to the bids x e , and pays each winning agent her bid x e . The first constraint captures individual rationality. The third constraint states that the bids x e are such that S still wins, and the final constraint states that for each winning agent, there is a tight set preventing her from bidding higher. That is, if e increases her bid, the buyer will select a set T excluding e instead of S. We say that a vector x is feasible if it satisfies all these constraints.
While this optimization problem is inspired by the analogy of Nash Equilibria, it should be noted that first-price auctions do not in general have Nash Equilibria due to tie-breaking issues (see a more detailed discussion in [16,18]).
Elkind et al. [11] and Chen and Karlin [8] observed that the quantity ν − (c) has several undesirable non-monotonicity properties. For instance, adding new feasible sets to the set system, and thus increasing the amount of competition between agents, can sometimes lead to higher values of ν − (c). Similarly, lowering the costs of losing agents, or increasing the costs of winning agents, can sometimes increase ν − (c). Furthermore, ν − (c) is NP-hard to compute even if the set system is the set of all s-t paths [8].
Instead, Elkind et al. [11] propose replacing the minimization by a maximization in the above optimization problem. An important advantage of this optimization problem is that the maximization objective ensures that for every e ∈ S, there is a tight set T . Thus, the maximization objective removes the need for the final constraint, and turns the optimization problem into an instance of Linear Programming, which can be solved in many cases. We thus obtain the following definition (which [11] refers to as NTU max ). Intuitively, this definition captures the bids in the most expensive Nash Equilibrium of a first-price auction, with the same caveat as before about the non-existence of equilibria.
Maximize ν(c) := e∈S x e subject to (i) x e ≥ c e
for all e (ii) x e = c e for all e / ∈ S (iii) e∈S x e ≤ e∈T x e for all T ∈ F (2)
As stated above, a consequence of this maximization is that, for every e in the winning set, there is a tight set T excluding e that prevents e from bidding higher:
∀e ∈ S : ∃T ∈ F : e / ∈ T and e ′ ∈S x e ′ = e ′ ∈T x e ′ .(3)
We will refer to the bounds ν − (c) and ν(c) as buyer-optimal and buyer-pessimal, respectively, throughout the paper. Moreover, due to the advantages discussed above, we will use the quantity ν(c) as a natural lower bound for this paper. Despite the preceding discussion, in order to emphasize the intuition behind the bounds, we will refer to the x e values of the LP (2) as the Nash Equilibrium bids of agents e, or simply the Nash bids of e.
Notice that ν(c) is defined for all monopoly-free set systems. We now formally define the frugality ratio of a mechanism M for a set system (E, F ), and the notion of a competitive mechanism.
Definition 2.1 (Frugality Ratio, Competitive Mechanism) Let M be a truthful mechanism for the set system (E, F ), and let P M (c) denote the total payments of M when the vector of actual costs is c.
The frugality ratio of M is
φ M = sup c P M (c) ν(c) .
The frugality ratio of the set system
(E, F ) is Φ (E,F ) = inf M φ M ,
where the infimum is taken over all truthful mechanisms M for (E, F ).
A mechanism M is κ-competitive for a class of set systems
{(E 1 , F 1 ), (E 2 , F 2 ), . . .} if φ M is within a factor κ of Φ (E i ,F i ) for all i.
Remark 2.2
The frugality ratio of a mechanism is defined as instance-based. The frugality ratio of a set system captures the inherent structural complexity of that instance, which can be "exploited" with careful worst-case choices of costs. Competitiveness, on the other hand, is defined over a class of set systems. If a single mechanism, such as the ones defined in this paper, is competitive, it does as well on each set system in the class as the best mechanism, which could possibly be tailored to this specific instance. The nomenclature "competitive" is motivated by the analogy with online algorithms.
The instance-based definition [18,11] allows us a more fine-grained distinction between mechanisms than earlier work (e.g., [3,24]), where a lower bound in terms of a worst case over all instances was used.
As discussed above, the motivation for the LPs (1) and (2) was that they provide "natural lower bounds" on the payments of any truthful mechanism. However, to the best of our knowledge, it was previously unknown whether the solutions do in fact provide lower bounds. Indeed, it is easy to define mechanisms that achieve arbitrarily lower payments for particular cost vectors, albeit at the cost of significantly higher payments on other cost vectors. Here, we establish that the objective value of the LP (2) indeed does give a lower bound in terms of the frugality ratio. This resolves an open question from the preliminary version of this paper [19].
Proposition 2.3 Let (E, F ) be an arbitrary set system and M a truthful and individually rational mechanism on
(E, F ). Then, φ M ≥ 1.
Proof. Let c be an arbitrary cost vector. Let S ∈ F be the set minimizing c(S), and x the solution to the LP (2). Let S ′ ∈ F be the winning set for M with cost vector x. Because M is truthful and individually rational, its payment P M (x) is at least e∈S ′ x e . By the third constraint of the LP (2), e∈S ′ x e ≥ e∈S x e . Finally, by construction, we have that ν(c) = ν(x). Taken together, this implies that
P M (x) ≥ e∈S ′ x e ≥ e∈S x e = ν(x).
By definition of the frugality ratio, this implies that φ M ≥ 1.
Vertex Cover Auctions
In this section, we describe and analyze a constant-competitive mechanism for Vertex Cover auctions. We then show how to use it as the basis for a methodology for designing frugal mechanisms for other set systems. The graph is denoted by G = (V, E), with n vertices. We
write u ∼ v to denote that (u, v) ∈ E.
Our mechanmism is based on certain modifications to the well-known Vickrey-Clarke-Groves (VCG) mechanism [29,9,15]. Recall that VCG always selects the cheapest feasible set S with respect to the submitted bids b e , and pays each agent her threshold bid.
Weighting the bids with an eigenvector
The important change to VCG in our mechanism is that each agent's bid is scaled by an agent-specific multiplier. The multipliers capture "how important" an agent is for the solution, roughly in the sense of how many other agents can be omitted by including this agent. They are computed as entries of the dominant eigenvector of a certain matrix K. As we will see, the computation of K is NP-hard itself, so the mechanism will in general not run in polynomial time unless P=NP.
As a first step, our mechanism removes all isolated vertices. We assume that the resulting graph G is connected. Let 1 v (for any vertex v) be the vector with 1 in coordinate v and 0 in all other coordinates. We define ν v = ν(1 v ) ≥ 1 to be the total "Nash Equilibrium" payment of the first-price auction in the sense of the LP (2) if agent v has cost 1 and all other agents have cost 0. Notice that in this case, v loses. We prove in Section 3.1 that ν v is exactly the fractional clique number of the graph induced by the neighbors of v, without v itself. This implies that unless ZPP=NP, ν v cannot be approximated to within a factor O(n 1−ǫ ) in polynomial time, for any ǫ > 0. Our inability to compute ν v is the chief obstacle to a constant-competitive polynomial-time mechanism.
Let A be the adjacency matrix of G (with diagonal 0). Define D = diag(1/ν 1 , 1/ν 2 , . . . , 1/ν n ), and K = DA. That is,
K u,v = 1/ν u if u ∼ v 0 if u ∼ v .
If we define K ′ = D −1/2 KD 1/2 = D 1/2 AD 1/2 , then K and K ′ have the same eigenvalues, and the eigenvectors of K are of the form D 1/2 · e, where e is an eigenvector of K ′ . Moreover,
K ′ u,v = 1/ √ ν u ν v if u ∼ v 0 if u ∼ v ,
so K ′ is symmetric and has non-negative entries. By the Perron-Frobenius Theorem, the eigenvalues of K' and K are real. Since we assumed G to be connected, the dominant eigenvector of K ′ is unique and has positive entries, and the same holds for K. Let α be the largest eigenvalue of K, and q the corresponding eigenvector. Notice that given K as input, α and q can be computed efficiently and without knowledge of the agents' bids or costs.
The mechanism EV (which stands for "Eigenvector Mechanism") is now as follows: after all nodes v submit their bids b v , the algorithm sets c ′ v = b v /q v , and computes a minimum cost vertex cover S with respect to the costs c ′ v (ties broken lexicographically). S is chosen as the winning set, and each agent in S is paid her threshold bid. Notice that the second step of the mechanism again requires the solution to an NP-hard problem.
EV is truthful since the selection rule is clearly monotone, and the payments are the threshold bids. Thus, we can assume without loss of generality that bids and costs coincide.
In the following, we analyze the frugality ratio of EV, and show that EV is competitive. Proof. We start by considering only cost vectors with exactly one non-zero entry, i.e., of the form c = c v · 1 v . For such a cost vector, the mechanism will choose a subset of V \ {v} as the winning set, and pay each u in that subset her threshold bid. We calculate the threshold bids of all these agents u.
First, consider any agent u ∼ v. If u were to raise her bid above (q u /q v ) · c v , while all agents besides u and v continued to bid 0, then the set V \ {u} would be cheaper than {u} with respect to the new bid vector c ′ . Therefore, u would not be part of the winning vertex cover. Thus u's threshold payment is at most
(q u /q v ) · c v .
Next, consider any agent u ∼ v. Because V \ {u, v} is a vertex cover, u cannot raise her bid above zero without losing, so her threshold bid is 0. Hence, the total payment of EV is at most P (c) = (1/q v ) · c v · u∼v q u . On the other hand, by the definition of ν v and linearity of ν, we have that ν(c) = c v ν v , so the frugality ratio for cost vectors of the form
c v · 1 v is (1/q v ) · c v · u∼v q u c v ν v = 1 q v · u∼v 1 ν v · q u = 1 q v · α · q v = α,
where the second equality followed because the vector q is an eigenvector of K with eigenvalue α. Thus, for any cost vector with only one non-zero entry, the frugality ratio is at most α. Now consider an arbitrary cost vector c, and write it as
c = v c v 1 v . We claim that P (c) ≤ v c v P (1 v ). For consider any vertex u ∈ S winning with cost vector c. If the cost vector were c v 1 v instead, u's payment would be (q u /q v ) · c v if u ∼ v and 0 otherwise.
On the other hand, when the cost vector is c, if u bids strictly more than v∼u q u /q v · c v , then u cannot be in the winning set, as replacing u with all its neighbors would give a cheaper solution with respect to the costs c ′ . Thus, each node u gets paid at most v∼u q u /q v · c v with cost vector c, and the total payment is at most
P (c) = u v∼u q u q v · c v = v c v · u∼v q u q v = v c v P (1 v ).
On the other hand, we have that
ν(c) ≥ v c v ν(1 v ) = v c v ν v ,
because of the following argument: for each v, let x (v) be a an optimal solution for the LP (2) with cost vector 1 v . Then, simply by linearity, the vector (2) with cost vector c, and achieves the sum of the payments. Thus, the optimal solution to (2) with cost vector c can have no smaller total payments. Combining the results of the previous two paragraphs, we have the following bound on the frugality ratio:
x = v c v x (v) is feasible formax c P (c) ν(c) ≤ max c v c v P (1 v ) v c v ν v ≤ max v P (1 v ) ν v ≤ α.
Next, we prove that no other mechanism can do asymptotically better.
Lemma 3.2 Let M be any truthful vertex cover mechanism on G. Then, M has frugality ratio at least α 2 .
Proof. We construct a directed graph G ′ = (V, E ′ ) from G by directing each edge e of G in at least one direction. Consider any edge e = (u, v) of G. Let c be the cost vector in which c u = q u , c v = q v , and c i = 0 for all i = u, v. When M is run on the cost/bid vector c, at least one of u and v must be in the winning set S; otherwise, it would not be a vertex cover. If u ∈ S, then add the directed edge (v, u) to E ′ . Similarly, if v ∈ S, then add (u, v) to E ′ . If both u, v ∈ S, then add both directed edges. By doing this for all edges e ∈ G, we eventually obtain a graph G ′ . Now give each node v a weight q v . Each node-weighted directed graph (V, E ′ ) contains at least one node v such that
u:(v,u)∈E ′ q u ≥ u:(u,v)∈E ′ q u ,
(see, e.g., the proof of Lemma 11 in [18]), and hence
u:(v,u)∈E ′ q u ≥ 1 2 u:u∼v q u .
Fix any such node v in G ′ with respect to the weights q v . Now consider the cost vector c with c v = q v and c i = 0 for all i = v. By monotonicity of the selection rule of M (which follows from the truthfulness of M), at least all nodes u such that (v, u) ∈ G ′ must be part of the selected set S of M, and must be paid at least q u . Therefore, the total payment of M is at least
u:(v,u)∈G ′ q u ≥ 1 2 u∼v q u = 1 2 ν v u∼v 1 ν v q u = 1 2 ν v · αq v ,
where the last equality followed from the fact that q is an eigenvector of the matrix K.
On the other hand, as in the proof of Lemma 3.1, ν(c) = ν v q v for our cost vector c, so the frugality ratio is at least 1 2 α, when the cost vector is c. Combining Lemma 3.1 and Lemma 3.2, we have proved the following theorem:
Theorem 3.3 EV is 2-competitive for Vertex Cover auctions.
Remark 3.4 The lower bound of 1 2 α on the frugality ratio of any mechanism can potentially be large. For instance, for a complete bipartite graph K n,n , we have α = Θ(n). Thus, such large overpayments are inherent in truthful mechanisms in general. However, truthful mechanisms may be much more frugal on specific classes of graphs.
Remark 3.5 EV in general does not run in polynomial time. For the final step, computing a minimum-cost vertex cover with respect to the scaled costs, we could use a monotone 2-approximation, as suggested by Elkind et al. [11]. The hardness of computing K is more severe. However, notice that for specific classes of graphs, such as degree-bounded or trianglefree graphs, K can be computed efficiently, giving us non-trivial polynomial-time mechanisms for Vertex Cover on those classes. This issue is discussed more in Section 6.
Nash Equilibria and the Fractional Clique Problem
In this section, we show that the Nash Equilibrium values ν v used for scaling of the matrix actually have a natural interpretation. To state the result, recall that the fractional clique number is the solution to the linear program
Maximize u x u subject to u∈I x u ≤ 1 for all independent sets I x u ≥ 0 for all u(4)
The fractional chromatic number is the solution of the dual problem, where we have a variable y I for each independent set I and a constraint I∋u y I ≥ 1 for each vertex u, and we minimize I y I . By LP duality, the fractional clique number and the fractional chromatic number are equal. Proof. Let x be any bid vector feasible for the LP (2). First, for all vertices u that do not share an edge with v, we must have x u = 0, because V \ {u, v} is a feasible set. So we can restrict our attention to G v .
For a set I, we write x(I) = u∈I x u for the total bids of the vertices in I. If I is an independent set in G v , then x(I) ≤ 1. The reason is that the set V \ I is also feasible, and would cost less than V \ {v} if x(I) exceeded 1. Thus, any feasible bid vector x induces a feasible solution to the LP (4), of the same total cost.
Conversely, if we have a feasible solution to the LP (4), we can extend it to a bid vector for all agents by setting x v = 1, and x u = 0 for all vertices u outside v's neighborhood. We need to show that each feasible set T , i.e., each vertex cover, has total bid x(T ) at least as large as the set V \ {v}. If T does not contain v, it must contain all of v's neighbors; it thus has the same bid as V \{v} by definition. Otherwise, because V \T is an independent set, the feasibility of x for the LP (4) implies that x(V \T ) ≤ 1. Thus, x(T ) ≥ x(V )−1 = x(V \{v}), and the two LPs 2 and (4) have the same value.
Standard randomized rounding arguments (see, e.g., [22]) imply that for any graph, the chromatic number and the fractional chromatic number are within a factor O(log n) of each other. Therefore, any approximation hardness results for Graph Coloring also apply to the Fractional Clique Problem with at most a loss of logarithmic factors. In particular, the result of Feige and Kilian [13] implies that unless ZPP=NP, ν v cannot be approximated to within a factor O(n 1−ǫ ) in polynomial time, for any ǫ > 0.
Composability and a General Design Approach
Vertex Cover auctions can be used naturally as a way to deal with other types of set systems. First, pre-process the set system by removing a subset of agents, turning the remaining set system into a Vertex Cover instance; then, run EV on that instance.
The important part is to choose the pre-processing rule to ensure that the overall mechanism is both truthful and competitive. A condition termed composability in [1, Definition 5.2] is sufficient to ensure truthfulness. We show here that a comparison between lower bounds is sufficient to show competitiveness. Definition 3.7 (Composability [1]) Let σ be a selection rule mapping bid vectors to subsets of (remaining) agents. We say that σ is composable if σ(b) = T implies that σ(b ′ e , b −e ) = T for any e ∈ T and b ′ e ≤ b e . In other words, not only can a winning agent not become a loser by bidding lower; she cannot even change which set containing her wins.
Formally, when we talk about "removing" a set of agents from a set system, we are replacing (E, F ) with (T, F | T ), where T = σ(b), and F | T := {S ∈ F | S ⊆ T }.
Theorem 3.8 Let σ be a composable selection rule with the following additional property: For all monopoly-free set systems (E, F ) in the class, and all cost vectors c, writing
(E ′ , F ′ ) := (σ(c), F | σ(c) ): 1. (E ′ , F ′ ) is a Vertex Cover instance, and 2. ν (E ′ ,F ′ ) (c) ≤ κ · ν (E,F ) (c).
Let the Remove-Cover Mechanism RCM consist of running EV on (E ′ , F ′ ). Then RCM is a truthful 2κ-competitive mechanism.
Proof. Truthfulness is proved in [1, Lemma 5.3]. The proof is short, and we include a version here for completeness. Consider any agent e, and a bid vector b −e for agents other than e. Because σ is composable, and thus also monotone, there is a threshold bid τ e such that e wins iff her bid is at most τ e . Furthermore, whenever b e ≤ τ e , the set σ(b) is uniquely determined, and independent of b e . Thus, whenever b e ≤ τ e , EV will be run on the same set system (σ(b), F | σ(b) ), and the selection rule of EV on this set system is monotone. Hence, the overall selection rule of RCM is monotone for e, implying directly that RCM is truthful.
The upper bound on the frugality ratio of RCM follows simply from Lemma 3.1 and the assumption of the theorem:
P RCM (c) ≤ α((E ′ , F ′ )) · ν (E ′ ,F ′ ) (c) ≤ α((E ′ , F ′ )) · κ · ν (E,F ) (c).
To prove the lower bound, let M be any truthful mechanism for (E, F ), and let (E ′ , F ′ ) be the Vertex Cover set system maximizing α((E ′ , F ′ )). We consider cost vectors c with c e = ∞ (or some very large finite values) for e / ∈ E ′ . For such cost vectors, we can safely disregard all elements e / ∈ E ′ altogether, as they will not affect the solutions to the LP (2), nor be part of any solution selected by M.
But then, M is exactly a mechanism selecting a feasible solution to the Vertex Cover instance (E ′ , F ′ ). By Lemma 3.2, M thus has frugality ratio at least α((E ′ , F ′ ))/2, completing the proof.
A simple general way to obtain a composable rule is to choose the set with the minimum total cost, from some subset of the feasible sets: Lemma 3.9 Let σ be any rule with consistent tie breaking selecting a set S minimizing b(S) over all sets S with a certain property P . Then σ is composable.
Proof. Consider any agent e who is part of the winning set S with respect to b. If e's bid decreases by ǫ, the cost of S decreases by ǫ, while the costs of all other sets decrease by at most ǫ. Thus, because ties are broken consistently, S will still be selected.
A Mechanism for Flows
We apply the methodology of Theorem 3.8 to design a mechanism F M for purchasing k edge-disjoint s-t paths. We are given a (directed) graph G = (V, E), source s, sink t, and target number k. As discussed earlier, the agents are edges of G. We assume that G is monopoly-free, which is equivalent to saying that the minimum s-t cut contains at least k + 1 edges. For convenience, we will refer to a set of k edge-disjoint s-t paths simply as a k-flow, and omit s and t.
To specify F M, all we need to do is describe a composable pre-processing rule σ. Our rule is simple: Choose (k + 1) edge-disjoint s-t paths, of minimum total bid with respect to b, breaking ties lexicographically. We call such a subgraph a (k + 1)-flow, where it is implicit that we are only interested in integer flows, and identify the flow with its edge set. Call the minimum-cost (k + 1)-flow H. (In Section 3.3, we generically referred to this set system as (E ′ , F ′ ).)
Theorem 4.1 The mechanism F M is truthful and 2(k + 1)-competitive and runs in polynomial time.
We show this theorem in three parts. First, we establish that the k-flow problem on H indeed forms a Vertex Cover instance (Lemma 4.2). By far the most difficult step is showing that the lower bound satisfies ν H (c) ≤ (k + 1) · ν G (c) for all cost vectors c (Lemma 4.4). The composability of σ follows from Lemma 3.9. Together, these three facts allow us to apply Theorem 3.8, and conclude that F M is a truthful 2(k + 1)-competitive mechanism. Finally, we verify that F M runs in polynomial time (Lemma 4.7).
Lemma 4.2
The instance (E ′ , F ′ ) whose feasible sets are all k-flows on H is a Vertex Cover set system.
Proof.
Recall that H is a minimal (k + 1)-flow, a fact that we exploit repeatedly in this proof. The edges of H are the vertices in the Vertex Cover instance. For clarity, consider explicitly the graph R, which contains a vertex u e for each edge e ∈ H, and an edge between u e , u e ′ if and only if removing e would create a monopoly for e ′ . This is the case iff there exists at least one minimum s-t cut in H containing both e and e ′ ; in particular, R is symmetric. The construction is illustrated for the case k = 2 in Figure 1. An alternative characterization of the edges in R is given in Proposition 4.3 below, and will be used as part of this proof. For any set of edges E ′ in H, let N(E ′ ) be the corresponding set of nodes in R. Thus, for any minimum s-t cut E ′ , the set N(E ′ ) forms a clique in R. If E ′ is a k-flow, then for any pair of edges e, e ′ that lie on a minimum s-t cut, E ′ must contain at least one of e, e ′ . Thus, N(E ′ ) is a vertex cover of R.
Conversely, let E ′ be a set of edges in H such that N(E ′ ) is a vertex cover of R. We will show that for every s-t cut F ⊆ E, at least k edges of E ′ cross F , i.e., |E ′ ∩ F | ≥ k. This will imply that E ′ is a k-flow. Assume for contradiction that |E ′ ∩ F | < k. Because N(E ′ ) is a vertex cover of R, there can be no edge between any pair of vertices in N(F \ E ′ ) in R. By definition, this means that for any pair e, e ′ ∈ F \ E ′ , there is no minimum s-t cut containing both e and e ′ . By Proposition 4.3 below, this is equivalent to saying that for each pair e, e ′ ∈ F \ E ′ , the graph H contains a path from e to e ′ or a path from e ′ to e.
Consider a directed graph whose vertices are the edges F \ E ′ , with an edge from e to e ′ whenever H contains a path from e to e ′ . By the above argument, this graph is a tournament graph, and thus contains a Hamiltonian path. That is, there is an ordering e 1 , . . . , e ℓ of the edges in F \ E ′ such that each e i+1 is reachable from e i in H. By adding a path from s to e 1 and from e ℓ to t, we thus obtain an s-t path P containing all edges in F \ E ′ . The graph H \ P is a k-flow, so the set E ′ ∩ F , having size less than k, cannot be an s-t cut in H \ P . Let P ′ be an s-t path in H \ P disjoint from E ′ ∩ F . By construction, P ′ is also disjoint from F \ E ′ . Thus, we have found an s-t path P ′ in H disjoint from F , contradicting the assumption that F is an s-t cut. Proof. Assume that there is a path from v to u ′ . Let P be a concatenation of an s-v path using e, the path from v to u ′ , and a path from u ′ to t using e ′ . Then, H \ P is a k-flow, and therefore has k edge-disjoint paths. Any s-t cut in H must thus contain at least k edges from H \ P , and no s-t cut with fewer than k + 2 edges can contain both e and e ′ .
Conversely, if there is no minimum cut containing both e, e ′ , then every minimum cut in H \ {e, e ′ } must contain k edges. Thus, H \ {e, e ′ } contains k edge-disjoint s-t paths. Removing these paths from H leaves us with a 1-flow, i.e., one s-t path. By construction, this path must contain e and e ′ ; thus, at least one is reachable from the other.
Lemma 4.4 ν H (c) ≤ (k + 1) · ν G (c) for all c.
Proof. Let S be the cheapest k-flow in G with respect to the costs c. Because H is a (k + 1)-flow, Corollary 4.6 below implies that ν H (c) = k · π H (c), where π H (c) is the cost of the most expensive s-t path in H.
Let x be a solution to the LP (2) with cost vector c on the graph G. Define a graph G ′ consisting of all edges in S, as well as all edges that are in at least one tight feasible set T (i.e., a set T for which the constraint (iii) is tight, meaning that x(T ) = x(S)).
By definition, G ′ contains at least k + 1 edge-disjoint s-t paths. Lemma 4.5 below (the key step) implies that all s-t paths in G ′ have the same total bid with respect to x. Let P be an s-t path in G ′ of maximum total cost c(P ). By individual rationality (Constraint (i) in the LP (2)), we have that x(P ) ≥ c(P ), and hence x(P ′ ) ≥ c(P ) for all s-t paths P ′ . In particular, ν G (c) = x(S) ≥ k · c(P ). Since H is a minimum-cost (k + 1)-flow, and because G ′ contains at least k + 1 edge-disjoint s-t paths, we have π H (c) ≤ (k + 1) c(P ). Thus,
ν G (c) ≥ k · c(P ) ≥ k k + 1 · π H (c) = 1 k + 1 · ν H (c),
which completes the proof. Let v be an arbitrary node in G, and P 1 , P 2 two paths from v to t. Then, x(P 1 ) = x(P 2 ).
Proof. Let E be the collection of all tight k-flows from s to t except S, i.e., the set of all F such that F = S, F consists of exactly k edge-disjoint s-t paths, and x(F ) = x(S). We define a directed multigraphG as follows: for each F ∈ E, we add toG a copy of each edge e ∈ F (creating duplicate copies of edges e which are in multiple flows F ). We call these edges forward edges. In addition, for each edge e = (u, v) ∈ S, we add |E| copies of the backward edge (v, u) toG, i.e., we direct e the other way.
In the resulting multigraph, each node v has an in-degree equal to its out-degree. For v = s, t this follows since each edge set we added constitutes a flow. For v = s, t, it follows since each F ∈ E adds k edges out of s and into t, while the |E| copies of S add k|E| edges into s and out of t. As a result,G is Eulerian, a fact we use below.
We define a mapping γ(e), which assigns to each edge e ∈G its "original" edge in G. As usual, we extend notation and write γ(R) = {γ(e) | e ∈ R} for any set R of edges.
We will be particularly interested in analyzing collections of cycles inG. We say that two cycles C 1 , C 2 are image-disjoint if γ(C 1 ) ∩ γ(C 2 ) = ∅. A cycle set is any set of zero or more image-disjoint cycles inG (which we identify with its edge set), and Γ denotes the collection of all cycle sets. For a cycle set C ∈ Γ, let C → and C ← denote the set of forward and backward edges in C, respectively. Then, we define φ(C) = S ∪ γ(C → ) \ γ(C ← ). It is easy to see that for each cycle set C, φ(C) is a k-flow in G ′ . Conversely, for every k-flow F in G ′ , there is a cycle set C ∈ Γ with φ(C) = F . We assign each edge e ∈G a weight w e . For forward edges e, we set w e = x γ(e) , while for backward edges e = (v, u), we set w e = −x γ(e) . Notice that because each copy of S contributes weight −x(S), and each set F ∈ E contributes x(F ) = x(S), the sum of all weights inG is 0.
Now, let C be any cycle set, and F = φ(C) its corresponding k-flow. We have
e∈C w e = x(F \ S) − x(S \ F ) = x(F ) − x(S),
Thus F is tight, i.e., x(F ) = x(S), if and only if e∈C w e = 0. We next show that for any cycle C inG, the k-flow φ(C) is tight, and therefore that C has total weight zero, i.e., e∈C w e = 0. Assume for contradiction that this is not the case, and let C be a cycle with e∈C w e = 0. Let F = φ(C) be the corresponding k-flow. Because we showed above that e∈C w e = x(F ) − x(S), we can rule out the possibility that e∈C w e < 0; otherwise, x(F ) < x(S), which would violate Constraint (iii) of the LP (2).
If e∈C w e > 0, consider the multigraphG ′ obtained by removing C fromG. Its total weight is e / ∈C w e < 0, because the sum of all weights inG is 0 (as shown above). Sincẽ G is Eulerian, so isG ′ , and its edges can be partitioned into a collection of edge-disjoint cycles {C 1 , . . . , C ℓ }. By the Pigeonhole Principle, at least one of the C i must have negative total weight. But then x(F i ) < x(S), where F i = φ(C i ), violating Constraint (iii) of (2) as in the previous case. This completes the proof that φ(C) is tight for any cycle C. By our observation above, the total weight of any cycle C is zero.
Finally, we prove the statement of the lemma by induction on a reverse topological sorting of the vertices v-that is, an ordering in which the index of v is at least as large as the index of any u such that (v, u) ∈ G ′ . Because G ′ is acyclic, such a sorting exists. The base case v = t is trivial. For v = t, let P 1 , P 2 be two v-t paths. We distinguish three cases, based on the first edges e 1 = (v, u 1 ), e 2 = (v, u 2 ) of the paths P 1 , P 2 .
1. IfG contains a forward edge (v, u 1 ) and a backward edge (u 2 , v) (or vice versa), then since every set of edges added toG is a flow,G must contain a v-t path P ′ 1 entirely consisting of forward edges and starting with e 1 , and a t-v path P ′ 2 entirely consisting of backward edges and ending with e 2 (backward). Applying the induction hypothesis to u 1 and u 2 , since P 1 and P ′ 1 share their first edges and similarly for P 2 and P ′ 2 , we have x(γ(P ′ 1 )) = x(P 1 ) and x(γ(P ′ 2 )) = x(P 2 ). Because P ′ 1 ∪ P ′ 2 forms a cycle, it has total weight 0. Then x(γ(P ′ 2 )) = −w P ′ 2 = w P ′ 1 = x(γ(P ′ 1 )), and so x(P 1 ) = x(P 2 ).
2. IfG contains forward edges (v, u 1 ) and (v, u 2 ), then it contains v-t paths P ′ 1 , P ′ 2 starting with (v, u 1 ) and (v, u 2 ), respectively, and consisting entirely of forward edges. Applying the induction hypothesis to u 1 and u 2 , we have x(γ(P ′ 1 )) = x(P 1 ) and x(γ(P ′ 2 )) = x(P 2 ). Since every set of edges added toG is a flow,G must contain an s-v path P consisting entirely of forward edges, andG must also contain a t-s path P ′ consisting entirely of backward edges. Because P ∪ P ′ ∪ P ′ i forms a cycle for each i ∈ {1, 2} and has total weight zero, we obtain x(P i ) = x(γ(P ′ i )) = −w P ∪P ′ for each i. In particular, x(P 1 ) = x(P 2 ).
3. Finally, ifG contains backward edges (u 1 , v) and (u 2 , v), we apply an argument similar to the previous case. By induction, x(γ(P ′ 1 )) = x(P 1 ), and x(γ(P ′ 2 )) = x(P 2 ). Again using the fact thatG consists of flows, it contains t-v paths P ′ 1 , P ′ 2 with respective last edges (u 1 , v) and (u 2 , v). In addition,G contains a v-s path P consisting entirely of backward edges, and an s-v path P ′ consisting entirely of forward edges. Then for each i ∈ {1, 2}, P ∪ P ′ ∪ P ′ i forms a cycle with total weight zero, so x(P 1 ) = x(P 2 ).
As a corollary, we can derive a characterization of Nash Equilibria in (k + 1)-flows.
Corollary 4.6
If G is a (k + 1)-flow, then a bid vector x is a Nash Equilibrium if and only if x(P ) = π G (c) for all s-t paths P . In particular, all Nash Equilibria have the same total cost x(S) = k · π G (c), where S is the winning set.
Proof. First, because G is a (k + 1)-flow, the graph G ′ defined in the proof of Lemma 4.4 actually equals G, since it must contain k + 1 edge-disjoint s-t paths. If x is a Nash Equilibrium, then by Lemma 4.5, all s-t paths P have the same total bid x(P ). LetP be an s-t path maximizing c(P ), i.e., c(P ) = π G (c). G \P is a k-flow, and clearly the cheapest k-flow by definition ofP . Therefore, all agents inP lose, and x(P ) = c(P ) by Constraint (ii) of the LP (2).
Finally, we show that the mechanism EV runs in polynomial time for the special case of graphs derived from k-flows. Lemma 4.7 For the Vertex Cover instance derived from computing a k-flow on a (k + 1)flow, the mechanism EV runs in polynomial time.
Proof. There are two steps which are of concern: computing the values ν v , and finding the cheapest vertex cover with respect to the scaled bids. The latter is exactly a Minimum Cost Flow problem by Proposition 4.2, and thus solvable in polynomial time with standard algorithms [2].
For the former, we claim that ν ue = k for all u e ∈ R. By Proposition 3.6 and LP duality, ν ue is upper bounded by the chromatic number of u e 's neighborhood, and lower bounded by its clique number. Since each edge e ∈ H is part of a minimum cut of size k + 1, and the edges of the minimum cut form a clique in R, the clique number of u e 's neighborhood is at least k. On the other hand, we can decompose H into k + 1 edge-disjoint paths, and color the vertices corresponding to each path with its own color in R. By Proposition 4.3, this is a valid coloring, and shows that u e 's neighborhood is k-colorable.
Remark 4.8
The factor 2 in the result of Theorem 4.1 comes from the factor 1 2 in the lower bound in Lemma 3.2. Using a more refined lower bound based on Young's Inequality, Chen et al. [7] showed that for an unscaled version of the Vertex Cover mechanism, the factor 1 2 in the lower bound is unnecessary. For the instances of Vertex Cover produced as a result of the pruning in this section, the mechanism from [7] coincides with EV, and hence F M is the same as the flow mechanim [7].
Chen et al. also showed that while F M is (k + 1)-competitive when compared against the buyer-optimal lower bound [18], it is in fact optimal compared to the buyer-pessimal version [11].
A Mechanism for Cuts
As a second application of our methodology, we give a competitive mechanism CM for purchasing an s-t cut, given a (directed) graph G = (V, E), source s, and sink t. Again, the agents are edges. Here, the necessary monopoly-freeness is equivalent to G not containing the edge (s, t).
As before, it suffices to specify and analyze a composable pre-processing rule σ. Our preprocessing rule is to compute a minimum-cost set E ′ of edges (with respect to the submitted bids b), such that E ′ contains at least two edges from each s-t path. We call such an edge set a double cut. We show below restricting the set system to E ′ gives a Vertex Cover instance, and at most increases the cost of the winning set by a factor of 2.
Restricting to a double cut
To restrict the set system to E ′ , we contract all edges in E \ E ′ . Since no such edge will be cut, contracting it ensures that its endpoints will always lie on the same side of the cut. Let H denote the resulting graph. We begin with a simple structural lemma about H.
Lemma 5.1 In H, all s-t paths have length exactly 2.
Proof. If there were an s-t path of length 1 in H, i.e., an edge (s, t), then consider the edge (u, v) in the original graph corresponding to (s, t). Because u was contracted with s, and v with t, there must be an s-u path and a v-t path in G using only edges from E \ E ′ . In that case, (u, v) is the only edge on this path contained in E ′ , so E ′ cannot have been a double cut. Similarly, if there were an s-t path P of length at least 3, then at least one edge (u, v) of P has neither s nor t as an endpoint. This edge could be safely contracted, i.e., removed from E ′ , in which case E ′ was not a minimum-cost double cut. Composability follows from Lemma 3.9, and the final conclusion then follows from Theorem 3.8 once we establish the other claims.
We can obtain a Vertex Cover instance by imposing a graph structure on H, treating each edge as a vertex and adding an edge (e, e ′ ) between any two edges that form an s-t path. A set of edges is an s-t cut if and only if it contains at least one of e, e ′ in each such pair, so it is a vertex cover of the resulting graph.
We can think of this in turn as a flow problem as follows. Lemma 5.1 implies that H is of the following form: in addition to s and t, there are vertices v 1 , . . . , v ℓ , and for each i = 1, . . . , ℓ, a set of parallel edges E i from s to v i , and a set of parallel edges E ′ i from v i to t. Any s-t cut has to include, for each i, all of E i or all of E ′ i . Thus, if we define a minimally 2-connected graph consisting of a series of vertices u 0 , u 1 , . . . , u ℓ with two vertex-disjoint paths of length |E i | and |E ′ i | between u i−1 and u i for each i, an s-t cut in H is a 1-flow from u 0 to u ℓ . We can then apply Lemma 4.2. Notice that this equivalence also establishes that EV runs in polynomial time on the instances produced by this selection rule.
As before, the key part is to analyze the increase in the lower bound. In particular, this implies that u ∈ S and v ∈ S. As stated above in equation (3), since x maximizes the LP (2), Constraint (iii) must be tight for some feasible set excluding e, since otherwise the bid x e could be increased. Let (T, T ) be the corresponding cut. Then x(E(T, T )) = x(E(S, S)), and e does not cross (T, T ). Thus, either both u and v are in T , or both are in T . Since (T, T ) ∈ C, this gives a contradiction. Now define G ′ := E(T − , T − ) ∪ E(T + , T + ). Because G ′ consists of two disjoint s-t cuts, it is a double cut, and the cost-minimality of H implies that c(G ′ ) ≥ c(H). These two cuts both have minimal cost, so ν G (c) = x(G ′ )/2. By the "individual rationality" LP constraint (i), x(G ′ ) ≥ c(G ′ ), and hence
ν G (c) = x(G ′ ) 2 ≥ c(G ′ ) 2 ≥ c(H) 2 ≥ ν H (c) 2 .
For the last inequality, notice that in the "Nash Equilibrium" on H, for each i, the cheaper of E i and E ′ i will collectively raise their bids to the cost of the more expensive one, so the total bid of the winning set will be ν H (c) = i max(c(E i ), c(E ′ i )) ≤ c(H).
A Primal-Dual Algorithm for Minimum Double-Cuts
Finally, we present a polynomial time algorithm to compute a minimum-cost double cut. The minimum-cost double cut is characterized by integer solutions to the following LP, where P denotes the set of all s-t paths in G.
Minimize e∈E c e x e subject to e∈P x e ≥ 2 for all P ∈ P x e ≤ 1 for all edges e ∈ E x e ≥ 0 for all e,
Remark 5.4 It is not difficult to show that the constraint matrix for this LP is totally unimodular. By a well-known theorem [27], because the right-hand sides of the constraints are integral, total unimodularity implies that all the vertices of the LP's polytope are integral.
Since there is a separation oracle for the LP (as well as an equivalent polynomial-sized LP formulation), an integer solution can be found in polynomial time, giving us a polynomialtime algorithm. However, the resulting algorithm is rather inefficient.
Here, we present a more efficient primal-dual algorithm generalizing the Ford-Fulkerson Max-Flow algorithm. The dual of the LP is Maximize 2 P ∈P f P − e r e subject to P :e∈P f P ≤ c e + r e for all e ∈ E f P , r e ≥ 0 for all P ∈ P and all e ∈ E .
We interpret the dual variables f P as describing a flow in the usual way. That is, the flow along each edge e is f e = P :e∈P
f P .
We say that e is saturated if f e = c e + r e . We call r e the relief on e: in order to send more flow on an edge e, we can increase its capacity, but we pay for it in the objective function. It is worth augmenting the flow along a path so long as at most one edge on the path is saturated, since increasing the flow and the relief of the saturated edge at the same time increases the dual objective. Our primal-dual algorithm is similar to the Ford-Fulkerson algorithm, and is based on the same concept of a residual graph. The residual graph contains forward edges for all edges e in the original graph, even when they are saturated, because it is possible to send more flow by adding relief. In addition, if e = (u, v) carries flow f e , then the residual graph, as usual, contains the backward edge (v, u) with capacity f e . To capture how much relief would have to be added to augment the flow along a path, we define, for each edge e in the residual graph, a length ℓ e as follows:
2. If (v, u) is a backward edge such that (u, v) has positive relief, then ℓ (v,u) = −1.
3. The lengths of all other edges are ℓ e = 0.
For paths P , we define ℓ(P ) = e∈P ℓ e . We give our primal-dual algorithm as Algorithm 1.
Algorithm 1 Flow computation for Minimum Double Cut 1: Flow Computation: 2: Let f be an arbitrary maximum flow on G. 3: while there is an s-t path P with ℓ(P ) ≤ 1 in the residual graph G f do 4:
Let P be such a path with minimum length ℓ(P ).
5:
Augment the flow on P by δ, while simultaneously increasing the relief of any saturated edge by δ, and decreasing the relief of any backward edge by δ, for the smallest value of δ such that this action increases ℓ(P ), i.e., the smallest δ such that either a new forward edge becomes saturated or the relief on a backward edge becomes zero.
Notice that for any path P of length at most 1, augmenting the flow increases the dual objective. This follows since the total number of saturated edges is at most one greater than the total number of backward edges with relief; for the latter, each unit of flow reduces the total relief by one unit, while for the former, each unit of flow increases the total relief by one unit. Thus, the total increase in relief for sending δ units of flow is at most δ, while the first term of the objective function, i.e., the value of the flow, increases by 2δ.
As with the Ford-Fulkerson algorithm, the running time could be pseudo-polynomial with a poor choice of the augmenting path P . But breaking ties for the smallest total number of edges in P gives strongly polynomial running time, as with the Edmonds-Karp algorithm.
When the algorithm terminates, we have a set T of saturated edges, and a subset R ⊆ T of relief edges e with r e > 0. We will pick two edge-disjoint s-t cuts (S 1 , S 1 ), (S 2 , S 2 ) with the properties that:
1. Only saturated edges cross either of the cuts.
2. S 1 ⊂ S 2 .
3. Each relief edge crosses one of the two chosen cuts.
This will naturally satisfy all complementary slackness conditions for the two LPs, and thus prove optimality of the cuts.
To define and compute the two cuts, we focus on the graph G ′ obtained from the residual graph by removing all forward edges e with f e = 0. Importantly, we use the same notion of length defined above. From now on, all references to reachability, distances, etc. are with respect to G ′ .
For each node v, let d v denote the minimum distance from s to v in G ′ . We show next that G ′ has no negative cycles, so these distances are well-defined.
Lemma 5.5 G ′ has no path from the sink t to the source s of length strictly less than −1. In particular, G ′ contains no negative cycles.
Proof. We show by induction that these properties hold for the residual graph in each iteration. Since G ′ is obtained from the residual graph only by deleting edges with zero flow (and thus length 0 or 1 only), distances can only increase in G ′ .
Initially, all edges have length 0 or 1, and there are no backward edges, so the claim clearly holds. If the residual graph contained a negative-length cycle C, then C would have to contain at least one flow-carrying edge e = (u, v). Since e has incoming flow from s and outgoing flow to t, the residual graph would contain a path of backward edges from u to s and one from t to v. Thus, any negative cycle would give arbitrarily negative-length paths from t to s. It is therefore enough to establish the first claim.
Consider an iteration when flow is augmented along a path P . Suppose that this generates a t-s path P ′ in the residual graph of length strictly less than −1. P ∪ P ′ gives a cycle. If we assign edges in P their length prior to the augmentation, and edges in P ′ their length after the augmentation, then the total length of the cycle P ∪ P ′ is negative. The only edges in P ′ whose length can have decreased through the augmentation are the backward versions of edges to which relief was added by P . They were saturated before the augmentation, so their forward length was 1, and their backward length after augmentation is −1.
Now consider removing edges that appear both forward and backward in P ∪ P ′ . We obtain a union of edge-disjoint cycles, such that all edges in these cycles were present in the residual graph prior to the flow augmentation. Of these edge-disjoint cycles, by the argument of the previous paragraph, at least one cycle C has negative length with respect to the previous paragraph's definition. The edges in P ′ \ P don't change their length, so C had negative length before the augmentation, contradicting the induction hypothesis.
For two nodes u, v, we write u → 0 v if there is a path of length at most 0 from u to v in G ′ . We now define the cuts. Let S 1 := {v | d v ≤ 0}. Define E ′ := {(u, v) ∈ R | d u > 0}. Now, let U be the set of all vertices lying on a v-t path for some edge (u, v) ∈ E ′ , and let S 2 be the set of all vertices y such that y → 0 w for some w ∈ U. Clearly, (S 1 , S 1 ) and (S 2 , S 2 ) define two s-t cuts using only saturated edges. Lemma 5.6 No edge crosses both cuts (S 1 , S 1 ) and (S 2 , S 2 ). Each relief edge e ∈ R crosses one of the cuts (S 1 , S 1 ) or (S 2 , S 2 ).
Proof. To prove the first claim, suppose that e = (u, v) crosses both cuts. By the definition of S 2 , that means that there is an edge e ′ = (u ′ , v ′ ) ∈ E ′ ⊆ R such that there is a path of length at most 0 from v to some node w on a v ′ -t path. Consider the path from s to u (of length at most 0), followed by e (of length at most 1), followed by the path from v to w, and then the path to v ′ backwards, followed by e ′ backwards. This is a path of length at most 0 from s to u ′ , meaning that u ′ should have been in S 1 , and contradicting that e ′ was in E ′ .
To prove the second claim, suppose that a relief edge e = (u, v) crosses neither of the cuts. We distinguish two cases:
2. If u / ∈ S 1 , then d u > 0 and e ∈ E ′ . Thus v ∈ U, and since v → 0 v, we have v ∈ S 2 . Since e does not cross either cut, we also have u ∈ S 2 . This means that there is a w ∈ U and e ′ = (u ′ , v ′ ) ∈ E ′ such that w lies on a v ′ -t path, and u → 0 w. Now consider the path from t to v (of length at most 0), then using e backwards (of length −1), then the length-0 path from u to w and the path from w to v backwards (of length at most 0), followed by e ′ backwards, and the path from u ′ to s backwards (of length at most 0). This gives a t-s path of total length at most −2, again contradicting Lemma 5.5. Lemma 5.6 implies that the set R of relief edges forms a double cut. Thus our algorithm finds a minimum-cost double cut in polynomial time.
Remark 5.7 As in Remark 4.8 for the flow mechanism F M, we can show that on instances derived from the pruning step, EV coincides with the mechanism of [7]. Thus, the tighter analysis shows that CM is in fact 2-competitive. We conjecture that CM is indeed optimal when compared against the buyer-pessimal lower bound of [11].
Directions for Future Work
We have presented novel truthful and competitive mechanisms for three important combinatorial problems: Vertex Covers, k-flows, and s-t cuts. The Vertex Cover mechanism was based on scaling the submitted bids by multipliers derived as components of the dominant eigenvector of a suitable matrix. Both the flow and cut mechanisms were based on pruning the input graph, and then applying the Vertex Cover mechanism to the pruned version. Besides the individual mechanisms, we believe that the methodology of reducing input instances to Vertex Cover problems may be of interest for future frugal mechanism design.
In general, the Vertex Cover mechanism does not run in polynomial time, due to two obstacles. First, computing the matrix K requires computing the largest fractional clique size in the neighborhood of each node v. Subsequently, computing the solution with respect to scaled costs requires finding a cheapest vertex cover. For the second obstacle, it seems quite likely that monotone algorithms such as the one in [11] could be adapted to our setting, and yield constant-factor approximations. However, the difficulty of computing the entries of K seems more severe. In fact, we conjecture that no polynomial-time truthful mechanism for Vertex Cover can be constant-competitive. This result would be quite interesting, in that it would show that the requirements of incentive-compatibility and computational tractability together can lead to significantly worse guarantees than either requirement alone. A positive resolution of this conjecture would thus be akin to the types of hardness results demonstrated recently for the Combinatorial Public Project Problem [26].
While our methodology of designing composable pre-processing algorithms will likely be useful for other problems as well, it does not apply to all set systems. It is fairly easy to construct set systems for which no such pruning algorithm is possible. Even when pruning is possible in principle, it may come with a large blowup in costs.
Thus, the following bigger question still stands: which classes of set systems admit constant-competitive mechanisms? The main obstacle is our inability to prove strong lower bounds on frugality ratios. To date, all lower bounds (here, as well as in [12,18]) are based on pairwise comparisons between agents, which can then be used to show that certain agents, by virtue of losing, will cause large payments. This technique was exactly the motivation for our Vertex Cover approach. In order to move beyond Vertex Cover based mechanisms, it will be necessary to explore lower bound techniques beyond the one used in this paper.
In recent joint work with the authors of [7], we have shown that the factor 1 2 in the lower bound of Lemma 3.2 can be removed, thus showing that EV is optimal. The proof of this result will be presented in a joint full version of the present paper with [7].
Lemma 3. 1
1EV has frugality ratio at most α.
Proposition 3. 6
6Let G v be the subgraph induced by the neighborhood of v but without v itself. Then, ν v is exactly the fractional clique number, and thus the fractional chromatic number, of G v .
Figure 1 :
1A minimally 3-connected graph (left) and the resulting vertex cover instance for k = 2 (right).
Proposition 4. 3
3Let H be a graph consisting of k + 1 edge-disjoint s-t paths, and let e = (u, v), e ′ = (u ′ , v ′ ) be two edges of H. Then, there is a minimum s-t cut containing both e and e ′ if and only if there is no path from v to u ′ and no path from v ′ to u.
Lemma 4. 5
5Let x be a solution to the LP (2), and G ′ as in the proof of Lemma 4.4.
Theorem 5. 2
2The double cut selection rule is composable and produces a Vertex Cover instance with ν H (c) ≤ 2ν G (c). Furthermore, both the selection rule and the subsequent Vertex Cover mechanism can be computed in polynomial time. Thus, CM is a polynomialtime 4-competitive mechanism.
Lemma 5. 3
3For all cost vectors c, ν H (c) ≤ 2ν G (c). Proof. Let (S, S) be the cheapest s-t cut in G with respect to the costs c, and x a solution to the LP (2) with cost vector c on the graph G. Let C be the set of all minimum s-t cuts (T, T ) with respect to the costs x; thus, each of these cuts has cost x(E(S, S)). Define T − = (T,T )∈C T , and T + = (T,T )∈C T . Then, both (T − , T − ) and (T + , T + ) are minimum s-t cuts as well (see, e.g., [2, Exercise 6.39]).Furthermore, the edge sets E(T − , T − ) and E(T + , T + ) are disjoint. For assume that there is an edge e = (u, v) in common between these sets. Then, u ∈ (T,T )∈C T and v ∈ (T,T )∈C T .
AcknowledgmentWe would like to thank Edith Elkind, Uriel Feige, Nick Gravin, Anna Karlin, Tami Tamir, and Mihalis Yannakakis for useful discussions and pointers, and anonymous reviewers for useful feedback. D.K. is supported in part by an NSF CAREER Award, an ONR Young Investigator Award and an award from the Sloan Foundation. C.M. is supported in part by the McDonnell Foundation.
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| [] |
[
"Pixelwise Instance Segmentation with a Dynamically Instantiated Network",
"Pixelwise Instance Segmentation with a Dynamically Instantiated Network",
"Pixelwise Instance Segmentation with a Dynamically Instantiated Network",
"Pixelwise Instance Segmentation with a Dynamically Instantiated Network"
] | [
"Anurag Arnab [email protected] \nUniversity of Oxford\n\n",
"Philip H S Torr [email protected] \nUniversity of Oxford\n\n",
"Anurag Arnab [email protected] \nUniversity of Oxford\n\n",
"Philip H S Torr [email protected] \nUniversity of Oxford\n\n"
] | [
"University of Oxford\n",
"University of Oxford\n",
"University of Oxford\n",
"University of Oxford\n"
] | [] | Semantic segmentation and object detection research have recently achieved rapid progress. However, the former task has no notion of different instances of the same object, and the latter operates at a coarse, bounding-box level. We propose an Instance Segmentation system that produces a segmentation map where each pixel is assigned an object class and instance identity label. Most approaches adapt object detectors to produce segments instead of boxes. In contrast, our method is based on an initial semantic segmentation module, which feeds into an instance subnetwork. This subnetwork uses the initial category-level segmentation, along with cues from the output of an object detector, within an end-to-end CRF to predict instances. This part of our model is dynamically instantiated to produce a variable number of instances per image. Our end-to-end approach requires no post-processing and considers the image holistically, instead of processing independent proposals. Therefore, unlike some related work, a pixel cannot belong to multiple instances. Furthermore, far more precise segmentations are achieved, as shown by our substantial improvements at high AP r thresholds. | 10.1109/cvpr.2017.100 | [
"https://arxiv.org/pdf/1704.02386v1.pdf"
] | 689,238 | 1704.02386 | 6d8a42dce4d79435c42bf8eefddbea0e38951f4e |
Pixelwise Instance Segmentation with a Dynamically Instantiated Network
Anurag Arnab [email protected]
University of Oxford
Philip H S Torr [email protected]
University of Oxford
Pixelwise Instance Segmentation with a Dynamically Instantiated Network
Semantic segmentation and object detection research have recently achieved rapid progress. However, the former task has no notion of different instances of the same object, and the latter operates at a coarse, bounding-box level. We propose an Instance Segmentation system that produces a segmentation map where each pixel is assigned an object class and instance identity label. Most approaches adapt object detectors to produce segments instead of boxes. In contrast, our method is based on an initial semantic segmentation module, which feeds into an instance subnetwork. This subnetwork uses the initial category-level segmentation, along with cues from the output of an object detector, within an end-to-end CRF to predict instances. This part of our model is dynamically instantiated to produce a variable number of instances per image. Our end-to-end approach requires no post-processing and considers the image holistically, instead of processing independent proposals. Therefore, unlike some related work, a pixel cannot belong to multiple instances. Furthermore, far more precise segmentations are achieved, as shown by our substantial improvements at high AP r thresholds.
Introduction
Semantic segmentation and object detection are wellstudied scene understanding problems, and have recently witnessed great progress due to deep learning [22,13,7]. However, semantic segmentation -which labels every pixel in an image with its object class -has no notion of different instances of an object (Fig. 1). Object detection does localise different object instances, but does so at a very coarse, bounding-box level. Instance segmentation localises objects at a pixel level, as shown in Fig. 1, and can be thought of being at the intersection of these two scene understanding tasks. Unlike the former, it knows about different instances of the same object, and unlike the latter, it operates at a pixel level. Accurate recognition and localisation of objects enables many applications, such as autonomous driving [9], image-editing [53] and robotics [17].
Many recent approaches to instance segmentation are based on object detection pipelines where objects are first localised with bounding boxes. Thereafter, each bounding box is refined into a segmentation [19,20,32,37,30]. Another related approach [12,56] is to use segment-based region proposals [10,41,42] instead of box-based proposals. However, these methods do not consider the entire image, but rather independent proposals. As a result, occlusions between different objects are not handled. Furthermore, many of these methods cannot easily produce segmentation maps of the image, as shown in Fig. 1, since they process numerous proposals independently. There are typically far more proposals than actual objects in the image, and these proposals can overlap and be assigned different class labels. Finally, as these methods are based on an initial detection step, they cannot recover from false detections.
Our proposed method is inspired by the fact that instance segmentation can be viewed as a more complex form of semantic segmentation, since we are not only required to label the object class of each pixel, but also its instance identity. We produce a pixelwise segmentation of the image, where each pixel is assigned both a semantic class and instance label. Our end-to-end trained network, which outputs a variable number of instances per input image, begins with an initial semantic segmentation module. The following, dynamic part of the network, then uses information from an object detector and a Conditional Random Field (CRF) model to distinguish different instances. This approach is robust to false-positive detections, as well as poorly localised bounding boxes which do not cover the entire object, in contrast to detection-based methods to instance segmentation. Moreover, as it considers the entire image when making predictions, it attempts to resolve occlusions between different objects and can produce segmentation maps as in Fig. 1 without any post-processing.
Furthermore, we note that the Average Precision (AP) metric [14] used in evaluating object detection systems, and its AP r variant [19] used for instance segmentation, considers individual, potentially overlapping, object predictions in isolation, as opposed to the entire image. To evaluate methods such as ours, which produce complete segmentation maps and reason about occlusions, we also evaluate using Our proposed method jointly produces both semantic and instance segmentations. Our method uses the output of an object detector as a cue to identify instances, but is robust to false positive detections, poor bounding box localisation and occlusions. Best viewed in colour.
the "Matching Intersection over Union" metric.
Our system, which is based on an initial semantic segmentation subnetwork, produces sharp and accurate instance segmentations. This is reflected by the substantial improvements we achieve over state-of-the-art methods at high AP r thresholds on the Pascal VOC and Semantic Boundaries datasets. Furthermore, our network improves on the semantic segmentation task while being trained for the related task of instance segmentation.
Related Work
An early work on instance segmentation was by Winn and Shotton [51]. A per-pixel unary classifier was trained to predict parts of an object. These parts were then encouraged to maintain a spatial ordering, that is characteristic of an instance, using asymmetric pairwise potentials in a Conditional Random Field (CRF). Subsequent work [54], presented another approach where detection outputs of DPM [15], with associated foreground masks, were assigned a depth ordering using a generative, probabilistic model. This depth ordering resolved occlusions.
However, instance segmentation has become more common after the "Simultaneous Detection and Segmentation" (SDS) work of Hariharan et al. [19]. This system was based on the R-CNN pipeline [16]: Region proposals, generated by the method of [1], were classified into object categories with a Convolutional Neural Network (CNN) before applying bounding-box regression as post-processing. A classspecific segmentation was then performed in this bounding box to simultaneously detect and segment the object. Numerous works [20,8,30] have extended this pipeline. However, approaches that segment instances by refining detections [19,20,8,11,30] are inherently limited by the quality of the initial proposals. This problem is exacerbated by the fact that this pipeline consists of several different modules trained with different objective functions. Furthermore, numerous post-processing steps such as "superpixel projection" and rescoring are performed. Dai et al. [12] addressed some of these issues by designing one end-to-end trained network that generates box-proposals, creates foreground masks from these proposals and then classifies these masks. This network can be seen as an extension of the end-to-end Faster-RCNN [44] detection framework, which generates box-proposals and classifies them. Additionally, Liu et al. [37] formulated an end-to-end version of the SDS network [19], whilst [32] iteratively refined object proposals.
On a separate track, algorithms have also been developed that do not require object detectors. Zhang et al. [57,58] segmented car instances by predicting the depth ordering of each pixel in the image. Unlike the previous detectionbased approaches, this method reasoned globally about all instances in the image simultaneously (rather than individual proposals) with an MRF-based formulation. However, inference of this graphical model was not performed end-toend as shown to be possible in [60,2,5,34]. Furthermore, although this method does not use object detections, it is trained with ground truth depth and assumes a maximum of nine cars in an image. Predicting all the instances in an image simultaneously (rather than classifying individual proposals) requires a model to be able to handle a variable number of output instances per image. As a result, [45] proposed a Recurrent Neural Network (RNN) for this task. However, this model was only for a single object category. Our proposed method not only outputs a variable number of instances, but can also handle multiple object classes.
Liang et al. [33] developed another proposal-free method based on the semantic segmentation network of [6]. The category-level segmentation, along with CNN features, was used to predict instance-level bounding boxes. The number of instances of each class was also predicted to enable a final spectral clustering step. However, this additional information predicted by Liang's network could have been obtained Figure 2: Network overview: Our end-to-end trained network consists of semantic-and instance-segmentation modules. The intermediate category-level segmentation, along with the outputs of an object detector, are used to reason about instances. This is done by instance unary terms which use information from the detector's bounding boxes, the initial semantic segmentation and also the object's shape. A final CRF is used to combine all this information together to obtain an instance segmentation. The output of the semantic segmentation module is a fixed size W × H × (K + 1) tensor where K is the number of object classes, excluding background, in the dataset. The final output, however, is of a variable W × H × (D + 1) dimensions where D is the number of detected objects (and one background label).
from an object detector. Arnab et al. [3] also started with an initial semantic segmentation network [2], and combined this with the outputs of an object detector using a CRF to reason about instances. This method was not trained endto-end though, and could not really recover from errors in bounding-box localisation or occlusion.
Our method also has an initial semantic segmentation subnetwork, and uses the outputs of an object detector. However, in contrast to [3] it is trained end-to-end to improve on both semantic-and instance-segmentation performance (to our knowledge, this is the first work to achieve this). Furthermore, it can handle detector localisation errors and occlusions better due to the energy terms in our end-to-end CRF. In contrast to detection-based approaches [19,20,12,37], our network requires no additional postprocessing to create an instance segmentation map as in Fig. 1(c) and reasons about the entire image, rather than independent proposals. This global reasoning allows our method to produce more accurate segmentations. Our proposed system also handles a variable number of instances per image, and thus does not assume a maximum number of instances like [57,58].
Proposed Approach
Our network (Fig. 2) contains an initial semantic segmentation module. We use the semantic segmentation result, along with the outputs of an object detector, to compute the unary potentials of a Conditional Random Field (CRF) defined over object instances. We perform mean field inference in this random field to obtain the Maximum a Posteriori (MAP) estimate, which is our labelling. Although our network consists of two conceptually different parts -a semantic segmentation module, and an instance segmentation network -the entire pipeline is fully differentiable, given object detections, and trained end-to-end.
Semantic Segmentation subnetwork
Semantic Segmentation assigns each pixel in an image a semantic class label from a given set, L. In our case, this module uses the FCN8s architecture [38] which is based on the VGG [47] ImageNet model. For better segmentation results, we include mean field inference of a Conditional Random Field as the last layer of this module. This CRF contains the densely-connected pairwise potentials described in [26] and is formulated as a recurrent neural network as in [60]. Additionally, we include the Higher Order detection potential described in [2]. This detection potential has two primary benefits: Firstly, it improves semantic segmentation quality by encouraging consistency between object detections and segmentations. Secondly, it also recalibrates detection scores. This detection potential is similar to the one previously proposed by [28], [48], [52] and [55], but formulated for the differentiable mean field inference algorithm. We employ this potential as we are already using object detection information for identifying object instances in the next stage. We denote the output at the semantic segmentation module of our network as the tensor Q, where Q i (l) denotes the probability (obtained by applying the softmax function on the network's activations) of pixel i taking on the label l ∈ L.
(a) Semantic Segmentation (b) Instance Segmentation Figure 3: Instance segmentation using only the "Box" unary potential. This potential is effective when we have a good initial semantic segmentation (a). Occlusions between objects of the same class can be resolved by the pairwise term based on appearance differences. Note that we can ignore the confident, false-positive "bottle" detections (b). This is in contrast to methods such as [8,19,20,30] which cannot recover from detection errors.
Instance Segmentation subnetwork
At the input to our instance segmentation subnetwork, we assume that we have two inputs available: The semantic segmentation predictions, Q, for each pixel and label, and a set of object detections. For each input image, we assume that there are D object detections, and that the i th detection is of the form (l i , s i , B i ) where l i ∈ L is the detected class label, s i ∈ [0, 1] is the confidence score and B i is the set of indices of the pixels falling within the detector's bounding box. Note that the number D varies for every input image.
The problem of instance segmentation can then be thought of as assigning every pixel to either a particular object detection, or the background label. This is based on the assumption that every object detection specifies a potential object instance. We define a multinomial random variable, V , at each of the N pixels in the image, and V = [V 1 V 2 . . . V N ] T . Each variable at pixel i, V i , is assigned a label corresponding to its instance. This label set, {0, 1, 2, ..., D} changes for each image since D, the number of detections, varies for every image (0 is the background label). In the case of instance segmentation of images, the quality of a prediction is invariant to the permutations of the instance labelling. For example, labelling the "blue person" in Fig. 1(c) as "1" and the "purple person" as "2" is no different to labelling them as "2" and "1" respectively. This condition is handled by our loss function in Sec. 3.4.
Note that unlike works such as [57] and [58] we do not assume a maximum number of possible instances and keep a fixed label set. Furthermore, since we are considering object detection outputs jointly with semantic segmentation predictions, we have some robustness to high-scoring false positive detections unlike methods such as [8,20,37] which refine object detections into segmentations.
We formulate a Conditional Random Field over our instance variables, V , which consists of unary and pairwise (a) Only Box term (b) Box and Global terms Figure 4: The "Global" unary potential (b) is particularly effective in cases where the input detection bounding box does not cover the entire extent of the object. Methods which are based on refining bounding-box detections such as [19,20,8,12] cannot cope with poorly localised detections. Note, the overlaid detection boxes are an additional input to our system.
energies. The energy of the assignment v to all the variables, V, is
E(V = v) = i U (v i ) + i<j P (v i , v j ).(1)
The unary energy is a sum of three terms, which take into account the object detection bounding boxes, the initial semantic segmentation and shape information,
U (v i ) = −ln[w 1 ψ Box (v i ) + w 2 ψ Global (v i )+ w 3 ψ Shape (v i )],(2)
and are described further in Sections 3.2.1 through 3.2.3. w 1 , w 2 and w 3 are all weighting co-efficients learned via backpropagation.
Box Term
This potential encourages a pixel to be assigned to the instance corresponding to the k th detection if it falls within the detection's bounding box. This potential is proportional to the probability of the pixel's semantic class being equal to the detected class Q i (l k ) and the detection score, s k .
ψ Box (V i = k) = Q i (l k )s k if i ∈ B k 0 otherwise (3)
As shown in Fig. 3, this potential performs well when the initial semantic segmentation is good. It is robust to false positive detections, unlike methods which refine bounding boxes [8,19,20] since the detections are considered in light of our initial semantic segmentation, Q. Together with the pairwise term (Sec. 3.2.4), occlusions between objects of the same class can be resolved if there are appearance differences in the different instances.
Global Term
This term does not rely on bounding boxes, but only the segmentation prediction at a particular pixel, Q i . It encodes the intuition that if we only know there are d possible instances of a particular object class, and have no further localisation information, each instance is equally probable, and this potential is proportional to the semantic segmentation confidence for the detected object class at that pixel:
ψ Global (V i = k) = Q i (l k ).(4)
As shown in Fig. 4, this potential overcomes cases where the bounding box does not cover the entire extent of the object, as it assigns probability mass to a particular instance label throughout all pixels in the image. This is also beneficial during training, as it ensures that the final output is dependent on the segmentation prediction at all pixels in the image, leading to error gradients that are more stable across batches and thus more amenable to backpropagation.
Shape Term
We also incorporate shape priors to help us reason about occlusions involving multiple objects of the same class, which may have minimal appearance variation between them, as shown in Fig. 5. In such cases, a prior on the expected shape of an object category can help us to identify the foreground instance within a bounding box. Previous approaches to incorporating shape priors in segmentation [23,8,50] have involved generating "shape exemplars" from the training dataset and, at inference time, matching these exemplars to object proposals using the Chamfer distance [46,36]. We propose a fully differentiable method: Given a set of shape templates, T , we warp each shape template using bilinear interpolation intoT so that it matches the dimensions of the k th bounding box, B k . We then select the shape prior which matches the segmentation prediction for the detected class within the bounding box, Q B k (l k ), the best according to the normalised cross correlation. Our shape prior is then the Hadamard (elementwise) product (⊙) between the segmentation unaries and the matched shape prior:
t * = arg max t∈T Q B k (l k ) ⊙ t Q B k (l k ) t (5) ψ(V B k = k) = Q B k (l k ) ⊙ t * .(6)
Equations 5 and 6 can be seen as a special case of maxpooling, and the numerator of Eq. 5 is simply a convolution that produces a scalar output since the two arguments are of equal dimension. Additionally, during training, we can consider the shape priors T as parameters of our "shape term" layer and backpropagate through to the matched exemplar t * to update it. In practice, we initialised these parameters (a) Without shape term (b) With Shape term Figure 5: The "Shape" unary potential (b) helps us to distinguish between the green and purple sheep, which the other two unary potentials cannot. Input detections are overlaid on the images.
with the shape priors described in [50]. This consists of roughly 250 shape templates for each of five different aspect ratios. These were obtained by clustering foreground masks of object instances from the training set.
Here, we have only matched a single shape template to a proposed instance. This method could be extended in future to matching multiple templates to an instance, in which case each shape exemplar would correspond to a part of the object such as in DPM [15].
Pairwise term
The pairwise term consists of densely-connected Gaussian potentials [26] and encourages appearance and spatial consistency. The weights governing the importance of these terms are also learnt via backpropagation, as in [60]. We find that these priors are useful in the case of instance segmentation as well, since nearby pixels that have similar appearance often belong to the same object instance. They are often able to resolve occlusions based on appearance differences between objects of the same class (Fig. 3).
Inference of our Dynamic Instance CRF
We use mean field inference to approximately minimise the Gibbs Energy in Eq. 1 which corresponds to finding the Maximum a Posteriori (MAP) labelling of the corresponding probability distribution,
P (V = v) = 1 Z exp (−E(v))
where Z is the normalisation factor. Mean field inference is differentiable, and this iterative algorithm can be unrolled and seen as a recurrent neural network [60]. Following this approach, we can incorporate mean field inference of a CRF as a layer of our neural network. This enables us to train our entire instance segmentation network end-to-end.
Because we deal with a variable number of instances for every image, our CRF needs to be dynamically instantiated to have a different number of labels for every image, as observed in [3]. Therefore, unlike [60], none of our weights are class-specific. This weight-sharing not only allows us to deal with variable length inputs, but class-specific weights (a) Original ground truth, G (b) Prediction, P (c) "Matched" ground truth, G * Figure 6: Due to the problem of label permutations, we "match" the ground truth with our prediction before computing the loss when training.
also do not make sense in the case of instance segmentation since a class label has no particular semantic meaning.
Loss Function
When training for instance segmentation, we have a single loss function which we backpropagate through our instance-and semantic-segmentation modules to update all the parameters. As discussed previously, we need to deal with different permutations of our final labelling which could have the same final result. The works of [57] and [58] order instances by depth to break this symmetry. However, this requires ground-truth depth maps during training which we do not assume that we have. Proposal-based methods [12,19,20,37] do not have this issue since they consider a single proposal at a time, rather than the entire image. Our approach is similar to [45] in that we match the original ground truth to our instance segmentation prediction based on the Intersection over Union (IoU) [14] of each instance prediction and ground truth, as shown in Fig. 6.
More formally, we denote the ground-truth labelling of an image, G, to be a set of r segments, {g 1 , g 2 , . . . , g r }, where each segment (set of pixels) is an object instance and has an associated semantic class label. Our prediction, which is the output of our network, P, is a set of s segments, {p 1 , p 2 , . . . , p s }, also where each segment corresponds to an instance label and also has an associated class label. Note that r and s may be different since we may predict greater or fewer instances than actually present. Let M denote the set of all permutations of the ground-truth, G. As can be seen in Fig. 6, different permutations of the ground-truth correspond to the same qualitative result. We define the "matched" ground-truth, G * , as the permutation of the original ground-truth labelling which maximises the IoU between the prediction, P, and ground truth:
G * = arg max m∈M IoU(m, P).(7)
Once we have the "matched" ground truth, G * , (Fig. 6) for an image, we can apply any loss function to train our network for segmentation. In our case, we use the common cross-entropy loss function. We found that this performed better than the approximate IoU loss proposed in [27,45]. Crucially, we do not need to evaluate all permutations of the ground truth to compute Eq. 7, since it can be formulated as a maximum-weight bipartite matching problem. The edges in our bipartite graph connect ground-truth and predicted segments. The edge weights are given by the IoU between the ground truth and predicted segments if they share the same semantic class label, and zero otherwise. Leftover segments are matched to "dummy" nodes with zero overlap.
Additionally, the ordering of the instances in our network are actually determined by the object detector, which remains static during training. As a result, the ordering of our predictions does not fluctuate much during training -it only changes in cases where there are multiple detections overlapping an object.
Network Training
We first train a network for semantic segmentation with the standard cross-entropy loss. In our case, this network is FCN8s [38] with a CRF whose inference is unrolled as an RNN and trained end-to-end, as described in [60] and [2]. To this pretrained network, we append our instance segmentation subnetwork, and finetune with instance segmentation annotations and only the loss detailed in Sec. 3.4. For the semantic segmentation subnetwork, we train with an initial learning rate of 10 −8 , momentum of 0.9 and batch size of 20. The learning rate is low since we do not normalise the loss by the number of pixels. This is so that images with more pixels contribute a higher loss. The normalised learning rate is approximately 2 × 10 −3 . When training our instance segmentation network as well, we lower the learning rate to 10 −12 and use a batch size of 1 instead. Decreasing the batch size gave empirically better results. We also clipped gradients (a technique common in training RNNs [40]) with ℓ 2 norms above 10 9 . This threshold was set by observing "normal" gradient magnitudes during training. The relatively high magnitude is due to the fact that our loss is not normalised. In our complete network, we have two CRF inference modules which are RNNs (one each in the semantic-and instance-segmentation subnetworks), and gradient clipping facilitated successful training.
Discussion
Our network is able to compute a semantic and instance segmentation of the input image in a single forward pass. We do not require any post-processing, such as the patch aggregation of [37], "mask-voting" of [12], "superpixel projection" of [19,20,30] or spectral clustering of [33]. The fact that we compute an initial semantic segmentation means that we have some robustness to errors in the object detector (Fig. 3). Furthermore, we are not necessarily limited by poorly localised object detections either (Fig. 4). Our CRF model allows us to reason about the entire image at a time, rather than consider independent object proposals, as done in [19,20,12,37,30]. Although we do not train our object detector jointly with the network, it also means that our segmentation network and object detector do not succumb to the same failure cases. Moreover, it ensures that our instance labelling does not "switch" often during training, which makes learning more stable. Finally, note that although we perform mean field inference of a CRF within our network, we do not optimise the CRF's likelihood, but rather a cross-entropy loss (Sec 3.4).
Experimental Evaluation
Sections 4.1 to 4.6 describe our evaluation on the Pascal VOC Validation Set [14] and the Semantic Boundaries Dataset (SBD) [18] (which provides per-pixel annotations to 11355 previously unlaballed images from Pascal VOC). Section 4.7 details results on Cityscapes [9].
Experimental Details
We first train a network for semantic segmentation, therafter we finetune it to the task of instance segmentation, as described in Sec. 3.5. Our training data for the semantic segmentation pretraining consists of images from Pascal VOC [14], SBD [18] and Microsoft COCO [35]. Finally, when finetuning for instance segmentation, we use only training data from either the VOC dataset, or from the SBD dataset. We train separate models for evaluating on the VOC Validation Set, and the SBD Validation Set. In each case, we remove validation set images from the initial semantic segmentation pretraining set. We use the publicly available R-FCN object detection framework [13], and ensure that the images used to train the detector do not fall into our test sets for instance segmentation.
Evaluation Metrics
We report the mean Average Precision over regions (AP r ) as defined by [19]. The difference between AP r and the AP metric used in object detection [14] is that the Intersection over Union (IoU) is computed over predicted and ground-truth regions instead of bounding boxes. Furthermore, the standard AP metric uses an IoU threshold of 0.5 to determine whether a prediction is correct or not. Here, we use a variety of IoU thresholds since larger thresholds require more precise segmentations. Additionally, we report the AP r vol which is the average of the AP r for 9 IoU thresholds ranging from 0.1 to 0.9 in increments of 0.1.
However, we also observe that the AP r metric requires an algorithm to produce a ranked list of segments and their object class. It does not require, nor evaluate, the ability of an algorithm to produce a globally coherent segmentation map of the image, for example Fig. 1c. To measure this, we propose the "Matching IoU" which matches the predicted image and ground truth, and then calculates the corresponding IoU as defined in [14]. This matching procedure is the same as described in Sec. 3.4. This measure was originally proposed in [54], but has not been used since in evaluating instance segmentation systems.
Effect of Instance Potentials and End-to-End training
We first perform ablation studies on the VOC 2012 Validation set. This dataset, consisting of 1464 training and 1449 validation images has very high-quality annotations with detailed object delineations which makes it the most suited for evaluating pixel-level segmentations.
In Tab. 1, we examine the effect of each of our unary potentials in our Instance subnetwork on overall performance. Furthermore, we examine the effect of end-to-end training the entire network as opposed to piecewise training. Piecewise training refers to freezing the pretrained semantic segmentation subnetwork's weights and only optimising the instance segmentation subnetwork's parameters. Note that when training with only the "Box" (Eq. 3) unary potential and pairwise term, we also have to add in an additional "Background" detection which encompasses the entire image. Otherwise, we cannot classify the background label.
We can see that each unary potential improves overall instance segmentation results, both in terms of AP r vol and the Matching IoU. The "Global" term (Eq. 4) shows particular improvement over the "Box" term at the high AP r threshold of 0.9. This is because it can overcome errors in bounding box localisation (Fig. 4) and leverage our semantic segmentation network's accurate predictions to produce precise labellings. The "Shape" term's improvement in the AP r vol is primarily due to an improvement in the AP r at low thresholds. By using shape priors, we are able to recover instances which were occluded and missed out. End-to-end training also improves results at all AP r thresholds. Training with just the "Box" term shows a modest improvement in the AP r vol of 1.3%. Training with the "Global" and "Shape" terms shows larger improvements of 2.1% and 2.3% respectively. This may be because the "Box" term only considers the semantic segmentation at parts of the image covered by object detections. Once we include the "Global" term, we consider the semantic segmentation over the entire image for the detected class. Training makes more efficient use of images, and error gradients are more stable in this case.
Results on VOC Validation Set
We then compare our best instance segmentation model to recent methods on the VOC Validation Set in Tab. 2. The fact that our algorithm achieves the highest AP r at thresholds above 0.7 indicates that our method produces more detailed and accurate segmentations.
At an IoU threshold of 0.9, our improvement over the previous state-of-the-art (MPA [37]) is 6.6%, which is a relative improvement of 36%. Unlike [37,19,8], our network performs an initial semantic segmentation which may explain our more accurate segmentations. Other segmentation-based approaches, [3,33] are not fully endto-end trained. We also achieve the best AP r vol of 57.5%. The relatively small difference in AP r vol to MPA [37] despite large improvements at high IoU thresholds indicates that MPA performs better at low IoU thresholds. Proposalbased methods, such as [37,19] are more likely to perform better at low IoU thresholds since they output more proposals than actual instances in an image (SDS evaluates 2000 proposals per image). Furthermore, note that whilst MPA takes 8.7s to process an image [37], our method requires approximately 1.5s on the same Titan X GPU. More detailed qualitative and quantitative results, including success and failure cases, are included in the supplementary material. [20] 56.5 37.0 --IIS [30] 60.1 38.7 --CFM [11] 60.7 39.6 --Hypercolumn rescore [20] 60.0 40.4 --MPA 3-scale rescore [37] 61.8 -52.0 -MNC [12] 63. 5
Results on SBD Dataset
We also evaluate our model on the SBD dataset, which consists of 5623 training and 5732 validation images, as shown in Tab. 3. Following other works, we only report AP r results at IoU thresholds of 0.5 and 0.7. However, we provide more detailed results in our supplementary material. Once again, we show significant improvements over other work at high AP r thresholds. Here, our AP r at 0.7 improves by 1.5% over the previous state-of-the-art [30]. Note that [30,37,20] perform additional post-processing where their results are rescored using an additional object detector. In contrast, our results are obtained by a single forward pass through our network. We have also improved substantially on the AP r vol measure (3.4%) compared to other works which have reported it. We also used the publicly available source code 1 , model and default parameters of MNC [12] to evaluate the "Matching IoU". Our method improves this by 8.3%. This metric is a stricter measure of segmentation performance, and our method, which is based on an initial semantic segmentation and includes a CRF as part of training therefore performs better.
Improvement in Semantic Segmentation
Finetuning our network for instance segmentation, with the loss described in Sec. 3.4 improves semantic segmentation performance on both the VOC and SBD dataset, as shown in Tab. 4. The improvement is 0.9% on VOC, and 1% on SBD. The tasks of instance segmentation and semantic segmentation are highly related -in fact, instance segmentation can be thought of as a more specific case of semantic segmentation. As a result, finetuning for one task improves the other.
Results on Cityscapes
Finally, we evaluate our algorithm on the Cityscapes road-scene understanding dataset [9]. This dataset consists of 2975 training images, and the held-out test set consisting of 1525 images are evaluated on an online server. None of the 500 validation images were used for training. We use an initial semantic segmentation subnetwork that is based on the ResNet-101 architecture [59], and all of the instance unary potentials described in Sec. 3.2.
As shown in Tab. 5, our method sets a new state-of-theart on Cityscapes, surpassing concurrent work [21] and the best previous published work [49] by significant margins.
Conclusion and Future Work
We have presented an end-to-end instance segmentation approach that produces intermediate semantic segmentations, and shown that finetuning for instance segmentation improves our network's semantic segmentations. Our approach differs from other methods which derive their architectures from object detection networks [12,37,20] in that our approach is more similar to a semantic segmentation network. As a result, our system produces more accurate and detailed segmentations as shown by our substantial improvements at high AP r thresholds. Moreover, our system produces segmentation maps naturally, and in contrast to other published work, does not require any post-processing. Finally, our network produces a variable number of outputs, depending on the number of instances in the image. Our future work is to incorporate an object detector into the endto-end training of our system to create a network that performs semantic segmentation, object detection and instance segmentation jointly. Possible techniques for doing this are suggested by [25] and [39]. Figure 7: Success cases of our method. First and second row: Our algorithm can leverage good initial semantic segmentations, and detections, to produce an instance segmentation. Third row: Notice that we have ignored three false-positive detections. Additionally, the red bounding box does not completely encompass the person, but our algorithm is still able to associate pixels "outside-the-box" with the correct detection (also applies to row 2). Fourth row: Our system is able to deal with the heavily occluded sheep, and ignore the false-positive detection. Fifth row: We have not been able to identify one bicycle on the left since it was not detected, but otherwise have performed well. Sixth row: Although subjective, the train has not been annotated in the dataset, but both our initial semantic segmentation and object detection networks have identified it. Note that the first three images are from the VOC dataset, and the last three from SBD. Annotations in the VOC dataset are more detailed, and also make more use of the grey "ignore" label to indicate uncertain areas in the image. The first column shows the input image, and the results of our object detector which are another input to our network. Best viewed in colour.
Input image
Semantic Segmentation
Instance Segmentation Ground truth Figure 8: Failure cases of our method. First row: Both our initial detector, and semantic segmentation system did not identify a car in the background. Additionally, the "brown" person prediction actually consists of two people that have been merged together. This is because the detector did not find the background person. Second row: Our initial semantic segmentation identified the table, but it is not there in the Instance Segmentation. This is because there was no "table detection" to associate these pixels with. Using heuristics, we could propose additional detections in cases like these. However, we have not done this in our work. Third row: A difficult case where we have segmented most of the people. However, sometimes two people instances are joined together as one person instance. This problem is because we do not have a detection for each person in the image. Fourth row: Due to our initial semantic segmentation, we have not been able to segment the green person and table correctly. Fifth row: We have failed to segment a bird although it was detected. Sixth row: The occluding cows, which all appear similar, pose a challenge, even with our shape priors. The first column shows the input image, and the results of our object detector which are another input to our network. Best viewed in colour.
MNC [12] Ours Ground truth Figure 9: Comparison to MNC [12] The above examples emphasise the advantages in our method over MNC [12]. Unlike proposal-based approaches such as MNC, our method can handle false-positive detections, poor bounding box localisation, reasons globally about the image and also produces more precise segmentations due to the initial semantic segmentation module which includes a differentiable CRF. Row 1 shows a case where MNC, which scores segment-based proposals, is fooled by a false-positive detection and segments an imaginary human (yellow segment). Our method is robust to falsepositive detections due to the initial semantic segmentation module which does not have the same failure modes as the detector. Rows 2, 3 and 4 show how MNC [12] cannot deal with poorly localised bounding boxes. The horizontal boundaries of the red person in Row 2, and light-blue person in Row 4 correspond to the limits of the proposal processed by MNC. Our method, in contrast, can segment "outside the detection bounding box" due to the global instance unary potential (Eq. 4). As MNC does not reason globally about the image, it cannot handle cases of overlapping bounding boxes well, and produces more instances than there actually are. The first column shows the input image, and the results of our object detector which are another input to our network. MNC does not use these detections, but does internally produce box-based proposals which are not shown. Best viewed in colour.
Input image MNC [12] Ours Ground truth Figure 9 continued: Comparison to MNC [12] The above examples show that our method produces more precise segmentations than MNC, that adhere to the boundaries of the objects. However, in Rows 3, 4 and 5, we see that MNC is able to segment instances that our method misses out. In Row 3, our algorithm does not segment the baby, although there is a detection for it. This suggests that our shape prior which was formulated to overcome such occlusions could be better. As MNC processes individual instances, it does not have a problem with dealing with small, occluding instances. In Row 4, MNC has again identified a person that our algorithm could not. However, this is because we did not have a detection for this person. In Row 5, MNC has segmented the horses on the right better than our method. The first column shows the input image, and the results of our object detector which are another input to our network. MNC does not use these detections, but does internally produce box-based proposals which are not shown. We used the publicly available code, models and default parameters of MNC to produce this figure. Best viewed in colour. Figure 10: Comparison to FCIS [31] The above images compare our method to the concurrent work, FCIS [31], which was trained on COCO [35] and won the COCO 2016 challenge. Unlike proposal-based methods such as FCIS, our method can handle false-positive detections and poor bounding-box localisation. Furthermore, as our method reasons globally about the image, one pixel can only be assigned to a single instance, which is not the case with FCIS. Our method also produces more precise segmentations, as it includes a differentiable CRF, and it is based off a semantic segmentation network. The results of FCIS are obtained from their publicly available results on the COCO test set (https://github.com/daijifeng001/ TA-FCN). Note that FCIS is trained on COCO, and our model is trained on Pascal VOC which does not have as many classes as COCO, such as "umbrella" and "suitcase" among others. As a result, we are not able to detect these objects. The first column shows the input image, and the results of our object detector which are another input to our network. FCIS does not use these detections, but does internally produce proposals which are not shown. Best viewed in colour. A. Detailed results on the VOC dataset Figure 12 shows a visualisation of the AP r obtained by our method for each class across nine different thresholds. Each "column" of Fig. 12 corresponds to the AP r for each class at a given IoU threshold. It is therefore an alternate representation for the results tables (Tables 7 to 9). We can see that our method struggles with classes such as "bicycle", "chair", "dining table" and "potted plant". This may be explained by the fact that current semantic segmentation systems (including ours) struggle with these classes. All recent methods on the Pascal VOC leaderboard 2 obtain an IoU for these classes which is lower than the mean IoU for all classes. In fact the semantic segmentation IoU for the "chair" class is less than half of the mean IoU for all the classes for 16 out of the 20 most recent submissions on the VOC leaderboard at the time of writing. Tables 7 to 9 show per-class instance segmentation results on the VOC dataset, at IoU thresholds of 0.9, 0.7 and 0.5 respectively. At an IoU threshold of 0.9, our method achieves the highest AP r for 16 of the 20 object classes. At the threshold of 0.7, we achieve the highest AP r in 15 classes. Finally, at an IoU threshold of 0.5, our method, MPA 3-scale [37] and PFN [33] each achieve the highest AP r for 6 categories.
B. Detailed results on the SBD dataset
Once again, we show a visualisation of the AP r obtained by our method for each class across nine different thresholds (Fig. 13). The trend is quite similar to the VOC dataset in that our algorithm struggles on the same object classes ("chair", "dining table", "potted plant", "bottle"). Note that our AP r for the "bicycle" class has improved compared to the VOC dataset. This is probably because the VOC dataset has more detailed annotations. In the VOC dataset, each spoke of a bicycle's wheel is often labelled, whilst in SBD, the entire wheel is labelled as a single circle with the "bicycle" label. Therefore, the SBD dataset's coarser labelling makes it easier for an algorithm to perform well on objects with fine details. Table 6 shows our mean AP r over all classes at thresholds ranging from 0.5 to 0.9. Our AP r at 0.9 is low compared to the result which we obtained on the VOC dataset. This could be for a number of reasons: As the SBD dataset is not as finely annotated as the VOC dataset, it might not be suited for measuring the AP r at such high thresholds. Additionally, the training data is not as good for training our system which includes a CRF and is therefore able to delineate sharp boundaries. Finally, as the SBD dataset has 5732 validation images (compared to the 1449 in VOC), it leaves less data for pretraining our initial semantic segmen-2 http://host.robots.ox.ac.uk:8080/leaderboard/ displaylb.php?challengeid=11&compid=6 tation module. This may hinder our network in being able to produce precise segmentations. Tables 10 and 11 show per-class instance segmentation results on the SBD dataset, at IoU thresholds of 0.7 and 0.5 respectively. We can only compare results at these two thresholds since these are the only thresholds which other work has reported. Figure 12: A visualisation of the AP r obtained for each of the 20 classes on the VOC dataset, at nine different IoU thresholds. The x-axis represents the IoU threshold, and the y-axis each of the Pascal classes. Therefore, each "column" of this figure corresponds to the AP r per class at a particular threshold, and is thus an alternate representation to the results tables. Best viewed in colour. Figure 13: A visualisation of the AP r obtained for each of the 20 classes on the SBD dataset, at nine different IoU thresholds. The x-axis represents the IoU threshold, and the y-axis each of the Pascal classes. Therefore, each "column" of this figure corresponds to the AP r per class at a particular threshold, and is thus an alternate representation to the results tables. Best viewed in colour.
Figure 1 :
1Object detection (a) localises the different people, but at a coarse, bounding-box level. Semantic segmentation (b) labels every pixel, but has no notion of instances. Instance segmentation (c) labels each pixel of each person uniquely.
Figure 11 :
11Sample results on the Cityscapes dataset The above images show how our method can handle the large numbers of instances present in the Cityscapes dataset. Unlike other recent approaches, our algorithm can deal with objects that are not continuous -such as the car in the first row which is occluded by a pole. Best viewed in colour.
[
Table 1 :
1The effect of the different CRF unary potentials, and end-to-end training with them, on the VOC 2012 Validation Set.AP r
AP r
vol
match
IoU
0.5
0.7
0.9
Box Term
(piecewise)
60.0 47.3 21.2
54.9
42.6
Box+Global
(piecewise)
59.1 46.1 23.4
54.6
43.0
Box+Global+Shape
(piecewise)
59.5 46.4 23.3
55.2
44.8
Box Term
(end-to-end)
60.7 47.4 24.6
56.2
46.9
Box+Global
(end-to-end)
60.9 48.1 25.5
56.7
47.1
Box+Global+Shape
(end-to-end)
61.7 48.6 25.1
57.5
48.3
Table 2 :
2Comparison of Instance Segmentation performance to recent methods on the VOC 2012 Validation SetMethod
AP r
AP r
vol
0.5
0.6
0.7
0.8
0.9
SDS [19]
43.8 34.5 21.3 8.7
0.9
-
Chen et al. [8]
46.3 38.2 27.0 13.5 2.6
-
PFN [33]
58.7 51.3 42.5 31.2 15.7
52.3
Arnab et al. [3]
58.3 52.4 45.4 34.9 20.1
53.1
MPA 1-scale [37] 60.3 54.6 45.9 34.3 17.3
54.5
MPA 3-scale [37] 62.1 56.6 47.4 36.1 18.5
56.5
Ours
61.7 55.5 48.6 39.5 25.1
57.5
Table 3 :
3Comparison of Instance Segmentation performance on the SBD DatasetMethod
AP r
AP r
vol
match
0.5
0.7
IoU
SDS [19]
49.7 25.3
41.4
-
MPA 1-scale [37]
55.5
-
48.3
-
Hypercolumn
Table 4 :
4Semantic Segmentation performance before and after finetuning for Instance Segmentation on the VOC and SBD Validation SetsDataset
Mean IoU [%] before
Instance finetuning
Mean IoU [%] after
Instance finetuning
VOC
74.2
75.1
SBD
71.5
72.5
Table 5 :
5Results on Cityscapes Test Set. Evaluation metrics and results of competing methods obtained from the online server. The "AP" metric of Cityscapes is similar to our AP r vol metric.Method
AP AP at 0.5 AP 100m AP 50m
Ours
20.0
38.8
32.6
37.6
SAIS [21]
17.4
36.7
29.3
34.0
DWT [4]
15.6
30.0
26.2
31.8
InstanceCut [24]
13.0
27.9
22.1
26.1
Graph Decomp. [29] 9.8
23.2
16.8
20.3
RecAttend [43]
9.5
18.9
16.8
20.9
Pixel Encoding [49]
8.9
21.1
15.3
16.7
R-CNN [9]
4.6
12.9
7.7
10.3
Table 6 :
6Comparison of Instance Segmentation performance at multiple AP r thesholds on the VOC 2012 Validation SetMethod
AP r
AP r
vol
0.5
0.6
0.7
0.8
0.9
Ours (piecewise)
59.1 51.9 42.1 29.4 12.0
52.3
Ours (end-to-end ) 62.0 54.0 44.8 32.3 13.8
55.4
Table 7 :
7Comparison of mean AP r , achieved by different published methods, at an IoU threshold of 0.9, for all twenty classes in the VOC dataset.Method
Mean
AP r
(%)
aero-
plane
bike
bird
boat
bot-
tle
bus
car
cat
chair
cow
table
dog
horse mbike
per-
son
plant sheep
sofa
train
tv
Our method
25.1
56.6
0.03
36.8
14.4
9.9
39.0
17.2
47.1
1.3
29.0
9.5
47.2
29.8
20.0
14.8
2.3
25.9
23.8
45.7
32.3
MPA 3-scale [37]
Table 8 :
8Comparison of mean AP r , achieved by different published methods, at an IoU threshold of 0.7, for all twenty classes in the VOC dataset.Method
Mean
AP r
(%)
aero-
plane
bike
bird
boat
bot-
tle
bus
car
cat
chair
cow
table
dog
horse mbike
per-
son
plant sheep
sofa
train
tv
Our method
48.6
69.6
1.4
68.2
45.1
25.2
61.1
38.7
72.1
11.2
56.3
Chen et al.3]
45.4
68.9
0.84
65.1
38.3
26.3
64.7
31.8
72.7
6.7
45.4
32.9
67.9
60.0
63.7
41.1
13.4
43.9
41.1
74.6
48.1
PFN [33]
42.5
68.5
5.6
60.4
34.8
14.9
61.4
19.2
78.6
4.2
51.1
28.2
69.6
60.7
60.5
26.5
9.8
35.1
43.9
71.2
45.6
[8]
27.0
40.8
0.07
40.1
16.2
19.6
56.2
26.5
46.1
2.6
25.2
16.4
36.0
22.1
20.0
22.6
7.7
27.5
19.5
47.7
46.7
SDS [19]
21.3
17.8
0
32.5
7.2
19.2
47.7
22.8
42.3
1.7
18.9
16.9
20.6
14.4
12.0
15.7
5.0
23.7
15.2
40.5
51.4
Table 9 :
9Comparison of mean AP r , achieved by different published methods, at an IoU threshold of 0.5, for all twenty classes in the VOC dataset. Arnab et al.Chen et al.Method
Mean
AP r
(%)
aero-
plane
bike
bird
boat
bot-
tle
bus
car
cat
chair
cow
table
dog
horse mbike
per-
son
plant sheep
sofa
train
tv
Our method
61.7
80.2
19.3
76.4
69.0
35.3
74.5
50.8
84.5
22.8
70.9
43.3
87.7
71.3
76.2
65.6
37.2
61.3
50.3
90.5
67.2
MPA 3-scale [37]
62.1
79.7
11.5
71.6
54.6
44.7
80.9
62.0
85.4
26.5
64.5
46.6
87.6
71.7
77.9
72.1
48.8
57.4
48.8
78.9
70.8
MPA 1-scale [37]
60.3
79.2
13.4
71.6
59.0
41.5
73.8
52.3
87.3
23.3
61.2
42.5
83.1
70.0
77.0
67.6
50.7
56.0
45.9
80.0
70.5
[3]
58.4
80.4
7.9
74.4
59.8
32.7
76.6
39.6
84.6
19.3
62.7
44.1
81.0
74.7
72.0
58.6
32.0
59.6
50.5
87.4
68.4
PFN [33]
58.7
76.4
15.6
74.2
54.1
26.3
73.8
31.4
92.1
17.4
73.7
48.1
82.2
81.7
72.0
48.4
23.7
57.7
64.4
88.9
72.3
[8]
46.3
63.6
0.3
61.5
43.9
33.8
67.3
46.9
74.4
8.6
52.3
31.3
63.5
48.8
47.9
48.3
26.3
40.1
33.5
66.7
67.8
SDS [19]
43.8
58.8
0.5
60.1
34.4
29.5
60.6
40.0
73.6
6.5
52.4
31.7
62.0
49.1
45.6
47.9
22.6
43.5
26.9
66.2
66.1
Table 10 :
10Comparison of mean AP r , achieved by different published methods, at an IoU threshold of 0.7, for all twenty classes in the SBD dataset.Method
Mean
AP r
(%)
aero-
plane
bike
bird
boat
bot-
tle
bus
car
cat
chair
cow
table
dog
horse mbike
per-
son
plant sheep
sofa
train
tv
Our method
44.8
69.0
27.4
52.7
26.4
22.4
70.3
46.0
74.7
9.6
46.8
16.9
71.6
48.4
46.3
40.3
14.8
47.6
36.5
69.7
58.2
IIS sp, rescore
[30]
43.3
61.9
35.1
44.4
26.4
29.6
74.0
48.7
66.8
10.9
48.4
13.6
64.0
53.0
46.8
33.0
19.0
51.0
23.7
62.2
53.9
IIS raw [30]
38.7
61.8
31.5
42.0
22.0
22.7
72.4
44.8
65.4
7.2
37.6
10.4
60.4
39.6
41.9
32.5
12.0
40.9
19.9
58.8
50.8
Table 11 :
11Comparison of mean AP r , achieved by different published methods, at an IoU threshold of 0.5, for all twenty classes in the SBD dataset.Method
Mean
AP r
(%)
aero-
plane
bike
bird
boat
bot-
tle
bus
car
cat
chair
cow
table
dog
horse mbike
per-
son
plant sheep
sofa
train
tv
Our method
62.0
80.3
52.8
68.5
47.4
39.5
79.1
61.5
87.0
28.1
68.3
35.5
86.1
73.9
66.1
63.8
32.9
65.3
50.4
81.4
71.4
IIS sp, rescore
[30]
63.6
79.2
67.9
70.0
47.9
45.3
81.6
68.8
84.1
30.4
65.5
31.8
83.6
75.5
74.5
66.6
37.7
70.6
44.7
77.7
68.7
IIS raw [30]
60.1
77.3
65.3
65.5
42.5
35.4
80.3
62.2
83.9
27.2
61.6
32.4
82.3
70.9
71.4
63.1
31.3
63.6
44.9
78.3
62.4
https://github.com/daijifeng001/MNC
Acknowledgements We thank Bernardino Romera-Paredes and Stuart Golodetz for insightful discussions and feedback. This work was supported by the EPSRC, Clarendon Fund, ERC grant ERC-2012-AdG 321162-HELIOS, EPRSRC grant Seebibyte EP/M013774/1 and EPSRC/MURI grant EP/N019474/1.AppendixIn this supplementary material, we include more detailed qualitative and quantitative results on the VOC and SBD datasets. Furthermore, we also show the runtime of our algorithm.Figures 7 and 8show success and failure cases of our algorithm.Figure 9compares the results of our algorithm to the publicly available model for MNC[12].Figure 10compares our results to those of FCIS[31], concurrent work which won the COCO 2016 challenge.Figure 11presents some qualitative results on the Cityscapes dataset.Section A shows more detailed results on the VOC dataset.Figure 12shows a visualisation of our results at different AP r thresholds, andTables 7 to 9show per-class AP r results at thresholds of 0.5, 0.7 and 0.9.Section B shows more detailed results on the SBD dataset.Table 6shows our mean AP r results at thresholds from 0.5 to 0.9, whilstTables 10 and 11show per-class AP r results at thresholds of 0.7 and 0.5 respectively.Input imageSemantic SegmentationInstance Segmentation Ground truth
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| [
"https://github.com/daijifeng001/",
"https://github.com/daijifeng001/MNC"
] |
[
"High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs",
"High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs"
] | [
"Bernard Deconinck [email protected] \nDepartment of Applied Mathematics\nUniversity of Washington\nCampusBox 35242098195SeattleWAUSA\n",
"Olga Trichtchenko [email protected] \nDepartment of Mathematics\nUniversity College London\nGower Street LondonWC1E 6BTUK\n"
] | [
"Department of Applied Mathematics\nUniversity of Washington\nCampusBox 35242098195SeattleWAUSA",
"Department of Mathematics\nUniversity College London\nGower Street LondonWC1E 6BTUK"
] | [] | Generalizing ideas of MacKay, and MacKay and Saffman, a necessary condition for the presence of high-frequency (i.e., not modulational) instabilities of small-amplitude periodic solutions of Hamiltonian partial differential equations is presented, entirely in terms of the Hamiltonian of the linearized problem. With the exception of a Krein signature calculation, the theory is completely phrased in terms of the dispersion relation of the linear problem. The general theory changes as the Poisson structure of the Hamiltonian partial differential equation is changed. Two important cases of such Poisson structures are worked out in full generality. An example not fitting these two important cases is presented as well, using a candidate Boussinesq-Whitham equation. | 10.3934/dcds.2017055 | [
"https://arxiv.org/pdf/1505.03688v1.pdf"
] | 16,236,804 | 1505.03688 | 8e6bfea911e1761230f733879028a5578a956784 |
High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs
May 15, 2015
Bernard Deconinck [email protected]
Department of Applied Mathematics
University of Washington
CampusBox 35242098195SeattleWAUSA
Olga Trichtchenko [email protected]
Department of Mathematics
University College London
Gower Street LondonWC1E 6BTUK
High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs
May 15, 2015
Generalizing ideas of MacKay, and MacKay and Saffman, a necessary condition for the presence of high-frequency (i.e., not modulational) instabilities of small-amplitude periodic solutions of Hamiltonian partial differential equations is presented, entirely in terms of the Hamiltonian of the linearized problem. With the exception of a Krein signature calculation, the theory is completely phrased in terms of the dispersion relation of the linear problem. The general theory changes as the Poisson structure of the Hamiltonian partial differential equation is changed. Two important cases of such Poisson structures are worked out in full generality. An example not fitting these two important cases is presented as well, using a candidate Boussinesq-Whitham equation.
Introduction
It is expected that much of the dynamics of the small-amplitude solutions of a partial differential equation (PDE), including their stability or instability, is dictated by the study of a linearized (about a trivial solution, say u = 0) problem. In this article, we focus specifically on the spectral stability of periodic traveling-wave solutions of Hamiltonian PDEs as they bifurcate away from a trivial solution. Our work follows earlier ideas of MacKay [35] and MacKay and Saffman [36]. We start from an autonomous Hamiltonian system of PDEs [2], i.e.,
u t = J δH δu .(1)
Here and throughout, indices involving x or t denote partial derivatives. Further, u = (u 1 (x, t), . . . , u M (x, t)) T is an M -dimensional vector function defined in a suitable function space, and J is a Poisson operator [2,3]. More details and examples are given below. Finally, H = D H(u, u x , . . .)dx is the Hamiltonian, whose density H depends on u and its spatial derivatives, defined for x ∈ D. We consider only the stability of periodic solutions, thus D is any interval of length L, the period. Note that for some of our examples H will depend on spatial derivatives of u of arbitrary order.
To investigate the stability of traveling wave solutions of this system, we reformulate (1) in a frame moving with speed c, using the transformationx = x − ct,t = t, and considering solutions u(x,t) = U (x) (successively omitting hats). This leads to
u t − cu x = J δH δu ⇔ u t = J δH c δu ,(2)
for a modified Hamiltonian H c . Traveling wave solutions are solutions of the ordinary differential system
− cU x = J δH δU ⇔ 0 = J δH c δU .(3)
Thus if J is invertible, traveling waves are stationary points of the Hamiltonian H c . The system (3) typically has the zero (trivial) solution for a range of c values. The small-amplitude solutions whose stability we investigate bifurcate away from these trivial zero-amplitude solutions at special values of the speed parameter c, as is schematically shown in Fig. 1. It is our goal to see to what extent anything can be said about the stability of the small-amplitude solutions (with amplitudes in the shaded regions of Fig. 1) from knowledge of the zero-amplitude solutions at the bifurcation point. An outline of the steps in this process is as follows.
1. Quadratic Hamiltonian. A linear system of equations is obtained by linearizing the system (3) around the zero solution: let u = v + o( ) and omit terms of order o( ). Alternatively, if J is independent of u and its spatial derivatives, one may expand the Hamiltonian H c as a function of and retain its quadratic terms. The resulting Hamiltonian H 0 c of the linearized system is the starting point for the next steps.
2. Dispersion relation. The linearized system has constant coefficients and is easily solved using Fourier analysis. The dispersion relation F (ω, k) = 0 governs the time dependence of the solutions. It is obtained by investigating solutions whose spatial and temporal dependence is proportional to exp(ikx − iωt). Here F (ω, k) = 0 is of degree M in ω. It is a fundamental assumption of our approach that all solutions ω j (k) (j = 1, . . . , M ) of F (ω, k) = 0 are real for k ∈ R. The dispersion relation can be expressed entirely in terms of the coefficients appearing in the quadratic Hamiltonian H 0 c . For periodic systems of period L, the values of k are restricted to be of the form 2πN/L, N ∈ Z.
3. Bifurcation branches. The values of the phase speed c j = ω j /k for which nontrivial solutions bifurcate away from the zero-amplitude solution are determined by the condition that the zero solution is not the unique solution to the Fourier transformed problem. In effect, this is the classical bifurcation condition that a Jacobian is singular. This simple calculation determines the bifurcation branch starting points explicitly in terms of the different solutions to the dispersion relation. In what follows, we follow the first branch, starting at c 1 , without loss of generality.
It is assumed that only a single non-trivial bifurcation branch emanates from a bifurcation point. Although more general cases can be incorporated, we do not consider them here. Further, we fix the period of the solutions on the bifurcation branch (usually to 2π). Other choices can be made. Instead of varying the amplitude as a function of the speed for fixed period, one could fix the speed and vary the period, etc. The methods presented can be redone for those scenarios in a straightforward fashion.
4. Stability spectrum. The spectrum of the linear operator determining the spectral stability of the zero solution at the bifurcation point on the first branch is calculated.
Since this spectral problem has constant coefficients, this calculation can be done explicitly. Again, this is done entirely in terms of the dispersion relation of the problem. Using a Floquet decomposition (see [15,31]), the spectrum is obtained as a collection of point spectra, parameterized by the Floquet exponent µ ∈ (−π/L, π/L]. Due to the reality of the branches of the dispersion relation, the spectrum is confined to the imaginary axis. In other words, the zero-amplitude solutions are spectrally stable. The use of the Floquet decomposition allows for the inclusion of perturbations that are not necessarily periodic with period L. Instead, the perturbations may be quasiperiodic with two incommensurate periods, subharmonic (periodic, but with period an integer multiple of L), or spatially localized [15,25,31]. 5. Collision condition. Given the explicit expression for individual eigenvalues λ, it is easy to find the conditions for which eigenvalues corresponding to different parameters (Floquet exponent, branch number of the dispersion relation, etc.) coincide on the imaginary axis. This is referred to as the collision condition. Once again, it is given entirely in terms of the dispersion relation.
It is a consequence of the Floquet theorem [11] that collisions need to be considered only for spectral elements corresponding to the same value of the Floquet exponent since the subspaces of eigenfunctions for a fixed Floquet exponent are invariant under the flow of the linearized equation.
6. Krein signature. Having obtained the stability spectrum at the starting point of the bifurcation branches, we wish to know how the spectrum evolves as we move up a bifurcation branch. One tool to investigate this is the Krein signature [32,33,34,35,38] collision of such eigenvalues does not result in the creation of unstable modes. In other words, it is a necessary condition for collisions to lead to instability that the Krein signature of the colliding eigenvalues is different. This scenario is illustrated in Fig. 2. That figure also illustrates the quadrifold symmetry of the stability spectrum of the solution of a Hamiltonian system: for each eigenvalue λ ∈ C, λ * , −λ and −λ * are also eigenvalues.
Here λ * denotes the complex conjugate of λ. It should be noted that the occurrence of a collision is required for eigenvalues to leave the imaginary axis, due to the quadrifold symmetry of the spectrum.
Thus we calculate the Krein signature of any coinciding eigenvalues, obtained in Step 5. If these Krein signatures are equal, the eigenvalues will remain on the imaginary axis as the amplitude is increased. Otherwise, the eigenvalues may leave the imaginary axis, through a so-called Hamiltonian Hopf bifurcation [45], resulting in instability. Thus we establish a necessary condition for the instability of periodic solutions of small amplitude. The Krein signature condition cannot be expressed entirely in terms of the dispersion relation, and the coefficients of H 0 c are required as well. Please refer to the next two sections for details.
Although all calculations are done for the zero-amplitude solutions at the starting point of a bifurcation branch, the continuous dependence of the stability spectrum on the parameters in the problem [26], including the velocity of the traveling wave or the amplitude of the solutions, guarantees that the stability conclusions obtained persist for solutions of small amplitude. Thus meaningful conclusions about solutions in the shaded regions of the bifurcation branches of Fig. 1 are reached, despite the fact that we are unable to say anything about the size of the maximal amplitude for which these conclusions are valid.
Remarks.
• Throughout this introduction, and in fact throughout much of this manuscript, our emphasis is on generality and usability as opposed to rigor. As a consequence some of the statements made above are necessarily vague: more precise statements would limit the generality aimed for. Within the context of more specific examples, more precision may be possible. Along the same lines, we have limited ourselves to the most generic case of two eigenvalues colliding on the imaginary axis. Lastly, since we are assuming an initial (i.e., for a starting value of 0 for the amplitude parameter) situation that is neutrally stable, our only interest is in collisions on the imaginary axis. If eigenvalues are present off the imaginary axis for the zero-amplitude solution, such solutions are already spectrally unstable, and the continuous dependence of the spectrum on its parameters [26] guarantees instability for solutions of small amplitude as a consequence.
• If eigenvalues collide at the origin, their Krein signature is zero (this follows immediately from its definition) and no conclusion can be drawn about whether or not colliding eigenvalues leave the imaginary axis or not. Because of this, the methods discussed here allow only for the study of so-called high-frequency instabilities. Indeed, if eigenvalues collide away from the origin, the unstable perturbations do not only grow in magnitude exponentially, they also display an oscillation of frequency ω that is equal to the non-zero imaginary part of the colliding eigenvalues. In this sense high frequency is more accurately described as non-zero frequency. We use the high-frequency name to distinguish the instabilities investigated here from the modulational instability [54], which is a consequence of eigenvalues colliding at the origin. Further, as seen in the examples below, often the presence of one high-frequency instability is accompanied by a sequence of such instabilities with increasing frequencies tending to infinity.
The study of the modulational instability requires a different set of techniques, see for instance [4,47,54] for different classical approaches. More recently, a general framework was developed by Bronski, Johnson and others (see e.g. [9,27]). Collisions at the origin are common, since any Lie symmetry of the underlying problem gives rise to a zero eigenvalue in the stability problem [23,31]. For this same reason, such collisions typically involve more than two eigenvalues (for instance, the original Benjamin-Feir instability involves four [4,8]), which is one of the reasons why their treatment is often complicated: its analysis is not only technical but also involves tedious calculations, as exemplified recently in a paper by Hur and Johnson [27] on the modulational instability for periodic solutions of the Whitham equation.
• Following the Floquet decomposition, we use eigenfunctions given by a single Fourier mode since the linear stability problem for the zero-amplitude solutions has constant coefficients. Lastly, the calculation of the Krein signature involves only the finite-dimensional eigenspace of the eigenvalue under consideration. Thus, in essence, all the calculations done in this paper are finite dimensional, as they are in [36], for instance. As was noted there, for equations with real-valued solutions, it is necessary to consider the eigenfunctions corresponding to Floquet exponent µ and −µ simultaneously, since the eigenfunctions with −µ are the complex conjugates of those with µ.
• We call an equation dispersive if all branches of its dispersion relation ω(k) are real valued for k ∈ R. It is easy to see that there exist linear, constant-coefficient Hamiltonian systems that are not dispersive. An explicit example is
q t = q xx , p t = −p xx .(4)
This (admittedly bizarre) system is Hamiltonian with canonical Poisson structure
J = 0 1 −1 0 ,(5)
and Hamiltonian H = − 2π 0 q x p x dx. The branches of its dispersion relation are given by
ω 1,2 (k) = ∓ik 2 .
It is an interesting question whether there exist dispersive systems that are not Hamiltonian. One may be tempted to consider an example like
q t = aq x , p t = bp x .(6)
For a, b ∈ R this system is dispersive (ω 1 = −ak, ω 2 = −bk), but it is not Hamiltonian with canonical structure. However, allowing for the noncanonical structure
J = ∂ x 0 0 ∂ x ,(7)
the system is easily found to have the Hamiltonian H = 1 2 2π 0 (aq 2 +bp 2 )dx. We conjecture that no systems exist that are dispersive but not Hamiltonian. As demonstrated by the example above, the question is difficult to analyze, since different forms of the Poisson operator have to be allowed.
• The investigation of the Hamiltonian structure of the zero-amplitude solution follows earlier work by Zakharov, Kuznetsov, and others. A review is available in [53]. The stability of the trivial solution is also investigated there, with stability criteria given entirely in terms of the branches of the dispersion relation. Using canonical perturbation theory, Hamiltonians containing higher-than-quadratic terms are considered. This is used to consider the impact of nonlinear effects on the dynamics of the trivial solution. At this point, connections to resonant interaction theory [5,24,40] become apparent, as they will in what follows. Although the physical reasoning leading to resonant interaction theory and that leading to the criteria established below is different, it is clear that connections exist.
Motivating example
Our investigations began with the study of the so-called Whitham equation [49], [50, page 368]. The equation is usually posed on the whole line, for which the equation satisfied by u(x, t) is
u t + N (u) + ∞ −∞ K(x − y)u y (y, t)dy = 0.(8)
The term N (u) denotes the collection of all nonlinear terms in the equation. It is assumed that lim →0 N ( u)/ = 0. The last term encodes the dispersion relation of the linearized Whitham equation. The kernel K(x) is the inverse Fourier transform of the phase speed c(k), where c(k) = ω(k)/k, with ω(k) the dispersion relation. Here c(k) is assumed to be real valued and nonsingular for k ∈ R. Thus
K(x) = 1 2π − ∞ −∞ c(k)e ikx dk,(9)
where − denotes the principal value integral. Depending on c(k), this equation may have to be interpreted in a distributions sense [20,43]. Letting u ∼ exp(ikx−ω(k)t), it is a straightforward calculation to see that the dispersion relation of the linear Whitham equation is ω(k). In fact, the linear Whitham equation is easily seen to be a rewrite of the linear evolution equation [1]
u t = −iω(−i∂ x )u,(10)
where ω(−i∂ x ) is a linear operator with odd symbol ω(k): ω(k) = −ω(−k), the dispersion relation considered. Indeed, letting ω(−i∂ x ) act on
u(x, t) = 1 2π ∞ −∞ e ikxû (k, t)dk,(11)
and replacingû(k, t) byû
(k, t) = ∞ −∞ e −iky u(y, t)dy,(12)
the linear part of (8) is obtained after one integration by parts. We restrict our considerations to odd dispersion relations, to ensure the reality of the Whitham equation.
One of Whitham's reasons for writing down the Whitham equation [49,50] was to describe waves in shallow water (leading to the inclusion of a KdV-type nonlinearity N (u) ∼ uu x ) that feel the full dispersive response of the one-dimensional water wave problem (without surface tension), for which ω 2 (k) = gk tanh(kh), (13) with g the acceleration of gravity and water depth h. It is common to choose c(k) = ω 1 (k)/k > 0 in (8), so that ω 1 (k) is the root of (13) with the same sign as k. In what follows, we refer to this choice as the Whitham equation. The stability of periodic traveling wave solutions of the Whitham equation has received some attention recently. Notable are [19], where the focus is on solitary waves, and [27], where the modulational instability of small-amplitude periodic solutions is emphasized, although the main result for the high-frequency instabilities discussed in Example 3.2 is included as well. Most recently, the spectral stability of periodic solutions of the Whitham equation was examined in [41]. The goal of considering an equation like (8) as opposed to the Korteweg-de Vries equation or other simpler models is to capture as much of the dynamics of the full water wave problem as possible, without having to cope with the main difficulties imparted by the Euler water wave problem [46] (e.g., it is a nonlinear free boundary-value problem, the computation of its traveling-wave solutions is a nontrivial task, etc). One of the important aspects of the dynamics of a nonlinear problem is the (in)stability of its traveling wave solutions. It was shown explicitly in [17] that periodic traveling wave solutions of the one-dimensional Euler water wave problem are spectrally unstable for all possible values of their parameters h, g, amplitude, and wave period. The nature of the instabilities depends on the value of these parameters. As is well known, waves in deep water are susceptible to the Benjamin-Feir or modulational instability (see [54] for a review). In addition, waves in both deep and shallow water of all non-zero amplitudes are unstable with respect to high-frequency perturbations: these are perturbations whose growth rates do not have a small imaginary part, resulting in oscillatory behavior in time, independent of the spatial behavior of the unstable modes. The work in [41] does not reveal any high-frequency instabilities for solutions of small amplitude in water that is shallow, in the context of the Whitham equation. Thus an important aspect of the Euler water wave dynamics is absent from (8). We will provide an analytical indication that the Whitham equation misses the presence of these instabilities, while explaining why they are missed. This explanation leads to a way to address this problem. For suitable N (u), the Whitham equation (8) is a Hamiltonian system. In fact, for our considerations, it matters only that the linearized Whitham equation is Hamiltonian. The Lagrangian structure with the dispersion relation given by ω 1 (k) as in (13) was already written down by Whitham in [48], from which the Hamiltonian structure easily follows. Explicitly, for the linearized Whitham equation posed on the whole line, for any odd ω(k) we have
H = − 1 2 ∞ −∞ ∞ −∞ K(x − y)u(x)u(y)dxdy,(14)
with J = ∂ x . Then
u t = ∂ x δH δu .(15)
If instead the linearized equation is posed with periodic boundary conditions u(x+L, t) = u(x, t), it follows immediately from (10) that we have
u t + L/2 −L/2 K(x − y)u(y)dy = 0,(16)
where we have used a Fourier series instead of a Fourier transform. Further,
K(x) = 1 L ∞ j=−∞ c(k j )e ik j x ,(17)
and k j = 2πj/L, j ∈ Z. The Hamiltonian formulation for the periodic Whitham equation (16) is also given by (15), but with This equation is shown to at least satisfy the same necessary condition for the presence of high-frequency instabilities as the water wave problem, and these high-frequency instabilities originate from the same points on the imaginary axes as they do for the Euler equations. However, it is easily seen that this equation is ill posed for solutions that do not have zero average. As such it is a poor candidate to replace the Whitham equation (8) as a shallow-water equation with the correct dispersive behavior.
H = − 1 2 L/2 −L/2 L/2 −L/2 K(x − y)u(x)u(y)dxdy.(18)
Remark. If ω(k) is not odd, then the function u solving the linear Whitham equation (10) is necessarily complex. The problem is Hamiltonian:
u t = −i δH δu * ,(19)
where * denotes the complex conjugate. The Poisson structure is
J = 0 −i i 0 ,(20)
although usually one writes only the first of the two evolution equations, omitting the equation that is the complex conjugate of (19). The Hamiltonian is given by
H = 2π 0 u * ω(−i∂ x )udx.(21)
A real formulation in terms of the real and imaginary parts of u is possible as well (using the canonical transformation u = (q + ip)/ √ 2), resulting in a canonical Hamiltonian structure. The linear Whitham equation (19) is often rewritten in Fourier space using the discrete Fourier transform, due to the periodic boundary conditions. This leads to the Hamiltonian
H = ∞ n=−∞ ω(n)z n z * n .
Scalar Hamiltonian PDEs
In this section, we investigate the stability of 2π-periodic traveling wave solutions of Hamiltonian systems of the form
u t = ∂ x δH δu ,(22)
where u(x, t) is a scalar real-valued function. Thus J = ∂ x . Since this Poisson operator is singular, all equations of this form conserve the quantity 2π 0 udx, which is the Casimir for this Poisson operator. Systems of this form include the Korteweg-de Vries equation [22,52] and its many generalizations, the Whitham equation (8), and many others. As mentioned above, our only interest is in the linearization of these equations around their trivial solution. We write the quadratic part H 0 of H as
H 0 = − 1 2 2π 0 ∞ n=0 α n u 2 nx dx,(23)
where the coefficients α n ∈ R. As before, indices on u denote partial derivatives. Specifically, u nx denotes ∂ n x u. (23) is finite, and all but a few of the coefficients α n are nonzero. For the Whitham equation (8), the number of nonzero terms is infinite, but convergence is easily established. Note that (23) is the most general form of a quadratic Hamiltonian depending on a single function. Indeed, a term in H 0 containing u mx u nx , with m and n positive integers, may be reduced to a squared term using integration by parts.
For most examples, the number of terms in
Using the notation (23), the linearized equation is
u t = − ∞ n=0 (−1) n α n u (2n+1)x .(24)
We proceed with the six steps outlined in the introduction.
H 0 c = c 2 2π 0 u 2 dx − 1 2 2π 0 ∞ n=0 α n u 2 nx dx.(25)
2. Dispersion relation. For equations of the form (22), the dispersion relation has only a single branch:
ω(k) = ∞ n=0 α n k 2n+1 .(26)
The absence of even powers of k in (26) is due to our imposition that (24) is a conservative equation, i.e., there is no dissipation. All integers are allowable k values, since we have equated the period to be 2π. The equation (24) may be written as
u t = −iω(−i∂ x )u.(27)
3. Bifurcation branches. Since (22) is scalar, only one branch can bifurcate away from the trivial solution. To find the corresponding value of c, we write (24) in a moving frame as
u t − cu x = iω(i∂ x )u.(28)
This equation has its own dispersion relation given by
Ω(k) = ω(k) − ck,(29)
obtained by looking for solutions of the form u = exp(ikx − iΩt). Letting u = ∞ n=−∞ exp(inx)û n , it follows that ∂ tûn = −iΩ(n)û n . Thus a nonzero stationary solution may exist provided Ω(N ) = 0, for N ∈ N, N = 0. We have used the oddness of Ω(N ) to restrict to strictly positive values of N . Thus the starting point of the bifurcation branch in the (speed, amplitude)-plane is (c, 0), where c is determined by
c = ω(N ) N ,(30)
for any integer N > 0. Choosing N > 1 implies that the fundamental period of the solutions is not 2π, but 2π/N . In practice, we choose N = 1. A Fourier series approximation to the explicit form of the small-amplitude solutions corresponding to this bifurcation branch may be obtained using a standard Stokes expansion [44,50].
4. Stability spectrum. In order to compute the stability spectrum associated with the zero-amplitude solution at the start of the bifurcation branch, we let u(x, t) = U (x) exp(λt)+c.c., where c.c. denotes the complex conjugate of the preceding term. As usual, if any λ are found for which the real part is positive, the solution is spectrally unstable [31]. All bounded eigenfunctions U (x) may be represented as
U (x) = ∞ n=−∞ a n e i(n+µ)x ,(31)
where µ ∈ (−1/2, 1/2] is the Floquet exponent. Such a representation for U (x) is valid even for solutions on the bifurcation branch of nonzero amplitude [15]. Since (28) is a problem with constant coefficients, only a single term in (31) is required. We obtain
λ (µ) n = −iΩ(n + µ) = −iω(n + µ) + i(n + µ)c, n ∈ Z.(32)
As expected, all eigenvalues are imaginary and the zero-amplitude solution is neutrally stable. For a fixed value of µ, (32) gives a point set on the imaginary axis in the complex λ plane. As µ is varied in (−1/2, 1/2], these points trace out intervals on the imaginary axis. Depending on ω(k), these intervals may cover the imaginary axis.
5. Collision condition. The most generic scenarios for two eigenvalues given by (32) to collide are that (i) two of them are zero, and they collide at the origin, and (ii) two of them are equal, but nonzero. We ignore the first possibility, since the next step proves to be inconclusive for this case, as discussed in the introduction. The second possibility requires λ
(µ) n = λ (µ) m , for some m, n ∈ Z, m = n, fixed µ ∈ (−1/2, 1/2], and λ (µ) n , λ (µ) m = 0.
This may be rewritten as
ω(n + µ) − ω(m + µ) n − m = ω(N ) N , m, n ∈ Z, m = n and µ ∈ (−1/2, 1/2].(33)
This equation has an elegant graphical interpretation: the right-hand side is fixed by the choice of N , fixing the bifurcation branch in Step 3. It represents the slope of a line through the origin and the point (N, ω(N )) in the (k, ω) plane. The left hand side is the slope of a line in the same plane passing through the points (n + µ, ω(n + µ)) and (m + µ, ω(m + µ)), see Fig. 3.
Even though the graph of the dispersion relation admits parallel secant lines, this is not sufficient for a solution of (33), as it is required that their abscissas are an integer apart. Nevertheless, the graphical interpretation can provide good intuition for solving the collision condition, which typically has to be done numerically.
6. Krein signature. The Krein signature of an eigenvalue is the sign of the Hamiltonian H 0 c evaluated on the eigenspace associated with the eigenvalue. We are considering two simple eigenvalues colliding, thus the eigenspace for each eigenvalue consists of multiples of the eigenfunction only. To allow for eigenfunctions of the form a n exp(i(n + µ)x + λ (µ) n t)+c.c, which are not 2π-periodic (unless µ = 0), it is necessary to replace the integral in (23) with a whole-line average. More details on this process are found, for instance, in [17]. A simple calculation shows that the contribution to H 0 c from the (n, µ) mode is proportional to the ratio of Ω(n + µ)/(n + µ):
H 0 c | (n,µ) ∼ −|a n | 2 Ω(n + µ) n + µ .(34)
Other terms are present in the Hamiltonian density, but they have zero average. The sign of this expression is the Krein signature of the eigenvalue λ m to leave the imaginary axis for solutions of non-zero amplitude is that the signs of (34) with (n, µ) and (m, µ) are different, contingent on µ, m and n satisfying (33). Explicitly, this condition is Figure 3: The graphical interpretation of the collision condition (33). The solid curve is the graph of the dispersion relation ω(k). The slope of the dashed line in the first quadrant is the right-hand side in (33). The slope of the parallel dotted line is its left-hand side.
sign ω(n + µ) n + µ − c = sign ω(m + µ) m + µ − c .(35)ω(k) k (n + µ, ω(n + µ)) (m + µ, ω(m + µ)) (N, ω(N ))
Alternatively, the product of the left-hand side and the right-hand side should be negative. Using (33), (35) becomes
(n + µ)(m + µ) < 0,(36)
or, provided mn = 0, and using that µ ∈ (−1/2, 1/2],
nm < 0.(37)
Remarks.
• It is clear from (34) why our methods do not lead to any conclusions about collisions of eigenvalues at the origin. If λ (µ) n = 0, then Ω(n + µ) = 0, and the contribution to the Hamiltonian of such a mode vanishes. As a consequence, the associated Krein signature is zero.
• When the theory of [32] is restricted to the case of solutions of zero-amplitude, so as to recover the constant coefficient stability problem, the graphical stability criterion given there coincides with the one presented here.
We conclude our general considerations of this section with the following summary.
Assume that the linearization of the scalar Hamiltonian system (22) is dispersive (i.e., its dispersion relation ω(k) is real valued for k ∈ R). Let N be a strictly positive integer. Consider 2π/N -periodic traveling wave solutions of this system of sufficiently small-amplitude and with velocity sufficiently close to ω(N )/N . In order for these solutions to be spectrally unstable with respect to high-frequency instabilities as a consequence of two-eigenvalue collisions, it is necessary that there exist n, m ∈ Z, n = m, µ ∈ (−1/2, 1/2] for which
ω(n + µ) n + µ = ω(N ) N , ω(m + µ) m + µ = ω(N ) N ,(38)
such that
ω(n + µ) − ω(m + µ) n − m = ω(N ) N ,(39)
and (m + µ)(n + µ) < 0.
Next we proceed with some examples.
The (generalized) Korteweg -de Vries equation
We consider the generalized KdV (gKdV) equation
u t + σu n u x + u xxx = 0,(41)
where we restrict n to integers 1 or greater. Here σ is a constant coefficient, chosen as convenient. Important special cases discussed below are the KdV equation (n = 1) and the modified KdV (mKdV) equation (n = 2). Many of the details below extend easily to more general nonlinearities, with the main requirement being that the linearized equation is u t + u xxx = 0. The stability of periodic solutions of the gKdV equation has received some attention recently [9,10,14,28]. For the integrable cases n = 1 and n = 2, more detailed analysis is possible, see [6,13,16]. We do not claim to add anything new to these discussions, but we wish to use this example to illustrate how the six-step process outlined in this section leads to easy conclusions before moving on to more complicated settings.
The modified Hamiltonian is given by
H 0 c = 1 2 2π 0 (u 2 x + cu 2 )dx.(42)
2. The dispersion relation is
ω = −k 3 .(43)
3. Bifurcation branches in the (c, amplitude)-plane start at (c, 0), with
c = ω(k) k = −k 2 .(44)
Since we desire 2π periodic solutions, we choose k = 1. Any choice k = N , where N is a non-zero integer is allowed. Choosing k = 1, bifurcation branches start at (−1, 0).
For the integrable cases n = 1 (KdV) and n = 2 (mKdV), these bifurcation branches may be calculated in closed form. For the KdV equation in a frame traveling with speed c, the 2π-periodic solutions are given by (with σ = 1) where cn denotes the Jacobian elliptic cosine function and K(κ) is the complete elliptic integral of the first kind [18,39]. Further,
u = 12κ 2 K 2 (κ) π 2 cn 2 K(κ)x π , κ ,(45)c(κ) = 4K 2 (κ) π 2 (2κ 2 − 1),(46)
resulting in an explicit bifurcation curve (c(κ), 12κ 2 K 2 (κ)/π 2 ), parameterized by the elliptic modulus κ ∈ [0, 1). This bifurcation curve is shown in Fig. 4a.
For the mKdV equation (n = 2), different families of traveling-wave solutions exist [16]. We consider two of the simplest. For σ = 3 (focusing mKdV), a family of 2π-periodic solutions is given by
u = 2 √ 2κK(κ) π cn 2K(κ)x π , κ ,(47)
with c(κ) given by (46), resulting in an explicit bifurcation curve (c(κ), 2 √ 2κK(κ)/π), parameterized by the elliptic modulus κ ∈ [0, 1). This bifurcation curve is shown in Fig. 4b. It should be noted that a solution branch exists where the solution is expressed in terms of the Jacobian dn function [18,39]: u = (2)K(κ)dn(K(κ)x/π, κ)/π, but this solution does not have a small-amplitude limit and our methods do not apply directly to it. Rather, the solutions limit to the constant solution u = 1/ √ 2 as κ → 0. A simple transformation v = u − 1/ √ 2 transforms the problem to one where our methods apply.
For σ = −3 (defocusing mKdV), a period 2π solution family is
u = 2 √ 2κK(κ) π sn 2K(κ)x π , κ ,(48)
with c(κ) = −4(1 + κ 2 )K 2 (κ)/π 2 . Here sn is the Jacobian elliptic sine function [18,39], resulting in an explicit bifurcation curve (c(κ), 2 √ 2κK(κ)/π), parameterized by the elliptic modulus κ ∈ [0, 1). This bifurcation curve is shown in Fig. 4c. 4. The stability spectrum is given by (32), with ω(k) = −k 3 and c = −1, resulting in
λ (n) µ = i(n + µ)(1 + (n + µ) 2 ).(49)
These eigenvalues cover the imaginary axis, as n and µ are varied. The imaginary part of this expression is displayed in Fig. 5a. For the sake of comparison with Fig. 2 in [6] we let µ ∈ [−1/4, 1/4), which implies that n is any half integer. The results of Fig. 2 in [6] are for elliptic modulus κ = 0.8, implying a solution of moderate amplitude. The comparison of these two figures serves to add credence to the relevance of the results obtained using the zero-amplitude solutions at the start of the bifurcation branch.
5. With n + µ = k and m + µ = k + l, for some l ∈ Z, the collision condition (33) is written as where the trivial solution l = 0 has been discarded. This is the equation of an ellipsoid in the (k, l) plane. It intersects lines of nonzero integer l in six integer points: ±(1, −2), ±(0, 1), ±(1, −1). Since for all of these, Ω(k) = 0, any collisions happen only at the origin λ (µ) n = 0. This is also illustrated in Fig. 5b. 6. The final step of our process is preempted by the results of the previous step. No Krein signature of colliding eigenvalues can be computed, since no eigenvalues collide.
l 2 + 3kl + 3k 2 − 1 = 0,(50)
Since eigenvalues do not collide away from the origin they cannot leave the imaginary axis through such collisions and no high-frequency instabilities occur for small amplitude solutions of the gKdV equation. This result applies to the KdV and mKdV equations as special cases. The absence of high-frequency instabilities for small amplitude solutions is consistent with the results in, for instance, [6,13,14].
The Whitham equation
As our second scalar example, we consider the Whitham equation (8). For this example, no analytical results exist, but the work of Sanford et al. [41] allows for a comparison with numerical results. Sanford et al. do not report the presence of high-frequency instabilities for solutions of any period. Their absence has been verified by us using the same methods, see Fig. 6. Hur & Johnson [27] consider periodic solutions, focusing on the modulational instability. However, they do include a Krein signature calculation of the eigenvalues of the zero-amplitude solutions, reaching the same conclusions obtained below. In what follows the nonlinear term N (u) does not contribute, as in the previous examples. We use V to denote the speed of the traveling wave, to avoid confusion with the phase speed c(k) in the kernel of the Whitham equation.
The modified Hamiltonian is
H 0 V = V 2 2π 0 u 2 dx − 1 2 π −π π −π K(x − y)u(x)u(y)dxdy. k ω(k) k Ω(k + n) (a) (b)
2. The dispersion relation is given by ω(k) = sign(k) gk tanh(kh).
3. The bifurcation branch starts at (V, 0) = ( g tanh(h), 0), where we have chosen N = 1 so that the minimal period of the solutions is 2π.
4. The elements of the stability spectrum are given by λ (µ) n = i(n + µ) g tanh(h) − isign(n + µ) g(n + µ) tanh(h(n + µ)). Fig. 7(a), together with the line through the origin with slope ω(1)/1. Since the dispersion relation is concave down (up) in the first (third) quadrant, the condition (33) is not satisfied. Thus collisions of eigenvalues away from the origin do not occur. This is also illustrated in Fig. 7(b), where the imaginary part of λ (µ)
The dispersion relation for the Whitham equation is plotted in
n is plotted for various integers n.
No
Krein signature calculation is relevant since eigenvalues do not collide away from the origin.
We conclude that periodic solutions of sufficiently small amplitude of the Whitham equation are not susceptible to high-frequency instabilities. This is consistent with the results presented in [41], see also Fig. 6. Thus, the Whitham equation is unable to replicate the instabilities found in the shallow depth water wave problem for solutions of small amplitude, despite having a dispersion relation that is identical to one branch of the water wave dispersion relation. We return to this in the next section.
∂ ∂t q p = J∇H ⇔ q t = δH δp p t = − δH δq ,(52)
where the Poisson operator J is given by (5). This Poisson operator is nonsingular, thus there are no Casimirs. Examples of systems of this form are the Nonlinear Schrödinger equation in real coordinates [7], the Sine-Gordon equation [21,29,30], and the water wave problem [51]. As before, our interest is in the linearization of this system around the zero-amplitude solution.
The quadratic Hamiltonian corresponding to this linearization can be written as
H 0 = 2π 0 1 2 ∞ n=0 c n q 2 nx + 1 2 ∞ n=0 b n p 2 nx + p ∞ n=0 a n q nx dx,(53)
with a n , b n , c n ∈ R. Typically the number of terms in the sums above is finite, but an example like the water wave problem requires the possibility of an infinite number of nonzero contributing terms in the Hamiltonian. As for the Whitham equation, convergence of the resulting series is not problematic. The form (53) is the most general form of a quadratic Hamiltonian depending on two functions q(x, t) and p(x, t). Indeed, any quadratic term of a form not included above is reduced to a term that is included by straightforward integration by parts. The linearization of (52) is given by
q t = ∞ n=0 a n q nx + ∞ n=0 (−1) n b n p 2nx ,(54a)p t = − ∞ n=0 (−1) n c n q 2nx − ∞ n=0
(−1) n a n p nx .
We proceed with the six step program outlined in the introduction.
1. Quadratic Hamiltonian. The modified Hamiltonian H 0 c is given by
H 0 c = 2π 0 cpq x + 1 2 ∞ n=0 c n q 2 nx + 1 2 ∞ n=0 b n p 2 nx + p ∞ n=0 a n q nx dx.(55)
This expression serves as a repository for the coefficients which are needed in what follows.
2. Dispersion relation. We look for solutions to (54a) of the form q =q exp(ikx − iωt), p =p exp(ikx − iωt). Requiring the existence of non-trivial (i.e., non-zero) solutions, we find that ω(k) is determined by
det iω + ∞ n=0 a n (ik) n ∞ n=0 b n k 2n − ∞ n=0 c n k 2n iω − ∞ n=0 a n (−1) n (ik) n = 0.(56)
This is a quadratic equation for ω(k), resulting in two branches of the dispersion relation, ω 1 (k) and ω 2 (k). Assuming that (54a-b) is dispersive, ω 1 (k) and ω 2 (k) are real-valued for k ∈ R. This is not easily translated in a condition on the coefficients a n , b n , c n and d n , since their reality is not assumed.
3. Bifurcation branches. Traveling wave solutions are stationary solutions of
q t = cq x + ∞ n=0 a n q nx + ∞ n=0 (−1) n b n p 2nx = δH 0 c δp , (57a) p t = cp x − ∞ n=0 (−1) n c n q 2nx − ∞ n=0 (−1) n a n p nx = − δH 0 c δq . (57b)
This system has the dispersion relations Ω 1,2 (k) = ω 1,2 (k) − ck. In Fourier space the stationary equations become
0 = ikcq + ∞ n=0 a n (ik n )q + ∞ n=0 (−1) n b n (ik) 2np ,(58a)0 = ikcp − ∞ n=0 (−1) n c n (ik) 2nq − ∞ n=0 (−1) n a n (ik) np . (58b)
Thus c is obtained from the condition that these equations have a nontrivial solution (q,p). This condition requires that the 2 × 2 determinant of the system above is zero. 4. Stability spectrum. To find the stability spectrum, we let q = Q(x) exp(λt), p = P (x) exp(λt). Next, using Floquet's Theorem,
Q = e iµx ∞ j=−∞ Q j e ijx , P = e iµx ∞ j=−∞ P j e ijx ,(59)
with µ ∈ (−1/2, 1/2]. Since (57a-b) has constant coefficients, it suffices to consider monochromatic waves, i.e., only one term of the sums in (59) is retained. It follows that λ satisfies (56) with iω replaced by −λ + i(n + µ)c. Thus λ (µ) n,l = i(n + µ)c − iω l (n + µ) = −iΩ l (n + µ), l = 1, 2.
As expected, the zero solution is neutrally stable since ω 1,2 (k) are real, assuming dispersive equations. The stability spectrum consists of two one-parameter point sets, one for l = 1, the other for l = 2.
5. Collision conditions. Ignoring collisions at the origin, we require λ (µ)
n 1 ,l 1 = λ (µ)
n 2 ,l 2 = 0 for some n 1 , n 2 ∈ Z, µ ∈ (−1/2, 1/2], l 1 , l 2 ∈ {1, 2}. This gives Figure 8: The graphical interpretation of the collision condition (61). The dashed curves are the graphs of the dispersion relations ω 1 (k) and ω 2 (k). The slope of the segment P 1 P 2 is the right-hand side in (61). The collision condition (61) seeks points whose abscissas are an integer apart, so that at least one of the segments P 3 P 4 , P 3 P 6 , P 5 P 4 or P 5 P 6 is parallel to the segment P 1 P 2 .
N ω(k) k n + µ m + µ ω 1 (k) ω 2 (k) P 1 P 2 P 3 P 4 P 5 P 6ω l 1 (n 1 + µ) − ω l 2 (n 2 + µ) n 1 − n 2 = ω 1 (N ) N .(61)
The right-hand side depends on ω 1 since we have chosen the first branch of the dispersion relation in Step 3. As before, this collision condition may be interpreted as a parallel secant condition, but with the additional freedom of being able to use points from both branches of the dispersion relation. This is illustrated in Fig. 8.
6. Krein signature. In the setting of a system of Hamiltonian PDEs as opposed to a scalar PDE, we use a different but equivalent characterization of the Krein signature [35]. The Krein signature is the contribution to the Hamiltonian of the mode involved with the collision. Since our Hamiltonians are quadratic, this implies that the Krein signature of the eigenvalue λ with eigenvector v is given by
signature(λ, v) = sign(v † L c v),(62)
where L c is the Hessian of the Hamiltonian H 0 c , and v † denotes the complex conjugate of the transposed vector. Since the Hessian L c is a symmetric linear operator, the argument of the sign in (62) is real. Recall that the linearization of the system (2) can be written as
∂ t q p = JL c q p ,(63)
which makes it easy to read off L c . For the case of (57a-b),
L c = ∞ n=0 c n (−1) n ∂ 2n x −c∂ x + ∞ n=0 a n (−1) n ∂ n x c∂ x + ∞ n=0 a n ∂ n x ∞ n=0 b n (−1) n ∂ 2n x .(64)
Next, the eigenvectors v are given by
q p = e λt+iµx+inx Q n P n ,(65)
where (Q n , P n ) T satisfies
λ Q n P n = JL c Q n P n .(66)
HereL c is the symbol of L c , i.e., the 2 × 2 matrix obtained by replacing ∂ x → i(n + µ) in (64):
L c = ∞ n=0 c n (n + µ) 2n −ic(n + µ) + ∞ n=0 a n (−1) n (in + iµ) n ic(n + µ) + ∞ n=0 a n (in + iµ) n ∞ n=0 b n (n + µ) 2n .
The determinant is imaginary, since interchanging the rows gives the complex conjugate result. Since λ (µ) n,l is imaginary, the result is real and the signature is well defined. Again, it is clear that no conclusions can be drawn if λ (µ) n,l = 0. Since we wish to examine whether signatures are equal or opposite, we consider the product of the signatures corresponding to λ (µ) n 1 ,l 1 and λ (µ) n 2 ,l 2 . Using (67) we find that signatures are opposite, provided that
∞ j 1 =0 c j 1 (n 1 + µ) 2j 1 ∞ j 2 =0 c j 2 (n 2 + µ) 2j 2 ω l 1 (n 1 + µ) + ∞ j 3 =0 a 2j 3 +1 (−1) j 3 (n 1 + µ) 2j 3 +1 × ω l 2 (n 2 + µ) + ∞ j 4 =0 a 2j 4 +1 (−1) j 4 (n 2 + µ) 2j 4 +1 < 0.(69)
The above condition is obtained by expressing the eigenvectors in (66) in terms of the entries of the first row of (67). An equivalent condition is obtained using the second row:
∞ j 1 =0 b j 1 (n 1 + µ) 2j 1 ∞ j 2 =0 b j 2 (n 2 + µ) 2j 2 ω l 1 (n 1 + µ) − ∞ j 3 =0 a 2j 3 +1 (−1) j 3 (n 1 + µ) 2j 3 +1 × ω l 2 (n 2 + µ) − ∞ j 4 =0 a 2j 4 +1 (−1) j 4 (n 2 + µ) 2j 4 +1 < 0.(70)
Depending on the system at hand, the condition (69) or (70) may be more convenient to use.
Remark. An important class of systems is those for which ω 1 (k) = −ω 2 (k). We refer to such systems as even systems. It follows immediately from (56) that for even systems a 2j+1 = 0, j = 1, 2, . . .. The Krein conditions (69) and (70) simplify significantly, becoming
ω l 1 (n 1 + µ)ω l 2 (n 2 + µ) ∞ j 1 =0 c j 1 (n 1 + µ) 2j 1 ∞ j 2 =0 c j 2 (n 2 + µ) 2j 2 < 0,(71)
or
ω l 1 (n 1 + µ)ω l 2 (n 2 + µ) ∞ j 1 =0 b j 1 (n 1 + µ) 2j 1 ∞ j 2 =0 b j 2 (n 2 + µ) 2j 2 < 0.(72)
We summarize our results.
Assume that the linearization of the Hamiltonian system (52) is dispersive (i.e., its dispersion relations ω 1 (k) and ω 2 (k) are real valued for k ∈ R). Let N be a strictly positive integer. Consider 2π/N -periodic traveling wave solutions of this system of sufficiently small-amplitude and with velocity sufficiently close to ω 1 (N )/N . In order for these solutions to be spectrally unstable with respect to high-frequency instabilities as a consequence of two-eigenvalue collisions, it is necessary that there exist l 1 , l 2 ∈ {1, 2}, n 1 , n 2 ∈ Z, n 1 = n 2 , µ ∈ (−1/2, 1/2] for which ω l 1 (n 1 + µ)
n 1 + µ = ω 1 (N ) N , ω l 2 (n 2 + µ) n 2 + µ = ω(N ) N ,(73)
such that
ω l 1 (n 1 + µ) − ω l 2 (n 2 + µ) n 1 − n 2 = ω 1 (N ) N ,(74)
and (69), or equivalently, (70) holds.
We proceed with examples.
The Sine-Gordon equation
As a first example, we consider the Sine-Gordon (SG) equation [42]:
u tt − u xx + sin u = 0.(75)
The stability of the periodic traveling wave solutions of this equation has been studied recently by Jones et al. [29,30]. Different classes of periodic traveling wave solutions exist, but only two of those can be considered as small-amplitude perturbations of a constant background state.
We consider the so-called superluminal (c 2 > 1) librational waves. The subluminal (c 2 < 1) librational waves require the use of the transformation v = u − π so that their small amplitude limit approaches the zero solution. We do not consider them here. The limits of the rotational waves are either soliton solutions or have increasingly larger amplitude. As such the rotational waves do not fit in the framework of this paper. An overview of the properties of these solutions as well as illuminating phase-plane plots are found in [29]. In contrast to [29,30], we fix the period of our solutions, as elsewhere in this paper. This makes a comparison of the results more complicated.
1. Quadratic Hamiltonian. With q = u, p = u t ,
H 0 c = 2π 0 cpq x + 1 2 p 2 + 1 2 q 2 + 1 2 q 2 x dx.(76)
Thus b 0 = 1, c 0 = 1, c 1 = 1 are the only non-zero coefficients.
2. Dispersion relation. Using (56),
ω 1,2 = ± 1 + k 2 .(77)
These expressions are real valued for k ∈ R, thus the SG equation is dispersive when linearized around the superluminal librational waves. Both branches of the dispersion relation are displayed in Fig. 9a. 3. Bifurcation branches. With N = 1, we obtain c = ω 1 (1)/1 = √ 2.
4. Stability spectrum. The stability spectrum is given by (60):
λ (µ)
n,l = −iΩ l (n + µ) = i(n + µ)
√ 2 ∓ i 1 + (n + µ) 2 ,(78)
with l = 1 (l = 2) corresponding to the − (+) sign. Here n ∈ Z, µ ∈ [−1/2, 1/2).
Collision condition. The collision condition (61) becomes
ω 1 (n 1 + µ) − ω 2 (n 2 + µ) n 1 − n 2 = √ 2.(79)
We have chosen ω l 1 = ω 1 and ω l 2 = ω 2 , since it is clear that the collision condition can only be satisfied if points from both dispersion relation branches are used. This is illustrated in Fig. 9a. In fact, many collisions occur, as is illustrated in Fig. 9b. One explicit solution is given by
n 1 = 3, n 2 = 0, µ = √ 10 − 3 2 ≈ 0.081138830.(80)
6. Krein signature. Since ω 2 (k) = −ω 1 (k), we may use the conditions (71) or (72). Since only one b j = 0, (72) is (slightly) simpler to use. We get that
ω 1 (n 1 + µ)ω 2 (n 2 + µ) < 0(81)
is a necessary condition for the presence of high-frequency instabilities of small-amplitude superluminal librational solutions of the SG equation. The condition is trivially satisfied k ω(k) Figure 9: (a) The two branches of the dispersion relation for the Sine-Gordon equation. The line segment P 1 P 2 has slope ω(1)/1, representing the right-hand side of (61). The slope of the parallel line segment P 3 P 4 represents the left-hand side of (61). (b) The two families of curves Ω 1 (k +n) (red, solid) and Ω 2 (k +n) (black, dashed), for various (integer) values of n, illustrating that many collisions occur away from the origin.
P 1 P 2 P 3 P 4 ω 1 (k) ω 2 (k) k Ω(k + n) (a) (b)
as it was remarked in the previous step that points from both dispersion relation branches have to be used to have collisions.
It follows that for the superluminal solutions of the SG equations the necessary condition for the occurrence of high-frequency instabilities is satisfied. Nevertheless, as the results of [29,30] show, such instabilities do not occur. This is illustrated in Fig. 10. The left panel illustrates an exact 2π-periodic superluminal solution of the SG equation with c ≈ 1.236084655663, obtained using elliptic functions. Computing the stability spectrum (right panel) of the solution using the Fourier-Floquet-Hill method [15] with 51 Fourier modes and 1000 Floquet exponents shows that no high-frequency instabilities are present to within the accuracy of the numerical method. This is consistent with the results of [29,30] where only the presence of a modulational instability is observed. Thus the example of this section illustrates that the necessary condition is not always sufficient.
The water wave problem
As a final example, we consider the water wave problem: the problem of determining the dynamics of the surface of an incompressible, irrotational fluid under the influence of gravity. For this example, the effects of surface tension are ignored and we consider only two-dimensional fluids, i.e., the surface is one dimensional. The Euler equations governing the dynamics are
φ xx + φ zz = 0, (x, z) ∈ D,(82a)φ z = 0, z = −h,(82b)η t + η x φ x = φ z , z = η(x, t),(82c)φ t + 1 2 (φ 2 x + φ z ) + gη = 0, z = η(x, t),(82d)
where x and z are the horizontal and vertical coordinate, respectively, see Fig. 11; z = η(x, t) is the free top boundary and φ(x, z, t) is the velocity potential. Further, g is the acceleration due to gravity and h is the average depth of the fluid.
The main goal of the water wave problem is to understand the dynamics of the free surface η(x, t). Thus it is convenient to recast the problem so as to involve only surface variables. Zakharov [51] showed that the water wave problem is Hamiltonian with canonical variables η(x, t) and ϕ(x, t) = φ(x, η(x, t), t). In other words ϕ(x, t) is the velocity potential evaluated at the surface. Following [12], the Hamiltonian is written as
H = 1 2 2π 0 ϕG(η)ϕ + gη 2 dx,(83)
where G(η) is the Dirichlet → Neumann operator: G(η)ϕ = (1 + η 2 x ) 1/2 φ n , at z = η(x, t). Here φ n is the normal derivative of φ. Using the water wave problem, G(η)ϕ = φ z − η x φ x = η t , which is the first of Hamilton's equations. The water wave problem for η(x, t) and ϕ(x, t) is
η t = δH δϕ , ϕ t = − δH δη .(84)
k ω(k)
ω 1 (k) ω 2 (k) k Ω(k + n) (a) (b)
h Ω (c) Figure 12: (a) The two branches of the dispersion relation for the water wave problem (g = 1, h = 1). The line through the origin has slope ω 1 (1)/1, representing the right-hand side of (61). (b) The two families of curves Ω 1 (k + n) (red, solid) and Ω 2 (k + n) (black, dashed), for various (integer) values of n, illustrating that many collisions occur away from the origin. (c) The origin of the high-frequency instability closest to the origin as a function of depth h.
5. Collision condition. The condition (61) is easily written out explicitly, but for our purposes it suffices to plot Ω 1 (k + n) and Ω 2 (k + n), for different values of n. This is done in Fig. 12b
ω l 1 (n 1 + µ)ω l 2 (n 2 + µ)g 2 < 0,(91)
and ω l 1 (n 1 + µ)ω l 2 (n 2 + µ)
∞ j 1 =1 α j 1 −1 h 2j 1 −1 (n 1 + µ) 2j 1 ∞ j 2 =1 α j 2 −1 h 2j 2 −1 (n 2 + µ) 2j 2 < 0,(92)
respectively. Here the coefficients α j are related to the Bernoulli numbers [18], as they are defined by the Taylor series
tanh(z) = ∞ j=0 α j z 2j+1 , |z| < π/2.(93)
Because of the finite radius of convergence of this series, (92) is only valid for small values of the wave numbers n 1 + µ and n 2 + µ, but it is possible to phrase all results in terms of tanh directly, avoiding this difficulty. For instance, using (93), (92) may be rewritten as Thus all collision points are potential origins of high-frequency instabilities. It appears from the numerical results in [17] that the bubble of non-imaginary eigenvalues closest to the origin contains the high-frequency eigenvalues with the largest real part. Thus for waves in shallow water kh < 1.363 (no Benjamin-Feir instability) [4,47,54], these are the dominant instabilities. For waves in deep water (kh > 1.363) the Benjamin-Feir instability typically dominates, although there is a range of depth in deep water where the high-frequency instabilities have a larger growth rate, see [17]. The dependence on depth h of the location on the imaginary axis from which the high-frequency bubble closest to the origin bifurcates is shown in Fig. 12(c), with g = 1. As h → ∞, the imaginary part of λ → 3/4. This asymptote is drawn in Fig. 12(c) for reference. This figure demonstrates that for all positive values of the depth h, the instabilities considered are not modulational as they do not bifurcate away from the origin as the amplitude increases.
ω l 1 (n 1 + µ)ω l 2 (n 2 + µ) ω 2 α (n 1 + µ) g ω 2 β (n 2 + µ) g < 0 ⇒ ω l 1 (n 1 + µ)ω l 2 (n 2 + µ) < 0,
It was remarked in Example 3.2 that no collisions are possible due to the concavity of the dispersion relation. As a consequence, all collisions away from the origin observed in Fig. 12b involve both branches of the dispersion relation, i.e., they involve a solid curve and a dashed curve. This is easily seen from Fig. 12a: a parallel cord with abscissae of the endpoints that are integers apart is easily found by sliding a parallel cord away from the cord ((0, 0), (1, ω 1 (1))) until the integer condition is met. This implies that ω l 1 (n 1 +µ) and ω l 2 (n 2 + µ) in the collision condition (61) have opposite sign and (91) is always satisfied. Thus colliding eigenvalues of zero-amplitude water wave solutions always have opposite Krein signature. As a consequence, the necessary condition for the presence of highfrequency instabilities is met. In fact, it was observed in [17] that all colliding eigenvalues give rise to bubbles of instabilities as the amplitude is increased.
Our general framework easily recovers the results of MacKay & Saffman [36]. There the set-up is for arbitrary amplitudes of the traveling wave solutions, but the results are only truly practical for the zero-amplitude case.
Remark. It follows from these considerations that the high-frequency instabilities present in the water wave problem are a consequence of counter-propagating waves as no such instabilities are present in the Whitham equation (8). Although it is often stated that the value of the Whitham equation lies in that it has the same dispersion relation as the water wave problem (see for instance [50]), this is in fact not the case as it contains only one branch of the dispersion relation. Thus the equation does not allow for the interaction of counter-propagating modes, and as such misses out on much of the important dynamics of the Euler equations.
A Boussinesq-Whitham equation
The goal of this section is the introduction of a model equation that has the same dispersion relation as the Euler equations (82a-d) at the level of heuristics that led Whitham to the model equation (8). In other words, we propose a bidirectional Whitham equation, so as to capture both branches of the water wave dispersion relation. We refer to this equation as the Boussinesq-Whitham (BW) equation. It is given by
q tt = N (q) + ∂ 2 x αq 2 + ∞ −∞ K(x − y)q(y)dy ,(94)
where
K(x) = 1 2π ∞ −∞ c 2 (k)e ikx dk,(95)
and c 2 (k) = g tanh(kh)/k, α > 0 for the water wave problem without surface tension. In (94), N (u) denotes the nonlinear terms, which are ignored in the remainder of this section. Since our methods focus on the analysis of zero amplitude solutions, the sign of α is not relevant in what follows. This equation is one of many that may stake its claim to the name "Boussinesq-Whitham equation". Equation (94) is a "Whithamized" version of the standard Bad Boussinesq equation and it may be anticipated that it captures at least the small-amplitude instabilities of the water wave problem in shallow water. It should be remarked that the Bad Boussinesq equation is ill posed as an initial-value problem [37], but it might be anticipated that the inclusion of the entire water-wave dispersion relation overcomes the unbounded growth that is present due to the polynomial truncation. We return to this at the end of this section. Before applying our method to examine the potential presence of high-frequency instabilities of small-amplitude solutions of the BW equation, we need to present its Hamiltonian structure. Further, since (94) is defined as an equation on the whole line, a periodic analogue is required, as in Section 2.
It is easily verified that (94) is Hamiltonian with (non-canonical) Poisson operator [37]
J = 0 ∂ x ∂ x 0 ,(96)
and Hamiltonian
H = ∞ −∞ 1 2 p 2 + α 3 q 3 dx + 1 2 ∞ −∞ dx ∞ −∞ dy K(x − y)q(x)q(y).(97)
Indeed, (94) can be rewritten in the form (1) with u = (q, p) T .
To define a periodic version of (94), let
K(x) = 1 L ∞ j=−∞ c 2 (k j )e ik j x ,(98)
where k j = 2πj/L, j ∈ Z. The periodic BW equation is obtained from (1), using (96) and (97), but with all ± infinities in the integration bounds replaced by ±L/2, respectively. Since (94) has a Poisson operator (96) that is different from those used in Sections 3 and 4, minor modifications to the use of the method are necessary.
1. Quadratic Hamiltonian. Ignoring the contributions of the nonlinear term, the quadratic Hamiltonian in a frame of reference moving with speed V is given by
H 0 V = 2π 0 V qp + 1 2 p 2 dx + 1 2 2π 0 dx 2π 0 dy K(x − y)q(x)q(y),(99)
where we have fixed the period of the solutions to be L = 2π. The inclusion of the first term in (99) is one place where the effect of the different form for J is felt, as its functional form is a direct consequence of the form of (96).
Dispersion Relation.
A direct calculation confirms that
ω 2 = gk tanh(kh),(100)
which is, by construction, identical to the dispersion relation for the full water wave problem (88). This gives rise to two branches of the dispersion relation (89), corresponding to right-and left-going waves.
3. Bifurcation Branches. Bifurcation branches for 2π-periodic solutions start at (V 1,2 , 0), where the phase speeds V 1,2 are given by V 1,2 = ± g tanh(h).
4. Stability Spectrum. The stability spectrum elements are, again by construction, identical to those for the water wave problem, given in (90a-b).
Collision Condition.
Given that the spectral elements are identical to those for the water wave problem, the collision condition is identical too. It is displayed in Fig. 12(a-b). Thus, collisions away from the origin occur. It remains to be seen whether these can result in the birth of high-frequency instabilities.
6. Krein Signature. As for the canonical case of Section 4, we use (62). Thus we calculate the Hessian L c of the Hamiltonian H 0 c . Let
c 2 (k) = ∞ j=0 γ j k 2j ,(101)
where γ j = gh 2j+1 a j , with the coefficients a j defined in (93). A direct calculation gives that the Hamiltonian (99) is rewritten as
H 0 V = 1 2 2π 0 p 2 + V qp + ∞ j=0 γ j q 2 jx dx.(102)
Using this form of the Hamiltonian, the calculation of the Hessian is straightforward, leading to
L V = ∞ j=0 γ j (−1) j ∂ 2j x V V 1 .(103)
Next, we compute the eigenvectors v = (q, p) T . We have
q p = e iλt Q(x) P (x) ,(104)
where (Q, P ) T satisfies
λ Q P = 0 ∂ x ∂ x 0 L V Q P .(105)
This is a second place where the Poisson operator J plays a crucial role as it affects the form of v = (q, p) T and thus the expression for the signature. One easily verifies that
Q P = e i(n+µ)x i(n + µ) λ − i(n + µ)V(106)
satisfies (105).
We need to evaluate the sign of
Q P † L V Q P = e −i(n+µ)x −i(n + µ) −λ + i(n + µ)V T ∞ j=0 γ j (−1) j ∂ 2j x V V 1 i(n + µ) λ − i(n + µ)V e i(n+µ)x = −i(n + µ) −λ + i(n + µ)V T ∞ j=0 γ j (n + µ) 2j V V 1 i(n + µ) λ − i(n + µ)V = −i(n + µ) iω(n + µ) T c 2 (n + µ) V V 1 i(n + µ) −iω(n + µ) = 2ω (ω − (n + µ)V ) .(107)
Let the signature associated with the first eigenvalue be the sign of 2ω j 1 (ω j 1 − (n 1 + µ)V ), where ω j 1 is a function of n 1 + µ. Similarly, for the second eigenvalue, the signature is the sign of 2ω j 2 (ω j 2 − (n 2 + µ)V ). Using the collision condition λ (µ) n 1 ,j 1 = λ (µ) n 2 ,j 2 , the product of these two expressions is 4ω j 1 ω j 2 (ω j 2 − (n 2 + µ)V ) 2 , which is less than zero since collisions can only occur for eigenvalues associated with opposite branches of the dispersion relation, see Fig. 12b. It follows that, as in the water wave case, the signatures of colliding eigenvalues are always opposite, and the necessary condition for spectral instability is met. Thus, unlike the Whitham equation (8), the BW model (94) does not exclude the presence of high-frequency instabilities of small-amplitude solutions.
The results obtained from the Krein signature calculations are confirmed by numerical results, see Fig. 13. Panel (a) shows a numerically computed traveling wave solution of the BW equation (94). This solution is computed using a cosine collocation method with 60 points, as for the Whitham equation, see Fig. 6 [41]. For the solution plotted, c ≈ 1.049815. The second panel in the first row displays the spectrum computed using Hill's method with 100 modes and 20000 values of the Floquet parameter, using an interpolation of the solution profile. This panel shows the presence of a large number of apparent instabilities, most with small growth rate, in the neighborhood of the imaginary axis. The third panel shows a zoom of the region around the origin, revealing a modulational instability. This is expected, since such an instability is also present for the Whitham equation, see Section 3.2. The fourth panel zooms in on the first bubble of instabilities centered on the positive imaginary axis, revealing a shape and location that is consistent with the Krein collision theory presented here. The wave form displayed in Fig. 13(a) does not have zero average, unlike the one shown in Fig. 6, for reasons explained here. Let us examine the stability of a flat-water state q = a (constant), p = 0. Thinking of the BW equation as an approximation to the water wave problem where the flat-water state is neutrally stable (spectrum on the imaginary axis), independent of the reference level of the water, the neutral stability of this state is desired in the context of the BW system as well. However, the BW system is easily checked to not be Galilean invariant, thus the average value of the solution may be important.
Linearizing the system around the flat water state (q, p) = (a, 0) results in a linear system with constant coefficients, whose dispersion relation is given by ω 2 = 2ak 2 +k 2 c 2 (k). This results in two branches for the dispersion relation: ω 1,2 = ±k c 2 (k) + 2a. It follows that if a > 0 then both ω 1 and ω 2 are real, resulting in neutral stability, since the stability eigenvalue and the frequency ω(k) are different by a factor of i. On the other hand, if a < 0, it follows that both ω 1 and ω 2 are imaginary for sufficiently large k, since lim |k|→∞ c(k) = 0. This leads to the dynamics of the flat-water state with a < 0 to not only be unstable, but to be ill-posed, as the growth rate of the instability → ∞ as |k| → ∞. Thus Whitham-izing the Bad Boussinesq equation and incorporating the full water wave dispersion relation does not remove the illposedness of the problem. Rather it alters it where negative constant solutions experience unbounded growth, unlike positive constant solutions.
It is observed numerically that this behavior of perturbed constant solutions is carried over to nonconstant solutions: solutions of negative average display the same illposed behavior described above, with stability spectra that have unbounded real part. In contrast, the spectra of solutions of positive average have bounded real part, as in Fig. 13. Annoyingly, the illposed behavior extends to numerical solutions constructed to have zero average. Presumably this is a consequence of numerical error, as higher accuracy numerical experiments display narrower spectra whose real part tends to infinity more slowly.
We may summarize our findings on the BW equation as follows. The equation was constructed as a bi-directional Whitham equation so as to truly have the same linear dispersion relation as the water wave problem. Even though the BW has a different Poisson structure than the water wave problem, we find that periodic solutions of the BW are susceptible to high-frequency instabilities originating from Krein collisions at the exact same locations on the imaginary axis as the water wave problem. On the other hand, we have not attempted to quantify whether the resulting growth rates are comparable to those for the water wave problem. Further, the illposedness of the equation for solutions of negative average is a significant strike against its potential use in applications. Nevertheless, it appears possible to design more equations like the BW equation, possessing the exact same dispersion relation as the water wave problem and with it all its high-frequency instabilities, without the equation dynamics being illposed for any important class of solutions.
Figure 1 :
1A cartoon of the bifurcation structure of the traveling waves for a third-order (M = 3) system: solution branches bifurcate away from the trivial zero-amplitude solution at specific values of the traveling wave speed c.
In fact, a formal L → ∞ immediately results in the recovery of the equation posed on the whole line. Thus the Whitham equation and its periodic solutions fit in to the framework developed in this manuscript. It is one of many examples we use below. Other notable examples are the Euler water wave problem (as expected, allowing us to check our results with those of MacKay & Saffman [36]), the KdV equation, the Sine-Gordon equation, etc. We are particularly interested in the comparison between the results for the Euler water wave problem and those for the Whitham equation. The results of Examples 3.2 and 4.2 show that the Whitham equation cannot possess the high-frequency instabilities present in the water wave problem. This leads us to propose a new model equation, a so-called Boussinesq-Whitham or bidirectional Whitham equation.
Figure 4 :
4The amplitude vs. c bifurcation plots for the traveling-wave solutions of the generalized KdV equation(41). (a) The KdV equation, n = 1, for the cnoidal wave solutions(45). (b) The mKdV equation, n = 2, for the cnoidal wave solutions(47). Lastly, (c) shows the bifurcation plot for the snoidal wave solutions (48) of mKdV, n = 2. Note that all bifurcation branches start at (−1, 0), as stated above.
Figure 5 :
5(a) The imaginary part of λ (µ) n ∈ (−0.7, 0.7) as a function of µ ∈ [−1/4, 1/4). (b) The curves Ω(k + n), for various (integer) values of n, illustrating that collisions occur at the origin only.
Figure 6 :
6(a) The profile of a 2π-periodic small-amplitude traveling wave solution of the Whitham equation(8)with c ≈ 0.7697166847, computed using a cosine collocation method with 128 Fourier modes, see[41]. (b) The stability spectrum of this solution, computed using the Fourier-Floquet-Hill method[15] with 128 modes and 2000 different values of the Floquet parameter µ. The presence of a modulational instability is clear, but no high-frequency instabilities are observed, in agreement with the theory presented. Note that the hallmark bubbles of instability were looked for far outside of the region displayed here.
Figure 7 :
7(a) The dispersion relation for the Whitham equation (curve), together with the line through the origin of slope ω(1)/1, representing the right-hand side of (33). (b) The curves Ω(k + n), for various (integer) values of n, illustrating that collisions occur at the origin only.
A simple comparison with (56) gives that there are two bifurcation points given by (ω 1 (N )/N, 0) and (ω 2 (N )/N, 0). Any positive integer value of N is allowed, but we usually choose N = 1 so that the fundamental period is 2π. In what follows, we examine the small-amplitude solutions starting from the branch (c, 0) = (ω 1 (N )/N, 0), without loss of generality.In many systems the two bifurcation branches are reflections of each other about the vertical axis. The corresponding solution profiles are identical to each other, moving to the right on one branch, moving to the left on the other. Examples are given below. Nonsymmetric bifurcation branches cannot be excluded, however, without imposing extra assumptions on the coefficients of (54a-b).
Figure 10 :Figure 11 :
1011(a) A small-amplitude 2π-periodic superluminal solution of the SG equation (c ≈ 1.236084655663). (b) A blow-up of the numerically computed stability spectrum in a neighborhood of the origin, illustrating the presence of a modulational instability, but the absence of high-frequency instabilities. The domain for the water wave problem. Here z = 0 is the equation of the surface for flat water, z = −h is the flat bottom.
with g = 1 and h = 1. Although only the first collision is visible in the figure (all intersection points are horizontal integer shift of each other and correspond to the same value of µ and λ (µ) n,j ), it is clear from the curves shown that many collisions occur. The figure is qualitatively the same for all finite values of depth. 6. Krein signature. The conditions (71) and (72) become
Figure 13 :
13(a) A small-amplitude traveling wave solution of the Boussines-Whitham equation (94) with c ≈ 1.0498515. (b) The numerically computer stability spectrum. (c) A blow-up of the stability spectrum in a neighborhood of the origin. (d) A blow-up of the stability spectrum around what appears as a horizontal segment visible in (b) immediately above the longest segment appearing horizontal. More detail is given in the main text.
. In essence, the Krein signature of an eigenvalue is the sign of the Hamiltonian of the linearized system evaluated on the eigenspace of the eigenvalue. Different characterizations are given below. If two imaginary eigenvalues of the same signature collide as a parameter changes, their collision does not result in them leaving the imaginary axis. Thus the Colliding eigenvalues in the complex plane as a parameter is increased. On the left, two eigenvalues are moving towards each other on the positive imaginary axis, accompanied by a complex conjugate pair on the negative imaginary axis. In the middle, the eigenvalues in each pair have collided. On the right, a Hamiltonian Hopf bifurcation occurs: the collided eigenvalues separate, leaving the imaginary axis (implying that the two Krein signatures were different).Parameter value
Imλ
0
Imλ
0
Imλ
0
Figure 2:
1 .
1Quadratic Hamiltonian. The modified Hamiltonian H 0 c is given by
Two-component Hamiltonian PDEs with canonical Poisson structureWe generalize the ideas of the previous section to the setting of two-component Hamiltonian PDEs with canonical Poisson structure. In other words, the evolution PDE can be written as
AcknowledgementsWe wish to thank Richard Kollar for interesting discussions. John Carter helped our understanding of the Whitham equation and Mat Johnson was a part of our initial investigations on the Boussinesq-Whitham equation. This work was supported by the National Science Foundation through grant NSF-DMS-1008001 (BD) and in part by the EPSRC under grant EP/J019569/1 and NSERC (OT). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding sources.The quadratic Hamiltonian H 0 c is given bygiving rise to the linearized equations in a frame moving with velocity c:2. Dispersion relation. The well-known dispersion relation[46]for the water wave problem is immediately recovered from the linearized system (87a-b) with c = 0 (no moving frame), resulting inNote that the right-hand side of this expression is always positive. Thus there are two branches to the dispersion relation:Thus ω 1 (ω 2 ) corresponds to positive (negative) phase speed, independent of the sign of k.3. Bifurcation branches. Branches originate from (c, amplitude) = (ω 1 (1)/1, 0) and (c, amplitude) = (ω 2 (1)/1, 0). Without loss of generality, we focus on the first branch for which the phase speed g tanh(h) is positive. This allows for a straightforward comparison of our results with those for the Whitham equation, in Example 3.2.4. Stability spectrum. The elements of the spectrum are given byn,2 = −i Ω 2 (n + µ) = i(n + µ) g tanh(h) + i sign(n + µ) g(n + µ) tanh(h(n + µ)).The sign(n+µ)'s may be omitted in these expressions, as the same set of spectral elements is obtained.
Since for our purposes, the linearization of (84-b) in a moving frame is required. Quadratic Hamiltonian, it suffices to evaluate the Dirichlet → Neumann operator G(η) at theQuadratic Hamiltonian. Since for our purposes, the linearization of (84-b) in a moving frame is required, it suffices to evaluate the Dirichlet → Neumann operator G(η) at the
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| [] |
[
"System-Level Predictive Maintenance: Review of Research Literature and Gap Analysis",
"System-Level Predictive Maintenance: Review of Research Literature and Gap Analysis"
] | [
"Kyle Miller ",
"Artur Dubrawski "
] | [] | [] | This paper reviews current literature in the field of predictive maintenance from the system point of view. We differentiate the existing capabilities of condition estimation and failure risk forecasting as currently applied to simple components, from the capabilities needed to solve the same tasks for complex assets. System-level analysis faces more complex latent degradation states, it has to comprehensively account for active maintenance programs at each component level and consider coupling between different maintenance actions, while reflecting increased monetary and safety costs for system failures. As a result, methods that are effective for forecasting risk and informing maintenance decisions regarding individual components do not readily scale to provide reliable sub-system or system level insights. A novel holistic modeling approach is needed to incorporate available structural and physical knowledge and naturally handle the complexities of actively fielded and maintained assets. | null | [
"https://arxiv.org/pdf/2005.05239v1.pdf"
] | 218,581,628 | 2005.05239 | be9f96bc84a24e84fa68a20879c1bc0871096a1f |
System-Level Predictive Maintenance: Review of Research Literature and Gap Analysis
October 31, 2019
Kyle Miller
Artur Dubrawski
System-Level Predictive Maintenance: Review of Research Literature and Gap Analysis
October 31, 2019
This paper reviews current literature in the field of predictive maintenance from the system point of view. We differentiate the existing capabilities of condition estimation and failure risk forecasting as currently applied to simple components, from the capabilities needed to solve the same tasks for complex assets. System-level analysis faces more complex latent degradation states, it has to comprehensively account for active maintenance programs at each component level and consider coupling between different maintenance actions, while reflecting increased monetary and safety costs for system failures. As a result, methods that are effective for forecasting risk and informing maintenance decisions regarding individual components do not readily scale to provide reliable sub-system or system level insights. A novel holistic modeling approach is needed to incorporate available structural and physical knowledge and naturally handle the complexities of actively fielded and maintained assets.
Introduction
Predictive maintenance describes an approach to equipment management that focuses on exploiting sensing, inspection, and maintenance data to forecast future degradation state, remaining-useful-life, or similar quantity characterizing expected future performance of the equipment. Such forecasts are then used to optimize maintenance planning, supply chain, and other maintenance, design, and engineering activities. As a conceptual framework, it has gained significant popularity in recent years. This is not least due the very attractive claim that predictive maintenance can significantly improve over the stateof-practice by more closely aligning maintenance effort with maintenance need, thereby saving significant amounts of money and time while decreasing unplanned downtime and uncertainty. In applications with limits on equipment availability and/or budgets, predictive maintenance promises to enable intelligent planing to effectively and efficiently satisfy such constraints.
Predictive maintenance is sometimes abbreviated PMx or PdM and sometimes referred to as predictive/prognostic health management (PHM). It is closely associated with condition based maintenance (CBM) and reliability centered maintenance (RCM). Figure 1 shows the number of predictive maintenance related academic publications by year 1 . Note the low count in 2019 is an artifact due to the date of the query. Table 1 lists the top 5 countries of origin by article count. Table 2 lists the top 10 funding agencies acknowledged by article count. These publication records demonstrate a growing interest in the field, which is likely correlated with advances in machine learning and artificial intelligence, and a reduction in data storage and processing costs over the past decade.
This vast amount of literature makes a comprehensive review challenging. Rather, we review the recent literature on predictive maintenance with a focus on complex equipment at the system and fleet/enterprise levels, examples of which include airlines, truck fleets, etc. The costs, repair time, number of components, variation in use/duty/load, and amount and scope of available data, are all dramatically higher in such scenarios as compared to analysis of individual components. With an increase in the problem complexity and scope, it may not be effective to craft data processing pipelines, data featurizations, and predictive models, for individualized components and failure modes, as is commonly demonstrated in the literature [Rögnvaldsson et al., 2018].
The remainder of this document is structured as follows. In Section 2 we provide our view on what differentiates predictive maintenance of complex assets from individual components. In Section 3 we review other relevant review articles, highlighting recent conceptual trends and industrial foci. In Section 4 we review primary research, organized by principal concepts in predictive maintenance; condition estimation, forecasting, planning/scheduling, and performance quantification. In Section 5 we characterize the gap between prior work and the needed capabilities for system-level PMx and review relevant existing work. In Section 6 we conclude with promising future directions. 1 Web of Science query (conducted on 2019-03-01): TS=("predictive maintenance" OR "condition estimation" OR "remaining useful life" OR "degradation model" OR "failure prediction") AND SU=(ENGINEERING OR COMPUTER SCIENCE OR SCIENCE TECHNOLOGY OTHER TOPICS OR MATHEMATICS OR MECHANICS OR ROBOTICS OR OPERATIONS RESEARCH MANAGEMENT SCIENCE )
Complex Assets
We differentiate between predictive maintenance applied to a single component with that applied to a complex asset. By complex asset we mean a system of several interacting components. In most cases, component interactions such as redundancies make application of predictive maintenance focused on each constituent component, an unsatisfying solution at the system level. Here, we itemize several fundamental distinctions between component level and system level problem elements to emphasize the importance of this differentiation.
Faults. In single component analysis, faults are typically not enumerated. Rather, failure is the only outcome. On the other hand, complex assets may present numerous and varied faults, to the degree that novel fault types may be wholly unobserved in training data. Additionally, for complex assets, faults are typically observed at the sub-system or system level. For example, it may be recorded that an engine fails to start, but not that a particular valve gasket has ruptured. Further, it may be that no individual component fails outright, but rather in their degraded state multiple components fail to work together.
Degradation state. For individual components, degradation state is synonymous with wear and tear, and is tightly connected with remaining useful life (RUL) 2 . It is typical that degradation state is modeled as a single-dimensional quantity that monotonically increases with use. For complex systems this notion must be extended, for example modeling degradation state as a multidimensional vector encoding the wear state of each constituent component. Additionally, the relationship between degradation state and failure can be complex and non-linear. This relationship is the object of study in the discipline of reliability analysis. The use of reliability models such as fault trees and Bayesian networks in predictive maintenance is briefly touched upon in Section 5.1.1.
Data. Unlike isolated components, it is typically not cost-effective or not feasible to conduct run-tillfailure experiments. As a result, observed data are collected from machines operating in a production environment or in the field, which likely includes significant variation in operating loads. These sources of variation may need to be accounted for in predictive models to achieve desired levels of predictive performance. Degradation state is almost surely not directly observed in complex assets. Direct observation is sometimes assumed for analysis of individual components, but the volumes of data that would be required to directly record degradation state of all of the individual components in a complex asset would likely be prohibitive to say nothing of large sensing array that would be required. Rather, degradation state will be indirectly observed or perhaps partially observed. Additionally, data will typically reflect sub-system behavior rather than individual component state.
Maintenance actions.
Maintenance can often be ignored in the context of individual components.
If training data consist of run-till-failure experiments, maintenance is not performed. In other cases maintenance may consist of replacing a component near failure, in which case the effect of maintenance is to return a part to like-new condition as is often modeled in the literature [Yildirim et al., 2016a, Yildirim et al., 2016b, Hao et al., 2017b, Feng et al., 2017a. In training data this can be viewed as censoring observations and maintenance type and effect need not be explicitly considered. In contrast, complex assets may be actively maintained with both repair and replacement of components, impacting degradation state and/or degradation rates, as well as their estimates, in non-trivial ways.
As a direct result of maintenance, examples of failure may be rare or absent in available training data. In safety-critical systems, components will be serviced or replaced prior to actual failure events. In down-time sensitive applications maintenance may be performed opportunistically during available maintenance windows rather than in correlation with impending failure. Therefore, it is important to account for the effects of maintenance during the development of predictive models.
Fleet.
A fleet of individual components is often treated as a set of identical pieces, the observations of which can be pooled into a single training set. However, with long lived complex assets, individual histories of maintenance, asset-specific usage histories and aging, customization, and modifications, may result in a set of similar but not identical assets. If the degree of similarity is moderate, due to say unit customization, model training procedures will have to be adapted to reflect the resulting subjectivity. Transfer learning or multi-task learning frameworks may be needed for sharing information across the fleet. Additionally, long lived assets may show additional forms of concept drift. For example, replacement parts may be sourced from a new supplier with slightly different tolerances or operational characteristics. In such a situations, historical data may not be perfectly reflective of the current reality.
Planning. As with fleets of individual components, fleet-level planning for systems requires taking all assets into consideration for making optimal maintenance decisions. This is typically because finite maintenance resources induce a coupling across maintenance decisions for each asset. However, with complex assets, additional couplings always exist between components of a given asset. If maintenance is to be performed on one component, it may induce or block a maintenance window for another component, and may impact the effective duration of jointly performed maintenance actions.
Critical Capabilities
Generally, predictive maintenance can be described as failure risk forecasting combined with maintenance planning. There are a number of sub-problems that must be solved to realize PMx capabilities. The importance of each of these sub-problems can vary significantly depending on the use case under consideration. The principal components of PMx are described in Figure 2. Data must be collected and curated for use, requiring infrastructure for data collection, storage, and analysis. One common consideration in the literature is how to facilitate data collection through cloud solutions and IoT technology [Meraghni et al., 2018, Chukwuekwe et al., 2016, although cloud solutions are not immediately applicable to some asset types or scenarios due to safety and security concerns. Once data is available, it can be used to estimate historical, current, and future condition or failure risk. As noted in Section 2, these are not synonymous, though often conflated. Given ones belief of the future risks, operational and maintenance plans can be formulated to optimize global objectives.
This viewpoint is fundamentally asset-centric. Figure 2 does not call out the need to estimate uncertainties in supply chain (e.g. shipping lead times, etc.) or in maintenance itself (e.g. time to repair). This focus on asset-centric capabilities is typical of the literature reviewed. The majority of academic research in the PMx field has focused on condition and failure risk forecasting. Maintenance scheduling has been addressed, but to a lesser extent. Operational planning, such as e.g. assigning vehicles to delivery routes [Biteus and Lindgren, 2017], has been briefly touched upon. Cost-benefit analysis of predictive capabilities (e.g. [Busse et al., 2018]) appears also to be currently understudied.
Review of Reviews
There exist several reviews of CBM and PMx. Most of these reviews walk the reader through the basic pipeline of data acquisition, processing and feature extraction, modeling and prediction, and finally decision support. Usual points under discussion are classes of data types, tools, and techniques that are commonly used. We give a brief overview of these reviews here to build out a description of the current state of the field. [Si et al., 2011] is one of the most frequently cited papers in the field. The authors review several families of RUL prediction approaches. The methods are stratified by whether (degradation) state is directly or indirectly observed. For directly observed state, reviewed approaches include regression based models, Wiener processes, Gamma processes, and Markovian models. For indirect observation, the authors describe filtering-based methods, hazard models, and hidden Markov models. [Lei et al., 2018] gives a recent review of data acquisition and RUL prediction. The authors identify four technical processes; data acquisition, health indicator construction, health stage segmentation, and RUL prediction. The authors review four commonly used public data sets for RUL prediction; The NASA turbofan dataset [Saxena and Goebel, 2008], the FEMTO bearing dataset [Nectoux et al., 2012], the IMS bearing dataset [Qiu et al., 2006, Lee et al., 2007, and a milling dataset [Agogino and Goebel, 2007]. For each dataset [Lei et al., 2018] give a description, list of important properties, and recounts applications. [Eker et al., 2012] additionally describes a Li-ion battery dataset [Saha and Goebel, 2007], a Insulated Gate Bipolar Transistor (IGBT) dataset [Celaya et al., 2009], and the Vickler dataset [Virkler et al., 1979]. Most of these datasets are from the NASA Ames prognostics data repository [NASA, 2019], which currently hosts 16 datasets. [Lei et al., 2018] summarizes performance metrics used in RUL prediction and concludes with future challenges including data volume (either limited or overwhelming), handling multiple failure modes, system level RUL prediction, and others.
Several reviews focus on Industry 4.0 and the "digital-twin" concepts. The digital-twin is meant to be a "living model" which can forecast effectively the behavior (including failure) of its real-world asset counterpart. 3 [Liu et al., 2018] describes the development of the digital-twin concept in aerospace, while [Daily and Peterson, 2017] review the potential benefits and necessary ingredients of this concept. These authors describe some of the logistical, engineering, and data-science challenges associated with realizing such an ideal. At the heart is the difficulty in fusing multi-physics models, sensing, and ML/data-science technology. Digital twin concepts appear to apply best to clean-sheet engineering efforts, where the digital twins can be built as part of the engineering process. For existing complex systems, the engineering effort to reverse engineer digital models can be formidable and cost prohibitive.
Industry 4.0 represents the thought that modern industrial environments ought to be fully connected and self-aware. This includes automatic fault reporting, self-diagnosis, and automatic control of maintenance and production quality, among other capabilities. [Chukwuekwe et al., 2016] gives a very high level description of several interacting trends surrounding Industry 4.0. The authors propose (at the conceptual level) a closed loop feedback system for data-driven predictive maintenance. They advise that predictive maintenance elements and capabilities be standardized with an emphasis on interoperability. The authors suggest predictive maintenance capabilities need to be developed early in the design phase of modern equipment. [Lee et al., 2015, Lee et al., 2014a contrast today's technology with Industry 4.0 in which smart sensors and fault detection (today) are replaced with degradation monitoring and RUL prediction in self-aware machines. The authors discuss a 5-level architecture that builds from condition monitoring, to prognostics, to fleet level (peer-to-peer monitoring), to decision support, and finally to resilient control systems. Aspects of each level are discussed. [Meraghni et al., 2018] proposes a cloud-based architecture for making use of RUL and other prognostic results in PMx applications.
Many predictive maintenance reviews are focused on a specific industry, which heavily influences the nature of available data as well as logistic and process constraints in implementation. Spinning/cutting, CNC, and related machinery is a common focal industry [Ferreiro et al., 2016, Yan et al., 2017, Vogl et al., 2019, Sakib and Wuest, 2018, Trodd, 1998, Lee et al., 2014b. [Merizalde et al., 2019] gives a recent review of predictive maintenance in the wind power industry. [Barajas and Srinivasa, 2008] gives an early review of best practices in predictive maintenance in the automotive industry. The authors describe important enterprise level concepts in the deployment of predictive maintenance and health monitoring strategies. They break down faults along the dimensions of cost/impact and frequency and identify relevant envelopes for reactive, preventative, and predictive maintenance strategies. [Enrico et al., 2019] reviews reliability technologies in the aviation industry, and [Gerdes et al., 2016] surveys potential impact on flight delays. [Blechertas et al., 2009] describes a conceptual and procedural map for condition based maintenance approach for rotorcraft. The authors focus on the value of non-destructive analysis techniques, specifically describing the value of vibration, temperature, acoustic emission, electrical signature analysis, and oil/oil debris analysis. [Rezvanizaniani et al., 2014] reviews RUL prediction for batteries. Less common industrial focuses include naval [Tambe et al., 2015] and hospital operations [Galán and Gómez, 2018].
Other reviews focus on data processing and the IT infrastructure (e.g. cloud infrastructure) needed to deploy fleet level or enterprise level predictive maintenance solutions. [Wagner et al., 2016] gives an overview of different data sources that are used in predictive maintenance and the value of each. [Schmidt et al., 2016, Schmidt andWang, 2018] review concepts for adding contextual information to condition estimation, using cloud-based technology to facilitate centralized data processing and sharing fleet information. Similarly, [Galar et al., 2015] proposes a hybrid system that considers contextual information for weighing cumulative damage. The objective of the authors' approach is to fuse the capabilities of physics-of-failure modeling with observed data modeling to shore up the weakness of each. For example, observed data and failure/maintenance events can reduce uncertainty in the historic state of a system which physical models can then forecast. [Patwardhan et al., 2016] focuses on the technical infrastructure and steps necessary for data preparation in a big data environment. [Dragomir et al., 2009] reviews the advantages and disadvantages of physical-model based and data-driven approaches to RUL prediction. [Liu and Goebel, 2018] discusses the challenges of transitioning predictive health management systems to complex systems such as the next generation national airspace system.
Our review differs from the reviews described above in that we aim to highlight the gap between common predictive maintenance strategies and tools, with those needed to achieve the promise of PMx on large scale complex assets.
Primary Literature
We now turn to a detailed review of literature highlighting each PMx task. Before jumping in however, we pause to discus the issues that arise when different types of data are or are not available. We categorize information as sensor data, which describe the current behavior of an asset, maintenance logs which describe the actions taken with the intent to extend the utility of an asset or set of assets, and fault records which describe observed failures.
When fault records are unavailable or insufficient in number to support statistical analysis, the PMx effort is called unsupervised in contrast to supervised. In the supervised setting direct RUL prediction is the most common approach. In the unsupervised setting anomaly detection is the principal approach. In contrast, most planning algorithms presume a failure risk forecast.
Condition Estimation and Forecasting
The bulk of the academic literature in the field focuses on condition estimation and forecasting, often specializing to application domain. For this reason, we structure these studies according to domain and primary PMx sub-task; condition estimation and fault detection or RUL prediction. We make note of whether techniques are supervised or unsupervised, grouping similar methods together. Finally, we note special cases where sensor data and/or fault records are not available.
Bearings, spinning, and cutting
Condition estimation [Jia et al., 2019] uses a WS-ZHT1 multifunctional rotor test rig to simulate faults, generating supervised data. The authors evaluate infrared thermography (IRT) for condition estimation of bearings. They conclude that IRT base condition estimation is more effective than traditional vibration based methods. Other authors working in the supervised setting focus on vibration data. [Sezer et al., 2018] fit a model to predict roughness from vibration and temperature data in CNC machines. [Kateris et al., 2014] performs condition monitoring of bearings in rotating machinery using vibration data. The authors use neural networks to identify and locate (inner/outer race) faults using fully labeled data collected on a test machine. [Ferreiro et al., 2016] describes unsupervised predictive maintenance in the spinning tool setting. The authors use a finger-print learning method using supervised data to train a fault detector and an envelope analysis to detect outliers.
RUL prediction Supervised RUL prediction methods for bearings or rotating machinery very often use data from run-till-failure bench-top experiments. Occasionally, partially damaged bearings will be used to accelerate failure in order to gather more failure examples or explore specific failure modes. [Yan et al., 2017] describes a data processing pipeline for predictive maintenance in the industrial setting. The authors demonstrate predicting tool wear and tool RUL on CNC machines. While the authors describe processes for creating structured data from semi-structured or un-structured data common in the industrial setting, their demonstration focuses on the use of featurized vibrations data via envelope analysis and similar strategies. [Li et al., 2019a] developed a state-space based RUL prediction algorithm that is robust to varying operating conditions. The approach uses a particle filter with linear drift term. The linear term is modulated by operating conditions. A pair of operating condition dependent jump coefficients are introduced to the observation model, to account for jump discontinuities or change points in the observed degradation signal. In earlier work, [Bian et al., 2015] model degradation in a randomlyevolving environment modeled as a continuous-time Markov chain. The authors argue that most hazard models and prior research considers only static environments, which can lead to model-mismatch and degraded model performance if environments do vary. [Hao et al., 2017a] consider a serial processing line in which tool wear impacts production quality and production quality of preceding steps effects tool wear rates of downstream steps. A linear relationship between burr size and tool wear is presumed. The approach is demonstrated on simulated data. [Fumeo et al., 2015] uses an online-support vector regression machine to efficiently learn/predict RUL on railway bearings. The authors use vibration and temperature as inputs from run-to-failure data. [Liao et al., 2016] develop feature extraction capabilities for improved RUL prediction on bearing systems, again using run-till-failure experiments. [Luo et al., 2019] demonstrates RUL and wear prediction on spinning tools. The authors use advanced dynamic identification techniques to process vibrations data coupled with deep learning methods to produce their final predictive model. , Fang et al., 2019 develops tensor-based methods for RUL prediction from streams of infrared images of bearings. When stacked, these images form a rank-3 tensor. The approach is to project the tensors to a low-dimensional tensor subspace and then apply a penalized location-scale regression using RUL as the dependent variable. [Guo et al., 2017] apply recurrent neural networks to RUL estimation on bearings in the supervised setting. The authors conclude that RNNs give superior performance to self organizing maps. [Kanawaday and Sane, 2017] analyzed industrial cutting tools. The authors used unsupervised techniques to establish outliers. They then trained supervised models to forecast the occurrence of these outliers.
Gearboxes
Condition estimation [Zhao et al., 2019] develops supervised methods for fusing wavelets and deep learning approaches. The authors demonstrated their method on planetary gearbox fault diagnosis. [Wade et al., 2017] use vibration data to estimate condition of nose gearboxes (NGBs). Authors cite prior work indicating that vibration exceedences are a sub-optimal condition estimator due to variation in individual aircraft and components. The authors develop aerospace specific metrics for model selection. Data represents 600 assets with 40 ground-truth faults. Authors describe several metrics including bookmakers informedness (TPR-FPR), historical based TNR, asset based TNR, in-sample informedness, cross-validation informedness, absolute difference between in-sample and cross-validation informedness, and position shuffle. [Wade et al., 2015] describes data preparation for health status prediction of engine output gearboxes and turbo shaft engines. Health and Usage Monitoring System (HUMS) data are used as predictors, including Outside Air Temperature (OAT), Turbine Gas Temperature (TGT), Torque, Compressor Speed (NG), Power Turbine Speed (NP), Anti-Ice, Indicated Airspeed (IAS), and Barometric altitude. The predictive target is engine removal events for reason of low power/low torque (LPLQ). [Oehling and Barry, 2019] suggest that the state-of-art for informing safety from flight data is to monitor for exceedences of established thresholds. The authors use unsupervised outlier detection to identify potentially safety-relevant occurrences from flight data and compare to the exceednce-based approach. Outlier detection is shown to have good utility.
RUL prediction [Martin-del Campo et al., 2019] demonstrates an unsupervised dictionary learning based approach for faults in wind farms. Data are gearbox vibration records for six turbines (publicly available). Condition evaluation is effected by building anomaly detection capability using learned dictionaries by means of a "dictionary distance." Dictionaries are realized as a sparse coding model.
Turbines
Condition estimation [Rahman et al., 2018] use a supervised signature based algorithm for detecting and characterizing faults. Fault signatures are produced by simulating different fault types. [Rausch et al., 2007] also used supervised learning to detect and classify faults and used these classifications to adjust flight parameters in real-time for improved flight safety. Training data was again produced using numeric simulations. [Yan, 2016] uses unsupervised anomaly detection of redundant (simultaneous) temperature measures to diagnose combustor issues in gas turbine engines. Data is sampled at 1/60 Hz and an extreme learning machine (ELM) is adapted for anomaly detection. [Michelassi et al., 2018] presents a very similar work. [Michau et al., 2018] uses deep-learning based anomaly detection methods to identify potential faults in gas turbine data. The authors also explore the use of "sub-fleets" creating appropriate cohorts for comparison. [Xue et al., 2008] developed a fuzzy-similarity based method for estimating RUL on aircraft turbine engines. Authors analyzed cases of high pressure turbine shroud burn faults. Their algorithm identifies peer groups based on exhausted gas temperature (EGT), fuel flow (WF), and core speed (N2) after correcting for flight envelopes. Observed RUL from from identified peers is then aggregated to estimate the RUL of a target engine. Many supervised RUL prediction studies use the NASA turbofan dataset [Saxena and Goebel, 2008] as benchmark. [Fang et al., 2017b] develop methods for improved multivariate RUL regression, including feature selection. [Cao et al., 2018] proposes a change point detection modeling a (linear) gradual degradation to a subset of sensor streams (p0), where observations before and after the change point k are assumed to be i.i.d. normal. The detection is based on a generalized likelihood ratio (GLR) statistic considering average run length (ARL) and expected detection delay (EDD). Several extensions of the technique are proposed e.g. non i.i.d. case and modelling adaptive subset of crushed sensors p0. The method is demonstrated on stock bidding trend detection as well as the NASA turbofan dataset. [Fang et al., 2017a uses functional Principal Component Analysis (FPCA) and location-scale regression are used to predict time to failure (RUL) of partially degraded equipment. A multivariate FPCA and hierarchical FPCA is used for data fusion on a massive dataset. One of the key contributions is that the scalability of (multivariate) FPCA is enhanced by exploiting Randomized Low-rank Approximation (RLA) without knowing the rank of the RLA in advance. [Zhang et al., 2018] use a 3-layer LSTM for gas turbine engine RUL prediction. The authors define a health index as the output of a single ReLU neuron, fit to predict 1 at the beginning of an engine's time series and 0 at time of failure, regularized against first differences. This regularization encourages smooth health index trajectories. Finally, the 3-layer LSTM is trained for a one-step forecast task. By repeated forecasts at test time, RUL is inferred. [Ragab et al., 2016] uses a discrete logic approach for RUL prediction given observed operating parameters and condition indicators. [Li et al., 2019b] describes an ensemble RUL prediction approach, using Random Forest, CART, RNNs, and several other algorithms as constituents of the ensemble. The authors show that the ensemble is able to predict RUL on the NASA turbofan dataset with high accuracy. [Coraddu et al., 2016] describe a simulation experiment for CBM on gas turbines in naval ship propulsion. A sophisticated physics model characterizing gas turbines and ship propulsion is used to simulate data. They then use ML models to estimate decay rates from observed data. [Baraldi et al., 2012] use ensemble methods to RUL of the turbine blades of generators within nuclear power plants using simulated mechanical stress (mechanical fatigue) of the turbine blades.
RUL prediction
Vehicles
Condition estimation [Atamuradov et al., 2018] describes supervised health indicator (HI) construction, assessment, and prognostics for railway applications.
[ Rögnvaldsson et al., 2018] describes a life-long learning approach to fault detection, arguing that it is economically infeasible to use human experts to build, evaluate, and field predictive models for each failure mode. This is especially true for novel failure modes or (potentially) occasionally modified equipment. The paper gives a good review of unsupervised methods. The authors remark that most of the prior work presumes high-quality features are provided (presumably by experts) and that little work in the unsupervised space accounts for inter-asset variation. The authors' approach is based on Consensus Self-organizing Models (COSMO), and the basic elements of the strategy are to first identify interesting (in an information theoretic sense) functional transformation of raw sensor data and then to compare these derived values across the fleet. One or a few outliers were there is general consensus otherwise in one or more derived signal suggests a fault. The authors conclude that there is a significant need to improve the quality of date in service records. [Dubrawski and Sondheimer, 2011] demonstrate detection of escalating maintenance issues by comparing event counts with historical counts as well as with similar cohorts. [Bonissone andVarma, 2005, Bonissone et al., 2005] present a fuzzy-similarity based approach to identify peer groups for a fleet of 1100 locomotives. Peers are similar in maintenance history, usage, and expected behavior. Observed RUL is aggregated across peers to estimate RUL for target locomotives. The authors use an evolutionary framework for model optimization to maintain an up-to-date similarity measure. [Teixeira et al., 2015] models the evolution of fault magnitude in components, using their supervised model to disregard apparent faults that do not follow expected evolutionary behavior. The result is that their model successfully disregards most cases of "no fault found." [Le et al., 2017] use ML models to predict RUL for engine oil in land based military vehicles. Data collected from VHUMS included engine RPM, temperature, throttle position, oil temperature, among others. Oil condition was measured by means of laboratory tests. Data included 16 vehicles with a total of 30 oil test results. Rule-learning gave very good performance in stratified cross-validation (number of folds was unspecified). [Nascimento and Viana, 2019] proposes an LSTM with monotonic damage accumulation. The utility of the model is demonstrated by synthetic simulation. Training data are "far field stresses," and labels are periodic inspections for cracks. [Magargle et al., 2017] gives and in-silico demonstration of digital-twin methodology in support of predictive maintenance for automotive breaks. By reference to the digital twin, wear rate is inferred from data and RUL predictions are made. [Prytz et al., 2015] describes the application of predictive maintenance to a fleet of trucks. Three years of data are used to demonstrate the approach. The authors describe common difficulties; data is co-opted for mining, maintenance records are incomplete and free-text based, etc. The authors note that the feature distribution used for predicting future faults is age dependent, and apply modeling strategies to correct for equipment age. They also discuss issues arising of dependence among observations in the data set and recommend a leave-one-vehicle out cross-validation approach. [Nixon et al., 2018] describes predictive maintenance analysis on diesel engines for military vehicles. Input data consists coarsely sampled measures from the engine management computer. Predictive targets were created by grouping unscheduled maintenance actions by failure mode. [Baptista et al., 2019] studies how Kalman filtering can be used to smooth RUL estimates over time, reducing noise and improving overall accuracy. [Cipollini et al., 2018] evaluate several ML approaches for engine health analysis on naval vessels. The authors conclude that unsupervised anomaly/outlier detection methods are the most appropriate as they can be realized with minimal ground-truth. A public dataset is available for this work.
RUL prediction
Industrial plant operations
Condition estimation [Amruthnath and Gupta, 2018] explores some unsupervised methods for fault detection using vibration data from a cooling fan. [Graß et al., 2019] proposes an unsupervised approach for anomaly detection in time-series data representing configuration-based electronics production lines. [Hendrickx et al., 2018] describes an unsupervised clustering approach for comparing similar machines in industrial environments. Anomalies are detected by means of monitoring similarity among machine equivalence classes. [Kroll et al., 2014] describes an anomaly detection strategy based on discretecontinuous hybrid automata. [Mattes et al., 2012] evaluates Bayesian networks, Random Forest, and linear regression for supervised RUL prediction using equipment specific sensor data. [Susto et al., 2015] presents a supervised model for predicting failure within m iterations using data from run-to-failure experiments for an ion implanter tool. [Susto and Beghi, 2016] explores the application of a time-series featurization approach to RUL prediction. [Bastos et al., 2014] describes an ML framework for predictive maintenance of a nuclear plant. Features consist of monitoring data reported at 1 Hz frequency. The prediction target are failures, recorded in maintenance records.
RUL prediction
Other
Condition estimation [Poosapati et al., 2019] proposes a rule-based strategy for processing anomalies in predictive maintenance applications. Their goal is to develop cognitive reasoning capabilities that can recognize patterns and suggest courses of action. [Cristaldi et al., 2016] evaluates a few models for supervised RUL prediction of a "fleet" of circuit breakers. Models have access to an observed health condition (HC), and forecast the point at which the HC reaches the end-of-life level using observed time-series as inputs. Fleet level data is used to learn probability distributions over HC variation. [Cline et al., 2017] review 19 years of inspection data for swivels and valves on oil and gas equipment. Authors noted that they were unable to compute residual life of the majority of components due to the fact that most components never failed. Failure within next year was selected as the most viable target. Features included wear-index, derived values thereof, and counts of previous failure or inspection events. [Bey-Temsamani et al., 2009] suggests that data engineering and feature selection through case studies can be used to facilitate later development of prognostic models. The authors demonstrate RUL prediction on copy machines. [Mishra et al., 2018] apply hierarchical Bayesian modeling to forecasting battery performance. The hierarchical modeling structure effects a peer-to-peer comparison, and can make predictions without sensing data based on a battery's peer group (i.e. its prior). [Xin et al., 2017] extends Bayesian hazard modeling for fire and industrial accidents to include dynamics.
RUL prediction
No sensors
In the absence of sensor data, researchers have used maintenance and/or failure data to uncover patterns that can forecast future failures. [Baptista et al., 2018] proposes an ARMA based model for supervised prediction of RUL/failure risk aimed at reducing unnecessary removals and avoiding failure. The authors use an ARMA model and PCA to featurize time-series of past removal/failure events and pass this through a predictive model which forecasts RUL. They demonstrate on a data set of 584 engine bleed valve removals. [Sipos et al., 2014] uses distribution-classification to predict upcoming failure from the distribution of observed fault codes in log-data collected from medical equipment. Service notifications are used to denote failure. [Korvesis et al., 2018] parse post-flight event logs to predict landing gear faults in aircraft. Druzdzel, 2016, Kraisangka andDruzdzel, 2018] show how Bayesian networks can be used to model hazard rates, leading to more powerful models. [Wang et al., 2017] demonstrate a classification based approach for predictive maintenance in automated teller machines (ATMs). The authors use statistics of error message occurrences, occurrences of temporal patterns of error messages, and individual machine characteristics (model, installation date, etc.). Error message type is extracted from error codes present in the ATM log files. Labels are determined by the occurrence of a maintenance ticket. [Salo et al., 2018] present a poster describing an NLP pipeline for extracting useful information from free-text maintenance write-ups in wind farm data. The approach cluster text descriptions into equivalence classes, grouping write-ups that describe the same/similar maintenance actions.
No sensors and no faults
If neither explicit failures nor sensor data are available, one can predict future maintenance using historical maintenance. For example, [Gardner et al., 2017] uses tensor decomposition to data-mine maintenance data for patterns. A rank-3 tensor is created out of vehicle ID, maintenance action type, and time. An LSTM is trained to forecast maintenance actions as well, treating each vehicle's time series as an observation.
Maintenance and Operational Planning
Most work in predictive maintenance does not consider variable workloads, operating conditions, or equipment use. Further, those that do explicitly take these issues into account [Hao et al., 2017b, Li et al., 2019a, Bian et al., 2015 generally do not forecast use and/or modify usage plans in consideration of degradation status. [Biteus and Lindgren, 2017] is an exception. The authors describe an end-to-end predictive maintenance program that predicts failure risks, schedules maintenance actions, and creates condition-aware plans of utilization (route planning) for a fleet of trucks. Maintenance actions are broken down into the smallest possible units and transformed into constraint rules. A random forest is used to predict failure risk, and constrained optimization strategies are used to produce maintenance and route plans. The approach is demonstrated on a fleet of 80,000 trucks and a single component (air dryer purge valve) for which failures are observed in 1.6% of records. Data are publicly available in UCI repository [Dua and Graff, 2017]. [Maillart, 2006] applies a POMDP framework, assuming that without maintenance, system state degrades stochastically, over discrete states, according a known transition function. Maintenance costs are differentiated according to whether they are preventive or reactive. Both types of actions are assumed to return the system to like-new condition. POMDP formulations are also explored by [Ghasemi et al., 2007, Jiang et al., 2015, Li and Pozzi, 2019. [Yildirim et al., 2016a, Yildirim et al., 2016b] represent a two-part paper. In part I, the authors assume the ability to observe a degradation signal which is given by a parametric degradation function plus additive noise. Observation of the degradation signal allows inference of the asset-specific degradation parameters, some of which are shared across a fleet. A Bayesian model is presumed, and the distribution of RUL for each asset is inferred from the observed degradation signal. A cost function relates RUL to cost by dictating a different (lower) cost for planned maintenance than for failure events. A maintenance action (planned or otherwise) is presumed to return the asset to "new" status (note assets are treated as singlecomponent systems). A mixed-integer program is defined for characterizing total maintenance costs. A constraint on labor capacity couples maintenance actions across assets. In part II, the mixed-integer program is extended to include constraints on asset commitments and loads. These can encode constraints on the number of up/down transitions, total availability or capacity, etc. Experiments demonstrate significant improvement in reliability and reduced costs over standard practice. [Yildirim et al., 2017] demonstrates a very similar approach to [Yildirim et al., 2016a, Yildirim et al., 2016b for a fleet of wind turbines. The authors add constraints that limit location visits on the part of the maintenance crew, constraints of maintenance effort, and constraints on turbine output which couples the maintenance effort across turbines encouraging concurrent maintenance actions. This leads to cost optimization.
Maintenance scheduling
[ Basciftci et al., 2018] extends the mixed-integer programming planning algorithm of [Yildirim et al., 2016a, Yildirim et al., 2016b to include a probabilistic constraint on availability. This constraint ensures that the likelihood of too many assets in maintenance simultaneously is low. The purpose of this constraint is to guard against the risks and costs of unexpected failures. [Moghaddass and Ertekin, 2018] solves the joint condition estimation and maintenance planning problem for single-component systems. The authors assume preventative maintenance is less expensive than failure and maintenance actions require a certain lead time. The approach is demonstrated with numerical simulations. [Yang et al., 2008] describes a genetic-algorithm optimization approach for scheduling maintenance actions based on noisy RUL predictions. [Rajora, 2018] is a dissertation largely focusing on solving hierarchical coupled constraint optimization problems that arise in maintenance scheduling and assembly planning problems. [Hao et al., 2017b] presumes that system degradation is a function of workload (increased workload increases degradation). The authors develop a control system that dynamically modulates workload between multiple machines, based on posterior degradation belief distributions. The controller seeks to guide failure of machines in such a way that they do not overlap, reducing risk of work-stoppage. The approach is demonstrated on simulated stamping machines. [Lin et al., 2018] argues that most CBM-oriented research in the aerospace domain focus on minimizing cost or maximizing availability of single aircraft in isolation, and rarely consider both objectives simultaneously much less that for an entire fleet. The authors propose a model for doing just that. The model assumes a simple deterministic damage function (of time) and cost function. The authors use support vector regression to effect the multi-objective optimization. [Feng et al., 2017a] describes a learning game-theoretic approach to fleet-level maintenance strategy aimed a minimizing cost under an availability constraint. The game is focused on learning strategies of when to replace line-replaceable modules, given failure probabilities. The authors also touch on the NP-hard nature of the fleet level CBM problem. [Feng et al., 2017b] extends this work to include dispatched and standby sets of aircraft. Again, game theory is used to search for optimal decision strategies.
Performance quantification
It is advisable to understand the level of predictive performance necessary for a predictive maintenance effort to yield positive utility. Such measures serve the important function of defining success both for proofs-of-concept predictive models and system performance while scaling solutions to the enterprise level. Toward that end, [Busse et al., 2018] demonstrates an a priori cost-benefit-analysis for predictive maintenance capabilities. This is significant, as such analyses can provide the aforementioned understanding. The authors use a Wiener process with linear drift to model the predictions of a hypothetical RUL prediction module. They then push sampled predictions through different maintenance planning strategies, and compute total costs using a hypothetical cost model. The demonstration is conducted for a single component machine with single failure mode. [Lei et al., 2018] reviews performance metrics for RUL prediction. The authors divide metrics into offline and online measures. Offline metrics measure accuracy of RUL estimations or failure risks for example. THe proposed online metrics, in contrast, do not require knowledge of future failures, comparing the current RUL estimate to its recent estimates. [Boev et al., 2019] sketch out a constrained optimization based approach for prescribing maintenance plans and spare part availability.
Supply planning
Gap Analysis
In the reviewed literature, the asset under study is sometimes simple such as a bearing or cutting tool and sometimes complex such as a gas turbine or automotive engine. However, when complex assets are considered, it is largely the case that either only a small number of simple components or a small number of failure modes are studied. As such, these complex assets are treated using methods analogous to those for individual components. This approach has the advantage that methods and insights developed using run-till-failure bench experiments on bearings say, can be utilized on larger systems where runtill-failure is not realistic. Further, it could be argued that one could repeat such a process for all the major components and/or failure modes of a complex asset. It has been pointed out however, that this approach could be prohibitively expensive due to the resources needed to build and maintain the numerous required models [Rögnvaldsson et al., 2018]. Further, if dependencies between failure modes are to be taken into account, then there is little justification for not starting with a comprehensive model approach.
In our view, the primary gaps between our view of PMx for complex assets and current literature, center on the handling of condition and failure risk estimation/forecasting. We identify two principal gaps: failure to incorporate inter-component interactions, and failure to address the effects of maintenance.
Modeling Interactions
Modeling interactions between components can enable sub-system or system level models of failure risk, facilitating PMx for fleets of complex assets. This is not a new concept. We review some initial work in detail below. But first, we highlight work from Dependability Modeling and Analysis, a closely related discipline that specializes in this area.
Dependability Modeling and Analysis
Reliability and dependability analysis is standard practice is product design. The term describes the problem of quantifying the risk and nature of failures of (typically complex) equipment. Generally, the goal of dependability modelling is to relate basic events, which often represent failure of individual components, to overall sub-system and system level behavior. Such models can be used to determine the criticality of different components, overall system robustness, as well as to diagnose, correct, and avoid failures. Common methods include Failure Mode and Effects Analysis (FMEA), Failure Modes, Effects and Criticality Analysis (FMECA), (dynamic) fault-trees, (dynamic) Bayesian networks, and stochastic Petri-nets. These methods are currently being integrated into the DoD digital engineering strategy [Boydston et al., 2015] on the Future Vertical Lift (FVL) program. Of particular note on the FVL efforts is the use of modeling at both the system and subsystem level.
[ Chemweno et al., 2018] gives a recent review of dependability modelling with a focus on the treatment of uncertainty, both uncertainty of predictions (aleatory) and uncertainty of the model (epistemic) [Fox andÜlkümen, 2011]. The authors find that dynamic fault-tree analysis and dynamic Bayesian networks are the most common methods, together accounting for 44% of the dependability modelling literature (as measured by count of articles). They note that while Bayesian methods are naturally suited for combining evidence from different sources, limited reliability data necessitates quantifying the epistemic uncertainty beyond typically analysis of posterior distributions. Toward that end, the authors review fuzzy analysis, interval analysis, and Dempster-Shafer evidence theory (DSTE) for quantifying epistemic uncertainty. DSTE is the most common such method accounting for 46% of articles that address epistemic uncertainty. Finally, the authors identify inclusions of predictive models of failure probability into reliability models as a key future research direction.
In that respect, there are several degrees of potential integration between these distinct modeling exercises. One may build RUL and/or failure risk forecasting capability for individual components, treating each as independent. The forecast risks can then be fed into a reliability model to more comprehensively inform risk assessment process. [Lee and Pan, 2019] can be viewed as a step in this direction. The authors combine estimates of failure probability via a Markov model with a reliability model using a tree-structured Bayesian network. If significant dependencies exist in the failure risks of basic events, they will have to be taken into account. Sub-system or system level faults may impact the degradation rates of components (e.g. adjusting workloads or operating conditions due to a fault). In such circumstances is may be desirable to model basic failure risk and system reliability jointly. This could be accomplished using dynamic Bayesian networks, for example. [Chiacchio et al., 2016a, Chiacchio et al., 2016b join stochastic hybrid automaton with dynamic fault-trees to jointly model age of components and failure risk under dynamic operating conditions. However, no learning is performed as the governing equations of the approach are given upfront.
Primary literature focused on complex assets
Some authors have begun to address the challenges that arise when considering complex assets. Often this means modeling the relationship between sensing and component state and component-component interactions. [Rodrigues, 2017] introduces a particle filter model wherein the observation function is informed by system architecture. Incorporating this system-level model allows the method to relate system level performance indicators to component health state. The authors model the component health state as a gamma process. The method is demonstrated on two simulated data sets; a simplified multi-pump hydraulic system and a multi-component air conditioning system. [Lee and Pan, 2019] assume the degradation state of each component is described by a discrete vector with hi ∈ {0, 1, . . . , fi}, where i = 1, . . . , N enumerates components and fi ∈ N is the failure state for component i. These so-called health states are presumed to be increasing in severity, until failure. Health state values are forecast n time steps into the future using a Markov model, for which the transition matrices Pi are known (or learned from historical data) for each component. The Pi also encode the assumption of non-decreasing state transitions, i.e. no spontaneous repair. Let hi be a one-hot vector encoding the current health state for component i, then P n i hi is the posterior health state distribution for component i. Finally, probability of the system or a sub-system level failure is computed by a treestructured Bayesian network, for which the parameters (conditional probability tables) are presumed known a priori.
[ Barde et al., 2019] demonstrates a classical reinforcement learning strategy for maintenance of a fleet of trucks. The authors consider 8 components, and use a model-free approach with tabular Q function to learn the optimal maintenance policy under different choices of reward function. This type of reinforcement learning does have optimality guarantees in the limit of sufficient state-action space exploration. However, the main advantage may be that it is easy to integrate complex logistics and incorporate the effects of multiple concurrent maintenance actions. Unfortunately, this kind of approach can only work if (i) the state-action space is discrete and of low enough arity that it can be sufficiently explored, (ii) ample observed data or realistic simulations of equipment histories are available, and (iii) failures are observed. If maintenance is largely preventative, the learning agent will not effectively be able to directly learn policy since it will not encounter penalties associated with failure. Additionally, in real-world complex equipment, the state-action space is likely to be at least partly continuous and complex, requiring function approximation techniques to learn the Q function. In practice, these conditions would require massive amounts of trials to find good policies. Further, current opinion in the field is that reinforcement learning using model approximation can be very difficult to tune properly and can produce sporadic unanticipated behavior. This is unacceptable in safety critical applications such as e.g. aerospace.
[ focuses on maintenance planning for a fleet of aircraft. The authors presume that a probability-of-failure model is given for each component, which is a function of the component's damage level. Aircraft failure probability is taken as the maximum component level failure probability. This assumption may be in error and the aircraft failure probability depends on the statistical dependency between components. In any case, the authors define a repair cost function, dependent on the damage level of a component and a wasted RUL function. Finally, they optimize a two-objective decision model under the constraint that failure probability is very small. [Hao et al., 2015] consider sub-system level sensing, e.g. vibration measurements, and study how one can isolate component-level degradation signals. The authors use independent component analysis (ICA) to separate the degradation signals for a known number of components and demonstrate RUL prediction on synthetic data. [Blancke et al., 2018] describes the use of Petri-nets for failure risk forecasting on complex systems. Their approach relies on expert knowledge of failure physics, and models fault propagation using a colored Petri net. Modeling the fault propagation allows for prescriptive diagnostic inference as well.
Maintenance
Maintenance of complex assets raises two primary issues. The first is that maintenance censors future failure events. Second, maintenance actions could alter the latent degradation state and its trajectory in non-trivial ways. Yet, little to no work has been put toward modeling the impact of maintenance on the latent degradation state. Figure 3: Illustration of state-space trajectories for 3 assets. Solid circles indicate measurement events, stars represent failure events. The dash arrow indicates a state transition due to maintenance action. Figure 3 illustrates this concept. The trajectories of three assets in the latent degradation space are shown. The failure boundary reflects the level of degradation that results in an observed failure. Asset 1 degrades and fails. For this asset, time-till-failure would be retrosepctively available. Asset 2 degrades, but transitions to another point of the state-space due to maintenance action, after which it degrades to failure along a different trajectory. For this asset time-till-failure would be misleading for early observations as they are confounded by the effect of the maintenance. No failure is observed for Asset 3, making it unusable for simple supervised RUL-based methods. Figure 3 represents a Markovian view-point. But the existence of such a latent state is well motivated by the predictive state representation (PSR) approach to partially observable Markov decision processes (POMDPs) [Littman and Sutton, 2002, Singh et al., 2003, Boots et al., 2011. Our perspective is that modeling the degradation process in this way neatly addresses the effects of maintenance on system evolution towards failure. We can structure this formulation as a representation learning task. The objective would be to learn an embedding function that would map an asset's history into a latent vector representation. The evolution of these vectors could be presumed (e.g. incremented by cumulative historical load), or modeled. Finally, the effect of each maintenance action could also be modeled. This approach naturally makes use of all available data whether or not failures are observed. It can be realized in many ways. For example, one could use deep recurrent networks to map histories to the latent state and model maintenance as additive functions of current state and action type. Finally, structural, physical, or reliability models of the assets can be incorporated into this modeling exercise to reduce data-driven model learning costs and improve accuracy.
Conclusion
We reviewed current literature in the field of predictive maintenance. We identified several fundamental differences between condition estimation and failure risk forecasting as applied to simple components such as bearings and cutting tools from the capabilities needed to solve the same tasks on complex assets. These differences stem from complex latent degradation states, active maintenance programs, increased coupling between maintenance actions, and higher monetary and safety costs for failures.
As a result, methods that are effective for forecasting risk and informing maintenance decisions for individual components do not readily scale to sub-system or system level insights. A holistic modeling approach is needed that incorporates available structural and physical knowledge and naturally handles the complexities of actively fielded and maintained assets.
[ Barajas and Srinivasa, 2008]
Figure 1 :
1Count of predictive maintenance related academic publications by year (as of March 1, 2019).
Figure 2 :
2Model components for fleet level predictive maintenance of complex equipment
[
Bian et al., 2015] Bian, L., Gebraeel, N., and Kharoufeh, J. P. (2015). Degradation modeling for realtime estimation of residual lifetimes in dynamic environments. IIE Transactions, 47(5):471-486. [Biteus and Lindgren, 2017] Biteus, J. and Lindgren, T. (2017). Planning flexible maintenance for heavy trucks using machine learning models, constraint programming, and route optimization. SAE International Journal of Materials and Manufacturing, 10(3):306-315. [Blancke et al., 2018] Blancke, O., Combette, A., Amyot, N., Komljenovic, D., Lévesque, M., Hudon, C., Tahan, A., and Zerhouni, N. (2018). A predictive maintenance approach for complex equipment based on petri net failure mechanism propagation model. In Proceedings of the European Conference of the PHM Society, volume 4. [Blechertas et al., 2009] Blechertas, V., Bayoumi, A., Goodman, N., Shah, R., and Shin, Y.-J. (2009). CBM fundamental research at the university of south carolina: a systematic approach to us army rotorcraft CBM and the resulting tangible benefits. In The American Helicopter Society Technical Specialists' Meeting on Condition Based Maintenance. Huntsville. [Boev et al., 2019] Boev, S., Kiryanov, D., and Matveeva, S. (2019). Adaptive management of system parameters of maintenance, as well as repair and optimization of spare parts, tools and accessories of the integrated radar system on the basis of reliability model and control data. In International Scientific Conference" Far East Con"(ISCFEC 2018). Atlantis Press. [Bonissone et al., 2005] Bonissone, P., Varma, A., and Aggour, K. (2005). A fuzzy instance-based model for predicting expected life: A locomotive application. In IEEE int. conf. on computational intelligence for measurement systems and applications, pages 20-25. [Bonissone and Varma, 2005] Bonissone, P. P. and Varma, A. (2005). Predicting the best units within a fleet: prognostic capabilities enabled by peer learning, fuzzy similarity, and evolutionary design process. In The 14th IEEE International Conference on Fuzzy Systems, 2005. FUZZ'05., pages 312-318. IEEE.[Boots et al., 2011] Boots, B., Siddiqi, S. M., and Gordon, G. J. (2011). Closing the learning-planning loop with predictive state representations. The International Journal of Robotics Research, 30(7):954-966. [Boydston et al., 2015] Boydston, A., Feiler, D. P., Vestal, D. S., and Lewis, B. (2015). Joint common architecture (JCA) demonstration architecture centric virtual integration process (ACVIP) shadow effort. In AHS 71st Annual Forum May 5-7, 2015. Copyright c 2015 by the American Helicopter Society International, Inc., page 12, Virginia Beach, VA. [Busse et al., 2018] Busse, A., Metternich, J., and Abele, E. (2018). Evaluating the benefits of predictive maintenance in production: A holistic approach for cost-benefit-analysis. In Congress of the German Academic Association for Production Technology, pages 690-704. Springer. [Cao et al., 2018] Cao, Y., Xie, Y., and Gebraeel, N. (2018). Multi-sensor slope change detection. Annals of Operations Research, 263(1-2):163-189. [Celaya et al., 2009] Celaya, J., Wysocki, P., and Goebel, K. (2009). IGBT accelerated aging data set. NASA Ames Prognostics Data Repository. [Chemweno et al., 2018] Chemweno, P., Pintelon, L., Muchiri, P. N., and Van Horenbeek, A. (2018). Risk assessment methodologies in maintenance decision making: A review of dependability modelling approaches. Reliability Engineering & System Safety, 173:64-77. [Chiacchio et al., 2016a] Chiacchio, F., D'Urso, D., Compagno, L., Pennisi, M., Pappalardo, F., and Manno, G. (2016a). SHyFTA, a stochastic hybrid fault tree automaton for the modelling and simulation of dynamic reliability problems. Expert Systems with Applications, 47:42-57. [Chiacchio et al., 2016b] Chiacchio, F., D'Urso, D., Manno, G., and Compagno, L. (2016b). Stochastic hybrid automaton model of a multi-state system with aging: Reliability assessment and design consequences. Reliability Engineering & System Safety, 149:1-13.
Table 1 :
1Top 5 leading countries by publication count.Country
Number of
publications
% of
publications
United States Of America
1578
25.402
People's Republic China
1544
24.855
France
421
6.777
United Kingdom
327
5.264
Canada
245
3.944
Table 2 :
2Top 10 leading funding agencies by publication count, after some de-duplication.Agency
Barajas, L. G. and Srinivasa, N. (2008). Real-time diagnostics, prognostics and health management for large-scale manufacturing maintenance systems. In Asme 2008 international manufacturing science and engineering conference collocated with the 3rd jsme/asme international conference on materials and processing, pages 85-94. American Society of Mechanical Engineers.[Baraldi et al., 2012] Baraldi, P., Mangili, F., and Zio, E. (2012). A Kalman filter-based ensemble approach with application to turbine creep prognostics. IEEE Transactions on Reliability, 61(4):966-977. [Barde et al., 2019] Barde, S. R., Yacout, S., and Shin, H. (2019). Optimal preventive maintenance policy based on reinforcement learning of a fleet of military trucks. Journal of Intelligent Manufacturing, 30(1):147-161. [Basciftci et al., 2018] Basciftci, B., Ahmed, S., Gebraeel, N. Z., and Yildirim, M. (2018). Stochastic optimization of maintenance and operations schedules under unexpected failures. IEEE Transactions on Power Systems, 33(6):6755-6765.[Bastos et al., 2014] Bastos, P., Lopes, I., and Pires, L. (2014). Application of data mining in a maintenance system for failure prediction. Safety, Reliability and Risk Analysis: Beyond the Horizon: 22nd European Safety and Reliability, 1:933-940. [Bey-Temsamani et al., 2009] Bey-Temsamani, A., Engels, M., Motten, A., Vandenplas, S., and Ompusunggu, A. P. (2009). A practical approach to combine data mining and prognostics for improved predictive maintenance. Data Min. Case Stud, 36.
[Zhang et al., 2018] Zhang, J., Wang, P.,Yan, R., and Gao, R. X. (2018). Deep learning for improved system remaining life prediction. Procedia CIRP, 72:1033-1038. [Zhao et al., 2019] Zhao, M., Kang, M., Tang, B., and Pecht, M. (2019). Multiple wavelet coefficients fusion in deep residual networks for fault diagnosis. IEEE Transactions on Industrial Electronics, 66(6):4696-4706. ARL Average Run Length ARMA Autoregressive Moving Average ATM Automated Teller Machine CART Classification And Regression Trees CBM Condition Based Maintenance CNC Computer Numerical Control COSMO Consensus Self-Organizing Models DET Digital Engineering Transformation DSTE Dempster-Shafer Evidence Theory EDD Expected Detection Delay EGT Exhausted Gas Temperature ELM Extreme Learning Machine EU European Union FEMTO Franche-Comté Electronics Mechanics Thermal Science and Optics FMEA Failure Mode and Effects Analysis FMECA Failure Modes, Effects and Criticality Analysis FPCA Functional Principal Component Analysis FPR False Postive Rate FVL Future Vertical Lift GLR Generalized Likelihood Ratio HUMS Health and Usage Monitoring System IAS Indicated Airspeed ICA Independent Component Analysis IGBT Insulated Gate Bipolar Transistor IMS Intelligent Maintenance Systems IRT Infrared Thermography LPLQ Low Power/Low Torque LSTM Long Short-Term Memory N2 Core Speed NASA National Aeronautics and Space Administration NG Compressor Speed NGB Nose Gearboxe NLP Natural Language Processing NP Power Turbine Speed OAT Outside Air Temperature PCA Principle Component Analysis PdM Predictive Maintenance PHM Predictive/Prognostic Health Management PMx Predictive Maintenance POMDP Partially Observable Markov Decision Process PSR Predictive State Representation RCM Reliability Centered Maintenance RLA Randomized Low-rank Approximation RNN Recurrent Neural Network RPM Rotations Per Minute RUL Remaining Useful Life TGT Turbine Gas Temperature TNR True Negative Rate TPR True Positive Rate UCI University of California, Irvine USA United States of America VHUMS Vehicle Health and Usage Monitoring System WF Fuel FLowA Acronyms
Perhaps measured in accumulated load and/or use, as opposed to wall clock time.
The idea of a digital twin is also found within the U.S. Depratment of Defense under the banner of the Digital Engineering Transformation (DET) and is spelled out in the Digital Engineering strategy by the [Office of the Deputy Assistant Secretary of Defense for Systems Engineering, 2018].
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Machine learning approaches for improving condition-based maintenance of naval propulsion plants. [ Coraddu, Proceedings of the Institution of Mechanical Engineers, Part M: Journal of Engineering for the Maritime Environment. the Institution of Mechanical Engineers, Part M: Journal of Engineering for the Maritime Environment230[Coraddu et al., 2016] Coraddu, A., Oneto, L., Ghio, A., Savio, S., Anguita, D., and Figari, M. (2016). Machine learning approaches for improving condition-based maintenance of naval propulsion plants. Proceedings of the Institution of Mechanical Engineers, Part M: Journal of Engineering for the Mar- itime Environment, 230(1):136-153.
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| [] |
[
"Towards an Optimal Contention Resolution Scheme for Matchings",
"Towards an Optimal Contention Resolution Scheme for Matchings"
] | [
"Pranav Nuti \nStanford University\n94305StanfordCA\n",
"Jan Vondrák \nStanford University\n94305StanfordCA\n"
] | [
"Stanford University\n94305StanfordCA",
"Stanford University\n94305StanfordCA"
] | [] | In this paper, we study contention resolution schemes for matchings. Given a fractional matching x and a random set R(x) where each edge e appears independently with probability x e , we want to select a matching M ⊆ R(x) such that Pr[e ∈ M | e ∈ R(x)] ≥ c, for c as large as possible. We call such a selection method a c-balanced contention resolution scheme.Our main results are (i) an asymptotically (in the limit as x ∞ goes to 0) optimal 0.544-balanced contention resolution scheme for general matchings, and (ii) a 0.509-balanced contention resolution scheme for bipartite matchings. To the best of our knowledge, this result establishes for the first time, in any natural relaxation of a combinatorial optimization problem, a separation between (i) offline and random order online contention resolution schemes, and (ii) monotone and non-monotone contention resolution schemes. We also present an application of our scheme to a combinatorial allocation problem, and discuss some open questions related to van der Waerden's conjecture for the permanent of doubly stochastic matrices. | 10.48550/arxiv.2211.03599 | [
"https://export.arxiv.org/pdf/2211.03599v1.pdf"
] | 253,384,641 | 2211.03599 | 426ffc09348c1212e29537a96a27626928284673 |
Towards an Optimal Contention Resolution Scheme for Matchings
7 Nov 2022
Pranav Nuti
Stanford University
94305StanfordCA
Jan Vondrák
Stanford University
94305StanfordCA
Towards an Optimal Contention Resolution Scheme for Matchings
7 Nov 2022
In this paper, we study contention resolution schemes for matchings. Given a fractional matching x and a random set R(x) where each edge e appears independently with probability x e , we want to select a matching M ⊆ R(x) such that Pr[e ∈ M | e ∈ R(x)] ≥ c, for c as large as possible. We call such a selection method a c-balanced contention resolution scheme.Our main results are (i) an asymptotically (in the limit as x ∞ goes to 0) optimal 0.544-balanced contention resolution scheme for general matchings, and (ii) a 0.509-balanced contention resolution scheme for bipartite matchings. To the best of our knowledge, this result establishes for the first time, in any natural relaxation of a combinatorial optimization problem, a separation between (i) offline and random order online contention resolution schemes, and (ii) monotone and non-monotone contention resolution schemes. We also present an application of our scheme to a combinatorial allocation problem, and discuss some open questions related to van der Waerden's conjecture for the permanent of doubly stochastic matrices.
Introduction
Suppose that there are n employees looking for jobs. Each employee likes a random set of jobs which, on average, has cardinality one. n jobs are available in total, and no job is especially popular amongst the employees, though some employees might have a strong preference for some particular jobs. We would like to match the employees to jobs.
We are immediately faced with many natural questions: On average, what fraction of employees can we match to a job they like? Can we match employees to jobs in a fair way, without partially favoring any particular employee? What if no employee has a strong preference for any particular job? Is it easier to match employees if we learn about their preferences all at once, rather than if we learn about them in an online fashion? Our paper provides answers to these questions, through the lens of contention resolution schemes.
Contention resolution schemes aim to solve the following problem: Given a family of feasible sets F ⊂ 2 E and a random set R sampled from a distribution on 2 E , how can we choose a feasible subset I ⊆ R, I ∈ F, so that each element from R is picked with some guaranteed conditional probability: Pr[e ∈ I | e ∈ R] ≥ c for some fixed c > 0 and all e ∈ E? We call such a scheme c-balanced. This balancedness is a kind of fairness constraint, ensuring every element e has a reasonable chance of making it into I.
In this paper, we think about E as the set of edges in a graph, and F as the set of matchings of the graph. The constant c is the conditional probability with which we can ensure an edge ends up in the matching I we pick, given it appears in R.
A natural assumption on the random set R is that it comes from a product distribution with marginal probabilities x e such that x is in a polytope corresponding to the family F (either the exact convex hull, or a suitable relaxation, depending on the application). For matchings on graphs, this corresponds to an assumption that each edge e appears in R independently with probability x e , and the vector (x e ) e∈E belongs to the matching polytope, i.e, is a fractional matching.
The formal notion of contention resolution was first investigated as a tool for randomized rounding. It was introduced by Feige [3], who developed a contention resolution scheme (CRS) for matchings on the restricted class of star graphs, in the context of an application to combinatorial auctions. CRSs were then investigated more systematically in [7] in the context of submodular optimization. In particular, an optimal (1 − 1/e)-balanced CRS was identified in [7] for the case where F forms a matroid. The 1 − 1/e factor is optimal even for F = {I : |I| ≤ 1}.
Contention resolution has also been studied in online settings with either adversarial or random ordering of elements [9,11,12,13,18]. For example, for matroids there is a 1/2-balanced adversarial order online CRS [12]. We do not investigate online contention resolution here, but we should mention that in prior results, random order online contention resolution schemes (RCRS) are able to match the best known offline results: For matroids, there is a (1 − 1/e)-balanced RCRS, due to an elegant LP duality connection with prophet inequalities [12].
The situation is much more complicated when F encodes constraints such as matchings and the optimal factors are generally unknown. The cases of bipartite and general matchings have attracted attention due to their fundamental nature and their frequent appearance in applications. We can think of matching constraints as an intersection of two matroid constraints, and for an intersection of k matroid constraints, there is a 1 k+1 -balanced RCRS [11]; in particular, this gives a 1/3-balanced RCRS (and hence also an offline CRS with the same factor) for bipartite matchings.
Recent work in both offline and online settings has significantly improved the factor of 1/3. In the offline setting, [10] gives a (1 − e −2 )/2 0.432-balanced scheme for general matchings, which can be improved slightly further [14]. Very interestingly, [14] identifies the optimal monotone scheme for bipartite matchings, which achieves a balancedness of 0.476. Here, monotonicity means that Pr[e ∈ I | R] is a non-increasing function of R (among those R containing e). This property is particularly relevant in the context of randomized rounding. Nevertheless, the optimal CRSs for bipartite and general matchings are still unknown. In terms of impossibility results, an upper bound of 0.544 follows from a classical paper of Karp and Sipser [1], as discussed in [10].
In the online setting, the best known CRSs are due to recent results in [16]: In the random order case, they provide a 0.474-balanced scheme for general matchings and a 0.476-balanced scheme for bipartite matchings, and in the bipartite case, they also establish an upper bound of 0.5. Notably, the 0.474-balanced scheme is in fact the best known CRS for general matchings, whether offline or online. In the adversarial order case, they provide a 0.344-balanced scheme for general matchings and a 0.349-balanced scheme for bipartite matchings
Our results
To explain our results, we start by formally setting up some notation. Given a graph G = (V, E), a fractional matching is a point x ∈ [0, 1] E in the matching polytope, i.e., a point in the convex hull of vectors 1 M for all matchings M in G. For a fractional matching x, let x uv be the component of x corresponding to the edge (u, v).
The problem we are interested in studying is:
Contention resolution for matchings. We are given a fractional matching x, and a random set R(x) where edges appear independently with probabilities x uv . Our goal is to choose a matching M ⊆ R(x) such that for every edge (u, v),
Pr[(u, v) ∈ M | (u, v) ∈ R(x)] ≥ c.
Such a scheme is called c-balanced, and we want to find a scheme with c as large as possible. The main questions we ask are:
(i) Is there a contention resolution scheme for matchings achieving the upper bound 0.544 of Karp and Sipser?
(ii) Is there a separation between the optimal c for online and offline contention resolution schemes?
(iii) Is there a separation between the optimal c for monotone and non-monotone contention resolution schemes?
In this paper, we prove the following results. The first result, which applies to both bipartite and non-bipartite matchings, is an attempt to answer (i).
Theorem 1.
Assuming that x is a fractional matching such that x ∞ ≤ , there is (γ − f ( ))balanced contention resolution scheme, where γ 0.544 is the impossibility bound of Karp and Sipser and lim →0 f ( ) = 0.
For fractional matchings without any assumption on their ∞ norm 1 , we present an improved CRS in the bipartite case.
Theorem 2. There is 0.509-balanced contention resolution scheme for bipartite matchings.
This theorem answers questions (ii) and (iii), since the optimal RCRS for bipartite matchings is at most 0.5-balanced, and the optimal monotone CRS for bipartite matchings is 0.476-balanced. Our theorem thus establishes separations that, to our knowledge, have not been demonstrated in any other natural relaxations of combinatorial optimization problems before. (Note that for matroids, the known optimal (1 − 1/e)-balanced schemes are monotone.)
Returning to the context we started this paper with, our results establish that we can match more than half of all the employees to jobs they like without partially favoring any particular employee, and in case no employee has a strong preference for any particular job, we can do better, and match 54% of employees to jobs. This is a significant improvement over what we can do if we learn the employees preferences in an online fashion.
We should also mention here the important concept of a correlation gap. In the context of bipartite matchings, the correlation gap is defined as the minimum possible ratio between E[max{ e∈M w e : M ⊆ R(x)}] and e∈E w e x e , where x is a fractional bipartite matching and w is any vector of weights. By LP duality (see [7]), Theorem 2 also provides (the best known) lower bound of 0.509 on the correlation gap for bipartite matchings.
In light of Theorem 1, we believe that the correlation gap for bipartite (and perhaps even non-bipartite) matchings is indeed the Karp-Sipser bound of γ 0.544, and the optimal CRS is γ-balanced. This conjecture has intriguing connections with van der Waerden's conjecture for the permanent of doubly stochastic matrices; we discuss this in Section 1.3.
Finally, we note that although monotonicity is usually needed in the applications of CRSs using randomized rounding, non-monotone CRSs also have applications we have not mentioned here in the introduction, like in the rounding of the Configuration LP for assignment problems. We present a sample application of this type in Section 4.
Our techniques
Our Theorem 1 follows from an improved and simplified analysis of Karp and Sipser's algorithm [1] for constructing matchings by adding random edges adjacent to a leaves. While we utilize many of the ideas from Karp and Sipser's paper, our analysis of the algorithm is an improvement in several ways:
• We obtain a contention resolution scheme, while Karp and Sipser only compute the expected size of the maximum matching. This yields the somewhat surprising conclusion that Karp and Sipser's algorithm works just as well for weighted matchings as it does for unweighted matchings.
• We avoid Karp and Sipser's (technically complicated) use of the so-called differential equation method. We also avoid the use of generating functions, another method used recently to calculate the expected size of the maximum matching in random graphs [8].
• We obtain results for any random graph R(x) constructed from a fractional matching satisfying x ∞ ≤ , unlike Karp and Sipser who only consider the Erdos-Renyi random graph G n,c/n .
Many previous results require that there be some kind of symmetry in the random graph to obtain bounds on the size of the matching. We stress that we do not need to make any such assumption on R(x).
We do need to assume that x ∞ ≤ . This assumption is useful because it ensures that the neighbourhood of any particular edge looks like a random tree. There exists a significant amount of literature using a closely related assumption ("local weak convergence") and recursive distributional equations to formalize various statistical mechanical heuristics regarding matchings in random graphs. Most related to our work is the work of Bordenave, Lelarge, and Salez [6]. Once again, the advantage of our method is that we obtain a CRS (as opposed to computing the expected size of the maximum matching) and we avoid the use of technically complicated tools.
These improvements come at a cost-we assume that the average degree of each vertex is less than or equal to 1. The theoretical and practical significance of this case, and the importance of contention resolution schemes, make this trade-off a good choice.
Our Theorem 2 requires several new techniques, although the basic idea can be traced back to Karp and Sipser as well: When deciding which edge incident to a vertex we should add to a matching, it is beneficial to pick an edge which is adjacent to a leaf, since it doesn't block us from adding other edges into the matching. It turns out that for general matchings, it is actually better not to follow this rule absolutely (at least in our analysis) but we still pick degree-1 edges with significant priority over other edges.
We present two different schemes using these ideas; the first one is simpler and achieves a factor 0.480 (already establishing the separation between monotone and non-monotone schemes). An interesting feature of this scheme is that it can be implemented as a parallel algorithm with each vertex independently making decisions about whether to include an edge adjacent to it in the matching by looking only at its immediate neighborhood. The best schemes known previously did not have this useful property. Our more complicated scheme achieves a factor 0.509 (thus demonstrating a separation between offline CRSs and RCRSs).
Both schemes rely on an extended version of contention resolution for choosing 1 element from a possibly correlated distribution, which we present in Section 3.1, and the 0.509-balanced scheme uses the FKG inequality to handle correlations between edges in the final stage.
Throughout this paper, even though we state our theorems for fractional matchings x, we will actually only need to assume that x satisfies the vertex constraints v x uv ≤ 1. Furthermore, we can always assume that x satisfies v x uv = 1 for every u. We can achieve this by adding vertices and edges with probabilities such that the edge probabilities at each vertex add up to 1; this only makes the task of designing a CRS more difficult.
Parallels with van der Waerden's conjecture
Recall the classical van der Waerden's conjecture (in fact a theorem):
van der Waerden's conjecture Given any doubly stochastic n × n matrix A (an array of coefficients (a ij ) n i,j=1 such that n i=1 a ij = 1 for every j and n j=1 a ij = 1 for every i), the permanent of A is at least n!/n n (which is the permanent of the matrix where all entries are equal to 1/n). Now every doubly stochastic matrix A corresponds to a fractional matching x of K n,,n in a standard way-If u is the i th vertex on the left, and v is the j th vertex on the right, x uv = a ij . Therefore, given any doubly stochastic matrix A, we can construct a random bipartite graph R(x).
In this interpretation, the permanent of the matrix is the expected number of perfect matchings in R(x). Hence, we have the following reformulation:
van der Waerden's conjecture, reforulated The expected number of perfect matchings in R(x), as a function over fractional bipartite matchings x, is minimized when x uv = 1 n for all edges (u, v).
Our first conjecture is therefore the following. Conjecture 1. The expected size of the maximum matching in R(x), as a function over fractional bipartite matchings x, is minimized when x uv = 1 n for all edges (u, v). In other words, if A is a doubly stochastic matrix and R is a random bipartite graph where the edge between the i th vertex on the left and the j th vertex on the right appears in R independently with probability a ij , then the expected size of the maximum matching in R is minimized when a ij = 1/n for all (i, j).
It is well known that the expected size of the maximum matching in R(x) ⊂ K n,n where x uv = 1/n is (γ + o(1))n where γ = 2(1 − λ) − λ 2 and λ = e −λ (λ 0.567 and γ 0.544) (though we also provide an explanation of this fact in as a consequence of Theorem 1). Our conjecture is that this is the worst case among all fractional matchings in K n,n .
By LP duality between CRSs and the quantities discussed in this section (see [7]), we would obtain an optimal γ-balanced CRS for bipartite matchings if the following weighted version of the conjecture were true.
Conjecture 2. Let A be a doubly stochastic matrix and W a matrix of weights such that n i,j=1 w ij a ij = 1. Then for a random bipartite graph R where the edge between the i th vertex on the left and the j th vertex on the right appears in R independently with probability a ij , the expected maximum-weight matching in R is minimized when a ij = w ij = 1/n for all (i, j).
An extension of van der Waerden's conjecture which is also known to be true is the following (see e.g. [2], Theorem 4.3): For any 1 ≤ k ≤ n, the expected number of k-matchings (matchings of k edges) in the random set R(x) as above is minimized again for x uv = 1/n for all edges (u, v). This however does not imply that the probability that a k-matching exists, a quantity useful to consider to establish the first conjecture, is minimized for the same matrix. The thresholds for the existence of a k-matching and the expected number of k-matchings are not the same. It can be verified by relatively straightforward computations that for x uv = 1/n, the expected number of k-matchings is large for k = 0.6n, but the probability that a 0.6n-matching exists is vanishingly small. This is the discrepancy between the existence and expectation thresholds, addressed by the Kahn-Kalai conjecture; however, the recent resolution of it [17] does not shed any light on our problem since the general bounds on the gap are logarithmic.
Nevertheless, we believe that these parallels with prominent problems and results in probabilistic combinatorics make our conjectures quite appealing.
2 An optimal contention resolution scheme when x ∞ → 0
In this section, we will analyze the Karp-Sipser algorithm to prove the following rephrasing and strengthening of Theorem 1:
Theorem 3. Suppose x ∈ [0, 1] E is such that for every vertex u, v x uv = 1 and x ∞ ≤ .
There is an algorithm A which when given the random graph R(x), selects a matching M with the property that
Pr
[(u, v) ∈ M | (u, v) ∈ R(x)] ≥ 2(1 − λ) − λ 2 − f ( )
where λ is the unique real root of the equation λ = e −λ , and lim →0 f ( ) = 0. Furthermore, this algorithm is optimal in the sense that
E[size of maximum matching in R(x)] ≤ ((1 − λ) − 1 2 λ 2 + g( ))|V (x)|.
where |V (x)| is the size of the set of vertices of the graph x is a fractional matching of, and lim →0 g( ) = 0.
The Karp-Sipser algorithm
The Karp-Sipser algorithm is a method to select a matching in a graph. Given a graph G, the algorithm deletes all the degree 0 vertices, selects a random degree 1 vertex (if one exists), and adds the edge adjacent to it to the matching. Then, it deletes all the edges adjacent to the edge just added to the matching, and recurses on the newly obtained graph G . Note that unlike in the paper of Karp and Sipser, we do not use a two stage process to generate the matching.. An attractive feature of the Karp-Sipser algorithm is that it doesn't "make any mistakes". This is because for any vertex v of degree 1 in a graph G, G has a maximum matching in which v is matched.
If an edge is deleted by the algorithm at some stage, we will say that it disappears. We also say that a vertex is added to the matching if an edge adjacent to it is added to the matching. Before we discuss the analysis of the algorithm, we take a brief detour.
Random trees
Consider the following method to generate a random tree in steps. Fix two special vertices, u and v, and draw an edge between them. In step i, for each vertex at the depth i − 1, independently sample a Poisson random variable with mean 1, and add as many children to the vertex as the obtained sample. Stop at step j if there are no vertices at depth j − 1. Let us call the random tree generated by this process T .
Since the two subtrees of u and v are independent copies of a Galton-Watson process with 1 expected child at each node, it is straightforward to prove that this process terminates with probability 1. So it is almost always true that this process produces a finite tree.
The following lemma, only true for graphs without parallel edges, explains why we care about the process T : Up to small errors, it describes the distribution of the connected component containing a given edge (u, v) in R(x). Lemma 1. Let x be a fractional matching with w x vw = 1 for every vertex v, and x ∞ ≤ . Let R(x) be the corresponding random graph. Let us condition on (u, v) ∈ R(x) and define N ((u, v)) to be the connected component in R(x) containing (u, v). Let T be a random tree produced by the process described above and T 0 be any finite realization of the process. Then
| Pr[N ((u, v)) = T 0 | (u, v) ∈ R(x)] − Pr[T = T 0 ]| = O( |T 0 | 2 ).
Proof. (Sketch) First, let us clarify-for a finite tree T 0 with two marked, adjacent vertices u and v, we say N ((u, v)) = T 0 if there is a graph isomorphism from N ((u, v)) to T 0 that fixes the special vertices u and v.
Let us build the connected component N ((u, v)) by revealing gradually the edges appearing in R(x), starting from the edges incident to u and v. At any point, we take a vertex of the component not processed so far, and generate the edges to vertices not visited so far. If the neighborhood at any point is not consistent with the respective neighborhood in T 0 , we fail. Let us estimate the probability that this process succeeds, i.e. generates a connected component equal to T 0 .
At any point, conditioning on the component generated so far, if we process a vertex w, we claim that the probability that w has any incident edge that creates a cycle is O( |T 0 |). This is because the number of vertices in the connected component C w generated so far is at most |T 0 | (unless the process failed already), and the probability of each edge appearing is at most .
The edges from w to vertices not visited so far appear independently, and their expected number is w / ∈Cw x ww . Since w x ww = 1 and |C w | ≤ |T 0 | and x ww ≤ , we have
w / ∈Cw x ww ∈ [1 − |T 0 |, 1]
. Thus the distribution of the numbers of new neighbors of w (outside of C w ) is a summation of independent Bernoulli variables with probabilities bounded by , and the total expectation is between 1 − |T 0 | and 1. By Le Cam's theorem, this distribution is close to Poisson with the same expectation, within total variation distance O( w / ∈Cw x 2 ww ) = O( ). Finally, a Poisson variable with expectation in [1 − |T 0 |, 1] is within total variation distance O( |T 0 |) of a Poisson variable of expectation 1.
Hence the processes in R(x) and in T can be coupled so that at each step, the probability that the neighborhood of w is different from the neighborhood in T , conditioned on the processes between identical so far, is O( |T 0 |). By induction, the probability that the two processes ever deviate in the first |T 0 | steps is O( |T 0 | 2 ). Hence the probability that N ((u, v)) = T 0 can differ from the probability that T = T 0 by at most O( |T 0 | 2 ).
The Karp-Sipser algorithm on trees
It is easy to prove by induction (using the fact that trees always have degree 1 vertices) that in an execution of the Karp-Sipser algorithm on a forest, an edge must eventually either disappear, or else, is added to the matching. Together with the fact that the Karp-Sipser algorithm does not make mistakes, this shows that the Karp-Sipser algorithm finds a maximum matching in a tree.
Given a tree, we would like to be able to analyze which vertices and edges end up in the matching the algorithm selects, independent of the random choices the algorithm makes. To that end, consider the following algorithm to label the vertices of a tree (this is similar to the scheme in [1]):
Root the tree at an arbitrary vertex. Starting at the maximum possible depth, look at all the vertices at a fixed depth. If a vertex has no L children (this can perhaps be true vacuously), label it L. Else, label it W. Iteratively label vertices higher in the tree, until the root of the tree receives a label.
The following claims are true (regardless of the chosen root, and regardless of the random choices the algorithm makes):
1. If an edge between a W parent and an L child disappears, it must be because the W vertex was added to the matching.
2. Every W vertex is added to the matching.
Every edge between two W vertices disappears.
Proof of claim 1. Suppose by way of contradiction that an edge between a W parent and L child disappears because the L vertex was added to the matching. Certainly, this does not happen in the first step of the execution of the algorithm. Consider the very first time it happens.
The L vertex must have been added to the matching through a W labelled child it has. This W vertex must have degree 1, and so the edge connecting it to an L child must have disappeared. This contradicts our assumption of the original edge being the first edge between a W parent and an L child that has disappeared because the L vertex was added to the matching.
Proof of claim 2. Every W vertex has an edge connecting it to an L child; either that edge disappears, and the claim follows by claim 1, or that edge is added to the matching and the claim still follows.
Proof of claim 3. Suppose by way of contradiction that an edge between two W vertices is added to the matching. Consider the state of the graph just before this edge is added. One of the vertices must have degree 1, so an edge connecting to its L child must have disappeared. But the only way such an edge can disappear is by the W vertex being added to the matching, contradiction!
The Karp-Sipser algorithm on random trees
We can now calculate the probability with which the Karp-Sipser algorithm, when executed on the random tree T , adds the special edge between u and v to the matching.
To this end, first label the trees rooted at u and v using the procedure described in the previous section (imagining the special edge connecting u and v does not exist, and we are just labelling two different rooted trees).
Let us first calculate the probability λ that u is labelled L: Second, let us calculate the probability that the edge between u and v is added to the matching, and v is labelled L. Imagine now rooting the random tree T at u. This does not change the label of any of the vertices except possibly u which is now labelled W.
λ =
This means that u must end up in the matching. None of the edges connecting u with any of its W children end up in the matching. All the edges connecting u with any of its L children, and the special edge between u and v are completely symmetric from the standpoint of the execution of the Karp-Sipser algorithm. Therefore,
Pr[(u, v) is added to the matching, v is labelled L] = ∞ k=0
Pr[(u, v) is added to the matching, v is labelled L, u initially has k L children]
= ∞ k=0 λ k + 1 ∞ r=k r k λ k (1 − λ) r−k e −1 r! = ∞ k=0 λ k+1 (k + 1)! · e −λ = e −λ · (e λ − 1) = 1 − λ
Third, note that if we initially labelled both u and v L, then (u, v) must end up in the matching. This is because if we imagine rooting the tree at u, u is labelled W, so ends up in the matching, but the only way this can happen is if (u, v) ends up in the matching since all of its other children are labelled W.
Fourth, note that if we initially labelled both u and v W, then (u, v) must disappear. This is because if we imagine rooting the tree at u, the labelling remains the same, and every edge between W vertices disappears.
Finally, we can compute the probability that the special edge (u, v) ends up in the matching selected by Karp-Sipser as the sum of the probabilities of the edge ending up in the matching when u and v are labelled (respectively) L and L, L and W, W and L, and W and W. This is equal to 2(1 − λ) − λ 2 .
Putting it all together
Let us consider a fractional matching x such that x ∞ ≤ . Lemma 1 immediately implies that for any set of finite trees F = {T 1 , T 2 , . . . , T m } that are realizations of the random process T ,
| Pr[N ((u, v)) ∈ F | (u, v) ∈ R(x)] − Pr[T ∈ F ]| = O |T i | 2
and hence it follows that
| Pr[N ((u, v)) / ∈ F | (u, v) ∈ R(x)] − (1 − Pr[T ∈ F ])| = O |T i | 2
and so, for any F we have
lim →0 Pr[N ((u, v)) contains a cycle | (u, v) ∈ R(x)] ≤ lim →0 Pr[N ((u, v)) / ∈ F | (u, v) ∈ R(x)] = 1 − Pr[T ∈ F ]
Since we know that T produces a finite tree with probability 1, we can take F larger and larger to prove that lim
|V (x)| = lim →0 Pr[(u, v) ∈ M | (u, v) ∈ R(x)] 2
It remains to show that
lim →0 Pr[(u, v) ∈ M | (u, v) ∈ R(x)] = Pr[(u, v) ∈ M | (u, v) ∈ T ]
In this direction, fix a set of finite trees F . Then,
lim →0 Pr[(u, v) ∈ M | (u, v) ∈ R(x)] − Pr[(u, v) ∈ M | (u, v) ∈ T ] = lim →0 Pr[(u, v) ∈ M | (u, v) ∈ R(x), N ((u, v)) / ∈ F ] Pr[N ((u, v)) / ∈ F | (u, v) ∈ R(x)] − Pr[(u, v) ∈ M | (u, v) ∈ T, T / ∈ F | Pr[T / ∈ F ] ≤ lim →0 Pr[N ((u, v)) / ∈ F | (u, v) ∈ R(x)] + Pr[T / ∈ F ] = 2(1 − Pr[T ∈ F ])
Since F was arbitrary, and the same calculation works for the absolute value of the quantity considered, the desired result follows.
Improved contention resolution schemes for bipartite matchings
Now we turn to contention resolution for bipartite matchings, without any assumption on the ∞ norm of the fractional matching.
Contention resolution for 1 element
A basic building block of our CRSs is the following theorem which establishes the existence of a scheme for choosing 1 out of n elements (historically the first CRS [3]).
Theorem 4. Suppose that D is a distribution on 2 E such that for every set S ⊆ E,
Pr R∼D [S ∩ R = ∅] ≥ i∈S β i .
Then there is a monotone contention resolution scheme for choosing one element e(R) from R ∼ D such that Pr[e(R) = i] ≥ β i for every i ∈ E.
This theorem can be proved using a max-flow/min-cut argument, as briefly discussed in [3]: We set up a bipartite graph where the left-hand side is 2 E and the right-hand side is E. We insert a directed edge for every pair (S, i) where i ∈ S, with unbounded capacity. Finally, we add a source vertex s which is connected to each vertex S ∈ 2 E by an edge (s, S) with a capacity of Pr D [S], and a sink vertex t which is connected to each vertex i ∈ E by an edge (i, t) of capacity β i . By our assumption, it can be verified that there is no s-t cut of capacity less than i∈E β i . Hence, there exists a flow from s to t of capacity i∈E β i , which can be interpreted as the desired contention resolution scheme by considering the flow through each vertex S ∈ 2 E . We remark that the resulting scheme can be assumed to be monotone: Denote π i (S) = Pr[e(R) = i | R = S], and consider a scheme as above minimizing the potential function
Φ = n i=1 S 1 ⊂S 2 ,i∈S 1 Pr[R = S 1 ] Pr[R = S 2 ](π i (S 2 ) − π i (S 1 )) + .
For any two sets S 1 ⊂ S 2 , |S 2 | = |S 1 | + 1, if i ∈ S 1 such that π i (S 1 ) < π i (S 2 ), there is another element i ∈ S 1 such that π i (S 1 ) > π i (S 2 ), and hence we can modify the scheme by transferring some survival probability from π i (S 1 ) to π i (S 1 ) and from π i (S 2 ) to π i (S 2 ), thereby reducing the potential function Φ.
A 0.480-balanced scheme for bipartite matchings
Before going to the proof of Theorem 2, we will show the existence of a simple 2(1 − e −1/e ) − e −2balanced contention resolution scheme for bipartite matchings 2 . We remark that 2(1 − e −1/e ) − e −2 ≥ 0.480 and hence this already beats the optimal monotone scheme for bipartite matchings. By necessity, our scheme is non-monotone and this result establishes a strict separation between monotone and non-monotone schemes for bipartite matchings.
The simple scheme:
1. For each edge (u, v) (with probability x uv of appearing), independently declare it active with probability 1−e −xuv xuv given it appears.
2. For each vertex u, call an active edge (u, v) "available at vertex u" if v has no other edges adjacent to it which are active.
3. Using a contention resolution scheme for 1 element, select one of the available edges at each vertex. Ensure that an edge (u, v) gets selected at vertex u with probability at least
x uv (1 − e −1/e ) − 1 2e 2 + e 2xuv −e xuv 2e 2 .
4. The set of edges selected at all the vertices form the matching M .
The first step of the scheme is a kind of pre-processing which ensures that edges with high probabilities of appearing don't destroy the chances of neighbouring edges getting picked by the scheme. This strategy has also been used in the literature previously [14]. The second step is a first attempt at using the idea that it is always useful to add vertices with degree 1 into the matching. This does not seem to have been explicitly exploited by previous CRSs.
To analyze the algorithm, note firstly that it is easy to see that the selected edges really do form a matching. Secondly, let us note that the probability that an edge (u, v) is available at a vertex u is Furthermore whether (u, v i ) is available at u is independent of whether (u, v j ) is available at u. Therefore, it follows from Theorem 4 that if the probability of (u, v i ) appearing is x i (short for x uv i ), the existence of the required CRS depends only on whether
F (x uv ) = 1 − e −xuv e −x uv = 1 − e −Pr[at least one of (u, v i ) is available] = 1 − n i=1 (1 − F (x i )) ≥ n i=1 x i (1 − e −1/e ) − 1 2e 2 + e 2x i − e x i 2e 2
Therefore, the desired result follows from the following lemma:
Lemma 2. If x i ≤ 1, x i ≥ 0, and F (x) = e x −1 e , then 1 − n i=1 (1 − F (x i )) ≥ n i=1 x i (1 − e −1/e ) − 1 2e 2 + e 2x i − e x i 2e 2
Proof. Let us assume without loss of generality that 0 ≤ x 1 ≤ x 2 ≤ · · · ≤ x n , and let us write
A i = e −x i /e , B i = 1 − F (x i ). Now we know that e − x i /e − n i=1 (1 − F (x i )) = A i − B i = n j=1 A j − B j B j i≤j B i i>j A i Claim 1. Assuming 0 ≤ x 1 ≤ x 2 ≤ · · · ≤ x n and x i ≤ 1, we have that i≤j B i i>j A i ≥ B 1/x j j .
Proof. Let us write S = i≤j x i . Then we know
log i>j A i = − x i − S e ≥ − 1 − S e Furthermore, since log(1−F (x)) x is a decreasing function, log B i x i ≥ log B j x j . It follows that log i≤j B i = i≤j log B i ≥ log B j x j S Finally, since − 1 e ≥ log B j x j (once again because log(1−F (x))
x is a decreasing function) it follows that
log B j x j S − 1 − S e ≥ log B j x j
We conclude that
e − x i /e − n i=1 (1 − F (x i )) ≥ n j=1 A j − B j B j B 1/x j j .
But we also know that since 0 ≤ x i ≤ 1, we must have
1 − x i · (1 − e −1/e ) ≥ e − x i /e
So the desired result follows upon checking that
A j −B j B j B 1/x j j ≥ e 2x j −e x j −x j 2e 2
, which we may check in a straight forward way since the difference of the two quantities is an increasing function of the single variable x j ∈ [0, 1].
There are many ways to improve this scheme. For example, the scheme completely ignored edges all of whose neighbours are edges associated with vertices of degree at least 2. We could use some other contention resolution scheme, for example, the one from [14] to add some of these edges to the matching generated by our scheme. However, we do not investigate this further here. Instead, we will consider a different approach.
A 0.509-balanced scheme for bipartite matchings
In this section we present our best scheme for bipartite matchings in general. This beats the best possible random order online contention resolution scheme.
A warm-up: the 2-stage scheme
First, we describe a simple scheme (due to Chandra Chekuri [15]) which achieves the factor of α = 1 − 1/e 1−1/e 0.468; i.e., weaker than the optimal monotone scheme. This scheme illustrates one of our main ideas. We call the set of vertices on the left V 1 , and on the right, V 2 .
The 2-stage scheme.
1. For each vertex u ∈ V 1 independently, perform contention resolution among the edges (u, v) ∈ R, to choose one edge incident to u; call these edges R 1 .
2. For each vertex v ∈ V 2 independently, perform contention resolution among the edges (u, v) ∈ R 1 , to choose one edge incident to v; call these edges M .
It is easy to see that M is a matching, since the two stages of contention resolution ensure that the degrees in M are at most 1 on both sides. Lemma 3. The first stage of contention resolution can be implemented so that for each edge,
Pr[(u, v) ∈ R 1 ] ≥ (1 − 1/e)x uv .
Also, these events are mutually independent for a fixed value of v.
Proof. For any subset of edges S incident to u,
Pr[S ∩ R = ∅] = 1 − (u,v)∈S (1 − x uv ) ≥ 1 − e − (u,v)∈S xuv ≥ (1 − 1/e) (u,v)∈S x uv ,
by the concavity of the function 1 − e −t . By Theorem 4, there is a scheme for each fixed u ∈ V 1 such that every edge (u, v) survives with probability at least (1 − 1/e)x uv . We do this independently for each u ∈ V 1 , so there is no correlation between different edges (u, v) for a fixed v.
Lemma 4. The second stage of contention resolution can be implemented so that for each edge,
Pr[(u, v) ∈ M ] ≥ (1 − 1/e 1−1/e )x uv .
Proof. Consider a fixed vertex v ∈ V 2 . We have Pr[(u, v) ∈ R 1 ] ≥ (1 − 1/e)x uv , and these events are independent for different edges incident to v. Hence, for any subset S of such edges,
Pr[S ∩ R 1 = ∅] ≥ 1 − (u,v)∈S (1 − (1 − 1/e)x uv ) ≥ 1 − e −(1−1/e) (u,v)∈S xuv .
Improving the 2-stage scheme
Next, we design a more complicated scheme which builds upon the 2-stage scheme. The main idea involves noticing that edges (u, v) such that v has degree 1 in R have no competition in the second stage and hence should be preferentially selected in the first stage, since if selected they will certainly survive in our matching.
This idea needs to be modified, since edges of high probabilities of appearing could destroy the chance of neighbouring edges getting picked. We therefore define the notion of being a preferred or "red" edge a bit more carefully.
Similar to our previous scheme, we start by letting an edge be"active" with probability 1−e −xuv . We mark the edges which are not active gray. Then, we mark active edges with degree 1 on the right-hand side red with a certain probability, so that Pr [(u, v) is red] = 1 − e −xuv/e . This is somewhat lower than the probability of having degree 1 on the right-hand side, but just enough to guarantee a probability of survival (1 − e −1/e )x uv among red edges, which is the optimal factor. We mark the remaining active edges blue and run a contention resolution scheme amongst them separately. At the end, all surviving red, blue and gray edges enter a contention resolution scheme at each vertex on the right-hand side.
The red/blue/gray scheme.
Recall that Pr
[(u, v) ∈ R] = x uv . Decide for each edge (u, v) ∈ R independently at random whether to mark it gray, so that Pr[(u, v) is gray] = x uv − (1 − e −xuv )
. We call the edges (u, v) ∈ R which are not gray active. 5. For each u ∈ V 1 , if there are no active (red or blue) edges incident to u, if there are some gray edges incident to u, perform contention resolution to include one of them in R 3 , so that
Pr[(u, v) ∈ R 3 ] ≥ 1 2e x 2 uv .
6. Finally, for each v ∈ V 2 , perform contention resolution among all edges in R 1 ∪ R 2 ∪ R 3 incident to v, to include one of them in M , so that
Pr[(u, v) ∈ M ] ≥ 0.509 x uv .
Implicit in each of steps 3, 4, 5, and 6 above is a claim that there exists a certain contention resolution scheme for choosing 1 out of n elements. The existence of such schemes can be proved by applying Theorem 4, if we can calculate the probability that at least one of a subset of edges "is available for consideration at that stage" (i.e., is red, is blue, etc).
For steps 3, 4, and 5, this quantity is fairly simple to calculate, because all the choices (to designate edges red or blue or gray) are made independently. The following lemmas are devoted to establishing the claimed existences in these cases.
Lemma 5. The definition of red edges is valid and there is a CRS among red edges such that
Pr[(u, v) ∈ R 1 ] = (1 − e −1/e )x uv for every edge (u, v).
Proof. The probability that (u, v) is the only active edge incident to v is
(1 − e −xuv ) u ∈V 1 \{u} e −x u v = (e xuv − 1) u ∈V 1 e −x u v = (e xuv − 1)e −1 .
By elementary inequalities, this is at least x uv e −1 ≥ 1 − e −xuv/e . Therefore, the notion of red edges is well defined.
For every subset of edges S incident to u, we have This happens independently for each edge incident to u (since (u, v) being red/blue depends only on edges incident to v), and so we obtain Pr[there is a blue edge among S | no red edges incident to u]
= 1 − (u,v)∈S Pr[(u, v) is not blue | no red edges incident to u] = 1 − e −(1−1/e) (u,v)∈S xuv ≥ (1 − e −(1−1/e) ) (u,v)∈S x uv .
Therefore, by Theorem 4 there is a CRS among blue edges which, conditioned on no red edges being incident to u, includes each edge in R 2 with probability (1 − e −(1−1/e) )x uv . The probability of no red edge being incident to u is v∈V 2 e −xuv/e = e −1/e . Therefore,
Pr[(u, v) ∈ R 2 ] = e −1/e (1 − e −(1−1/e) )x uv = (e −1/e − e −1 )x uv . Corollary 1. Pr[(u, v) ∈ R 2 | (u, v) is blue] ≥ β = e −1/e −e −1 1−e −1 .
Proof. We have Pr [(u, v) is blue] = e −xuv/e − e −xuv , and Pr[(u, v) ∈ R 2 ] = (e −1/e − e −1 )x uv . Recall that only blue edges can appear in R 2 . Hence,
Pr[(u, v) ∈ R 2 | (u, v) is blue] = (e −1/e − e −1 )x uv e −xuv/e − e −xuv ≥ (e −1/e − e −1 )x uv 1 − e −(1−1/e)xuv ≥ e −1/e − e −1 1 − e −1 .
Lemma 7. There is a CRS among gray edges such that Pr[(u, v) ∈ R 3 ] ≥ 1 2e x 2 uv for every edge (u, v).
Proof. Consider a fixed u ∈ V 1 and let's condition on no active edges being incident to u; this happens with probability v∈V 2 e −xuv = e −1 .
= (u,v)∈S (1 − x uv ) (u,v)∈S e −xuv = e (u,v)∈S (xuv+ln(1−xuv))
. Consider the Taylor expansion of the exponent:
x uv + ln(1 − x uv ) = − ∞ k=2 x k uv k .
We observe that this is a concave function, not only as a function of x uv but even as a function of x 2 uv . Therefore, if we fix (u,v)∈S x 2 uv = σ 2 , the maximum value that the exponent could achieve is attained for x uv = 1 √ |S| σ for all (u, v) ∈ S, in which case
(u,v)∈S (x uv + ln(1 − x uv )) = |S|σ + |S| ln(1 − σ |S| ).
Again, we want to determine the maximum possible value of the right-hand side for given σ.
Observe that since (u,v)∈S x uv = σ |S| ≤ 1, we have |S| ≤ 1/σ 2 . The right-hand side is an increasing function of |S|, hence it is maximized when |S| = 1/σ 2 (possibly not an integer, but certainly this gives an upper bound):
(u,v)∈S (x uv + ln(1 − x uv )) ≤ 1 + 1 σ 2 ln(1 − σ 2 ).
Further, we can bound
1 + 1 σ 2 ln(1 − σ 2 ) ≤ ln(1 − 1 2 σ 2 )
for example by comparing the Taylor series. We conclude that Pr[no gray edge among S | no active edge incident to u] = e (u,v)∈S (xuv+ln(1−xuv)) .
≤ 1 − 1 2 σ 2 = 1 − 1 2 (u,v)∈S x 2 uv .
By Theorem 4, we obtain the desired CRS.
To finish, we need to analyze Step 6 of the algorithm, which is contention resolution among all the surviving edges on the right-hand side. Here, there can be at most one red edge incident to a vertex v ∈ V 2 , possibly multiple gray edges which appear independently, and possibly multiple blue edges whose survival up to this stage of the scheme is correlated. This correlation causes the main trouble in our analysis of this final step, because it makes it harder to calculate the probability that at least one of a subset of edges incident at a vertex v is in R 1 ∪ R 2 ∪ R 3 . Ideally, we would like to prove that the appearance of blue edges satisfies some form of negative correlation. At the moment, we are able to prove only pairwise negative correlation which is sufficient to achieve the factor of 0.509. A stronger correlation result (for example negative cylinder dependence) would lead to an improved factor. Lemma 8. For any two incident edges (u, v) and (u , v),
Pr[(u, v) ∈ R 2 & (u , v) ∈ R 2 | (u, v), (u , v) are blue] ≤ Pr[(u, v) ∈ R 2 | (u, v), (u , v) are blue] · Pr[(u , v) ∈ R 2 | (u, v), (u , v) are blue].
Proof. Define Γ(u) = {v : (u, v ) active} and Γ(u ) = {v : (u , v) active}. Note that conditioning on (u, v), (u , v) being blue edges is the same as conditioning on v ∈ Γ(u) ∩ Γ(u ), because edges (u, v), (u , v) being active also means that they must be blue.
We claim that conditioned on Γ(u), Γ(u ) such that v ∈ Γ(u) ∩ Γ(u ), the probability that (u, v) ∈ R 2 is decreasing in Γ(u) and increasing in Γ(u ), while conversely the probability that (u , v) ∈ R 2 is increasing in Γ(u) and decreasing in Γ(u ).
We prove this by considering a fixed choice of the active edges incident to V 1 \ {u, u }, and at the end averaging over these choices. Consider Γ(u), Γ(u ) where v ∈ Γ(u) ∩ Γ(u ). For (u, v) to be selected in R 2 , there cannot be any red edge incident to u. The only candidates for such red edges are (u,ṽ) whereṽ ∈ Γ(u) \ Γ(u ), because edges incident to Γ(u) \ Γ(u ) are blue by definition. For eachṽ ∈ Γ(u) \ Γ(u ), (u,ṽ) is red ifṽ does not have any other incident active edges (and an additional independent coin flip succeeds, as defined in Step 2). Clearly, the event of no red edge incident to u is monotonically decreasing in Γ(u) \ Γ(u ).
In case there is no red edge incident to u, we perform contention resolution among the blue edges incident to u, which are all the edges (u, v ), v ∈ Γ(u). Since this scheme is monotone, the probability of survival is monotonically decreasing in Γ(u). This monotonicity property also remains preserved when we average over the choices of active edges incident to V 1 \ {u, u }. Overall, the probability of (u, v) surviving in R 2 is monotonically decreasing in Γ(u) and increasing in Γ(u ). Symmetrically, the probability of (u , v) surviving in R 2 is monotonically decreasing in Γ(u ) and increasing in Γ(u).
Given this monotonicity property, we use the FKG inequality to prove our result. The appearances of vertices in Γ(u) and Γ(u ) are independent. Let us define γ ∈ {0, 1} 2n where γ i = 0 if i ∈ Γ(u) and γ n+i = 1 if i ∈ Γ(u ). As we argued, conditioned on v ∈ Γ(u)∩Γ(u ), Pr[(u, v) ∈ R 2 | γ] is increasing in γ and Pr[(u , v) ∈ R 2 | γ] is decreasing in γ. Therefore, by the FKG inequality,
Pr[(u, v) ∈ R 2 & (u, v) ∈ R 2 | v ∈ Γ(u) ∩ Γ(u )] ≤ Pr[(u, v) ∈ R 2 | v ∈ Γ(u) ∩ Γ(u )] · Pr[(u, v) ∈ R 2 | v ∈ Γ(u) ∩ Γ(u )].
The main takeaway from this lemma is that if we let β be a lower bound on the probability that an edge survives in R 2 given that it is blue (as in Corollary 1, then, conditioned on having at least two active (and hence blue) edges at a vertex v in V 2 in step 6, the probability that one of them survives in R 2 is at least 2β − β 2 . In the final analysis of step 6, if there are more than 2 blue edges at a vertex v, we only use two of them. This allows us to establish our desired conclusion.
Theorem 5. There is a CRS in Step 6 which achieves a factor of 0.509.
Proof. We consider a vertex v ∈ V 2 and all the edges incident to v which are in R 1 ∪ R 2 ∪ R 3 (i.e. survived contention resolution on the left-hand side). We want to perform contention resolution among all these edges, to choose 1 edge to remain in the final matching.
Again, we consider a subset of edges S incident to v. What is the probability that at least one of them survives? We divide into cases as follows:
Pr[S ∩ (R 1 ∪ R 2 ∪ R 3 ) = ∅] = Pr[S ∩ R 1 = ∅] + Pr[S ∩ R 1 = ∅ & S ∩ R 2 = ∅] + Pr[S ∩ (R 1 ∪ R 2 ) = ∅ & S ∩ R 3 = ∅].
The first case is a red edge surviving in S: since there can be only one red edge incident to v, these are disjoint events for each (u, v) ∈ S:
Pr[S ∩ R 1 = ∅] = (u,v)∈S (1 − e −1/e )x uv .
The second case is no red edge surviving in S, and some blue edge surviving S. We divide this further into two subcases: (a) there is only one active edge in S, this edge is blue, and it survives; (b) there are at least two active edges in S (and hence blue), and at least one of them survives.
The probability that (u, v) is the only active edge in S is (1 − e −xuv ) (u ,v)∈S,u =u e −x u v = (e xuv − 1)e −x(S) (where we use the notation x(S) = (u,v)∈S x uv . The probability that this edge is red is 1 − e −xuv/e , hence the probability that (u, v) is the only active edge in S, and it is blue, is as above. Conditioned on having at least two active (and hence blue) edges, the probability that one of them survives in R 2 is at least 2β − β 2 , because the survivals of two blue edges are negatively correlated due to Lemma 8. (If there are more than 2 blue edges, we are only using two of them in this argument -this bound could be improved if we had a correlation inequality for more than 2 blue edges.) Hence, case (b) contributes at least (1 − e −x(S) − (u,v)∈S (e xuv − 1)e −x(S) )(2β − β 2 ). Combining these contributions, we obtain
Pr[S ∩ R 1 = ∅ & S ∩ R 2 = ∅] ≥ (u,v)∈S (e xuv − 1)e −x(S) − (1 − e −xuv/e ) β + 1 − e −x(S) − (u,v)∈S (e xuv − 1)e −x(S) (2β − β 2 ).
The third case is no red or blue edges surviving in S and some gray edge surviving in S. This case certainly happens if there are no active edges in S and some gray edge survives in S. These two events are positively correlated, since having no active edges in S can only increases the probability that gray edges are considered in Step 5, but we ignore this benefit here; we analyze the two events as independent. We have Pr[no active edges in S] = e −x(S) and Pr[no gray edge survives in S] ≤
(u,v)∈S (1 − 1 2e x 2 uv ) ≤ e − 1 2e (u,v)∈S x 2
uv , because different gray edges incident to v survive independently. Hence, this case contributes
Pr[S ∩ (R 1 ∪ R 2 ) = ∅ & S ∩ R 3 = ∅] ≥ e −x(S) 1 − e − 1 2e (u,v)∈S x 2 uv ≥ e −x(S) (1 − e − 1 2e ) (u,v)∈S x 2 uv .
Combining cases the different contributions, we obtain
Pr[S ∩ (R 1 ∪ R 2 ∪ R 3 ) = ∅] ≥ (u,v)∈S (1 − e −1/e )x uv + (u,v)∈S ((e xuv − 1)e −x(S) − (1 − e −xuv/e ))β + 1 − e −x(S) − (u,v)∈S (e xuv − 1)e −x(S) (2β − β 2 ) + e −x(S) (1 − e − 1 2e ) (u,v)∈S x 2 uv = (1 − e −1/e )x(S) − (u,v)∈S (1 − e −xuv/e )β + (e xuv − 1)e −x(S) (β − β 2 ) +(1 − e −x(S) )(2β − β 2 ) + e −x(S) (1 − e − 1 2e ) (u,v)∈S x 2 uv .
We simplify this expression using some elementary estimates: e xuv − 1 ≤ x uv + (e − 2)x 2 uv , and 1−e −xuv/e ≤ 1 e x uv +(1−1/e−e −1/e )x 2 uv (for 0 ≤ x uv ≤ 1). Also, we observe that 1−1/e−e −1/e < 0, so we can write 1 − e −xuv/e ≤ 1 e x uv + (1 − 1/e − e −1/e )e −x(S) x 2 uv (which simplifies the following bound). We obtain Hence, by Theorem 4, there is a CRS which preserves each edge with probability at least γx uv ≥ 0.509x uv .
An application to a combinatorial allocation problem
In this section, we consider a problem that can arise in a scheduling setting. Consider a set of customers X each of whom needs to be scheduled to attend one of m possible events, over a time period of n hours, each event one hour long. Naturally, each customer can attend only one event at a time, and wants to attend each event at most once. What is the "best" possible way to schedule the customers? One way to formalize the problem is as follows:
Allocation with row/column disjointness constraints. Here, we are thinking of the s index as denoting the events, the t index as denoting time, and then S st is the set of customers attending event s at time t. The welfare of a set of customers S attending event s at time t is given by v st (S). The goal is maximize the total welfare of a schedule.
We note that this is a special case of maximization subject to a bipartite matching constraint, due to the following reduction: We define a bipartite "conflict graph" G a ⊂ K m,n for each a ∈ X. The interpretation of using edge (s, t) in G a is that we include element a in S st . Since the sets S st should be disjoint for each fixed s and each fixed t, this corresponds to the constraint that we should choose a matching in each graph G a . The objective function is s,t v st (S st ) where S st = {a : we choose edge (s, t) in G a }. Hence, this is a maximization problem over matchings in the disjoint union of the graphs G a , a ∈ X.
If the valuation functions v ij are monotone submodular, this is a monotone submodular maximization problem over bipartite matchings, for which there is a known (1/2 − )-approximation due to [5]. Here we show an improved approximation for the above allocation problem (but not for submodular maximization over matchings) in the demand oracle model. This LP is exponentially large but it can be solved in polynomial time using demand oracles for v st , since a separation oracle for the dual LP is exactly a sequence of calls to the demand oracles for v st . This is very similar e.g. to the approach in [4].
Given a fractional solution x s,t,S , we can round it as follows: For each s, t independently, sample a random set R st which is equal to S with probability x s,t,S . For each item a ∈ X, form a bipartite conflict graph G a where edge (s, t) appears if a ∈ R st . This is a random graph where edge (s, t) appears with probability S:a∈S x s,t,S and hence due to our constraints this is in expectation a fractional matching in G a .
Using our CRS for bipartite matchings, we can select a matching M a ⊆ G a , such that Pr[(s, t) ∈ M a | (s, t) ∈ G a ] ≥ 0.509. Finally, we define sets S st = {a ∈ X : (s, t) ∈ M a }, which form our solution. Observe that S st ⊆ R st and each element of R st survives in S st with conditional probability at least 0.509 (the survivals of different elements are correlated, but this is not a problem). By a basic property of submodular functions (see e.g. Lemma 1.7 in [4]),
E[v st (S st )] ≥ 0.509 · E[v st (R st )] = 0.509 S v st (S)x s,t,S .
In our analysis, we do not require the monotonicity of the contention resolution scheme, just like it is not required in [4].
λ) k = e −1 · e 1−λ = e −λλ is thus the unique real number which solves the equation x = e −x .
→0
Pr[N ((u, v)) contains a cycle | (u, v) ∈ R(x)] = 0 and hence we know that lim →0 E[# of edges in R(x) whose connected components contain cycles] 2|V (x)| = 0 Now, since we know the Karp-Sipser algorithm finds a maximum matching in a tree, it follows that lim →0 E[size of maximum matching in R(x)] |V (x)| = lim →0 E[size of matching selected by Karp-Sipser in R(x)]
xuv e −(1−xuv) = e xuv − 1 e and similarly, the probability that an edge (u, v) is isolated amongst active edges is 1 − e −xuv e −(1−xuv) 2 = e 2xuv − e xuv e 2 Hence, if a CRS of the sort used in step 3 exists, the desired result follows since Pr[(u, v) ∈ M ] = 2 Pr[(u, v) is selected at u] − Pr[(u, v) is selected at both u, v] = 2 Pr[(u, v) is selected at u] − Pr[(u, v) is isolated amongst active edges] = 2x uv (1 − e −1/e ) − 1 2e 2 + e 2xuv − e xuv e 2 − e 2xuv − e xuv e 2 = 2x uv (1 − e −1/e ) − 1 2e 2
Again by the concavity of the function 1−e −(1−1/e)t , we have Pr[S∩R 1 = ∅] ≥ (1−e −(1−1/e) ) (u,v)∈S x uv . Hence by Theorem 4, there is a scheme such that each edge incident to v survives with probability Pr[(u, v) ∈ M ] ≥ (1 − e −(1−1/e) )x uv .
2 .
2For each (u, v) such that (u, v) the only active edge incident to v, decide independently at random whether to mark (u, v) red, so that Pr[(u, v) is red] = 1 − e −xuv/e . Mark all other active edges blue. We have Pr[(u, v) is blue] = e −xuv/e − e −xuv . 3. For each u ∈ V 1 , if there are any red edges incident to u, perform contention resolution to include one of them in R 1 , so that Pr[(u, v) ∈ R 1 ] = (1 − e −1/e )x uv .4. For each u ∈ V 1 , if there are no red edges incident to u, and there are some blue edges incident to u, perform contention resolution to include one of them in R 2 , so that Pr[(u, v) ∈ R 2 ] = (e −1/e − e −1 )x uv .
e
−xuv/e = 1 − e − (u,v)∈S xuv/e ≥ (1 − e −1/e ) (u,v)∈S x uv . By Theorem 4, there is a CRS among the red edges incident to u such that each edge survives as a red edge with probability (1 − e −1/e )x uv . Lemma 6. There is a CRS among blue edges such that Pr[(u, v) ∈ R 2 ] = (e −1/e − e −1 )x uv for every edge (u, v). Proof. Each edge is active with probability 1 − e −xuv and marked red with probability 1 − e −xuv/e ; otherwise it is blue, hence Pr[(u, v) is blue] = e −xuv/e − e −xuv . Consider a set of edges S incident to a vertex u ∈ V 1 and condition on the event that there are no red edges incident to u. Since the events of different edges incident to u being red or blue are independent, Pr[(u, v) is blue | no red edges incident to u] = Pr[(u, v) is blue | (u, v) is not red] = e −xuv/e − e −xuv e −xuv/e = 1 − e −(1−1/e)xuv .
Conditioned on this, (u, v) is not a gray edge if and only if (u, v) / ∈ R. Therefore, for a set of edges S incident to u, Pr[no gray edge among S | no active edge incident to u] = Pr[S ∩ R = ∅ | no active edge incident to u]
(e xuv − 1)e −x(S) − (1 − e −xuv/e ). Finally the probability that this edge survives is β = e −1/e −e −1 1−e −1 . Hence, case (a) contributes (u,v)∈S ((e xuv − 1)e −x(S) − (1 − e −xuv/e ))β. The probability that there are at least two active edges in S is 1 − Pr[no active edges ∈ S] − Pr[exactly one active edge in S], where Pr[no active edges ∈ S] = e −x(S) and Pr[exactly one active edge in S] = (u,v)∈S (e xuv − 1)e −x(S)
Pr[S ∩ (R 1 ∪ R 2 ∪ R 3 ) = ∅] ≥ (1 − e −1/e )x(S) − β e x(S) − (β − β 2 )e −x(S) x(S) + (2β − β 2 )(1 − e −x(S) ) + (e −1/e − 1 + 1/e)β − (e − 2)(β − β 2 ) + (1 − e − 1 2e ) e −x(S) Recall that β = e −1/e −e −1 1−e −1 .Plugging numerical values into the last line, we find that the constant in front of e −x(S) (u,v)∈S x 2 uv is positive ( 0.019). Hence we can ignore the last line and writePr[S ∩ (R 1 ∪ R 2 ∪ R 3 ) = ∅] ≥ (1 − e −1/e − β/e)x(S) − e −x(S) x(S)(β − β 2 ) + (1 − e −x(S) )(2β − β 2 ).It can be verified that this is a concave function of x(S) ∈ [0, 1], which is lower-bounded by γ x(S) where γ is the value of the right-hand side when x(S) = 1, γ = (1 − e −1/e ) − e −1 (4β − 2β 2 ) + (2β − β 2 ) ≥ 0.509.
Given a finite set X and valuations v st : 2 X → R + for s ∈ [m], t ∈ [n], choose sets S st ⊆ X, s ∈ [m], t ∈ [n] such that for every s, {S st : t ∈ [n]} are disjoint and for every t, {S st : s ∈ [m]} are disjoint, in order to maximize s,t v st (S st ).
Theorem 6 .
6For submodular valuations v st given by demand oracles (returning argmax S⊆X v st (S)− a∈S p a for given prices p a ), there is a 0.509-approximation for the problem of Allocation with row/column disjointness constraints.Proof. We use the "Configuration LP" which is a standard tool for allocation problems of this typet ∈ [n]; ∀S; x s,t,S ≥ 0.
It might appear from the work of Bruggmmann and Zenklusen (see lemma 7 in[14]) that the assumption of x ∞ ≤ should be easy to drop from Theorem 1. This would be the case if our theorem applied to graphs with parallel edges, which unfortunately, it does not.
Note the similarity in the expression to 2(1 − λ) − λ 2 = 2(1 − e −λ ) − λ 2 , the constant from our previous analysis. This similarity is not a coincidence, and we can think of the scheme we describe as first-order approximation to Karp-Sipser.
Acknowledgements.We would like to thank Chandra Chekuri for stimulating discussions.
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| [] |
[
"NECESSARY CONDITIONS ON THE SUPPORT OF RP-MEASURES",
"NECESSARY CONDITIONS ON THE SUPPORT OF RP-MEASURES"
] | [
"\nLINUS BERGQVIST\n\n"
] | [
"LINUS BERGQVIST\n"
] | [] | We give necessary conditions for when a subset of T n can contain the support of some non-zero RP-measure. Among other things we show that the support of a positive RP-measure cannot be contained in reflections of inverse images of half-planes by certain functions in A(D n ), sets of linear measure zero, and when n = 2, graphs of strictly increasing functions. For n = 2 we also prove that failure to contain the support of any positive RP-measure implies that restrictions of functions in a subspace of A(D 2 ) are uniformly dense among the continuous functions on a related set.2010 Mathematics Subject Classification. 28A25, 28A35 (primary); 32A10 (secondary). | null | [
"https://export.arxiv.org/pdf/2304.03072v1.pdf"
] | 257,985,136 | 2304.03072 | 5a8fc54324897fd163b7a5dd80dab8c2aebc0fd4 |
NECESSARY CONDITIONS ON THE SUPPORT OF RP-MEASURES
6 Apr 2023
LINUS BERGQVIST
NECESSARY CONDITIONS ON THE SUPPORT OF RP-MEASURES
6 Apr 2023and phrases RP-measuresPolydiscsAnalytic functions with positive real part
We give necessary conditions for when a subset of T n can contain the support of some non-zero RP-measure. Among other things we show that the support of a positive RP-measure cannot be contained in reflections of inverse images of half-planes by certain functions in A(D n ), sets of linear measure zero, and when n = 2, graphs of strictly increasing functions. For n = 2 we also prove that failure to contain the support of any positive RP-measure implies that restrictions of functions in a subspace of A(D 2 ) are uniformly dense among the continuous functions on a related set.2010 Mathematics Subject Classification. 28A25, 28A35 (primary); 32A10 (secondary).
Background and preliminaries
In the theory of analytic functions on the unit disc D it is extremely useful that every harmonic function on D is the real part of some analytic function, and that every non-negative harmonic function is the Poisson integral of some positive Borel measure on T (see for example the Theorem on page 33-34 in [7]). When analysing analytic functions on the unit polydisc D n however, one faces a new difficulty. Even though the Poisson integral of every positive Borel measure on T n is a non-negative n-harmonic function, not every n-harmonic function is the real part of some analytic function on D n . Here the Poisson integral of µ is defined as
P [dµ](z) := T n n j=1 (1 − |z j | 2 ) |ζ j − z j | 2 dµ(ζ), z ∈ D n .
We denote by M (T n ) the set of complex Borel measures on T n of bounded variation, and we denote by RP (T n ) the set of real measures in M (T n ) whose Poisson integrals are the real parts of analytic functions on D n . We will call such measures RP-measures since we will use a lot of results from the book [10] in which this terminology is used. Note however that many authors instead call such measures pluriharmonic measures.
In many ways RP (T n ) is not well understood for n > 1. Among other things, it is not invariant under certain operations one would wish it to be: for example if µ ∈ RP (T n ) and f ∈ L 1 (dµ), then f (z)dµ will generally not be an RP-measure, and it's not clear when it will be. Despite some efforts, see for instance [9], the extreme points of RP (T n ) have not been completely characterized. Also, the support of RP-measures is restricted in some ways that are not entirely understood. Recently, some necessary conditions on the support of RP-measures were obtained in [8]. Among other things, the authors proved that for n ≥ 2, if a positive RP-measure µ has support which doesn't intersect n j=1 {z ∈ [0, 2π) n : α j < z j < β j } for any choice of constants 0 ≤ α j < β j < 2π, for j = 1, . . . , n, then µ ≡ 0 (see Corollary 4.7 of [8]). Here T n has been identified with [0, 2π) n .
In this paper we will continue investigating which subsets cannot contain the support of any non-zero RP-measure. We will mainly, but not exclusively, be interested in positive RP-measures. The positive RP-measures are precisely the measures corresponding to positive harmonic functions, and such measures appear naturally in many different contexts. For example, Doubtsov recently initiated the study of Clark measures on T n in [4], and all such measures are positive RP-measures. For further results on Clark measures, see also [2] and [1], where among other things properties of the supports of Clark measures on T 2 corresponding to Rational inner functions were studied. We begin with some notation and preliminaries. Let X be a subspace of C(T n ) equipped with the supremum norm. A measure µ ∈ M (T n ) is said to be an annihilating measure for X if T n f (z)dµ(z) = 0 for all f ∈ X. The set of all annihilating measures of X will be denoted by X ⊥ . We will mainly be interested in the cases where X is the polydisc algebra A(D n ) (or A for short, when the dimension is contextually clear) or the space of functions of the form z 1 z 2 · · · z n f (z) where f (z) ∈ A(D n ). The space of functions of the form z 1 z 2 · · · z n f (z) where f (z) ∈ A(D n ) will be denoted by A 0 (D n ), or A 0 for short.
If S is a compact subset of T n , then we denote by X| S the subspace of C(S) consisting of all restrictions to S of functions f ∈ X. That is
X| S = {f | S : f ∈ X}.
Throughout this paper we will use the facts that every finite Borel measure on T n is a Radon measure, and that if µ is a Radon measure on a compact subset S ⊂ T n , then the space of continuous functions on S equipped with the supremum norm, which we will denote by C(S), is dense in L p (µ) for 1 ≤ p < ∞. See for example Theorem 7.8. and Proposition 7.9. in [5] for more details.
Finally, we will make extensive use of Theorem 6.1.2. from [10]. The theorem describes five different properties a compact subset of T n can have, and states that it has one of these properties if and only if it has all of the others. We will in particular use that a compact subset S ⊂ T n has the property that it is a null set for the annihilators of A, or more concretely that |µ|(S) = 0 for every complex measure µ ∈ A ⊥ if and only if it is an interpolation set for A. That a compact subset S is an interpolation set for A means that for every continuous function f on S there exists a function g ∈ A such that g| S = f . In particular, we will use that the same equivalence holds if we replace A with A 0 . Since this will be used several times, we formulate it as a theorem. Proof. We first show that S is an interpolation set for A if and only if it is an interpolation set for A 0 . Clearly an interpolation set for A 0 is an interpolation set for A since A 0 ⊂ A, and conversely, if S is an interpolation set for A, then for every f ∈ C(S) there is a function g ∈ A such that g| S = z 1 · · · z n f (z), and thus z 1 · · · z n g(z) ∈ A 0 and (z 1 · · · z n g(z))| S = f (z).
Secondly, S is a null set for all the annihilators of A if and only if it is a null set for all the annihilators of A 0 . By using that
T 2 z 1 · · · z n f (z)dµ(z) = T 2 f (z)(z 1 · · · z n dµ(z))
for every f (z) ∈ A, we see that µ is an annihilating measure for A 0 if and only if (z 1 · · · z n )dµ(z) is an annihilating measure for A. Since |z 1 · · · z n | = 1 on T n , these measures have the same null sets, and thus S is a null set for all annihilators of A if and only if it is a null set for all annihilators of A 0 .
The equivalence of the two properties now follow from Theorem 6.1.2. in [10] One of the five properties from Theorem 6.1.2. in [10] is that S ⊂ T n is a zero set for A. However, a zero set in that context is a set S on which a function f ∈ A vanishes, but such that f is non-zero everywhere in D n \ S.
In the next section we will consider level sets of functions f ∈ A 0 , but we will make no assumptions on f outside of T n .
Necessary conditions on the support of RP-measures
In this section we establish a correspondence between RP-measures and annihilators of A 0 , and then find necessary conditions for possible support sets of the latter class by: (1) using that certain sets corresponding to a function f ∈ A 0 can't possibly support measures that annihilate f , and (2) by applying known results about annihilators of the polydisc algebra from [10]. This will enable us to find necessary conditions on support sets for RP-measures.
The following simple lemma imposes a necessary condition on the support of positive measures in A ⊥ 0 and provides the idea for our later results. Lemma 1. The support of a non-zero positive measure µ ∈ A ⊥ 0 cannot be contained in the level set corresponding to any α = 0 of any function f ∈ A 0 .
Proof. If µ ∈ A ⊥ 0 is a non-zero positive measure and supp(µ) ⊂ {z ∈ T n : f (z) = α} then 0 = T n f (z)dµ(z) = {z∈T n :f (z)=α} f (z)dµ(z) = αµ(T n ) = 0,
which contradicts the assumption on µ.
We will show that by a simple transformation we can send any given RPmeasure to a real measure in A ⊥ 0 in a way that preserves positivity and whose effect on the support is easy to understand. Thus the above lemma can be used to obtain necessary conditions on the support of positive RP-measures.
By Theorem 2.4.1. in [10], a real measure µ is an RP-measure if and only if all of its mixed Fourier coefficients vanish. That is, µ is an RP-measure if and only if T n z k 1 1 · · · z kn n dµ(z) = 0 unless all k j ∈ Z satisfy k j ≥ 0 for j = 1, . . . , n, or k j ≤ 0 for j = 1, . . . n.
Note that for real measure µ
T n z k 1 1 · · · z kn n dµ(z) = 0 ⇐⇒ T n z −k 1 1 · · · z −kn n dµ(z) = 0.
In particular a real measure µ ∈ M (T 2 ) is an RP-measure if and only if
T 2 z n 1 z 2 m dµ(z 1 , z 2 ) = 0
for all m, n ≥ 1.
By making the change of variables T : (z 1 , . . . , z n ) → (z 1 , . . . , z n−1 , z n ) we see that if µ is an RP-measure, then the pushforward measure T * (µ), which is defined by T * (µ)(S) := µ(T −1 (S)) for every Borel set S, has the property that
T n z k 1 1 · · · z kn n dT * (µ)(z) = T n z k 1 1 · · · z k n−1 n−1 z n kn dµ(z) = 0 for all k j ≥ 1, j = 1 . . . n.
Since T is a bijection on T n and T 2 = Id (so T = T −1 ), this implies that if µ is an RP-measure then T * (µ) is a real measure that annihilates all monomials z k 1 1 · · · z kn n where k j ≥ 1, for j = 1, . . . n. By continuity of T * (µ) as a linear functional on C(T n ) this means that T * (µ) annihilates all f ∈ C(T n ) for whichf (k 1 , . . . , k n ) = 0 whenever k j < 1 for some j = 1 . . . , n. This is precisely the space of all functions of the form
z 1 · · · z n f (z) where f ∈ A(D n ), i.e. A 0 .
Remark. If n = 2 the above implications are in fact equivalences. So in this case µ is an RP-measure if and only T * (µ) is a real measure in A ⊥ 0 .
This means that if no real measure in A ⊥ 0 has support contained in S ⊂ T n then no RP-measure has support contained in T −1 (S). If µ is a positive RP-measure, then T * (µ) is also positive, and thus the corresponding statement also holds if we restrict our attention to positive measures. If S is a null set for all real measures in A ⊥ 0 , then T −1 (S) is a null set for all RPmeasures. Finally, note that if n = 2, then all of these implications become equivalences. We formulate this as a lemma.
Lemma 2. If a compact subset S ⊂ T n has the property that no non-zero real (positive) measure in
A ⊥ 0 has support contained in S, then no non-zero (positive) RP-measures has support contained in T −1 (S). Furthermore, if a compact subset S ⊂ T n has the property that µ(S) = 0 for all real (positive) µ ∈ A 0 (D n ) ⊥ , then ν(T −1 (S)) = 0 for all (positive) ν ∈ RP (T n ).
If n = 2 the above implications are equivalences.
For n = 2 one can thus completely understand the supports and null sets of RP-measures by understanding the corresponding sets for real measures in A 0 (D 2 ). For larger n, it is a lot more difficult for a measure µ to be an RP-measure than it is for T * (µ) to annihilate A 0 . Nevertheless, Lemma 2 can still be used to find necessary conditions. If µ is a positive RP-measure, then by Lemma 1 the support of T * (µ) can't be contained in any set of the form
{z ∈ T n : f (z) = α},
for any f ∈ A 0 and α = 0, and so by Lemma 2 the support of µ cannot be contained in the image of such a set under T −1 .
However, by noting that the only property a set S containing the support of µ needs to have for the proof of Lemma 1 to work is that the range of some suitable f ∈ A 0 on this set is such that the integral S f (z)dµ(z) cannot vanish, we can extend the above result to a much larger class of sets. As will soon be shown, every set that is mapped by f into some closed half-plane
H θ ǫ := {e iθ z ⊂ C : Re(z) ≥ ǫ},
will have this property whenever ǫ > 0 and θ ∈ [0, 2π). This is essentially a consequence of the fact that the range of f is then, by assumption, a subset of a convex set that does not contain the origin.
Note that H θ ǫ is the maximal closed convex subset of C with the property that ǫe iθ is the unique element in the subset which is closest to the origin: if we add any additional element outside H θ ǫ to the set, then convexity will imply that the line connecting ǫe iθ to this new point also lies in the set, and this line will contain points of smaller distance to the origin.
The following elementary lemma illustrates the above point, and will be used to give necessary conditions on the supports of RP-measures.
Lemma 3. Let f ∈ C(T n ) and let µ be a non-zero positive measure in M (T n ) such that supp(µ) ⊂ f −1 (H θ ǫ ). Then T n f (z)dµ(z) = 0.(1)
Proof. By setting g(z) = f (z)e −iθ if necessary, we can without loss of generality assume that θ = 0.
We have that
Re T n f (z)dµ(z) = Re f −1 (H 0 ǫ ) f (z)dµ(z) = f −1 (H 0 ǫ ) Re(f (z))dµ(z) ≥ ǫµ(T n ) > 0,
and thus the integral on the left of (1) is not 0.
We are now ready to prove the main lemma of this section. Proof. Suppose on the contrary that there is a non-zero positive measure µ ∈ M (T n ) which annihilates all functions in A 0 , but supp(µ) ⊂ f −1 (H θ ǫ ) for some choice of f ∈ A 0 , constant ǫ > 0, and angle θ ∈ [0, 2π). Then by Lemma 3
T 2 f (z)dµ(z) = 0,
which contradicts the assumption that µ annihilates A 0 .
By combining Lemma 4 and Lemma 2 we obtain the following theorem. Note that the above theorem also prohibits supports contained in T −1 (S) where S is the level set of a function f ∈ A 0 corresponding to any α = 0.
When dealing with concrete subsets of T n it is often more convenient to describe these sets in terms of the covering map Φ : R n → T n given by Φ(x 1 , . . . , x n ) = (e ix 1 , . . . , e ixn ). For example when working with subsets of the form Φ(f (t), g(t)) with t ∈ R, in which case we can employ terminology like positive and negative slope and so on.
= Φ(v 1 , −v 2 ) : e i(nv 1 +mv 2 ) = e iω , ω ∈ [θ − w, θ + w] ⊂ T 2
for any choice of 0 ≤ w < π/2, θ ∈ [0, 2π), and n, m ≥ 1.
In particular, by setting m = 1, we see that no positive RP-measure has support contained in the set
{Φ(v 1 , nv 1 − ω) : v 1 ∈ [0, 2π), ω ∈ [θ − w, θ + w]}
for any 0 ≤ w < π/2, θ ∈ [0, 2π), and n ≥ 1.
For n = 2, the above example already excludes the possibility of positive RP-measures being supported on plenty of curves of strictly positive slope. In fact, by using the results of Chapter 6 in [10], we can show that no RPmeasure can be supported on a curve of strictly positive slope. To show this, we will need the following lemma, which is obtained by combining Theorem 6.3.5. and Theorem 6.1.2. in [10].
Lemma 5. Let F ⊂ R 2 be the graph of a strictly decreasing function ψ with domain R and let K ⊂ T 2 be a compact subset of Φ(F ). Then K is an interpolation set for A(D 2 ).
Note that ψ is not even assumed to be continuous.
Theorem 3. If F is the graph of a strictly increasing function ψ, then every compact subset K ⊂ Φ(F ) is a null set for RP (T 2 ).
In particular no non-zero RP-measure has support contained in Φ(F ) ⊂ T 2 where F is the graph of a strictly increasing function ψ.
Proof. By combining Lemma 5 and Theorem 1 we see that T (K) is a null set for A ⊥ 0 . The theorem now follows from Lemma 2.
That a result corresponding to Lemma 5 can't hold in general for functions of positive slope is clear, since if this was the case Theorem 3 could be extended to strictly decreasing functions as well. But this is impossible since, for example, Lebesgue measure on
Φ({(v, −v) : v ∈ [0, 2π)}) = Φ[T ({(v, v) : v ∈ [0, 2π)})]
is an RP-measure. In fact, by combining Theorem 6.3.4. and 6.1.2. of [10] one sees that Lemma 5 is false for every strictly increasing function ψ which can be extended to a holomorphic function in some neighborhood of the interval of R on which it is defined. This covers the example above since f (v) = v is strictly increasing on R and can be extended to a holomorphic function.
The results and examples established so far show that it is generally difficult for a "small" set -like a finite collection of points -to support any positive RP-measure, since a small set will be contained in f However, by using results from Chapter 6 in [10] we can prove that no set of linear measure zero -i.e. of Hausdorff dimension strictly smaller than one -can support any RP-measure (positive or not). Recall that a set F ⊂ R n has linear measure zero if to every ǫ > 0 there corresponds an open cover {V j } of F such that the sum of the diameters of the sets V j is smaller than ǫ.
By combining Theorem 6.1.2. in [10] and the corollary on page 149, we obtain the following lemma.
Lemma 6. If K ⊂ T n is a compact set of linear measure zero, then K is an interpolation set for A.
By combining this lemma with our previous results we can prove the following.
Theorem 4. If K ⊂ T n is a compact subset of linear measure zero, then K is a null set for RP (T n )
In particular no non-zero RP-measure on T n has support of linear measure zero, or equivalently the support of a non-zero RP-measure has Hausdorff dimension greater than or equal to one.
Proof. If K has linear measure zero, then so does T (K). Thus combining Lemma 6 and Theorem 1 shows that T (K) is a null set for A ⊥ 0 . The theorem now follows from Lemma 2.
This generalizes Remark 4.3. in [8] in which it is noted that all points must have zero mass for an RP-measures.
For n = 2 the above Theorem is sharp in the sense that there are RPmeasures supported on curves, i.e. on sets of Hausdorff dimension 1. One might ask if it is in fact the case that the dimension of the support of an RP-measure is at least n − 1.
Results related to interpolation for A 0
As a consequence of Lemma 4 we see that no positive measure that annihilates A 0 can have support contained in {z ∈ T 2 : f (z) = 1} for any f ∈ A 0 . The present section is concerned with exploring to what extent the converse holds.
This question is related to Bishop's theorem from [3] (Theorem 6.1.3. in [10]). From this theorem we can conclude that if S not only supports no positive measure in A ⊥ 0 , but even has the stronger property that |µ|(S) = 0 for every complex measure in A ⊥ 0 , then S is an interpolation set for A 0 (in fact S even has the stronger property that it is a peak-interpolation set, see Definition 6.1.1. of [10] for details). Clearly in this case there must be some function f ∈ A 0 such that f | S = 1, and thus S is contained in the level set of some function in A 0 .
By using an approach inspired by Helson and Lowdenslager (see [6]) we can show a partial converse to that statement: if a compact set S ⊂ T 2 has the property that no positive measure in A ⊥ 0 has support contained in S, then 1 lies in the L 2 (dµ) closure of A 0 for every positive Borel measure µ whose support is contained in S. But we can go a lot further. It turns out that if no positive measure in A ⊥ 0 has support contained in S, then A 0 | S is uniformly dense in C(S).
In order to get simpler notation, we will only study the case n = 2 in the current section. It should be noted however, that apart from the last theorem -which requires equivalence in Lemma 2 -none of the results require that n = 2.
The following lemma is inspired by, and resembles the presentation of Helson and Lowdenslager's ideas given in Chapter 4 of [7]. Proof. Denote by B the closed subspace of L 2 (dµ) spanned by functions in A 0 . Clearly B is contained in the closed subspace of L 2 (dµ) spanned by the functions in A since A 0 ⊂ A. We will prove the other inclusion, and thus the theorem, by showing that if any function of the form z n j is not contained in B, where n ≥ 0 and j = 1, 2, then there is a real measure supported on S which annihilates A 0 . This contradiction shows that a uniformly dense subspace of A is contained in B, and thus the closure of A is also contained in B.
Denote by F the orthogonal projection of z n j onto B. By the defining property of F the expression z n j − F = 0 is orthogonal to B, and hence to A 0 .
In fact, z n j − F is orthogonal to every function of the form f · (z n j − F ) where f ∈ A 0 . To see this, note that there is a sequence of functions
{f n } ∞ n=1 such that f n → F in L 2 (dµ), that f · (z n j − f n ) ∈ A 0 for every fixed choice of f ∈ A 0 , and that f · (z n j − f n ) → f · (z n j − F ).
The statement now follows from continuity of µ as a linear functional on C(S).
That z n j − F is orthogonal to f · (z n j − F ) for every f ∈ A 0 means that T 2 f |z n j − F | 2 dµ = 0
for all f ∈ A 0 . But this means that |z n j −F | 2 dµ is a real measure with support contained in S which annihilates A 0 . This contradicts the assumption on S.
Remark. The above lemma is sharp in the sense that if S is a set which supports a measure µ such that some function f ∈ A \ A 0 does not lie in the L 2 (dµ) closure of A 0 then by the same argument as above |f − F | 2 dµ will annihilate A 0 , be supported on S, and by assumption it is not the zero measure.
Thus for such sets the above construction can be used to generate nonzero positive measures in A ⊥ 0 with support contained in S, and thus for n = 2 it can be used to generate non-zero positive RP-measures with support contained in T −1 (S). Lemma 8. Let S ⊂ T 2 be a compact subset with the property that no nonzero positive Borel measure in A ⊥ 0 has support contained in S. Then A 0 is a dense subspace of L 2 (dµ) for every positive measure µ with support contained in S.
Proof. Denote by B the closed subspace of L 2 (dµ) spanned by functions in A 0 . We will prove that B = L 2 (dµ) by showing that B contains z α 1 z β 2 for all α, β ∈ Z. This will be done by applying Lemma (7) Note that S 1 = A 0 . By Lemma (7) we have that S 0 ⊂ B. Now suppose that S k is contained in the closed subspace spanned by the functions in S k+1 for some integer k ≤ 0. Since multiplication by a monomial is an isometry on L 2 (dµ) for every Borel measure µ on T 2 , we have, for every choice of
α, β ≥ k, that inf f ∈S k T 2 |z α−1 1 z β−1 2 − f (z)| 2 dµ = inf f ∈S k T 2 |z α 1 z β 2 − z 1 z 2 f (z)| 2 dµ = inf f ∈S k+1 T 2 |z α 1 z β 2 −f (z)| 2 dµ = 0,
where the last equality holds by the induction hypothesis. It follows that S k−1 is contained in the closed subspace spanned by functions in S k , and thus by induction we have that all functions of the form z α 1 z β 2 with α, β ∈ Z are contained in B. Thus C(S) ⊂ B by the Stone-Weierstrass theorem. Since µ is a Radon measure, we have that C(S) is dense in L 2 (dµ), and thus it also follows that L 2 (dµ) ⊂ B, which finishes the proof.
Remark. If we would have assumed that S has the stronger property that it supports no complex measure that annihilates A 0 , then the above result would follow a lot easier by using the same idea as in the proof of Lemma 7. In that case on could simply take any f ∈ C(S), let F denote the L 2 (dµ) projection of f onto B, and then note that the measure (f − F )dµ annihilates A 0 , contradicting the assumption on S unless the measure is 0. The difficulty is thus in obtaining a positive measure in A ⊥ 0 . In fact, by the Hahn-Banach theorem we immediately see that A 0 | S is uniformly dense in C(S) if every complex measure on S that annihilates A 0 also annihilates C(S), since if it is not dense there should be a non-trivial bounded linear functional on C(S) -that is, a complex measure with support contained in S -which annihilates A 0 | S . This idea will be used to prove the main theorem of this section. Proof. We begin by proving that the only complex measure in A ⊥ 0 with support contained in S is the zero measure.
Let µ be an arbitrary complex measure in A ⊥ 0 . Then by the Radon-Nikodym theorem dµ(z) = U (z)d|µ| for some measurable function U (z) with |U | = 1 everywhere. Thus
0 = S f (z)dµ(z) = S f (z)U (z)d|µ|(z) for all f (z) ∈ A 0 , so U (z) is orthogonal to A 0
Since the total variation measure |µ| is a positive measure with support contained in S, we know by Lemma 8 that A 0 is dense in L 2 (d|µ|). Thus we can find a sequence of functions f n ∈ A 0 such that f n → U in L 2 (d|µ|). It follows that
0 = lim n→∞ S f n U d|µ|(z) = S |U | 2 d|µ|(z) = |µ|(S),
so |µ| is the zero measure and thus µ is also the zero measure. Since µ was arbitrary it follows that the zero measure is the only complex measure with support contained in S that annihilates A 0 .
This implies that A 0 | S is a dense subspace of C(S) with respect to the supremum norm. To see this, note that if this was not the case, then by the Hahn-Banach theorem, there should be a non-trivial element in C(S) * that annihilates all functions in A 0 | S . That is, a non-trivial complex Borel measure µ on S such that S f (z)dµ(z) = 0 for all f ∈ A 0 . But we have proved that such a measure cannot exist, and thus A 0 | S is a dense subspace of C(S).
Finally, by combining Theorem 5 with Lemma 2 we get the following result for compact sets that support no positive RP-measures. Theorem 6. Let S ⊂ T 2 be a compact subset with the property that no positive RP-measure has support contained in S. Then A 0 | T (S) is a uniformly dense subspace of C(T (S)).
We end this paper with some further observations and questions.
Theorem 5 shows that if S ⊂ T 2 is a compact subset with the property that no non-zero positive Borel measure that annihilates A 0 has support contained in S, then A 0 | S is uniformly dense in S. One might ask if it is possible to go even further and conclude that such a set S ⊂ T 2 is an interpolation set for A 0 . In general this will not be possible, as will be explained below.
As mentioned earlier, Bishop's theorem (a more general version of one part of Theorem 1) shows that S is indeed an interpolation set for A 0 if S has the stronger property that |µ|(S) = 0 for every complex measure in A ⊥ 0 , and as Theorem 1 shows, this implication is in fact an equivalence. Thus if S is a compact subset with the property that no non-zero positive measure in A ⊥ 0 has support contained in S; then asking under what extra assumptions on S we can conclude that S is an interpolation set for A 0 is equivalent to asking under what extra assumptions such a set is a null set for all annihilators of A 0 .
Consider the situation for functions of one variable. By the theorem of F. and M. Riesz the only complex measures that annihilate the space A 0which consists of all functions in A(D) that vanish at the origin -are the measures of the form f (z)|dz| where |dz| denotes Lebesgue measure on T and f is an element of the Hardy space H 1 . This means that the only compact subset of T that contains the support of any annihilator -either complex, real or positive -is all of T. Note that a generalization of the observation that we get the same possible support sets for both positive and complex measures appears in the proof of Theorem 5.
So if a compact subset K does not support any real annihilator of A 0 then this only means that K = T, and thus this alone does not imply that it is a null set for all complex annihilators. For example no annihilator of A 0 has support contained in a half-circle, but the half circle is not a null set for all complex annihilators. This shows that Theorem 5 cannot (without further assumptions) be generalized to conclude that such a set K is an interpolation set.
However, if we add the assumption that K does not support any real annihilator of A 0 and has Hausdorff dimension less than the dimension of T (i.e. we assume it has measure 0), then in fact this does imply that it is a null set for all complex annihilators (again by the F. and M. Riesz theorem). One might ask if something similar holds on T 2 . For example if certain restrictions on the dimension and location of K combined with the assumptions from Theorem 5 are enough to conclude that K is an interpolation set.
Lemma 4 .
4No non-zero positive measure in A ⊥ 0 can have support contained in f −1 (H θ ǫ ) for any choice of f ∈ A 0 , constant ǫ > 0, and angle θ ∈ [0, 2π).
Theorem 2 .
2No positive RP-measure has support contained inT −1 ({z : f (z) ∈ H θ ǫ }) = {(z 1 ,.. . , z n−1 , z n ) : f (z 1 , . . . , z n ) ∈ H θ ǫ } for any choice of f ∈ A 0 , constant ǫ > 0, and angle θ ∈ [0, 2π).
Example 1 .
1Let n = 2. Every function of the form f n,m (z) = z n 1 z m 2 , n, m ≥ 1 lies in A 0 , and f n,m : T 2 → T. By choosing ǫ small enough, every set of the form {e iω : ω ∈ [θ − w, θ + w]} ⊂ C is contained in some H θ ǫ whenever 0 ≤ w < π/2. Thus no non-zero positive RP-measure has support contained in the set S m,n :
− 1 (
1H θ ǫ ) for some choice of f ∈ A 0 , constant ǫ > 0 and angle θ ∈ [0, 2π). However, it is for example not yet clear that there are no RP-measures supported on a Cantor like subset of the line Φ({(v, −v) : v ∈ [0, 2π)}), since the latter set does support positive RP-measures. Note that this set is not excluded by Corollary 4.7 in [8] either.
Lemma 7 .
7Let S ⊂ T 2 be a compact subset with the property that no nonzero positive Borel measure in A ⊥ 0 has support contained in S. Then the closed subspace of L 2 (dµ) spanned by A 0 equals the closed subspace spanned by A for every positive Borel measure µ with support contained in S.
and induction. Now define the spaces S k := {f ∈ C(T 2 ) :f (n, m) = 0 if n < k or m < k}.
Theorem 5 .
5Let S ⊂ T 2 be a compact subset with the property that no non-zero positive Borel measure that annihilates A 0 has support contained in S. Then the restrictions of functions in A 0 to S is a uniformly dense subspace of C(S).
Theorem 1 .
1Let S be a compact subset of T n . Then S is an interpolation set for A if and only if it is an interpolation set for A 0 , and it is a null set for A ⊥ 0 if and only if it is a null set for A ⊥ .Thus S has the property that for every continuous function f ∈ C(S) there
exists a function g ∈ A 0 such that g| S = f if and only if S is a null set for
A ⊥
0 .
Email address: [email protected]
AcknowledgementsThe author thanks Bartosz Malman for several valuable discussions and ideas, especially regarding the last section, and Lemma 8 and Theorem 5 in particular.
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| [] |
[
"Co distribution in ferromagnetic rutile Co-doped TiO 2 thin films grown by laser ablation on silicon substrates",
"Co distribution in ferromagnetic rutile Co-doped TiO 2 thin films grown by laser ablation on silicon substrates"
] | [
"Nguyen Hoa \nParc de Grandmont\nLaboratoire LEMA\nUMR 6157 CNRS-CEA\nUniversité F. Rabelais\n37200ToursFrance\n",
"Hong ",
"Joe Sakai \nSchool of Materials Science\nJAIST\nAsahidai 1-1, Tatsunokuchi-machi923-1292IshikawaJapan\n",
"W Prellier \nLaboratoire CRISMAT\nUMR 6508\nCNRS\nENSICAEN\n\n",
"Awatef Hassini \nParc de Grandmont\nLaboratoire LEMA\nUMR 6157 CNRS-CEA\nUniversité F. Rabelais\n37200ToursFrance\n",
"\nBd du Maréchal Juin\n14050CaenFrance\n"
] | [
"Parc de Grandmont\nLaboratoire LEMA\nUMR 6157 CNRS-CEA\nUniversité F. Rabelais\n37200ToursFrance",
"School of Materials Science\nJAIST\nAsahidai 1-1, Tatsunokuchi-machi923-1292IshikawaJapan",
"Laboratoire CRISMAT\nUMR 6508\nCNRS\nENSICAEN\n",
"Parc de Grandmont\nLaboratoire LEMA\nUMR 6157 CNRS-CEA\nUniversité F. Rabelais\n37200ToursFrance",
"Bd du Maréchal Juin\n14050CaenFrance"
] | [] | Pure rutile Co-doped TiO 2 films were fabricated successfully by the conventional pulsed laser deposition technique on silicon substrates from a ceramic target.Under the right fabrication conditions, Co concentration in the films could be almost the same as in the synthesized target, and films under various conditions all are ferromagnetic well above room temperature. Even though Rutherford backscattering spectroscopy measurements show that Co atoms seem to be mostly localized near the surface of the films and less exist in deeper levels, other experimental evidences show that the ferromagnetism does not come from Co segregations but from the Co-doped TiO 2 matrix. Rutile Ti 1-x Co x O 2 thin films grown by a very simple technique on low-price silicon substrates showing Curie temperature (T C ) above 400 K appear to be very attractive to applications. | 10.1063/1.1619227 | [
"https://export.arxiv.org/pdf/cond-mat/0308494v1.pdf"
] | 118,898,540 | cond-mat/0308494 | 59ab59dbb787e359fffe275147de170aa58d0962 |
Co distribution in ferromagnetic rutile Co-doped TiO 2 thin films grown by laser ablation on silicon substrates
Nguyen Hoa
Parc de Grandmont
Laboratoire LEMA
UMR 6157 CNRS-CEA
Université F. Rabelais
37200ToursFrance
Hong
Joe Sakai
School of Materials Science
JAIST
Asahidai 1-1, Tatsunokuchi-machi923-1292IshikawaJapan
W Prellier
Laboratoire CRISMAT
UMR 6508
CNRS
ENSICAEN
Awatef Hassini
Parc de Grandmont
Laboratoire LEMA
UMR 6157 CNRS-CEA
Université F. Rabelais
37200ToursFrance
Bd du Maréchal Juin
14050CaenFrance
Co distribution in ferromagnetic rutile Co-doped TiO 2 thin films grown by laser ablation on silicon substrates
Pure rutile Co-doped TiO 2 films were fabricated successfully by the conventional pulsed laser deposition technique on silicon substrates from a ceramic target.Under the right fabrication conditions, Co concentration in the films could be almost the same as in the synthesized target, and films under various conditions all are ferromagnetic well above room temperature. Even though Rutherford backscattering spectroscopy measurements show that Co atoms seem to be mostly localized near the surface of the films and less exist in deeper levels, other experimental evidences show that the ferromagnetism does not come from Co segregations but from the Co-doped TiO 2 matrix. Rutile Ti 1-x Co x O 2 thin films grown by a very simple technique on low-price silicon substrates showing Curie temperature (T C ) above 400 K appear to be very attractive to applications.
Since the discovery of Matsumoto et al. 1 about 2 years ago, Co-doped TiO 2 (Ti 1-x Co x O 2 ) thin films have attracted many research groups due to their exhibition of ferromagnetism well above room temperature which is very useful for applications. Growth of this diluted magnetic semiconductor (DMS) by thin film techniques, such as molecular beam epitaxy (MBE) or pulsed laser deposition (PLD)…, provides excellent control of the dopant concentration and the ability to grow single-layered film. However, there are certain issues in this research field at the moment: how to control the concentration of dopant more easily, how to improve the ferromagnetism, and how to clarify the nature of magnetism in those films. So far, Co-doped TiO 2 films were deposited from two targets, Ti and Co or TiO 2 and Co-doped TiO 2 with a very high concentration of Co in order to get very few percents of Co incorporated in the films, by using very sophisticated methods such as combinatorial laser ablation (using the rotation of combinatorial masks), MBE laser ablation, oxygen plasma assisted MBE or co-sputtering. [1][2][3][4] Some paper reported about films which were ablated from a ceramic target but it was said that Co did not get into the structure but remained as Co metal 5 .
The average magnetic moments per Co atom reported so far are still very modest (as 0.32 µ B for laser ablated films 1 and about 1.1 to 1.3 µ B for films grown by oxygen plasma assisted MBE 3,4 ), and the nature of ferromagnetism was claimed to be caused by Co or cobalt oxide clusters [4][5][6][7] . In order to look forward to solving some of those problems, in this work, we have tried to fabricate Co-doped TiO 2 films from a ceramic target on silicon substrates by using a conventional PLD system. It is believed that if we use a well-done ceramic target, and control correctly the growth conditions, the films whose dopant concentrations are almost the same as they are in the fixed target with ferromagnetism above room temperature can be obtained.
A polycrystalline target of Co-doped TiO 2 with Ti: Co ratio as 0.88 : 0.12 was synthesized by an organic gel-assisted citrate process. The films were deposited by the PLD techniques (248 nm KrF excimer laser, pulses of 5 Hz) on un-etched (100) Si substrates. We applied various conditions: the oxygen partial pressure (P O2 ) was kept as 1×10 -6 Torr or 1×10 -5 Torr, and the energy density was 1.5 J/cm 2 or 3 J/cm 2 . Hereafter, 4 main conditions will be marked as LL (low P O2 , low energy density), LH (low P O2 , high energy density), HL (high P O2 , low energy density) and HH (high P O2 , high energy density). The temperature on the substrates was kept as 700 o C. After deposition, all films were cooled down slowly to room temperature under the oxygen pressure of 20 mTorr. The typical thickness of the films was 230 nm. The crystalline structure was studied by X-ray diffraction (XRD) with Cu Kα radiation (λ=1.5406Å), using a Seifert for the Θ-2Θ scan and an X'Pert Philips MRD for the in-plane measurements (Φ-scans). The magnetization measurements were performed by a Quantum Design superconducting quantum interference device (SQUID) system from 0 to 0.5 T in the range of temperature from 400 K down to He temperature, the film morphology was checked by a scanning electron microscope (SEM), and the chemical composition was determined by both energy dispersive X-ray (EDX) and Rutherford backscattering spectroscopy (RBS) methods. The RBS measurements were performed with an incident energy of He + as of 3.049 MeV, a scattering angle of 170 o and an accumulation charge for each measurement as of 2 µC).
X-ray measurements confirmed that all film are single phased rutile with only rutile peaks appeared in the spectra (for an example, see Fig. 1 for X-ray patterns of the HL film). The films are highly epitaxial with the c-axis of the rutile (around 2.96Å) perpendicular to the substrate plane. Neither Co nor cobalt oxide phase was found in the spectra. Films on Si substrate are mostly c-axis oriented but other diffraction peaks, indexing on the basis of the rutile phase are present, indicating that the film grows with several orientations (probably due to the large lattice mismatch). However, the Φ-scan recorded around the 110 reflection (see the inset of Fig. 1a) shows 90° separated peaks that gives an evidence of in-plane texture of the rutile phase. Similar scans taken on the 220 reflection of Si revealed that the TiO 2 rutile layer grows epitaxially, cube-on-cube on Si substrates. In fact, from SEM images (to be discussed later in this report), the HL sample whose X-ray pattern are shown, is the one which has the worst morphology with the presence of some alien parts which are thought to be due to Co segregations but we found no peaks of cobalt or cobalt oxide, and the film is pure rutile.
For other better films, the same results are obtained, only rutile peaks appear in the spectra. It is not possible to say very confirmatively that there is no segregation of Co in the films if it is below the detection limit, but it is sure that the Co-doped films on Si are well-established rutile. Although rutile Co-doped films on Si substrates have been already fabricated successfully by co-sputtering from Co and Ti targets 8 but the present study is the first case they are done by a conventional PLD from one ceramic target. Fig. 2(a) shows the magnetization as a function of magnetic field taken at 300 K for the LL film. Hysteresis was observed showing that the film is ferromagnetic even at room temperature. M(T) curve taken at 0.2 T in Fig. 2(b) shows that the film has Curie temperature (T C ) higher than 400 K.
As mentioned above, a big issue in the field at the moment is how to control the dopant concentration, and to know Co distribution in the films and the nature of ferromagnetism as well. EDX measurements show that 4 films with 4 different conditions have Co content as of 12%, the same as in the synthesized target. SEM images are shown in Fig. 3. The LL and HH films' morphology are similar to each other, rather homogenous among all, but films are full of particles. The surface of LH film seems to be smoothest, even if with few alien particles on it (white parts). The morphology of the HL film is the worst with very big white parts which is believed to be excess Co, CoO or Co 3 O 4 since those white parts do not have the spherical shape of normal droplets of thin films, but they look like outgrowths. Basically, black and white parts are detected from parts which have different electric conductivities, therefore they are thought to show different compositions. However, we failed to distinguish the difference in compositions of those parts from EDX measurements (The detection limit of EDX technique does not support to specifically determine the composition of nanometer-sizeparticles). Since seeing neither peaks of cobalt nor cobalt oxides in the XRD does not rule out completely the possibility that excess Co or cobalt oxides exist, then transmission electron microscopy (TEM) measurements must be done in the near future. To know more precisely about the compositions of thin films, RBS measurements were performed. RBS spectra of Codoped TiO 2 films are shown in Fig. 4. Based on the obtained data, the Ti : Co ratio for each case can be estimated to be 90.8 : 9.2 for the LL film, 92.4 : 7.6 for the LH film, 90.0 : 10.0 for the HL film and 93.5 : 6.5 for the HH film. From RBS data, the highest Co concentration is 10% in the HL film whose SEM picture shows some outgrowth of excess Co or cobalt oxide. No reasonable explanation could be given to the SEM pictures of the LL film (with 9.2% Co) and of the HH film (6.5% Co) since they are quite the same and the HH film with lower Co concentration even has some alien particle on it. Then it is not very simple to say that when the amount of Co in the target is large, it gives some excess on the film which leads to Co or cobalt oxide particles/clusters, on the other hand, one must say that the way how Co atoms distribute in the films depends very much on the growth conditions. As seen in Fig. 4, Co atoms were not distributed uniformly in the films: while Ti peaks have simple rectangular shapes, Co peaks have larger height at the right hand side (shallower levels, taken from the surface) and smaller height at the left hand side (deeper levels). Detailed calculations give concrete information, for example, for the LL film, Ti : Co ratio in the depth from 0 up to 40nm is 70 : 30 while in the layer of from 40nm to 230nm-thick, it is 94 : 6, and as the results the averaged ratio of Ti : Co for the film will be 90.8 : 9.2 (as mentioned above). It means that Co atoms are localized mostly near to the surface of the films while they exist less in the deeper levels. This RBS result explains why by EDX the Co content was found to be 12% since signals from atoms near the surface are more sensitive in EDX.
It is known that the saturated magnetization of Co metal is 1.7 µ B /Co. It was confirmed by the experimental evidence of Co-doped TiO 2 films with Co clusters 5 . The value of M s as 0.31µ B /Co in our films shows that the ferromagnetism does not come from Co particles or clusters. This is also confirmed by magnetic force microscopy (MFM) measurements: we found no contrast on the surface of the film with LL conditions, for example, or in other words, no particles or clusters were observed and the film is very homogeneous. According to the theory for dopants in Co-doped TiO 2 of Sullivan and Erwin 9 , it seems that our films were grown in the "rich oxygen condition" and Co dopants were formed primarily in neutral substitutional form, but not interstitial ("poor oxygen condition" makes Co concentrations of substitutional and interstitial Co roughly equal, and M s must be in between 1 and 2µ B /Co as in the report of Chamber 4 ). It is thought that the magnitude of M s can be enhanced very much in an appropriate oxygen environment, on the other hand, the homogeneity of the film surely depends strongly on the growth conditions 10 .
In conclusion, we have achieved to fabricate rutile Co-doped TiO 2 films on silicon substrates by the conventional PLD technique from a ceramic target. Films with various conditions all show ferromagnetism far above room temperature. Co concentration in the film can be almost the same as in the synthesized target if we apply suitable growth conditions. Even though the distribution of Co is not uniform with Co atoms lie mostly near to the surface of the films, the ferromagnetism in our Co-TiO 2 films seemingly does not come from Co metals or clusters. Co-TiO 2 films with very high T C (above 400 K) fabricated by a very simple technique on low-price silicon substrates are useful for applications. However, the magnitude of saturated magnetization is still modest and a higher homogeneity is still looked forward.
All films with our chosen growth conditions showed ferromagnetic behaviors at room temperature. The magnetization loops are quite similar, except the difference in magnitude of saturated magnetization (M s ) and coercivity (H C ). The highest M s achieved in our films was 0.31 µ B /Co, almost the same as of the Co x Ti 1-x O 2 film with x = 7% on a SrTiO 3 substrate reported by Matsumoto et al. 1 .
Improvements can be done by changing the concentration of dopant in the target and adjusting growth conditions.
Figure
Captions
oxygen pressure of 1×10 -6 Torr and the fluence of 3J/cm 2 . The bump around 2Θ = 40° results from various diffraction peaks arising from different out-of-plane orientations of the TiO 2 phase. Only diffraction peaks corresponding to the rutile phase were observed. The inset depicts the Φ-scan recorded for the (110) reflection of the rutile TiO 2 . 2. Magnetization of a Ti 0.908 Co 0.092 O 2 thin film deposited on a Si substrate under the oxygen pressure of 1×10 -6 Torr and the fluence of 1.5J/cm 2 (a) versus magnetic field at 300 K and (b) versus temperature under 0.2 T. 3. SEM images of Co-doped TiO 2 films with 4 different conditions: a) LL (P O2 of 1×10 -6 Torr and fluence of 1.5J/cm 2 , b) LH (1×10 -6 Torr, 3J/cm 2 , c) HL (1×10 -5 Torr, 1.5J/cm 2 and d) HH (1×10 -5 Torr, 3J/cm 2 ). 4. RBS spectra of Co-doped TiO 2 films with 4 different conditions: a) LL, b) LH, c
AcknowledgementsThe authors would like to thank Dr. A. Ruyter for MFM measurements.Electronic mail: [email protected]
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| [] |
[
"Seasonality Effects on Consumers' Preferences Over Quality Attributes of Different Beef Products",
"Seasonality Effects on Consumers' Preferences Over Quality Attributes of Different Beef Products"
] | [
"Ali Ardeshiri [email protected] \nSchool of Civil and Environmental Engineering\n-Research Centre for Integrated Transport Innovation (rCITI)\nUniversity of New South Wales (UNSW) Sydney\n2052NSWAustralia\n",
"Spring Sampson \nHarvest Insights\nSouth Melbourne, VIC 3205Australia\n",
"Joffre Swait \nJDS Behavior Insights\n29687TaylorsSCUSA\n"
] | [
"School of Civil and Environmental Engineering\n-Research Centre for Integrated Transport Innovation (rCITI)\nUniversity of New South Wales (UNSW) Sydney\n2052NSWAustralia",
"Harvest Insights\nSouth Melbourne, VIC 3205Australia",
"JDS Behavior Insights\n29687TaylorsSCUSA"
] | [] | Using discrete choice modelling, the study investigates 946 American consumers' willingnessto-pay and preferences for diverse beef products. A novel experiment was used to elicit the number of beef products that each consumer would purchase. The range of products explored in this study included ground, diced, roast, and six cuts of steaks (sirloin, tenderloin, flank, flap, New York and cowboy/rib-eye).The outcome of the study suggests that US consumers vary in their preferences for beef products by season. The presence of a USDA certification logo is by far the most important factor affecting consumer's willingness to pay for all beef cuts, which is also heavily dependent on season. In relation to packaging, US consumers have mixed preference for different beef products by season. The results from a scaled adjusted ordered logit model showed that after price, safety-related attributes such as certification logos, types of packaging, and antibiotic free and organic products are a stronger influence on American consumers' choice.2 | P a g e Furthermore, US consumers on average purchase diced and roast products more often in winter "slow cooking season", than in summer; whereas New York strip and flank steak are more popular in the summer "grilling season".This study provides valuable insights for businesses as well as policymakers to make inform decisions while considering how consumers relatively value among different labelling and product attributes by season and better address any ethical, safety and aesthetic concerns that consumers might have. | 10.1016/j.meatsci.2019.06.004 | [
"https://arxiv.org/pdf/1902.02419v1.pdf"
] | 85,529,864 | 1902.02419 | c3c65dd289f5143848dde5a3f36a2b8635c1521e |
Seasonality Effects on Consumers' Preferences Over Quality Attributes of Different Beef Products
Ali Ardeshiri [email protected]
School of Civil and Environmental Engineering
-Research Centre for Integrated Transport Innovation (rCITI)
University of New South Wales (UNSW) Sydney
2052NSWAustralia
Spring Sampson
Harvest Insights
South Melbourne, VIC 3205Australia
Joffre Swait
JDS Behavior Insights
29687TaylorsSCUSA
Seasonality Effects on Consumers' Preferences Over Quality Attributes of Different Beef Products
1 | P a g e * -Corresponding author,Discrete Choice ExperimentsSeasonality EffectLabelling InformationProduct AppearanceInformation CuesBeef Preference
Using discrete choice modelling, the study investigates 946 American consumers' willingnessto-pay and preferences for diverse beef products. A novel experiment was used to elicit the number of beef products that each consumer would purchase. The range of products explored in this study included ground, diced, roast, and six cuts of steaks (sirloin, tenderloin, flank, flap, New York and cowboy/rib-eye).The outcome of the study suggests that US consumers vary in their preferences for beef products by season. The presence of a USDA certification logo is by far the most important factor affecting consumer's willingness to pay for all beef cuts, which is also heavily dependent on season. In relation to packaging, US consumers have mixed preference for different beef products by season. The results from a scaled adjusted ordered logit model showed that after price, safety-related attributes such as certification logos, types of packaging, and antibiotic free and organic products are a stronger influence on American consumers' choice.2 | P a g e Furthermore, US consumers on average purchase diced and roast products more often in winter "slow cooking season", than in summer; whereas New York strip and flank steak are more popular in the summer "grilling season".This study provides valuable insights for businesses as well as policymakers to make inform decisions while considering how consumers relatively value among different labelling and product attributes by season and better address any ethical, safety and aesthetic concerns that consumers might have.
INTRODUCTION
In 2016, the United States (US) was Australia's second largest export destination. With 24% of Australia's beef exports and a value of AUS$1.7 billion, the US beef market plays an important role in Australia's export economy (Meat & Livestock Australia, 2017). A better understanding of American demand for beef is important as Australia faces strong competition from Canada and New Zealand in the US beef market. In 2016, it was reported that Australia was ranked number one in exporting beef and veal to the US. However, in the first-half 2017, Australia has exported relatively less beef than its competitors and the exported carcass weight has declined by 34% compared to the first half of 2016. A better understanding of US consumers' preferences is crucial if Australia is to maintain their position in the US beef import market. This study explores US consumers decisions over series of repeated hypothetical scenarios and evaluates how their value judgment for beef products are formed and may differ by season.
Beef demand, in its simplest form, is influenced by price and the ongoing evolving consumer preferences for taste, health benefits, convenience, etc. Preference for different cuts of beef are not only based on intrinsic and extrinsic cues, but also the context in which it is eaten.
One context that has been recognized as important by the econometricians is modelling seasonality effects on customer demand for goods and services (Lusk, Marsh, Schroeder, & Fox, 2001;Moskowitz & Beckley, 2009). Studies have looked at the importance of seasonality as a factor in beef purchasing habits, such as the seasonality effect on beef price (Capps, Farris, Byrne, Namken, & Lambert, 1994), quality grade cues (Farris & Holloway, 1990;Hogan Jr & Ward, 2003;Lusk et al., 2001), and hedging wholesale beef cuts (Namken, Farris, & Capps Jr, 1994;Schroeder & Yang, 2001). However, there has been no systematic analysis of the effect seasonality has on purchasing behaviour and preference for different beef cuts. Beef industries can benefit from understanding the seasonality effects on consumers decision and reduce the high failure rate of their new developed products (Dijksterhuis, 2016).
The study makes a significant theoretical and methodological contribution to the literature as the first paper to systematically test seasonality effects on preferences for beef cuts using a novel Discrete Choice Experiment (DCE). The study includes attributes related to the appearance such as meat and fat colour, fat content, packaging type as well as labelling information, including price, brand, origin, traceability, weight, type of feed, certification logos, expiry date, and claims regarding organic, angus, pasture raised, non-GMO, and natural beef. By systematically manipulating these variables through the DCE, the study recognizes the main factors that US consumer consider when purchasing beef products. The DCE is also innovative in replacing the typical "pick one" choice with quantities, i.e., how many units (including zero) of each product would you buy?
The significant empirical findings of this study also contribute to the management and practice of developing new products as well as improving the existing products. Furthermore, the findings will inform different functional departments within the food industry to effectively meet consumer needs (Ardeshiri & Rose, 2018;Jacobsen et al., 2014) The paper, therefore, concludes with a discussion of the implications of the findings for industry.
BEEF LABELLING AND CONSUMER PREFERENCES
In the US, labelling of meat products is closely regulated by the United States Department of Agriculture (USDA) and Food Safety and Inspection Service (FSIS). The FSIS has firm guidelines concerning the appearance and content of meat product labels. These strict labelling requirements, including country of origin labelling (COOL), attempts to provide the knowledge needed for consumers to make informed decisions (Ikenson, 2004;Jin, Skripnitchenko, & Koo, 2004;Umberger, 2004). The USDA has eight specific requirements for each product label which are: (1) product name, (2) inspection legend and establishment number, (3) handling statement, (4) net weight statement, (5) ingredients statement, (6) address line, (7) nutrition facts, and (8) safe handling instructions (US Department of Agriculture & Service, 2005). The restricted space available on the front of the package urge the producers to know exactly what additional information to the mandatory ones should be provided to maximise consumers preference for that product.
Consumers' value judgements related to product quality is based on intrinsic and extrinsic beef attributes (Asioli et al., 2017). Examples of intrinsic beef cues are, but not limited to, beef colour, cut of the meat, visible fat, and nutritional attributes (Acebrón & Dopico, 2000) and examples of extrinsic attributes are price, health related claims, brand, packaging layout (Jaeger, 2006;Lähteenmäki, 2013).
Extensive research has been conducted on the elicitation of US consumers' preferences for different beef products information cues. Intrinsic, quality traits such as the freshness, colour, and fat content of beef have been shown to influence US consumers' purchasing decisions.
Numerous studies assessing aesthetic appeal and taste evaluations have shown that US, Italian and Spanish consumers prefer and believe red beef tastes superior to purple and brown cuts (Carpenter, Cornforth, & Whittier, 2001;Grebitus, Jensen, & Roosen, 2013;Realini et al., 2014;Zanoli et al., 2013). However, preferences differ between US consumers and consumers from other countries for other intrinsic attributes. Specifically, US consumers are more accepting of beef that has been fed genetically modified corn or has growth hormones added, compared to French, German and UK consumers (Lusk, Roosen, and Fox (2003). While US consumers may be willing to purchase corn-fed beef, evidence suggests they prefer grass-fed (Abidoye, Bulut, Lawrence, Mennecke, & Townsend, 2011;Lim & Hu, 2013).
Other studies have focused on the impact of extrinsic product information on consumer preferences, such as food safety, traceability and COOL. Studies have shown all three types of information have a significant impact on consumer preferences (Loureiro & Umberger, 2007).
US consumers are willing to pay a premium for food safety information to be included on the package, such as the food borne disease risk (Lim & Hu, 2013), and whether the beef has been tested for bovine spongiform encephalopathy (BSE) (Lim & Hu, 2013;Lim, Hu, Maynard, & Goddard, 2012). Furthermore, the importance of food safety information is magnified for consumers with strong perceptions and attitudes towards risk (Lim et al., 2012). Traceability information and COOL also contributes to safety and quality perceptions, with various studies
showing the presence of traceability information increases consumer preference for a beef product (Abidoye et al., 2011;Lim et al., 2012;Loureiro & Umberger, 2007).
US consumers are also willing to pay a premium for national/domestic produced beef products.
Multiple studies have shown that US beef is preferred to beef imported from Canada or Mexico (Lim & Hu, 2013;Tonsor, Schroeder, & Lusk, 2013). This premium for US-labelled steak is as high as 20 percent in one study (Umberger, Feuz, Calkins, & Sitz (2003) and is likely due to poor food safety perceptions of imported beef Loureiro and Umberger (2003).
DISCRETE CHOICE EXPERIMENT
To understand consumer behaviour, DCE were used to analyse consumer choices. In these studies consumers behaviour is elucidated using either the Lancaster's (1966) consumer utility maximization model or McFadden (1974) random utility theory framework. Using the random utility theory, DCEs measure the relative importance of studied attributes by asking participants to repeatedly choose among given alternatives with different combination of attributes (J.
Louviere & Hensher, 1982;J. Louviere, Hensher, & Swait, 2000). DCEs open up the possibility to explore the complex relationship between individual preferences and their values for different sets of attributes when making a decision and can be expanded across the populations of consumers (McEachern, Seaman, Padel, & Foster, 2005).
EXPERIMENT DESIGN AND MATERIALS
The study uses the novel choice modelling framework presented in Ardeshiri & Rose (2018) to uncover the effect of seasonality on consumer preferences for beef products. Participants were asked to select how many (including none) of each beef product would they most likely purchase. Nine different beef cuts were studied according the Institutional Meat Purchase Specifications (USDA, 2014) and North American Meat Processors Association (2007) descriptions. The attributes of ground, diced, roast, sirloin, tenderloin, flank, flap, New York strip and cowboy (rib eye) 1 were investigated in a DCE (refer to Figure 1 for a brief description of each cut). Feeding from the list of attributes presented in Ardeshiri & Rose (2018) study and conducting focus groups and interviews with experts in the US market and also looking at the literature review of similar studies conducted in the US, for each beef cut, this study used five discrete attributes (fat & meat Colour , marbling, type of packaging and brand) , three continues attributes (best before, weight and price attributes) and ten binary attributes (claim attributes) as presented in Table 1. Although the list of attributes selected is extensive and in addition to the restriction of having limited space available to present the information, one can argue that individuals may not asses all the information in their decision process, however, in the focus group, unanimously, it was agreed all the mentioned attributes is of importance in the decision process and should be investigated. A comparison among all attribute will also enable the producers to know exactly what additional information to the mandatory ones should be provided to maximise consumers preference for that product.
A key aspect in DCEs is the design of the set of choice alternatives that are presented to the respondents. DCEs consist of a sample of choice sets selected from the universal set of all possible choice sets that satisfy certain statistical properties. There is no consensus in the literature about the 'appropriate' number of choice sets per respondent. Louviere et al. (2000) noted that most studies ask respondents to evaluate between one and sixteen choice sets, with the average being somewhere around eight choice scenarios per respondent. In a later paper, (J. J. Louviere, 2004) argued that 'in contrast to the equivalent of widely held 'academic urban myth' in marketing and transport research, there is considerable evidence that humans will 'do' dozens (even hundreds) of T's [choice sets]' (p. 16). A number of studies have investigated the impact of the number of choice sets given to each respondent (Bech, Kjaer, & Lauridsen, 2011;Caussade, de Dios Ortúzar, Rizzi, & Hensher, 2005;David A Hensher, 2006;David A Hensher, Stopher, & Louviere, 2001;Hess, Hensher, Daly, & practice, 2012;Stopher & Hensher, 2000). Hess et al. (2012) argued that the concerns about fatigue in the literature are possibly overstated, with no clear decreasing trend in scale across choice tasks in any of their studies. They further stated that while the work by Bradley & Daly (1994) has become a standard reference in this context of reducing respondent engagement as a result of fatigue or boredom in large number of choice tasks, it should be recognised that not only was the fatigue part of the work based on a single dataset, but the state-of-the-art and the state-of-practice in stated choice survey design and implementation has moved on significantly since their study. Bech et al. (2011) studied the effect of 5,9 and 17 choice sets presented to each respondent and they found no differences in response rates and no systematic differences in the respondents' self-reported perception of the uncertainty of their DCE answers. Although there were some differences in willingness to pay (WTP) estimates suggesting that estimated preferences are to some extent context-dependent, but no differences in standard deviations for WTP estimates or goodness-of-fit statistics. Hensher et al. (2001)and Stopher and Hensher (2000) found that the number of choices had little impact on response rate, no impact on respondent fatigue and simplification of response strategies, minimal impact on the goodness of-fit statistics, and finally, little impact on the mean WTP estimates. Hensher (2006) concluded that the number of choice sets presented to the respondents did not influence the aggregate mean WTP estimates. Caussade et al. (2005) investigated five design parameters (number of alternatives in a choice set, number of attributes defined in each alternative, number of levels, range of attribute levels and number of choice sets) and concluded that all five parameters affected variance but did not systematically affect mean WTP estimates. Furthermore, as cited in Adamowicz, et al. (1998), a study conducted by Bunch and Batsell in 1989 demonstrated that with as few as six respondents per choice scenario, the asymptotic properties for maximum likelihood-based inference are satisfied (p. 15). For this study we used the Ngene software and applied an orthogonal main effects experimental design (J. Louviere et al., 2000), to select 200 choice task from the universal set for the experimental design. The 200 choice tasks were divided into 50 blocks allowing each participant to complete 4 repeated choice tasks. Four random alternatives were presented in each task that vary in the attribute levels. As shown in Table 1, the various attributes presented in all alternatives included; beef and fat colour, marbling (ground and diced beef were excluded), types of packaging, origin/brand 2 , claims, weight and price. As illustrated in Figure 3, the traditional underlying mechanism of "pick a product" in traditional choice experiment surveys was substituted with "how many" (including zero) products would you purchase, to represent the real-life ordered logit structure when purchasing a food product (Ardeshiri & Rose, 2018). Each respondent was assigned to a winter or summer scenario and was asked to complete four sets. As shown in Figure 2, based on the season that was assigned to the individual, a current weather widget appeared next to the task to remind the individual about the hypothetical season they are shopping in.
DATA COLLECTION
Data for our analysis derived from an online survey distributed through a panel company in April 2017 and was completed by 946 American residents of north-eastern US. The studied regions included Connecticut, Maine, Massachusetts, New Hampshire, New Jersey, New York, Rhode Island and Vermont. From the 946 respondents, 468 were allocated a summer scenario and 478 were assigned to the winter scenario. Any US residents aged 18 years and above and responsible for their household grocery shopping, including the household's meat purchases were eligible to participate in the survey. The majority of respondents (39%) purchase beef once a week and only 16% purchase beef once a month or less. A summary of the sample demographics is provided in Table 2.
DATA ANALYSIS
It can reasonably hypothesize that each individual has a continues preference with different strength that underlines "how many" beef products they will purchase. This strength of preference is labelled as "utility" (V) and the utility value can have the following range of values:
−∞ < ≤ ∞
where i is referred to the individual and m presents the beef product. In this experiment the individuals had the option to select a value from 0 to 10 as an indication of "how many" of the product they would like to purchase. The underlying utility will then be translated to a rating and viewed as a censoring of the underlying utility,
= 0 − ∞ < ≤ 1 = 1 1 < ≤ 2 = < ≤ ∞
The thresholds, , are specific to the beef cut and number (J-1) where J is the number of possible ratings (here, eleven) J-1 values are needed to divide the range of utility into J cells.
(For a more detailed explanation of the threshold elements we encourage the readers to read Ardeshiri & Rose, 2018;Greene & Hensher, 2010).
The utility function includes individual preferences for beef attributes which we denote 1 , 2 , … , . It also embraces individual sociodemographic variables 1 , 2 , … , , to allow for covariate specific heterogeneity. And finally, an aggregate of unmeasured and unmeasurable (by the statistician) idiosyncrasies, denoted . For conventional reasons, we
(1) assume these features have a linear format in the utility function. The described utility function is presented as below:
= + + +
Initially two separate ordered logits were estimated for summer and winter scenarios. In order to determine whether the preferences stated in both methods were proportionally similar, the coefficients derived from the two models were plotted (D. Hensher, Louviere, & Swait, 1998;Huynh, Coast, Rose, Kinghorn, & Flynn, 2017;Swait & Bernardino, 2000). This offers a preliminary suggestion as to whether the data is likely to be possible to be pooled together. In other words, the respondents are using similar cognitive processes, although the decision context is changing. A Chow test for data pooling was conducted using the Swait and Louviere (1993) approach. This test compares the sum of the log likelihood from the ordered logit models of each season, to a log likelihood that allows for scale differences across the data sources (Swait, 2006). This will allow testing the hypothesis that there is homogeneity in preferences across the attributes included in the utility per seasons, whilst allowing for error variance differences between the unobserved factors in the two datasets. Passing this test at the 1% significance level indicates that it would be suitable to combine the datasets as long as the different scale factors are accounted for. The combined data were analysed using a seasonality scale 3 adjusted ordered logit model in Python Biogeme 2.4. (Bierlaire, 2016). The final utility function to be estimated for each decision maker i is:
= exp( ) [ + + + ]
Where represents the scale value for cut b and season j. Alternative specific beef attributes as well as season specific parameters were measured. However, some attributes entered the final model as generic across all or a range of beef products. The DCE attributes with qualitative levels are effects coded to ensure the systematic utility effects are uncorrelated with the intercept (Bech & Gyrd-Hansen, 2005; J. Louviere et al., 2000). Any coefficients that were not significant at the 90% level of accuracy have been removed using Louviere et al. (2000) log likelihood ratio test.
WILLINGNESS TO PAY ESTIMATION
Increasingly, discrete-choice models are being used to derive estimates of money an individual is willing to pay (or willing to accept) to obtain some benefit (or avoid some cost) from a specific action (J. Louviere et al., 2000). In the presented model where each attribute in a utility expression is associated with a single taste weight, the WTP for a unit change in a certain attribute can be computed as the marginal rate of substitution between cost and the quantity expressed by the attribute, holding all other potential influences constant.
= − WTP estimates for categorical attributes refer to the benefit obtained by having that level and for continues attributes, it represents the benefit of a change in a unit of the attribute.
(4)
RESULTS
Estimation results from the scaled adjusted ordered logit model are presented in several tables.
Tables 3 and 4 present seasonal parameter estimates for specific beef cuts. With regards to the labelling information, the largest difference in preferences between seasons is for certified logos representing the authority which has approved and certified the beef.
Consumers reacted strongly to packages with no certification logo in summer, compared to in winter. The highest negative WTP for no program or logo in winter was calculated as -$28.42
for New York strip, whereas in summer this value was calculated at -$87.05.
Consumer preferences for product claim information is homogenous for both seasons.
Heterogeneity in preferences appears only in claims related to specific cuts of beef. Grebitus, Jensen, & Roosen (2013), this study produced a lower WTP for longer use-by dates in summer than in winter.
US consumers are also willing to pay more for larger cuts of roast and flap in winter than in summer. Flank has a negative estimated coefficient for weight in both seasons, demonstrating that consumers prefer flank steaks in smaller portions. As anticipated, the price coefficient was negative for all beef products. US consumer are more sensitive to a price increase for diced products. Tables 7 and 8 provide all the WTP calculated values. couple family with no kids' household and being in the lower age spectrum will increase the probability of purchasing beef products. Moreover, a quadratic form of income became significant with a negative value representing that middle-income US consumer will more likely purchase beef products rather than individuals belonging to both ends of the income range. The majority of the scale parameters are not statistically different from one at the 95% confidence level: only the scale estimates for ground, diced and roast in summer. This is to be expected as the scale differences have been captured through introducing season and beef specific parameters in the utility function. Table 6 provides the estimation of the threshold properties. The thresholds are an important element of the model; they divide the range of utility into cells that are then identified with the observed ratings. One of the admittedly unrealistic assumptions in many applications is that these threshold values are the same for all individuals. Importantly, difference on a rating scale (e.g., 0 compared to 1, 1 compared to 2) are not equal on a utility scale; hence we have a strictly nonlinear transformation captured by the thresholds, which are estimable parameters in an ordered choice model (Ardeshiri & Rose, 2018;Greene & Hensher, 2010). The threshold parameters are all incrementally increasing, in order to preserve the positive signs of all of the probabilities. For example, in Table 6 the ground cut threshold (1) is equal to 0, threshold (2) is 1.57, threshold (3) is 2.74, etc.
DISCUSSION
Beef industries need to be ensured that their products match consumer demand and they retain their market share. The study outlined a choice experiment and attempt to differentiate between internally variety-seeking behaviour and seasonality as an external motivator.
The outcome of the study suggest that consumers vary in their preferences for beef products by season. The presence of a USDA certification logo is by far the most important factor affecting consumer's willingness to pay for all beef cuts, except flank. The value of USDA certification to consumers is heavily dependent on season, with customers willing to pay up to an additional $58 for USDA certification in summer for New York strip beef than in winter.
Similarly, consumers are highly averse to purchasing beef with no certification (USDA or otherwise). This effect is stronger in summer, indicating that consumers are concerned about food safety and may be aware of the opportunities for foodborne bacteria to thrive in warm weather ( Figure 4).
In relation to packaging, US consumers have mixed preference for different beef products by season. For ground and diced beef, consumers prefer to purchase the beef fresh over the counter in both seasons. Cowboy cut is the only product that consumers prefer to have in trey packed in both seasons. Although US consumers prefer New York strip beef in vacuum packed the most but they still value purchasing the product fresh over the counter in winter whereas in summer they only prefer in vacuumed packed.
The results also show that for some products, consumers differ in their preferences by season for product net weight. For example, the bigger the flap and roast products in winter the better.
US consumers are willing to pay an extra $3.9 and $1.75 for flap and roast products in winter than in summer respectively. However, the opposite applies for Sirloin steak where US consumers are willing to pay an extra $2.9 for a higher product weight in summer than in winter. A higher product weight for flank is perceived negatively for both seasons and it intensifies by -$1.35 in summer relative to winter. Finally, having a longer used by date for Sirloin, NY strip and Flap is more preferable in winter than in summer. Moreover, for Flap cuts with having a longer used-by-date in summer is seen as a disadvantage.
Regardless of seasonality effect, the results of this study also support the findings of previous work in a number of ways. Claims are important to US consumers. Antibiotic-free is the most important claim for the roast ($10.59), sirloin ($47.50) and flank ($10.79), but it has no effect on willingness to pay for tenderloin beef ($0). Although no added hormones had a relatively small effect on willingness to pay, it had an effect on preferences across all cuts. Table 9 summarises the impact of each claim on the willingness to pay and for which cuts.
With regards to product appearance, white fat colour was preferred more than light yellow fat colour and consumers were WTP a premium of $1.61 for diced beef and up to $5.26 for sirloin.
Similar to Zanoli et al. (2013) and Carpenter et al. (2001) study, red coloured beef was the most preferred beef colour. Sirloin steak with $3.66 has the highest premium for having a red coloured meat while diced beef with $1.12 has the lowest premium. Intramuscular fat content also known as marbling has been considered as one of the main determinants in the beef quality grading system. Parallel to the findings of Lusk et al. (2003), US consumers preferred beef products (not applicable to ground and diced beef products) with the least amount or no marbling. Sirloin steak with $16.46, received the highest premium for having no intramuscular whereas the lowest was for Cowboy cut with -$17.92.
Other non-DCE studies have indicated that consumers' preferences for purchasing beef products is also shaped by other variables such as the location; frequencies of eating beef (by specific cut) in restaurants, cafés, bars, etc.; cultural beliefs; level of knowledge; religious beliefs; environmental sustainability preferences and so on. Thus, a limitation of this study that should be addressed in future research is to explore and identify consumers segments to improve marketing strategies for beef producers based on other heterogeneity features available in segments of the population.
MANAGERIAL AND POLICY IMPLICATIONS
The current study benefits from inclusive range of policy implication influencing consumers, businesses and policy makers. Authorities and policy makers require information about consumer preferences to make informed decisions and avoid market failures. One of the reasons market failures occur is that the intrinsic and extrinsic attributes of a product do not meet consumers' needs, or, due to labelling, are not perceived by consumers to meet their needs. Consumers are faced with a large number of food choices on the daily bases and it is unlikely to allocate significant cognitive effort and time to each of their decisions. Therefore, it is important that products and labels reflect the information consumers value when purchasing beef. The findings of this study will help managers design labels with useful and meaningful information for consumers, as well as help consumers to navigate product choices more efficiently. Industry firms can profit from similar research with their product differentiation strategy, cost-benefit assessment and product design by season. The study also highlights the importance of seasonality effects on consumers purchasing behaviour.
HYPOTHETICAL SCENARIO
This section provides an example of how the output of this research can benefit business firms with the decision making and market assessment on a newly developed product by season.
For this reason, Figures 5 and 6 presents purchase probabilities for all the nine cuts in both seasons. The product details (i.e. fat colour, beef colour, claims, brands and the used-by date) are constrained to be the same among all nine cuts. The average value for price per pound and package size differed based on the beef cut, however, they were assumed to be equal in both seasons. Figures 6 and 7 presents the probability of purchasing different quantities (including none) for winter and summer. Roast and diced have the biggest change in units purchased between seasons. In summer, roast and diced respectively have a 15% and 16% probability of having zero units purchased, compared to 10% and 11% in winter. This result suggests that winter is the slow cooked season. However, the simulation also shows that winter competes strongly with summer as the grilling season, as there are few differences between the summer and winter steak purchase probabilities: only New York strip has a higher probability of not being purchased in winter compared to summer (78% purchase probability of at least one unit in summer, compared to 74% in winter).
The average purchase quantity for each product supports the underlying comparison made earlier by only looking at the zero purchase probabilities. On average, diced beef has the highest purchase quantity in winter with more than 2 quantities and, as expected, tenderloin has the lowest purchase quantity in summer as it is the most expensive cut. Although flank steak had the same probability of zero purchases (10%), on average it is being purchased more in summer. This results in the New York strip and flank steaks to be the only grilling products to have a higher average purchase quantity in summer than in winter.
The above scenarios present a simple example of the benefit of this study to the beef industry.
By creating hypothetical products, managers can observe and predict the attractiveness of a new product. Furthermore, by using the product weights, the market share of these new products can be estimated based on total pounds sold for each beef cut. The purchase probability trends can be drawn for each cut to observe the different probabilities for an incremental change in price values. These graphs can help to find the optimal price value that maximises revenue. This is a simple, yet powerful, predictive tool for beef producers, processors and policymakers to strengthen their decision-making capabilities.
CONCLUSION
This study provides information regarding the importance and effect of different information cues to US consumers when selecting beef products in winter and summer.
This study represents how choice experiments can offer valuable insights for businesses as well as policymakers to make inform decisions while considering how consumers relatively value among different labelling and product attributes by season and better address any ethical, safety and aesthetic concerns that consumers might have. Finally, the findings of this study can assist policymakers and stockholders with estimating the economic benefits of a given policy measure by season.
A limitation to this study and as a further research stream is to investigate consumers processing resources and cognitive efforts for each decision. Given the sheer number of decisions involved across the many facets of people's lives, it seems unlikely that individuals allocate substantial cognitive effort and time to each decision. Indeed, decisions regarding small budget items like food or consumer packaged goods would seem more likely to be relegated to some form of habitual choice behaviour (W. L. Adamowicz & Swait, 2012). Considering a large number of attributes included boredom and fatigue effects may be observed if respondents are presented large numbers of complex choice tasks. While literature supports the notion that increasing complexity over a number of attributes changes choice behaviour toward strategies that employ less attribute, this may not always be the case, of course (Swait & Adamowicz, 2001).
Finally, an alternative suggestion for future directions of research is to investigate and compare current market data and the observed market behaviour with the finding of this study to increase the robustness of the results. (3) Marbling*
1-Not marbled (0) 2-Somewhat marbled (4)
Type of Packaging
1-Tray Packed (TP)
Tray packed meat is when the meat is packed into an open container or tray and covered with a film. This is mainly used in smaller primal cuts or portioned meat.
2-Vacuum Packed (VP)
Vacuum Packed involves the removal of air and oxygen from the packaging. This creates a vacuum and assists in the preservation of meat and improvement in meat quality due to the lack of oxygen around the meat that promotes bacterial growth. $30, $42, $54, $66 *Note that we used the handbook of Australian meat standards to present the levels for these attributes. The number between the parentheses refers to the reference standard score. Please see the following link for more information. https://www.ausmeat.com.au/custom-content/cdrom/Handbook-7th-edition/English/DA71F4DE-F68A-11DA-AA4B-000A95D14B6E.html Ground Beef (IMPS 136), unless otherwise specified, ground beef may be derived from boneless meat which has been frozen and stockpiled.
3-Fresh from the butcher.
Diced beef (IMPS/135) shall be prepared from any portion of the carcass which yields product that meets the end item requirements.
Roast beef (IMPS/109) is that portion of the forequarter remaining after removal of the cross-cut chuck and short plate.
Sirloin steak (IMPS/181) is the posterior section of the full loin.
NY strip steak (IMPS/158B) consists of the round (top, bottom, heel, rump, and shank) excluding the full sirloin tip (knuckle).
Cowboy Steak (IMPS/1103B) may be prepared from any IMPS bone-in rib item. Each steak must be cut between the rib's bones.
Beef Tenderloin (IMPS/191) consist of the sirloin butt portion of the tenderloin.
Flank (IMPS/193) is a single flat muscle cut from the flank region of the carcass.
Flap steak (IMPS/185A) comes from a bottom sirloin butt cut of beef and is generally a very thin steak.
Source: Institutional Meat Purchase Specifications, 2014 https://www.ams.usda.gov/sites/default/files/media/IMPS_100_Fresh_Beef%5B1%5D.pdf
Figure 1 :
1Beef cuts descriptions
Figure 2 :
2The use of weather widget to create hypothetical season.
Figure 3 : 45 Figure 4 :
3454Sample choice experiment task.Representing winterRepresenting summerNo certification USDA certification -$6.30 --$28.40 -$25.00 --$87.05 +$6.30 -+$23.30 +$20.00 -+$75.Willingness to pay ranges for certifications in winter and summer.
Figure 5 :Figure 6 :
56Probability of purchase quantity for each beef products in winter *Pi represents the product probability of purchase and the number in the brackets presents the quantity of the product. * Probability of purchase quantity for each beef products in summer *Pi represents the product probability of purchase and the number in the brackets presents the quantity of the product.
*
Table 5
5presents the covariate estimates andTable 6presents threshold estimates.Consumers showed similar taste preferences for beef colour, fat colour and marbling in both seasons. A generic parameter was used to measure beef and fat colour preferences across all beef cuts and both seasons. US consumers prefer white fat in comparison to light yellow, and red beef over pink. Consumers are willing to pay $3.14 and $2.18 respectively for beef cuts with white fat and red beef. An alternative specific parameter was used to represent marbling preferences across all beef cuts, however, the beef specific parameters are the same across both seasons (marbling does not apply to ground or diced beef). No differences were observed between no marbling and somewhat marbled levels for tenderloin, flank and New York steak.No marbling was preferred for roast, sirloin and flap, with a respective WTP of $10.18, $16.46 and $3.89. Somewhat marbled is more preferred for cowboy cuts and US consumers are willing to pay $17.92 for this.An alternative specific parameter representing both beef cuts and season were used to present heterogeneity in preferences for packaging type. For all beef cuts, except cowboy, tray packed was less preferred relative to vacuumed packed and sold fresh. While this finding holds across both seasons, the magnitudes differ. For example, US consumers have a stronger negative preference for tray packed diced, tenderloin and flank beef in summer relative to winter. In general, consumers value fresh sold ground and diced beef; vacuum packed roast, sirloin, flank and New York steak; and tray packed cowboy steaks in both seasons. Vacuum packed tenderloin is preferred in winter and fresh in summer. Finally, no difference in packaging types were found for flap cuts in winter, although consumers prefer fresh in summer. For WTP estimates please refer to Tables 7 and 8. These results are in line and contribute (by observing preference for different beef cuts) to results fromCarpenter et al. (2001) where they observed U.S. consumers preferred vacuum skin packaging.
Grass fed is preferred over grain fed and US consumers have the highest WTP for grass fed cowboy cuts at $11.60. Furthermore, US consumer are willing to pay a premium for beef products that are traceable back to farm, with the highest WTP for New York strip at $15.89. The impact ofantibiotic free claims on WTP have the highest variance, ranging from $4.53 for ground beef
up to $47.52 for sirloin steak. On the contrary, WTP for hormone free claims have the lowest
variance, ranging from $0.69 for diced beef to $2.27 for sirloin steak. US consumers also prefer
beef that is Angus, organic, non-GMO, pasture and natural (please view Tables 7 and 8 for all
the WTP calculated values). Different to
Table 5
5presents the covariate coefficient estimates. The results from this table demonstrate that covariates such as having a graduate degree, owning a dwelling, larger household size, being a male, living in New York state, purchasing beef 2 plus times a week, belonging to a
Table 1 .
1Attributes and levels in the choice experiment.Attribute
Levels
Fat
Colour*
1-White (0)
2-Light yellow (4)
Meat
Colour*
1-Pink (1A)
2-Red
Table 2 .
2Demographic variables and summary statistics of choice experiment participants.Variable
Definition
Statistics
Gender
Male
50.4%
Female
49.6%
Total participants
946
Age
Modal age group
25-34 years (28.1%)
State of origin
New York
72%
Other
28%
Household type
Couple family with no children
16%
Couple family with children
33.2%
One parent family
6.7%
Single person household
15.6%
Household size
Average household size
3.2
Household income
Modal income bracket
$75,000 to $100,000
(33.2%)
Employment
Full-time
58.8%
Part-time
15.1%
Retired
13.9%
Un-employed
12.2%
Education
Graduate degree
30%
Bachelor's degree
32%
Associate's degree
8%
College graduate or less
30%
Dwelling status
Owned
71.6%
Renting
28.4%
Beef consumption frequency
2+ per week
24%
Once a week
39%
2 or 3 a month
21%
Once a month
16%
Table 3
3Ordered logit parameter estimate for winter season.Attribute
Level
Ground Diced Roast Sirloin Tenderloin Flank
Flap
New
York
Cowboy
Scale
( )
1.18
1.32
1.413
1
1
1
1
1
1
ASC
ASC
0.278
0.237 -0.193 -0.178
0.341
0.957 -0.772 -0.806
0.136
Fat colour White
0.022
0.022
0.022
0.022
0.022
0.022
0.022
0.022
0.022
Light Yellow*
-0.022
-0.022 -0.022 -0.022
-0.022
-0.022 -0.022 -0.022
-0.022
Meat
colour
Table 4
4Ordered logit parameter estimates for summer seasonAttribute
Level
Ground Diced
Roast Sirloin Tenderloin Flank
Flap
New
York
Cowboy
Scale
( )
1
1
1
1
1
1
1
1
1*
ASC
ASC
0.175
0.132
-0.284
-0.39
0.0713
0.978
-0.0582 -0.6340
0.0099
Fat colour
White
0.022
0.022
0.022
0.022
0.022
0.022
0.022
0.022
0.022
Light Yellow*
-0.022
-0.022 -0.022 -0.022
-0.022
-0.022
-0.022
-0.022
-0.022
Meat colour
Pink*
-0.015
-0.015 -0.015 -0.015
-0.015
-0.015
-0.015
-0.015
-0.015
Red
0.015
0.015
0.015
0.015
0.015
0.015
0.015
0.015
0.015
Marbling
Not marbled
-
-
0.065
0.068
-
-
0.041
-
-0.140
Somewhat
marbled*
-
-
-0.065 -0.068
-
-
-0.041
-
0.140
Packaging
type
Vacuum
Packed
-
0.033
0.033
0.114
0.077
0.242
-
0.043
-0.037
Tray packed
-0.024
-0.111 -0.041 -0.096
-0.201
-0.457
-0.056
-
0.090
Fresh*
0.024
0.078
0.008
-0.018
0.124
0.215
0.056
-0.043
-0.054
Feed type
Grass
-
0.043
0.038
-
0.034
0.033
0.048
-
0.091
Grain*
-
-0.043 -0.038
-
-0.034
-0.033
-0.048
-
-0.091
Traceable back
to farm
Yes
0.041
0.041
0.055
0.032
-
0.039
0.065
0.070
0.055
No*
-0.041
-0.041 -0.055 -0.032
-
-0.039
-0.065
-0.070
-0.055
Antibiotic
free
Yes
0.031
0.068
0.068
0.196
-
0.075
0.052
-
0.045
No*
-0.031
-0.068 -0.068 -0.196
-
-0.075
-0.052
-
-0.045
Hormone
added
Yes*
-0.009
-0.009 -0.009 -0.009
-0.009
-0.009
-0.009
-0.009
-0.009
No
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
Organic
Yes*
-
0.055
0.064
0.057
-
0.067
0.026
0.075
0.065
No
-
-0.055 -0.064 -0.057
-
-0.067
-0.026
-0.075
-0.065
Angus
Yes
0.034
0.068
0.062
0.036
0.147
-
-
0.055
0.048
No*
-0.034
-0.068 -0.062 -0.036
-0.147
-
-
-0.055
-0.048
Non-GMO
Yes*
0.029
0.029
0.029
0.029
0.029
0.029
0.029
0.029
0.029
No
-0.029
-0.029 -0.029 -0.029
-0.029
-0.029
-0.029
-0.029
-0.029
Pasture
Yes
Table 5
5Covariate estimates in the ordered logit model.Demographics
Description
Values
Education
Graduate degree
0.023
Bachelor's degree
-0.047
Associate's degree*
0.024
Dwelling
Owned
0.033
Renting*
-0.033
Household Size
Continues
0.014
Income
Continues (Quadratic form)
-0.008
Origin
New York state
0.035
Other states*
-0.035
Frequency of
purchase
2+ per week
0.059
Once a month*
-0.059
Gender
Female
-0.021
Male*
0.021
Age
Continues
-0.034
Household type
Couple family with no children
0.099
One parent family
-0.040
Couple family with children*
-0.059
*Represents the base level
Table 6
6Estimation of threshold properties in the ordered logit model.Threshold's
(τ)
Ground
Diced
Roast
Sirloin Tenderloin Flank
Flap
New
York
Cowboy
Threshold 1
0
0
0
0
0
0
0
0
0
Threshold 2
1.57
1.74
1.61
1.9
3.04
1.83
1.87
1.8
1.63
Threshold 3
2.74
2.96
3.09
3.31
4.1
3.54
3.11
3.18
2.95
Threshold 4
3.61
3.88
4.01
4.21
4.94
4.84
4.13
3.94
3.81
Threshold 5
4.56
4.71
4.92
5.03
5.85
5.87
5
4.68
4.59
Threshold 6
5.28
5.49
5.99
5.77
6.24
6.79
5.48
5.39
5.42
Threshold 7
5.78
6.22
6.32
6.38
6.62
7.36
6
5.73
5.85
Threshold 8
6.35
6.98
6.8
6.99
7.07
7.91
6.54
6.49
6.58
Threshold 9
6.76
7.39
7.21
7.55
7.63
8.25
7.01
7.19
6.99
Threshold 10
7.46
8.09
7.9
8.25
8.33
9.17
7.77
8.11
7.68
Estimation Report
Final log likelihood
-18789.548
Number of parameters
257
Sample size
3784
Table 7
7Willingness to pay estimates for winter season.Attribute
Level
Ground Diced
Roast Sirloin Tenderloin Flank
Flap
New
York
Cowboy
Fat colour
White
$3.14
$1.61
$3.40
$5.26
$2.48
$3.14
$2.05
$4.93
$2.78
Light Yellow*
-$3.14
-$1.61
-$3.40
-$5.26
-$2.48
-$3.14
-$2.05
-$4.93
-$2.78
Meat colour
Pink*
-$2.18
-$1.12
-$2.37
-$3.66
-$1.73
-$2.18
-$1.43
-$3.43
-$1.93
Red
$2.18
$1.12
$2.37
$3.66
$1.73
$2.18
$1.43
$3.43
$1.93
Marbling
Not marbled
-
-
$10.18
$16.46
-
-
$3.89
-
-$17.92
Somewhat
marbled*
-
-
-$10.18 -$16.46
-
-
-$3.89
-
$17.92
Packaging type
Vacuum
Packed
-
$2.44
$5.11
$33.21
$8.78
$33.69
-
$13.82
-$4.68
Tray packed
-$3.47
-$5.32
-$6.36
-$30.30
-$16.01
-$60.43
-
-$16.41
$11.57
Fresh*
$3.47
$2.88
$1.25
-$2.91
$7.23
$26.75
-
$2.59
-$6.89
Feed type
Grass
-
$3.17
$5.89
-
$3.93
$4.71
$4.52
-
$11.60
Grain*
-
-$3.17
-$5.89
-
-$3.93
-$4.71
-$4.52
-
-$11.60
Traceable back to
farm
Yes
$5.91
$3.00
$8.68
$7.81
-
$5.62
$6.14
$15.89
$6.98
No*
-$5.91
-$3.00
-$8.68
-$7.81
-
-$5.62
-$6.14
-$15.89
-$6.98
Antibiotic free
Yes
$4.53
$5.07
$10.59
$47.52
-
$10.79
$4.87
-
$5.80
No*
-$4.53
-$5.07
-$10.59 -$47.52
-
-$10.79
-$4.87
-
-$5.80
Hormone
added
Yes*
-$1.35
-$0.69
-$1.47
-$2.27
-$1.07
-$1.35
-$0.88
-$2.13
-$1.20
No
$1.35
$0.69
$1.47
$2.27
$1.07
$1.35
$0.88
$2.13
$1.20
Organic
Yes*
-
$4.07
$10.09
$13.92
-
$9.74
$2.47
$17.02
$8.33
No
-
-$4.07
-$10.09 -$13.92
-
-$9.74
-$2.47
-$17.02
-$8.33
Angus
Yes
$4.87
$5.04
$9.65
$8.68
$16.81
-
-
$12.45
$6.16
No*
-$4.87
-$5.04
-$9.65
-$8.68
-$16.81
-
-
-$12.45
-$6.16
Non-GMO
Yes*
$4.19
$2.15
$4.54
$7.03
$3.32
$4.19
$2.74
$6.59
$3.71
No
-$4.19
-$2.15
-$4.54
-$7.03
-$3.32
-$4.19
-$2.74
-$6.59
-$3.71
Pasture
Yes
-
-
-
$16.90
$10.05
$8.34
$5.43
$19.66
$4.94
No*
-
-
-
-$16.90
-$10.05
-$8.34
-$5.43
-$19.66
-$4.94
Natural
Yes
-
-
-
-
$6.21
-
-
$10.70
$8.06
No*
-
-
-
-
-$6.21
-
-
-$10.70
-$8.06
Certified logo
USDA
Verified*
$19.23
$14.67
$13.47
$23.27
$14.53
$22.84
$6.32
$16.82
$18.43
Global Animal
Partnership
-$5.78
-
-
-
-
-
-
-
-
Self-Assurance
Program
-
-
-
-
-
-$9.83
-
$11.59
-$5.76
No
Program/Logo
-$13.59
-$14.67 -$13.47 -$23.27
-$14.53
-$13.01
-$6.32
-$28.41
-$12.67
Brands
Brand 1*
$7.62
$9.46
$1.97
-$15.10
-$12.94
-$5.65
-$0.67
$10.55
$6.98
Brand 2
-$7.62
-$11.48
-
-
-$7.10
-
-$7.22
-$10.55
$7.23
Brand 3
-
-$5.13
-$15.51
-
$7.46
-
-
-
-
Brand 4
-
$9.19
-
$15.10
$12.58
-
$7.89
-
-
Brand 5
-
-
-
-
-
-
-
-
-$14.21
Brand 6
-
$3.86
-
-
-
-
-
-
-
Brand 7
-
-$5.90
-
-
-
$5.65
-
-
-
Brand 8
-
-
$13.53
-
-
-
-
-
-
Use-By date
Continues
$1.81
$1.15
$2.85
$5.70
$1.17
$2.50
$6.82
$7.82
$2.05
Net weight
Continues
-
-
$10.15
$5.05
$2.02
-$11.84
$6.08
$2.27
-
Table 8
8Willingness to pay estimates for summer season.Attribute
Level
Ground Diced
Roast Sirloin Tenderloin Flank
Flap
New
York
Cowboy
Fat colour
White
$3.14
$1.61
$3.40
$5.26
$2.48
$3.14
$2.05
$4.93
$2.78
Light Yellow*
-$3.14
-$1.61
-$3.40
-$5.26
-$2.48
-$3.14
-$2.05
-$4.93
-$2.78
Meat colour
Pink*
-$2.18
-$1.12
-$2.37
-$3.66
-$1.73
-$2.18
-$1.43
-$3.43
-$1.93
Red
$2.18
$1.12
$2.37
$3.66
$1.73
$2.18
$1.43
$3.43
$1.93
Marbling
Not marbled
-
-
$10.18
$16.46
-
-
$3.89
-
-$17.92
Somewhat
marbled*
-
-
-$10.18 -$16.46
-
-
-$3.89
-
$17.92
Packaging type Vacuum
Packed
-
$2.44
$5.11
$27.64
$8.78
$34.99
-
$9.75
-$4.68
Tray packed
-$3.47
-$8.22
-$6.36
-$23.20
-$22.99
-$66.07
-$5.28
-
$11.57
Fresh*
$3.47
$5.79
$1.25
-$4.44
$14.21
$31.08
$5.28
-$9.75
-$6.89
Feed type
Grass
-
$3.17
$5.89
-
$3.93
$4.71
$4.52
-
$11.60
Grain*
-
-$3.17
-$5.89
-
-$3.93
-$4.71
-$4.52
-
-$11.60
Traceable back to
farm
Yes
$5.91
$3.00
$8.68
$7.81
-
$5.62
$6.14
$15.89
$6.98
No*
-$5.91
-$3.00
-$8.68
-$7.81
-
-$5.62
-$6.14
-$15.89
-$6.98
Antibiotic free
Yes
$4.53
$5.07
$10.59
$47.52
-
$10.79
$4.87
-
$5.80
No*
-$4.53
-$5.07
-$10.59 -$47.52
-
-$10.79
-$4.87
-
-$5.80
Hormone
added
Yes*
-$1.35
-$0.69
-$1.47
-$2.27
-$1.07
-$1.35
-$0.88
-$2.13
-$1.20
No
$1.35
$0.69
$1.47
$2.27
$1.07
$1.35
$0.88
$2.13
$1.20
Organic
Yes*
-
$4.07
$10.09
$13.92
-
$9.74
$2.47
$17.02
$8.33
No
-
-$4.07
-$10.09 -$13.92
-
-$9.74
-$2.47
-$17.02
-$8.33
Angus
Yes
$4.87
$5.04
$9.65
$8.68
$16.81
-
-
$12.45
$6.16
No*
-$4.87
-$5.04
-$9.65
-$8.68
-$16.81
-
-
-$12.45
-$6.16
Non-GMO
Yes*
$4.19
$2.15
$4.54
$7.03
$3.32
$4.19
$2.74
$6.59
$3.71
No
-$4.19
-$2.15
-$4.54
-$7.03
-$3.32
-$4.19
-$2.74
-$6.59
-$3.71
Pasture
Yes
-
-
-
$16.90
$10.05
$8.34
$5.43
$19.66
$4.94
No*
-
-
-
-$16.90
-$10.05
-$8.34
-$5.43
-$19.66
-$4.94
Natural
Yes
-
-
-
-
$6.21
-
-
$10.70
$8.06
No*
-
-
-
-
-$6.21
-
-
-$10.70
-$8.06
Certified logo
USDA
Verified*
$40.77
$20.00
$34.93
$50.91
$29.05
$37.30
$25.01
$75.45
$32.64
Global Animal
Partnership
-$5.78
-
-
-
-
-
-
-
-
Self-Assurance
Program
-
-
-
-
-
-$9.83
$0.00
$11.59
-$5.76
No
Program/Logo
-$34.99
-$20.00 -$34.93 -$50.91
-$29.05
-$27.47 -$25.01 -$87.05
-$26.88
Brands
Brand 1*
$7.62
$5.55
$1.97
-$15.10
-$5.48
-$5.65
$4.57
$10.55
$6.98
Brand 2
-$7.62
-$9.41
-
-
-$7.10
-
-$12.46 -$10.55
$7.23
Brand 3
-
-
-$15.51
-
-
-
-
-
-
Brand 4
-
-
$15.10
$12.58
-
$7.89
-
-
Brand 5
-
-
-
-
-
-
-
-$14.21
Brand 6
-
$3.86
-
-
-
-
-
-
-
Brand 7
-
-
-
-
-
$5.65
-
-
-
Brand 8
-
-
$13.53
-
-
-
-
-
-
Use-By date
Continues
$1.81
$1.15
$2.85
$2.74
$1.17
$2.50
-$1.67
$2.11
$2.05
Net weight
Continues
-
-
$8.41
$7.95
$2.02
-$13.19
$2.19
$2.27
-
Table 9 :
9Willingness to pay range between cuts for each product claim. Diced, Flank, Flap, Ground, NY Strip, Roast, Sirloin, Tenderloin * Cuts refers to the cuts with a willingness to pay above $0.00 for the presence of this claim.Claims
Willingness to Pay
range
Cuts*
Antibiotic free
$0.00-$47.50
Cowboy, Diced, Flank, Flap, Ground, Roast,
Sirloin
Pasture raised/not
confined
$0.00-$19.66
Cowboy, Flank, Flap, Sirloin, Tenderloin
Organic
$0.00-$17.00
Cowboy, Diced, Flank, Flap, NY Strip, Roast,
Sirloin
Angus
$0.00-$16.80
Cowboy, Diced, Ground, NY Strip, Roast, Sirloin,
Tenderloin
Grass Fed
$0.00-$11.60
Cowboy, Diced, Flank, Flap, Roast, Tenderloin
Natural
$0.00-$10.70
Cowboy, NY Strip, Roast, Tenderloin
Non-GMO
$0.00-$7.05
Cowboy, Diced, Flank, Flap, Ground, NY Strip,
Sirloin, Tenderloin
Traceable back to
farm
$0.00-$5.90
Cowboy, Diced, Flank, Flap, Ground, NY Strip,
Roast, Sirloin,
Hormone Free
$0.70-$2.30
Cowboy,
The selection of the beef cuts was developed in consultation with the industry partner involved in the ARC grant associated with this research.
For the design of this study country of origin and brands have been constrained and are coincident.
In this study eighteen scales (holding one constant at one) represented the nine studies cuts in two different seasons.(2)(3)
Brands have been de-identified for confidentiality reason. Brands 1 & 7 are from Australia only, brands 2, 4 & 6 are from United States only and other brands are sourced from a mixture of countries.
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| [] |
[
"A maximum likelihood based technique for validating detrended fluctuation analysis (ML-DFA)",
"A maximum likelihood based technique for validating detrended fluctuation analysis (ML-DFA)"
] | [
"Maria Botcharova \nCoMPLEX\nCentre for Mathematics and Physics\nLife Sciences and Experimental Biology/University College London\nLondonUK\n\nInstitute of Neurology\nUniversity College London\nLondonUK\n",
"Simon F Farmer †[email protected] \nInstitute of Neurology\nUniversity College London\nLondonUK\n",
"Luc Berthouze *[email protected] \nCentre for Computational Neuroscience and Robotics\nUniversity of Sussex\nFalmerUK\n\nInstitute of Child Health/University College London\nLondonUK\n"
] | [
"CoMPLEX\nCentre for Mathematics and Physics\nLife Sciences and Experimental Biology/University College London\nLondonUK",
"Institute of Neurology\nUniversity College London\nLondonUK",
"Institute of Neurology\nUniversity College London\nLondonUK",
"Centre for Computational Neuroscience and Robotics\nUniversity of Sussex\nFalmerUK",
"Institute of Child Health/University College London\nLondonUK"
] | [] | Detrended Fluctuation Analysis (DFA) is widely used to assess the presence of long-range temporal correlations in time series. Signals with long-range temporal correlations are typically defined as having a power law decay in their autocorrelation function. The output of DFA is an exponent, which is the slope obtained by linear regression of a log-log fluctuation plot against window size. However, if this fluctuation plot is not linear, then the underlying signal is not self-similar, and the exponent has no meaning. There is currently no method for assessing the linearity of a DFA fluctuation plot. Here we present such a technique, called ML-DFA. We scale the DFA fluctuation plot to construct a likelihood function for a set of alternative models including polynomial, root, exponential, logarithmic and spline functions. We use this likelihood function to determine the maximum likelihood and thus to calculate values of the Akaike and Bayesian information criteria, which identify the best fit model when the number of parameters involved is taken into account and over-fitting is penalised. This ensures that, of the models that fit well, the least complicated is selected as the best fit. We apply ML-DFA to synthetic data from FARIMA processes and sine curves with DFA fluctuation plots whose form has been analytically determined, and to experimentally collected neurophysiological data. ML-DFA assesses whether the hypothesis of a linear fluctuation plot should be rejected, and thus whether the exponent can be considered meaningful. We argue that ML-DFA is essential to obtaining trustworthy results from DFA. | null | [
"https://arxiv.org/pdf/1306.5075v1.pdf"
] | 17,079,537 | 1306.5075 | 6761d1e188fd8c01351acb8a566c4f2f9d77380c |
A maximum likelihood based technique for validating detrended fluctuation analysis (ML-DFA)
21 Jun 2013
Maria Botcharova
CoMPLEX
Centre for Mathematics and Physics
Life Sciences and Experimental Biology/University College London
LondonUK
Institute of Neurology
University College London
LondonUK
Simon F Farmer †[email protected]
Institute of Neurology
University College London
LondonUK
Luc Berthouze *[email protected]
Centre for Computational Neuroscience and Robotics
University of Sussex
FalmerUK
Institute of Child Health/University College London
LondonUK
A maximum likelihood based technique for validating detrended fluctuation analysis (ML-DFA)
21 Jun 20131
Detrended Fluctuation Analysis (DFA) is widely used to assess the presence of long-range temporal correlations in time series. Signals with long-range temporal correlations are typically defined as having a power law decay in their autocorrelation function. The output of DFA is an exponent, which is the slope obtained by linear regression of a log-log fluctuation plot against window size. However, if this fluctuation plot is not linear, then the underlying signal is not self-similar, and the exponent has no meaning. There is currently no method for assessing the linearity of a DFA fluctuation plot. Here we present such a technique, called ML-DFA. We scale the DFA fluctuation plot to construct a likelihood function for a set of alternative models including polynomial, root, exponential, logarithmic and spline functions. We use this likelihood function to determine the maximum likelihood and thus to calculate values of the Akaike and Bayesian information criteria, which identify the best fit model when the number of parameters involved is taken into account and over-fitting is penalised. This ensures that, of the models that fit well, the least complicated is selected as the best fit. We apply ML-DFA to synthetic data from FARIMA processes and sine curves with DFA fluctuation plots whose form has been analytically determined, and to experimentally collected neurophysiological data. ML-DFA assesses whether the hypothesis of a linear fluctuation plot should be rejected, and thus whether the exponent can be considered meaningful. We argue that ML-DFA is essential to obtaining trustworthy results from DFA.
Introduction
Detrended Fluctuation Analysis (DFA) is a technique commonly applied to time series as a means of approximating the Hurst exponent, which indicates the degree of long-range temporal correlations present [1][2][3][4]. Long-range temporal correlations (LRTCs) occur in time series with an autocorrelation function that decays as a power law function of the lag [5]. The presence of LRTCs suggests that the underlying signal is governed by non-local behaviour, with all scales contributing to system behaviour. LRTCs have been detected in various biological time series and natural phenomena [1][2][3][6][7][8][9], see a review in [10]. In neurophysiological signals, it has been argued that LRTCs facilitate essential functions such as memory formation, rapid information transfer, and the efficient neural network reorganisation that promotes learning [11][12][13][14][15][16][17].
DFA produces estimates of the magnitude of detrended fluctuations at different scales (window sizes) of a time series and assesses the scaling relationship between estimates and time scales. Estimation of the Hurst exponent through DFA assumes self-similarity in the time series. If the signal is self-similar, then the detrended fluctuations will increase as a power law function of window size, and the relationship between the two can be visualised as a straight line on a log-log fluctuation plot [1,2]. DFA returns the slope of the plot as its exponent with no check as to whether the self-similarity of the time series is supported by there being a linear fluctuation plot. At present there is no method which establishes the linearity of a DFA plot and an important shortcoming of the typically used method (see below) is that unless there is gross violation of linearity that can be detected by visual inspection then DFA exponents can be used for data that are not self-similar. In fact, Maraun and colleagues went as far as suggesting that DFA results are sensitive but not specific concerning long-range correlations [18].
Previous studies have described non-linear characteristics in DFA fluctuation plots for signals constructed by independent superposition of a number of processes with specific characteristics. When a noise time series contains a linear, sinusoidal, or a power law trend, the DFA plot will contain several linear segments, joined at crossover points [19]. Studies have also looked at noise time series with sections of silence, concatenations of noisy signals with different amplitude standard deviations, and of noisy signals with varying levels of temporal correlation [20,21]. These fluctuation plots show different combinations of linear and quasi-linear fragments.
At present the standard approach used to characterise the fit of the linear regression is to calculate an R 2 value (for example [22]). However, the R 2 value is a very insensitive measure [23]. An alternative technique may be to assume that the errors around a linear fit have a χ 2 distribution, but this assumption cannot be made for a DFA fluctuation plot because the magnitude of detrended fluctuation is dependent on the window length so that the fluctuation plot suffers from heteroscedasticity [24]. Namely, this approach would not allow one to distinguish between a self-similar signal yielding non χ 2 -distributed regression errors and a non self-similar signal. Another approach may be to compute the probability of the fluctuation plot taking the form of a specific function for signals with different self-similarity properties, based on the probability distribution of the innovations. A paper by Bardet [25] formulates such a distribution for the scale-invariant process that would give rise to a perfectly linear plot in its DFA fluctuation plot, i.e., fractional Gaussian noise. However, this approach would restrict the technique to being able to identify only a limited set of signals, and furthermore, one would need to know a priori the nature of the signal in order to employ the appropriate distribution.
Here, we propose a maximum likelihood based technique to assess the validity of the assumption of linearity through model selection.
Our technique, referred to as ML-DFA henceforth, is rooted in likelihood theory. We calculate a loglikelihood function for both a linear model and a number of alternative models. This requires formulating the DFA fluctuation plot as a probability density, which we do by normalising the fluctuation magnitudes. We use this to compute the Akaike and Bayesian Information Criterion (AIC and BIC, respectively) [26,27] which reveal the best-fitting model to the fluctuation plot, while compensating for over-fitting. If no model amongst the set of alternative models is a better fit than the linear model, then we accept (or more accurately, we do not reject) the hypothesis that the fluctuation plot is linear.
In the following sections, we apply the method to simulated time series for which we can control the expected outcome, and to neurophysiological data for which no ground truth is available.
Synthetic time series are generated by an Autoregressive Fractionally Integrated Moving Average (FARIMA) process [28] (also referred to as ARFIMA or AFRIMA). We use FARIMA because it provides an easily tunable algorithm for constructing time series with a combination of short-term and long-term correlations, which we will show influence the DFA fluctuation plot. A FARIMA process in its simplest form can be used to generate fractional Gaussian noise, which has been shown analytically to produce linear DFA fluctuation plots in their asymptotic limit [25,29]. However, by gradually introducing shortterm correlations through smoothing the data and enforcing autoregression, it is possible to destroy the self-similarity of the time series, and a statistically robust method should capture this. We note that a FARIMA process has also been used to model neurophysiological signals such as EEG, which have the properties of being stationary and whose amplitude fluctuations follow Gaussian statistics [30]. A FARIMA process therefore provides an efficient and malleable method of generating and manipulating time series. We further apply ML-DFA to a sinusoidal signal and a sinusoidal signal with independent additive noise, whose DFA fluctuation plots take known forms [19].
Finally, we apply the method to EEG data recorded from a group of twenty human subjects at rest to demonstrate the technique on experimentally acquired data. We also study how the choice of window lengths over which DFA is calculated affects the linearity or otherwise of the fluctuation plot.
Results
Simulated Data
We applied ML-DFA to simulations of a FARIMA process. We first used FARIMA to generate self-similar fractional Gaussian noise with varying Hurst exponents, and then we altered its parameters to generate a more general non self-similar FARIMA signal. For a more complete discussion of the FARIMA process parameters, the reader is referred to the Methods and Materials section below. We further applied ML-DFA to sinusoidal signals with three different periods, and to sinusoidal signals with independently added noise.
We fitted the DFA fluctuation plots for 1000 simulations of each of the generated time series with the set of alternative models listed in the Methods and Materials section. We report the proportion of best-fits for each model as determined by the AIC and BIC measures.
Fractional Gaussian Noise
Fractional Gaussian noise can be generated by a FARIMA(0,d,0) process with 0 < d < 0.5. The case of d = 0 is called white Gaussian noise, however, we will here refer collectively to FARIMA(0,d,0) processes with 0 ≤ d < 0.5 as fractional Gaussian noise. Fractional Gaussian noise has been proved to be asymptotically scale-invariant, and therefore its associated DFA fluctuation plot should be linear with a slope α given by d + 0.5 [25,29]. The value of α is an approximation to the Hurst exponent of the data, H, where H = 0.5 indicates Gaussian white noise and H = 1 indicates pink noise. We demonstrate ML-DFA on three simulations of fractional Gaussian noise, spanning the possible range of d values.
In Figure 1A we show DFA plots for three FARIMA time series with Hurst exponents of 0.5, 0.7 and 1.0. The slopes of the DFA plots recover estimates of the Hurst exponents of 0.50, 0.71 and 1.01 respectively. Figure 1B-D shows that the results of ML-DFA confirm that a linear model is appropriate for each of the time series, thus validating the results of standard DFA.
Tables 2 and 1 report the proportion of times out of 1000 simulations that each of the alternative models was found by ML-DFA to be the best fit, according to the AIC and BIC values respectively. We found that the AIC and BIC were both successful at identifying the linear model as the best fit in over 95% of the simulations. The mean slopes of those fluctuation plots that were not rejected were the same to 3 decimal places for both the AIC and BIC measures, and were 0.500, 0.696 and 0.995 respectively for expected Hurst parameters of 0.5, 0.7 and 1.0. The standard deviations for all slopes were 0.01.
FARIMA processes
The FARIMA(1,d,1) process is one which includes a single φ and a single θ coefficient, indicated by the parameter values of 1. It is possible to include a greater number of φ and θ coefficients, but we consider only a single addition for simplicity. We vary φ and θ in the range 0 < φ < 1, 0 < θ < 1, which satisfies the conditions |φ| < 1, |θ| < 1 for convergence [28]. Throughout the manuscript, FARIMA([φ],d,[θ]) will denote the FARIMA process with φ 1 = φ and θ 1 = θ.
In the general case, a FARIMA(1,d,1) time series is not expected to be self-similar and therefore, the associated DFA fluctuation plots should not necessarily be linear. Variations in the φ and θ parameters contribute to a range of fluctuation plots, with examples in Figure 2 illustrating a number of cases in which different alternative models were found by ML-DFA to be the best fit.
In all cases, ML-DFA showed sensitivity to more or less subtle deviations from the linear model. Tables 2 and 1 provide the proportion of times out of 1000 simulations of FARIMA(1,d,1) time series that each of the alternative models were found by ML-DFA to be the best fit, using the AIC and BIC values respectively. In most cases (6/9 pure FARIMA scenarios), the results of AIC and BIC were compatible. Specifically, 4 of the 9 scenarios showed almost identical results (scenarios 1, 2, 6, 9) whilst 2 were qualitatively similar in that the same models were involved albeit with varying percentages (scenarios 3 and 8). Three scenarios showed substantial differences with different models being involved (scenarios 4, 5 and 7 Figure 3 shows that when φ = 0.4 the DFA fluctuation plot could certainly be considered linear on visual inspection. However, closer examination (we assist the reader by providing a log-log plot of the autocorrelation function -it is helpful to remember that the DFA exponent is directly linked to the exponent of the power law in the autocorrelation function) reveals otherwise. Application of the runs test [31] on the residuals of the regression shows that the residuals are not independent (p < 1e − 5) which confirms that the BIC results are false positives. For comparison, the runs test for the fractional Gaussian noise returns p > .2. With increasing values of φ, the distortion of the fluctuation plot (and associated autocorrelation function) becomes readily available to visual inspection and agreement between AIC and BIC is strong. Specifically, 100% of the simulations reject the linear model hypothesis and AIC and BIC return the same set of alternative models in more than 99% of the simulations for both φ = 0.6 and φ = 0.8.
Sinusoidal signals
From [19], the DFA fluctuation plot of a pure sine will have a crossover at a window size corresponding to the period of the oscillation, with a slope of 2 for low window sizes, and a slope of zero after the crossover point. We reproduce these fluctuation plots and demonstrate that they are best fit by a two-segment spline model, with crossovers as predicted by theory. In Figure 4A-C, we present results for three pure sine curves with periods of 1000, 100 and 30 respectively. We observe that the crossover points in each plot are at 3, 2 and 1.48, which are the base-10 logarithms of 1000, 100 and 30, respectively. ML-DFA therefore recovers both the spline function and at its point of inflection the period of the original sine Table 1. ML-DFA results on synthetic data using AIC. From 1000 simulations of noise time series, the table gives the proportion of times that each of the alternative models was found to be the best fit, according to AIC values when ML-DFA was applied to fractional Gaussian noise, FARIMA(1,d,1) processes and noisy sinusoidal signals.
Linear Non-linear Model x x 2 x 3 x 4 n √ x 4-x 3-x 2-x log e FARIMA([0],0.5,[0]) 96.6 3 - - - - - - - 0.4 FARIMA([0],0.2,[0]) 96.3 2.9 - - - - - - - 0.8 FARIMA([0],0,[0]) 95.9 2.8 - - 0.1 - - - 0.1 1.1 FARIMA([0.4],0.2,[0]) 2.1 7.5 - - 56.0 - - - 14.5 19.9 FARIMA([0],0.2,[0.4]) - 77 19.3 0.1 - - - 1.2 - 2.4 FARIMA([0.8],0.2,[0]) - - 0.3 - 90.3 - - - 7.1 2.3 FARIMA([0],0.2,[0.8]) - 47.7 2.8 2.8 - - - 45.5 - 1.2 FARIMA([0.4],0.2,[0.8]) - 64.7 - - - - - 22.1 - 13.2 FARIMA([0.8],0.2,[0.4]) - 0.4 - - 76.5 - - - 13.5 9.6 FARIMA([0],-0.2,[0])+sin( 2πt 200 ) - - - - - 7.4 92.6 - - - FARIMA([0],0,[0])+sin( 2πt 100 ) - - - 2 - 98 - - - -
The fitted models are listed in the top row, alongside the proportion of best fits assigned to each one by the value of the AIC measure. The shorthand n-x is used to denote a n-segment spline. The shorthand n √
x combines results for n = 2, 3, 4. The signals whose DFA fluctuation plots are analysed are described in the left-hand side column.
signal.
The addition of independent noise to sinusoidal signals has also been studied [19]. The DFA fluctuation plot of a sine signal with anti-correlated noise (Hurst exponent H ∈ [0, 0.5)) will have two crossover points, and therefore three segments. One will be located at the window size corresponding to the period of the oscillation, and one at a smaller window length. We demonstrate in Figure 4D that ML-DFA identifies a three-segment spline as the best fitting model for such a fluctuation plot.
A sine curve with independent, additive white or correlated noise will show three crossovers, or four segments in its DFA fluctuation plot. One crossover is again at the period of the sine curve. Figure 4E demonstrates ML-DFA alongside its resulting best-fit four-segment spline.
Tables 2 and 1 provide the proportion of times out 1000 simulations of two sets of sines with added noise that each of the alternative models was found to be the best fit by the AIC and BIC measures, respectively. No data are provided for repeated simulations of sines without added noise since these would produce rigorously identical fluctuation plots. Compared to the BIC measure, we found that the AIC measure assigned a greater proportion of the DFA fluctuation plots obtained from the sine with FARIMA([0],0,[0]) noise to the four-segment spline model (98% vs 84.9% for AIC and BIC, respectively), as predicted by theory [19]. The BIC measure returned a higher proportion of quartic model because of the reduced number of parameters. However, both AIC and BIC performed similarly in identifying the three-segment spline as the best fit for fluctuation plots of the sine with FARIMA([0],-0.2,[0]) noise, as expected.
Physiological Data
We applied ML-DFA to EEG data, according to the method set out in Linkenkaer-Hansen et al. [16]. Specifically, we took the power spectrum of the EEG, found the peak corresponding to the alpha rhythms and bandpass filtered the signal to isolate the corresponding range. Following this, we obtained the Table 2. ML-DFA results on synthetic data using BIC. From 1000 simulations, the table gives the proportion of times that each of the alternative models was found to be the best fit, according to BIC values when ML-DFA was applied to fractional Gaussian noise, FARIMA(1,d,1) processes and noisy sinusoidal signals.
Linear
Non-linear Model
x x 2 x 3 x 4 n √ x 4-x 3-x 2-x log e FARIMA([0],0.5,[0]) 96.6 3 - - - - - - - 0.4 FARIMA([0],0.2,[0]) 96.3 2.9 - - - - - - - 0.8 FARIMA([0],0,[0]) 96.8 2.3 - - 0.1 - - - 0.1 0.7 FARIMA([0.4],0.2,[0]) 31.1 4.6 - - 40.6 - - - 11.4 12.3 FARIMA([0],0.2,[0.4]) 0.1 93.9 1.6 - - - - 0.5 - 3.9 FARIMA([0.8],0.2,[0]) - - - - 90.4 - - - 7.1 2.5 FARIMA([0],0.2,[0.8]) - 71.2 1.7 - - - - 23.6 - 3.5 FARIMA([0.4],0.2,[0.8]) - 86.3 - - - - - 0.4 - 13.3 FARIMA([0.8],0.2,[0.4]) - 0.4 - - 76.5 - - - 13.5 9.6 FARIMA([0],-0.2,[0])+sin( 2πt 200 ) - - - - - 7.4 92.6 - - - FARIMA([0],0,[0])+sin( 2πt 100 ) - - - 15.1 - 84.9 - - - -
The fitted models are listed in the top row, alongside the proportion of best fits assigned to each one by the value of the BIC measure. The shorthand n-x is used to denote a n-segment spline. The shorthand n √
x combines results for n = 2, 3, 4. The signals whose DFA fluctuation plots are analysed are described in the left-hand side column.
amplitude envelope by using the Hilbert transform, and applied standard DFA and ML-DFA.
The amplitude envelope was obtained by first applying the Hilbert transform to the time series s(t) in order to obtain its analytic signal s a , which is a corresponding unique complex representation of a real-valued time series:
s a (t) = s(t) + H {s(t)}
where the Hilbert transform is represented by H {}. The time-varying envelope A(t) is then the amplitude of the analytic signal, given by:
A(t) = s(t) 2 + H {s(t)} 2 .
We demonstrate these steps in Figure 5. DFA and ML-DFA were applied to A(t), the amplitude envelope of an EEG time series filtered between 8 and 13 Hz. The minimum box size for applying DFA was 1 second of data, in order to include at least 8 oscillations at the minimum frequency of the band-pass filter. The largest window size was set to one tenth of the full length of the data for each subject, as suggested by [16]. This allows a sufficient number of windows to provide a robust measure of the average fluctuation magnitude for a large window length, thus correcting for the variability of root mean square fluctuations from one window to the next. Note that, in [16], the window sizes are determined by inspecting a fluctuation plot that spans across all possible window sizes, and then the range of windows that adhere to a power law is selected for further analysis. We will return to this in the Discussion.
Human EEG Data
We report the best fit models as determined by ML-DFA for the amplitude envelope of the EEG of 20 human subjects tested, which had previously been filtered between 8 and 13 Hz. For each subject, an EEG time series from the Cz electrode was used after artefact removal because of its central location on the scalp, leading to fewer potential artefacts caused by muscle movements or eye-blinks. If the best fit model, as assessed by the BIC value, is linear, then we also report the DFA exponent in Table 3. Figure 6 shows 4 examples of each of the ML-DFA fit types obtained from the 20 subjects. These data were selected to illustrate both the linear fitting by standard DFA and a range of model fits that led to the rejection of the linear model hypothesis. In total, the linear model hypothesis was not rejected in 12/20 (AIC) and 16/20 (BIC) of the subjects (see Table 3). Figure 5. Preprocessing of the time series on an example EEG data set. Panel A shows the raw EEG signal. This is filtered between 8 − 13 Hz for Panel B, and the amplitude envelope, derived from the real part of the Hilbert transform, is plotted above the filtered data.
Minimum and Maximum Window Sizes
In neurophysiological data, the choice of window sizes over which DFA is calculated is an important consideration. Using the data from one subject for which both the DFA fluctuation plot was best fit by a linear model according to both AIC and BIC, we explore how the choice of minimum and maximum window sizes affects the linearity of the DFA fluctuation plot. We demonstrate that using a minimum window length smaller than a minimum oscillatory period of the data examined gives rise to DFA fluctuation plots for which the linear model hypothesis is rejected. We also show that taking a maximum window length larger than N 10 gives rise to DFA fluctuation plots for which the linear model hypothesis may be rejected. Figure 7 shows the application of standard DFA to the fluctuation plots of the EEG signal subject number 7, alongside the best-fitting model determined by ML-DFA using AIC. In Panel A, the minimum and maximum window sizes are set as suggested by [16]. In Panel B, the minimum window length is set to 1 second of data as previously in Figure 6, while the maximum window length is N . The magnitude of detrended fluctuations grows more slowly for large window sizes due to the finite length of the data, Figure 6. DFA fluctuation plots for 4 example signals from the Cz electrode of an EEG recording. Specifically, the 4 rows A-D correspond to subjects 2,3,7,13. In each row, the left-hand side panel shows a representative 3000 innovations of the time series, which corresponds to approximately 15 seconds, the middle panel shows the fluctuation plot fitted using standard DFA with the DFA exponent α given above each plot, and the right-hand side panel shows the best fit model as determined by ML-DFA using AIC. and this gives rise to a two-segment spline as the best fit model such that the DFA exponent should not be trusted. In Panel C of Figure 7, the minimum window size is set to 8 samples of the recording (≈ 0.31 seconds), while the maximum window length is held constant at a tenth of the length of the time series, N 10 as before. The linear model hypothesis is rejected by the AIC method, because the best-fit model is logarithmic. This is consistent with the fact that, as the signal was filtered in the α range of 8-13 Hz, a minimum window length less than f s 8 , is less than a single cycle of the slowest 8Hz frequency present, which will certainly produce a crossover, as shown in Figure 7. In order to select a suitable minimum Figure 6. The minimum window size is 1 second, and the maximum is N 10 , or 187 seconds, both following [16]. The arrows in each plot indicate the range over which the fluctuation plot is calculated to obtain the DFA exponents in Tables 3, which is the full range of the fluctuation plot in Panel A. In Panel B, the minimum window length is also 1 second of data, and the maximum window length is N , which is the full length of the signal, or 1865 seconds (31 minutes) in this case. In Panel C, the minimum window length is 8 samples ( 8 256 ≈ 0.31 seconds) and the maximum window length is N 10 , or 187 seconds. In each row, the left-hand side panel shows the fluctuation plots fitted using standard DFA with the DFA exponent α given above each plot, the right-hand side panel shows the best-fit model as determined by ML-DFA using AIC.
window size for a signal, its characteristic frequency should be known. In this case, the characteristic frequency is not a single value, but a range between 8 and 13 Hz, so that the crossover in the fluctuation plot is not a single point (as with previously studied pure sine curves), but rather a range of points. We suggest that this is why the best fit model is the smoother logarithmic model rather than a spline.
This analysis was applied to all 20 subjects. When the minimum window size was taken to be 8 samples Each subject is identified by number in the left-hand side column, alongside the best fit model determined by ML-DFA using AIC and BIC. The † symbol indicates those subjects whose fluctuation plots were rejected as not being linear by at least one of the AIC or BIC measures. When the fluctuation plot is rejected by BIC, it is also rejected by AIC in all cases. The exponent provided in column 2 was obtained using standard DFA.
(while keeping the maximum window size at N 10 ), the fluctuation plots of data for all 20 subjects were rejected as not being linear by both AIC and BIC. When the maximum window size was set to N , and the minimum kept at 1 second, application of ML-DFA resulted in 4/20 (BIC) and 10/20 (AIC) fluctuation plots for which the linear model hypothesis was rejected. Once again, BIC shows less sensitivity than AIC in identifying the loss of linearity due to a strong bias towards the simplest (linear) model.
Discussion
In this paper, we have presented a technique (ML-DFA) to determine whether a DFA exponent can be trusted based on whether the linear model hypothesis for its associated fluctuation plot is accepted or rejected by a model selection approach. We have validated ML-DFA by applying it to DFA fluctuation plots obtained from FARIMA(0,d,0) time series, which have been shown to be asymptotically linear [25,29]. We have explored ML-DFA in relation to DFA plots obtained from time series generated by FARIMA (1,d,1) processes, which allow flexible combinations of long and short correlations in the time series, and which we expect will produce fluctuation plots that are rejected as not being linear. We have recovered the piecewise linear form of the DFA fluctuation plot for sinusoidal signals, and sinusoidal signals with additive independent noise, as previously documented [19]. Finally, we applied the method to the amplitude envelopes of filtered EEG time series as in [16], showing that 12 out of 20 recorded signals were not rejected by AIC and 16 out of 20 by BIC. Without the use of a test such as ML-DFA, the value of a DFA exponent could be meaningless and we suggest that it may be valuable to re-examine previously published results.
We have stated the values of both the AIC and BIC measures throughout. It has been argued that the BIC is the most reliable information criterion [32,33]. However in this study, the AIC shows fewer false-positive results. This is demonstrated by the fact that a greater proportion of time series generated by FARIMA (1,d,1) processes are rejected as not being linear, which is a result that we would expect. Furthermore, the AIC was more successful at correctly identifying the fluctuation plots that we expected to be four-segment splines because they were obtained from a sinusoidal curve with independent, additive anti-correlated noise. The BIC often selected a quartic model instead, because it has fewer parameters. For this reason, we suggest that AIC should be used to determine the best-fitting model.
It is important to stress that ML-DFA does not verify or demonstrate the linearity of a plot. Merely, it concludes that a linear model is the best choice given the set of alternative models considered. For this reason, it is important to carefully select the set of alternative models. Since fluctuation plots should always be monotonic because the fluctuations of a time series will yield an error of at least equal size for windows of greater length, we have only considered models that (a) can capture the monotonicity of a DFA fluctuation plot and (b) are informed by experience and previous studies of non-linear DFA fluctuation plots. Note that the necessarily finite set of alternative models means that there is always a possibility that a different model could prove a better fit and therefore one should be very cautious of drawing conclusions about the nature of a time-series based on the best-fit model. Further, because the best fitting models are calculated from initial parameters that are set randomly, using a set of closelyrelated models with an equal number of parameters may result in different best-fit models for different runs of ML-DFA. To address this concern, initial parameters for the polynomial models were set to those that best fitted the fluctuation plots in a least square sense; however, this remains an open issue for models of arbitrary functional form. It is for these two above reasons that our focus has been primarily on whether the linear model hypothesis is rejected (which determines whether the DFA exponent can be trusted) and not on interpreting or explaining why a particular functional form was the best fitted model.
Several papers have discussed non-linear DFA fluctuation plots for specific time series. A DFA fluctuation plot which flattens out with increasing box size typically reflects a periodic signal, such as a sine [19]. Increasing fluctuations at large window sizes may be consistent with a noise process with segments removed, one with spikes added, one using concatenated segments of different standard deviations, or else with a power law trend [19][20][21]. Finite-size effects cause smaller windows to always have fluctuation magnitudes below the expected regression line [34].
Additionally, a fluctuation plot can be non-linear if the DFA scaling exponent is not a single value, but comes from a distribution. In this case, it may be relevant to apply multi-fractal DFA [35]. If the scaling behaviour of a time series is not constant across time, then a suitable technique is Adaptive Timevarying DFA [36], which uses optimal filtering to track changes in DFA exponent over the record. Any of these considerations may help elucidate a DFA fluctuation plot for which the linear model hypothesis is rejected.
We also varied the minimum and maximum window sizes used in the course of DFA application to highlight the fact that an inappropriate window size may affect the validity of the DFA exponent. A preliminary inspection of the whole fluctuation plot (as done by [16]) can be instructive for gaining a broad idea of the scales over which long-range correlations may be located. However, we stress that good practice should be to establish a priori the range of scales over which LRTCs are expected -taking into account the constraints of both the nature of the data (e.g., sampled oscillatory data) and a statistically appropriate number of maximum window sizes -and to accept the result returned by ML-DFA. It would be inappropriate to use this technique to identify the range of scales over which LRTCs exist. Indeed, it will always be possible to find a range of scales over which the linear model hypothesis will be accepted.
For neurophysiological data, the minimum window size should include several oscillations of the lowest frequency, and we took 1 second of the recording to ensure this. The frequencies present are determined by the range of the bandpass filter used. FARIMA signals do not have a characteristic time scale, so the minimum window size can be smaller, and we took 8 innovations (a smaller window size of 1 or 2 innovations would have given an artefactual result because 1 or 2 samples can always be fitted perfectly by a line and the fluctuation magnitude will thus always be zero, a minimum window size of 4 samples can cause inaccuracies due to finite-size effects [34]). For the sinusoidal curves, we also used 8 innovations for the minimum window, which was smaller than the cycle period, precisely to allow us to demonstrate the crossovers in the fluctuation plot. The maximum window size was set to a tenth of the length of the time series for all signals considered to allow a sufficient number of values for a robust estimate of an average fluctuation size. In order to obtain a reliable fluctuation plot for larger time scales, a longer data series would typically be required [16]. In general, we recommend the use of these or similar guidelines for correct application of DFA and ML-DFA. Interestingly, ML-DFA makes it possible to approach the question of the maximum box size in a more systematic manner. The length of a neurophysiological time series will depend on numerous considerations, many of them experimental, and using a tenth of the data length as a maximum box size may lead to confusion when trying to infer meaning in time series of different length. Depending on the strictness of the model selection criteria between 50 and 80% of EEG time series did not reject the linear model hypothesis even when the entire record length, i.e. ∼ 20 minutes, was considered. When the linear model hypothesis was rejected at large window sizes, the window size above which loss of scaling occurred could be identified (see Figure 7). We suggest therefore that ML-DFA can be used to validate relaxing a conservative choice of maximum window size (i.e., to extend the length of meaningful correlations) to help with heterogeneous lengths of time series.
Materials and Methods
Scaling and Fit
DFA is used to assess the self-similarity in a signal [1,2]. The application of DFA returns the value of an exponent α, which is an estimate of the Hurst parameter, H, which in turn reveals the degree of long-range temporal correlation (LRTC) in the time series [3]. DFA can be applied to both stationary and non-stationary data, avoiding artefactual dependencies [37].
To calculate the DFA exponent, the time series is first de-meaned and then cumulatively summed. After being divided into non-overlapping windows of a given size (i.e., a scale), it is detrended (linearly for 1-DFA, non-linearly for higher-order DFA) yielding a fluctuation calculated as the root-mean-square deviation over every window at that scale. The process is repeated for different window sizes.
For oscillatory signals, the smallest window size should be large enough to avoid errors in local root mean square fluctuations, and is typically taken to be three or four times the length of a cycle at the characteristic frequency in the time series. If the minimum window size is significantly smaller than this, then the fluctuation plot will typically contain a crossover at the window length of a single period [19]. In the case of non-oscillatory signals such as those from a FARIMA process, there is no characteristic time scale and a smaller window size may be used. The maximum window size should be small enough to provide a robust average for the fluctuation magnitude across the time series. It is typically taken to be N/10 where N is the length of the data [16], however, a maximum window size of N/4 has previously also been used and shown to provide a sufficiently good estimate of the average fluctuations in some circumstances [38].
We call ns the vector of window sizes and F the vector of corresponding root mean square fluctuations. We label the number of distinct window sizes n, which are taken as the maximum possible to allow each window to be non-overlapping. The base 10 logarithm of these two vectors are labelled lns and lF respectively.
If the signal is self-similar, then the log-log plot of fluctuation sizes against window sizes, referred to as DFA fluctuation plot throughout the manuscript, will be linear and the DFA exponent is obtained by determining the slope of the best fitting regression line. A DFA exponent in the range 0.5 < α < 1 indicates the presence of long-range temporal correlations. An exponent of 0 < α < 0.5 is obtained when the time series is anti-correlated and α = 1 represents pink noise. Gaussian white noise has an exponent of α = 0.5. For a tutorial, see [39].
However, since there is no a priori means of confirming that a signal is indeed self-similar, an exponent can always be obtained even though the DFA fluctuation plot may not necessarily be linear -the only certainty being that it will be increasing (albeit not necessarily monotonously so) with window sizes. The models used by ML-DFA are listed below, with the a i parameters to be found. The number of parameters ranges between 2 for the linear model, and 8 for the four-segment spline model.
Polynomial -f (x) = K i=0 a i x i for K = {1, ..., 5} Root -f (x) = a 1 (x + a 2 ) 1/K + a 3 for K = {2, 3, 4} Logarithmic -f (x) = a 1 log(x + a 2 ) + a 3 Exponential -f (x) = a 1 e a2x + a 3
Spline with 2, 3 and 4 linear sections.
We first normalise the fluctuation magnitudes with:
lF scaled = 100 × lF − lF min lF max − lF min
where lF min and lF max are the minimum and the maximum values of vector lF respectively. We define a likelihood function:
L = n i=1 p(lns(i)) lF scaled (i)
which is a product across all windows i, where p(lns) represents the function:
p(lns) = |f (lns)| n i=1 |f (lns)|
where f (lns) is the fitted model. Absolute values are used in order to ensure that p(lns) remains in the range [0, 1], so that we reject a likelihood function if it falls below 0. The log-likelihood is then defined as:
logL = n i=1 lF scaled (i)logp(lns(i))
We maximise this function to find the parameters a i necessary for f (lns). The largest log-likelihood is the model which best fits the data, however, no consideration of the number of parameters used is taken when comparing log-likelihoods. To address this, we compute both the AIC and BIC measures which are designed to prevent over-fitting, which should in general be avoided [33].
It should be noted that the scaling step implies that DFA exponents cannot be recovered from the parameters of the linear or spline models following ML-DFA. For this reason, if a spline model is found to be the best-fitting model and the user is interested in the value of the exponents at each scale -as is sometimes used in clinical studies of heart beat variability [40] -then the user should apply standard DFA to each segment separately to obtain the corresponding exponents.
Akaike's Information Criterion
Akaike's Information Criterion (AIC) is used to compare the goodness-of-fit of probability distributions [26]. The AIC can only be used to compare models, but gives no information on how good the model is at fitting the data. This means that only the relative values of this measure, for different models, are important.
For a model using k parameters, with likelihood function logL, the Akaike Information Criterion is calculated using the following expression:
AIC = 2k − 2logL + 2k(k + 1) n − k − 1
where k is the number of parameters that the model uses. Note that we are using the formula proposed by [41] which accounts for small sample sizes, as advocated by [42][43][44] amongst others. The model which provides the best fit to the data is that with the lowest value of AIC.
Bayesian Information Criterion
The Bayesian Information Criterion was developed by Akaike and Schwartz [27]. It puts harsher restrictions on the parameter number required for the model:
BIC = −2logL + klog(n).
The lowest BIC indicates the best fit model. There is considerable discussion regarding which of the AIC or BIC measure is more effective at selecting the 'correct' model, and indeed it is possible to simulate situations in which one and the other is optimal [42]. [42] suggests that the BIC is effective primarily when the number of observations n is large enough, which may not be the case with DFA calculations with a typical number of window sizes of 50. On the other hand, the BIC is considered more reliable because it is by construction an approximation to the Bayes factor, which is considered by many to be the only possible approach to model selection (see Chapter 1 of [32], and [45] who tries to combine the two measures).
In the analysis here, we output both the AIC and BIC measures, but ultimately base our conclusions on the AIC when the two disagree. The BIC is the stricter approach in selecting a model with the least number of parameters, however, this will lead to an undesired bias toward choosing the linear model. Our results will actually show that the AIC is more reliable in determining the best fit to fluctuation plots for signals whose functional form has previously been studied and is known.
Signal Simulation
Self-similarity is a property of signals belonging to the class of signals with long-range dependence (LRD) [10]. In order to demonstrate and test our methodology, we apply it to signals simulated using an Autoregressive Fractionally Integrated Moving Average model (FARIMA) [28], which provides a process that can easily be manipulated to include a variable level of short and long-term correlations within a signal, which in turn provide a broad range of DFA fluctuation plots.
To construct a FARIMA process a sequence of zero-mean white noise is first generated, which is typically taken to be Gaussian, and necessarily so to produce fractional Gaussian noise. The FARIMA process, X t , is then defined by parameters p, d and q and given by:
1 − p i=1 φ i B i (1 − B) d X t = 1 + q i=1 θ i B i ε t .(1)
B is the backshift operator operator, so that BX t = X t−1 and B 2 X t = X t−2 . Terms such as (1 − B) 2 are calculated using ordinary expansion, so that (1 − B) 2 X t = X t − 2X t−1 + X t−2 . While the parameter d must be an integer in the ARIMA model, the FARIMA can take fractional values for d. A binomial series expansion is used to calculate the result:
(1 − B) d = ∞ k=0 d k (−B) k .
The left hand sum deals with the autoregressive part of the model where p indicates the number of back-shifted terms of X t to be included, φ i are the coefficients with which these terms are weighted. The right hand sum represents the moving average part of the model. The number of terms of white noise to be included are q, with coefficients θ i . In the range |d| < 1 2 , FARIMA processes are capable of modelling long-term persistence [28]. As we will only consider p = 1 and q = 1 throughout the manuscript, we will refer to φ 1 as φ and θ 1 as θ. We set |φ| < 1, |θ| < 1 to ensure that the coefficients in Equation 1 decrease with increasing application of the backshift operator, thereby guaranteeing that the series converges, and X t is finite [28].
A FARIMA(0,d,0) is equivalent to fractional Gaussian noise with d = H − 1 2 [28]. This produces a time series with a DFA fluctuation plot that has been shown to be asymptotically linear [25,29]. By manipulating the φ and θ parameters, the DFA fluctuation plots can also be distorted.
In a FARIMA(1,d,0) process, the φ parameter is non-zero, and an autoregressive term is added to the process. In general, an increase in the φ coefficient at constant θ induces a decrease in fluctuations for small window sizes, and a concavity is seen in the fluctuation plot [46]. The value of φ increases the short-range exponent with an exponential relationship [46]. This means that the process at a given time point depends linearly on the previous values in the series, so that a single impulse would affect the rest of the process infinitely far into the future. The process is expected to behave like a FARIMA(0,d,0) time series in the long-term, but the short term behaviour will have short-term correlations, depending on the size of φ [28].
For a FARIMA(0,d,1) time series, the θ parameter is non-zero, which indicates exponential smoothing and a time series with noisy fluctuations around a slowly-varying mean. The resulting DFA fluctuation plots have fluctuation levels that are above the expected regression line at large box sizes. An increase in θ for φ = 0 induces convexity in the fluctuation plot.
Neurophysiological Data
A total of twenty healthy subjects were recruited from the workforce at the Royal Hospital for Neurodisability 6 males, age range 24-59 years, of mean age 39.94 years, ±10.2. All subjects gave informed consent. Recording procedures were carried out in accordance with the declaration of Helsinki. None of the subjects had previous history of blackouts, faints, or psychiatric illness. None of the subjects were on any medication known to have centro-encephalic effects. All subjects were right handed.
The EEG recordings were conducted as part of a study exploring EEG changes occurring during music therapy. The subjects were seated in a comfortable chair with arm rests. A total of 23 Ag/AgCl electrodes (Unimed Electrodes, Surrey, UK) were applied individually to the scalp in accordance with the 10 − 20 system of electrode placement [47]. Electrodes were fixed in place using Ten20 conductive paste (Weaver and Company, USA). Electrode impedances were maintained below 5KΩ. The EEG was acquired using an XLTEK Video-EEG monitoring system (Optima Medical, Putney, UK) which incorporated a 50 channel amplifier. The EEG signals were acquired using a sampling rate of 256Hz, and filter settings between 0.5 − 70Hz without mains suppression. The montage regime used for on-line acquisition was common average reference [48]. Recordings were taken over a period of 40 − 60 minutes. The initial 5 minutes of the recording was designated the baseline silence period (background noise 34dB) Here, the subjects were instructed to close their eyes on hearing a series of clicks. The initial 2.5 minutes of the baseline recording during the silence period were taken with the eyes open. Across the remainder of the session, subjects listened to different sounds/music the order of delivery having been randomly selected.
The recorded EEG signals were converted off-line to Laplacian derivation [49,50]. Artefact rejection was performed through visual inspection of the EEG and Independent Component Analysis in EEGlab [51]. For this reason, the length of the continuous signals subjected to analysis varied from subject to subject but had a minimum length of approximately 20 minutes.
Matlab Code
Full code for ML-DFA is available from the corresponding author upon request. It will be made freely available upon acceptance of the manuscript. The data from FARIMA processes were generated using Matlab code published by [52].
Figure 1 .
1Time series and corresponding DFA fluctuation plots for signals obtained by FARIMA(0,d,0) processes with φ and θ set to 0 and with d = 0, and d = 0.2, and d = 0.5 to produce fractional Gaussian noise. Panel A shows the three DFA fluctuation plots fitted using standard DFA. Values d = 0, d = 0.2, and d = 0.5 will produce time series with Hurst exponents 0.5 (white noise, blue diamonds), 0.7 (correlated noise, green crosses) and 1 (pink noise, pink circles) respectively. The slopes estimated by application of standard DFA are stated at the top, and correspond closely to these theoretical values. Panels B-D show the best fit model according to the AIC measure in ML-DFA. The best-fit model is linear in all cases.
Figure 2 .
2Time series and corresponding DFA fluctuation plots for signals obtained by FARIMA(1,0.2,1) processes with d = 0.2 taken as an representative value, and varying values of φ and θ. Each row A-C corresponds to a different set of φ and θ coefficients, which alter the resulting DFA fluctuation plots. In each row, the left-hand side panel shows a representative 3000 innovations of the time series, the middle panel shows the fluctuation plot fitted using standard DFA with the resulting exponent α given above, and the right-hand side panel shows the best-fit model determined by ML-DFA using AIC.
Figure 3 .
3Time series, DFA fluctuation plots and autocorrelation functions for signals obtained by FARIMA([p],0.2,[0]) processes with varying values of p. Each row A-D corresponds to φ taking values from 0, 0.4, 0.6, 0.8. In each row, the left-hand side panel shows a representative 3000 innovations of the time series, the middle panel shows the fluctuation plot fitted using standard DFA with the resulting exponent α given above, and the right-hand side panel shows the autocorrelation function of the (complete) signal in log-log coordinates.
Figure 4 .
4Time series and corresponding DFA fluctuation plots for 5 sinusoidal signals with varying levels of independent, additive noise. Each row A-E corresponds to a different sinusoidal function. In each row, the left-hand side panels show a representative 3000 innovations of the time series, the middle panel shows the fluctuation plots fitted using standard DFA, and the right-hand side panel shows the best-fit model as determined by ML-DFA using AIC.
Figure 7 .
7DFA fluctuation plots when different window lengths are used to analyse the Cz electrode signal of an EEG recording for subject number 7. Panel A shows the DFA and ML-DFA analysis performed for Subject 7 in
Table 3 .
3Results of ML-DFA with the EEG signal obtained from the Cz electrode in 20 healthy subjects.Subject Number
Slope
AIC
BIC
1
0.7861 †
Square Root
Linear
2
0.6204 †
Cube Root
Cube Root
3
0.7798 †
Quadratic
Linear
4
0.7504
Linear
Linear
5
0.8593 † Two-segment spline
Linear
6
0.9231 † Two-segment spline Two-segment spline
7
0.8496
Linear
Linear
8
0.8450
Linear
Linear
9
0.7654
Linear
Linear
10
0.7249
Linear
Linear
11
0.7795
Linear
Linear
12
0.6856
Linear
Linear
13
0.9595 † Two-segment spline
Quadratic
14
0.9093 † Two-segment spline Two-segment spline
15
0.8762 † Two-segment spline
Linear
16
0.8578
Linear
Linear
17
0.7833
Linear
Linear
18
0.7631
Linear
Linear
19
0.7350
Linear
Linear
20
0.9120
Linear
Linear
AcknowledgmentsWe would like to thank Dr Cédric Ginestet for useful discussions, and Dr Leon James and Dr Agnieszka Kempny for allowing us to use their recordings of EEG data, which were made at the Royal Hospital of Neuro-disability, and all EEG subjects for their willing participation. Simon F. Farmer was supported by University College London Hospitals Biomedical Research Centre (BRC). Maria Botcharova thanks the Centre for Mathematics and Physics in the Life Sciences and Experimental Biology (CoMPLEX), University College London for their funding and continuing support.
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| [] |
[
"Cooperative Cognitive Relaying Protocol for an Energy Harvesting Cognitive Radio User",
"Cooperative Cognitive Relaying Protocol for an Energy Harvesting Cognitive Radio User",
"Cooperative Cognitive Relaying Protocol for an Energy Harvesting Cognitive Radio User",
"Cooperative Cognitive Relaying Protocol for an Energy Harvesting Cognitive Radio User"
] | [
"Ahmed El \nWireless Intelligent Networks Center (WINC)\nNile University\nGizaEgypt\n",
"Shafie † ",
"Tamer Khattab \nElectrical Engineering\nQatar University\nDohaQatar\n",
"Ahmed El \nWireless Intelligent Networks Center (WINC)\nNile University\nGizaEgypt\n",
"Shafie † ",
"Tamer Khattab \nElectrical Engineering\nQatar University\nDohaQatar\n"
] | [
"Wireless Intelligent Networks Center (WINC)\nNile University\nGizaEgypt",
"Electrical Engineering\nQatar University\nDohaQatar",
"Wireless Intelligent Networks Center (WINC)\nNile University\nGizaEgypt",
"Electrical Engineering\nQatar University\nDohaQatar"
] | [] | In this paper, we investigate the maximum secondary throughput of a saturated rechargeable secondary terminal sharing the spectrum with a primary terminal. The secondary transmitter (ST) harvests energy packets from the environment with a certain harvesting rate. All transmitters are assumed to have data buffers. In addition to its own traffic buffer, the ST has a buffer for storing the admitted primary packets for relaying; and a buffer for storing the energy packets harvested from the environment. We propose a new cooperative cognitive relaying protocol that allows the ST to relay a fraction of the undelivered primary packets. We consider an interference channel model (or a multipacket reception (MPR) channel model), where concurrent transmissions can survive with certain probability characterized by the complement of channel outages. The proposed protocol exploits the primary queue burstiness and receivers' MPR capabilities. In addition, it efficiently expends the secondary energy packets. We derive formulas of misdetection and false alarm probabilities for the proposed cognitive setting. Our numerical results show the benefits of the cooperation, the receivers' MPR capabilities, and the secondary energy queue arrival rate on the system performance from network layer standpoint.Index Terms-Cognitive radio, relaying, protocol design, cooperation, throughput analysis, queue stability. | 10.1109/pimrc.2014.7136325 | [
"https://arxiv.org/pdf/1401.3387v3.pdf"
] | 11,131,347 | 1401.3387 | 0ac76eb8480065cfeac6bc6382284ccf2e9f68ca |
Cooperative Cognitive Relaying Protocol for an Energy Harvesting Cognitive Radio User
14 Jan 2014
Ahmed El
Wireless Intelligent Networks Center (WINC)
Nile University
GizaEgypt
Shafie †
Tamer Khattab
Electrical Engineering
Qatar University
DohaQatar
Cooperative Cognitive Relaying Protocol for an Energy Harvesting Cognitive Radio User
14 Jan 20141Index Terms-Cognitive radiorelayingprotocol designcoop- erationthroughput analysisqueue stability
In this paper, we investigate the maximum secondary throughput of a saturated rechargeable secondary terminal sharing the spectrum with a primary terminal. The secondary transmitter (ST) harvests energy packets from the environment with a certain harvesting rate. All transmitters are assumed to have data buffers. In addition to its own traffic buffer, the ST has a buffer for storing the admitted primary packets for relaying; and a buffer for storing the energy packets harvested from the environment. We propose a new cooperative cognitive relaying protocol that allows the ST to relay a fraction of the undelivered primary packets. We consider an interference channel model (or a multipacket reception (MPR) channel model), where concurrent transmissions can survive with certain probability characterized by the complement of channel outages. The proposed protocol exploits the primary queue burstiness and receivers' MPR capabilities. In addition, it efficiently expends the secondary energy packets. We derive formulas of misdetection and false alarm probabilities for the proposed cognitive setting. Our numerical results show the benefits of the cooperation, the receivers' MPR capabilities, and the secondary energy queue arrival rate on the system performance from network layer standpoint.Index Terms-Cognitive radio, relaying, protocol design, cooperation, throughput analysis, queue stability.
I. INTRODUCTION
Secondary utilization of a licensed system can efficiently enhance the electromagnetic spectrum usage. Secondary users (SUs) can use the spectrum under certain quality of service for the primary users (PUs). High performance wireless communication networks relies, among other technologies, on cooperative communications.
In many practical situations and applications, the secondary transmitter (ST) is a rechargeable device. The secondary operation, which involves spectrum sensing and access, is accompanied by energy consumption. Consequently, an energyaware ST must optimize its sensing and access decisions to efficiently invest the available energy.
Energy harvesting technology is an emerging technology for energy-constrained terminals which allows the transmitter to collect (harvest) energy from its environment. For a comprehensive overview of the different energy harvesting technologies, the reader is referred to [1] and the references therein.
Data transmission by an energy harvester with a rechargeable battery has got a lot of attention recently [2]- [10]. In [2], the optimal online policy for controlling admissions into the data buffer is derived using a dynamic programming framework. In [3], energy management policies which stabilize the data queue are proposed for single-user communication and some delay-optimal properties are derived. In [4], the optimality of a variant of the back-pressure algorithm using energy queues is shown.
In a cognitive setting, the authors of [5] considered the scenario two different priority nodes share a common channel. The higher priority user (PU) has a rechargeable battery, whereas the lower priority user (SU) is plugged to a reliable power supply and therefore has energy each time slot without limitations. In [6], the authors investigated a cognitive setting with one PU and one rechargeable SU. The ST randomly accesses and senses the primary channel and can possibly leverage primary feedback. Receivers are capable of decoding under interference as they have multipacket reception (MPR) capabilities. The authors investigated the maximum secondary throughput under stability and delay constraints on the primary queue. In [7], the ST randomly accesses the channel at the beginning of the time slot to exploit the MPR capability of receivers. The ST aims at maximizing its throughput under stability and queueing delay constraints on the primary queue. In [8], El Shafie et al. investigated the maximum stable throughput of an energy harvesting ST under stability of an energy harvesting primary transmitter (PT). The SU selects a sensing duration each time slot from a predefined set such that its stable throughput is maximized under the stability of the primary queue.
Cooperative cognitive relaying, where cooperation among primary and secondary nodes, has got extensive attention recently [9]- [14]. In [11], Sadek et al. proposed cognitive protocols for a multiple access system with a single relay aids the transmitting nodes transmission. The proposed cooperative protocols enable the relaying node to aid the transmitters operating in a time-division multiple access network in their silent periods due to source burstiness. The secondary throughput of the proposed protocol as well as the delay of symmetric nodes were investigated. The authors of [12] investigated a network composing of one primary transmitterreceiver pair and one secondary transmitter-receiver pair. The cognitive radio transmitter aims at maximizing its throughput via optimizing its transmit power such that the primary and the relaying queues are kept stable. In [13], the authors considered a cognitive setting with one PU and one SU. The relaying process is managed using an admitting coefficient which controls the flow of arrivals to the relaying queue. A priority of transmission is given to the relaying packets over secondary own packets. The authors of [14] proposed a cluster of SUs helping the PT with a single relaying queue accessible by all the SUs.
Emerging cooperative communications and energy harvesting technologies has been considered in [9], [10]. In [9], the authors investigate the effects of network layer cooperation in a wireless three-node network with energy harvesting nodes and bursty data traffic. The authors derived the maximum stable throughput of the source as well as the required transmitted power for both a non-cooperative and an orthogonal decodeand-forward cooperative schemes. In [10], the authors studied the impact of the energy queue on the maximum stable throughput of an energy harvesting SU utilizes the spectrum whenever the primary queue being empty and capable of relaying the undelivered primary packets. The authors assumed an energy packet consumption in data decoding and data transmission. Inner and outer bounds on the secondary throughput were proposed.
In this work, we investigate the maximum secondary throughput for an energy harvesting ST in presence of a PT. In contrast to [9], [10], in this paper, we consider a generalized MPR channel model and propose a new cooperative protocol which exploit the MPR capabilities of the receivers. In addition, we assume a finite capacity energy queue. In the proposed cooperative cognitive relaying protocol, the ST cooperatively relays a certain fraction of undelivered primary packets when primary destination fails in decoding them. The flow of the primary packet at the ST is controlled using some designable parameters which depend on the channels quality and other queues state. The proposed cooperative cognitive relaying protocol allows the ST to transmit simultaneously with the PT at a fraction of the time slots to exploit the MPR capabilities of the receiving nodes. If the ST has energy packets, it may access the channel at the beginning of the time slot or decides to receive the primary packet. After τ seconds relative to the beginning of the time slot, which we refer to as decision duration, the ST uses the gathered primary samples to declare the state of the PT based on their energy and then the ST decides whether to resume decoding the primary packet or access the channel. The decision duration is designed to render the detection errors probabilities of the PT negligible. If the PT is inactive, the ST accesses with probability one, whereas if the PT is active, the ST may access simultaneously with the PT with a certain probability. If in a given time slot the secondary energy queue is empty at the beginning of the time slot, the operation of the ST becomes a decision on receiving the primary packet or not. In all cases (with and without the availability of secondary energy packets), when the PT is active, if the primary destination fails in decoding it, the ST decides at the end of the time slot whether to accept or reject the correctly decoded primary packet. The proposed protocol is simple and doesn't require continuous estimation of the channel state information (CSI) at the transmitting terminals.
The contributions of this paper can be summarized as follows:
• We propose a new cooperative cognitive relaying protocol which exploits primary queue burstiness and receivers MPR capabilities and efficiently expends the secondary energy packets.
• We assume a finite buffer energy queue, and investigate the impact of buffer size on the secondary throughput. • We investigate the impact of the MPR capabilities of the receivers and the secondary energy queue on the secondary throughput. To make the characterization of the secondary throughput feasible, we consider three approximated systems: two of them are inner bounds for the original system, whereas the third is an outer bound. This paper is structured as follows: Next we describe the system model adopted in this paper. We explain the proposed cooperative cognitive relaying protocol and provide the analysis of the queues rates and the problem formulation in Section III. In Section IV, we provide some numerical results. The conclusions are drawn in Section V.
II. SYSTEM MODEL We consider a simple configuration comprising of one rechargeable secondary transmitter 's', one secondary destination 'd s ', one primary transmitter 'p' and one primary destination 'd p '. The primary transmitter-receiver pair operates over slotted channels. Time is slotted and a slot is T seconds in length. Each transmitter has an infinite buffer (queue) to store its own incoming fixed-length data packets, denoted by Q ℓ , ℓ ∈ {p, s}. In addition to its own traffic queue, the cognitive user has a finite capacity queue with buffer size of K < ∞ energy packets to store energy harvested from the environment, and an infinite capacity relaying queue to store the accepted primary packets for relaying. Let Q r denote the secondary relaying queue and Q e denote the secondary energy queue with mean arrival rate 0 ≤ λ r ≤ 1 packets/slot and 0 ≤ λ e ≤ 1 energy packets/slot, respectively. The secondary queue is assumed to be saturated (always backlogged). Arrivals at queues Q p and Q e are assumed to be Bernoulli random variables [15], [16] with means λ p ∈ [0, 1] packets/slot and λ e ∈ [0, 1] energy packets/slot, respectively. The arrivals at each queue are assumed to be independent and identically distributed (i.i.d.). The Bernoulli model is simple, but it captures the random availability of ambient energy sources. More importantly, in the analysis of discrete-time queues, Bernoulli arrivals see time averages (BASTA), which is equivalent to the Poisson arrivals see time averages (PASTA) property in continuoustime systems [7]. Arrivals are also independent from queue to queue. All data packets are of size B bits. The energy queue has energy packets each of e energy units. The primary and secondary queues and links are shown in Fig. 2. For similar assumptions of infinite size of data buffers and modeling the arrivals of data and energy queues as Bernoulli arrivals, the reader is referred to [5]- [10] and the references therein.
The proposed cooperative cognitive relaying protocol and the theoretical development in this work can be readily generalized to networks with more than one PU and more than one cognitive radio user, where several PUs may choose one or more cognitive radio users or the best cognitive radio user for cooperation. 1 All wireless links exhibit a stationary non-selective Rayleigh block fading, which means that the instantaneous gain of link j → k, connecting nodes j and k, remains constant during a given time slot T ∈ {1, 2, 3, . . . , } with valueζ T jk , but changes independently from one slot to another according to a circularly symmetric complex Gaussian distribution with zero mean and variance σ jk . Received signals are corrupted by additive white Gaussian noise (AWGN) with zero mean with variance N • Watts. Hereinafter, we omit the time superscript of the symbols. Let ζ jk = |ζ jk | 2 denote the fading coefficient of link j → k. We do not assume the availability of CSI at the transmitters. Since the PT transmits at the beginning of the time slot over the whole slot duration if its queue is nonempty, the spectral efficiency of the primary terminal is R p = B/(T W ) bits/sec/Hz, where W is the channel bandwidth. The cognitive radio user may transmit either at the beginning of the time slot or after τ seconds relative to the beginning of the time slot. Hence, the secondary transmission time is T = B/((T − τ )W ) bits/sec/Hz for i = 0 and i = 1, respectively. Note that the decision duration, τ , should be long enough to justify the perfect detection of the primary state assumption, as discussed in the next Section. The PT transmits data with a fixed power P p Watts, whereas the ST transmits with power P Outage of a link occurs when the instantaneous capacity of that link is lower than the transmitted spectral efficiency rate [6], [7], [9].
Assume that node j transmits a packet to node k and at the same time node ν transmits to its respective receiver. Due to the broadcast nature of the wireless communication channel, the signal transmitted by node ν arrives at node k and causes interference with the signal transmitted by node j. Let us assume that node j starts transmission at t = iτ , whereas node ν starts transmission at t = nτ , where i, n ∈ {0, 1}. Under this setting, the probability that a transmitted packet by node j being successfully received at node k is P being successfully decoded at k is P jk,i . The physical layer is explained with details in Appendix A.
At the far end of each time slot, a feedback acknowledgement/negative-acknowledgement (ACK/NACK) signal is sent from the destination to inform the respective transmitter about the decodability status of its packet. The feedback message is overheard by all nodes in the network due to the wireless channel broadcast nature. Decoding error of the feedback messages at the transmitters is negligible, which is reasonable for short length packets as low rate and strong codes can be employed in the feedback channel [11], [15]. If a packet is received correctly at its destination, it is then removed from the respective transmitter's queue.
For the primary packets, if the primary destination can decode the transmitted packet, it sends an ACK and both the PT and the ST drops that packet. If the ST can decode the packet and the packet is being admitted (accepted) for relaying while the primary destination cannot, the ST sends an ACK and the PT drops that packet. If the ST cannot decode or decides to reject the primary packet and the primary destination fails in decoding the packet, the PT retransmits that packet at the following time slot. We note that the feedback signals sent by the ST and the primary destination are separated either in time or frequency.
A fundamental performance measure of a communication network is the stability of the queues. Stability can be defined rigorously as follows: For every queue represented by an irreducible and aperiodic Markov chain with countable number of states, the chain and its associated queue are called stable if and only if there is a positive probability for the queue, represented by the chain, to become empty. Denote by Q (t) the length of queue Q at the beginning of time slot t. Queue Q is said to be stable if lim κ→∞ lim t→∞ Pr{Q (t) < κ} = 1 [11], where Pr{.} denotes the probability of the argument. In a multiqueue system, the system is stable when all queues are stable. We can apply the following theorem to check the stability of a queue [11]. Loynes theorem: if the arrival process and the service process of a queue are strictly stationary, and the average arrival rate is greater than the average service rate of the queue, then the queue is unstable. If the average arrival rate is lower than the average service rate, then the queue is stable [11].
III. PROPOSED COOPERATIVE COGNITIVE RELAYING
PROTOCOL
In this section, we analyze the proposed cooperative cognitive relaying protocol, denoted by S. The time slot structure is shown in Fig. 1. The operation of the ST can be summarized as follows. At the beginning of the time slot, if the secondary energy queue is nonempty, the ST may decide to receive the primary packet with probability f or decide to access the channel using one of its queues with probability f , where φ = 1−φ. Accessing the channel at the beginning of the time slot is motivated by the following facts. First of all, it may be the case that using the whole time in data transmission provides better throughput than wasting τ seconds for channel sensing, specially at low primary arrival rate as the PT will be inactive most of the time slots. Recall that the probability of outage of certain link decreases with the total time used in data transmission. This fact is discussed and proved in Appendix A. Secondly, the presence of MPR capability at the receiving nodes allows packets decoding under interference with nonzero probability, which can be exploited by the ST to boost its throughput. Thirdly, as will be explained in details later, due to the fixed energy transmission property of the energy harvesting ST, secondary delays of channel access may increase the interference at the primary destination due to increasing of the secondary transmitted power, which in turn reduces the probability of successful decoding of the primary packets at the primary destination. Based on these observations, channel accessing at the beginning of the time slot may be useful for certain scenarios and under specific system and channels parameters. If the ST decides to receive the primary packet in a time slot, it takes another action/decision after τ seconds of primary packet reception. The decision duration τ is designed such that the probability of detecting the state of the primary activity is one. This is important for designing an efficient access protocol on the basis of the actual state of the time slot, i.e., busy/free.
Based on the gathered samples of the primary transmission, the ST perfectly detects the state of the PT. 2 If the PT is active, the ST decides whether to resume primary packet reception, which occurs with probability ω; or to access the channel concurrently with the PU using one of its queues, which occurs with probability ω. In this case, accessing the channel simultaneously with the PT is motivated by the presence of the MPR capabilities at receivers. If the PT is inactive and the secondary energy queue is nonempty, the ST accesses with probability 1 using one of its queues.
If at the beginning of the time slot the ST has no energy packets in its energy queue, it decides whether to receive the primary packet, which occurs with probability α, or not. Note that since there is no energy in the secondary energy queue, there is no need to take another decision at t = τ seconds. This is because the ST is incapable of establishing any data transmission due to the lack of energy. In such cases, the probability of receiving the primary packet is α, whereas the probability of remaining silent till the end of the current time slot is α. We would like to emphasize here the importance of having different parameters associated with the different state of the queues in the system. Having such parameters enhance the system performance and help in achieving the optimal performance of the network under investigation.
At the far end of the time slot, the ST decides, on the basis of its ability to decode the primary packet and the status of primary packet decoding at the primary destination, whether to accept or reject the admission of the primary packet to the relaying queue. The acceptance probability of a primary packet is β, whereas the rejection probability is β = 1−β.
If the relaying queue is nonempty, the ST selects one of its packets for transmission with probability Γ = 1 − Γ; or selects one of the relaying packets with probability Γ. If the relaying queue is empty, the ST accesses using its own packets with probability 1. The selection probability Γ represents the relative importance of the primary relaying packets and is used for controlling the throughput of the relaying queue. Choosing Γ = 1 gives full priority to the relaying packets over the secondary packets, while Γ = 0 favors the secondary packets (i.e., no selection for the relaying packets). By varying Γ between 0 and 1, we can maximize the secondary throughput under stability of the other queues.
It should be noted that the probability of outage of a certain link depends on the time available for data transmission. Hence, the probability of outage when the ST transmits at the beginning of the time slot is less than the outage probability when it starts the transmission at t = τ . Although using lower transmission time raises the secondary transmitted power, e/(T − τ ), the channel outage raises as well [6], [8] (see Appendix A for proof). We should note that the interference caused by the ST on the PT's transmission increases with delaying in secondary transmission. This happens because the secondary transmit power increases as mentioned earlier. The reader is referred to Appendix A for the proof.
A. Queues Service and Arrival Processes
Let us first consider the packets of the primary queue, Q p . A packet at the head of the primary queue is served in either one of the following events. If the link p → d p is not in outage; or if the link p → d p is in outage, the link p → s is not in outage, and the ST decides to admit the packet to the relaying queue. A successfully received packet by either the primary destination or the ST will be dropped form the primary queue. The mean service rate of the primary queue is then given by µ p = P pdp,0 (Pr{Q e = 0}+f Pr{Q e = 0}ω) +Pr{Q e = 0}(δ pdp,00 f + δ pdp,01 f ω)
+P pdp,0 P ps,0 (αPr{Q e = 0} + f Pr{Q e = 0}ω)β(1)
where δ pdp,00 and δ pdp,01 denote the reduction in P pdp,0 due to concurrent transmission when the ST accesses at t = 0 and t = τ , respectively. The definition and derivation of P jk,i and δ jk,in are provided in Appendix A. It should be pointed out here that without cooperation the maximum mean service rate for the primary queue is P pdp,0 , whereas with cooperation the maximum achievable primary mean service rate is P pdp,0 + P pdp,0 P ps,0 , which attained when the ST sets β = α = f = ω = 1. Thus, the maximum achievable throughput of the PT is increased by P pdp,0 P ps,0 packets per time slot.
A packet from Q s is served if the secondary energy queue is nonempty, the ST decides to access the channel using Q s , and the link s → d s is not in outage. The mean service rate of Q s is given by
µ s = P sds,0 f Pr{Q p = 0, Q e = 0}δ sds,00 +Pr{Q p = 0, Q e = 0}
+δ sds f ωPr{Q p = 0, Q e = 0}δ sds,10
+Pr{Q p = 0, Q e = 0} × ΓPr{Q r = 0} + Pr{Q r = 0} (2) whereδ jk = P jk,1 P jk,0 is defined in Appendix B.
Similarly, the mean service rate of Q r is given by
µ r = P sdp,0 Γ f Pr{Q p = 0,Q e = 0}δ sdp,00 +Pr{Q p = 0,Q e = 0}
+δ sdp f ωPr{Q p = 0, Q e = 0}δ sdp,10
+Pr{Q p = 0, Q e = 0}(3)
The mean arrival rate of the relaying queue is obtained directly from (1). That is,
λ r = P pdp,0 P ps,0 (αPr{Q e = 0} + f Pr{Q e = 0}ω)βPr{Q p = 0}(4)
where Pr{Q p = 0} in (4) means that the arrival of a primary packet at Q r occurs when the primary queue is nonempty. An energy packet is consumed from the secondary energy queue in a time slot if the ST decides to transmit a data packet from one of its queues. The mean service rate of Q e is then given by
µ e =f +Pr{Q p = 0}f ω + f Pr{Q p = 0} = 1−Pr{Q p = 0}f ω(5)
In (5), f means that the ST accesses the channel at t = 0; Pr{Q p = 0}f ω means that the ST decides to access the channel at t = τ seconds, which occurs with probability ω when {Q p = 0}; and f Pr{Q p = 0} means that the ST decides to access the channel after τ seconds with probability one when {Q p = 0}. Relaying the primary packets by the ST may seem to waste the time slots that could be used for its own packets. However, it turns out that the ST is indeed gaining since opportunistic relaying of primary packets results in emptying (servicing) the primary queue faster as the service process of the primary queue increases; in return, more network resources can be utilized for delivering the secondary packets. As a result, all users simultaneously achieve performance gains.
B. Radio Sensing and Design of τ
We note that the choice of τ depends on the required values of misdetection and false alarm probabilities. In this paper, we assume perfect detection of the state of PT's activity. This can be obtained via adjusting the decision duration, τ , such that the detection errors are kept below certain threshold I. The detection problem at slot T ∈ {1, 2, 3, . . . , } (assuming that τ F s is an integer, where F s is the sampling frequency of spectrum sensing [17]) is described as follows:
H 1 : s(k) =ζ ps x(k) + ε(k) H 0 : s(k) = ε(k) (6) T (s) = 1 F s τ Fsτ k =1 |s(k)| 2(7)
where |ζ ps | 2 = ζ ps is channel gain of link p → s, hypotheses H 1 and H 0 denote the cases where the PT is active and inactive, respectively, τ F s is the total number of used samples for primary activity detection, ε is the noise instantaneous value at time slot T, x is the primary transmitted signal at slot T with variance P p , x(k) is thekth sample of the primary transmit signal, s(k) is thekth received sample of the primary signal at the ST and T (.) is the test statistic of the energy detector.
The quality of the sensing process outcome is determined by the probability of detection, P D , and the probability of false alarm, P FA , which are defined as the probabilities that the sensing algorithm (technique) detects a PT under hypotheses H 1 and H 0 , respectively. Obviously, for a good detection algorithm, the probabilities of misdetection (complement of detection) and false alarm should be as low as possible. The lower the probability of misdetection, the better protection the PT receives specially when the MPR capability of the primary receiver is low. Hence, secondary decisions on accessing the channel under the constraint on stability of primary and relaying queues can be more efficient when the ST knows the exact state of the PT. The exact knowledge of the primary state at a slot would, on the average, increase both the primary and the secondary throughput as the primary queue gets emptied faster due to the increasing of its queue service rate. This may also allow more interference-free time slots for the ST to be used for its own packets transmission and the relaying packets transmission as well. In addition, the lower the probability of false alarm, there are more chances for the ST to use almost all the free time slots alone, which provide a better throughput for the secondary queues relative to case of transmissions under interference. From the above, we can conclude that both detection errors probabilities affect the primary and the secondary rates. Hence, designing and controlling such probabilities can make the users exploit the channel resources in a better way.
Using the central limit theorem (CLT), the test statistic T for hypothesis H θ , θ ∈ {0, 1}, can be approximated by Gaussian distributions [17] with parameters
Λ θ = θζ ps P p + N • , σ θ = (θζ ps P p + N • ) 2 F s τ(8)
where Λ θ and σ θ denote the mean and the variance of the Gaussian distribution for the hypothesis H θ , where θ ∈ {0, 1}.
Since ζ ps is Exponentially distributed random variable with parameter 1/σ ps , the probabilities P FA and P D can be written as (for proof see Appendix C)
P D = Pr{T (s) > ǫ|H 1 } = exp( N• σpsPp ) σ ps P p ∞ N• Q( F s τ [ ǫ Z −1]) exp(− Z σ ps P p )dZ(9)P FA = Pr{T (s) > ǫ|H 0 } = Q( F s τ [ ǫ N • − 1])(10)
where exp(.) denotes the exponential function, ǫ is the energy threshold and Q(Y) =
1 √ 2π ∞ Y exp(−z 2 /2)dz is the Q- function.
For a targeted false alarm probability,P FA , the value of the threshold ǫ is given by
ǫ = N • Q −1 (P FA ) √ F s τ + 1(11)
Thus, for a targeted false alarm probability,P FA , the probability of misdetection is given by substituting Eqn. (11) into Eqn. (9). That is,
P MD = 1− 1 σ ps P p exp( 1 σ ps P p N • ) × ∞ N• Q( F s τ N • Q −1 (PFA) √ Fsτ +1 Z −1 ) exp(− Z σ ps P p )dZ(12)
where Q −1 (.) is the inverse of Q-function. The argument of Q-function in (12) can be rewritten as
N • Q −1 (P FA ) Z + F s τ N • Z −1(13)Since Z ≥ N • , N• Z − 1 is definitely a non-positive value. Hence, √ F s τ ( N• Z −1)
is monotonically decreasing with τ . Since the Q-function is decreasing with increasing of the argument, increasing τ decreases the probability of misdetection for a targeted false alarm.
The minimum decision period τ which results in a negligible detection errors probabilities is obtained via solving the following optimization problem:
min . 0<τ <T τ, s.t. P FA (τ ), P MD (τ ) ≤ I(14)
where I is chosen such that the detection probabilities defined as being sufficiently negligible. It is assumed here to be in order of 10 −3 . Substituting (12) into (14) and puttingP FA = I, we get the formula (15) at the top of the following page. Note that as we mentioned earlier, as τ increases, the primary actual state detection probabilities decrease, hence the minimum value of the decision duration, τ , over all feasible values is attained when the inequality of the constraints in (14) and (15) hold to equality. Furthermore, as the primary transmit power increases, P p , Λ 1 increases as well. Consequently, from (54) in Appendix C, the detection probability, P D , increases due to decreasing of the argument of the Q-function. This reduction would make the required P D achievable with lower decision duration τ . The problem in (15) can be solved numerically or via grid search over τ . The optimal τ is taken as the lowest value of 0 < τ < T which satisfies the constraint P D ≥ 1−I for a targetedP FA = I. As discussed above, detection reliability depends on the decision duration, τ . That is, as τ increases, primary detection at a slot becomes more reliable at the expense of reducing the time available for secondary transmission either for transmitting its own packets or retransmitting the primary relaying packets which, in turn, increase the outage probabilities of the links s → sd and s → pd as shown in Appendix A. This is the essence of the sensing-throughput tradeoff in cognitive radio systems [17].
C. Approximated Systems
The service processes of the primary data queue and the secondary energy queue are coupled, i.e., interacting queues. This means that the departure of a packet at any of them depends on the state of the other. Hence, we cannot analyze the system performance or compute the service process of each queue directly. For this reasoning, we study three approximated systems. Two of them are inner bounds for the actual performance of the original system and the third is an outer bound.
In the first approximated system, we assume that the PT will transmit dummy packets when its queue is empty. These packets may interfere with the ST in case of concurrent transmission, but do not contribute the throughput of the PT. The essence of such assumption is to cause a constant interference with the ST to decouple the queues interaction and to render the computation of nodes' service rates feasible. Under such assumption, the probability of the primary queue being empty is zero; that is, Pr{Q p = 0} = 0 and Pr{Q p = 0} = 1. 3 Since the PT is always backlogged (has at least one packet at its queue each time slot), the probability of the ST finds a free time slot is zero. Thus, all time slots that the ST decides to access in are occupied by the a primary transmission. Hence, the service rates of the secondary queues, Q s and Q r , are reduced relative to the original system in which the PT's queue may be emptied in some time slots and the ST can access the channel alone. 4 Accordingly, this system is an inner bound for the original system.
min . 0<τ <T τ, s.t. 1 σ ps P p exp( 1 σ ps P p N • ) ∞ N• Q( F s τ N • Q −1 (I) √ Fsτ +1 Z −1 ) exp(− 1 σ ps P p Z)dZ ≥ 1 − I(15)
In the second approximated system, we assume an energy packet consumption each time slot, which implies that µ e = 1 energy packets per time slot. Under such assumption, the probability of the energy queue being empty is increased relative to the original system. 5 Consequently, the secondary packets get service less frequently. Furthermore, the relaying packets get service in a lower rate, hence the event of primary queue being empty decreases. Thus, the possibility of having a free time slot or an interference-free time slot for the ST is reduced as well. Accordingly, this system is an inner bound for the original system.
In the third approximated system, we assume that the departure of the energy queue is almost zero, or equivalently, the probability of having an energy packet stored in the secondary energy queue in any time slot is one. This system is an outer bound for the original system as the ST will always be able to access the channel for transmitting its own packets or retransmitting the relayed primary packets each time slot, if there is a chance for the SU to access the channel. Hence, all data queues service rates will be increased simultaneously.
1) First Approximated System: Inner Bound:
In this case, denoted by S 1 , the PT is always backlogged. If the ST decides not to access the channel at the beginning of the time slot, it will not access later at t = τ . This is because the PT is always active and wasting τ seconds for knowing the activity state of the PT will not lead to any gains in terms of secondary queues throughput. Therefore, the optimal ω is ω * = 1. Moreover, the decision on accessing the channel or receiving of the primary packet is taken at the early beginning of the time slot, specifically at t = 0. If the secondary energy queue is nonempty, the ST decides to access the channel by one of its queue with probability f or decides to receive the possible primary transmission with probability f . If the secondary energy queue is empty, the ST cannot transmit data and its decision becomes whether to receive of the possible primary transmission with probability α or remain idle with probability 1−α. At the end of the time slot, the ST decides whether to admit the primary packet or to reject it, as explained earlier.
Under the first approximated system, the mean service rate of the energy queue is given by
µ e = 1 − f(16)
Using the results provided in Appendix D (setting µ = µ e = 1−f ), the probability of the energy queue being empty is given by
ν • = η η f f /λ e − η K = f /λ e − 1 f /λ e − η K(17)
where η = f λe f λe . The probability of nonempty energy queue is given by
ν • = 1 − η K f /λ e − η K(18)
Based on this, the relaying queue departure and arrival mean rates are given by
µ r = P sdp,0 Γf ν • δ sdp,0(19)λ r = P pdp,0 P ps,0 (αν • +f ν • )β (20)
The probability of the relaying queue being nonempty is given by 6 Pr{Q r = 0} = π r = λ r µ r (21)
The mean service rate of Q s becomes µ s = P sds,0 f ν • δ sds,00 Γπ r + π r (22)
The primary queue mean service rate is given by
µ p =P pdp,0 (1−ν • f )+δ pdp,00 f ν • +P pdp,0 P ps,0 (αν • +f ν • )β(23)
We note that the queues are not interacting anymore. Hence, we can apply Loynes theorem to check the stability of the queues and obtain the maximum stable throughput based on the first approximated system via solving the following constrained optimization problem.
max .
β,f,α,Γ µ s , s.t. λ r ≤ µ r , λ p ≤ µ p(24)
where µ r , λ r , µ s and µ p are in (19), (20), (22) and (23), respectively.
For a given f and β, we can get a closed-form expressions for Γ and α, then we solve a family of convex optimization problems parameterized by β and f . Specifically, the optimal solutions of Γ and α are a set of points which satisfies the stability constraint of the primary and relaying queues stability, respectively. Using (23), the optimal α for a fixed f and β is given by
α * ≥ λp−P pdp,0 (1−ν• f )+δ pdp ,00 f ν• P pdp ,0 Pps,0β −f ν • ν • (25)
Using the constraint on the stability of the relaying queue, 6 The expression in (21) is obtained via solving the Markov chain modeling the relaying queue when its arrival and service processes are decouple of the other queue and become computable. This formula is exactly 1 − ν• in Appendix D with the relevant mean service and arrival rates of the relaying queue and with setting the buffer size K to infinity. the optimal Γ is given by
Γ * ≥ P pdp,0 P ps,0 (α * ν • +f ν • )β P sdp,0 f ν • δ sdp,0(26)
where α * is given in (25). The optimal β and f are obtained via grid search and are selected as the pair of parameters that yields the highest objective function in (24). From (26), we note that the optimal selection probability of the relaying queue for transmission, Γ * , increases with increasing the acceptance probability of the primary undelivered packets, β, and the flowing rate to the relaying queue P pdp,0 P ps,0 (α * ν • +f ν • )β. This is because the ST should increase the selection of Q r for transmission to maintain the relaying queue stability. In addition, Γ * increases with decreasing of P sdp,0 f ν • δ sdp,0 . This is because P sdp,0 f ν • δ sdp,0 determines the probability of certain transmitted packet from the relaying queue being correctly received at the primary destination and therefore if this term is high, the ST will not need several transmission for the same packet each time slot. Hence, the ST can reduce the probability of choosing the relaying queue for transmission at a time slot and rather it could use that time slot for the transmission of its own packets.
2) Second Approximated System:Inner Bound:
In this approximated system, denoted by S 2 , we assume that an energy packet is consumed per time slot. That is, µ e = 1 energy packets per time slot. Using the results in Appendix D, if µ = µ e = 1, η = 0 and the probability of the energy queue being empty is given by
ν • = 1−λ e(27)
We can interpret the probability λ e as the fraction of time slots that can be used by the ST for data transmission. It should be pointed out here that the buffer size does not change the state probabilities. Hence, does not have any impact on the queues' rates. The Markov chain modeling the energy queue in this case is composing of two states only: state 0 where the energy queue has no packets, and state 1 where the energy queue has only one packet. The probability of the energy queue having more than one packet, ν k , k ≥ 2, is zero. This case is discussed in Appendix D. The primary throughput is given by µ p = P pdp,0 (λ e + f λ e ω)+λ e (δ pdp,00 f + δ pdp,01 f ω) +P pdp,0 P ps,0 (αλ e + f λ e ω)β
(28)
The probability of the primary queue being nonempty is given by
Pr{Q p = 0} = π p = λ p µ p(29)
The relaying queue mean service and arrival rates are given by µ r = λ e P sdp,0 Γ f π p δ sdp,00 +π p +fδ sdp δ sdp,10 ωπ p +π p (30) λ r = P pdp,0 P ps,0 (αλ e +f λ e ω)βπ p (31)
The mean service rate of Q s is then given by µ s = P sds,0 λ e f π p δ sds,00 +π p +δ sds f ωπ p δ sds,10 +π p × Γπ r +π r
Since the queues are decoupled in the second approximated system, the maximum secondary throughput is given by solving the following problem.
max .
β,f,α,ω,Γ µ s , s.t. λ r ≤ µ r , λ p ≤ µ p (33)
where µ p , µ r , λ r and µ s are in (28), (30), (31), and (32), respectively.
3) Third Approximated System: Outer Bound:
In this case, denoted by S 3 , we consider a backlogged energy queue. This means that there exists at least one energy packet each time slot in Q e . This case can happen if the energy arrival rate is greater than or equal to the energy departure rate or when λ e = 1 regardless of the value of µ e . In this case, the probability of the energy queue being nonempty approaches the unity. The mean service and arrival rates of the queues are then given by µ p = P pdp,0 f ω +(δ pdp,00 f + δ pdp,01 f ω) +P pdp,0 P ps,0 f ωβ,
µ r = P sdp,0 Γ f π p δ sdp,00 +π p + fδ sdp ωπ p δ sdp,10 +π p ,
λ r = P pdp,0 P ps,0 f ωβπ p ,(35)
where π p in (35) and (36) follows (29) with µ p in (34), and µ s = P sds,0 f π p δ sds,00 + π p +fδ sds ωπ p δ sds,10 +π p × Γπ r +π r
where π r follows (21) with µ r and λ r in (35) and (36), respectively. The outer bound can be computed by solving the following problem max .
β,f,ω,Γ µ s , s.t. λ r ≤ µ r , λ p ≤ µ p(38)
with µ p , µ r , λ r and µ s are in (34), (35), (36), and (37), respectively. The optimization problems (24), (33) and (38) are solved numerically at the ST for a given channels and system parameters. Specifically, for a given parameters, the ST solves the optimization problem and use the optimal parameters for the system's operation.
D. Some Important Remarks
Following are some important remarks.
1) First Remark:
Using the results in Appendix A, the complement of outage probability of link p → d p when the ST starts transmission at the beginning of the time slot is given by
P (c) pdp,00 = 1 1+ 2 B W T −1 γ sdp,0 σ sdp γ jk σ jk exp − 2 B W T −1 γ pdp,0 σ pdp (39)
while the probability of that link being not in outage when the ST starts transmission at t = τ is given by
P (c) pdp,01 = 1 1+ 2 B W T −1 γ sdp,1 σ sdp γ jk σ jk exp − 2 B W T −1 γ pdp,0 σ pdp(40)
The ratio of (39) to (40) is given by
ρ = P (c) pdp,01 P (c) pdp,00 = 1 + 2 B W T − 1 γ sdp,0 σ sdp γ pdp ,0 σ pdp 1 + 2 B W T − 1 γ sdp,1 σ sdp γ pdp ,0 σ pdp = 1 + a 1 + a 1−τ /T
We note that γ sdp,1 = γ sdp,0 /(1 − τ /T ) and a = 2
B W T − 1 γ sdp,0 σ sdp γ pdp ,0 σ pdp .
If a ≫ 1, the reduction of the primary packet correct reception probability due to secondary access delay (when the ST accesses at t = τ ) is ρ ≈ 1−τ /T . Therefore, if the secondary decides to access after τ seconds of primary packet reception based on the primary activity, the probability of the primary packet decoding reduces by a factor 1 − τ /T relative to the case when the ST accesses at the beginning of the slot. The reduction of the primary packet correct reception is a linear function of τ . If the decision time, τ , is high, the primary packet decoding will be reduced significantly.
Assume that the primary transmits with a very low power. This makes a much greater than 1. Thus, we can approximate the reduction, due to secondary access delay, of the probability of the primary channel not being in outage by ρ ≈ 1 − τ /T . At the same time, since the primary transmit power is low, the required τ for perfect primary detection is high, as discussed beneath (14). This means that the reduction of the primary packet decoding at the primary destination due to concurrent transmissions is significantly high. In this case, the secondary access probability at t = 0 is definitely higher than the access probability at t = τ when the PT is detected to be active and the ST decides to accesses the channel. Moreover, it may be better for the ST to access the channel at t = 0 to use the whole slot time in data transmission; and at t = τ if the PT is declared to be inactive, if the PT is declared to be active, it may be better to resume receiving the primary packet because concurrent transmission would be harmful for the PT as explained earlier.
2) Second Remark: Assume that the current primary arrival rate is λ p = λ ⋆ p . Increasing the primary arrival rate to λ ⋆ p + ∆ λp , ∆ λp ≥ 0, increases the probability of the primary queue being nonempty. This is because the probability of having an arrival at a certain time slot is increased. Consequently, the number of empty time slots that the ST can detect or access alone decreases as well. In addition, the probability of relaying queue selection, Γ, must be increased to maintain the stability of the relaying queue as the arrival rate of the relaying queue is increased due to the increasing of λ p . These two observations lead to the fact that the achievable secondary rate is increased relative to the case of λ p = λ ⋆ p . This means that the secondary service rate, µ s , is a non-increasing function of the primary arrival rate λ p .
3) Third Remark: From the expressions of the service rates of the queues, the service processes are functions of channel outages probabilities. Based on the formulas of the channel outage in Appendix A, the outage probability of a certain link is a decreasing function of R p = B/(T W ). Therefore, increasing the targeted primary spectral efficiency rate, R p , decreases all queues service rates. This leads to a reduction in the maximum achievable secondary throughput, µ s . This means that the secondary service rate, µ s , is a nonincreasing function of the primary targeted spectral efficiency rate R p = B/(T W ).
The following proposition summarize the main observations in the second and the third remarks.
Proposition 1: For a given channel and system parameters, let µ * s (λ p , R p ) be the maximum secondary throughput at the pair (λ p , R p ). The optimal secondary throughout satisfies the following properties:
• µ * s (λ p , R p ) ≥ µ * s (λ p + ∆ λp , R p ), ∆ λp ≥ 0. • µ * s (λ p , R p ) ≥ µ * s (λ p , R p + ∆ Rp ), ∆ Rp ≥ 0.
IV. NUMERICAL RESULTS
In this section, we provide some numerical results for the optimization problems presented in this paper. The decision duration τ is chosen such that P MD , P FA ≤ I = 10 −3 . We define here the conventional scheme, denoted by S c , where the ST senses the channel for τ seconds and if the primary data queue and the secondary energy queue are simultaneously empty and nonempty, respectively, the ST accesses with probability 1 using one of its queues probabilistically if the relaying queue is nonempty. In addition, if the PT is transmitting a packet to its destination, the ST accepts with probability one to relay and admit the transmitted packet if the primary destination fails in decoding that packet. The secondary throughput of the conventional system is obviously a subset of the proposed cooperative system, S, and can be obtained from S via setting β = 1, α = 1, f = 1 and ω = 0. The other parameters are optimized over their domain to achieve the maximum secondary throughput.
Figs. 3 and 4 represent the maximum secondary throughput of the approximated systems of system S. The figures are generated using the following common parameters: P sd p , 0 = 0.8, δ sdp,00 = 0.3, P sds,0 = 0.7, P ps,0 = 0.8, δ sds,00 = 0.3, K = 60, P pdp,0 = 0, P (c) pdp,00 = P (c) pdp,01 = 0,δ sdp = 0.7, δ sds = 0.7, δ sdp,10 = 0.2, δ sds,10 = 0.2. In Fig. 3, the maximum secondary throughput under the approximated systems is plotted versus λ p . This figure is plotted with λ e = 0.9 energy packets per time slot. The figure shows that the second approximated system provides throughput higher than the first approximated system, hence the union, which represents an inner bound on the actual performance of system S, is the second approximated system. The outer bound which represents the case of backlogged energy queue is close to the inner bound. Fig. 4 reveals two important observations. First, the figure reveals the impact of the arrival rate of the secondary energy queue on the system's inner bound. Precisely, as the energy arrival rate increases, the inner and the outer bound become close to each other and they overlap for all λ p when λ e = 1 energy packets/slot. Second, the figure reveals that the inner bound of the proposed system can outperform the outer bound of the conventional cooperation protocol with reliable energy source plugged to the ST, system S c is plotted with λ e = 1 energy packets per time slot (outer bound on S c ).
We note that for Figs. 3 and 4, without cooperation the primary packets outage probability is 1 − P pdp,0 = 1 which implies that the probability of a primary packet being served at an arbitrary time slot is zero. Hence, the primary queue is always backlogged and will never be emptied. On the other hand, with cooperation the maximum feasible primary arrival rate is 0.3 packets per time slot. Fig. 5 demonstrates the impact of buffer size, K, on the first approximated system performance. As expected, increasing the buffer size boosts the secondary throughput. This is because when the buffer size is high, the ST can store more packets to be used in future for transmitting its own packets or retransmitting the primary packets. The case of finite buffer capacity is obviously a subset of the case of infinite or large buffer capacity. The figure is generated with λ e = 0.8 energy packets per time slot, P sd p , 0 = 0.5, δ sdp,00 = 0.5, P sds,0 = 0.5, P ps,0 = 0.8, δ sds,00 = 0.1, P pdp,0 = 0, P (c) pdp,0 = 0, δ sdp = 0.1,δ sds = 0.1, δ sdp,10 = 0.1, δ sds,10 = 0.1.
Finally, the impact of MPR capabilities is shown in Fig. 6. The figure reveals the gains of the MPR capability on achieving higher throughput for both users. The parameters are chosen to be: λ e = 0.8, P sd p , 0 = 0.8, P sds,0 = 0.7, P ps,0 = 0.8, P pdp,0 = 0.6,δ sdp = 0.5,δ sds = 0.5, and P (c) pdp,00 /P pdp,0 = δ sds,00 = δ sdp,00 = δ sds,10 = δ sdp,10 = X , which represents the MPR strength. At strong MPR, we can achieve orthogonal channels for terminals over most λ p range. The plot also shows that the inner and the outer bounds coincide for high λ p . This happens because the energy queue is backlogged under the used parameters.
From the figures, it is noted that cooperation boosts both primary and secondary throughput. Furthermore, the energy arrival rate increases the probabilities of the secondary packets and the relayed primary packets being served which, in turn, boost both primary and secondary throughput. The figures also show that the increasing of λ p decreases the maximum achievable secondary throughput.
V. CONCLUSION In this paper, we have proposed a new cooperative cognitive relaying protocol, where the ST relays some of the undelivered primary packets. We have considered a generalized MPR channel model, and investigated the impact of the receivers' MPR capabilities on the users throughout. We also have investigated the impact of the secondary energy queue on the system performance. We have provided two inner bounds and an outer bound on the secondary throughput, and showed that the bounds are coincide when the secondary energy queue is always backlogged. The proposed protocol is designed such that the ST exploits the MPR capability and manages its energy packets to maximize its throughput under stability of the primary and the relaying queues. We have derived formulas of misdetection and false alarm probabilities for the giving setting under fading channels. We also designed the decision duration τ such that the detection errors probabilities are kept negligible.
A possible extension of this work can be directed to span the case of having an SU equipped with multiple antennas and with the availability of CSIs at the transmitting antennas to achieve the maximum rates for the queues.
APPENDIX A
We derive here a generic expression for the outage probability at the receiver of transmitter j (node k) when there is concurrent transmission from the transmitter v. Assume that node j starts transmission at iτ and node j starts transmission at nτ . Outage occurs when the spectral efficiency R
(i) j = B W T (i) j exceeds the channel capacity P (c) jk,in = Pr R (i) j > log 2 1 + γ jk,i ζ jk γ vk,n ζ vk + 1(41)
where the superscript c denotes concurrent transmission, Pr{.} denotes the probability of the argument, γ jk,i = P
(i) j /N k , P (i) j
is the used transmit power by node j when it starts transmission at t = iτ , γ vk,n = P
γ jk,i ζ jk γ vk,n ζ vk + 1 < 2 R (i) j − 1(42)
Since ζ jk and ζ vk are independent and exponentially distributed (Rayleigh fading channel gains) with means σ jk and σ vk , respectively, we can use the probability density functions of these two random variables to obtain the outage as
P (c) jk,in = 1 − 1 1 + 2 R (i) j − 1 γ vk,n σ vk γ jk,i σ jk exp − 2 R (i) j − 1 γ jk,i σ jk(43)
We note that from the outage probability (43), the numerator is increasing function of R (i) j and the denominator is a decreasing function of R
jk,i = 1 − P (c)
jk,i is thus given by
P (c) jk,in = P jk,i 1 + 2 B T W ( 1− iτ T ) − 1 γ vk,n σ vk γ jk,i σ jk = δ jk,in P jk,i (44) where P jk,i = exp − 2 R (i) j −1 γ jk,i σ jk
is the probability of correct packet reception when node j transmits alone (without interference) and δ jk,in ≤ 1. As is obvious, the probability of correct reception is lowered in the case of interference. Based on (44), we note that
P (c) jk,in P (c) jk,im = δ jk,in δ jk,im(45)
Following are some important notes. First, note that if the PT's queue is nonempty, the PT transmits the packet at the head of its queue at the beginning of the time slot (at t = 0) with a fixed transmit power P p and data transmission time T p = T . Accordingly, the superscript i which represents the instant that a transmitting node starts transmission in is removed in case of PU.
Second, for the ST, the formula of probability of complement outage of link s → k when the PT is active is given by
P (c) sk,i0 = exp − 2 B T W ( 1− iτ T ) −1 γ sk,i σ sk 1 + 2 B T W ( 1− iτ T ) − 1 γ pk,0 σ pk γ sk,i σ sk(46)
where n = 0 because the PT always transmits at t = 0 and γ sk,i = e/(T (1 − iτ /T )) = γ sk,0 /(1 − iτ /T ). The denominator of (46) is proportional to 2 B T W ( 1− iτ T ) − 1 (1 − i τ T ), which in turn monotonically decreasing with iτ . Using the first derivative with respect to iτ , the numerator of (46),
P sk,i = exp − 2 B T W ( 1− iτ T ) −1 e T (1−i τ T ) σ sk
, can be easily shown to be decreasing with iτ as in [6], [8]. Since the numerator of (46) is monotonically decreasing with iτ and the denominator is monotonically increasing with i, P
sk,i0 is monotonically decreasing with iτ . Therefore, the delay in the secondary access causes reduction in the probabilities of the secondary packets correct reception and the primary relayed packets correct reception at their destinations.
APPENDIX C
In this Appendix, we prove the misdetection and false alarm probabilities provided in (9) and (10). Using the mean and variance of each hypothesis provided in (8), we have
Λ 0 = N • , σ 0 = N 2 • F s τ = Λ 2 0 F s τ , Λ 1 = ζ ps P p + N • , σ 1 = (ζ ps P p + N • ) 2 F s τ = Λ 2 1 F s τ(51)
Given hypothesis H 1 , a correct detection of the primary activity occurs when the test function, T (s), exceeds the threshold ǫ. That is,
P D = Pr{T (s) > ǫ|H 1 } = ∞ 0 1 2π Λ 2 1 Fsτ ∞ ǫ exp(−(F s τ) (z − Λ 1 ) 2 2Λ 2 1 )dz G(ζ ps )dζ ps(52)
where G(ζ ps ) is the probability density function of ζ ps , which is Exponential with parameter 1/σ ps in case Rayleigh fading. Making the change of variableỸ = √ F s τ ( z−Λ1 Λ1 ) and substituting into (52), we get
P D = Pr{T (s) > ǫ|H 1 } = ∞ 0 1 √ 2π ∞ √ Fsτ ( ǫ−Λ 1 Λ 1 ) exp(−Ỹ 2 )dỸ G(ζ ps ) dζ ps(53)
The probability of detection can be rewritten as
P D = Pr{T (s) > ǫ|H 1 } = ∞ 0 Q( F s τ ( ǫ − Λ 1 Λ 1 )) G(ζ ps ) dζ ps (54) where Q(Y) = 1 √ 2π ∞ Y exp(−z 2 /2) dz is the Q-function.
Rewriting (54) in terms of Λ 1 , we get P D = Pr{T (s) > ǫ|H 1 }
= 1 σ ps P p exp( N • σ ps P p ) ∞ N• Q( F s τ[ ǫ Λ 1 −1]) exp(− Λ 1 σ ps P p )dΛ 1(55)
In a similar fashion, using the hypothesis H 0 , the false alarm probability is given by
P FA = Pr{T (s) > ǫ|H 0 } = Q( F s τ [ ǫ N • − 1])(56)
APPENDIX D When the arrival and departure of the secondary energy queue become decoupled from all other queues in the network as in the approximated systems, we can construct and solve its Markov chain. The Markov chain is shown in Fig. 7, where the mean arrival rate is λ e and the mean service rate is µ. Solving the state balance equations of the Markov chain modeling the secondary energy queue, it is straightforward to show that the probability that the energy queue has 1 ≤ ϑ ≤ K packets, ν ϑ , is
ν ϑ = ν • 1 µ λ e µ λ e µ ϑ = ν • η ϑ µ , ϑ = 1, 2, . . . , K(57)
where η = λeµ λeµ . Since the sum of all states' probabilities is the unity, K ϑ=0 ν ϑ = 1. The probability of the secondary energy being empty is obtained via solving the following linear equation:
ν • + K ϑ=1 ν ϑ = ν • + ν • K ϑ=1 1 µ η ϑ = 1(58)
After some mathematical manipulations, ν • is given by
ν • = µ λe − 1 µ λe − η K(59)
with λ e < µ. If λ e ≥ µ, the energy queue saturates, i.e., always backlogged. Thus, v • = 0, which boosts the secondary rate. If the buffer size is large, i.e., K → ∞, the probability of the primary energy queue being empty is 1 − λ e /µ. The blocking probability of an arrived packet to the secondary energy queue is equal to the probability that the queue is full, i.e., storing K packets in its buffer. Thus,
P B = ν K = µ λe − 1 µ λe − η K η K µ(60)
The blocking probability represents the amount of energy packets that will be rejected at the energy queue due to storage limitations. As the buffer size, K, increases, the blocking probability vanishes. If µ = 1, η = 0 and the probability that the energy queue having more than one packet is zero. The states probabilities in such case are given by ν 0 = 1 − λ e , ν 1 = λ e , ν ϑ = 0, ϑ = 2, . . . , K
Fig. 1 .
1Time slot structure.
T −iτ , where i = 0 if the ST transmits at t = 0, and i = 1 if the ST transmits at t = τ . The spectral efficiency of the secondary transmission is either R
e/T i Watts, i ∈ {0, 1}. The secondary transmit power is a function of the time instant in which the ST starts data transmission within the time slot.
Fig. 2 .
2,in = 1−P (c) jk,in (see Appendix A for the exact expression). If transmitter j sends its packet alone (without interference) to node k, and starts transmission at t = iτ , the probability of that packet Primary and secondary queues and links. In the figure, the solid lines are the communication channels while the dashed lines are the interference channels. The primary and secondary receivers are denoted by PR and SR, respectively.
Fig. 3 .Fig. 4 .
34The maximum secondary throughput of the approximated systems for each λp. The maximum secondary throughput of the conventional cooperative protocol and the second and third approximated systems for each λp and for different values of λe.
Fig. 5 .Fig. 6 .
56The maximum secondary throughput for each λp and for different values of energy buffer size K. The maximum secondary throughput for each λp and for different values of MPR capabilities.
transmit power by node v when it starts transmission at t = nτ . The outage probability can be written as P (c) jk,in = Pr
APPENDIX B
APPENDIXIn this Appendix, we compute the ratioP
(c)
jk,1n
P jk,0
. Let
δ jk =
P jk,1
P jk,0
(47)
From (44), we have
P jk,1 =
P
(c)
jk,1n
δ jk,1n
(48)
Thus,δ
jk =
P jk,1
P jk,0
=
P
(c)
jk,1n
δ jk,1n P jk,0
(49)
After some mathematical manipulations, the ratio
P
(c)
jk,1n
P jk,0
is
given by
P
(c)
jk,1n
P jk,0
= δ jk,1nδjk
The considered network can be seen as part of a larger network with multiple primary nodes assigned to orthogonal frequency bands or different time slots via employing frequency division multiple-access or time division multiple-access, respectively.
Perfect declaration of the PT due to the fact that τ is designed for negligible detection errors as will be explained and justified later.
This is actually the stochastic dominance approach extensively investigated in the literature, see for example[6],[7],[9],[11],[18],[19].4 Accessing the channel alone (without interference) provides a successful packet decoding at the relevant receiver higher than the case of concurrent transmission as is obvious. The reader is referred to Appendix A for proofs and further details.
This is actually the highest probability for a queue to be empty as the service rate is 1 packet/slot.
Fig. 7. The Markov chain modeling the secondary energy queue when its service rate is independent of the other queues and has a mean service rate 0 ≤ µ ≤ 1.
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| [] |
[
"Fast Approximate Clearance Evaluation for Rovers with Articulated Suspension Systems",
"Fast Approximate Clearance Evaluation for Rovers with Articulated Suspension Systems"
] | [
"Kyohei Otsu ",
"Guillaume Matheroǹ [email protected] ",
"Sourish Ghosh [email protected] ",
"Olivier Toupet ",
"Masahiro Ono ",
"\nJet Propulsion Laboratory\nCalifornia Institute of Technology Pasadena\nEcole Normale Supèrieure de Paris Paris\n91109CAFrance\n",
"\nJet Propulsion Laboratory California Institute of Technology Pasadena\nIndian Institute of Technology Kharagpur\n91109CAIndia\n"
] | [
"Jet Propulsion Laboratory\nCalifornia Institute of Technology Pasadena\nEcole Normale Supèrieure de Paris Paris\n91109CAFrance",
"Jet Propulsion Laboratory California Institute of Technology Pasadena\nIndian Institute of Technology Kharagpur\n91109CAIndia"
] | [] | We present a light-weight body-terrain clearance evaluation algorithm for the automated path planning of NASA's Mars 2020 rover. Extraterrestrial path planning is challenging due to the combination of terrain roughness and severe limitation in computational resources. Path planning on cluttered and/or uneven terrains requires repeated safety checks on all the candidate paths at a small interval. Predicting the future rover state requires simulating the vehicle settling on the terrain, which involves an inverse-kinematics problem with iterative nonlinear optimization under geometric constraints. However, such expensive computation is intractable for slow spacecraft computers, such as RAD750, which is used by the Curiosity Mars rover and upcoming Mars 2020 rover. We propose the Approximate Clearance Evaluation (ACE) algorithm, which obtains conservative bounds on vehicle clearance, attitude, and suspension angles without iterative computation. It obtains those bounds by estimating the lowest and highest heights that each wheel may reach given the underlying terrain, and calculating the worst-case vehicle configuration associated with those extreme wheel heights. The bounds are guaranteed to be conservative, hence ensuring vehicle safety during autonomous navigation. ACE is planned to be used as part of the new onboard path planner of the Mars 2020 rover. This paper describes the algorithm in detail and validates our claim of conservatism and fast computation through experiments. | 10.1002/rob.21892 | [
"https://arxiv.org/pdf/1808.00031v2.pdf"
] | 195,766,936 | 1808.00031 | 5c68561ffcd240f5153d4a1cf0d1e79feea591a1 |
Fast Approximate Clearance Evaluation for Rovers with Articulated Suspension Systems
Kyohei Otsu
Guillaume Matheroǹ [email protected]
Sourish Ghosh [email protected]
Olivier Toupet
Masahiro Ono
Jet Propulsion Laboratory
California Institute of Technology Pasadena
Ecole Normale Supèrieure de Paris Paris
91109CAFrance
Jet Propulsion Laboratory California Institute of Technology Pasadena
Indian Institute of Technology Kharagpur
91109CAIndia
Fast Approximate Clearance Evaluation for Rovers with Articulated Suspension Systems
We present a light-weight body-terrain clearance evaluation algorithm for the automated path planning of NASA's Mars 2020 rover. Extraterrestrial path planning is challenging due to the combination of terrain roughness and severe limitation in computational resources. Path planning on cluttered and/or uneven terrains requires repeated safety checks on all the candidate paths at a small interval. Predicting the future rover state requires simulating the vehicle settling on the terrain, which involves an inverse-kinematics problem with iterative nonlinear optimization under geometric constraints. However, such expensive computation is intractable for slow spacecraft computers, such as RAD750, which is used by the Curiosity Mars rover and upcoming Mars 2020 rover. We propose the Approximate Clearance Evaluation (ACE) algorithm, which obtains conservative bounds on vehicle clearance, attitude, and suspension angles without iterative computation. It obtains those bounds by estimating the lowest and highest heights that each wheel may reach given the underlying terrain, and calculating the worst-case vehicle configuration associated with those extreme wheel heights. The bounds are guaranteed to be conservative, hence ensuring vehicle safety during autonomous navigation. ACE is planned to be used as part of the new onboard path planner of the Mars 2020 rover. This paper describes the algorithm in detail and validates our claim of conservatism and fast computation through experiments.
Introduction
Future planetary missions will require long-distance autonomous traverse on challenging, obstacle-rich terrains. For example, the landing site for the NASA/JPL Mars 2020 (M2020) mission will be the Jezero crater, a 49 [km]-wide crater considered to be an ancient Martian lake produced by the past water-related activities (Goudge et al., 2015). Autonomous driving in this crater is expected to be challenging due to its high rock abundance. The state-of-the-art on-board path planner for Mars rovers called GESTALT (Maimone et al., 2006), which has successfully driven Spirit, Opportunity, and Curiosity rovers (Fig. 1), is known to suffer from high rock density due to its highly conservative design. More specifically, GESTALT frequently fails to find a feasible path through a terrain with 10% Cumulative Fractional Area (CFA) (cumulative fraction of area covered by rocks), where CFA is a commonly used measure of rock abundance on Mars (Golombek et al., 2008). The Jezero crater has significantly higher rock density than any landing sites of previous Mars rover missions, where the CFA is up to 15-20% based on the orbital reconaissance (Golombek et al., 2015). For this reason, a significant improvement in the autonomous driving capability was demanded by the Mars 2020 rover mission.
Conservatism is both a virtue and a limitation for spacecraft software. In general, any on-board algorithms must be conservative by design because no one can go to Mars to fix rovers if something goes wrong. In case of collision check in path planning, for example, false positives (a safe path is incorrectly assessed to be unsafe) are acceptable but false negatives (an unsafe path is incorrectly assessed to be safe) are unacceptable. However, excessive conservatism (i.e., too frequent false positives) results in reduced efficiency (e.g., unnecessarily winding paths) or inability to find solutions. Therefore, we have two adversarial objectives: guaranteing safety and reducing the algorithmic conservatism 1 .
We found a source of excessive conservatism in GESTALT is collision-checking. Most tree-or graph-based path planners, including GESTALT, need to check if each arc (edge) of the path tree or graph is safe by running a collision check at a certain interval (typically tens of cm). The collision check algorithm estimates multiple safety metrics, such as the ground clearance, tilt, and suspension angles, and check if all of them are within pre-specfied safety ranges. In GESTALT, the rover state is simply represented by a point in the 2D space, which represents the geometric center of the 2D footprint of the rover. Roughly speaking, GESTALT expands potentially colliding obstacles by the radius of the rover such that any part of the rover is guaranteed to be safe as long as the center point is outside of the expanded obstacles, as in Fig. 2(a) 2 . Densely populated rocks may block a significant portion of the state space, resulting in a failure of finding a feasible path. In particular, this approach does not allow the rover to straddle over a rock even if it does not hit the belly pan. This approach is safe and computationally simple, but often overly conservative, particularly on a terrain with high rock density or undulation.
The main idea of the proposed approach in this paper, called Approximate Clearance Evaluation (ACE), is to check collision without expanding obstacles. Instead of representing the rover state just by a 2D Figure 2: Conceptual illustration of the collision checking approach in (a) GESTALT, the state-of-the-art Mars rover autonomous navigation algorithm used in Spirit, Opportunity, and Curiosity rovers, and (b) ACE, the proposed approach used in the Mars 2020 rover. The red triangles represent obstacles. GESTALT conservatively expands obstacles by the radius of the rover while ACE assesses the collision in consideration of the wheel footprints. point, ACE considers the wheel footprint and resulting suspension angles to evaluate the safety metrics such as ground clearance and tilt, as illustrated in Fig. 2(b). This approach can significantly mitigate the level of conservatism while still guaranteeing safety. For example, path planning with ACE allows straddling over rocks as long as there is sufficient clearance to the belly pan. However, precisely evaluating these metrics requires solving an iterative kinematics problem, which is not tractable given the very limited computational resources of the planetary rovers. Besides the nonlinear kinematics equations associated with the suspension mechanisms, a rough terrain profile makes it difficult to precisely predict wheel-terrain contact (Sohl and Jain, 2005). There are no known analytic solutions in general, and the problem is typically approached by iterative numerical methods at the cost of computational efficiency. Therefore, we turn to a conservative approximation. That is, instead of running an iterative kinematic computation, ACE computes the worst-case bounds on the metrics. This approach is a practical compromise for our Mars rovers as it has guaranteed conservatism with acceptable computational requirement. This claim will be empirically supported by simulations (Section 4.1) and hardware experiments (Section 4.2). Furthermore, as we will empirically show in Section 4.3, ACE-based path planning is significantly less conservative than GESTALT.
There have been a significant body of works in literature, but none of these were sufficient for our application in terms of speed, path efficiency, and safety guarantee. Most of the motion planning methods for generic ground vehicles do not explicitly consider suspension articulation. In non-planar terrain, it is very common to model the terrain as a 2.5D or 3D map and fit a robot-sized planar patch to terrain models to obtain geometric traversability (Gennery, 1999, Chilian and Hirschmuller, 2009, Ishigami et al., 2013, Wermelinger et al., 2016, Krüsi et al., 2017. It is also common to add other criteria such as roughness and step hazards to capture obstacles and undulations. Similar to GESTALT, those planners will suffer from extremely less path options in a highly cluttered environment such as the surface of Mars. Without reasonable vehicle state prediction, it is difficult to utilize the greater body-ground clearance of off-road vehicles.
To enable more aggressive yet safe planning, pose estimation on uneven terrains has been used together with path planners. Kinematics and dynamics are two major categories which account for the state of articulated models on uneven terrain. With kinematics-based approach, the problem is to find contact points between the wheel and the terrain under the kinematic constraints of the vehicle. Generic kinematics modeling is introduced for articulated rovers such as NASA/JPL's rocker-bogie rovers (Tarokh andMcDermott, 2005, Chang et al., 2006). These kinematic models are used to compute vehicle settling on uneven terrain by minimizing the wheel-ground contact errors (Tarokh and McDermott, 2005,Howard and Kelly, 2007,Ma and Shiller, 2019. The terrain settling technique is used in the current ground operation of Mars rovers to check safety before sending mobility commands (Yen et al., 2004,Wright et al., 2005. The kinematic settling is also effective for other types of vehicles, such as tracked vehicles (Jun et al., 2016). The kinematics approach is generally faster than dynamics-based approach, but still computationally demanding for onboard execution on spacecraft computers. Dynamics simulation is typically performed with general purpose physics engines. Due to its popularity, many works use Open Dynamics Engine (ODE) for simulating robot motion during planning (Papadakis and Pirri, 2012). Rover Analysis Modeling and Simulation (ROAMS) (Jain et al., 2003, Jain et al., 2004 is simulation software developed at Jet Propulsion Laboratory (JPL), which models the full dynamics of flight systems including Mars rovers. ROAMS was used to predict the high-fidelity rover behavior in rough terrain (Huntsberger et al., 2008, Helmick et al., 2009. Another faster dynamics simulator is proposed in (Seegmiller and Kelly, 2016), which runs simulation over 1000 times faster than real time in decent computing environment. Although these methods can provide high-fidelity estimate in clearance, vehicle attitude, and suspension angles, they cannot directly be deployed onto the rovers due to its intractable computational cost. Moreover, for on-board path planning in planetary missions, conservatism in safety is more important than accuracy: a single collision can terminate a mission as it is not possible to repair a damaged vehicle on another planet at least for the foreseeable future.
The main contribution of this paper is to introduce a novel kinematics solution named ACE and its empirical validation based on simulations and hardware experiments. The key concept of ACE is to quickly compute the vehicle configuration bounds, instead of solving the full kinematic rover-terrain settling. Knowing the bounds of certain key states, ACE can effectively produce a conservative estimation of the rover-terrain clearance, rover attitude, and suspension angles in a closed form. ACE is being developed as part of the autonomous surface navigation software of NASA/JPL's M2020 mission. The initial idea of ACE appears in (Otsu, 2016), and a probabilistic extension of this work is reported in (Ghosh et al., 2019). This paper introduces improved mathematical formulation and extensive Verification and Validation (V&V) work.
The remainder of this paper is structured as follows: Section 2 formulates the kinematics models of articulated suspension systems, Section 3 describes the ACE algorithm, Section 4 provides experimental results including benchmarking, and Section 5 concludes the paper.
Suspension Models
Our approach is to use kinematic equations to propagate the bounds on the height of wheels to the bounds on vehicle configuration. While this approach is applicable to many vehicles with articulated suspension systems used in the planetary rover domain, this section particularly focuses on the rocker and rockerbogie suspensions. The latter is the suspension system of choice for the successful NASA/JPL's Mars rover missions (Harrington and Voorhees, 2004).
Frames
We first introduce the reference frames used in the paper, which are illustrated in Fig. 3 and 4. Following the aerospace convention, the forward-right-down coordinate system is employed for the body frame of the rover. The origin is set to the center of middle wheels at the height of ground contact point when the rover is stationary on the flat ground. In this frame, the wheel heights are described in z-axis pointing downward (i.e., A greater wheel "height" indicates that the wheel is moved downward).
A global reference frame is defined as a north-east-down coordinate system. The terrain geometry, which can be specified in any format such as a point cloud or a Digital Elevation Map (DEM), is expressed in an arbitrary frame. The rover path planning is conventionally performed in 2D or 2.5D space based on the nature of rover's mobility systems. A path is typically represented as a collection of poses containing 2D position and heading angle (x, y, ψ). A path is regarded as safe if all poses along the path satisfies the safety constraints.
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Rocker Suspension
The rocker suspension in Fig. 3 (a) is a simpler variant of the rocker-bogie suspension, which will be discussed in the next section. The rocker suspension usually consists of four wheels, where the two wheels on the same side are connected with a rigid rocker link. The left and right suspensions are related through a passive differential mechanism, which transfers a positive angle change on one side as a negative change to the other side.
The kinematic relation of the rocker suspension is represented by a simple triangular geometry in Fig. 5. Consider a triangle ABC with a known shape parameterized by two side lengths and the angle between them (l ca , l ab , ϕ a ). Given the height of A and B (i.e., z a , z b ), there are up to four solutions for z c , but other constraints such as vehicle orientation uniquely specifies a single solution given by:
z c = z a − l ca sin κ(z a , z b )(1)
where κ(·) denotes an angle of link AC with respect to the reference line (i.e., κ c ) and is defined as
κ(z a , z b ) = ϕ a + sin −1 z a − z b l ab .(2)
The solution only exists if |z a − z b | ≤ l ab .
The rocker suspension model is formulated using the triangular geometry in (1) and (2). Given two wheel heights z f and z r , the rocker joint height is given as
z d = z f − l df sin κ d (z f , z r )(3)
where l df is the length of front rocker link and κ d (·) is an instance of (2) with the rocker suspension parameters (l df , l f r , ϕ f ). Due to the differential mechanism, the left and right rocker angles in relative to the body, δ l , δ r , have the same absolute value with the opposite sign. They can be computed from link angles as:
δ l = −δ r = κ d (z fr , z rr ) − κ d (z f l , z r l ) 2 .(4)
The body attitude is a function of left and right rocker joint states. The roll angle of the body is computed from the difference of joint heights:
φ = sin −1 z dr − z d l 2y od ,(5)
where y od is the lateral offset from the center of body to a differential joint. The pitch angle is computed as
θ = κ d0 − κ d (z f l , z r l ) + κ d (z fr , z rr ) 2 (6)
where the first term represents an angle offset of front link when the rover is on a flat ground (κ d0 = ϕ f in this example).
Finally, the body frame height in the global frame can be obtained as
z o = z d l + z dr 2 + x od sin θ cos φ − z od cos θ cos φ(7)
where x od and z od are offsets from the body frame origin to a differential joint. Since the belly pan is rigidly attached to the body frame, the rover-terrain clearance can be derived from these height and attitude information.
Rocker-bogie Suspension
The rocker-bogie suspension ( Fig. 3 (b)) is a rocker suspension with an additional free joint on each side. According to the previous Mars rover conventions, we assume that the front wheels of the six-wheeled rover are connected directly to the rocker suspension while the middle and rear wheels are attached to the bogie suspension. The inverse kinematics of the rocker-bogie suspension can be derived by extending that of the rocker suspension, described in the previous subsection.
We first determine the state of bogie link. The bogie joint heights can be estimated from middle and rear wheel heights (z m , z r )
z b = z m − l bm sin κ b (z m , z r )(8)
where l bm is the length of bogie front link and κ b (·) denotes the triangular geometry for the bogie triangle.
Using the height of bogie joint z b , the rocker joint height can be computed as
z d = z f − l df sin κ d (z f , z b ) .(9)
Given the heights of the wheels and joints, rocker and bogie angle changes are computed as
δ l = −δ r = κ d (z fr , z br ) − κ d (z f l , z b l ) 2 (10) β l = κ d (z f l , z b l ) − κ b (z m l , z r l ) − κ d0 + κ b0 (11) β r = κ d (z fr , z br ) − κ b (z mr , z rr ) − κ d0 + κ b0(12)
where κ d0 and κ b0 denote the initial angles of rocker and bogie joints. The attitude and height of the body are derived as:
φ = sin −1 z dr − z d l 2y od (13) θ = κ d0 − κ d (z f l , z b l ) + κ d (z fr , z br ) 2 (14) z o = z d l + z dr 2 + x od sin θ cos φ − z od cos θ cos φ .(15)
Algorithm
Remember that ACE is designed to quickly compute conservative bounds on vehicle states. Unlike the full kinematics settling that relies on iterative numerical methods, our approach computes the bounds in a closed form. The ACE algorithm is summarized as follows:
1. For a given target rover pose (x, y, ψ), find a rectangular wheel box in x-y plane in the body frame that conservatively includes the footprint of each wheel over any possible rover attitude and suspension angles.
2. Find the minimum and maximum terrain heights in each of the wheel boxes (see Fig. 6).
3. Propagate the bounds on wheel heights to the vehicle configuration with the kinematic formula derived in the previous section.
4. Assess vehicle safety based on the worst-case states.
In 3), all possible combinations are considered to obtain the worst-case bounds. Due to the monotonic nature of suspension, the bounds can be obtained via the evaluation of extreme configurations. For example, the bounds on the rocker/bogie states are obtained by finding the worst cases among the eight extreme combinations of the min/max heights of three wheels, as illustrated in Fig. 7. This propagation process is visually presented in the supplemental video.
To precisely describe the algorithm, we first introduce the interval arithmetic as a mathematical framework in our method. We then describe how we apply it to solve our problem with a case study using the M2020 rover.
Notation
In the interval arithmetic (Hickey et al., 2001), an interval is defined as follows Figure 6: Ranges of possible wheel configurations computed from terrain geometry and mechanical constraints.
[x − , x + ] = {x ∈ R * | x − ≤ x ≤ x + }(16)
where a pair [x − , x + ] represents the all reals between two. The symbol R * denotes an extended real defined as R * = R∪{−∞, ∞}. Elementary arithmetic operations on reals can be extended to intervals, such as
[x − , x + ] + [y − , y + ] = [x − + y − , x + + y + ](17)[x − , x + ] − [y − , y + ] = [x − − y + , x + − y − ].(18)
For a continuous function f (x), we can extend its input and output space to intervals
f ([x − , x + ]) = min x∈[x − ,x + ] f (x), max x∈[x − ,x + ] f (x) .(19)
Computing the minimum and maximum is trivial if the function f is monotonic, or special non-monotonic functions such as trigonometric functions.
In the rest of the paper, we use the following abbreviation to represent an interval unless explicitly stated
[x] ≡ [x − , x + ] .(20)
Wheel Height Intervals
Firstly, we estimate the wheel height intervals based on terrain measurements from sensors (e.g., stereo vision). The span of wheel heights can be computed from the highest and lowest terrain points within the wheel boxes (see green boxes in Fig. 6). The x and y dimensions of the wheel boxes are derived from the vehicle's mechanical properties such as wheel size, suspension constraints, and vehicle tip-over stability. We can determine a conservative range of wheel contact locations for all possible suspension angles and stable attitude.
In the rest of this paper, we represent the bound for i-th wheel by [z i ]. It is important to estimate these bounds conservatively to make the final state bounds to be complete, since the uncertainty in wheel heights is directly propagated to other states. For the conservative estimate, we may need to include the dynamic effect such as terrain deformation, wheel slips and sinkage, depending on the environment to explore. In addition, it is important to consider the effect of perception error as detailed in the experiment section.
(a) (z + f , z + m , z + r ) (b) (z − f , z + m , z + r ) (c) (z + f , z − m , z + r ) (d) (z − f , z − m , z + r ) (e) (z + f , z + m , z − r ) (f) (z − f , z + m , z − r ) (g) (z + f , z − m , z − r ) (h) (z − f , z − m , z − r )
Figure 7: Extreme configurations for state bound computation. Three elements in a tuple represent the front, middle and rear wheel heights, respectively. The superscripts + and − represent the maximum and minimum for the state variable.
Suspension Intervals
Since sin −1 (x) is monotonically increasing in x ∈ [−1, 1], we can extend the concept of intervals to the function κ(·) in (2) κ
([z a ], [z b ]) = [κ(z − a , z + b ), κ(z + a , z − b )].(21)
On the other hand, (1) is a convex function which has a global minimum if z b = z − b and the rear link CB is aligned with z-axis. In case of the M2020 rover, the minimum is located outside of the mechanical limits. Therefore, in practice, we can assume the monotonicity and use the following interval for the height
[z c ] = [z − a − l ab sin κ b (z − a , z − b ), z + a − l ab sin κ b (z + a , z + b )] .(22)
Based on the above intervals, the suspension state intervals are computed for joint heights
[z b ] = [z − m − l bm sin κ b (z − m , z − r ), z + m − l bm sin κ b (z + m , z + r )](23)[z d ] = [z − f − l df sin κ d (z − f , z − b ), z + f − l df sin κ d (z + f , z + b )](24)
and for joint angles
[δ l ] = −[δ r ] = κ d (z − fr , z + br ) − κ d (z + f l , z − b l ) 2 , κ d (z + fr , z − br ) − κ d (z − f l , z + b l ) 2 (25) [β l ] ⊆ κ d (z − f l , z + b l ) − κ b (z + m l , z − r l ) − κ d0 + κ b0 , κ d (z + f l , z − b l ) − κ b (z − m l , z + r l ) − κ d0 + κ b0 (26) [β r ] ⊆ κ d (z − fr , z + br ) − κ b (z + mr , z − rr ) − κ d0 + κ b0 , κ d (z + fr , z − br ) − κ b (z − mr , z + rr ) − κ d0 + κ b0 .(27)
For the sake of simplicity, we use loose bounds for the bogie angles., The boundary configurations may be impossible in reality. In this example, the lower bound of β requires (z − f , z + m , z − r , z + b ) but z + b requires (z + m , z + r ), which is inconsistent in z r (except the case where z − r and z + r are identical).
Attitude Intervals
Similarly, the intervals for body attitude can be derived from wheel height intervals. Using the kinematics equations (13) and (14) yields
[φ] = sin −1 z − dr − z + d l 2y od , sin −1 z + dr − z − d l 2y od(28)[θ] = κ d0 − κ d (z + f l , z − b l ) + κ d (z + fr , z − br ) 2 , κ d0 − κ d (z − f l , z + b l ) + κ d (z − fr , z + br ) 2 .(29)
Clearance Intervals
Since the vehicle body has connection to the world only through its suspension and wheel systems, its configuration is fully determined by the suspension state. The body height bound in the world frame is computed with (15):
[z o ] ⊆ z − d l + z − dr 2
− z od cos |θ| + cos |φ| + + x od min(sin θ − cos |φ| − , sin θ − cos |φ| + ),
z + d l + z + dr 2 − z od cos |θ| − cos |φ| − + x od max(sin θ + cos |φ| − , sin θ + cos |φ| + )(30)
using the intervals of absolute roll/pitch angles [|φ|], [|θ|]. Note that the trigonometric functions in the equations are monotonic since we can assume |φ|, |θ| ∈ 0, π 2 for typical rovers. Assuming the belly pan is represented as a plane with width w p and length l p at nominal ground clearance c 0 , a loose bound for the maximum (lowest) height point in belly pan is computed as
[z p ] ⊆ [z − o − c 0 cos |θ| − cos |φ| − + l p 2 sin |θ| − cos |φ| + + w p 2 sin |φ| − , z + o − c 0 cos |θ| + cos |φ| + + l p 2 sin |θ| + cos |φ| − + w p 2 sin |φ| + ](31)
Let's define the rover-ground clearance as a height gap between the lowest point of the belly pan and the highest point of the ground. This is a conservative definition of clearance. Given the intervals of ground point height under the belly pan [z m ], the clearance is computed as
[c] ≡ [z − m − z + p , z − m − z − p ] .(32)
Safety Metrics
We use the above state intervals to test if a given pose has chance to violate safety conditions. Different safety conditions can be applied to different rovers. For example, the following metrics are considered for the M2020 rover.
• Ground clearance must be greater than the threshold.
• Body tilt (computed from roll and pitch angles) must be smaller than the threshold.
• Suspension angles must stay within the predefined safety ranges.
• Wheel drop (defined as a span of wheel height uncertainty) must be smaller than the threshold.
Experiments
Recall that ACE is designed to be conservative, and at the same time, to reduce the conservatism compared to the state-of-the-art. In Sections 4.1 and 4.2, we will show that ACE is conservative, hence safe, through simulation and hardware experiments. Then, Section 4.3 compares the algorithmic conservatism with the state-of-the-art. Our tests involve rovers with different sizes to observe the performance difference due to mechanical system configurations.
Simulation Study
We first tested ACE with simulation to validate the algorithm in noise-free scenarios. On these tests, terrain topological models are directly loaded from the simulator to ACE. Therefore, the presented results are not contaminated by measurement noise from perception systems. The algorithm is tested on different terrain configurations, from artificial quadratic functions to a realistic Martian terrain model. We use simulatorreported rover state as ground truth, which is originally computed with iterative numeric methods based on rover and terrain models.
Simulation with a single ACE run
We placed a simulated rover on simple geometric terrains and run ACE to compute bounds on the belly pan clearance, the attitude, and the suspension angles. Their ground-truth values were also obtained from In addition, these results were compared against a simple, planefit-based estimation approach. More specifically, we fitted a plane to a given topography over a window with a 1.25 m radius and approximately estimate the rover's pose by assuming that the rover is placed on this plane. The ground clearance was estimated by computing the difference between the highest point of the terrain in the window and the belly pan height based on the estimated rover pose. The planefit-based approximation gives the exact ground clearance when the terrain is flat. We chose plane fitting as the point of comparison because, as we shall see shortly, it provides an insight to the cause of the conservatism of GESTALT, the state-of-the-art autonomous rover navigation algorithm used for the three existing Mars rovers.
We used simple terrains represented by z = ax 2 in the body frame with varying a for this test. Tests with more complex, realistic terrains will follow. The ground-truth settling was obtained via a numeric optimization method. Fig. 8 shows the results. Note that the z-axis points downwards, meaning that the terrain is convex with a negative a and concave with a positive a. The brown solid lines represent ground-truth states, with ACE bounds denoted by orange shaded areas. As expected, ACE bounds always provided conservative estimate. Compared to attitude and suspension angles, the clearance estimation resulted in a greater conservatism in general. This is because the clearance is the last estimated property propagated from terrain heights and hence accumulates uncertainty.
In contrast, the planefit-based approach consistently gave an optimistic estimate of the ground clearance and the pitch angle. In addition, since the rover is always placed on a plane, the bogie angle is always estimated to be zero. GESTALT does not explicitly computes the ground clearance; instead, it computes the "goodness" of each cell on the terrain from multiple factors including roughness (i.e., residual from the planefit) and step obstacles (i.e., maximum height difference between adjacent cells), where the weights on each factor are manually tuned such that the conservatism is guaranteed for the worst cases. The fact that the planefit-based clearance estimation is optimistic for non-zero a implies that the weights on roughness and the step obstacle must be sufficiently great for the worst-case a. This in turn makes the algorithm overly conservative when the terrain is nearly flat (i.e., smaller a), which is the case for most of the time of driving on Mars. In contrast, ACE gives tighter bounds for a smaller a. This illustrates a desirable behavior of ACE; that is, it adjusts the level of conservatism depending on the terrain undulation. It results in the exact estimation on a perfectly flat terrain and increases conservatism for undulating terrains. Overall, the ground truths are always conservatively bounded. We also note that ACE becomes overly conservative on a highly undulating terrain. We believe the impact of this issue in practical Mars rover operation is relatively limited because we typically avoid such terrains when choosing a route. Having said that, even though ACE enables the rover to drive on significantly more difficult terrains than GESTALT, this conservatism is one of the remaining limitations. Mitigating the conservatism of ACE on a highly undulating terrain is our future work.
Simulation with multiple ACE runs
We then drive a rover with a pre-specified path on various terrains in simulation while calling ACE multiple times at a fixed interval to check collisions.
Flat Terrain with a Bump
The test environment is a simple flat terrain with a 0.2 [m] height bump. A Curiosity-sized rover is driven over the bump with three different trajectories shown in Fig. 9. The rover drives at the nominal speed of Curiosity on Mars (∼0.04 [m/s]). We collected data in 8 [Hz], including ground-truth pose from a numeric method. The ranges of six wheel heights are extracted directly from the base map using the ground-truth pose reported by the simulator. Fig. 10 shows the time-series profiles of suspension and body states for three trajectories. The solid lines denote the ground-truth states computed by the numeric method, and the shaded regions represent the state bounds computed by ACE. All the ground-truth states are always within the bounds, meaning ACE bounds are conservative as expected. It is interesting to observe how the algorithm evaluates rover states for its worst-case configurations. With trajectory (a), the rover approaches perpendicularly to the bump. The ground-truth roll angle stays zero for the entire trajectory since the left and right wheels interact with the ground exactly at the same time in this noise-free simulation. However, this is unlikely in the real-world settings where small difference in contact time, or difference in wheel frictions, can disturb the symmetry and cause rolling motion. ACE computes the state bounds based on the worst-case configurations. Therefore, the algorithm captures such potential perturbation and conservatively estimate the state bounds, as presented in the top left figure of Fig. 10.
Martian Terrain Simulation Next, we tested the ACE algorithm in a Mars-like environment. We imported a DEM of Jezero crater into the ROAMS simulator (Jain et al., 2003). Since no spacecrafts have landed on Jezero, we only have a limited resolution of terrain model from satellite measurements. To create an environment closer to the actual, we populate rocks based on the empirically created Martian rock sizedistribution model (Golombek et al., 2008). We populate rocks assuming 10% CFA. For the hardware platform, we used the Rocky 8 rover which is a mid-sized rover similar to Mars Exploration Rovers (MERs). We drove the rover at a speed of 0.15 [m/s] with an autonomous hazard avoidance mode. The data are taken at 10 [Hz] including ground-truth pose reported by the simulator.
The state estimation result is shown in Fig. 11. The figure only reports the body states including roll, pitch, and minimum clearance, but similar results are obtained for the other suspension states. Again, all the ground-truth states are always within the ACE bounds, successfully confirming the algorithmic conservatism. The level of conservatism varies from time to time. For most of the time, the span between upper and lower bounds were within a few degrees. However, at 55 [s] in Fig. 11, for example, the upper bound on the pitch angle was about 10 [deg] while the actual angle is around 1 [deg]. Such false alarms typically occur when a large rock is in one of the wheel boxes but the rover did not actually step on it. This behavior is actually beneficial because it helps the path planner to choose less risky paths if the planner uses the bounds as a part of its cost function. Of course, such conservatism may result in a failure of finding a feasible path. However, we reiterate that conservatism is an objective of ACE because safety is of supreme importance for Mars rovers. Furthermore, as we will demonstrate in subsection 4.3, ACE significantly reduces the conservatism compared to the state-of-the-art. An additional idea that can further mitigate the conservatism is to introduce a probabilistic assessment, as proposed by (Ghosh et al., 2019).
Hardware Experiments
We deployed ACE on actual hardware systems and validated through the field test campaign in the JPL Mars Yard. ACE is deployed on two rover testbeds: the Athena rover whose size is compatible to the MER rovers, and the Scarecrow rover, a mobility testbed for MSL. In both systems, terrain height measurements are obtained by the stereo camera attached to the mast. Therefore, the heightmap that ACE receives involves noise from the camera and stereo processing. As we will report shortly, the stereo noise results in occasional bound violations. Adding an adequate margin to account for the noise restores the conservatism of ACE.
Athena Rover
The first experiment is performed with the Athena rover developed at JPL (see Fig. 12). The platform is designed for testing Mars rover capabilities on Earth and is comparable in size to MER. The navigation system is primarily vision-based using a stereo-camera pair consisting of two PointGrey Flea2 cameras rigidly mounted on a movable mast. The mast is at a height of 1.4 [m], and the baseline of the stereo-camera pair is 0.30 [m]. The images are captured at a resolution of 640 × 480 from wide field-of-view lenses. The groundtruth pose is obtained from OxTS xNAV system, which reports 6DoF pose from integrated GPS and inertial measurements. The suspension angles are not measured on this platform.
We manually drove Athena on a slope of 6 to 12 [deg] in the Mars Yard. The slope consists of multiple terrain materials including cohesive/cohesion-less sands and bedrocks. We evaluated ACE by comparing the estimated state bounds from the algorithm and the ground-truth state. ACE was applied to a past few image sets prior to the driving time. Unlike the rover autonomous drive software that prevents the placement of wheels in unknown terrain, our dataset collected by manual commanding contains samples in which the point clouds do not cover the terrain under all six wheels. We do not report state estimation results for such incomplete data. 13 shows the ground truth, as well as the upper and lower bounds from ACE, of the roll and pitch angles and the ground clearance for two drives. Each run consists of about 40 [m] traverse including level, uphill, downhill, and cross-slope drives. As expected, the ground truth is within the bounds for most of the time. However, unlike the simulation results reported in the previous subsection, occasional bound violations were observed, as shown by the red crosses on the plots. This was due to perception errors, such as stereo matching error and calibration error. The positional error in point clouds from the stereo camera is propagated to the rover states through the kinematic equations, causing the error in state bounds. A practical approach to restore the conservatism is to add a small margin to the perceived height of the ground. More specifically, the maximum and minimum height of each wheel box, z + w and z − w (w ∈ {f, m, r}) in (21)-(32), are replaced with z + w + and z − w − , where is the estimated upper bound of the height error. A downside of this approach is an increased conservatism. Although it is rare that these small estimation error contributes to the hazard detection miss which is critical to the mission, extra conservatism is preferred for planetary applications. The conservatism is fully restored (i.e., 100% success rate) with = 15 [mm], which roughly corresponds with the worst-case height perception error.
Scarecrow Rover
We deployed ACE on JPL's mobility testbed called Scarecrow and performed a series of experiments in JPL's Mars Yard. The purpose of the experiments is to test ACE with hardware and software that is close to the Mars 2020 rover. The mobility hardware of Scarecrow, including the rocker-bogie suspension system and wheels, are designed to be nearly identical to that of Curiosity and Mars 2020 rovers. The vehicle's mass is about one third of Curiosity and Mars 2020 rovers, simulating their weight under the Martian gravity. In terms of software, ACE is re-implemented in C and integrated with the Mars 2020 flight software. Since Scarecrow was originally designed for mobility experiments, it does not have the identical processor as the real Mars rovers. Instead, we compiled the software for Linux and run on a laptop computer placed on the top deck of the vehicle. Therefore, this experiment does not replicate the run time of the software. We evaluated the run time of ACE in a hardware-in-the-loop simulation using RAD750, as described in Section 4.2.3. The original design of Scarecrow also lacks cameras. Therefore, we retrofitted a pair of Baumer cameras, from which height map is created on-the-fly via on-board stereo processing. The Mars Yard is configured in a way to represent some of the most difficult conditions in the Mars 2020 landing site, including 30 degree of slope and 15% CFA (Golombek et al., 2008) of rock density. Fig. 14 shows a typical set up of the Mars Yard.
Our extensive test campaign consisted of 42 days of experiments in the Mars Yard using Scarecrow. The analysis of the test results were largely qualitative rather than quantitative or statistical for a few reasons. First, we are unable to keep the exactly same set up of the Mars Yard as it is shared by many teams. It is also slightly altered by precipitation and wind. Second, the driving speed of Scarecrow is only 0.04 [m/s] (same as Curiosity and Mars 2020 rover), therefore it typically takes 20 to 30 minutes to complete a single Mars Yard run. Repeating a statistically significant number of runs with the same set up is difficult. Third, the software implementation was continuously improved throughout the test campaign. Fourth, the ground truth of belly pan clearance is difficult to measure directly. Fifth and finally, the tests were performed as a part of the software development for the Mars 2020 rover mission, where the main purpose of the tests were the verification and validation of the integrated software capabilities rather than the quantitative assessment of the performance of ACE alone.
Qualitatively, through the test campaign, the algorithm and implementation were matured to the point where the vehicle can drive confidently over ∼ 40 [m] through a high rock density (15% CFA) terrain. Since the path planner solely rely on ACE for collision check, the fact that the rover reliably avoids obstacles without hitting the belly pan is an indirect and qualitative evidence that ACE is working properly. For example, Fig. 15 shows the 3D reconstruction of the terrain and the vehicle configuration from the Scarecrow test data.
A limited quantitative assessment is possible because a few intermediate and derivative variables in ACE were directly measured and recorded. These variables include rocker angle, left and right bogie angles, and the vehicle's tilt angle. Figure 16 shows the ground-truth measurement of rocker and right bogie angles as well as the bounds computed by ACE on three long Scarecrow drives in the Mars Yard. There are a few observations from the results. Firstly, the bounds successfully captured the ground-truth trends. For example, the negative spike in the rocker angle at ∼ 1100 [s] in Figure 16(a) is correctly predicted by ACE, indicated by the reduced lower bound around that time. Secondly, the bounds were almost always respected. Thirdly, however, we observed occasional violations of the bounds as shown in red crosses on the plots. Our investigation concluded that the main causes of bound violations are the error in encoders and the error in perceived height map. The height map error is a result of two factors: 1) error in stereo processing (i.e., feature extraction and matching, error in camera model, noise in images, etc) and 2) "smoothing effect" due to re-sampling (3D point cloud from stereo processing is binned and averaged over a 2D grid). This conclusion was derived by using simulations in the following steps: 1) assured that ACE bounds are always respected when running ACE on a ground-truth height map, 2) reproduced the stereo error by using simulated camera images, and 3) ACE bound violations occur with comparable frequency and magnitude with the simulated stereo error. As in the Athena rover experiment, adding an adequate margin on the perceived height can restore the conservatism.
Run-time Performance
The run-time performance is important for space applications where the computational resources are severely limited. ACE has a significant advantage on this regard, compared to other alternatives that depend on iterative numeric methods. In the following analysis, we chose plane fitting as a point of comparison because it is a light-weight approximations for estimating rover state on rough terrain and used as the basis of GESTALT, the state-of-the-art autonomous navigation algorithm being used for the existing Mars rovers.
The computation of ACE is very fast due to its closed-form formulation. On the NVIDIA Jetson TK1 board on the Athena rover, ACE takes 11.2 [µs] for a single pose evaluation while plane fitting takes 26.1 [µs] over ∼100 points and 68.2 [µs] over ∼200 points. ACE runs faster than the naive plane-fit approach using least squares, as well as providing richer information about the vehicle state. For reference, the average run-time of ACE on a 2.8GHz Intel Core i7 machine is 2 [µs], which enables a robot to evaluate 500k poses at a second, whereas plane-fit is 5 times slower with 200 points. Next, perhaps more importantly for spacecraft applications, the computational time of ACE is constant. Thanks to the analytic formulation of ACE, the computational time is always the same regardless of terrain patterns. This is not the case for numeric methods, which require more iterations for complex terrain before converging.
We also evaluate the performance of ACE on the RAD750 CPU, which is used for the Curiosity and Mars 2020 rovers. While the precise timing is difficult due to the specialized configuration of the flight software, the typical run-time was 10-15 [ms] with a 10 [cm] resolution DEM. This is sufficient run-time as a collision checker to support the ambitious traversal plans on the M2020 mission.
Comparison with State-of-the-Art
Finally, we directly compared the performance of ACE-based path planning with the state-of-the-art in simulation. The point of comparison was a variant of GESTALT implemented in MATLAB 3 , which computes slope, roughness, and step hazards from plane fitting, and creates a goodness map by inflating hazards by the rover radius. In addition, we also compared against the "ideal" path planner that uses the ground-truth collision check (no conservatism). Such a planner is computationally unacceptable for the practical Mars rovers, but the comparison gives us an insight about how close the ACE-based paths are to the strictly optimal paths. The path planning algorithm is the same for all the three planners; a depth-five tree search was used for path selection with 1.5 [m] edge for each depth, while the collision check was run at every 0.25 [m]. The only difference is the collision check method.
The terrains we tested are flat, 30-by-40 meters in size, randomly populated with rocks at four different CFA levels (5, 10, 15, 20%). We created 20 terrains for each CFA levels (80 terrains in total). Three planners were run on each of the 80 terrains. A Curiosity-sized rover was commanded to go to the point 20 [m] away. Two quantitative metrics were used for the comparison. The first is the path inefficiency, defined as the fraction of the generated path length and the straight-line distance. Intuitively, the over-conservatism of collision checking should result in an increased path inefficiency because it is more likely that the paths heading straightly towards the goal is incorrectly judged unsafe, resulting in a highly winding path. The second metric is the success rate, defined as the number of runs the planner successfully arrived in the goal divided by the total number of runs. An excessive conservatism may result in a failure to reach the goal because no feasible path is found to move forward. Fig. 17 shows representative examples of paths generated by the three methods. The top, middle, and bottom rows are the state-of-the-art, ACE-based, and ideal path planners. As expected, the state-of-the-art paths were most winding (greater path inefficiency) while the ideal paths were the most straight. Notably, the state-of-the-art approach failed to find a path to the goal at 15 and 20% CFA, while the ACE-based planner were able to find a way to the goal. The ACE-based planner was more capable of finding paths through cluttered environments mainly because it allows straddling over rocks if sufficient clearance is available. However, the ACE-based paths are less efficient compared to the ideal ones. This result is again expected, because ACE conservatively approximates the rover states for the sake of significantly reduced computation (as reported in Section 4.2.3) compared to the exact kinematic solution. Fig. 18 shows the statistical comparison over the 20 randomly generated maps for each CFA level in terms of the two quantitative metrics. According to Fig. 18(a), the ACE-based planner was capable of driving reliably (≥ 95% success rate) up to 15% CFA, but the success rate drops significantly at 20% CFA. In comparison, the state-of-the-art path planning had only 40 % success rate at rather benign 10% CFA terrains. The ideal path planner was always be able to find a path to the goal for all the tested CFA values. Next, the results on path inefficiency in Fig. 18(b) clearly shows the difference in algorithmic conservatism. For example, at 10% CFA, the state-of-the-art planner resulted in 33% path inefficiency while it was nearly zero for the ACE-based and the ideal planners. At 15% CFA, the ACE-based planner resulted in 12% path inefficiency while that of the ideal planner is still nearly zero. The path inefficiency of the state-of-the-art planner was not computed for 15 and 20% CFA because the success rate was zero. Finally, at 20% CFA, the path inefficiency of the ACE-based planner went up to 33% while that of the ideal planner was at 3%. The CFA of the landing site of the Mars 2020 Rover (Jezero Crater) is typically less than 15%, while we can almost surely find a round to go around the fragmented spots with ≥ 15% CFA. Therefore, with these results, ACE allows us to confidently drive the Mars 2020 rover autonomously for the majority of the drive.
Conclusions
In this paper, we presented an approximate kinematics solver that can quickly, albeit conservatively, evaluate the state bounds of articulated suspension systems. The proposed method provides a tractable way of determining path safety with the limited computational resources available to planetary rovers. ACE avoids expensive iterative operations by only solving for the worst-case rover-terrain configurations. The algorithm is validated using simulations and actual rover testbeds, giving satisfactory results in all experiments including 42 days of field test campaign.
The experimental results indicate that the ACE-based planner successfully navigates the rover in environments with similar complexity to the planned landing site of Mars 2020 mission; however, one of the remaining algorithmic limitations is over-conservatism in estimated state bounds. Especially, the conservatism becomes greater on highly undulating terrain. An excessive conservatism may result in path inefficiency or a failure to find a path to the goal. Mitigating the extra conservatism is deferred to our future work.
Although the algorithm is primarily designed for planetary rover applications, the work is applicable to other domains where fast state estimation is needed but the fidelity of estimation is not demanded. Examples include trajectory planning of manipulators and path planning of ground/aerial/maritime vehicles. The importance of this method is in how we incorporate environmental uncertainty into the planning problem, without redundant computation or unsafe approximation. With the proper bounds of uncertainty, the robot state is guaranteed to be safe within well-defined intervals.
The ACE algorithm was successfully integrated with the surface navigation software of M2020 rover mission. ACE will enable faster and safer autonomous traverse on more challenging terrains on the red planet.
Figure 1 :
1Self-portrait of the MSL Curiosity rover, taken on Sol 1065. Sufficient clearance between the vehicle and the ground is necessary to safely traverse on rough terrain with outcrops. (Credit: NASA/JPL-Caltech/MSSS)
Figure 3 :Figure 4 :
34Articulated suspension systems on flat ground, viewed from the left side. Rocker-bogie suspension model, viewed from top. For visibility, wheels do not represent actual positions.
Figure 5 :
5Simple triangular geometry that models rocker/bogie systems.
Figure 8 :
8Analysis on artificial terrains with varying undulation levels the simulation to check if they are conservatively confined by the ACE bounds.
Figure 9 :
9Flat terrain with a smooth bump. Six wheel trajectories are shows in different color. The initial positions of the wheels are marked with circles. (a) Linear approach to the bump. (b) Curved approach to the bump. (c) Climb over the bump only with the right wheels.
Figure 10 :
10ACE estimation results for body roll φ, body pitch θ, left rocker angle δ l , left and right bogie angles β l and β r , and body height z o . The solid lines represent the ground-truth state computed by a numeric method. The shaded regions represent the ranges between the ACE upper/lower bounds.
Figure 11 :
11ACE estimation result in a Mars-like environment. The Rocky 8 rover was deployed on a synthetic Martian terrain. The base terrain is an orbit-based DEM for the Jezero crater. The rocks are randomly populated according to the Martian size-frequency distribution model.
Figure 12 :Figure 13 :
1213Athena rover driving on a slope of JPL Mars Yard. ACE estimation result for Athena's two different drives. The ground-truth (GT) state is within the ACE bounds except occasional violations.
Figure 14 :
14Scarecrow test in JPL's Mars Yard on July 17, 2018, showing a typical set-up of the Yard for the experiments.
Figure 15 :
15Visualization of a path planned in JPL's Mars Yard on July 23, 2018 with Caspian visualizer.
Figure 16 :
16Recorded ground-truth (GT) rocker and right bogies angles as well as the predicted bounds by ACE on three Scarecrow tests performed on Sept 12, 2018, in JPL's Mars Yard.
Figure 17 :Figure 18 :
1718Comparison of safety assessment methods in 20 [m] path planning with varying CFA levels. a) Conventional method that checks slope and step hazards with rover-sized inflation; b) Assessment with worst-case state from ACE bounds; c) Assessment with ground-truth state. Statistical result of path planning over 20 different maps in each CFA level.
Table 1 :
1Error statistics from cumulative 130 [m] drive by Athena.Max State Violation
Table 1
1shows a statistical result from cumulative 130 [m] drive by Athena. The total success rate is computed by counting samples that all state variables are within the ACE bounds. The success rate was 98.74% without the perception error margin, with ∼ 3 [deg] maximum attitude error and 0.012 [m] clearance error.
Conceptually, it is analogues to solve an inequality-constrained optimization such as minx x s.t x ≥ 0. 2 In reality, the terrain assessment of GESTALT is not binary; the terrain assessment map (called the goodness map) is pre-processed by taking the worst value within the diameter of the rover from each grid of the map (i.e., dilation in image processing). This is equivalent to obstacle expansion in case of binary goodness value.
We did not use the flight implementation of GESTALT because porting a part of spacecraft flight software is difficult due to technical and security reasons.
AcknowledgmentsThis research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. Copyright 2019 California Institute of Technology. Government sponsorship acknowledged.A Variable DefinitionsThe following table introduces a list of variables used in this paper.Variable Definition (x, y, z, φ, θ, ψ) 6 DoF pose l df Link length between differential joint and front wheel l dr Link length between differential joint and rear wheel (Rocker) l bm Link length between bogie joint and middle wheel (Rocker-bogie) l br Link length between bogie joint and rear wheel (Rocker-bogie) z {f,m,r} Height of front, middle, and rear wheels z d Height of differential joint z b Height of bogie joint δ {l,r} Angle change of left and right differential joints (δ l = δ r ) β {l,r} Angle change of left and right bogie joints (x od , y od , z od ) Translational offset from the body frame origin to differential joint z o Height of body frame origin κ d0 Angle between horizontal line and differential-front link on flat plane κ b0 Angle between horizontal line and bogie-middle link on flat planeB Video AttachmentThe supplement movie visually presents the state bound propagation process from terrain heights to vehicle's attitude through rocker-bogie suspensions.
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"JOINT MODELLING OF SPOKEN LANGUAGE UNDERSTANDING TASKS WITH INTEGRATED DIALOG HISTORY",
"JOINT MODELLING OF SPOKEN LANGUAGE UNDERSTANDING TASKS WITH INTEGRATED DIALOG HISTORY"
] | [
"Siddhant Arora \nCarnegie Mellon University\n\n",
"Hayato Futami \nSony Group Corporation\nJapan\n",
"Emiru Tsunoo \nSony Group Corporation\nJapan\n",
"Brian Yan \nCarnegie Mellon University\n\n",
"Shinji Watanabe \nCarnegie Mellon University\n\n"
] | [
"Carnegie Mellon University\n",
"Sony Group Corporation\nJapan",
"Sony Group Corporation\nJapan",
"Carnegie Mellon University\n",
"Carnegie Mellon University\n"
] | [] | Most human interactions occur in the form of spoken conversations where the semantic meaning of a given utterance depends on the context. Each utterance in spoken conversation can be represented by many semantic and speaker attributes, and there has been an interest in building Spoken Language Understanding (SLU) systems for automatically predicting these attributes. Recent work has shown that incorporating dialogue history can help advance SLU performance. However, separate models are used for each SLU task, leading to an increase in inference time and computation cost. Motivated by this, we aim to ask: can we jointly model all the SLU tasks while incorporating context to facilitate low-latency and lightweight inference? To answer this, we propose a novel model architecture that learns dialog context to jointly predict the intent, dialog act, speaker role, and emotion for the spoken utterance. Note that our joint prediction is based on an autoregressive model and we need to decide the prediction order of dialog attributes, which is not trivial. To mitigate the issue, we also propose an order agnostic training method. Our experiments show that our joint model achieves similar results to task-specific classifiers and can effectively integrate dialog context to further improve the SLU performance. 1 | 10.1109/icassp49357.2023.10095055 | [
"https://export.arxiv.org/pdf/2305.00926v1.pdf"
] | 258,426,270 | 2305.00926 | bc3690edd40cc9946f8162727b357b926d1127bc |
JOINT MODELLING OF SPOKEN LANGUAGE UNDERSTANDING TASKS WITH INTEGRATED DIALOG HISTORY
Siddhant Arora
Carnegie Mellon University
Hayato Futami
Sony Group Corporation
Japan
Emiru Tsunoo
Sony Group Corporation
Japan
Brian Yan
Carnegie Mellon University
Shinji Watanabe
Carnegie Mellon University
JOINT MODELLING OF SPOKEN LANGUAGE UNDERSTANDING TASKS WITH INTEGRATED DIALOG HISTORY
Most human interactions occur in the form of spoken conversations where the semantic meaning of a given utterance depends on the context. Each utterance in spoken conversation can be represented by many semantic and speaker attributes, and there has been an interest in building Spoken Language Understanding (SLU) systems for automatically predicting these attributes. Recent work has shown that incorporating dialogue history can help advance SLU performance. However, separate models are used for each SLU task, leading to an increase in inference time and computation cost. Motivated by this, we aim to ask: can we jointly model all the SLU tasks while incorporating context to facilitate low-latency and lightweight inference? To answer this, we propose a novel model architecture that learns dialog context to jointly predict the intent, dialog act, speaker role, and emotion for the spoken utterance. Note that our joint prediction is based on an autoregressive model and we need to decide the prediction order of dialog attributes, which is not trivial. To mitigate the issue, we also propose an order agnostic training method. Our experiments show that our joint model achieves similar results to task-specific classifiers and can effectively integrate dialog context to further improve the SLU performance. 1
INTRODUCTION
Spoken dialogue systems aim to enable dialogue agents to engage in a more natural conversation with humans. They have commonly represented a possible dialogue by a series of frames [1,2]. Each frame represents the type of task the user seeks and has attributes representing the information that can help the system to complete the task. These spoken dialog systems aim to automatically identify the topic of conversations as well as other dialog frame attributes (e.g., dialogue act, emotion) to engage in conversation with the user.
Conventional Spoken Language Understanding (SLU) [3,4] systems independently process each utterance in a conversation. However, the meaning of an utterance in a spoken dialog depends on the context. Prior work has shown dialogue context to be particularly useful in resolving ambiguities and co-references [5,6]. As a result, there has been extensive work on incorporating context to improve Natural Language Understanding (NLU) performance [7][8][9]. Prior work [10][11][12] on conversational speech has also shown strong improvements in ASR performance by incorporating dialog context. Thus, several approaches [13][14][15][16] have looked into incorporating dialog history in pipeline based SLU systems. Recently, there has been some work [17][18][19] in incorporating context to improve endto-end (E2E) SLU performance. One such approach [17] integrates 1 Our code & models are publicly available as part of ESPnet-SLU toolkit. dialog history into an E2E SLU system by using ASR transcripts of previous spoken utterances. There has also been an effort [18] to incorporate context directly from the audio of previous utterances.
However, these works mostly focus on building a single model for each SLU task like intent classification, dialog act classification, and emotion recognition. To jointly predict all the SLU tasks, we have to execute all models separately. Consequently, these models not only have a large memory footprint but also have high latency, which can affect the naturalness of spoken conversations [20] when these systems are deployed in commercial applications like voice assistants. Prior work has shown that jointly modeling different speech processing tasks together in a united framework can perform comparable to task-specific models [21,22] while reducing latency [23]. Motivated by this work, we ask the following questions: (i) Can we jointly model all SLU tasks while incorporating context in a single unified implementation without much loss in performance? (ii) Does incorporating the SLU tags predicted at previous spoken utterances by the joint model help in better modeling spoken dialogue context?
We seek to answer these questions by proposing a novel E2E SLU architecture that jointly models all the SLU tasks while effectively learning context from previous spoken utterances. This joint model uses an autoregressive decoder to predict dialog attributes one by one and the order in which the model predicts the SLU attributes may impact model performance. Hence, we investigate (iii) if we can use order agnostic training to make the model automatically predict in the optimal ordering during inference. Inspired by prior work on ASR [24], we also investigate (iv) if an intermediate CTC loss advances SLU performance further. We conduct extensive experiments on the recently released HarperValley [25] Bank dataset, which consists of dialogs between users and consumer bank agents. Our results show that our joint model performs at par with the individual classifiers for each SLU task, and incorporating dialog context and order agnostic training can further lead to significant improvements in performance. Our code and models are made publicly available as part of the ESPnet-SLU [26] toolkit.
The key contributions of our work are summarised below.
• We propose a novel joint model that uses dialog context to jointly predict the intent, dialog act, speaker role and emotion. • We propose order agnostic training of the joint model and show an improvement in performance across all SLU tasks. • We investigate the efficacy of using SLU tags predicted from previous utterances to model dialog context. • We show that incorporating the usage of intermediate CTC loss can advance SLU performance.
BACKGROUND: DIALOG CONTEXT IN SLU
The formulation for SLU with integrated dialog context extends the well-studied framework of NLU systems [7][8][9]. For NLU systems, dialog context sequence is represented as sequence of C utterances, i.e., S = {sc|c = 1, . . . , C}. Each utterance in the dialog context sequence is represented as sc = {wcn ∈ V|n = 1, . . . , Nc}, with length Nc and vocabulary V. Each utterance has a tag for each of the NLU tasks. In this work, each utterance has a tag from label sets L da , L ic ,L sr and L er indicating dialogue act classification, intent classification, speaker role prediction, and emotion recognition, respectively. This produces a label sequence of the same length C for each task, for instance, Y da = {y da c ∈ L da |c = 1, . . . ,C}. Using the maximum a posteriori theory, NLU models seek to out-putŶ da ,Ŷ ic ,Ŷ sr , andŶ er that maximise the posterior distribution P (Y da |S),P (Y ic |S),P (Y sr |S) and P (Y er |S) given S, respectively.
SLU introduces an additional complexity of modeling dialog context from the spoken utterance. Dialog context sequence is formed by C spoken utterances, i.e., X = {xc|c = 1, . . . , C}. Each spoken utterance xc = {xct ∈ R d |t = 1, . . . , Tc} is a sequence of d dimensional speech feature of length Tc frames. Similar to the NLU formulation, SLU systems seek to estimate the label se-quenceŶ da ,Ŷ ic ,Ŷ sr andŶ er that maximise the posterior distribution P (Y da |X),P (Y ic |X),P (Y sr |X) and P (Y er |X) given X, respectively. We can model these posterior distributions as described in the subsections below.
Seperate E2E model with dialog context
Prior work [17] models each posterior, e.g., P (Y da |X), using a sequence of transcripts S by applying the Viterbi approximation:
P (Y da |X) = S P (Y da |S,X)P (S|X) (1) ≈ max S P (Y da |S,X)P (S|X)(2)
Their approach then assumes the conditional independence of y da c |s1:c−1 from x1:c−1,y da 1:c−1 to simplify the Eq 2:
P (Y da |X) ≈ max S C c=1 P (y da c |s1:c−1, xc)P (S|X)(3)
Transcripts are computed using a separate ASR module that seeks to estimateŜ that maximises P (S|X). UsingŜ, we can modify Eq 3:
P (Y da |X) ∼ C c=1 P (y da c |ŝ1:c−1, xc)(4)
Prior work [17] models P (y da c |ŝ1:c−1, xc) in Eq. 4 by passing ASR transcriptsŝ1:c−1 to a pretrained language model (LM) like BERT [27] and then concatenating these context embeddings to the acoustic embedding obtained from xc. They focus on building a separate model for each of the SLU tasks, which in the above description is dialogue act classification. Thus, to predict all the SLU tasks, all the separate models that estimate P (Y da |X),P (Y ic |X),P (Y sr |X), and P (Y er |X) independently need to be executed which can increase latency and computational cost and also does not consider the dependency between SLU tasks.
PROPOSED JOINT E2E MODEL W/ DIALOG CONTEXT
In this work, we extend the prior work [17] on dialog integration discussed in section 2.1 and propose to jointly model all SLU tasks. We denote a single target containing all the SLU tags as
R = (Y da ,Y ic ,Y sr ,Y er )
and modify Eq. 4 with rc = (y da c ,y ic c ,y sr c ,y er c ) as shown below:
P (R|X) ∼ C c=1 P (rc|r1:c−1,ŝ1:c−1, xc)(5)
The SLU tags predicted for the previous spoken utterances i.e. r1:c−1 = (y da 1:c−1 ,y ic 1:c−1 ,y sr 1:c−1 ,y er 1:c−1 ) may be incorporated (Eq. 5) to better model the dialog context unlike in prior work [17] which assumes conditional independence of y da c |s1:c−1 from y da 1:c−1 . In §5.1, we confirm experimentally whether this previous SLU tag condition is helpful. Further, by jointly modeling all the SLU tasks, we expect significantly lower latency and lightweight inference.
To realize this formulation, we propose a joint model architecture shown in Fig. 1. The input speech signal for each utterance i.e., xc in Eq. 5, is passed through an acoustic encoder (Encoderaco) to generate acoustic embeddings caco.
caco = Encoderaco(xc)(6)
We concatenate ASR transcriptsŝ1:c−1 and SLU tags r1:c−1 for all previous spoken utterances and pass them through semantic encoder (Encodersem) like a pretrained LM to encode the dialog history:
ccont = Encodersem(concat(ŝ1:c−1, r1:c−1))(7)
The output of the semantic encoder is also passed to a linear layer to ensure that context embeddings ccont have the same hidden dimension as acoustic embeddings caco. The acoustic and context embeddings are concatenated together (concat(caco, ccont)) and attended by a joint encoder Encoderjoin to produce the joint embedding cjoin:
cjoin = Encoderjoin(concat(caco, ccont))(8)
The model is trained using joint CTC-attention training [28], where the CTC objective function is used to train the attention model encoder as an auxiliary task. Because we use an autoregressive decoder to predict tags one by one (Eq. 11), the likelihood P (rc|xc,ŝ1:c−1, r1:c−1) is dependent on the order of SLU tags in the target sequence rc. This has been referred to as label ambiguity (or permutation) problem in prior work [29,30]. Inspired by prior work on permutation invariant training [30][31][32], we use CTC objective function to perform permutation-free training as shown in Eq. 9, which is referred to as order agnostic training in this work. Let z be the output sequence variable computed from the joint embedding cjoin, then the optimal permutation orderπ is computed as:
π = argmin π∈P LossCTC(z,r π c ),(9)
where P is the set of 4! possible permutations of SLU tags (da,ic,sr,er), π is one such permutation and r π c is the reference target with the order of SLU tags indicated by π. Later, the optimal For each pair of optimal decoding order and reference target (π,rc), the attention likelihood is calculated as shown below.
h l = Decoder(cJOIN, (rπ c ) 1:l−1 )(10)
PAttn((rπ c ) l |xc,ŝ1:c−1, r1:c−1, (rπ c ) 1:l−1 ) = SoftmaxOut(h l ),
where SoftmaxOut denotes a linear layer followed by the softmax function and (rπ c ) l is the l th term in target sequence. The joint likelihood PAttn(rπ c |xc,ŝ1:c−1, r1:c−1) can then be computed as a product of the individual likelihood for each of the SLU tags (i.e., from l in [1,4]). It is important to note that this "order agnostic training" does not add any new model parameters. During inference, the model automatically picks the tag order, i.e., unlike training, we do not enforce the predicted tag order to have the minimum CTC loss. Further, our joint models can also be trained with an auxiliary ASR objective [26,33] by making the model generate both the SLU tags rπ c and ASR transcript sc.
EXPERIMENT SETUP
Datasets
To show the effectiveness of our joint model, we conducted experiments on publicly available HarperValleyBank [25] spoken dialog corpus, where dialogs are simulated conversations between bank employees and customers. The corpus consists of 1,446 conversations with 23 hours of audio and 25,730 utterances with human annotated transcripts. The utterances are spoken by 59 unique speakers and have annotations for the dialog act, intent of the conversation, speaker role, and emotional valence.
In this work, we trained a joint model that performs intent classification, dialog act recognition, emotion recognition and speaker attribute prediction. Dialogue act recognition is a multi-label multiclass classification whereas all other tasks are single-label multiclass classification. We followed the setup in prior work [18] and split the conversations into train, valid and test set 3 . Similar to [17], 2 We use the CTC loss to compute optimal order instead of decoder loss for lightweight modeling. 3 However, the prior work [17,18] also uses an off the shelf ASR model to realign the audio with transcripts making their results not directly comparable to results obtained by our setup. . We report macro F1 for dialog act prediction and accuracy for intent classification, speaker role prediction and emotion recognition as recommended by prior work [17,25]. The latency of our system is reported using two metrics: (1) Real time factor (RTF), which is the average time taken to process an input audio file as a ratio of the duration of input and (2) Endpoint latency which is the elapsed time from the utterance end to getting predictions for SLU tasks. We also show the training time per epoch for all our models.
Architecture details and training
Our approach is compared to state-of-the-art SLU task-specific baselines referred to as "Separate E2E models without context" (optimizes P (y da c |xc) instead of P (y da c |ŝ1:c−1, xc) in Eq. 4). We also compared with our proposed model without context, i.e., "Joint E2E model without context" which optimizes P (rc|xc) instead of P (rc|r1:c−1, xc,ŝ1:c−1) in Eq. 5. This baseline joint E2E model does not incorporate order agnostic training defined in Eq. 9 and instead is trained using a fixed order (y da , y ic , y sr , y er ) of SLU tags. To understand the efficacy of our proposed modifications to [17], we first trained a joint model (referred to as "Joint E2E model with context") that does not use SLU tags of previous spoken utterances (i.e. optimize P (rc|xc,ŝ1:c−1) instead of P (rc|r1:c−1, xc,ŝ1:c−1) in Eq. 5) and also does not utilize order agnostic training. In our ablation study, we further trained proposed joint model that incorporates "order agnostic training" as defined in Eq. 9 and another joint model that incorporates "previous SLU tag condition" i.e. tags predicted for previous spoken utterances (r1:c−1 in Eq. 5).
Our models were implemented in pytorch [34] and experiments were conducted through the ESPNet-SLU [26,35] toolkit. All the models were trained using an auxiliary ASR objective. Our taskspecific baselines consist of 12 layer conformer [36,37] encoder that inputs features extracted by a strong self-supervised speech model WavLM [38] and 6 layer transformer [39,40] decoder. Our "Joint E2E model without context" has a similar architecture as task specific baselines. Prior work [24] used an intermediate CTC loss attached to an intermediate layer in the encoder network to regularize CTC training and improve performance. Inspired by this work, we experimented with adding intermediate CTC loss at layers 4, 6 and 8 of our conformer encoder. The "Joint E2E model without context" is used to generate dialog context 4ŝ 1:i−1. We incorporated the usage of Transformers library [41] to get BERT-base-uncased [27] as semantic encoder (encoderSEM in Eq. 7). The conformer architecture with intermediate CTC loss is leveraged as the joint encoder (encoderJOIN in Eq. 8) in our proposed models. Teacher forcing is used for all our models with integrated dialog context i.e., the context was created using ground truth transcripts during training. The training and inference of our models were performed using a single NVIDIA Tesla V100-32GB GPU. The inference was computed using 4 parallel jobs and we use the time when the utterance has been processed by all the jobs (i.e. time taken by slowest job) as well as the sum of time taken by all the jobs to compute latency metrics. All hyperparameters were selected based on validation performance. Table 1 shows the results of our joint model both with and without using dialog context. The performance of our "Separate E2E models without context" (section 2.1) is similar to the baseline results reported in prior work [18], though these results are not directly comparable because of different data preparation setups. Our "Joint E2E model without context" performs at par with these taskspecific models while significantly reducing latency 5 and the number of trainable model parameters, showcasing the utility of jointly modeling all SLU tasks. We investigated integrating dialog context in the joint model ("Joint E2E model with context" in Table 1) using formulation described in section 3 and observe a significant improvement in performance, particularly for intent, dialog acts, and speaker role identification, providing evidence that our proposed methodology can effectively encode the context. We further observe a performance gain using "order agnostic training", as defined in Eq. 9, which confirms our hypothesis that the ordering of SLU tags while training can impact model performance. While our order agnostic training framework increases the training time as it requires the computation of CTC loss over all possible permutations of SLU tags (Eq. 9), this increase in training time is not very significant. We also experimented with "previous SLU tag condition" (i.e. attending to r1:c−1 in Eq. 5) using only utterance-level annotated tags, i.e., dialog act, emotion, and speaker role, to encode context. This model achieves a performance gain in intent classification and dialog act prediction with comparable results on predicting other SLU tasks. We plan to further investigate the utility of SLU tags from previous spoken utterances in future work. Our joint E2E model with context also achieved similar performance to "Separate E2E model with context" [18] with no knowledge distillation from a text-based system, although the two models have different data preparation setups.
RESULTS
Main Results
To better understand our joint model, we also perform an ablation study in Table 2 and infer that using intermediate CTC loss can stabilize training and improve the performance of our joint model.
Using oracle context
To understand the impact of errors in ASR transcripts , we compute results of our joint E2E model with oracle context, i.e., using ground truth transcripts asŝ1:c−1 in Eq. 5. Table 3 shows only a slight increase in performance compared with the model that uses ASR transcripts to encode context, indicating that it is robust to ASR errors. 4 The Word Error Rate (WER) of ASR transcripts is 11.7. 5 Task-specific baselines are executed one after other in sequential order. Table 4: Results showcasing the optimal order of SLU tags found during training using Eq. 9 and predicted order of SLU tags during inference. We report sample training utterance for each tag order.
Predicted ordering of SLU tags
We analyze the optimal order of SLU tags found during training using Eq. 9 in Table 4 and observe that optimal permutation sequencê π are among only 5 of 4! possible permutations of SLU tags. We compare it to the tag order predicted by our joint E2E model with context and order agnostic training. The predicted order of SLU tags during inference has a similar trend to the optimal order of SLU tags found during training, with (sr, ic, da, er) and (da, ic, sr, er) being the two most common orders. Further the training utterances with optimal order (sr, da, ic, er) are usually spoken at the start of the conversation and are mainly greetings, as shown in Table 4. We hypothesize that this tag order is optimal for these utterances as it is challenging to detect the intent of the conversation from these greeting utterances, and hence intent is predicted after the model decodes dialogue act and speaker role tags. This gives model the ability to incorporate the dependency on these tags to better extract the intent of current utterance. Our analysis is similar for other tag orders and we infer that we can validate the optimal tag order found for training utterances. After a more fine-grained analysis, we observe that test utterances that are similar to training utterances with optimal tag order π are also predicted by the joint model in the same tag order π during inference. This finding provides initial evidence to the hypothesis that the joint model learns to automatically predict in the optimal tag order during inference using order agnostic training. Based on this interesting insight and performance gains observed in Table 1, we recommend future studies on jointly modeling different SLU tasks to incorporate order agnostic training in their framework.
CONCLUSION
We propose a novel model architecture that can jointly model intent classification, dialogue act prediction, speaker role identification and emotion recognition with full integration of dialog history in spoken conversations. Our results show that the joint model achieves comparable performance to task-specific models with the additional benefits of low latency and lightweight inference. Our joint model can also successfully capture dialog context to improve the prediction performance of all SLU tasks significantly. We experimentally confirm that order agnostic training can further enhance performance.
In future work, we plan to explore E2E integration of dialog context as well as knowledge distillation from a text-based system.
ACKNOWLEDGEMENT
This work used the Extreme Science and Engineering Discovery Environment (XSEDE) [42], which is supported by NSF grant number ACI-1548562. Specifically, it used the Bridges system [43], which is supported by NSF award number ACI-1445606, at the Pittsburgh Supercomputing Center (PSC).
Table 2 :
2Results showing the impact of intermediate CTC to advance SLU performance.permutationπ is used for computing the attention decoder loss 2 .
Table 3 :
3Results showing the robustness of using the ASR transcripts instead of the oracle transcripts.we also removed non-lexical tokens such as [noise],[laughter] from
the transcript. Our training set contains 1,174 conversations (9.2
hours of audio, 15,424 utterances), valid set contains 73 conversa-
tions (0.6 hours of audio, 964 utterances) and our test set contains
199 conversations (1.6 hours of audio, 2903 utterances)
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| [] |
[
"Tube geometry controls protein cluster conformation and stability on the endoplasmic reticulum surface",
"Tube geometry controls protein cluster conformation and stability on the endoplasmic reticulum surface"
] | [
"Liam T Kischuck \nDepartment of Physics\nToronto Metropolitan University\nM5B 2K3TorontoOntarioCanada\n",
"Aidan I Brown \nDepartment of Physics\nToronto Metropolitan University\nM5B 2K3TorontoOntarioCanada\n"
] | [
"Department of Physics\nToronto Metropolitan University\nM5B 2K3TorontoOntarioCanada",
"Department of Physics\nToronto Metropolitan University\nM5B 2K3TorontoOntarioCanada"
] | [] | The endoplasmic reticulum (ER), a cellular organelle that forms a cell-spanning network of tubes and sheets, is an important location of protein synthesis and folding. When the ER experiences sustained unfolded protein stress, IRE1 proteins embedded in the ER membrane activate and assemble into clusters as part of the unfolded protein response (UPR). We use kinetic Monte Carlo simulations to explore IRE1 clustering dynamics on the surface of ER tubes. While initially growing clusters are approximately round, once a cluster is sufficiently large a shorter interface length can be achieved by 'wrapping' around the ER tube. A wrapped cluster can grow without further interface length increases. Relative to wide tubes, narrower tubes enable cluster wrapping at smaller cluster sizes. Our simulations show that wrapped clusters on narrower tubes grow more rapidly, evaporate more slowly, and require a lower protein concentration to grow compared to equal-area round clusters on wider tubes. These results suggest that cluster wrapping, facilitated by narrower tubes, could be an important factor in the growth and stability of IRE1 clusters and thus impact the persistence of the UPR, connecting geometry to signaling behavior. This work is consistent with recent experimental observations of IRE1 clusters wrapped around narrow tubes in the ER network. | null | [
"https://export.arxiv.org/pdf/2305.00971v1.pdf"
] | 258,426,291 | 2305.00971 | f4e7981208b046f070f62a62d76ce74e322d4afe |
Tube geometry controls protein cluster conformation and stability on the endoplasmic reticulum surface
Liam T Kischuck
Department of Physics
Toronto Metropolitan University
M5B 2K3TorontoOntarioCanada
Aidan I Brown
Department of Physics
Toronto Metropolitan University
M5B 2K3TorontoOntarioCanada
Tube geometry controls protein cluster conformation and stability on the endoplasmic reticulum surface
(Dated: May 2, 2023)
The endoplasmic reticulum (ER), a cellular organelle that forms a cell-spanning network of tubes and sheets, is an important location of protein synthesis and folding. When the ER experiences sustained unfolded protein stress, IRE1 proteins embedded in the ER membrane activate and assemble into clusters as part of the unfolded protein response (UPR). We use kinetic Monte Carlo simulations to explore IRE1 clustering dynamics on the surface of ER tubes. While initially growing clusters are approximately round, once a cluster is sufficiently large a shorter interface length can be achieved by 'wrapping' around the ER tube. A wrapped cluster can grow without further interface length increases. Relative to wide tubes, narrower tubes enable cluster wrapping at smaller cluster sizes. Our simulations show that wrapped clusters on narrower tubes grow more rapidly, evaporate more slowly, and require a lower protein concentration to grow compared to equal-area round clusters on wider tubes. These results suggest that cluster wrapping, facilitated by narrower tubes, could be an important factor in the growth and stability of IRE1 clusters and thus impact the persistence of the UPR, connecting geometry to signaling behavior. This work is consistent with recent experimental observations of IRE1 clusters wrapped around narrow tubes in the ER network.
I. INTRODUCTION
Protein clustering can be important to the health and function of a living cell. While many instances of protein clustering in cell biology occur in three-dimensional volumes [1], protein clusters can also form on the twodimensional surface of the cell membrane [2], such as those of nephrin [3] and GPCR proteins [4]; and on various intracellular membranes, such as DRP1 and MAVS proteins on mitochondria [5][6][7] and NBR1 proteins on peroxisomes [8,9]. Coarsening dynamics, including cluster coalescence and exchange of material between clusters, often describe cluster evolution following formation [1,8].
Clustering of the protein IRE1 on the endoplasmic reticulum (ER) membrane activates the unfolded protein response (UPR) that maintains protein folding homeostasis inside the ER [10][11][12]. The ER is an organelle composed of a cell-spanning network of tubes and sheets. A substantial fraction of proteins produced by the cell transit through the ER, and many are sent from ER exit sites to the Golgi for further processing [13]. While chaperones and other ER factors assist nascent proteins in folding into functional conformations and the ER contains quality control pathways to remove misfolded or excess unfolded proteins [12], unfolded proteins can accumulate in the ER and impede the function of the ER and the cell [14]. Accumulation of unfolded proteins in the ER network, known as ER stress, triggers the UPR, which induces changes to mitigate the unfolded protein stress.
There are multiple UPR signaling pathways, and the pathway mediated by IRE1 proteins is the most an- * [email protected] cient and widely conserved, shared across yeast, plants, and mammals [15]. IRE1 is activated by removal of an attached BiP chaperone by an unfolded protein, followed by autophosphorylation that enables the formation of IRE1 dimers and higher order oligomers [16,17]. IRE1 oligomers splice mRNA which then traffics into the nucleus to modify gene expression [16] and mRNA turnover [14,18]. Under conditions of prolonged activity, IRE1 can play a role in the activation of apoptosis (programmed cell death) [19]. The IRE1 activation from ER stress causes IRE1 clustering on the membrane of ER tubes [11,17,20,21]. This clustering behavior is thought to be essential to the UPR signaling of IRE1 [10,17]. The proteins at the cluster periphery diffuse in and out of the cluster, and evaporating clusters appear to break apart at their edges [11].
With IRE1 present on the ER membrane in relatively low concentrations (∼ 1/µm 2 in both mammals [11] and yeast [22][23][24]), the factors that control diffusive encounters between IRE1 proteins and clusters are expected to impact IRE1 clustering behavior. While mean search times can be calculated on spatial networks such as the ER [25,26], the diffusive search by activated IRE1 proteins for other IRE1 monomers and clusters is expected to exhibit large variation [27]. IRE1 clusters form in yeast in ∼10 minutes [22] and in mammals in ∼2 hours [11]. With IRE1 diffusivity of D = 0.24 µm 2 /s [11], diffusive search times for a target in the mammalian ER would be on the scale of an hour [25], consistent with diffusive search playing a role in determining cluster formation times. IRE1 clusters continue to evolve in both yeast and mammals over ∼10 hours [11,22,28]. This slow decrease of cluster number [11,28] and increase in cluster size after cluster formation [11], and large majority of IRE1 localized to clusters [22] are consistent with control of cluster evolution by Ostwald ripening interactions arXiv:2305.00971v1 [physics.bio-ph] 1 May 2023 between clusters. IRE1 clusters are confined to the two-dimensional ER membrane surface. It is expected that geometric confinement and search dimension will affect diffusive search and [25,[29][30][31][32] and phase separation behavior [8,9,33,34], key aspects of clustering activity. IRE1 clusters have been observed with diverse morphologies, including those that appear wrapped around ER tubes [11,35] and localize to narrow ER tubes [35].
Given the novel IRE1 clustering behavior under ER stress conditions, we use kinetic Monte Carlo simulations to explore how ER tube geometry impacts IRE1 clustering behavior. Because of the unique periodic geometry of a tube, sufficiently large clusters can 'wrap around' a tube, and reduce the interface length compared to an equal-area round cluster. The focus of this work is to describe how cluster wrapping impacts cluster dynamics. This 'wrapped' cluster morphology is directly impacted by tube geometry, as narrower tubes allow smaller clusters to become wrapped. We find that tube radius, by controlling the transition between wrapped and circular IRE1 clusters, impacts cluster growth rate and stability (evaporation rate and threshold between cluster growth and evaporation). This geometric effect on cluster dynamics is expected to be a significant factor for IRE1 cluster behavior and the persistence of intracellular UPR signaling.
II. RESULTS
A. Transition between circular and wrapped IRE1 clusters IRE1 protein dynamics on a two-dimensional lattice representing a single tube region of the ER are simulated using the kinetic Monte Carlo method [36][37][38]. This approach is similar to previous lattice-gas models of protein clustering on cellular membranes [39]. All IRE1 in this region are assumed activated, in line with the global IRE1 activation observed experimentally [11,17] or with local IRE1 activation [40]. The simulation includes (see Fig. 1) diffusion of both individual IRE1 proteins and IRE1 clusters, with each lattice site permitted to be occupied by one protein or not occupied. Following experiments showing diffusivity in a membrane scaling with inverse particle radius [41] and consistent with the Stokes-Einstein relation [42], cluster diffusivity is scaled as N −1/2 , where N is the number of proteins in the cluster. The two-dimensional lattice is periodic in one direction to represent the tubular geometry. In the other direction, IRE1 that diffuses past either end of the twodimensional lattice has left the tube region under consideration. Proteins enter both tube ends proportional to an external concentration c ext . Nearest-neighbor IRE1 proteins have a favorable interaction energy, with the energy Connection to the rest of the ER network is represented by a constant external concentration cext, with proteins able to both enter and leave the tube section. IRE1 interactions are favored by nearest-neighbor interaction energy J (see Eq. 1).
E of the system
E = −J ij n i n j ,(1)
where J > 0, ij representing the sum over nearestneighbor lattice sites, and n i ∈ [0, 1] is the occupation of lattice site i. Diffusive moves of individual IRE1 proteins to unoccupied sites occur at rate k = k 0 min[1, exp(−∆E/(k B T ))], with k 0 the rate in the absence of other proteins, and ∆E the change in E (Eq. 1) from the move, following the Metropolis criterion [43]. Clusters can additionally take collective diffusive steps to unoccupied sites with k 0 reduced by a factor √ N . While IRE1 clusters that are small compared to the ER tube circumference are expected to be approximately circular in shape ('round'), experiments show that IRE1 proteins can form clusters that wrap around an ER tube [11,35]. The inset of Fig. 2A shows a schematic of these two types of cluster shape. The interface length (circumference) of a round cluster is L round = 2π N a/π, where a is the membrane area occupied by one IRE1 protein. For a cluster that is sufficiently large to wrap around the tube, the interface length of a wrapped cluster is L wrap = 4πr tube , where r tube is the ER tube radius. Note that this wrapped cluster interface length is independent of the number of proteins in the cluster, in contrast to the protein number-dependent interface length of the round cluster. These interface lengths are shown in Fig. 2A cluster will have a shorter interface length. Because cluster energy is from nearest-neighbor interactions (Eq. 1), the cluster energy E cluster ≈ −2JN + LJ/ √ a is approximately the bulk interactions expected if all proteins were surrounded by nearest neighbors (−2JN ) plus the 'missing' interactions from proteins at the interface rather than surrounded by other proteins in the bulk (LJ/ √ a). Thus the energy of a cluster of a given protein number N is proportional to the interface length L, and a shorter-interface cluster of the same protein number will have a lower energy. Corresponding to cluster sizes for which the round cluster interface length will be longer than the wrapped cluster interface length, the round cluster energy will exceed the energy of a wrapped cluster for
N > 4πr 2 tube a .(2)
To explore transitions between round and wrapped clusters, we simulated IRE1 cluster dynamics on a closed tube with proteins unable to leave or enter (see Appendix for snapshots of a sample transition). While for a given cluster size and tube radius, either a wrapped or round cluster is energetically favored, there may not be a large enough energy difference for the cluster to indefinitely settle into either the wrapped or round conformation. The frequency of transitions between wrapped and round clusters is maximized at an intermediate protein concentration, with round clusters favored at lower concentrations and wrapped clusters at higher concentrations (Fig. 2B). The transition frequency peak indicates a phase transition between round and wrapped conformations. Figure 2C shows how the probability of finding a cluster in the the wrapped conformation increases with protein concentration (and thus cluster size), transitioning from a low wrapped probability to high over a relatively narrow range of protein concentrations. The transition concentration from round to wrapped clusters increases as the tube radius increases, as increased tube radii require larger clusters for a wrapped conformation to be favored (Eq. 2). Figure 2D plots the cluster size N at which wrapped clusters become favored, described by Eq. 2, and com- Cluster sizes over 20 are indicated as 20. Each tube initially contains a single IRE1 protein, as the maximum diameter and concentration sampled correspond to less than a single IRE1 protein in the tube region.
pares to the cluster size for this transition observed in simulations. While the simulation transition from round to wrapped follows a similar trend to the prediction, the simulation transitions occur for slightly higher cluster sizes. This is expected due to fluctuations in cluster size from proteins joining/leaving, and from fluctuations in cluster conformation as a wrapped cluster temporarily unwraps. Clusters wrap around the tube for substantially smaller cluster sizes than would naturally wrap through growth of a round cluster, i.e., at smaller cluster size than for cluster diameter = tube circumference (Fig. 2D).
B. Cluster growth depends on tube diameter
We now move from exploring IRE1 cluster dynamics in a closed system with a pre-formed cluster to cluster formation and growth behavior. In these simulations the external IRE1 concentration is held fixed, with proteins able to enter and leave the tube region under consideration. IRE1 clusters form through monomer dimerization followed by further growth and merging of small multimers. At low external IRE1 concentrations, clusters do not form. Cluster formation occurs above a concentration threshold of approximately 1/µm 2 independent of tube diameter for J = 5.3 k B T , shown in Fig. 3 after a period of 6 hours. Similar results are found at earlier times, suggesting cluster formation at different tube diameters and concentrations is no longer changing with time (see Appendix). The independence of the cluster formation concentration threshold on tube diameter is attributed to the lack of control of the diameter on the formation and encounter kinetics of monomers and small multimers.
We next examine the growth of IRE1 clusters over time, in these same open tubes exposed to a constant external concentration. When the cluster is small, the cluster has a round conformation as it grows. Once sufficiently large, a cluster will fluctuate into a wrapped conformation. The average cluster growth rate can substantially increase following this transition from a round to a wrapped conformation (Fig. 4A). The cluster growth rate distribution before the wrapping transition is distinct from the distribution after the transition (Fig. 4B), with the likelihood of these growth distributions being drawn from the same underlying distribution approximately 10 −7 according to the Kolmogorov-Smirnov test. While the choice of parameters in Fig. 4A,B emphasize the cluster growth rate difference, other parameter choices yield a smaller but consistent growth advantage for clusters following the wrapping transition (see Appendix). The increased growth rate of wrapped clusters, compared to round clusters on the same tube, is attributed to an interface that no longer grows with size (no further increase in cluster escape), a flat interface compared to round clusters (reduces cluster escape), and the encounter of all proteins with a wrapped cluster, compared to a round cluster that a protein can diffuse past without encountering the cluster. Figure 4C shows the cluster size at which wrapped clusters become energetically favored (Eq. 2) and the cluster size at which the wrapping transition occurs in simulations with a constant external concentration. The wrapping transition occurs in the simulated growing clusters at a significantly larger cluster size than when the wrapped conformation becomes favorable compared to the round conformation (compare to Fig. 2D). This widening of the gap for growing clusters (Fig. 4C) compared to a closed system ( Fig. 2D) is expected, as the fluctuation from a round to a wrapped conformation requires finite time, and if the cluster is continuing to grow the cluster size will be larger when the transition finally occurs.
We then investigated whether this increase in cluster growth rate after the wrapping transition leads to tuberadius dependent cluster growth rates. Figure 4D shows that clusters grow more rapidly on wider tubes compared to smaller tubes. With the external IRE1 concentration per membrane area held constant, wider tubes (with larger circumferences) will have a larger number of IRE1 enter per unit time, leading to faster cluster growth.
While clusters on a given tube diameter grow faster after wrapping (Fig. 4A,B), clusters reach a certain size more quickly on wider tubes that disfavor the wrapping transition (Fig. 4D). More proteins enter wider tubes because of their larger circumference exposed to the same constant area concentration of proteins as narrower tubes. The additional proteins provided to wide tubes al- lows clusters on wider tubes to more quickly reach a certain size, compared to narrow tubes with clusters that wrap at smaller cluster sizes. The time for a cluster to grow to a certain size incorporates delivery rates, which are larger for wider tubes; as well as cluster conformation, with enhances growth after the transition to a wrapped cluster, which occurs earlier in cluster growth on narrower tubes.
C. Wrapped clusters decay more slowly than round clusters
We now move from exploring cluster growth to cluster decay by setting the external IRE1 concentration to zero. Therefore, during these cluster decay processes, IRE1 proteins can exit the tube region under consideration, but do not enter. Figure 5A shows cluster decay for round and wrapped clusters. All clusters initially have 200 proteins, with round clusters on a wider (96 nm diameter) tube and wrapped clusters on a narrower (32 nm) tube. Wrapped clusters decay much more slowly, with 200 protein clusters in a wrapped conformation decaying to 50 proteins after approximately 9× (J = 5 k B T ), 5× (J = 4 k B T ), and 3.5× (J = 3 k B T ) longer time periods compared to round clusters. The cluster decay rates for wrapped and round clusters are compared across interaction energies in Fig. 5B. While wrapped clusters consistently decay more slowly than round clusters, the ratio between the wrapped and round cluster decay rates rises as the IRE1 interaction energy J increases (Fig. 5B inset).
Proteins evaporating from a cluster increase in energy by n n J, where n n is the typical number of neighbors of a protein on the cluster surface, such that the protein escape rate is proportional to e −nnJ/(kBT ) . n n is estimated from the cluster decay (see creased interface length and curvature for round clusters relative to the flatter wrapped cluster interface, which leads to increased protein escape rates from the surface of round clusters relative to wrapped clusters [33]. The ratio of the decay rates is an exponential, e −n n,round J/(kBT ) e −nn,wrapJ/(kBT ) = e ∆nJ/(kBT ) ,
with ∆n = n n,wrap − n n,round as shown in the inset of Fig. 5B. By selecting the cluster conformation (wrapped or round), tube width controls the speed of cluster evaporation.
D. Wrapped clusters have lower threshold protein concentrations between growth and decay
We determine the threshold external IRE1 concentration c ext at which simulated IRE1 clusters switch from growth to decay by finding the mean cluster size vs time for different c ext (Fig. 6A). The threshold concentration between cluster growth and decay varies with tube width (Fig. 6B), with the threshold suddenly increasing when the tube width increases above a particular diameter. The relatively low threshold concentrations for tube widths below this sudden increase in Fig. 6B correspond to wrapped clusters, and the relatively high threshold concentrations for tube widths above this sudden increase correspond to round clusters. The tube diameter at which the threshold concentration changes suddenly increases approximately corresponds to the diameter at which clusters transition between wrapped and round conformations (Fig. 2C). Below and above this diameter at which the threshold concentration suddenly changes (at which clusters switch from round to wrapped), the threshold concentration changes less with tube diameter.
For an IRE1 cluster at the threshold concentration c 0 , the inward flux of proteins from outside the tube region under consideration to the cluster is proportional to the external concentration c ext = c 0 , the tube radius r tube , and the probability of encountering the cluster at its axial position p e , and inversely proportional to the length of the tube (proportional to the probability of a protein at one end of the tube reaching the other end without first leaving by the initial end). The inward flux is Φ in = αc 0 r tube p e / , with α a proportionality constant that includes factors other than concentration and geometry, such as the IRE1 diffusivity and protein interaction strength. For wrapped clusters, p wrap e = 1, because the cluster wraps around the tube, such that proteins cannot reach the axial position of the cluster without encountering the cluster. For round clusters, p e = 2r cluster /(2πr tube ) = r cluster /(πr tube ) as a round cluster only covers a fraction of the tube circumference.
The flux of proteins escaping from the cluster will be proportional to the cluster interface length: Φ wrap out = β wrap L wrap and Φ round out = β round L round , with β a proportionality constant including IRE1 diffusivity, interaction strength, and the interface curvature (curvature will differ between wrapped and round conformations). L wrap = 4πr tube and L round = 2πr cluster .
At the threshold concentration c 0 , the cluster is not growing or decaying, and Φ in = Φ out . This yields c wrap 0 = 4πβ wrap /α and c round 0 = 2π 2 β round /α. The difference between β wrap and β round is that the round clusters are expected to have a more curved interface than wrapped clusters. However, this curvature difference is expected to become small for larger clusters, such that β wrap ≈ β round . With this approximation, the ratio between threshold concentrations is
c round 0 c wrapped 0 ≈ π 2 ,(4)
approximately 1.57. In Fig. 6B, this ratio is approximately 1.64 for clusters with 100 proteins and 1.48 for clusters with 200 proteins (comparing the threshold concentrations at the smallest and largest tube diameters in Fig. 6B). Round clusters on wider tubes require a higher external IRE1 concentration to avoid cluster decay compared to wrapped clusters on narrow tubes. This suggests that tube geometry (diameter) could play an important role in IRE1 cluster stability, and impact the location of persistent clusters as cluster coarsening occurs via Ostwald ripening. Figure 6 suggests that stability of clusters of a certain size is step-like, with clusters above a certain size gaining considerable stability on sufficiently narrow tubes.
Overall, our results indicate hysteretic behavior for IRE1 cluster dynamics on ER tubes (see Fig. 7). Under ER protein stress, the concentration of activated IRE1 will rise. Once the concentration has sufficiently risen (to approximately 1/µm 2 for the parameters of Fig. 7, comparable to physiological IRE1 concentrations of approximately 1/µm 2 [11,22,23]), IRE1 clusters can form. The available concentration of IRE1 in the ER network will decrease as IRE1 proteins join clusters and the clusters undergo Ostwald ripening. With time, as the actions stimulated by the unfolded protein response (UPR) path-way begin to mitigate ER protein stress, IRE1 may begin to deactivate, also reducing the pool of active IRE1 available for clustering. As the concentration of active IRE1 outside of clusters decreases, this concentration will first fall below the threshold concentration between cluster growth and decay for round clusters (∼ 0.4/µm 2 in Fig. 7), which is higher than this threshold concentration for wrapped clusters (∼ 0.25/µm 2 in Fig. 7) by approximately a factor π/2; and then fall below the threshold concentration between wrapped cluster growth and decay. Wrapped clusters will be the last to begin to decay as the active IRE1 concentration decreases. There is a gap between the threshold concentration between cluster growth and decay (low) and the concentration required for cluster formation (high). As concentration of active IRE1 proteins is rising from nearly zero, the concentration must become relatively high to cause cluster formation. Once formed, clusters will continue to grow at concentrations below the cluster formation threshold.
III. DISCUSSION
IRE1 protein clusters form on the surface of the endoplasmic reticulum (ER) under conditions of unfolded protein stress in the ER. As the ER is a network of tubes and sheets, many of these IRE1 clusters form on the tube surfaces. The tubular geometry allows a growing cluster to 'wrap' around the tube and grow without further increases to the cluster interface length. Using kinetic Monte Carlo simulations, we have shown that tube di- Fig. 6B). Clusters do not form until a relatively high IRE1 concentration is reached -as concentration decreases, clusters can grow below this high cluster formation threshold. Clusters will decay on wider tubes at a higher concentration than on narrower tubes, because clusters on narrower tubes are in a wrapped configuration and are more stable, compared to round clusters on wider tubes. Right: Schematic of cluster growth and decay trajectory. Concentration begins near zero, with no cluster. As the concentration increases past the cluster formation threshold, clusters will form and grow (black solid curve). As the concentration begins to decrease, the cluster growth diminishes (dashed black curve). Once the concentration is sufficiently low, first round clusters (blue solid curve) and then wrapped clusters (red solid curve) will decay. J = 5.3 kBT for both panels.
ameter affects cluster dynamics, as narrower tubes encourage cluster wrapping, affecting cluster growth and stability.
The cluster growth rate increases when the cluster is sufficiently large that the cluster has transitioned from a round to a wrapped conformation. When clusters are wrapped around the tube, they gain several advantages for growth relative to round clusters. Wrapped clusters have an interface that does not grow with increased cluster size, in contrast to round clusters which have an interface length that increases with the square root of the cluster area, and protein escape from clusters will increase with a longer interface. Wrapped clusters have a flat interface, while round clusters have a curved interface -lower interface curvature decreases protein escape from clusters [33]. Wrapped clusters are also able to encounter all proteins traversing the tube, in contrast to round clusters. However, clusters on wider tubes grow faster than those on narrower tubes, although clusters transition from round to wrapped at a smaller size on narrower tubes, due to the higher rate of proteins entering wider tubes exposed to the same external concentration, but with greater circumference, compared to narrower tubes. Overall we find that tube geometry is important to determining cluster growth rate, both because narrow tubes allow a wrapped cluster conformation that provides a growth increase relative to a round conformation, and because clusters on wider tubes grow more quickly.
We find that for clusters of a given size, wrapped clusters on narrower tubes decay significantly more slowly than round clusters on wider tubes. By examining cluster decay under conditions with no incoming proteins to grow the cluster, we isolated the impact of protein escape from clusters, showing that proteins escape from wrapped clusters more slowly than from round clusters. We attribute this slower escape from wrapped clusters to a shorter interface length and less curved interface compared to round clusters.
We find that the threshold IRE1 concentration for IRE1 cluster formation does not depend on tube diameter. However, the growth and decay of existing clusters depends on tube diameter, particularly through the control of cluster conformation (wrapped vs round) by the tube diameter. We show that wrapped clusters on narrower tubes are more stable than round clusters on wider tubes, by demonstrating that the threshold protein concentration between cluster growth and decay is higher for the round clusters on wide tubes than for equal-sized wrapped clusters on narrow tubes. This lower threshold concentration between growth and decay for wrapped clusters is attributed to less protein escape due to a shorter and less curved interface, and to the encounter of all proteins traversing the tube with the cluster. We predict that wrapped clusters will switch from growth to decay at protein concentrations that are lower by approximately a factor π/2, which is similar to our simulation results.
We also find that the threshold concentrations between growth and decay of existing clusters (for both round and wrapped clusters) are substantially lower than the cluster formation threshold. Cluster dynamics are thus hysteretic. Clusters do not form until a high concentration is reached, but once the clusters have formed, they can grow at concentrations substantially below the cluster formation concentration threshold. Cluster growth will decrease the concentration of available IRE1, evaporating all but the largest clusters; and the unfolded protein response (UPR) will also act to reduce the active IRE1 concentration. The cluster dynamics thus have a form of 'memory', because the cluster formation concentration threshold is higher than the threshold between the growth and decay of existing clusters.
It is known that Ostwald ripening will lead small clusters to decay prior to larger clusters [34]. Our results suggest that cluster wrapping adds an additional effect, where larger IRE1 clusters are able to wrap around the ER tube and decrease the threshold concentration be-tween cluster growth and decay. For wider ER tubes, clusters must be larger to achieve a wrapped conformation. As the cluster numbers decrease through Ostwald ripening, our results suggest that wrapped clusters, perhaps more likely to be found on narrow ER tubes, will be among the clusters that grow most persistently and will be among the last to begin to decay. The slower decay dynamics, at lower IRE1 concentration, of IRE1 clusters wrapped on narrow ER tubes thus control the persistence of IRE1 clustering and downstream signaling.
Our simulation results showing that sufficiently large clusters prefer to wrap around the ER tube align with experimental observations of IRE1 clusters. IRE1 clusters have been in observed with diverse morphologies, including those that appear wrapped around ER tubes [11,35] and localize to narrow ER tubes [35].
We have shown that domain geometry can affect cluster behavior, with cluster wrapping around tubes permitting narrower tubes to harbor more stable clusters. Previous work investigating autophagy receptor protein cluster formation on peroxisomes has shown that clusters are more likely to form and grow on larger spheres [8,9]. While both represent local geometric influence over cluster behavior, clusters on tubes enhance cluster stability by altering the cluster interface compared to round clusters, while cluster growth on larger spheres is mediated by initial formation of larger clusters with an Ostwald ripening growth advantage and the arrival of more proteins to larger spheres. Clusters forming on a finite two-dimensional surface have a minimum stable cluster size that increases as the finite two-dimensional area grows [33], similar to our finding that a smaller domain can increase the stability of smaller clusters.
Yeast ER tubes (mean diameter of 38 nm [23]) are significantly narrower than mammalian ER tubes (mean diameter of 96 nm [44]), suggesting IRE1 cluster wrapping could occur for smaller clusters and be more stable in yeast compared to mammalian cells. Recent experiments suggest that mammalian ER tubes have narrow (∼25 nm diameter) and wide (∼100 nm diameter) regions along their length, and that ER tubes in different cell types may be primarily the narrow or wide type [45]. Tube radii in both yeast (20 -50 nm diameter [23]) and mammalian (50 -140 nm diameter [44]) cells also vary widely. These observations, in combination with our results, suggest that IRE1 clusters could localize to or be more stable on narrow ER tube regions or narrower ER tubes. Clusters located in narrow regions or on narrow tubes would experience an energy difference if moved from a region where the cluster could wrap to where the cluster must be round (in addition to the energy barrier of cluster conformation change). It also suggests that cell types with largely narrow tubes could be more conducive to IRE1 cluster stability and persistence.
Many of the experiments that observe IRE1 clustering overexpressed IRE1. Despite work indicating that IRE1 clusters are important for UPR signaling [10,11,17,35], recent work [46,47] indicates that endogenous IRE1 lev-els in certain mammalian cells (approximately 1/µm 2 ) are significantly lower than when IRE1 is overexpressed, and that large IRE1 clusters may not be required for UPR signaling. However, it is noted that IRE1 concentration can vary between cells and for cells in pathological states [46,48,49]. For yeast, experiments demonstrate that IRE1 cluster formation upon ER stress in yeast is essential for UPR signaling [20,22,50] -we are not aware of any work suggesting otherwise. If there is a difference in IRE1 clustering behavior between yeast and mammalian cells, it is possible that the smaller ER tubes in yeast cells (∼40 nm diameter in yeast [23] vs ∼100 nm diameter in mammals [44]) combined with the increased cluster stability provided by cluster wrapping on narrower tubes could contribute to more IRE1 clustering behavior in yeast. Although large IRE1 clusters may not be required for UPR signaling to occur [46,47], there are IRE1 signaling modes that require larger IRE1 oligomers [18], and our work explores how the persistence of these signaling modes will be affected by IRE1 clustering and ER geometry.
IRE1 clustering on the ER may be tied to human health. Prolonged UPR activation (including IRE1 signaling) is associated with neurodegenerative diseases, suggesting a possible pathological role for dysfunctional IRE1 and other UPR signaling [51]. Insulin production in pancreatic beta cells induces ER stress and UPR (including IRE1) activation, and it has been proposed that UPR dysregulation may predispose individuals to diabetes [52]. Protein clustering on other organelles, such as mitochondria, suggests that further understanding of intracellular receptor clustering dynamics may be important for human health. Mitochondrial MAVS proteins aggregate as part of viral immunity. Typically dispersed on the membranes of many mitochondria, MAVS proteins aggregate on the membranes of a few mitochondria as part of antiviral innate immune response when a cell detects viral RNA, and are required for RLR viral infection signaling [7,53]. Persistent MAVS aggregation may play a role in lupus pathology [54].
With Monte Carlo simulations, we have found that narrow ER tubes facilitate an IRE1 cluster conformation that wraps around the tube, influencing cluster growth and increasing cluster stability and persistence. As IRE1 is an important protein in signaling of the unfolded protein response, our work shows that geometry can be an important factor for modulating cell signaling and maintaining cell health. To simulate IRE1 cluster dynamics on ER tubes we used the widely-applied kinetic Monte Carlo (kMC) algorith [36][37][38] with IRE1 proteins diffusing on a twodimensional lattice.
The diffusivity on a two-dimensional lattice with spacing ∆x is D = ∆x 2 /(4∆τ ), with τ the mean time between subsequent steps. The IRE1 diffusivity on the ER membrane has been measured as D = 0.24 µm 2 /s [11]. For the lattice spacing ∆x we will use the diameter of an IRE1 protein, which is approximately 10 nm [35]. This sets τ ≈ 10 −4 s.
While ER tube lengths have significant variation, many tube lengths fall in the range of 0.5 -3 µm in mammalian cells [55] and 250 -750 nm in yeast [23], and we use a tube length of 1 µm for all simulations. ER tube diameter has a wide range, in mammals largely falling from 50 -140 (mean 96 nm) [44] with narrower examples with diameters as small as 25 nm observed [35,45], and in yeast largely from 20 -50 nm (mean 38 nm) [23]. In our simulations, we explore diameters from 28 nm to 121 nm.
In the direction of the axis perpendicular to the length of the tube, a diffusing protein has periodic boundary conditions. While the results of Fig. 2 are for a closed tube without IRE1 protein exchange beyond the tube, all other results allow proteins to diffuse into and out of the tube ends. IRE1 proteins enter lattice sites at tube ends subject to a constant IRE1 protein concentration c ext boundary condition. With c ext expressed in units of proteins/µm 2 , and a concentration of 1/µm 2 = 10 −4 /lattice site, proteins enter empty edge lattice sites at a rate 10 −4 µm 2 c ext /(4τ ).
The IRE1 copy number in a mammalian cell has been estimated at 10 4 , and the ER surface area estimated as 10× the cell membrane area or 10 4 µm 2 , giving an IRE1 concentration of 1/µm 2 [11]. This corresponds to ∼0.31 IRE1 proteins on a tube with a 100 nm diameter and 1 µm length. The IRE1 copy number in a yeast cell has been estimated at 250 [22]. The ER volume in a 1 µm diameter bud has been estimated at 0.025 µm 3 [23], and an approximately spherical yeast cell has volume 42 µm 3 [24] corresponding to a 4.32 µm diameter, so scaling the ER volume by the cell/bud volume ratio, the ER volume in the mother cell is approximately 2 µm 3 . The ratio of volume to surface area in yeast ER is approximately 8 nm [23], so the yeast ER surface area is approximately 250 µm 2 . Thus the approximate IRE1 concentration on the ER surface is 1/µm 2 , very similar to the mammalian concentration. We explore IRE1 concentrations of zero to see how clusters will decay; within an order of magnitude of the physiological concentration 1/µm 2 ; and up to 34/µm 2 to encourage fast cluster growth.
IRE1 interaction strengths J in the range 3 -7 k B T were used. J = 5 -7 k B T lead to cluster formation near the cellular IRE1 concentration and J = 3 -4 k B T provide faster dynamics for efficient simulations.
The simulation allows IRE1 proteins forming a cluster to collectively diffuse, as IRE1 clusters freely diffuse on the ER membrane, rather than experiencing constrained diffusion or active transport [11]. Following experiments showing diffusivity in a membrane scaling with inverse radius [41] and consistent with the Stokes-Einstein relation [42], cluster diffusivity is scaled as 1/R where R is the cluster radius or N −1/2 , where N is the number of proteins in the cluster. In the simulation, IRE1 proteins can join clusters as multimers, but only escape as individual proteins. Clusters are also not permitted to leave the tube on the ends, with such Monte Carlo steps not allowed.
A cluster is considered wrapped if it has at least one protein in every row of the lattice. In Fig. 2B, the cluster conformation transition frequency is calculated by simulating for a fixed time, and then dividing the number of transitions by the simulation time. One transition is counted for each switch from round to wrapped conformation or vice versa. In Fig. 4A the round-to-wrapped transition time is the average of the first time the cluster transitions from round to wrapped and the final time the cluster transitions from round to wrapped, as the cluster can fluctuate between the two conformations. In Fig. 5B, the cluster decay rate is the slope between the first and last data point in a decay curve averaged over 30 simulations. Figure 8A,B,C shows the transition from a round cluster to a wrapped cluster. Figure 9A,B,C shows cluster formation with open boundary conditions for different times, after 2 hours of cluster development from a tube with a single protein (Fig. 9A), 4 hours (Fig. 9B), and 6 hours (Fig. 9A, which is shown in Fig. 3). Cluster formation, shown for various concentrations and tube diameters, does not appear to meaningfully change across these times 2 -6 hours. Figure 9D,E,F shows the probability of cluster decay for clusters having reached 10 proteins (Fig. 9D), 20 proteins (Fig. 9E), and 30 proteins (Fig. 9F). A cluster of size 10 is likely to decay, particularly at lower concentrations. However, clusters that have reached size 20 or 30 are very unlikely to decay, even at lower concentrations. This allows us to consider that clusters that have reach a size of 20 are unlikely to decay and the tube will continue to harbor a cluster. Figure 10A,B shows the change in cluster growth rates before and after a wrapping transition, as in Fig. 4B. Cluster growth rate increases after the wrapping transition.
Appendix B: Additional Simulations
Appendix C: Derivation
In the derivation of Eq. 4 there is an inverse length of the tube factor for the inward flux Φ in . Here we consider a protein taking steps on a discrete lattice away from an absorbing boundary, to represent the probability that a protein that has just entered the tube region under consideration (and is inside by a distance equal to a single lattice spacing) will reach a distance into the tube region under consideration without first reaching the absorbing boundary.
The probability that a protein a single lattice site into the tube will move to the second lattice site into the tube without hitting the absorbing boundary is 1/2. More generally, P n−1→n, w/o→0 is the probability that a protein n − 1 sites from the absorbing boundary will reach site n without first reaching the absorbing boundary at site 0, P n−1→n, w/o→0 = 1 2 + 1 2 P n−2→n−1, w/o→0 1 2 + 1 2 P n−2→n−1, w/o→0 2 1 2 + . . .
The first term in Eq. C1 represents a protein that immediately steps from site n − 1 to site n without other intervening steps. The second term in Eq. C1 represents a protein that first steps from site n−1 to site n−2 (probability 1/2) and then follows a trajectory that returns to site n − 1 without hitting the absorbing boundary (probability P n−2→n−1, w/o→0 ) and then takes a step to site n (probability 1/2). Subsequent terms involve m of these steps to site n − 2, and then trajectories that return to site n − 1 without hitting the absorbing boundary. Equation C3 is a sequential formula for P n−1→n, w/o→0 . We can start with P 1→2, w/o→0 = 1/2 and find P 2→3, w/o→0 = 2/3, and similarly that P 2→3, w/o→0 = 3/4. Generally, for P n−2→n−1, w/o→0 = (n − 2)/(n − 1) then P n−1→n, w/o→0 = 1 2 1 − 1 2
n − 2 n − 1 −1 (C4) = n − 1 n .(C5)
The probability of reaching a site n lattice sites from the absorbing boundary is then
The number of sites n = /∆x, so the probability of reaching a distance into the tube is ∆x/ . In the derivation of Eq. 4, the ∆x factor is absorbed into α. Mean growth rate before wrapping is 1.01 proteins/s, and after wrapping it is 1.14 proteins/s. The Kolmogorov-Smirnov (K-S) test returned a p-value of 4.27 × 10 −8 between the two distributions. (B) Tube diameter of 46 nm., Mean growth rate before wrapping is 1.47 protein/s, and after wrapping it is 1.75 proteins/s. The K-S test returned a p-value of 6.20 × 10 −10 between the two distributions. For both panels, J = 5.3 kBT as in Fig. 3, tube length is 1µm, cext = 1/µm 2 , and means averaged over 80 runs.
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FIG. 1 .
1Schematic diagram of the kinetic Monte Carlo simulation algorithm for IRE1 protein and cluster dynamics on an endoplasmic reticulum tube. Green circles are individual activated IRE1 proteins, which diffuse on a two-dimensional lattice as individual proteins and as clusters. Periodic boundary conditions (orange arrow) represent the tubular geometry.
FIG. 2 .
2Cluster conformation transitions. (A) Schematics of round (left inset) and wrapped (right inset) cluster conformations. Cluster interface lengths for round (blue), and wrapped (red and green) clusters on tubes (diameters 32 nm and 41 nm). (B) Frequency of transitions between round and wrapped clusters on a 32 nm diameter and 1 µm closed long tube, averaged over 24 runs; and corresponding probability of wrapped cluster conformation. (C) Probability of cluster conformation (wrapped or round) on a closed tube (no proteins enter or exit) for various protein concentrations, with 38, 48 and 57 nm tube diameter and 1 µm length. Simulations begin with all proteins in a round cluster at tube center. J = 3 kBT , average of 24 runs. (D)Cluster size at which the interface energy of wrapped clusters becomes energetically favored compared to round clusters (magenta, Eq. 2), cluster size at which 50% of clusters are in a wrapped conformation in simulations from (C) (cyan), and cluster size at which round cluster diameter is equal to tube circumference (blue).
FIG. 3 .
3Cluster formation with open boundary conditions. The IRE1 external (to the tube region under consideration) concentration cext and the tube diameter are varied, with color map indicating mean number of proteins in a cluster 6 hours after initializing with one protein in the tube, averaged over 20 runs, tube length of 1 µm, and J = 5.3 kBT .
E
round = E wrap FIG. 4. Cluster conformation affects cluster growth rate. (A) Individual cluster growth trajectories (thin lines) vs time, as well as linear fits of 80 cluster size trajectories before (thick blue) and after (thick red) round-to-wrapped transition, which is set at t = 0. J = 3 kBT , cext = 34/µm 2 , 30 nm tube diameter, and tube length of 1 µm. (B) Violin plot of the cluster growth rates before and after the cluster wrapping transition, with same parameters as (A). (C) Cluster size at wrapping transition vs tube diameter, with cluster size at which the interface energy of wrapped clusters becomes energetically favorable (purple, Eq. 2) and cluster size at which wrapping transition occurs during cluster growth simulations (cyan). Tube length 1 µm, J = 5.3 kBT , averaged over 30 samples. (D) Mean time period for a cluster to grow to a specific size (see legend) vs tube diameter. J = 5.3 kBT , cext = 1/µm 2 , tube length of 1 µm, averaged over 30 samples with initial cluster size of 30 proteins.
FIG. 5 .
5Fig. 5B), with 200protein wrapped clusters exhibiting n n,wrap = 2.44 and 200-protein round clusters n n,round = 1.95. This lower number of neighbors for round clusters aligns with the in-Cluster decay. (A) Mean cluster size vs time with external concentration cext = 0. Red lines are a wrapped cluster on a wide (96 nm diameter) tube and blue lines are a round cluster on a narrow (32 nm diameter) tube. J = 5 kBT (solid lines), J = 4 kBT (dashed), and J = 3 kBT (dotted). Cluster size is mean over 30 samples, with clusters initially 200 proteins, 1 µm tube length. (B) Cluster decay rates from decay curves in (A). Indicated n n,round and nn,wrap found by linear regression. Inset shows ratio of round cluster (wide tube) decay rate to wrapped cluster (narrow tube) decay rate, k decay,r /k decay,w .
FIG. 6 .
6Threshold concentration between cluster growth and decay. (A) Mean trajectories for clusters in tube with different external concentrations cext (each curve is a different cext). The lowest concentration is 0.1/µm 2 , highest concentration is 0.4/µm 2 , with concentration intervals of approximately 0.021/µm 2 , with 15 concentrations shown. Each concentration averaged over 20 runs. Clusters begin with 200 proteins, and begin as round unless tube is too narrow, in which case the cluster starts wrapped. (B) Threshold concentration c0 between cluster growth and decay as tube diameter is varied. Threshold concentration determined by linear fit of cluster growth rates near zero cluster growth. For diameters less than the diameter at which the threshold concentration transitions (suddenly jumps), clusters are wrapped; for diameters greater than the transition, clusters are round. Tube length 1 µm, J = 5.3kBT as in the phase diagram of Fig. 3.
FIG. 7 .
7Summary of IRE1 concentrations important for cluster dynamics. Left: Threshold concentrations for cluster formation (blue, seeFig. 3), decay of 100-and 200-protein clusters (cyan and magenta, respectively, from
ACKNOWLEDGMENTS
This work was supported by a Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant (A.I.B.) and by start-up funds provided by the Toronto Metropolitan University Faculty of Science (A.I.B.), and was enabled by computational resources provided by the Digital Research Alliance of Canada (al-
FIG. 8 .
8Snapshots of cluster wrapping transition. The lattice has periodic boundary conditions in the direction long the tube circumference. (A) Cluster is in an approximately round state. (B) Cluster is growing long in the direction of the circumference o the tube. (C) The cluster has wrapped around the tube. Closed boundary conditions at tube edges, J = 3kBT , and 35 proteins in approximately 32 nm diameter and length 500 nm tube.
FIG. 9 .
9Cluster formation with open boundary conditions. (A,B,C) mean IRE1 cluster size with open boundary conditions at the indicated times. The IRE1 external (to the tube region under consideration) concentration cext and the tube diameter are varied, with color map indicating mean number of proteins in a cluster. Clusters of sizes greater than 20 are binned with cluster size 20. (D,E,F) Cluster decay probability for clusters of different size with different external IRE1 concentrations cext. The left axis (magenta) is the probability that a cluster of size 10 (D), 20 (E), and 30 (F) will decay. The right axis is the number of clusters included in each decay probability calculation. Tube length 1 µm and J = 5.3 kBT . A B FIG. 10. Violin plot of the cluster growth rates before and after the cluster wrapping transition. (A) Tube diameter of 32 nm.
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for a 32 nm and 41 nm tube radius as the cluster size changes. Depending on the radius of the ER tube, if the cluster can wrap around the tube, a wrappedA
B
C
D
Tube diameter (nm)
Cluster
size
(proteins)
Cluster size (proteins)
Wrapped (d tube = 32 nm)
Wrapped (d tube = 41 nm)
Round
d tube = 38 nm
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| [] |
[
"Online Platt Scaling with Calibeating",
"Online Platt Scaling with Calibeating"
] | [
"Chirag Gupta ",
"Aaditya Ramdas "
] | [] | [] | We present an online post-hoc calibration method, called Online Platt Scaling (OPS), which combines the Platt scaling technique with online logistic regression. We demonstrate that OPS smoothly adapts between i.i.d. and non-i.i.d. settings with distribution drift. Further, in scenarios where the best Platt scaling model is itself miscalibrated, we enhance OPS by incorporating a recently developed technique called calibeating to make it more robust. Theoretically, our resulting OPS+calibeating method is guaranteed to be calibrated for adversarial outcome sequences. Empirically, it is effective on a range of synthetic and real-world datasets, with and without distribution drifts, achieving superior performance without hyperparameter tuning. Finally, we extend all OPS ideas to the beta scaling method. | 10.48550/arxiv.2305.00070 | [
"https://export.arxiv.org/pdf/2305.00070v1.pdf"
] | 258,426,330 | 2305.00070 | a679cddf372560009483d09b967d66b23062fc81 |
Online Platt Scaling with Calibeating
Chirag Gupta
Aaditya Ramdas
Online Platt Scaling with Calibeating
We present an online post-hoc calibration method, called Online Platt Scaling (OPS), which combines the Platt scaling technique with online logistic regression. We demonstrate that OPS smoothly adapts between i.i.d. and non-i.i.d. settings with distribution drift. Further, in scenarios where the best Platt scaling model is itself miscalibrated, we enhance OPS by incorporating a recently developed technique called calibeating to make it more robust. Theoretically, our resulting OPS+calibeating method is guaranteed to be calibrated for adversarial outcome sequences. Empirically, it is effective on a range of synthetic and real-world datasets, with and without distribution drifts, achieving superior performance without hyperparameter tuning. Finally, we extend all OPS ideas to the beta scaling method.
Introduction
In the past two decades, there has been significant interest in the ML community on post-hoc calibration of ML classifiers (Zadrozny and Elkan, 2002;Niculescu-Mizil and Caruana, 2005;Guo et al., 2017). Consider a pretrained classifier f : X Ñ r0, 1s that produces scores in r0, 1s for covariates in X . Suppose f is used to make probabilistic predictions for a sequence of points px t , y t q T t"1 where y t P t0, 1u. Informally, f is said to be calibrated (Dawid, 1982) if the predictions made by f match the empirically observed frequencies when those predictions are made: for all p P r0, 1s, Averagety t : f px t q « pu « p.
(1)
In practice, for well-trained f , larger scores f pxq indicate higher likelihoods of y " 1, so that f does well for accuracy or a ranking score like AUROC. Yet we often find that f does not satisfy (some formalized version of) condition (1). The goal of post-hoc calibration, or recalibration, 1 Carnegie mellon University, Pittsburgh PA, USA. Correspondence to: Chirag Gupta <[email protected]>. is to use additional held-out data to learn a low-complexity mapping m : r0, 1s Ñ r0, 1s so that mpf p¨qq retains the good properties of f -accuracy, AUROC, sharpness-as much as possible, but is better calibrated than f . The focus of this paper is on a recalibration method proposed by Platt (1999), commonly known as Platt scaling (PS). The PS mapping m is a sigmoid transform over f parameterized by two scalars pa, bq P R 2 : m a,b pf pxqq :" sigmoidpa¨logitpf pxqq`bq.
(
This set of mappings includes the identity mapping m 1,0 that recovers f . Figure 1a has additional illustrative m a,b plots; these are easily interpreted-if f is overconfident, that is if f pxq values are skewed towards 0 or 1, we can pick a P p0, 1q to improve calibration; if f is underconfident, we can pick a ą 1; if f is systematically biased towards 0 (or 1), we can pick b ą 0 (or b ă 0). The counter-intuitive choice a ă 0 can also make sense if f 's predictions oppose reality (perhaps due to a distribution shift). Given a batch of heldout data points, pa, bq is usually learnt by minimizing logloss over calibration data or equivalently maximizing loglikelihood under the model y i iid " Bernoullipm a,b pf px i qqq.
Although a myriad of recalibration methods now exist, PS remains an empirically strong baseline. In particular, PS is effective when few samples are available for recalibration (Niculescu-Mizil and Caruana, 2005;Gupta and Ramdas, 2021). Scaling before subsequent binning has emerged as a useful methodology (Kumar et al., 2019;Zhang et al., 2020). Multiclass adaptations of PS, called temperature, vector, and matrix scaling have become popular (Guo et al., 2017). Being a parametric method, however, PS comprises a limited family of post-hoc corrections-for instance, since m a,b is always a monotonic transform, PS must fail even for i.i.d. data for some data-generating distributions (see Gupta et al. (2020) for a formal proof). Furthermore, we are interested in going beyond i.i.d. data to data with drifting/shifting distribution. This brings us to our first question, (Q1) Can PS be extended to handle shifting or drifting data distributions?
A separate view of calibration that pre-dates the ML posthoc calibration literature is the online adversarial calibration framework (DeGroot and Fienberg, 1981; Foster and Vohra, Initialize weights w1 P R d " X At time t " 1, 2, . . . , T
• Observe features xt P R d • Predict pt " p1`e´w T t x t q´1
• Observe yt P t0, 1u
• Compute updated weight wt`1 P R d Goal: minimize regret ř T t"1 lpyt, ptq, where lpy, pq "´y log p´p1´yq logp1´pq.
(b) Online logistic regression
Expert (OPS)
The probability of rain is 75%.
Hedging (HOPS) Tracking (TOPS)
When you said 75% in the past, it has rained 85% of the time. So I will forecast 85%.
Calibeating I will play a hedging game on the instances where you said 75% so an adversary cannot fool me. 1998). Through the latter work, we know that calibration can be achieved for arbitrary y t sequences without relying on a pretrained model f or doing any other modeling over available features. This is achieved by hedging or randomizing over multiple probabilities, so that "the past track record can essentially only improve, no matter the future outcome" (paraphrased from Foster and Hart (2021)). For interesting classification problems, however, the y t sequence is far from adversarial and informative covariates x t are available. In such settings, covariate-agnostic algorithms achieve calibration by predicting something akin to an average ř t s"1 y s {t at time t`1 (see Appendix D). Such a prediction, while calibrated, is arguably not useful. A natural question is:
(Q2) Can informative covariates (features) be used to make online adversarial calibration practical?
We find an answer to (Q1) and (Q2) by leveraging the recently developed framework of calibeating (Foster and Hart, 2023), which is illustrated in Figure 1c. Calibeating shows how to perform certain corrections on top of pre-existing expert forecasts to improve (calibeat) them. The key calibeating idea relevant to our study is that adversarial calibration techniques can be used to simultaneously beat an expert and be provably calibrated.
For the sequential problem of being calibrated for a stream px t , y t q tě1 , where is one to find an expert forecaster that is both covariate-based and time-adaptive? One possibility is a time-series model-a sequence of predictors pf t : X Ñ r0, 1sq tě1 that are updated as more data becomes available, potentially using autoregression. Our forthcoming proposal is simpler and arguably requires less (domain) expertise.
Online adversarial post-hoc calibration
The proposal, summarized in Figure 2, is as follows. First, train any probabilistic classifier f on some part of the data. Then, perform online post-hoc calibration on top of f to get online adaptivity. In effect, this amounts to viewing f px t q as a scalar "summary" of x t , and the post-hoc mapping pm t : r0, 1s Ñ r0, 1sq tě1 becomes the time-series model over the scalar feature f px t q. Finally, apply calibeating on the post-hoc predictions m t pf px t qq to obtain adversarial validity. Figure 2 highlights our choice to do both post-hoc calibration and calibeating simultaneously on the streaming test data px t , y t q tě1 .
Such an online version of post-hoc calibration has not been previously studied to the best of our knowledge. We show how one would make PS online, to obtain Online Platt Scaling (OPS). OPS relies on a simple but crucial observation: PS is a two-dimensional logistic regression problem over "pseudo-features" logitpf px t qq. Thus the problem of learning OPS parameters is the problem of online logistic regression (OLR, see Figure 1b for a brief description). Several regret minimization algorithms have been developed for OLR (Hazan et al., 2007;Foster et al., 2018;Jézéquel et al., 2020). We consider these and find an algorithm with optimal regret guarantees that runs in linear time. These regret guarantees imply that OPS is guaranteed to perform as well as the best fixed PS model in hindsight for an arbitrarily distributed online stream px t , y t q tě1 , which includes the entire range of distribution drifts-i.i.d. data, data with covariate/label drift, and adversarial data. We next present illustrative experiments where this theory bears out impressively in practice.
Then, Section 2 presents OPS, Section 3 discusses calibeating, Section 4 presents baseline experiments on synthetic and real-world datasets. Section 5 discusses the extension of all OPS ideas to a post-hoc technique called beta scaling. The histogram of X t values in the corresponding time window is plotted with maximum height normalized to 1. Also plotted is the true curve for PrpY " 1 | X " xq and two predictive curves: a base model trained on t " 1 to t " 1000, and OPS-calibrated models with parameter values fixed at the start of the time window. The base model is accurate for the training data which is mostly in r´5, 10s, but becomes inaccurate and miscalibrated with the covariate-shifted values for larger t (bottom two subplots). OPS adapts well, agreeing with the base model in the top-right subplot, but flipping the base model predictions in the bottom-right subplot.
Illustrative experiments with distribution drift
Covariate drift. We generated data as follows. For t " 1, 2, . . . , 6000,
X t " N ppt´1q{250, 4q; Y t |X t " " Berp0.1q if modptX t {5u , 2q " 0, Berp0.9q if modptX t {5u , 2q " 1.(3)
Thus the distribution of Y t given X t is a fixed periodic function, but the distribution of X t drifts over time. The solid yellow line in Figure 3 plots PrpY " 1 | X " xq against x. We featurized x as a 48-dimensional vector with the components sin´x freq`t ranslation¯, where freq P t1, 2, 3, 4, 5, 6u and translation P t0, π{4, π{2, . . . 7π{4u.
A logistic regression base model f is trained over this 48dimensional representation using the points pX t , Y t q 1000 t"1 , randomly permuted and treated as a single batch of exchangeable points, which we will call training points. The points pX t , Y t q 6000 t"1001 form a supervised non-exchangeable test stream: we use this stream to evaluate f , recalibrate f using OPS, and evaluate the OPS-calibrated model.
Figure 3 displays f and the recalibrated OPS models at four ranges of t (one per plot). The training data has most x t -values in the range r´5, 10s as shown by the (heightnormalized) histogram in the top-left plot. In this regime, f is visually accurate and calibrated-the dotted light blue line is close to the solid yellow truth. We now make some observations at three test-time regimes of t:
(a) t " 1501 to t " 2000 (the histogram shows the distribution of px t q 2000 t"1501 ). For these values of t, the test data is only slightly shifted from the training data, and f continues to perform well. The OPS model recognizes the good performance of f and does not modify it much.
(b) t " 3500 to t " 4000. Here, f is "out-of-phase" with the true distribution, and Platt scaling is insufficient to improve f by a lot. OPS recognizes this, and it offers slightly better calibration and accuracy by making less confident predictions between 0.2 and 0.4.
(c) t " 5500 to t " 6000. In this regime, f makes predictions opposing reality. Here, the OPS model flips the prediction, achieving high accuracy and calibration.
These observations are quantitatively supported by the accuracy and 1 -calibration error (CE) values reported by the table in Figure 3a. Accuracy and CE values are estimated using the known true distribution of Y t | X t and the observed X t values, making them unbiased and avoiding some well-known issues with CE estimation. More details are provided in Appendix A.2.
Label drift. For t " 1, 2, . . . , 6000, data is generated as:
Y t " Bernoullip0.95p1´α t q`0.05α t q,
where α t " pt´1q{6000q;
X t |Y t " 1 tY t " 0u N p0, 1q`1 tY t " 1u N p2, 1q.(4)
Thus, X t | Y t is fixed while the label distribution drifts. We follow the same training and test splits described in the covariate drift experiment, but without sinusoidal featurization of X t ; the base logistic regression model is trained directly on the scalar X t 's. The gap between f and the true model increases over time but OPS adapts well ( Figure 4a).
Regression-function drift. For t " 1, 2, . . . , 6000, the data is generated as follows: α t " pt´1q{5000,
X t " N p0, 10q and Y t |X t " Bernoullipp t q, where (5) p t " " 0.1p1´α t q`0.5α t if modptX t {5u , 2q " 0, 0.9p1´α t q`0.5α t if modptX t {5u , 2q " 1.
Thus the distribution of X t is fixed, but the regression function PrpY t " 1 | X t q drifts over time. We follow the same training and test splits described in the covariate drift experiment, as well as the 48-dimensional featurization and logistic regression modeling. The performance of the base model worsens over time, while OPS adapts ( Figure 4b).
Online Platt scaling (OPS)
In a batch post-hoc setting, the Platt scaling parameters are set to those that minimize log-loss over the calibration data. If we view the first t instances in our stream as the calibration data, the fixed-batch Platt scaling parameters are,
pp a t , p b t q " arg min pa,bqPR 2 t ÿ s"1 lpm a,b pf px s qq, y s q,(6)
where lpp, yq "´y log p´p1´yq logp1´pq and m a,b is defined in (2). Observe that this is exactly logistic regression over the dataset plogitpf px s qq, y s q t s"1 . The thesis of OPS is that as more data is observed over time, we should use it to update the Platt scaling parameters. Define p OPS t :" m at,bt pf px t qq, where pa t , b t q depends on tpf px 1 q, y 1 q, . . . , pf px t´1 q, y t´1 qu. 1 One way to compare methods in this online setting is to consider regret R T with respect to a reference 2 -ball of radius B, B :" tpa, bq P R 2 : a 2`b2 ď B 2 u:
R T " T ÿ t"1 lpp OPS t , y t q´min pa,bqPB T ÿ t"1 lpm a,b pf px t qq, y t q. (7)
R T is the difference between the total loss incurred when playing pa t , b t q at times t ď T and the total loss incurred when playing the single optimal pa, bq P B for all t ď T . Typically, we are interested in algorithms that have low R T irrespective of how px t , y t q is generated.
Logarithmic worst-case regret bound for OPS
OPS regret minimization is exactly online logistic regression (OLR) regret minimization over "pseudo-features" logitpf px t qq. Thus our OPS problem is immediately solved (Hazan et al., 2007) e B log T T AIOLI (Jézéquel et al., 2020) B logpBT q T log T Aggregating Algorithm (AA) (Vovk, 1990;Foster et al., 2018) logpBT q B 18 T 24 Table 1: Asymptotic regret and running times of online logistic regression (OLR) algorithms for OPS as functions of the radius of reference class B and time-horizon T . For general OLR, regret and running times also depend on the dimension of X . However, OPS effectively reduces the dimensionality of X to 2, so that a second-order method like ONS runs almost as fast as a first-order method like OGD. Also note that B " ? a 2`b2 is small if the base model f is not highly miscalibrated. ONS with fixed hyperparameters was chosen for all OPS experiments; see Section 2.2 for implementation details.
using OLR methods. A number of OLR methods have been proposed, and we consider their regret guarantees and running times for the OPS problem. These bounds typically depend on T and two problem-dependent parameters: the dimension (say d) and B, the radius of B.
1. In our case, d " 2 since there is one feature logitpf pxqq and a bias term. Thus d is a constant. 2. B could technically be large, but in practice, if f is not highly miscalibrated, we expect small values of a and b which would in turn lead to small B. This was true in all our experiments.
Regret bounds and running times for candidate OPS methods are presented in Table 1, which is an adaptation of Step (ONS) as the best candidates for implementing OPS, since they both have Oplog T q regret and r OpT q running time. In the following theorem, we collect explicit regret guarantees for OPS based on ONS and AIOLI. Since the logloss can be unbounded if the predicted probability equals 0 or 1, we require some restriction on f px t q. Theorem 2.1. Suppose @t, f px t q P r0.01, 0.99s, B ě 1, and T ě 10. Then, for any sequence px t , y t q T t"1 , OPS based on ONS satisfies R T pONSq ď 2pe B`1 0Bq log T`1,
(8) while OPS based on AIOLI satisfies R T pAIOLIq ď 22B logpBT q.(9)
The ONS result follows from Hazan (2016, Theorem 4.5) and the AIOLI result follows from Jézéquel et al.
(2020, Theorem 1), plugging in the appropriate values for problem-dependent parameters; more details are in Appendix E. Since log-loss is a proper scoring rule (Gneiting and Raftery, 2007), minimizing it has implications for calibration (Bröcker, 2009). However, no "absolute" calibration bounds can be shown as discussed shortly in Section 2.4.
Hyperparameter-free ONS implementation
In our experiments, we found ONS to be significantly faster than AIOLI while also giving better calibration. Further, ONS worked without any hyperparameter tuning after an initial investigation was done to select a single set of hyperparameters. Thus we used ONS for experiments based on a verbatim implementation of Algorithm 12 in Hazan (2016), with γ " 0.1, ρ " 100, and K " tpa, bq : }pa, bq} 2 ď 100u. Algorithm 1 in the Appendix contains pseudocode for our final OPS implementation.
Follow-The-Leader as a baseline for OPS
The Follow-The-Leader (FTL) algorithm sets pa t , b t q " pp a t´1 , p b t´1 q (defined in (6)) for t ě 1. This corresponds to solving a logistic regression optimization problem at every time step, making the overall complexity of FTL ΩpT 2 q. Further, FTL has ΩpT q worst-case regret. Since full FTL is intractably slow to implement even for an experimental comparison, we propose to use a computationally cheaper variant, called Windowed Platt Scaling (WPS). In WPS the optimal parameters given all current data, pp a t , p b t q, are computed and updated every Op100q steps instead of at every time step. We call this a window and the exact size of the window can be data-dependent. The optimal parameters computed at the start of the window are used to make predictions until the end of that window, then they are updated for the next window. This heuristic version of FTL performs well in practice (Section 4).
Limitations of regret analysis
Regret bounds are relative with respect to the best in class, so Theorem 2.1 implies that OPS will do no worse than the best Platt scaling model in hindsight. However, even for i.i.d. data, the best Platt scaling model is itself miscalibrated on some distributions (Gupta et al., 2020, Theorem 3). This latter result shows that some form of binning must be deployed to be calibrated for arbitrarily distributed i.i.d. data. Further, if the data is adversarial, any deterministic predictor can be rendered highly miscalibrated (Oakes, 1985;Dawid, 1985); a simple strategy is to set y t " 1 tp t ď 0.5u. In a surprising seminal result, Foster and Vohra (1998) showed that adversarial calibration is possible by randomizing/hedging between different bins. The following section shows how one can perform such binning and hedging on top of OPS, based on a technique called calibeating.
Calibeating the OPS forecaster
Calibeating (Foster and Hart, 2023) is a technique to improve or "beat" an expert forecaster. The idea is to first use the expert's forecasts to allocate data to representative bins. Then, the bins are treated nominally: they are just names or tags for "groups of data-points that the expert suggests are similar". The final forecasts in the bins are computed using only the outcome (y t ) values of the points in the bin (seen so far), with no more dependence on the expert's original forecast. The intuition is that forecasting inside each bin can be done in a theoretically valid sense, irrespective of the theoretical properties of the expert.
We will use the following " -bins" to perform calibeating:
B 1 " r0, q, B 2 " r , 2 q, . . . , B m " r1´ , 1s. (10)
Here ą 0 is the width of the bins, and for simplicity we assume that m " 1{ is an integer. For instance, one could set " 0.1 or the number of bins m " 10, as we do in the experiments in Section 4. Two types of calibeating-tracking and hedging-are described in the following subsections. We suggest recalling our illustration of calibeating in the introduction ( Figure 1c).
Calibeating via tracking past outcomes in bins
Say at some t, the expert forecasts p t P r0.7, 0.8q. We look at the instances s ă t when p s P r0.7, 0.8q and computē y b t´1 " Averagety s : s ă t, p s P r0.7, 0.8qu. Suppose we find thatȳ b t´1 " 0.85. That is, when the expert forecasted bin r0.7, 0.8q in the past, the average outcome was 0.85. A natural idea now is to forecast 0.85 instead of 0.75. We call this process "Tracking", and it is the form of calibeating discussed in Section 4 of Foster and Hart (2023). In our case, we treat OPS as the expert and call the tracking version of OPS as TOPS. If p OPS
t P B b , then p TOPS t :" Averagety s : s ă t, p OPS s P B b u.(11)
The average is defined as the mid-point of B b if the set above is empty.
Foster and Hart (2023) showed that the Brier-score of the TOPS forecasts p TOPS t , defined as 1 T ř T t"1 py t´p TOPS t q 2 , is better than the corresponding Brier-score of the OPS forecasts p OPS t , by roughly the squared calibration error of p OPS t (minus a log T term). In the forthcoming Theorem 3.1, we derive a result for a different object that is often of interest in post-hoc calibration, called sharpness.
Segue: defining sharpness of forecasters
Recall the -bins introduced earlier (10). Define N b " |tt ď T : p t P B b u| and p
y b " 1 N b ř tďT,ptPB b y t if N b ą 0, else p y b " 0.
Sharpness is defined as,
SHPpp 1:T q :" 1 T m ÿ b"1 N b¨p y 2 b . 2 (12)
If the forecaster is perfectly knowledgeable and forecasts p t " y t , its SHP equals ř T t"1 y t {T ":ȳ T . On the other hand, if the forecaster puts all points into a single bin b, its SHP equals p ř T t"1 y t {T q 2 "ȳ 2 T . The former forecaster is precise or sharp, while the latter is not, and SHP captures this-it can be shown thatȳ 2 T ď SHPpp 1:T q ďȳ T . We point the reader to Bröcker (2009) for further background. One of the goals of effective forecasting is to ensure high sharpness (Gneiting et al., 2007). OPS achieves this goal by relying on the log-loss, a proper scoring rule. The following theorem shows that TOPS suffers a small loss in SHP compared to OPS.
Theorem 3.1. The sharpness of TOPS forecasts satisfies
SHPpp TOPS 1:T q ě SHPpp OPS 1:T q´ ´ 2 4´l og T`1 T .(13)
The proof (in Appendix E) uses Theorem 3 of Foster and Hart (2023) and relationships between sharpness, Brierscore, and a quantity called refinement. If T is fixed and known, setting « a log T {T (including constant factors), or equivalently, the number of bins B « a T { log T gives a rate of r Op a 1{T q for the SHP difference term. While we do not show a calibration guarantee, TOPS had the best calibration performance in most experiments (Section 4)
Calibeating via hedging or randomized prediction
All forecasters introduced so far-the base model f , OPS, and TOPS-make forecasts p t that are deterministic given the past data until time t´1. If the y t sequence is being generated by an adversary that acts after seeing p t , then the adversary can ensure that each of these forecasters is miscalibrated by setting y t " 1 tp t ď 0.5u.
Suppose instead that the forecaster is allowed to hedgerandomize and draw the forecast from a distribution instead of fixing it to a single value-and the adversary only has access to the distribution and not the actual p t . Then there exist hedging strategies that allow the forecaster to be arbitrarily well-calibrated (Foster and Vohra, 1998). In fact, Foster (1999, henceforth F99) showed that this can be done while hedging between two arbitrarily close points in r0, 1s.
In practice, outcomes are not adversarial, and covariates are available. A hedging algorithm that does not use covariates cannot be expected to give informative predictions. We verify this intuition through an experiment in Appendix on historical rain data D-F99's hedging algorithm simply predicts the average y t value in the long run.
A best-of-both-worlds result can be achieved by using the expert forecaster to bin data using x t values, just like we did in Section 3.1. Then, inside every bin, a separate hedging algorithm is instantiated. For the OPS predictor, this leads to HOPS (OPS + hedging). Specifically, in our experiments and the upcoming calibration error guarantee, we used F99:
p HOPS t :" F99py s : s ă t, p s P B b q.(14)
A standalone description of F99 is included in Appendix C. F99 hedges between consecutive mid-points of the -bins defined earlier (10). The only hyperparameter for F99 is .
In the experiments in the main paper, we set " 0.1. To be clear, p t is binned on the -bins, and the hedging inside each bin is again over the -bins.
The upcoming theorem shows a SHP lower bound on HOPS.
In addition, we show an assumption-free upper bound on the ( 1 -)calibration error, defined as
CEpp 1:T q :" 1 T m ÿ b"1 N b¨| p p b´p y b | ,(15)
where N b , p y b were defined in Section 3.2, and p
p b " 1 N b ř tďT,ptPB b p t , if N b ą 0, else p p b " mid-pointpB b q.
Achieving small CE is one formalization of (1). The following result is conditional on the y 1:T , p OPS 1:T sequences. The expectation is over the randomization in F99. Theorem 3.2. For adversarially generated data, the expected sharpness of HOPS forecasts using the forecast hedging algorithm of Foster (1999) is lower bounded as
E " SHPpp HOPS 1:T q ‰ ě SHPpp OPS 1:T q´ˆ `l og T`1 2 T˙,(16)
and the expected calibration error of HOPS satisfies,
E " CEpp HOPS 1:T q ‰ ď {2`2 a 1{ T .(17)
The proof in Appendix E is based on Theorem 5 of Foster and Hart (2023) and a CE bound for F99 based on Blackwell approachability (Blackwell, 1956). With " r ΘpT´1 {3 q, the difference term in the SHP bound is r OpT´1 {3 q and the CE bound is also r OpT´1 {3 q. High-probability versions of (17) can be derived using probabilistic Blackwell approachability lemmas, such as those in Perchet (2014).
Experiments
We perform experiments with synthetic and real-data, in i.i.d. and distribution drift setting. Code to reproduce the experiments can be found at https://github.com/aigen/ df-posthoc-calibration (see Appendix A.4 for more details). All baseline and proposed methods are described in Collection 1 on the following page. In each experiment, the base model f was a random forest (sklearn's Collection 1. Proposed and baseline methods for online post-hoc calibration. Final forecasts are identified in blue.
Input: f : X Ñ r0, 1s, any pre-learnt model Input: px 1 , y 1 q, px 2 , y 2 q, . . . , px T , y T q P Xˆt0, 1u
Input: calibration-set-size T cal ă T , window-size W Fixed Platt scaling: pa FPS , b FPS q Ð pp a Tcal , p b Tcal q (eq. 6) Windowed Platt scaling: pa WPS , b WPS q Ð pa FPS , b FPS q Online Platt scaling: pa OPS 1 , b OPS 1 q Ð p1, 0q for t " 2 to T do pa OPS t , b OPS t q Ð ONSppx 1 , y 1 q, . . . , px t´1 , y t´1 qq (ONS is Algorithm 1 in the Appendix) end for for t " T cal`1 to T do p BM t Ð f px t q p FPS t Ð sigmoidpa FPS¨l ogitpf px t qq`b FPS q p WPS t Ð sigmoidpa WPS¨l ogitpf px t qq`b WPS q p OPS t Ð sigmoidpa OPS t¨l ogitpf px t qq`b OPS t q p TOPS t
is set using past py s , p OPS s q values as in (11) p HOPS t is set using past py s , p OPS s q values as in (14) If mod pt´T cal , W q " 0, pa WPS , b WPS q Ð pp a t , p b t q end for implementation). All default parameters were used, except n estimators was set to 1000. No hyperparameter tuning on individual datasets was performed for any of the recalibration methods.
Metrics. We measured the SHP and CE metrics defined in (12) and (15) Reading the plots. The plots we report show CE values at certain time-stamps starting from T cal`2 W and ending at T (see third line of Collection 1). T cal and W are fixed separately for each dataset ( Table 2 in Appendix). We also generated SHP plots, but these are not reported since the drop in SHP was always very small.
Experiments on real datasets
We worked with four public datasets in two settings. Links to the datasets are in Appendix A.1.
Distribution drift. We introduced synthetic drifts in the data based on covariate values, so this is an instance of covariate drift. For example, in the bank marketing dataset (leftmost plot in Figure 5), the problem is to predict which clients are likely to subscribe to a term deposit if they are targeted for marketing, using covariates like age, education, and bank-balance. We ordered the available 12000 rows roughly by age by adding a random num- Fetal Health Figure 5: Drifting data. CE (calibration error) values over time of considered models on four datasets with synthetically induced drifts. The plots have invisible error bars since variation across the 100 runs was small. OPS consistently performs better than BM, FPS, and WPS, while TOPS is the best-performing among all methods across datasets and time. All methods had roughly the same SHP values at a given time-step, so the SHP plots are delayed to Appendix A (Figure 8).
ber uniformly from t´1, 0, 1u to age and sorting all the data. Training is done on the first 1000 points, T cal " 1000, and W " 500. Similar drifts are induced for the other datasets, and T cal , W values are set depending on the total number of points; further details are in Appendix A.1.
All simulations were performed 100 times and the average CE and SHP values with˘std-deviation errorbars were evaluated at certain time-steps. Thus, our lines correspond to estimates of the expected values of CE and SHP, as indicated by the Y-axis labels. We find that across datasets, OPS has the least CE among non-calibeating methods, and both forms of calibeating typically improve OPS further (Figure 5). Specifically, TOPS performs the best by a margin compared to other methods. We also computed SHP values, which are reported in Appendix A (Figure 8). The drop in SHP is insignificant in each case (around 0.005).
IID data. This is the usual batch setting formed by shuffling all available data. Part of the data is used for training and the rest forms the test-stream. We used the same values of T cal and W as those used in the data drift experiments (see Appendix A.1). In our experiments, we find that the gap in CE between BM, FPS, OPS, and WPS is smaller ( Figure 6). However, TOPS performs the best in all scenarios, typically by a margin. Here too, the change in SHP was small, so those plots were delayed to Appendix A (Figure 9).
Synthetic experiments
In all experiments with real data, WPS performs almost as good as OPS. In this subsection, we consider some synthetic data drift experiments where OPS and TOPS continue performing well, but WPS performs much worse. Covariate drift. Once for the entire process, we draw random orthonormal vectors v 1 , v 2 P R 10 ( v 1 2 " v 2 2 " 1, v T 1 v 2 " 0), a random weight vector w P t´1, 1u 10`p 10 2 q with each component set to 1 or´1 independently with probability 0.5, and set a drift parameter δ ě 0. The data is generated as follows:
u t " v 1 cospδtq`v 2 sinpδtq, X t " N p0, I 10`1 0u t u T t q, Y t |X t " Bernoullipsigmoidpw T r X t qq, where r X t " rx 1 , . . . , x 10 , x 1 x 2 , x 1 x 3 , .
. . , x 9 x 10 s P R 10`p 10 2 q .
Thus the distribution of Y t given X t is fixed as a logistic model over the expanded representation r X t that includes all cross-terms (this is unknown to the forecaster who only sees X t ). The features X t themselves are normally distributed with mean 0 and a time-varying covariance matrix. The principal component (PC) of the covariance matrix is a vector u t that is rotating on the two-dimensional plane containing the orthonormal vectors v 1 and v 2 . The first 1000 points are used as training data, the remaining T " 5000 form a test-stream, and W " 500. We report results in two settings: one is i.i.d., that is δ " 0, and the other is where the u for the first and last point are at a 180˝angle (Figure 7a).
Label drift. Given some δ ą 0, pX t , Y t q is generated as:
Y t " Bernoullip0.5`δtq, X t |Y t " 1 tY t " 0u N p0, R 10 q`1 tY t " 1u N pe 1 , R 10 q.
Thus P pY 1 " 1q " 0.5`δ and for the last test point, P pY 6000 " 1q " 0.5`6000δ. This final value can be set to control the extent of label drift; we show results with no drift (i.e., δ " 0, Figure 7b left) and δ set so that final bias 0.5`6000δ " 0.9 (Figure 7b right). The number of training points is 1000, T " 5000, and W " 500.
Changing , histogram binning, beta scaling
In Appendix A.3, we report versions of Figures 5, 6 with " 0.05, 0.2 (instead of " 0.1) with similar conclusions (Figures 12, 13). We also perform comparisons with a windowed version of the popular histogram binning method (Zadrozny and Elkan, 2001) and online versions of the beta scaling method, as discussed in the forthcoming Section 5.
Online beta scaling with calibeating
A recalibration method closely related to Platt scaling is beta scaling (Kull et al., 2017). The beta scaling mapping m has three parameters pa, b, cq P R 3 , and corresponds to a sigmoid transform over two pseudo-features derived from f pxq: logpf pxqq and logp1´f pxqq, m a,b,c pf pxqq :" sigmoidpa¨logpf pxqqb¨l ogp1´f pxqq`cq.
Observe that enforcing b "´a recovers Platt scaling since logitpzq " logpzq´logp1´zq. The beta scaling parameters can be learnt following identical protocols as Platt scaling: (i) the traditional method of fixed batch post-hoc calibration akin to FPS, (ii) a natural benchmark of windowed updates akin to WPS, and (iii) regret minimization based method akin to OPS. This leads to the methods FBS, WBS, and OBS, replacing the "P" of Platt with the "B" of beta. Tracking + OBS (TOBS) and Hedging + OBS (HOBS) can be similarly derived. Further details on all beta scaling methods are in Appendix B, where we also report plots similar to Figures 5, 6 for beta scaling (Figure 14). In a comparison between histogram binning, beta scaling, Platt scaling, and their tracking versions, TOPS and TOBS are the best-performing methods across experiments ( Figure 15).
Summary
We provided a way to bridge the gap between the online (typically covariate-agnostic) calibration literature, where data is assumed to be adversarial, and the (typically i.i.d.) post-hoc calibration literature, where the joint covariateoutcome distribution takes centerstage. First, we adapted the post-hoc method of Platt scaling to the online setting, based on a reduction to logistic regression, to give our OPS algorithm. Second, we showed how calibeating can be applied on top of OPS to give further improvements.
The TOPS method we proposed has the lowest calibration error in all experimental scenarios we considered. On the other hand, the HOPS method which is based on online adversarial calibration provably controls miscalibration at any predefined level and could be a desirable choice in sensitive applications. The good performance of OPS+calibeating lends further empirical backing to the thesis that scaling+binning methods perform well in practice, as has also been noted in prior works (Zhang et al., 2020;Kumar et al., 2019). Our theoretical results formalize this empirical observation.
We note a few directions for future work. First, online algorithms that control regret on the most recent data have been proposed (Hazan and Seshadhri, 2009;Zhang et al., 2018). These approaches could give further improvements on ONS, particularly for drifting data. Second, while this paper entirely discusses calibration for binary classification, all binary routines can be lifted to achieve multiclass notions such as top-label or class-wise calibration (Gupta and Ramdas, 2022b). Alternatively, multiclass versions of Platt scaling (Guo et al., 2017) such as temperature and vector scaling can also be targeted directly using online multiclass logistic regression (Jézéquel et al., 2021
A Experimental details and additional results
Some implementation details, metadata, information on metrics, and additional results and figures are collected here. Figure 11: The adaptive behavior of OPS for the simulated regression-function drift scenario described in Section 1.2. Table 2 contains metadata for the datasets we used in Section 4.1. T train refers to the number of training examples. The "sort-by" column indicates which covariate was used to order data points. In each case some noise was added to the covariate in order to create variation for the experiments. The exact form of drift can be found in the python file sec 4 experiments core.py in the repository https://github.com/AIgen/ df-posthoc-calibration/tree/main/Online%20Platt%20Scaling%20with%20Calibeating.
A.2 Additional plots and details for label drift and regression-function drift experiments from Section 1
Figures 3, 10, and 11 report accuracy (Acc) and calibration error (CE) values for the base model and the OPS model in the three dataset drift settings we considered. The Acc values are straightforward averages and can be computed without issues. However, estimation of CE on real datasets is tricky and requires sophisticated techniques such as adaptive binning, debiasing, heuristics for selecting numbers of bins, or kernel estimators (Kumar et al., 2019;Roelofs et al., 2022;Widmann et al., 2019). The issue typically boils down to the fact that PrpY " 1 | X " xq cannot be estimated for every x P X without making smoothness assumptions or performing some kind of binning. However, in the synthetic experiments of Section 1, PrpY " 1 | Xq is known exactly, so such techniques are not required. For some subset of forecasts p s , p 2 , . . . , p t , we compute
CE " 1 t´s`1 t ÿ i"s |p i´P rpY i " 1 | X i " x i q| ,
on the instantiated values of X s , X s`1 , . . . , X t . Thus, what we report is the true CE given covariate values.
A.3 Additional results with windowed histogram binning and changing bin width
Comparison to histogram binning (HB). HB is a recalibration method that has been shown to have excellent empirical performance as well as theoretical guarantees (Zadrozny and Elkan, 2001;Gupta and Ramdas, 2021). There are no online versions of HB that we are aware of, so we use the same windowed approach as windowed Platt and beta scaling for benchmarking (see Section 2.3 and the second bullet in Section B). This leads to windowed histogram binning (WHB), the fixed-batch HB recalibrator that is updated every Op100q time-steps. We compare WHB to OPS and OBS (see Section 5). Since tracking improves both OPS and OBS, we also consider tracking WHB. Results are presented in Figure 15.
We find that WHB often performs better than OPS and OBS in the i.i.d. case, and results are mixed in the drifting case. However, since WHB is a binning method, it inherently produces something akin to a running average, and so tracking does not improve it further. The best methods (TOPS, TOBS) are the ones that combine one of our proposed parametric online calibrators (OPS, OBS) with tracking.
Changing the bin width . In the main paper, we used " 0.1 and defined corresponding bins as in (10). This binning reflects in three ways on the experiments we performed. First, -binning is used to divide forecasts into representative bins before calibeating (equations (11), (14)). Second, -binning is used to define the sharpness and calibration error metrics. Third, the hedging procedure F99 requires specifying a binning scheme, and we used the same -bins.
Here, we show that the empirical results reported in the main paper are not sensitive to the chosen representative value of " 0.1. We run the same experiment used to produce Figures 5 and 6 but with " 0.05 ( Figure 12) and " 0.2 ( Figure 13). The qualitative results remain identical, with TOPS still the best performer and hardly affected by the changing epsilon. In fact, the plots for all methods except HOPS are indistinguishable from their " 0.1 counterparts at first glance. HOPS is slightly sensitive to : the performance improves slightly with " 0.05, and worsens slightly with " 0.2.
A.4 Reproducibility
All results in this paper can be reproduced exactly, including the randomization, using the IPython notebooks that can be found at https://github.com/aigen/df-posthoc-calibration in the folder Online Platt scaling with Calibeating. The README page in the folder contains a
Algorithm 1 Online Newton
Step for OPS (based on Hazan (2016, Algorithm 12)) Input: K " tpx, yq : }px, yq} 2 ď 100u, time horizon T , and initialization parameter pa OPS 1 , b OPS 1 q " p1, 0q ": θ 1 P K Hyperparameters: γ " 0.1, ρ " 100 Set A 0 " ρI 2 for t " 1 to T do Play θ t , observe log-loss lpm θt pf px t qq, y t q and its gradient ∇ t :" ∇ θt lpm θt pf px t qq, y t q A t " A t´1`∇t ∇ T t Newton step: r θ t`1 " θ t´1 γ A´1 t ∇ t Projection: pa OPS t`1 , b OPS t`1 q " θ t`1 " arg min θPK p r θ t`1´θ q T A t p r θ t`1´θ q end for B Online beta scaling This is an extended version of Section 5, with some repetition but more details. A recalibration method closely related to Platt scaling is beta scaling (Kull et al., 2017). The beta scaling mapping m has three parameters pa, b, cq P R 3 , and corresponds to a sigmoid transform over two pseudo-features derived from f pxq: logpf pxqq and logp1´f pxqq: m a,b,c pf pxqq :" sigmoidpa¨logpf pxqq`b¨logp1´f pxqq`cq.
Observe that enforcing b "´a recovers Platt scaling since logitpzq " logpzq´logp1´zq. The beta scaling parameters can be learnt following identical protocols as Platt scaling.
• The traditional method is to optimize parameters by minimizing the log-likelihood (equivalently, log-loss) over a fixed held-out batch of points.
• A natural benchmark for online settings is to update the parameters at some frequency (such as every 50 or 100 steps). At each update, the beta scaling parameters are set to the optimal value based on all data seen so far, and these parameters are used for prediction until the next update occurs. We call this benchmark windowed beta scaling (WBS); it is analogous to the windowed Platt scaling (WPS) benchmark considered in the main paper.
• Our proposed method for online settings, called online Beta scaling (OBS), is to use a log-loss regret minimization procedure, similar to OPS. Analogously to (7), R T for OBS predictions p OBS t " m at,bt,ct pf px t qq is defined as
R T pOBSq " T ÿ t"1 lpp OBS t , y t q´min pa,b,cqPB T ÿ t"1 lpm a,b,c pf px t qq, y t q,(19)
where B :" tpa, b, cq P R 3 : a 2`b2`c2 ď B 2 u for some B P R, and l is the log-loss. We use online Newton step (Algorithm 1) to learn pa t , b t , c t q, with the following initialization and hyperparameter values:
-K " tpx, y, zq : }px, y, zq} 2 ď 100u, pa OBS 1 , b OBS 1 , c OBS 1 q " p1, 1, 0q; γ " 0.1, ρ " 25, A 0 " ρI 3 .
These minor changes have to be made simply because the dimensionality changes from two to three. The empirical results we present shortly are based on an implementation with exactly these fixed hyperparameter values that do not change across the experiments (that is, we do not do any hyperparameter tuning).
Due to the additional degree of freedom, beta scaling is more expressive than Platt scaling. In the traditional batch setting, it was demonstrated by Kull et al. (2017) that this expressiveness typically leads to better (out-of-sample) calibration performance. We expect this relationship between Platt scaling and beta scaling to hold for their windowed and online versions as well. We confirm this intuition through an extension of the real dataset experiments of Section 4.1 to include WBS and OBS (Figure 14). In the main paper we reported that the base model (BM) and fixed-batch Platt scaling model (FPS) perform the worst by a margin, so these lines are not reported again. We find that OBS performs better than both OPS and WBS, so we additionally report the performance of calibeating versions of OBS instead of OPS. That is, we replace OPS + tracking (TOPS) with OBS + tracking (TOBS), and OPS + hedging (HOPS) with OBS + hedging (HOBS).
A regret bound similar to Theorem 2.1 can be derived for OBS by instantiating ONS and AIOLI regret bounds with d " 3 (instead of d " 2 as done for OPS). The calibeating theorems (3.1 and 3.2) hold regardless of the underlying expert, and so also hold for OBS.
C F99 online calibration method
We describe the F99 method proposed by Foster (1999), and used in our implementation of HOPS (Section 3.3). The description is borrowed with some changes from Gupta and Ramdas (2022a). Recall that the F99 forecasts are the mid-points of the -bins (10): B 1 " r0, q, B 2 " r , 2 q, . . . , B m " r1´ , 1s. For b P rms :" t1, 2, . . . , mu and t ě 1, define:
(mid-point of B b ) m b " pb´0.5q{m " b ´ {2, (left end-point of B b ) l b " pb´1q{m " pb´1q , (right end-point of B b ) r b " b{m " b ,
F99 maintains some quantities as more data set is observed and forecasts are made. These are,
(frequency of forecasting m b ) N t b " |t1 tp s " m b u : s ď tu| , (observed average when m b was forecasted) p t b " # ř t s"1 y s 1 tp s " m b u {N t b if N t b ą 0 m b if N t b " 0, (deficit) d t b " l b´p t b , (excess) e t b " p t b´rb .
The terminology "deficit" is used to indicate that p t b is smaller l b similarly. "Excess" is used to indicate that p t b is larger than r b similarly. The F99 algorithm is as follows. Implicit in the description is computation of the quantities defined above. .
These randomization probabilities are revealed before y t`1 is set by the agent that is generating outcomes, but the actual p t value is drawn after y t`1 is revealed.
D Forecasting climatology to achieve calibration
Although Foster and Vohra's result (1998) guarantees that calibrated forecasting is possible against adversarial sequences, this does not immediately imply that the forecasts are useful in practice. To see this, consider an alternating outcome sequence, y t " 1 tt is odd u. The forecast p t " 1 tt is odd u is calibrated and perfectly accurate. The forecast p t " 0.5 (for every t) is also calibrated, but not very useful.
Thus we need to assess how a forecaster guaranteed to be calibrated for adversarial sequences performs on real-world sequences. In order to do so, we implemented the F99 forecaster (described in Appendix C), on Pittsburgh's hourly rain data from January 1, 2008, to December 31, 2012. The data was obtained from ncdc.noaa.gov/cdo-web/. All days on which the hourly precipitation in inches (HPCP) was at least 0.01 were considered as instances of y t " 1. There are many missing rows in the data, but no complex data cleaning was performed since we are mainly interested in a simple illustrative simulation. F99 makes forecasts on an -grid with " 0.1: that is, the grid corresponds to the points p0.05, 0.15, . . . , 0.95q. We observe (Figure 16) that after around 2000 instances, the forecaster always predicts 0.35. This is close to the average number of instances that it did rain which is approximately 0.37 (this long-term average is also called climatology in the meteorology literature). Although forecasting climatology can make the forecaster appear calibrated, it is arguably not a useful prediction given that there exist expert rain forecasters who can make sharp predictions for rain that change from day to day.
E Proofs
Proof of Theorem 2.1. The regret bounds for ONS and AIOLI depend on a few problem-dependent parameters.
• The dimension d " 2.
• The radius of the reference class B.
• Bound on the norm of the gradient, which for logistic regression is also the radius of the space of input vectors. Due to the assumption on f px t q, the norm of the input is at most a logitp0.01q 2`12 " a logitp0.99q 2`12 ď 5.
The AIOLI bound (9) follows from Theorem 1, equation (4) In writing the proofs of the results in Section 3, we will use an object closely connected to sharpness called refinement. For a sequence of forecasts p 1:T and outcome sequence y 1:T , the refinement R is defined as
Rpp 1:T q :" 1 T m ÿ b"1 N b¨p y b p1´p y b q,(20)
where p y b is the average of the outcomes in every -bin; see the beginning of Section 3.2 where sharpness is defined. The function xpP r0, 1sq Þ Ñ xp1´xq is minimized at the boundary points t0, 1u and maximized at 1{2. Thus refinement is lower if p y b is close to 0 or 1, or in other words if the bins discriminate points well. This is captured formally in the following (well-known) relationship between refinement and sharpness.
Lemma E.1 (Sharpness-refinement lemma). For any forecast sequence p 1:T , the refinement R defined in (20) and the sharpness SHP defined in (12) are related as:
Rpp 1:T q "ȳ T´S HPpp 1:T q,
whereȳ T " 1 T ř T t"1 y t .
Proof. Observe that
Rpp 1:T q " 1 T B ÿ b"1 N b p y b´1 T B ÿ b"1 N b p y 2 b " 1 T B ÿ b"1 N b p y b´S HPpp 1:T q.
The final result follows simply by noting that
B ÿ b"1 N b p y b " B ÿ b"1˜ÿ tďT,ptPB b y t¸" T ÿ t"1 y t .
We now state a second lemma, that relates R to the Brier-score BS defined as
BSpp 1:T q :" ř T t"1 py t´pt q 2 T .
Unlike R and SHP, BS is not defined after -binning. It is well-known (see for example equation (1) of FH23) that if refinement is defined without -binning (or if the Brier-score is defined with -binning), then refinement is at most the Brier-score defined above. Since we define R defined with binning, further work is required to relate the two.
Lemma E.2 (Brier-score-refinement lemma). For any forecast sequence p 1:T and outcome sequence y 1:T , the refinement R and the Brier-score BS are related as
Rpp 1:T q ď BSpp 1:T q`
2 4` ,(22)
where is the width of the bins used to define R (10).
Proof. Define the discretization function disc : r0, 1s Ñ r0, 1s as discppq " mid-pointpB b q ðñ p P B b . Note that for all p P r0, 1s, |p´discppq| ď {2. Based on standard decompositions (such as equation (1) of FH23), we know that
Rpp 1:T q ď ř T t"1 py t´d iscpp TOPS t qq 2 T .(23)
We now relate the RHS of the above equation to BS T ÿ t"1 py t´d iscpp t qq 2 " T ÿ t"1 py t´pt`pt´d iscpp t qq 2 " T¨BSpp 1:T q`T ÿ t"1 pp t´d iscpp t qq 2`2 T ÿ t"1 py t´pt qpp t´d iscpp t qq ď T¨BSpp 1:T q`T p {2q 2`2 T ÿ t"1 |y t´pt | p {2q.
ď T¨BSpp 1:T q`T p {2q 2`T .
The result of the theorem follows by dividing by T on both sides.
Proceedings of the 40 th International Conference on Machine Learning, Honolulu, Hawaii, USA. PMLR 202, 2023. Copyright 2023 by the author(s).
Figure 1 :
1The combination of Platt scaling and online logistic regression yields Online Platt Scaling (OPS). Calibeating is applied on top of OPS to achieve further empirical improvements and theoretical validity.
+Figure 2 :
2Online adversarial post-hoc calibration.
Figure 3 :
3The adaptive behavior of Online Platt scaling (OPS) for the covariate drift dataset described in Section 1.2. The title of each panel indicates the time-window that panel corresponds to.
Pr(Y = 1 X = x) (a) OPS with label drift.
Pr(Y = 1 X = x) (b) OPS with regression-function drift.
Figure 4 :
4The adaptive behavior of OPS for the simulated label shift and regression-function drift datasets described in Section 1.2. For more details on the contents of the figure, please refer to Figure 3. The improvement in calibration and accuracy of OPS over the base model is visually apparent, but for completeness, {Acc, CE} values are reported in the Appendix as part of Figures 10 and 11.
respectively. Although estimating population versions of SHP and CE in statistical (i.i.d.) settings is fraught with several issues (Kumar et al. (2019); Roelofs et al. (2022) and several other works), our definitions target actual observed quantities which are directly interpretable without reference to population quantities.
[
Calibration-error (CE) at t] Angle between intial and final PC = 180 (a) Left plot: i.i.d. data, right plot: covariate drift.
plot: i.i.d. data, right plot: label drift.
Figure 7 :
7Experiments with synthetic data. In all cases, TOPS has the lowest CE across time.
Figure 8 :Figure 9 :
89Sharpness results with drifting data. SHP values over time of considered models on four datasets with synthetically induced drifts (Section 4.1). The plots have invisible error bars since variation across the 100 runs was small. The drop in expected sharpness is below 0.005 at all times except on the Fetal Health Dataset. Sharpness results with i.i.d. data. SHP values over time of considered models on four shuffled (ie, nearly i.i.d.) datasets (Section 4.1). The drop in expected sharpness is less than 0.005 in all cases except for the HOPS forecaster on the Fetal Health dataset, where it is 0.01.
Figure 10 :
10The adaptive behavior of OPS for the simulated label drift scenario described in Section 1.2.
predicted Pr(Y = 1 X = x)
F99
: the online adversarial calibration method of Foster (1999)• At time t " 1, forecast p 1 " m 1 .• At time t`1 (t ě 1q, if condition A: there exists an b P rms such that d t b ď 0 and e t b ď 0, is satisfied, forecast p t`1 " m b for any i that verifies condition A. Otherwise, condition B: there exists a b P rm´1s such that e t b ą 0 and d t b`1 ą 0, must be satisfied (see Lemma 5 (Gupta and Ramdas, 2022a)). For any index b that satisfies condition B
Figure 16 :
16Foster (1999)'s -calibrated forecaster on Pittsburgh's hourly rain data(2008)(2009)(2010)(2011)(2012). The forecaster makes predictions on the grid p0.05, 0.15, . . . , 0.95q. In the long run, the forecaster starts predicting 0.35 for every instance, closely matching the average number of instances on which it rained (« 0.37).
of Jézéquel et al. (2020), setting d " 2 and R " 10. The ONS bound (8) follows from Theorem 4.5 of Hazan (2016), plugging in G " 5, D " 2B, and α " e´B which is the known exp-concavity constant of the logistic loss over a ball of radius B (Foster et al., 2018).
IID data. CE values over time of considered models with four randomly shuffled (ie, nearly i.i.d.) datasets. The plots have invisible error bars since variation across runs was small. TOPS achieves the smallest values of CE throughout.Time t
0.025
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t ]
Base model (BM)
Fixed-batch Platt scaling (FPS)
Windowed Platt scaling (WPS)
Online Platt scaling (OPS)
OPS + tracking (TOPS)
OPS + hedging (HOPS)
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Figure 6:
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Name
T train
T cal
W
Sort-by
Link to dataset
Bank marketing 1000 1000 500
Age
https://www.kaggle.com/datasets/kukuroo3/
bank-marketing-response-predict
Credit default
1000 1000 500
Sex
https://www.kaggle.com/datasets/uciml/
default-of-credit-card-clients-dataset
Customer churn 1000 1000 500
Location
https://www.kaggle.com/datasets/
shrutimechlearn/churn-modelling
Fetal health
626
300 100 Acceleration
https://www.kaggle.com/datasets/andrewmvd/
fetal-health-classification
Table 2 :
2Metadata for datasets used in Section 4.1. The sort-by column indicates which covariate was used to order data points. All datasets are under the Creative Commons CC0 license.600 800 1000 1200 1400
Time t
0.025
0.050
0.075
0.100
0.125
0.150
E[CE
t ]
Base model (BM)
Fixed-batch Platt scaling (FPS)
Windowed Platt scaling (WPS)
Online Platt scaling (OPS)
OPS + tracking (TOPS)
OPS + hedging (HOPS)
2000 4000 6000 8000 10000
Time t
0.090
0.095
0.100
0.105
0.110
[Sharpness (SHP) at t]
Bank marketing response
5000
15000
25000
Time t
0.0775
0.0800
0.0825
0.0850
0.0875
0.0900
[Sharpness (SHP) at t]
Credit default
2000 4000 6000 8000
Time t
0.065
0.070
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0.095
[Sharpness (SHP) at t]
Customer churn modeling
600 800 1000 1200 1400
Time t
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0.20
[Sharpness (SHP) at t]
Fetal Health
table describing which notebook to run to reproduce individual figures from this paper. Time tFigure 14: Performance of online beta scaling (OBS) and its calibeating variants on real datasets with and without distribution drift. OBS further improves upon OPS in most cases. In each plot, TOBS is the best-performing method.[Calibration-error (CE) at t] Fetal Health(b) Calibration error for drifting data streams.Figure 15: Comparing the performance of windowed histogram binning (WHB), online Platt scaling (OPS), online beta scaling (OBS), and their tracking variants on real datasets with and without distribution drifts. Among non-tracking methods (dotted lines), WHB performs well with i.i.d. data, while OBS performs well for drifting data. Among tracking methods (solid lines), TOBS and TOPS are the best-performing methods in every plot. Tracking typically does not improve WHB much since WHB is already a binning method (so tracking is implicit).0.025
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(b) Calibration error for drifting data streams.
Figure 12: Results for the same experimental setup as Figures 5 and 6, but with " 0.05.
600 800 1000 1200 1400
Time t
0.025
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E[CE
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]
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(a) Calibration error for i.i.d. data streams.
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(b) Calibration error for drifting data streams.
Figure 13: Results for the same experimental setup as Figures 5 and 6, but with " 0.2.
Time t
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[Calibration-error (CE) at t]
Windowed beta scaling (WBS)
Online beta scaling (OBS)
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(a) Calibration error for i.i.d. data streams.
2000 4000 6000 8000 10000
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(b) Calibration error for drifting data streams.
2000 4000 6000 8000
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Windowed histogram binning (WHB)
WHB + tracking (TWHB)
Online Platt scaling (OPS)
OPS + tracking (TOPS)
Online beta scaling (OBS)
OBS + tracking (TOBS)
2000 4000 6000 8000 10000
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Bank marketing response
5000
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(a) Calibration error for i.i.d. data streams.
2000 4000 6000 8000 10000
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A variant of this setup also allows pat, btq to depend on f pxtq(Foster et al., 2018).
The original definition of sharpness(Murphy, 1973) was (essentially):´T´1 ř m b"1 N b p y b p1´p y b q, which equals SHPpp1:T qý T . We add the forecast-independent termȳT on both sides and define the (now non-negative) quantity as SHP.
AcknowledgementsWe thank Youngseog Chung and Dhruv Malik for fruitful discussions and comments on the paper, and the anonymous ICML reviewers for valuable feedback. CG was supported by the Bloomberg Data Science Ph.D. Fellowship. For computation, we used allocation CIS220171 from the Advanced Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS) program, which is supported by National Science Foundation (NSF) grants 2138259, 2138286, 2138307, 2137603, and 2138296. Specifically, we used the Bridges-2 system (Towns et al., 2014), which is supported by NSF award number ACI-1928147, at the Pittsburgh Supercomputing Center (PSC).Proof of Theorem 3.1. The calibeating paper(Foster and Hart, 2023)is referred to as FH23 in this proof for succinctness.We use Theorem 3 of FH23, specifically equation(13), which gives an upper bound on the Brier-score of a tracking forecast (B c t in their notation) relative to the refinement (20) of the base forecast. In our case, the tracking forecast is TOPS, the base forecast is OPS, and FH23's result gives,Using the Brier-score-refinement lemma E.2 to lower bound BSpp TOPS 1:T q givesFinally, using the sharpness-refinement lemma E.1, we can replace each R withȳ T´S HP. Rearranging terms gives the final bound.Proof of Theorem 3.2. The calibeating paper(Foster and Hart, 2023)is referred to as FH23 in this proof for succinctness.Sharpness bound (16). Theorem 5 of FH23 showed that the expected Brier-score for a different hedging scheme (instead of F99), is at most the expected refinement score of the base forecast plus 2`log T`1 2 T . In our case, the second term remains unchanged, but because we use F99, the 2 needs to be replaced, and we show that it can be replaced by next.Let us call the combination of OPS and the FH23 hedging method as FH23-HOPS, and the calibeating forecast as p FH23-HOPS 1:T . The source of the 2 term in Theorem 5 of FH23 is the following property of FH23-HOPS: for both values of y t P t0, 1u,where E t´1 r¨s is the expectation conditional on py 1:t´1 , p FH23-hedging 1:t´1, p OPS 1:t´1 q (all that's happened in the past, and the current OPS forecast). For HOPS, we will show thatfor y t P t0, 1u, which would give the required result.At time t, the F99 forecast falls into one of two scenarios which we analyze separately (see Appendix C for details of F99 which would help follow the case-work).• Case 1. This corresponds to condition A in the description of F99 in Section C. There exists a bin index b such that q " mid-pointpB b q satisfiesIn this case, F99 would set p HOPS t " q (deterministically) for some q satisfying the above. Thus,irrespective of y t , since q P r {2, 1´ {2s.• Case 2. This corresponds to condition B in the description of F99 in Section C. If Case 1 does not hold, F99 randomizes between two consecutive bin mid-points m´ {2 and m´ {2, where m is one of the edges of the -bins (10). Define n 1 :" Averagety s : s ă t, p OPS s " p OPS t , p HOPS s " m´ {2u and n 2 :" Averagety s :The choice of m in F99 guarantees that n 2 ă m ă n 1 , and the randomization probabilities are given byOnline Platt Scalingwhere P t´1 is the conditional probability in the same sense as E t´1 . We now bound Q t . If y t " 1,A 1 and A 2 simplify as follows.A 1 " pm´n 2 qpn 1´m qp2´pn 1`m qq`pn 1´m qpn 2´m qp2´pn 2`m qq n 1´n2 " pm´n 2 qpn 1´m qpn 2´n1 q n 1´n2 ă 0, since n 2 ă m ă n 1 .A 2 " ¨p m´n 2 qp1´mq n 1´n2` ¨p m´n 1 qp1´mq n 1´n2 ă ¨p m´n 2 qp1´mq n 1´n2 (since m ă n 1 ) ă p1´mq.Overall, we obtain that for y t " 1,where the final inequality holds since m is an end-point between two bins, and thus m ě . We do the calculations for y t " 0 less explicitly since it essentially follows the same steps:pm´n 2 qpm´n 1 qpm`n 1 q`pn 1´m qpm´n 2 qpm`n 2 q n 1´n2` ¨p n 2´m qm`pn 1´m qm n 1´n2`Finally, by Proposition 1 of FH23 and the above bound on Q t , we obtain,Using the sharpness-refinement lemma E.1, we replace each R withȳ T´S HP. Rearranging terms gives the sharpness result.Calibration bound(17). Recall that the number of bins is m " 1{ . For some bin indices b, b 1 P t1, 2, . . . , mu, let S bÑb 1 " tt ď T : p OPSOnline Platt ScalingNow for any specific b, consider the sequence py t q tPS b . On this sequence, the HOPS forecasts correspond to F99 using just the outcomes (with no regard for covariate values once the bin of p OPS t is fixed). Thus, within this particular bin, we have a usual CE guarantee that F99's algorithm has for any arbitrary sequence:This result is unavailable in exactly this form in Foster (1999) which just gives the reduction to Blackwell approachability, after which any finite-sample approachability bound can be used. The above version follows from Theorem 1.1 of Perchet (2014). The precise details of the Blackwell approachability set, reward vectors, and how the distance to the set can be translated to CE can be found in Gupta and Ramdas (2022a, Section 4.1).Jensen's inequality can be used to lift this CE guarantee to the entire sequence:(by(26))as needed to be shown. The inequality p‹q holds because, by Jensen's inequality (or AM-QM inequality),
An analog of the minimax theorem for vector payoffs. David Blackwell, Pacific Journal of Mathematics. 61David Blackwell. An analog of the minimax theorem for vector payoffs. Pacific Journal of Mathematics, 6(1):1-8, 1956.
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] | [] | Recent studies provide evidence that reducing structure in courses in accordance with active or inquiry-based learning strategies, while pedagogically more effective overall, can exacerbate inequities. In this study we extend the results of previous studies to a new context by conducting a mixed-methods empirical examination of gender inequitable group dynamics in two inquiry-based physics labs for non-majors at a large public university. We administered a pre-semester survey of preferences, coded video recordings of lab sessions, and implemented a re-validated post-semester survey of student perceptions of group work. We find similar patterns of gendered role-taking noted in prior studies and that they are not likely reducible to differences in student preferences. We do not find any corresponding gender differences in student's perceptions of their group work. We further report on the efficacy of an intervention that employed partner agreement forms, with the goal of reducing the predicted inequities. The coded video recordings suggest the intervention had a mixed or even counterproductive impact on student behavior. The post-semester survey showed the intervention had no discernible impact on student perceptions of group work. We provide analysis of these outcomes as motivation for future, potentially more effective interventions. Our results emphasize the challenges faced by instructors dedicated to both best practice pedagogical methods and efforts to promote diversity, equity, and inclusion. arXiv:2305.00609v1 [physics.ed-ph] 1 May 2023 | null | [
"https://export.arxiv.org/pdf/2305.00609v1.pdf"
] | 258,426,481 | 2305.00609 | 8ab93a99db2a07daff3a6d74e66f6e75765f5b91 |
Group Dynamics in Inquiry-based Labs: Gender Inequities and the Efficacy of Partner Agreements
Andrew Loveridge
Department of Physics
The University of Texas at Austin
78712AustinTexasUSA
Matthew Dew
Laboratory of Atomic and Solid State Physics
Cornell University
14853IthacaNew YorkUSA
Emma Hunt
Department of Physics
The University of Texas at Austin
78712AustinTexasUSA
Gregorio Ponti
Department of Physics
Harvard University
02138CambridgeMassachusettsUSA
Jonathan Perry
Department of Physics
The University of Texas at Austin
78712AustinTexasUSA
Viranga Perera
Department of Physics
The University of Texas at Austin
78712AustinTexasUSA
Group Dynamics in Inquiry-based Labs: Gender Inequities and the Efficacy of Partner Agreements
(Dated: May 2, 2023)
Recent studies provide evidence that reducing structure in courses in accordance with active or inquiry-based learning strategies, while pedagogically more effective overall, can exacerbate inequities. In this study we extend the results of previous studies to a new context by conducting a mixed-methods empirical examination of gender inequitable group dynamics in two inquiry-based physics labs for non-majors at a large public university. We administered a pre-semester survey of preferences, coded video recordings of lab sessions, and implemented a re-validated post-semester survey of student perceptions of group work. We find similar patterns of gendered role-taking noted in prior studies and that they are not likely reducible to differences in student preferences. We do not find any corresponding gender differences in student's perceptions of their group work. We further report on the efficacy of an intervention that employed partner agreement forms, with the goal of reducing the predicted inequities. The coded video recordings suggest the intervention had a mixed or even counterproductive impact on student behavior. The post-semester survey showed the intervention had no discernible impact on student perceptions of group work. We provide analysis of these outcomes as motivation for future, potentially more effective interventions. Our results emphasize the challenges faced by instructors dedicated to both best practice pedagogical methods and efforts to promote diversity, equity, and inclusion. arXiv:2305.00609v1 [physics.ed-ph] 1 May 2023
I. INTRODUCTION
Active and inquiry-based learning strategies have been found to be overall more pedagogically effective than traditional lectures or lab courses [1,2]. Implementing these instructional frameworks involves allowing students increased agency by reducing some forms of course structure. However, recent studies have found that these same strategies have the potential to exacerbate certain inequities. For example, Quinn et al. [3] observed that incorporating inquiry-based instructional practices into laboratory courses, compared directly with traditional labs, can result in an increase in the well-documented effect of gendered role-taking to the disadvantage of women [4][5][6]. In the context of lecture courses, Gordon et al. [7] found that a flipped classroom had a negative impact on learning and achievement for low-income, systemically non-dominant race/ethnicity [8], and first-generation students when compared with an interactive lecture. This corroborates results of other works, which show thatabsent proactive efforts by an instructor -systemically non-dominant groups engage less in active-learning components of lecture courses [9][10][11][12][13]. Collectively, these observations suggest that the removal of structures from a course may invite social-psychological forces into the newly opened spaces. This presents instructors with a serious tension: pedagogical methods which research suggests are best for overall student learning can be worse for diversity, equity, and inclusion. * [email protected] One major source of these inequities is problematic group dynamics. The value of group work, its potential for inequities, and frameworks for promoting fair and effective group work, have been important topics in physics education research for several decades, dating back to a multiyear study at the University of Minnesota [14,15]. More recent research has explored this issue in depth in the context of laboratory courses, finding cross-cultural evidence for gendered division of labor [16], documenting the likelihood of women to adopt secretary or project manager roles [6,17], and assessing how the frequency of inter-group interactions is affected by lab design and group gender composition [18]. However, these studies are limited to two institutions and it remains unclear what strategies can be used to resolve the inequities they identify.
In this work, we present a mixed-methods empirical examination of inequitable group dynamics and of a possible remediation strategy. The context of our work is two introductory physics lab courses for non-majors at a major public university. Importantly, our work is at the intersection of aforementioned results [3,7], given our courses' recent redesign implementing much of the inquiry-based framework advocated in other studies [2,19] at our institution. The size and diversity of these lab courses make them excellent testing grounds. The expository in The Inequality Machine: How College Divides Us by Paul Tough [20] suggests the observed student body may be of unique interest given its diversity, including dimensions explicitly mentioned as limitations in Quinn et al [3,21].
We choose to focus this work on the gendered aspect of group dynamics for three reasons: One, a number of previous works have also focused on gender [e.g., 3,22], so this allows direct comparison. Two, gender represents a subdivision of students with large populations in each category. Three, if inequities are observed, this can provide additional general evidence that identity-based inequities exist and can be detected by this analysis protocol, even if more work is needed to identify which others actually manifest themselves.
The first goal of this work is to extend previous results from Cornell University [3,22] to the context of inquiry-based labs for non-physics STEM majors at a large public, research-intensive institution in the southern U.S., The University of Texas at Austin. We administered a pre-semester preferences survey [22] and used a similar video recording coding procedure as Quinn et al [3] to study student behavior throughout the semester. This allowed us to observe any inequities, check if they are explained by differences in preferences, and make comparisons between our student body and that of Cornell University. Evidence for similar inequities despite the demographic differences will be taken to imply that they are likely inherent to the shared curriculum design framework. Evidence for differences in preferences or in inequities will be treated as motivation for institutionspecific curriculum development and intervention strategies.
The second goal of this work is to provide an assessment of an intervention meant to remediate the anticipated inequitable dynamics. The intervention involved students completing an individual reflection, partner agreement forms, and partner reflection forms (see Section III). Half of the lab sections examined in this study were treated as controls, while the other half were administered the intervention, allowing a side-by-side comparison. 1 Our third goal is to measure students' perceptions of group work and whether they differ by gender or are affected by the intervention. For that purpose we modified a prior survey from Kouros et al. [23]. That survey was originally designed to assess student perceptions of group work across four dimensions (see Section VI), and was delivered to high school and college biology students. In adapting the survey for our purposes, we modified the questions, re-validated the survey, and present an analysis of the design. The survey was administered at the end of the course. Including a post-semester survey of student perceptions is another way in which we extend previous results, allowing us to determine if and how students perceive any inequities we observe.
We choose to employ the pre-semester preferences survey, coded video recordings, and a post-semester experiences survey collectively as sub-components of a single mixed-methods study because the results of each motivate and illuminate each other in important ways. This allows us to triangulate to assess inequities and the intervention, and the role students' preferences and perceptions influence or are affected by them in a holistic way. We will comment on these relationships in our analysis.
In addition to the aforementioned data, we conducted in-depth interviews with a subset of students. We leave a systematic study of these interviews to future work, but we will include select quotes when they provide useful context. This paper is organized as follows: In Section II we explain the instructional context. In Section III we explain the motivation and form of our partner agreement intervention. The next three sections present the methods, results, and analysis of the three major data sets collected. In Section IV, we provide our methods, results and analysis of the preferences survey. In Sections V and VI we do the same for the video recordings and the group work survey, respectively. We conclude by synthesizing and discussing our results and their implications both for instruction and for future research on the dynamics of group work in physics lab courses in Section VII.
II. INSTRUCTIONAL CONTEXT
We investigated two introductory physics lab courses which took place during the Fall 2022 semester at The University of Texas at Austin. The two courses are sequential, and together constitute a two-semester introductory sequence. Each course is a single credit hour taken by students from one of three introductory lecture sequences, including algebra-based physics, calculusbased physics for life science majors, and calculus-based physics for engineering majors. This setup mixes students from all three tracks into the same lab sections, in roughly equal proportions, and provides an important dimension of diversity in these lab courses.
The first course, which we will refer to here as Physics I Lab, covers standard topics in mechanics. The second, which we will refer to as Physics II Lab, continues with optics, electromagnetism, and some modern physics. Both courses are designed to implement the Structured Quantitative Inquiry framework [24][25][26], which has some similarities to the Investigative Science Learning Environment [27] and Scientific Community [28] approaches. The Structured Quantitative Inquiry framework provides students with genuine investigative freedom, supported by research-based scaffolding (i.e., invention activities [29]), and requires students to make fully quantitative comparisons of models with data.
Both courses are very large (∼ 1, 000 students per course) and diverse along several dimensions. At the university level, approximately 20% of students are firstgeneration college students [30] and a similar percentage are Pell-grant eligible students [31]. Our institution is also designated a Hispanic Serving Institution [32]. The students in both lab courses are a representative crosssection of this student body (see the demographic breakdown in Section VI).
Lab sections are taught by graduate student teaching assistants (TAs), occasionally with assistance from undergraduate learning assistants (LAs). Each course is supervised by a faculty instructor of record and two to three graduate student assistant instructors, or "head TAs." Head TAs collectively are responsible for curriculum development, running weekly instructional meetings, resolving grade disputes, and supporting the other TAs.
Each course includes nine lab sessions. Each session is three hours long and meets about once a week. During a given lab session, students work in groups of usually two, but occasionally three. In rare instances students work in a larger group or on their own, but that practice is discouraged. Lab activities typically involve designing an experiment to test how well a given model describes a physical system. Student groups collectively turn in a single set of informal, but structured, "lab notes," at the start of the following lab session which document their procedure, analysis, results, and conclusions. This gives students a week to work on analysis and writing outside of class. Because of this, most students spend class time collecting and analyzing data, and leave the write-up for outside of class.
Students are allowed to pick their own partners throughout the semester. They change partners/groups every three labs so that they work with three distinct partners or groups in a given semester. This is occasionally complicated by absences or students dropping the class, in which case groups may be slightly shuffled.
In addition to lab assignments, students individually complete pre-lab activities and a final capstone quiz or project. The pre-lab activities are completed before lab sessions as quizzes on the Canvas Learning Management System and are graded upon completion. In Physics I Lab, the final assignment is a Lab Practical Quiz that is meant to test student's mastery of essential measurement methods, analysis tools, and familiarity with equipment. In Physics II Lab, the final assignment involves students proposing and executing their own experiment on a topic of their choice and turning in a scientific poster. The inclusion of these end-of-semester individual assignments, which are worth 20% of students' final grade, are meant to motivate individual responsibility, an important best practice for group work [33].
III. INTERVENTION
Given the similar pedagogical framework and cultural context, we anticipated observing comparably inequitable group dynamics similar to previous works [3]. As such, we designed an intervention aimed at remediating expected inequities. Below we explain the motivation, form, and implementation of this intervention.
There already exists some well-studied prescriptions for improving the equity of group work. The Cooperative Groups framework advocates a highly-structured approach to designing and managing groups [14,15]. In this framework, students are assigned specific roles which are regularly rotated, are prompted to write reflections on their experiences with their group, and are given group assignments that avoid placing women in the minority. In a summary of effective group work practices for college courses, Rosser suggests encouraging students to select their own leader, initially select their own roles but ensure rotation throughout the semester, and not isolating women in groups [34]. We seek alternative solutions for a few reasons. First, we believe that teaching students to work effectively in groups is an important learning goal for lab courses in itself. This is consistent with the recommendation of the American Association of Physics Teachers, which categorizes working in small groups as a component of scientific collaboration [35]. We expect that a highly-structured, top-down approach to group management does not leave students with enough agency to learn to resolve problematic group dynamics. In the spirit of the "structured inquiry" philosophy, which has proven effective for teaching students experimental physics, we seek solutions which enhance student agency. This allows students to learn to resolve inequities and establish more effective group dynamics. Second, research has shown that highly structured approaches to group work are often met with resistance from students [36]. Third, we prefer to avoid explicit role assignment and rotation because evidence suggests it is best for students to share, not split, work, even if the splitting is equitable with respect to gender [37]. Fourth, although avoiding isolating women in groups may be sufficient to prevent inequities in collective problem solving [15], it is unclear if this is also sufficient to prevent inequitable divisions of labor (e.g., in equipment use).
We therefore designed an intervention that was meant to enhance students' agency and opportunity to resolve problematic group dynamics themselves. This intervention has three components:
• Individual Reflections: Students were given a one-time, individual writing assignment, to be completed outside of class, before any lab sections met. The assignment asked students to reflect on their values and experiences with group work and to write about them. This component of the intervention is inspired heavily by the values affirmation intervention [38,39], which has been shown to reduce gender achievement gaps on high-stakes exams by combating stereotype threat. The primary purpose of this exercise was to prime students' awareness of what was important for them in group work, so that they would be better equipped to recognize when their lab experience did not conform to their values for group work and learning. We expected that by providing students with an opportunity to reflect on their values, we would induce more dialogue between group members in lab and through partner agreement or reflection forms. We also speculated it would reduce stereotype threat or other identitybased issues which could play a role in group dynamics.
• Partner Agreement Forms: Each time students were put into a new group, they were tasked with collectively filling out a partner agreement form. This form required students to introduce themselves and to have an explicit conversation about how work would be split or shared. It was deliberately worded not to bias students towards any particular way of sharing or splitting work, while giving them an opportunity to express the preferences that were primed by the individual reflection assignment. This component of the intervention was motivated in part by results suggesting reduced inequities due to explicit conversations about equipment usage [40].
• Partner Reflection Forms: Each time students returned to the same group, they were tasked with collectively filling out a reflection form. This gave them an opportunity to discuss their experience working together the previous week. This element was borrowed from the Cooperative Groups Framework [15], since it fit with our preferred approach. Importantly, our partner reflection forms differed in that they were done collectively, not individually.
The individual reflection assignment and partner reflection and agreement forms can be found in the supplementary material. Students were given participation credit for completing the individual reflection assignment. They were not given any points for completing the partner agreement nor reflection forms to minimize differences in grading across all sections. Instead, TAs required students to complete these at the start of class before proceeding with lab activities.
Partner agreements forms have been implemented and studied before, typically in courses with a project [36,[41][42][43]. These studies have found group contracts often improve communication [41][42][43]. However, few studies discuss role-taking. Students in one study were assigned a role at the start of the class [41]. In another study by Chang and Brickman of group work in an introductory biology class [36], students were instructed to rotate roles explicitly as well as to write and follow group contracts. However, students did not assign or rotate roles explicitly and they often disregarded their group contracts. These implementations differ from ours as we wanted students to share their roles in a context where their work varied week-to-week.
Our primary considerations when assessing the success of the intervention are the effects on equitable dynamics, as observed in the video recordings, and on student perceptions of their group work on the post-semester survey, as compared to the control group. We expect that the intervention will reduce gendered role-taking as well as any differences found in student satisfaction with their group work. We also expect that it will improve student satisfaction overall across all genders.
IV. PREFERENCES SURVEY
A. Methods
Prior to the first week of lab, students were asked to indicate their preferences for different lab activities, forms of role distribution, and leadership styles. The questions regarding each preference appeared in one of the mandatory pre-lab quizzes that students completed electronically before each lab (see Section II). All three preferences questions were closed response. Multiple preferences could be selected on the activity preferences question. Only one preference could be selected on the role distribution and leadership preferences questions.
We examined survey results for differences by gender 2 and course, and also compared to previous studies [22]. For survey questions where students selected only one option, we used the Chi-Square Test of Independence [44] and quantified effect size with Cramér's V [45]. For the survey question pertaining to role preferences, where students could select multiple responses, we used the first order Rao-Scott Chi-Square Test of Independence [46]. Since there is no corresponding effect size for the Rao-Scott Chi-Square Test, we do not present effect sizes. These Chi-Squared tests do not compare individual answer choices, rather, they compare the full distributions of answer choices between populations. We choose this instead of comparing individual answer choices to minimize the chances of type I errors. We used 95% confidence Wilson score intervals [47], which is better suited for proportions close to 0 or 1 than the more ubiquitous Wald interval [48].
B. Results Figure 1 shows student's reported preferences for certain roles in lab. We find that women and men indicate different preferences at the beginning of the semester [χ 2 (5) = 110.4, p = 3.39 × 10 −22 ]. Women more often prefer notes and managing, while men prefer analysis or have no preferred role. Importantly, there is no difference at the start of the lab courses between men and women in their preference for working with equipment, which was the most common role selected (69.6% overall). Figure 2 shows students' initial preference for FIG. 1. Fraction of students that preferred a given role across both the Physics I and Physics II Labs. Students could select as many or as few roles as they wanted. The error bars represent 95% confidence Wilson score intervals.
how to distribute work. While we notice a significant difference between men and women [χ 2 (4) = 20.9, p = 3.29 × 10 −4 , V Cramér = 0.106], this is attributable to men being slightly more likely to have no preference compared to women. Figure 3 shows reported preferences for group leadership. Here, we observe a significant difference [χ 2 (4) = 30.6, p = 3.69 × 10 −6 , V Cramér = 0.128]. Women more often report a preference for taking turns in leadership, while men more frequently report no preference.
Although it is not shown in Figs. 1, 2, and 3, there are some differences in student preferences when responses from the beginning of Physics I and Physics II Labs (see Tables VI, VII, and VIII) are compared. For preferred roles, Physics II Lab students have different preferences than Physics I Lab students [χ 2 (5) = 48.1, p = 3.40×10 −9 ]. Students in Physics II Lab are more likely to prefer notes and less likely to prefer managing (see Table VI). We similarly see a significant difference in preferred methods of role distribution between the two courses [χ 2 (4) = 17.3, p = 1.69 × 10 −3 , V Cramér = 0.096]. That is, Physics II Lab students are less likely to prefer working together (see Table VII). Finally, we find differences in leadership preferences between Physics I Lab and Physics II Lab students [χ 2 (4) = 30.6, p = 3.71 × 10 −6 , V Cramér = 0.128]. Students in Physics II Lab are less likely to prefer a rotating leader and slightly more likely to have no leadership preference than students in Physics I Lab (see Table VIII).
C. Analysis
When surveyed at the beginning of a semester, the most popular role among students was equipment use and there was no meaningful difference between men and women. The popularity of equipment use was echoed in student interviews, with one student saying, "everyone's favorite part I guess was actually doing the lab and taking data, I mean taking measurements and dealing with the data." This provides important context for the coded video data in the next section because it suggests observed differences in equipment use are not reducible to a difference in preferences. This corroborates what was found by Holmes et al [22].
There does appear to be an initial gender difference in preferences for note-taking and managing, with women slightly more likely to prefer these roles than men. This may be related to the "Hermione" and "secretary" archetypes from [6], suggesting that gendered division of labor may be driven in part by student preferences. Indeed, one woman interviewee said "I'm fine with group work, but I also am really picky about the work. . . I submit. . . so I like to have, I guess, a little more control in the group". Although the coded video data does not capture management activities, it does include note-taking (through 'Laptop,' 'Calculator,' and 'Notes'), so this may provide context for the next section as well.
There are no salient differences by gender in how students prefer to distribute roles. Overall, students prefer working on different tasks or working together on the same tasks nearly equally. To borrow language from Doucette et al. [37], students nearly equally prefer "splitting" and "sharing" the work. This is notably different from the findings of Holmes et al. [22] where both men and women preferred working on the same task together in a laboratory class targeted towards physics majors. This comparatively strong preference among students in our study for splitting the work may be because our students, not being physics majors as was the case in [22], prioritize efficiently completing lab work over content mastery.
Another difference with previous results appears in the leadership preferences. In Holmes et al. [22], students were unreceptive to a singular leader and more likely to prefer having no leader. A large fraction (66.6%) of our students want some form of leadership, whether that is a rotating or singular leader. This difference could again be a product of students' expressed desire for efficiency.
The observations of the last two paragraphs have two important implications. First, since student preferences inevitably intermix with course structures and interventions to produce outcomes, differences in student preferences between populations suggest best practices for group work may require some institution-specific or course-specific tailoring. Second, if large subsets of students prefer modes of group work, such as splitting the work, which are not optimal for student learning, then interventions meant to promote effective group work which rely on student agency may be of limited efficacy. This already suggests that the intervention we designed for this study may have limited or mixed results.
These observations also frame some of the results in the remainder of the study beyond suggesting limitations on the intervention effectiveness. Since there are no significant differences in role preferences by gender at the beginning of the semester, any observations of gendered role taking in the video recordings must represent outcomes which conflict with student's desired style of group work. In such cases we may expect this to affect student satisfaction with their group work as reported on the post-semester survey, for example on questions related to frustrations with group work or group members. Meanwhile, the small differences in preferences for roledistribution and leadership, though perhaps not causing a detectable effect on video recording observations, may impact measures on the post-semester survey related to aspects of collective learning.
We would also like to consider the differences between Physics I Lab and Physics II Lab, since these are interesting for two reasons. First, the preferences indicated by students in the former were likely a lot more speculative than the students who returned for the second course. Differences may indicate the impact of Physics I Lab on the students perceptions of and preferences for group work. Second, they provide context for the video recording data, which also shows some splits by course.
Students in the Physics II Lab exhibit stronger preferences for notes and weaker preferences for managing. This could be due to students becoming more familiar with the course structure and understanding what those roles mean in the context of this lab sequence. Students may enjoy the note-taking approach that more closely resembles experimental physics than the formal lab reports they often work with in other classes and may have been anticipating before taking Physics I Lab. As for managing, students in Physics I Lab may have negative experiences with managing or determine little need for a manager, and become less likely to prefer it.
Students in Physics II Lab are less likely to prefer working on tasks together. This could possibly be due to their experiences in Physics I Lab. Negative experiences in the form of domineering group members who make sharing difficult for women has been documented in the literature. Students may observe increased efficiency and improved grades in Physics I Lab when splitting rather than sharing roles, which leads them to rethink their group work strategy in the second course of the sequence.
Physics II Lab students have less preference than Physics I Lab students for rotating leadership and are more likely to have no preference. This could suggest that students that tried rotating leadership in Physics I Lab had difficulties, but may not think the alternatives are much better. These small differences may play a role in explaining, or should at least be compared to, any observed differences in the video recording data or found in the postsemester survey. See Sections V and VI for further detail.
V. VIDEO CODING
To evaluate how students actually divide tasks in the setting of a laboratory course, we observed four sections and recorded their activities over the course of a semester. This section details the methods, results, and analysis of task division of student groups across all meetings of the Physics 1 Lab and Physics 2 Lab courses.
A. Methods
Video Data Acquisition and Coding Scheme
Video recordings of four out of the 93 total sections were used to analyze student activity during class. All four recorded sections were taught by "head TA" assistant instructors, as opposed to TAs. These sections were chosen for analysis under the assumption that they would be the most uniform subset of sections, as well as most adherent to the course objectives. The four sections took place at the same time and weekday, and included two sections of the Physics I Lab and two sections of the Physics II Lab, with a control and partner agreement intervention section for each.
Labs started with a brief lecture from the TAs that range from roughly 10 minutes to 30 minutes. During this period, students listened and took notes and did not start on lab work until the lecture concluded. We coded the video recordings subsequent to the lecture in five minute intervals until the end of a particular lab period.
We used a coding scheme similar to Quinn et al [3]. Table I describes each code. The 'Other' code covered a broad range of activities including students being offtask (e.g., using their phone, leaving the room, talking to peers) as well as on-task (thinking, discussing with peers, LAs or TAs). Apparently passive activities such as looking at a computer screen without scrolling or typing, holding a pencil or piece of equipment but not using it, were coded as the closest relevant activity rather than the 'Other' code with the assumption that a student was actively engaging in that activity either before or after the relevant time-stamp. A student who watched another student do an activity was coded as 'Other.' TABLE I. Coding scheme used for video observations. The 'Laptop,' 'Calculator,' and 'Paper' codes were later collapsed.
Code Description Equipment
Student was handling the equipment. This includes handling objects (e.g. a phone) when it was explicitly obvious the materials were being used to conduct the experiment (e.g. timing a pendulum's period).
Desktop
Student was operating a lab desktop computer.
Laptop
Student was using a personal computer. This includes iPad or tablet use but excludes cell phone use. Calculator Student was using a calculator. This includes cell phone use when it was explicitly obvious the phone was being used as a calculator. Paper Student was using pen and paper. Other
All other student activities.
In our analysis, we chose to combine the 'Laptop', 'Calculator', and 'Paper' codes as the distinction between these activities was unsubstantial in two important ways. First, students often used their personal computers to do calculations and take notes. Second, all three activities were associated with analysis or report-writing and required technical understanding but not physical engagement with lab materials. They were thus functionally similar to one another, but distinct from conducting the experiment itself. The 'Desktop' code was not included in the grouping of 'Laptop', 'Calculator', and 'Paper' because the desktop computer had mixed uses. Students often used the desktop computers to collect data and could therefore be linked to the 'Equipment' code. Nevertheless, desktop computers were also often used to read lab instructions, conduct data analysis, or write lab notes which could align desktop computers more with the 'Laptop,' 'Calculator,' and 'Paper' codes. Due to this conflict, we left 'Desktop' as an independent code.
Coders and Inter-rater Reliability
To establish inter-rater reliability, three researchers coded 23 students in a single 3-hour recorded class session. For that purpose, we chose the second lab session in the Physics I Lab course. We chose that lab because we consider it an especially faithful implementation of the Structured Quantitative Inquiry framework with one of the most complete representations of the intended activities found throughout both courses.
We obtained a Fleiss' Kappa [49] of 0.80 and Kappa's over 0.75 signify excellent agreement [50]. When we combined the 'Laptop,' 'Calculator,' and 'Paper' codes, our Fleiss' Kappa increased to 0.84. After coding, the researchers discussed their disagreements. The researchers then coded separate sections. Two researchers (MD and AL) each coded one section of the Physics I Lab and one researcher (EH) coded both sections of the Physics II Lab.
Video Data Quantitative Analysis
We employed two distinct quantifications of student behavior in presenting our results. While our data was coded in counts of intervals, students spent varying amounts of time in the lab. We were interested in what students were doing within the lab session proportionally, so we normalized our data to how long students spent within the lab course. Additionally, since our research questions relate to how students work in groups, we also present results normalized to a student's group. We refer to these two types of data presentation as a student's "Individual Fraction" and "Group Fraction," explained in more detail below.
A student's Individual Fraction for a coded activity is the fraction of observations we have for that activity out of their total number of observations in one class session. For a given student and class session, the individual fraction for an activity f activity , is the number of observations of that student for that activity N activity divided by the total number of observations of the student that class session, N total , as given in Equation 1,
f activity = N activity N total(1)
For example, if a student was in a class session for 30 observations and we observed them using equipment 15 times that session, their equipment individual fraction would be 0.5.
A student's Group Fraction for a coded activity is the fraction of observations we have of that activity out of the group's total number of observations for that activity. In a former study, Day et al. referred to this as "normalized participation" [5]. For a given student and class session, the group fraction for an activity g activity , is the number of observations of that student for that activity N activity divided by the sum of the total number of observations of that activity N activity over all the students in that group, as given in Equation 2,
g activity = N activity group N activity(2)
For example, if we observed a student using equipment twice and their partner using it 8 times, the former had an equipment group fraction of 0.2 and the latter 0.8. If we did not observe a group doing an activity for an entire class session, no student in that group was assigned a group fraction for that activity. While this is infrequent for codes such as 'Equipment,' the appearance of codes such as 'Desktop' varied by group and lab.
For our analysis, a data point represents a student's participation in a given activity in a given lab session. This means that each student can appear on each plot multiple times.
A summary of our student population and the total number of class observations can be found in Table II. Most student genders were obtained in a survey where students were able to self-identify. A total of 14 students did not fill out this survey, so we supplemented our data with university enrollment information. None of the students who filled out the survey identified as non-binary. Therefore, our data set only considers men and women.
B. Results
We first discuss trends across both courses and intervention scenarios. We then discuss how roles were split, accounting for different group compositions of men and women. Finally, we discuss effects of the intervention in both courses separately, accounting for different group gender compositions. We use the word "group" for any set of students. Additionally, "equal gender groups" means groups with one man and one woman.
We present our data as combined violin and scatter plots rather than box plots. We chose to do this to more transparently show the shape of the distributions, since several are not even approximately normal.
To make assessments of statistical significance, we chose to examine only the mean values of the distributions of data. This is a convenient simplification since the observed distributions were not normal nor were they all even of the same type. This did not provide a full picture, since other aspects of the distribution may be meaningful. Given a comparison of means and plots of the distributions are sufficient to answer our research questions here, we leave a more sophisticated statistical analysis of the distributions to future work.
When two distinct populations of data are to be compared (e.g., individual fraction), we performed a twosample t-test to compare means. This is typically considered a good approximation as long as N > 25 − 30 [51] and most comparisons in this study exceed that threshold. We chose to employ the same analysis for smaller groups for simplicity and due to the lack of a viable alternative statistical test. We will note where this occurs in the results. In cases where the data were not distinct, independent distributions (e.g., group fraction), we chose to compare the mean of a distribution to an ideal (equitable) value using a one-sample t-test. For example, we compared the group fraction for equipment usage to 0.5 for groups of two or 0.33 for groups of 3. This avoided comparing two distributions which were not truly independent, such as group fractions for men vs. women in gender split groups, since each data point in a given distribution corresponded to another data point in the other identically.
To provide an estimate of effect size, we offer calculations of Cohen's h, in the absence of a better option given the complex distributions, which is appropriate for variables which are proportions. We will not use Cohen's d as our data is not normally distributed, which is one of its key assumptions [45]. However we provide these quantities with caution to the reader, since as noted above the distributions are not of known form. This means their precise values, and their standard categorization as "small", "medium", and "large" are likely less informative than, say, their relative values. We prefer to provide them as reference with this caveat rather than provide no measure of effect size at all. We will not rely on quantitative measures or comparisons of effect sizes in assessing the effect of the interventions. We provide them to follow standard practice and because they may be of interest to some readers.
The statistics we present here, such as p-values and effect sizes, are meant to be complementary to, rather than entirely representative of, our plots. We aim to provide a holistic picture with both the plotted distributions and statistical measures. Statistically significant differences in mean values may be thought of as proxies for more holistic differences in the full distributions which are visually obvious but difficult to precisely quantify in the absence of a better understanding of the family of distributions they belong to.
Overall Trends
Across all four course sections, we found several differences in how men and women spent their fraction of Intervention Control Intervention Control Intervention Control Intervention Men 8 13 10 9 61 101 74 67 Women 15 12 8 10 127 103 65 69 Total 24 25 18 19 197 204 139 136 a Note that one student preferred not to share their gender.
time in class sessions throughout the semester (see Fig. 4). There were more overall observations of men using their in-class time on 'Equipment' (p = 1. In groups of two students, different trends are apparent in the fraction of a group's time that a student spends on each activity (see Fig. 5). First, 'Desktop' is heavily bimodal. This implies that one student is typically in charge of the desktop computer each class and students do not frequently share it within a class session. We also notice a slightly gendered skew in the observations for 'Equipment' and for 'Laptop,' 'Calculator,' and 'Paper.' Indeed, the distributions for both men and women are significantly different from a mean group fraction of 0.5. That is, men are more often responsible for equipment usage in their groups (p = 1.75 × 10 −2 , h = 0.178) and less often responsible for 'Laptop,' 'Calculator,' and 'Paper' usage (p = 2.41 × 10 −2 , h = 0.166) (see Fig. 5). The women's distribution of equipment usage is more concen-trated near 0. Figure 5 only includes students from groups of two students. There is a degeneracy in our data points because the group fractions of each student in a given group sum to one. That is, one student's value fixed the other student's value. If a student in a group of two had a group fraction of 0.5 for an activity, that meant the group split the activity 50/50 for a class.
When considering groups of three, two students' group fraction values fixed the third student's value. An even split mean each member had a group fraction of 0.33. Because the group fraction for an equitable distribution is different for groups of three, we do not show data from groups of three in Fig. 5. These two different data sets with different ideal distributions would obfuscate any trends (or lack thereof).
Group Gender Composition
Next, we analyzed how students divided tasks in groups with different gender compositions (see Fig. 6).
Here and in the next subsection we only discuss differences between 'Equipment' observations and 'Laptop,' 'Calculator,' and 'Paper' observations as this study is intended to examine the intervention's impact on gendered differences in work distribution. We omit further comment on the 'Desktop' and 'Other' codes, as no significant differences were observed for the intervention data set.
We found statistically significant differences in observed usage between men and women in equal-gender partnerships and groups of three with one man and two women. In equal-gender partnerships, we found men used 'Equipment' more (p = 4.93 × 10 −4 , h = 0.427) and women used 'Laptop,' 'Calculator,' and 'Paper' more (p = 4.78 × 10 −5 , h = 0.390). In groups with two women and one man, men used 'Equipment' more (p = 2.28 × 10 −2 , h = 0.486), but we noticed no difference in 'Laptop,' 'Calculator,' and 'Paper' usage.
We probe how students share or split different tasks by analyzing the spread of their distributions. The 'Equipment' distributions for men in all-men partnerships (σ = 0.289) and women in all-women partnerships (σ = 0.301) are similar. However, for 'Laptop,' 'Calculator,' and 'Paper', the all-men partnership distribution is far larger (σ = 0.337) and the all-women partnership distribution is smaller (σ = 0.222). For 'Equipment' usage in gender-homogeneous groups of three, we find that groups of three men have a similarly compact distribution (σ = 0.243) to groups of three women (σ = 0.247). For 'Laptop,' 'Calculator,' and 'Paper', groups of three women have a more compact distribution (σ = 0.189) than groups of three men (σ = 0.301).
Intervention and Control
In the Physics I Lab, the equipment observations from mixed-gender groups in the control case are gendered (see Fig. 7). For pairs with one man and one woman, this difference has a medium effect size (p = 3.47 × 10 −3 , h = 0.702), while for groups with two women and one man there is a strong effect size (p = 1.11 × 10 −3 , h = 1.117).d 24 students (8 men, 16 women) in groups with two women and one man, which is of note when using the t-test. In the partner agreements section there are no significant differences.
For the 'Laptop,' 'Calculator,' and 'Paper' in the Physics I Lab, however, we see different trends. In the control section, we find women have a higher usage of 'Laptop,' 'Calculator,' and 'Paper' than men only in the two-women, one-man groups (p = 1.71×10 −2 , h = 0.408) (see Fig. 7). Again, we only have 24 students (8 men, 16 women) for this group gender composition in the Physics II Lab with no partner agreements. For the partner agreement section, there was no significant gendered differences for two-women, one-man groups. However, a significant gendered difference was introduced for one-man, one-woman pairs (p = 6.68 × 10 −3 , h = 0.358).
In the control Physics II Lab section (see Fig. 8)
there are no statistically significant differences between men and women for 'Equipment' usage or 'Laptop,' 'Calculator,' and 'Paper' usage in any gender composition. However, in the partner agreements section, women use 'Equipment' less than men and 'Laptop,' 'Calculator,' and 'Paper' more in equal-gender groups of two. These are statistically significant differences with a strong effect size for 'Equipment' (p = 2.77 × 10 −3 , h = 0.813) and a medium effect size for 'Laptop,' 'Calculator,' and 'Paper' (p = 3.97 × 10 −3 , h = 0.524).
C. Analysis
Overall Trends
Across all sections and throughout the semester, we found that men spent more of their class time using equipment and were responsible for more of their group's equipment usage than women. Similarly, we found that women spent more of their class time on laptop, calculator, and paper and were responsible for these activities more often than their group mates. Students primarily used their laptops for data analysis and note-taking, while they near-exclusively use calculators for analysis and paper for notes. This suggests that, in terms of roles, men were more likely to be a group's equipment user and women were more likely to be a data analyst and/or note-taker.
These results echo results of previous studies that have examined student roles in physics labs. In observations of similarly structured inquiry-based physics labs, Quinn et al. found that men were more often responsible for equipment usage while women used laptops more [3]. Another study of the labs at the same university found that equipment usage was similarly gendered for in-person courses, but that online courses with fixed groups across the semester resolved this inequity [40].
We found differences compared with a study by Day et al. who analyzed role distribution among mixed-gender pairs of students [5]. While their coding system differed from ours in that they had codes just for equipment, computer, and everything else (also called 'Other'), they found that men and women used equipment a similar amount. However, they observed men more frequently using computers and women were more frequently coded 'Other.' Since students submitted notes on paper in that course, this suggests that men in the course did more data analysis while women did more note-taking.
Our findings in this section regarding the observed gendered division of roles can be compared to student preferences discussed earlier (see Section IV) and shown in Fig. 1. The role preferences that students indicated at the beginning of the semester (see Fig. 1) were not entirely consistent with the findings regarding the gendered division of roles observed during lab sessions over the course of the term. So it does not appear that the observed differences can be explained by differences in preferences. For example, among students enrolled in these labs, men and women preferred equipment at similar rates, while we found differences in actual usage. While it is true that a higher fraction of women preferred notes, a higher fraction of men preferred data analysis, and given these were both subsumed into the same category of codes, this does not seem to explain the observed differences in role-taking either. 3 That gendered division of labor is not reducible to preferences in our study matches the findings that motivated this one [22]. This prompts questions regarding if and to what extent students recognize this mismatch between their preferences and actual outcomes. If they do, we would expect gender differences in students' perceptions in their group work which should be detectable in the post-semester survey. Absence of any corresponding difference on the survey, on the other hand, would suggest that students are unaware of, or unfazed by, these inequities. Given the intervention relies on students to resolve inequities themselves by leveraging the partner agreement and reflection forms, it is important to determine the impact on their perceptions in assessing the intervention. This will be commented upon in Section VI.
Group Gender Composition
When we split our data by group gender composition, we find that both equal-gender pairs and groups of three with only one man divide equipment inequitably. Equal-gender pairs also have gendered task division for 'Laptop,' 'Calculator,' and 'Paper' usage. Interestingly, men in gender-homogeneous groups and women in gender-homogeneous groups shared equipment similarly, but not note-taking and/or data analysis. In genderhomogeneous groups, women tend to share note-taking and/or data analysis, while men split the roles, as indicated by their more dispersed distributions.
Previous research recommended against isolating women in groups with multiple men [6,15]. Nevertheless, our results indicate that roles can be divided inequitably with respect to gender in equal gender as well as groups with men in the minority. These observations also pro- vide additional credence to the gendered "tinkerer" and "secretary" archetypes, as these are linked to the equipment and notes roles, respectively [6]. This provides further motivation for interventions that go beyond those in e.g. [15].
The smaller sample size of groups with two men and one woman make the gendered task division in those types of groups less clear. However, following the recommendations of previous research [6,15], these group compositions likely should be avoided. Additionally, women that are isolated in their groups are affected negatively outside of only considering roles [52].
In their study of inquiry-based labs, Quinn et al. found that women in mixed-gender groups were more likely to be a laptop user than men in mixed-gender groups, although they found no difference in equipment usage [3]. In this case, laptop usage mostly coincided with notetaking or data analysis. They did observe in mixedgender groups that men more often fell into the 'Other' category than women, where their 'Other' code is similar to ours. In a study of only mixed-gender groups, Day et al. found the earlier mentioned difference in 'Other' and 'Computer' [5]. By their coding scheme, it suggests that men more often took on the data analysis role while women more often did 'Other' activities (possibly notetaking, managing, or something not encompassed by the four roles).
Across studies, gendered patterns in roles often arise in mixed-gender groups but which roles are affected seem to vary. It seems a common possibility that women are more likely than men to be the group's note-taker, or "secretary" to use previous language [6]. Other studies seem to find a mix as to whether men are then more use equipment, do analysis, or fall into some separate category (slacking, discussing, managing, some combination of these, etc.). Alternatively, which gendered differences arise could depend upon the dominant preferences and goals of students. As seen comparing the student preference data in Section IV and a study of another institution [22], the two student bodies have different preferences despite similar course frameworks. Differences in lab curricula could also lead students to value certain roles more or less. For example, a lab curriculum that emphasizes data analysis skills might have different gendered effects from a lab curriculum that emphasizes hands-on time with equipment, diligent note-taking, or
Intervention Effectiveness
Recall that in Section III we provided a consideration for assessing the effectiveness of the intervention. It is effective if it resolves or at least mitigates any gender inequitable division of roles. By this metric, the results are mixed regarding the effectiveness of the partner agreement intervention. Our results, summarized in Table III, are consistent with the possibility that the partner agreements made students in mixed-gender groups share the equipment more equitably in the Physics I Lab. For 'Laptop,' 'Calculator,' and 'Paper' in Physics I Lab, the partner agreements seem to have made the distributions less equitable for pairs of one man and one woman but made groups of one man and two women more equitable. When we consider Physics II Lab, however, the intervention seems to make the mixed-gender pairs less equitable in all cases.
Given this mixed -if not mostly negative -effectiveness of partner agreements, it is of note that the 'Equipment' distribution is more equitable in Physics I Lab.
There are a few possible explanations for this. In the Physics I Lab, students are required to complete a practical quiz at the end of the course which tests, among other things, skills with equipment. Physics II Lab, however, does not have a practical quiz; it has a final project. While students may gain familiarity with note-taking and data analysis in various other courses, the equipment they use in Physics I Lab is likely not present in their other courses. Therefore, getting experience with equipment to prepare for the practical quiz might make that role particularly salient for students. As one student said in their interview, "if everyone could do the data collection that might help them with just understanding the actual technique better, and especially for the lab practical." When completing partner agreements, students might make sure everyone gets equal experience with the equipment. When they are not concerned with getting enough preparation in a skill, they may be less resistant to falling into more gendered roles. In fact, the partner agreements may make task division more inequitable if students agree to a set of role distributions that are gendered for sake of efficiency.
It is also possible that students are not interested in an equitable division of labor but rather an efficient one. We speculated that efficiency may have driven student preferences, and it may be that this priority affected the way students approached the partner agreement forms. It could be that facilitating explicit conversations about preferences for group work encourages rather than disrupts inequities for students whose primary consideration is efficient completion of the lab. This suggests a barrier to solutions to inequities rooted in student agency, at least for certain subsets of students.
Among students inclined to prioritize efficiency, more advanced students (i.e., those who have either taken Physics I Lab or are further along in their degree program) may have a better sense of their preferences and abilities with respect to tasks. This could enhance any tendency for explicit conversations regarding role assignments to result in less equitable divisions. This may explain why the intervention was less effective in Physics II lab than in Physics I Lab.
Zhang et al.'s findings on the effects of formal contracts and competence trust on collaboration might offer an alternative or additional lens on the complex outcomes observed here [53]. Competence trust is how confident a student is in their partners' experience and ability to complete a task at a high level [54]. Zhang et al.'s findings show maximal benefit to collaboration when group contracts are present and groups have mild competence trust, where neither partner is perceived to be notably more or less capable by the other. In the course studied here, competence trust may be activity-specific and differ between the Physics I and Physics II Labs. This is plausible since the equipment used in Physics II Lab often involves sophisticated electronics compared with the higher frequency of everyday or otherwise familiar items in Physics I Lab.
Student perceptions could shed additional light on differences in impact between the two courses. If group work functions differently in one course vs. the other, then we may expect a corresponding split in student responses on the end-of-semester survey. It could also be that the intervention has a more consistent impact on student perceptions across courses, given prior research indicating partner agreements can improve students' satisfaction with group work, or else given students perceptions were mixed in a corresponding way.
Limitations
Since the interventions were applied to separate sections and implemented for an entire semester, the data compares sections with different initial conditions. Though we attempted to minimize the differences in initial conditions between sections by analyzing sections taught by head TAs at the same time of day, uncontrollable factors can cause otherwise identical sections to differ in significant ways. This means differences in initial conditions other than the intervention itself could have affected the apparent intervention effectiveness. For example, student personalities, gender composition, etc. all differed between each section.
TA behavior and identity also differed between sections. This could have an effect that is as large as an intervention. However, previous research finds little impact due to instructor gender, suggesting this is unlikely [55][56][57][58], although this remains an open question.
Although the intervention wasn't necessarily consistently effective for equitable task division, it is possible that it still improved certain aspects of group work for students. The results of the post-semester survey in Section VI will provide a measure of this.
In this analysis we have neglected to discuss a "group manager" role. This is because observations were conducted on video that lacked audio. This put identifying a group manager outside the possible scope of this study. Other studies have analyzed the positive [17] and negative [6] aspects group management can have on a woman's experience in physics labs. Similarly to the above, we may find affects of this on student perceptions as reported on the end-of-semester survey.
VI. STUDENT ATTITUDES SURVEY
A. Methods
Survey
Our goal was to measure student perceptions of small group learning in physics labs at the end of our two lab courses, to complement the pre-semester survey and observations of students throughout the semester. Specifically we wanted a measure of group effectiveness, equity and fairness of group work, sense of belonging, and social, psychological and other benefits. To this aim, we sought a survey that could easily be administered to all students in our courses that would probe attitudes of students towards their group environments.
We found a survey [23] that was relevant to our goal and has been used in other studies examining group work [59,60]. The Student Attitudes towards Group Environments (SAGE) questionnaire was introduced by Kouros et al. to investigate how student attitudes influence learning. The SAGE questionnaire was validated and used as a pre-post assessment in the context of high school and junior college students in North America for group work in biology courses. SAGE is a 54-item Likertstyle questionnaire produced from existing classroom climate measures designed to measure student attitudes towards group work in four factors: quality of product and process, peer support, student interdependence, and frustrations with group members. To better measure our intended constructs, we elected to modify this survey, removing some questions and adding others related to mindset, and leadership and task division preferences.
Due to these changes, we elected to iteratively refine and re-validate our survey by following a standard sequence of survey validation steps:
1. Run a pilot version of the measure the semester prior to this study with additional free response questions for student comments.
2. Make any necessary changes based on the received comments, preliminary results in conjunction with comments from informal interviews, observations of group work, and other emergent research questions.
3. Distribute the survey to 40 experts (physics faculty, research scientists, post-docs, graduate students, and a survey expert) for responses and feedback and make any final changes.
After completion of the process outlined above, the survey consists of 32 statements: 23 attitude statements from SAGE, 2 leadership preference statements, 4 task division preference statements, and 3 mindset statements (see Appendix B for prompts). Students responded to attitude and preference statements using a five-point Likert scale (strongly agree to strongly disagree) with the exception of the mindset questions which were on a six-point Likert scale to remove a neutral response [61]. Three items (8,10,16) had negatively-worded alternatives that were reverse-coded and tested for reliability of responses (Mann-Whitney U p = 7.00 × 10 −3 , 1.02 × 10 −2 , 0.00, respectively). For our analysis, all Likert scales were converted to an equivalent scale where -1 indicates strong disagreement and 1 indicates strong agreement. Additionally, students provided background information at the end of the survey. Each student was required to submit course and section (denoting intervention) information and could optionally provide demographic information: gender (including non-binary/other options), racial/ethnic identity [62], and parents' highest level of education [63]. To compare the racial and ethnic subpopulations, we collapse these into two classifications based on systemic non-dominance in STEM fields. For parents' highest level of education, responses were classified as a measure of a student's first-generation college status.
The survey was administered anonymously to students in both Physics I and Physics II Labs in the final two weeks of the semester using an online survey tool, Qualtrics, with a 66.8% response rate. For the studentdriven factor structure analysis, only students that answered all 32 of the attitude and preference items were analyzed (N = 1273). Subsequently, for comparisons between sub-populations, partial survey responses were included if all items related to that section of the analysis were completed even if the full survey was not complete.
Factor Analysis
Since we both removed and added items to the original SAGE questionnaire, we chose to conduct Confirmatory Factor Analysis (CFA) on the associated subset of our survey to compare to prior results. CFA was performed using the four factors extracted by Kouros at al.: quality of product and process, peer support, student interdependence, and frustrations with group members. These four factors are used as a model for the student response data. The "lavaan" package in R [64] estimates the loadings of the relevant items to their factors using the correlation matrix of the data and produces goodness-of-fit parameters used to evaluate agreement.
Exploratory Factor Analysis (EFA) was conducted on the 32 attitude and preference statements to determine student-driven factor structures using an algorithm, explained in detail in Section III A in Ref. [65] and outlined in Appendix C. EFA analyzes the student responses and groups statements which students answered similarly into independent factors. These independent factors are considered to be indicative of underlying themes in student responses. Therefore, when presenting the factors, we will name the collection of items by what we consider to be the connecting theme.
First, we must determine if EFA is appropriate for our data. To accomplish this, we used Bartlett's test of sphericity and the Kaiser-Meyer-Olkin (KMO) measure of sampling adequacy. The former checks whether the correlation matrix can be reduced by measuring the difference between a correlation matrix to the identity matrix. We found that the data can be reduced into categories (χ 2 (496) = 11502.2, p = 0.00). The latter which measures the proportion of variance of items against a common variance also determines that factor analysis can be conducted (KMO = 0.906).
Next, to be able to conduct EFA and produce themes of student perceptions, one needs to determine the amount of factors to be used. Using the Kaiser Criterion ( > 1), there is an ambiguous number of factors. Namely, the eigenvalues of several number of factors (n = 4 − 9) are near 1 (see Fig. 9a). Therefore, we need another metric to know the ideal number of factors. To find these, we use an algorithm outlined in Appendix C. We use CFA in the final step to find parsimonious fit statistics such as the Akaike information criterion (AIC), which measures the likelihood of the model with added penalty from degrees of freedom of the fit, to compare between different models. The preferred model is one that minimizes the parsimonious fit statistics.
Intervention Effects
Once we determine the number of factors that best models our data, we can create a factor rating for each student to allow for comparison across student subpopulations. To obtain the factor ratings, each student's responses are dotted by the item loading for each factor and normalized by the sum of the factor item loadings (using only significant loadings). For example, let's take a factor that contains items 5 through 7 with loadings 0.5, -0.6, 0.7, respectively, and a student responds Strongly Agree (1), Disagree (-0.5), and Agree (0.5). Their factor rating would be
1 · 0.5 + (−0.5) · (−0.6) + 0.5 · 0.7 |0.5| + | − 0.6| + |0.7| = 0.64
Using these factor ratings, we performed multiple linear regression to calculate the main effects for course, intervention, gender, race/ethnicity, and parents' highest level of education. We also compared student ratings by course and by intervention to determine if the interventions have differing effects for each course. For these, we compare against our most common student: Woman, enrolled in Physics I Lab, of systemically dominant race/ethnicity, and not a first-generation college student (see Table IV).
B. Results
Factor Structures
First we compare our results to Kouros et al. [23] using CFA. The goodness-of-fit parameters indicates moderate agreement: CFI = 0.876, RMSEA = 0.066, SRMR = 0.058. In particular, items 17 ('I let the other students do most of the work.'), 21 ('We cannot complete the assignment unless everyone contributes.'), and 27 ('When I work with other students, we spend too much time talking about other things.') do not load on their intended constructs. These differences could be attributed to the different student demographic. Students could prefer to be efficient and divide their work (item 17) for courses not in their intended field of study but required to graduate. It is often the case that a student does not do their fair share of work and the others end up making the difference (item 21). Also, university students might not view socialization as a negative behavior (item 27).
CFA failed to provide a satisfactory model for our data. Because we included additional or tweaked questions that cannot be analyzed by CFA, we proceed to EFA to determine the appropriate factors.
To determine the number of factors used in EFA, we analyzed AIC as a function of the number of factors (Fig. 9b). We only consider factor counts that were near the Kaiser Criterion ( > 1). Figure 9a shows that factor counts from 4 to 9 match this condition. The 6-, 8-, and 10-factor models are local minima. Determining the ideal number of factors is a subjective process based on interpretability of the themes. We believe that the smaller number of items which contained more items in each factor was more appropriate. Therefore, we choose to use the 6-factor EFA model to discuss the student-driven factor structures for better interpretation. Ultimately, 23 of the items had significant loadings and are shown in Table V.
FIG. 9.
a) The scree plot for the factor models. b) The AIC fit statistic for factor models determined in the procedure outlined in Appendix C. The 6-, 8-, and 10-factor models result in local minima. Table V: 1. Items in the quality of process factor pertain to how the perceived quality of a student's learning experience is affected by the process of working in groups (e.g., "The work takes more time to complete when I work with other students."). A more negative response means that students view group work to improve the quality of their learning process.
The six factors indicated by this survey are shown in
2. The collective learning factor includes statements that indicate a perception of increased learning due to working in groups (e.g., "I also learn when I teach the material to my group members.").
3. The individual belonging factor gathers items that show a student's perceived inclusion and sense of belonging in their groups (e.g., "My group members make me feel that I am not as smart as they are.").
4. The mindset factor contains the three added questions to determine a fixed vs. growth mindset of students (e.g., "You can learn new things, but you can't really change your basic physics intelligence.").
5. Items in the impact on individual factor pertain to how working in groups affects their perceived academic success (e.g., "When I work in a group, I do higher quality work.").
6. The frustrations factor contains items that indicate a frustration in having to work with other students who they perceive as inadequate to their academic level (e.g., "I have to work with students who are not as smart as I am.").
There is one item (item 10) that cross loads onto the Collective Learning and Individual Belonging factors. We assume when running EFA that the factors produced are orthogonal to each other, meaning that the underlying themes are independent of each other. In reality, the produced factors describing learning and group work are complex and intertwined. This particular statement ("I feel I am part of what is going on in the group.") shows the crossover between a student's perceived inclusion in their group and their perception of increased learning from working in groups.
Here we discuss items that did not load onto any factor. Items 17, 21, and 27 did not load onto any factors. These items, all taken from SAGE, failed to load onto their respective factors in CFA as well. Further explanations can be found at the beginning of this section. Items on leadership preference (items 11 and 12) and items on role distributions (items 22, 28, and 29) do not load onto any factor. This is likely because these answers do not correlate with attitudes towards group work, but rather towards preferences students have as discussed in Section IV B. Item 4 ("My group members help explain things that I do not understand.") was expected to load onto the Collective Learning factor, but did not. It is possible that some students interpret this as being a receiver of knowledge, rather than the increased learning from working in groups.
Overall, students indicated that they have a generally positive outlook on the group experiences at the end of these two courses. Students' sense of belonging in their groups were high (µ = 0.586, ∼Agree) and generally had a growth mindset about physics (µ = 0.507, ∼Agree). Working in groups was beneficial for their learning experience (Collective learning µ = 0.499, ∼Agree) and their academic success (Impact on Individual µ = 0.374, Neither agree nor disagree−Agree). They also had a slight disagreement that the process of working in groups negatively affects their learning experience (Quality of process µ = −0.224, Disagree−Neither agree nor disagree) and that they didn't experience frustrations (µ = −0.259, Disagree−Neither agree nor disagree).
Correlations and Intervention Effects
Having determined the ideal number of factors, we can compare different student sub-populations (course, intervention, gender, systemically non-dominant race/ethnicity, first-generation status) by their responses using their factor ratings. We perform multiple linear regression (results tabulated in Appendix D) to determine size and statistical significance of the differences, if any. For the most part, we do not see any differences in student perceptions between demographics other than in gender (Table IX).
Comparing across gender, men are more likely to be frustrated while working with other students. Nonbinary/other gendered students have lower perceptions of group work than their binary-gendered counterparts as they have lower ratings on their perceptions of increased learning (Collective Learning) and academic success (Impact on Individual) due to group work. They also have less of a physics growth mindset. Figure 10 shows the distribution of student ratings by gender and by intervention.
Overall, there is no apparent correlation between the partner agreements intervention and student perceptions of group work. Associated regression statistics are shown in Tables X and XI.
C. Analysis
Our iteration on the SAGE survey still does not provide a CFI value (CFI = 0.893) meeting the criteria. This calls for further refinement of the survey. However, this has not precluded us from drawing some conclusions from our results.
Overall, we find that students had an overall positive attitude towards group work. Specifically, working in groups had a positive effect on their learning and academic success, their sense of belonging in their groups was high, and they had few frustrations with other group members. Students appreciate group work despite the many documented problems associated with group work in active-learning or inquiry-based course settings. Unfortunately though, contrary to our expectation, we find that the intervention did not meaningfully improve overall perceptions. This is yet another way in which the intervention must be considered ineffective.
We find that there is no appreciable difference based on gender. This marks an important contrast with the results of Section V where we found experiences do differ. This implies that students are unaware of, or unfazed by, the identified inequities. In the interviews, students were directly asked if they thought gender played a role in task division. Most said no, which fits with this interpretation. Interestingly one woman said ". . . no, not really" but also that ". . . I ended up doing kind of the same work that I did for each group."
This has important implications for the design of interventions. If students do not perceive problems in group dynamics, they are not equipped to resolve them on their own. This offers some additional perspective on the mixed impact of the intervention on equitable group dynamics.
In Section V we found that there was a difference in the effectiveness of the intervention between both courses, with Physics I having more positive outcomes (Table III). When we compare student perceptions between Physics I and Physics II labs, we find that there were differences in perceived quality of process and individual belonging (see tables in Appendix D). Students felt like they had an increased sense of belonging and increased quality in their learning experience in Physics II Lab. One might expect that these differences would correlate with decreased inequities but the video observation data does not show this. These results are consistent with the idea that students may be more settled into particular modes of group work in the later course. The intervention may then be reifying habits previously established rather than disrupting inequitable patterns.
In that sense, it comes as no surprise that perceptions of group work are not correlated with our intervention. This is another measure by which we must judge it as either ineffective or of mixed effectiveness. And so our null results in this section regarding differences based on identity and intervention are compatible with other aspects of our study [66].
VII. DISCUSSION AND CONCLUSIONS
In this section we discuss our findings holistically in light of our research goals, draw some conclusions, indicate implications for instruction, and suggest avenues for future work.
The first goal of this study was to apply the same methods as previous work [3,22] in the context of our courses. Given the similarity in course framework, we expected we would observe similar inequities. We did, indeed, observe inequities [3] and they do not appear to be reducible to differences in preferences among our students [22]. Given the differences in student populations between those studies and ours, we take this as evidence that they are likely inherent to inquiry-based labs or lab courses more broadly.
This emphasizes the tension between best practice pedagogical methods and efforts to promote diversity, equity, and inclusion. An important point of Quinn et al. [3] is that these inequities are not merely background effects of culture on all physics courses, they are consequences of particular choices of curriculum design.
Interestingly, although the pre-semester (mostly absent) gender differences in preferences matches Holmes et al. [22], we did find some differences in preferences between the two student populations. Our students are more likely to prefer dividing tasks and a single leader. Since student preferences play a role in driving outcomes, this means solutions to inequities which prove effective in one context may be less effective in another. For example, if our interpretation is correct that our students (comparatively) value efficiency, this should be considered when designing an intervention. Students may require additional motivation to master the material before they are willing to expend effort to resolve inequitable group dynamics.
Importantly, in this study we have only examined inequities based on student gender. There may be inequities based on other demographic criteria such as race/ethnicity or students' academic backgrounds [7]. We plan to follow up with future work using our data set to explore these possibilities.
The second goal of this study was to implement and evaluate the impact of an intervention meant to reduce inequitable task division. This intervention was intended to work by increasing student agency to resolve problematic group dynamics on their own. From multiple points of view, our results show that the intervention had mixed results or was ineffective. Observations show mixed or even counterproductive trends in equitable task division while our student attitudes survey shows no impact on student perceptions of their group work. This fits with some interpretations of the results of our preferences survey which suggested students may prefer more efficient modes of group work. It also may be that students simply do not perceive these inequities, which would also explain why there was no gender splits in perceptions of group work even though differences were observed in videos. If students are not aware of or are unconcerned with inequities, we would not expect our intervention to be successful, since it relies on students to use the intervention as an opportunity to address them. Important corroborating evidence for this interpretation was provided by our post-semester survey. The mismatch between the equality of pre-semester student preferences and the inequities observed in actual lab activities does not appear to have driven any differences in student perceptions of their group work as captured by our iteration of SAGE.
This calls for new intervention strategies. One possibility is to alter the partner agreement forms to include explicit prescriptions for research-based best practices of groups for sake of learning (sharing rather than splitting) and to meanwhile better communicate and further instantiate the students' self-interest in mastering all roles in the lab course. Students who are more aware of best practices and more motivated to implement them may make better use of agreement and reflection forms.
Our third goal was to provide a validated survey for use in courses to evaluate student perceptions of group dynamics at the end of a lab course. We have iteratively improved upon an existing survey, but further work is needed. A better understanding of the identified factors may lead to tweaks of questions, or suggestions for additional questions, that could produce a more robust survey that meets the CFI criteria. Nevertheless, we were able to draw some important conclusions using this version of the survey. Specifically, we find that the intervention had no impact on overall student satisfaction. We also did not find salient gender differences in either the control or intervention sections.
We hope that this or future versions of this survey can join the preferences survey [22] and the video analysis procedure [3] in beginning to provide researchers and instructors with a standard toolbox of measures of group dynamics. Given our study provides some evidence of institution-specific effects, it may be that the appropriate attitude is not to assume particular forms of inequities or student dispositions, but to measure them case-bycase. If these "diagnostic tools" can be matched with a corresponding toolbox of research-based interventions meant to resolve a wide variety of potential problems in group dynamics, instructors would be well position to identify and "cure" these problems on an institution-byinstitution basis.
Developing such a toolkit may be a difficult undertaking. But providing effective ways to implement inquirybased and active learning pedagogical methods while also improving inclusivity would be well worth future research efforts. 19. I have to work with students who are not as smart as I am. 20. When I work with other students the work is divided equally. 21. We cannot complete the assignment unless everyone contributes. 22. I prefer to take on tasks that I'm already good at. 23. I prefer to take on tasks that will help me better learn the material. 24. I also learn when I teach the material to my group members. 25 You're almost done! The final questions below ask you to share a little bit about your identity.
• Which gender(s) do you most identify (select all that apply)? • What is your race/ethnicity (select all that apply)? • What is the highest level of education either of your parents have achieved?
FIG. 2 .
2Fraction of students that preferred a given method of role distribution across both the Physics I and Physics II Labs. Students could select only one answer. The error bars represent 95% confidence Wilson score intervals.FIG. 3. Fraction of students that preferred a given leadership style across both the Physics I and Physics II Labs. Students could select only one answer. The error bars represent 95% confidence Wilson score intervals.
03×10 −0.5 ) and in the 'Other' category (p = 2.35 × 10 −3 ). Additionally, we observed that women spent more of their in-class time on 'Laptop,' 'Calculator,' and 'Paper' (p = 4.98×10 −10 ). The 'Other' (h = 0.0941) and 'Equipment' (h = 0.146) effect sizes were very small, while the 'Laptop,' 'Calculator,' and 'Paper' effect size was small (h = 0.251).FIG. 4. Individual fraction (see Equation 1) of each coded activity throughout the semester for all class sections separated by gender. Each data point is one student in one class session. The shaded areas are sideways histograms that represent the density of the data points.
FIG. 5 .
5Group fraction (seeEquation 2) of each coded activity throughout the semester for groups of two in all class sections separated by gender. Each data point is one student in one class session. The shaded areas are sideways histograms that represent the density of the data points. Note there are less data points for desktop because not every group used the desktop every class session.
FIG. 6 .
6Group fraction (see Equation 2) of a) 'Equipment' usage and b) combined 'Laptop,' 'Calculator,' and 'Paper' usage by group gender composition.
FIG. 7 .
7Group fraction of a) 'Equipment' usage and b) 'Laptop,' 'Calculator,' and 'Paper' usage in Physics I Lab. This data is separated by each group's gender composition and split by control (Ctrl.) and intervention (Int.).
FIG. 8 .
8Group fraction of a) 'Equipment' usage and b) 'Laptop,' 'Calculator,' and 'Paper' usage in Physics II Lab. This data is separated by each group's gender composition and split by control (Ctrl.) and intervention (Int.). management experience.
FIG. 10 .
10Comparison of distribution of factor ratings by gender separated by intervention, Control and Partner Agreements. Factors are: QP = Quality of Process, CL = Collective Learning, IB = Individual Belonging, M = Mindset, II = Impact on Individual, F = Frustrations.
TABLE II .
IIDemographics of students and number of observations for each of the recorded sections. aStudents
Observations
Physics I Lab
Physics II Lab
Physics I Lab
Physics II Lab
Control
TABLE III .
IIISummary of the effectiveness of the intervention in both courses and the two sets of codes.Equipment
Laptop, Calculator,
and Paper
Physics I Lab
Effective
Mixed
Physics II Lab Counterproductive
Counterproductive
TABLE IV .
IVDemographic breakdown of students from student attitudes survey data set across courses. Racial/ethnic groups
were not considered mutually exclusive. Counts may not equal the total as students not may have answered all background
questions or preferred to not disclose.
Survey Responses
Student-level variables
Full sample
Physics I Lab
Physics II Lab
All
1316
788
528
Intervention
Control
501
325
176
Partner Agreements
427
243
184
Gender
Women
751
485
266
Men
479
255
224
Non-binary/Other
32
18
14
Race/ethnicity
American Indian or Alaska Native
11
8
3
Asian
462
262
200
Black or African American
87
61
26
Hispanic, Latino, or Spanish
325
197
128
Middle Eastern or North African
42
26
16
Native Hawaiian or Other Pacific Islander
3
1
2
White
452
273
179
Some other race/ethnicity
6
4
2
Parents' highest level of education
High school
129
73
56
Some college but no degree
82
61
21
Associate's or technical degree
50
33
17
Bachelor's degree
359
211
148
Master's degree or above
597
353
244
TABLE V. Factor Loadings of the survey for a 6-factor EFA model. All items loading onto a single factor above 0.4 were kept.
Items dropped for reasons discussed in the text: 4, 11, 12, 17, 21, 22, 27, 28, 29.
Factor 1
Factor 2
Factor 3
Factor 4
Factor 5
Factor 6
Quality of process Collective Learning Individual Belonging
Mindset
Impact on Individual Frustrations
Item
Loading
Item
Loading
Item
Loading
Item Loading Item
Loading
Item Loading
2
0.748
10
0.478
5
0.432
30
0.822
1
0.423
19
0.541
3
0.490
14
0.535
6
-0.605
31
0.436
7
0.476
25
0.483
8
-0.582
15
0.500
9
0.710
32
0.847
13
0.470
23
0.470
10
0.454
18
0.477
24
0.554
16
-0.516
20
-0.662
26
0.522
. I become frustrated when my group members do not understand the material. 26. Everyone's ideas are needed if we are going to be successful. 27. When I work with other students, we spend too much time talking about other things. 28. My group did higher quality work when my group members worked on tasks together. 29. My group did higher quality work when group members worked on different tasks at the same time.The items on this page are probing your personal agreement/disagreement with each statement. Read each statement below and select the choice that most closely corresponds to how much you agree with it. There are no right or wrong answers.30. You have a certain amount of physics intelligence,and you can't really do much to change it. 31. Your physics intelligence is something about you that you can change. 32. You can learn new things, but you can't really change your basic physics intelligence.
TABLE VI .
VIPercentage of students that preferred a given role. Students could select as many or as few roles as they wanted. The errors represent 95% confidence Wilson score intervals.Preferred Role (%)
Student-level variables
N
Equipment
Notes
Analysis
Managing
No Pref
Overall
1871
69.6 +2.0
−2.1
38.7 +2.1
−2.2
45.6 +2.2
−2.2
54.1 +2.2
−2.3
12 +2.3
−2.2
Gender
Men
792
72.7 +3.0
−2.7
30.8 +3.3
−3.1
51.7 +3.5
−3.5
45.6 +3.5
−3.4
14.9 +2.7
−2.3
Women
1079
67.4 +2.7
−2.9
44.5 +3.0
−2.9
41.1 +3.0
−2.9
60.3 +2.9
−3.0
9.9 +1.9
−1.6
Course
Physics I Lab
1078
71.0 +2.6
−2.8
35.1 +2.9
−2.8
44.2 +3.0
−2.9
58.5 +2.9
−3.0
10.1 +1.9
−1.7
Men
401
75.5 +4.0
−4.4
24.8 +4.5
−4.0
51.3 +4.9
−4.9
49.0 +4.9
−4.9
13.0 +3.6
−2.9
Women
677
68.4 +3.4
−3.6
41.2 +3.7
−3.6
40.0 +3.7
−3.6
64.1 +3.5
−3.7
8.4 +2.3
−1.9
Physics II Lab
793
67.7 +3.2
−3.3
43.6 +3.5
−3.4
47.5 +3.5
−3.5
48.2 +3.5
−3.5
14.6 +2.6
−2.3
Men
391
69.8 +4.3
−4.7
37.1 +4.9
−4.6
52.2 +4.9
−4.9
42.2 +4.9
−4.8
16.9 +4.0
−3.4
Women
402
65.7 +4.5
−4.8
50.0 +4.9
−4.9
43.0 +4.9
−4.8
54.0 +4.8
−4.9
12.4 +3.6
−2.9
TABLE VII .
VIIPercentage of students that preferred a given method of role distribution. Students could select only one answer. The errors represent 95% confidence Wilson score intervals.Preferred Role Distribution (%)
Student-level variables
N
Take Turns
Different Tasks
Work Together
No Preference
Other
Overall
1871
10.5 +1.5
−1.3
36.0 +2.2
−2.1
40.8 +2.2
−2.2
11.7 +1.5
−1.4
1.0 +0.6
−0.4
Gender
Men
792
9.0 +2.2
−1.8
34.3 +3.4
−3.2
40.2 +3.5
−3.4
15.4 +2.7
−2.3
1.1 +1.0
−0.5
Women
1079
11.7 +2.1
−1.8
37.2 +2.9
−2.8
TABLE VIII .
VIIIPercentage of students that preferred a given leadership style. Students could select only one answer. The errors represent 95% confidence Wilson score intervals.Preferred Leadership Distribution (%)
Student-level variables
N
No Leader
Take Turns
One Leader
No Preference
Other
Overall
1871
13.9 +1.6
−1.5
42.7 +2.3
−2.2
23.9 +2.0
−1.9
17.5 +1.8
−1.7
1.9 +0.7
−0.5
Gender
Men
792
16.3 +2.7
−2.4
36.2 +3.4
−3.3
24.2 +3.1
−2.9
21.3 +3
−2.7
1.9 +1.2
−0.7
Women
1079
12.2 +2.1
−1.8
47.5 +3.0
−3.0
23.7 +2.6
−2.4
14.6 +2.2
−2.0
1.9 +1.0
−0.7
Gender and Course
Physics I Lab
1078
12.2 +2.1
−1.8
47.8 +3.0
−3.0
22.9 +2.6
−2.4
15.1 +2.3
−2.0
2.0 +1.0
−0.7
Men
401
14.5 +3.8
−3.1
41.4 +4.9
−4.7
23.7 +4.4
−3.9
18.2 +4.1
−3.5
2.2 +2.0
−1.1
Women
677
10.8 +2.6
−2.1
51.6 +3.7
−3.8
22.5 +3.3
−3.0
13.3 +2.8
−2.4
1.9 +1.3
−0.8
Physics II Lab
793
16.4 +2.7
−2.4
35.8 +3.4
−3.3
25.3 +3.1
−2.9
20.7 +3
−2.7
1.8 +1.2
−0.7
Men
391
18.2 +4.1
−3.5
30.9 +4.7
−4.4
24.8 +4.5
TABLE IX .
IXResults from the linear models of the extracted factors from the 6-factor EFA model controlling for student demographics, course, and intervention. β are the standardized regression coefficients with standard error SE. Total N = 899. * p < 0.05, ** p < 0.01, *** p < 0.001, **** p < 0.0001 TABLE X. Results from the linear models of the extracted factors from the 6-factor EFA model separated by intervention for Physics I Lab, controlling for student demographics. β are the standardized regression coefficients with standard error SE.Dependent variable
Quality of process Collective Learning Individual Belonging
Mindset
Impact on Individual
Frustrations
β
SE
β
SE
β
SE
β
SE
β
SE
β
SE
Intercept
-0.1542 0.0256
0.5080
0.0189
0.5810
0.0210
0.4668 0.0253
0.4028
0.0257
-0.1944 0.0296
Course
Physics I Lab
Physics II Lab
-0.0571* 0.0270
-0.0238
0.0200 0.0450*
0.0222
0.0317 0.0267
0.0194
0.0271
-0.0322 0.0312
Intervention
Control
Partner Agreements
-0.0268 0.0263
-0.0018
0.0194
-0.0022
0.0216
-0.0102 0.0260 -0.0113
0.0264
-0.0509 0.0304
Gender
Women
Men
-0.0454 0.0275
-0.0197
0.0204
0.0081
0.0226
0.0756 0.0272 -0.0565*
0.0276
0.0658* 0.0319
Non-binary/Other
0.0645 0.0784 -0.1715** 0.0580
-0.0643
0.0644
-0.0064** 0.0774 -0.2195** 0.0787
0.0273 0.0907
Systemically dominant race/ethnicity?
Yes
No
-0.0655* 0.0314
0.0329
0.0233
-0.0413
0.0258
-0.0333 0.0311 -0.0115
0.0316
-0.1056** 0.0364
First-Generation Student?
No
Yes
-0.0110 0.0337
0.0285
0.0249
0.0309
0.0277
0.0856* 0.0333
0.0132
0.0338
-0.0871* 0.0390
Physics I Lab
Dependent variable
Quality of process Collective Learning Individual Belonging
Mindset
Impact on Individual
Frustrations
Intervention
β
SE
β
SE
β
SE
β
SE
β
SE
β
SE
Control
Intercept
-0.1777
0.035
0.5324
0.024
0.5721
0.029
0.4886 0.034
0.4335
0.033
-0.1816 0.041
Gender
Women
Men
-0.0010
0.047
-0.0421
0.032
0.0392
0.038
0.0681 0.045 -0.1417*** 0.044
0.0421 0.054
Non-binary/Other
0.0491
0.122
-0.2158** 0.082
-0.0663
0.099
0.0622 0.116
-0.1577
0.114
-0.0328 0.140
Systemically dominant race/ethnicity?
Yes
No
-0.0971* 0.047
0.0296
0.032
0.0060
0.038
-0.0044 0.045
0.0264
0.044
-0.1892*** 0.054
First-Generation Student?
No
Yes
0.0481
0.053
-0.0201
0.036
-0.0493
0.044
-0.0005 0.051
-0.0072
0.050
0.0384 0.061
Partner Agreements
Intercept
-0.1358
0.039
0.4968
0.029
0.6039
0.035
0.4233 0.039
0.3905
0.040
-0.2619 0.045
Gender
Women
Men
-0.0800
0.056
0.0042
0.041
0.0218
0.050
0.1551** 0.055
-0.0198
0.057
0.0971 0.064
Non-binary/Other
-0.1855
0.196
0.0181
0.146
0.0093
0.175
-0.0450 0.195
-0.0851
0.202
0.0566 0.225
Systemically dominant race/ethnicity?
Yes
No
-0.0381
0.054
-0.0245
0.040
-0.1148*
0.048
-0.0902 0.053
-0.0975
0.055
-0.0322 0.061
First-Generation Student?
No
Yes
0.1543* 0.065
0.1053*
0.048
0.0513
0.057
0.1648* 0.064
0.0723
0.066
-0.1787* 0.074
TABLE XI .
XIResults from the linear models of the extracted factors from the 6-factor EFA model separated by intervention for Physics II Lab, controlling for student demographics. β are the standardized regression coefficients with standard error SE.Physics II Lab
Dependent variable
Quality of process Collective Learning Individual Belonging
Mindset
Impact on Individual Frustrations
Intervention
β
SE
β
SE
β
SE
β
SE
β
SE
β
SE
Control
Intercept
-0.2216
0.051
0.4781
0.038
0.6237
0.035
0.5147 0.048
0.4087
0.054
-0.2448 0.058
Gender
Women
Men
-0.0692
0.068
-0.0215
0.051
0.0177
0.046
0.0128 0.064
-0.0218
0.071
0.0690 0.077
Non-binary/Other
0.0939
0.196
-0.1026
0.147
-0.2056
0.134
-0.3217 0.186
-0.0213
0.207
0.0776 0.224
Systemically dominant race/ethnicity?
Yes
No
0.0321
0.073
0.0370
0.054
-0.0135
0.050
-0.0715 0.069
-0.0652
0.077
-0.0082 0.083
First-Generation Student?
No
Yes
-0.0021
0.084
0.0074
0.063
0.0375
0.057
0.1520 0.080
-0.0058
0.089
-0.1114 0.096
Partner Agreements
Intercept
-0.2410
0.043
0.4716
0.036
0.6152
0.038
0.4878 0.046
0.3824
0.043
-0.2536 0.052
Gender
Women
Men
-0.0619
0.056
-0.0256
0.047
-0.0666
0.050
0.0562 0.061
0.0087
0.057
0.0350 0.068
Non-binary/Other
0.2407
0.152
-0.3075*
0.127
-0.0006
0.134
0.1090 0.164 -0.6237**** 0.153
0.0210 0.184
Systemically dominant race/ethnicity?
Yes
No
-0.1038
0.062
0.1037*
0.052
0.0194
0.055
0.0828 0.067
0.0916
0.062
-0.1285 0.075
First-Generation Student?
No
Yes
0.0503
0.077
0.0121
0.064
0.1184
0.067
0.0192 0.083
-0.0379
0.077
-0.1868* 0.093
Another, roughly equal size set of sections of the course were treated with a different intervention, but this is beyond the scope of this study. We do not include data from those sections.
In this section we use university-supplied data on gender, since the survey of Section VI, which allowed students to self-report their own gender, was anonymous.
An important caveat is that different activities like note-taking or analysis may take different amounts of time and may not be directly comparable
* p < 0.05, ** p < 0.01, *** p < 0.001, **** p < 0.0001
ACKNOWLEDGEMENTSWe would like to thank Natasha Holmes, Kathryn Hendren, Jane Huk, Vernita Gordon, and John Yeazell for many useful discussions and feedback. We would like to thank Ben Costello (and his supply room workers) for implementing the video recording hardware and managing the software and servers. We would like to thank Laura Costello for feedback on our post-semester survey. We would also like to thank The College of Natural Sciences at the University of Texas at Austin for providing funding for this project through the 21st Century Curriculum Redesign Effort, and Kristin Patterson and Keely Finkelstein for their assistance with applying for and administering these funds. We would also like to thank the many students who agreed to participate in this study and the graduate student teaching assistants and undergraduate learning assistants who assisted with teaching the courses.Appendix A: Preferences ResultsWe present the full results of the preferences survey from Section IV.Table VI provides the roles students preferred, Table VII provides the role distributions students preferred, and Table VIIIprovides the leadership distributions students preferred.Appendix B: SurveyWhat we will ask you to do: This survey asks about your attitudes toward small group learning, or group work, in the classroom. When responding to each statement, please draw on your experiences completing coursework with groups in lecture and/or lab classes. Whenever there is a statement about group members, other students, etc., think of the students who have been in your groups during physics lab. This should take about 10 minutes to complete.• Which course are you currently enrolled in?• What is your section's unique number?Based on your group experiences this semester, please select the response that best reflects your level of agreement.1. When I work in a group, I do higher quality work. 2. When I work in a group, I end up doing most of the work. 3. The work takes more time to complete when I work with other students. 4. My group members help explain things that I do not understand. 5. When I work in a group, I am able to share my ideas. 6. My group members make me feel that I am not as smart as they are. 7. The material is easier to understand when I work with other students. 8. The workload is usually less when I work with other students. 9. My group members respect my opinions.10. I feel I am part of what is going on in the group. 11. I prefer when one student regularly takes on a leadership role. 12. I prefer when the leadership role rotates between students. 13. I do not think a group grade is fair. 14. I try to make sure my group members learn the material. 15. I learn to work with students who are different from me. 16. My group members do not care about my feelings.17. I let the other students do most of the work.18. I feel working in groups is a waste of time.Appendix C: Exploratory Factor AnalysisIn this section, we outline the specific algorithm for conducting EFA. using the specified number of factors.4. Calculate the commonalities, which are the proportion of the item's variance explained by the factors. If any item is below the cutoff (Commonality< 0.20), then the item with the lowest value is dropped and then restart at Step 1.5.Calculate the item loadings. If there are items that fail to load to any factor (Loading< 0.40), then the item with the smallest max loading is removed and then restart at Step 1.6. Create a model for CFA by placing each item onto the factor that contains the item's largest loading. If any items load equally onto more than one factor, then the items are added to all factors onto which they load equally.7. Fit this model using CFA and extract a fit statistic (Akaike information criterion) to be used as a comparison for the ideal number of factors.8. Change the number of factors and repeat the above steps.9. Plot the fit statistic against the number of factors. The model with the local minimum index is the preferred model.Appendix D: Linear RegressionWe present results of the multiple linear regression models from Section VI B 2.Table IXincludes all students, and Tables X and XI show results of linear regression models separated by intervention for Physics I and Physics II Labs, respectively.
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| [] |
[
"Temporal control of high-order harmonic cutoffs in periodic crystals",
"Temporal control of high-order harmonic cutoffs in periodic crystals"
] | [
"Nivash R \nDepartment of Physics\nBirla Institute of Technology and Science -Pilani\n333031RajasthanIndia\n",
"Amol R Holkundkar Id \nDepartment of Physics\nBirla Institute of Technology and Science -Pilani\n333031RajasthanIndia\n",
"IDJayendra N Bandyopadhyay \nDepartment of Physics\nBirla Institute of Technology and Science -Pilani\n333031RajasthanIndia\n"
] | [
"Department of Physics\nBirla Institute of Technology and Science -Pilani\n333031RajasthanIndia",
"Department of Physics\nBirla Institute of Technology and Science -Pilani\n333031RajasthanIndia",
"Department of Physics\nBirla Institute of Technology and Science -Pilani\n333031RajasthanIndia"
] | [] | A theoretical study on the high-harmonic generation (HHG) in solids by a synthesized driver field is carried out. The vector potential of the driver field is instrumental in determining the Bloch oscillations of the electrons in a periodic crystal, which eventually reflects in the high-order harmonic spectra. To this end, the interaction of the sinc-shaped driver with the periodic crystal is studied. A typical temporal profile of the associated vector potential manifests in extending the harmonic cutoffs compared to the standard sin 2 envelopes of similar intensity and duration. It is also observed that the harmonic cutoffs can be controlled in a temporal manner by varying the delay parameter introduced in the proposed sinc-shaped driver, with a well-defined scaling that depends on the energy bands of the periodic crystal under study. Furthermore, it is also observed that the emission time of the cutoff harmonics and even the harmonic cutoff energy can also be controlled by varying the delay parameter systematically. An optimum delay parameter for maximum harmonic cutoff energy and harmonic yield is also deduced. | null | [
"https://export.arxiv.org/pdf/2305.00752v1.pdf"
] | 258,426,641 | 2305.00752 | d2b7b90ec037201da51a6b3e00d6027de8aecc9c |
Temporal control of high-order harmonic cutoffs in periodic crystals
Nivash R
Department of Physics
Birla Institute of Technology and Science -Pilani
333031RajasthanIndia
Amol R Holkundkar Id
Department of Physics
Birla Institute of Technology and Science -Pilani
333031RajasthanIndia
IDJayendra N Bandyopadhyay
Department of Physics
Birla Institute of Technology and Science -Pilani
333031RajasthanIndia
Temporal control of high-order harmonic cutoffs in periodic crystals
(Dated: May 2, 2023)
A theoretical study on the high-harmonic generation (HHG) in solids by a synthesized driver field is carried out. The vector potential of the driver field is instrumental in determining the Bloch oscillations of the electrons in a periodic crystal, which eventually reflects in the high-order harmonic spectra. To this end, the interaction of the sinc-shaped driver with the periodic crystal is studied. A typical temporal profile of the associated vector potential manifests in extending the harmonic cutoffs compared to the standard sin 2 envelopes of similar intensity and duration. It is also observed that the harmonic cutoffs can be controlled in a temporal manner by varying the delay parameter introduced in the proposed sinc-shaped driver, with a well-defined scaling that depends on the energy bands of the periodic crystal under study. Furthermore, it is also observed that the emission time of the cutoff harmonics and even the harmonic cutoff energy can also be controlled by varying the delay parameter systematically. An optimum delay parameter for maximum harmonic cutoff energy and harmonic yield is also deduced.
I. INTRODUCTION
The last three decades have witnessed tremendous development in the field of high-order harmonic generation (HHG) by atomic and molecular targets and made it possible to have highly tunable extreme-ultraviolet (XUV) pulses in the attosecond regime [1]. The low conversion efficiency in gaseous HHG is one of the major bottlenecks, which is remedied after the advent of the HHG by the Bloch oscillations in the solids [2][3][4][5][6]. The HHG from the solids is a very lucrative venture as it promises a compact source of the XUV radiations and attosecond spectroscopy [7][8][9][10][11]. Various other aspects are also unique for the HHG obtained from solids compared to the atomic HHG. For example, the cutoff energy scales linearly with the field amplitude [12], multiple plateaus of higher energies [13], etc. The transition from the higher conduction bands to the valence band causes the generation of multiple plateaus, which can be understood from the quasi-classical model [14]. Some studies are also aimed toward extending the secondary plateaus of higher energies [15][16][17]. The inter-band and intraband dynamics of the Bloch electrons and the corresponding transitions give rise to rich features in the HHG from solids. The detailed account of the HHG from solids can be found in the following seminal review articles [8,9,11]. Extension of the harmonic cutoffs [18], understanding the real-space collision dynamics [19] in HHG from solids, study of HHG from periodic optical lattices [20], time-dependent band population imaging [21] via HHG in solids, the effect of vacancy defects [22,23], HHG in mono-layer and bi-layer Graphene [24], and many more interesting studies in the field of the HHG from solids are reported recently.
The HHG from the solids can be understood from the quasiclassical model [14], wherein some fraction of the electrons near the k 0 = 0 of valence band makes the transition to the conduction band, followed by the intraband dynamics wherein in the momentum space the temporal evolution of the crystal momentum is related to the driving vector potential as * E-mail: [email protected] k(t) = k 0 + A(t). Once the electron reaches the edge of the Brillouin zone (BZ), then the transition to the next higher conduction band is feasible, resulting in an increased band population of the higher conduction bands. Eventually, an interband transition causes the emission of high-energy photons, bringing back the electron to lower bands. When the electron reaches the edge of BZ, it can also undergo a Bragg's reflection in the same band, and as we mentioned, it can also tunnel to the neighboring conduction band if the bandgap at the BZ boundary is small, this tunneling is referred as the Zener tunneling, and collectively the dynamics are referred as the Bloch-Zener oscillations [25]. The vector potential associated with the laser pulse is instrumental in populating the higher-conduction bands in a step-by-step manner. As a consequence, the multiple plateaus in HHG can be understood. The detailed analysis regarding the role of the driver vector potential on the HHG by periodic crystals is studied in numerous journal articles, e.g., [13,15,26,27].
In this work, we study the HHG by an 1D periodic lattice potential driven by the sinc-shaped driver field [28][29][30]. A broad frequency distribution and a single relatively strong field amplitude make the sinc driver's field profile fascinating for generating and controlling the higher-order harmonics [28]. The optical frequency combs are routinely used for pulse shaping, which in principle might be a viable source for generating tailor-made field profiles [31][32][33][34]. In this work, we relied on the synthesized sinc shaped driver; it is observed that by changing the delay parameter in the driver field, the harmonic cutoff can be controlled temporally in a systematic manner. Furthermore, harmonic cutoff extension is also seen with the sinc shaped driver as compared to the standard sin 2 envelopes, and an optimum value for the delay parameter is observed. In principle, the delay parameter used in defining the driver field can be controlled by moving the mirror assembly on an optical bench [28].
This paper is organized as follows. The theoretical and numerical aspects of the work are presented in Sec. II, followed by the results and discussion in Sec. III, and finally conclusion in Sec. IV.
II. THEORY AND NUMERICAL METHODS
We have studied the laser-solid interaction using the linearly polarized laser propagates along the optical axis of the thin crystal, and the laser polarization direction lies in the crystal plane. The 1D periodic potential used throughout the manuscript is a Mathieu-type potential given as [35]:
V (x) = −0.37[1 + cos(2πx/d)] a.u.,(1)
with d = 8 a.u. being the lattice parameter. These types of potential are frequently used in describing optical lattices [36,37]. In this study, we have used N = 60 lattice sites, and hence our simulation domain in real space ranges from −L to +L where L = Nd/2. The typical band structure associated with this periodic potential is calculated and presented in Fig. 1.
We have verified that, for both sin 2 and sinc-shaped driver the results are converged with the inclusion of the 15 bands only [38]. Though, in Fig. 1, we have only shown nine bands, wherein VB and CB respectively mean 'Valence Band' and 'Conduction Band.' In the following, we present the details of the theoretical and numerical methods employed in this study.
A. Laser Pulse
In the HHG fraternity, it is pervasive to rely on the sin 2 field profiles as given by:
E sin2 (t) = F 0 f (t) sin(ω 0 t)x(2)
where, F 0 is the field amplitude, f (t) = sin 2 (πt/T ) is the envelope function with pulse duration T , and ω 0 is the fundamental driver frequency with one optical cycle being τ = 2π/ω 0 . In this work, we use a sinc shape driver pulse, such that the temporal profile of the electric field of the synthesized pulse is written as [28]:
E sinc (t) = F 0 f (t) sin[ω 0 (t − t 0 − t d )] ω 0 (t − t 0 − t d ) − sin[ω 0 (t − t 0 )] ω 0 (t − t 0 ) x,(3)
where t d is the delay between the pulses and t 0 introduces some constant phase. Note that t d = 0 corresponds to the outof-phase addition of the pulses, which results in E(t) = 0. The delay t d can easily be tuned by varying the mirror assembly on an optical bench [28]. It should be noted that the associated vector potential A sinc (t) ∼ − E sinc (t)dt is almost like a sinc function, which can be responsible for vibrant non-linear electron dynamics. The field profiles E sinc (t) (with t d = 0.7τ) and E sin2 (t) for some representative parameters are presented in Fig. 2(a). The temporal profile of the respective vector potentials is also shown in Fig. 2(b). We have used a pulse duration T = 5τ and constant phase factor t 0 = 2.14τ [so that the pulse peak is temporally at the center] throughout the manuscript. As can be seen from Fig. 2(b), the vector potential for the sinc driver is almost like a sinc function [sin(x)/x], wherein for most of the duration, the A(t) is positive only.
B. k-space HHG calculation
We solve the TDSE numerically in velocity gauge [38]. The electron wave function can be expanded in Bloch state basis |φ n k for a particular value of the crystal quasimomentum k and band index n. The Bloch states are evaluated by solving the single-electron stationary Schrödinger equation with fieldfree HamiltonianĤ o =p 2 /2 +V (x):
H o |φ n k = E n k |φ n k .(4)
In position basis, the Bloch states can be written as:
x|φ n k ≡ φ n k (x) = N max ∑ =1 C n k, e i(k+2π /d)x ,(5)
where N max = 15 are used throughout the work. After the evaluation of the Bloch states, the TDSE can be solved for electronic wavefunction |ψ k (t) as :
i ∂ ∂t |ψ k (t) = [Ĥ o +Ĥ int ] |ψ k (t) ,(6)
where,Ĥ int = A(t) ·p and A(t) is the vector potential associated with the laser pulse under dipole approximation. Furthermore,
|ψ k (t) = N max ∑ n=1 α n k (t) |φ n k ,(7)
where α n k (t) are the time dependent expansion coefficients. Using Eq. (7) in Eq. (6), we have coupled differential equations as [38]:
i ∂ α s k (t) ∂t = E s k α s k (t) + A(t) N max ∑ u=1 p su k α u k (t), .(8)
Here, p su k is the matrix element of the momentum operator, which can be calculated as:
p su k = φ s k |p|φ u k = N max ∑ =0 (k + 2π /d) C s k, * C u k, .(9)
If we consider the electron initially in the band q, then the initial condition for solving Eq. (8) is α s k (0) = δ qs . Finally, the single electron current density for a particular channel k can be calculated as:
j ks (t) = −Re[ ψ ks |p + A(t)|ψ ks ].(10)
In Eq. (10), the subscript 's' denotes that the electron was in the band s before the interaction. Total current density can be calculated by summing over all the bands and integrating over BZ as:
j(t) = ∑ s∈V B j ks (t)dk.(11)
In this Bloch state basis based formulation, there is no coupling between different values of k, which makes it very straightforward to implement. However, for k-space-based calculation of inter and intra-band currents, one must rely on the Houston state basis [5]. The spectra of the emitted harmonics can be estimated by doing the Fourier transform of the current density and are given as S(ω) = j(t)e iωt dt 2 . The harmonic yield Y for the frequency range ω 1 to ω 2 is calculated as:
Y = T −1 ω 2 ω 1 S(ω)dω.
III. RESULTS AND DISCUSSIONS
Here, we first compare the HHG spectra for the sin 2 and sinc-shaped driver, then the delay parameter's effect on the HHG spectra is explored.
FIG. 3. The HHG spectra for the sin 2 and sinc drivers with the same parameters as in Fig. 2(a) are presented (b) along with the Gabor transform for sinc driver (c) and sin 2 driver (d). The respective energy difference of different bands is also illustrated (a) to understand the observed cutoffs.
A. Comparing HHG by sinc and sin 2 driver In Fig. 3, we have compared the harmonic spectra as obtained by the 3.2 µm laser pulse with peak intensity 6 × 10 11 W/cm 2 and having the sin 2 envelope [Eq. (2)] and sinc shaped field profile with t d = 0.7τ [Eq. (3)]. It can be seen that the harmonic cutoff in the case of the sinc driver is extended well beyond the one obtained using the sin 2 driver. The corresponding energy difference of different bands is also shown in Fig. 3(a), and the multi-plateau structure with the respective harmonic cutoff for sinc shaped driver can easily be mapped on the energy difference between the bands. For the sinc-shaped driver, we see 4 plateaus respectively at ∼ 14 eV, ∼ 26 eV, ∼ 47 eV, and ∼ 79 eV, which corresponds to the transition from CB1→VB1, CB2→VB1, CB3→VB1, and CB4→VB0 respectively [ Fig. 3(a,b)]. The Gabor transform (time-frequency analysis) for sinc, and sin 2 shaped driver are respectively presented in Fig. 3(c) and 3(d). The higher harmonic cutoff for the sinc driver is also seen in the timefrequency analysis. Furthermore, as can be seen from Fig. 3(b), for the sinc driver, the harmonics near the ∼ 47 eV and ∼ 79 eV exhibit very smooth modulation. This can be understood from the single trajectory contribution to the harmonic spectra, which is corroborated by Fig. 3(d). In this figure, the higher-energy harmonics emitted only once because the interference from the multiple trajectories is absent, giving very smooth single trajectory contribution. The single trajectory contribution is always sought as a single attosecond pulse can be synthesized using such harmonics instead of the attosecond-pulse train [39]. The extension of the harmonic cutoff in the case of the sinc driver can be attributed to the vector potential associated with the same [ Fig. 2(b)]. The sinc-shaped vector potential does not have a very rapid oscil- latory part; as a result, rapid oscillations of the electron in the band are suppressed. The approximate 'uni-directional' nature of the vector potential drives the electron in the same direction for most of the pulse duration. As the electron reaches the edge of the BZ, then with Zeneer tunneling, it can climb up the bands [26]. Furthermore, in Fig. 4, the temporal evolution of the band population [40] using the sinc and sin 2 driver are presented. The time dependent population of the n th band is given by P n (t) = BZ |α n k (t)| 2 dk such that ∑ N max n=1 P n (t) = 1. The laser parameters are the same as Fig. 3. Initially, the electron is considered to be in the VB1 in both cases. As can be seen, using the sinc driver, the higher bands are populated, and an appreciable population of even the CB4 band is observed, which in turn results in the emission of the harmonics till ∼ 79 eV. However, for sin 2 pulse, the bands beyond CB2 are hardly populated, resulting in lower cutoff energies. In the following, we study the effect of delay parameter t d of the sinc driver on the harmonic cutoff and the harmonic yield.
We observed that the harmonic efficiency beyond ∼ 47 eV is drastically reduced, which is understandable as the probability of electron tunneling to higher conduction bands in a step-by-step manner reduces. In view of this, we now focus on the HHG spectra till ∼ 50 eV, which covers the transition from the CB3→VB1 maximally.
B. Temporal control of HHG cutoff
The delay parameter associated with the sinc pulse [Eq.
(3)] is very instrumental in determining the field profile and so the temporal profile of the vector potential [28]. From the experimental perspective, the variation of the delay parameter is equivalent to moving the mirror assembly on an optical bench, which brings us to the question, is it possible to control the band population and so the harmonic cutoff energies and the harmonic yield by simply changing this delay parameter? Given this, we studied the HHG by varying the delay parameter t d and connected the same to the temporal evolution of the band population.
In Fig. 5(a) we present the temporal evolution of CB3 band-population for t d = 0.5τ, 0.7τ and 1τ cases. The respective vector potential for these parameters is also shown in Fig. 5(b). The HHG spectra for these three cases are illustrated in Fig. 5(d). Please note that in Fig. 5(d), we have intentionally shifted the HHG spectra for 0.7τ and 1τ for better visual representation. In Fig. 5
(E CB3 − E V B1 ) eV].
As can be seen from Fig. 5(d) and (e) that the harmonic cutoff energies for all three cases can be mapped on the transitions from the CB3 to VB1 band. As in this figure, we are focusing on the harmonic cutoff caused by the transition from the CB3 to VB1, and hence in Fig. 5(c), we present the temporal evolution of the crystal momentum k(t) = k 0 + A(t), where k 0 is chosen from the classical trajectories such that for t d = 0.5τ ; k(2τ) = 0, t d = 0.7τ ; k(2.134τ) = 0 and t d = 1τ ; k(2.2τ) = 0. The electron is expected to be in CB3 at k = 0 (lowest energy of CB3); however, how and when it will make transition to CB3 is highly unpredictable. These time instances at which the electron is considered to oscillate in CB3 are back-traced by the knowledge of the time-frequency analysis of the emitted harmonics [refer Fig. 7], the classical trajectory calculations, and also noting the appropriate time for the transition from the CB3 to VB1, resulting in the harmonic cutoff. The time instants at which maximum harmonic energy (cutoff energy) is emitted for t d = 0.5τ, 0.7τ, and1τ are shown in Fig. 5(c) with solid circles at 3τ, 3.1τ and 3.37τ respectively, which also corroborates the time instants when and crystal momentum k(t) (c), and the HHG spectra (d) is presented for different delay parameter of sinc driver. The bands CB3 and VB1 and their respective energy difference is also shown (e). The HHG spectra for t d = 0.7τ and t d = 1τ cases are intentionally shifted downward by 4 and 8 units, respectively, for better representation. In (c), the black, orange, and blue circles at 3τ, 3.1τ, and 3.37τ show the time instants when the maximum harmonic energy is observed for different delay parameters. the CB3 band population is lowest, signifying that after the transition from CB3 to VB1 would seize to exist and hence the abrupt cutoff is observed. In order to further confirm these findings, in Fig. 6(a,b,c), we have presented the Gabor transforms for t d = 0.5τ, 0.7τ, and1τ cases. It can be observed that the maximum harmonic energy is emitted at respectively 3τ, 3.1τ, and 3.37τ. The variation of the time instant at which maximum energy is emitted (t peak ) with the sinc pulse delay parameter t d is also shown in Fig. 6(d). The red dots in Fig. 6(d) are the observed values from the simulation, and the solid line represents the scaling with the delay parameter, which is found to be t peak ∝ t 5/2 d . This very well-defined non-linear scaling with the delay parameter translates to merely moving the mirror assembly (from an experimental perspective); one can have fine control over the emission time of maximum or cutoff harmonic energy. This particular scaling with the delay parameter ∝ t 5/2 d is a property of the periodic lattice and the corresponding band-structure. For example, in Appendix-A we have obtained different scaling for the same periodic lattice with different lattice constant. This is so, because the driving of the electron in a band and the follow-up phenomenon of Zener tunneling to higher bands strongly depend on the respective band structure and the minimum band gap near the edge of BZ or at the center of BZ.
Next, in Fig. 7, we compare the temporal evolution of the total current (a), population of the conduction band CB3 (b) and CB2 (c), using sin 2 pulse and different delay parameters of the sinc-shaped pulse. It can be seen from Fig. 7(a) that the sinc driver can drive a very strong current in the lattice, closely mimicking the vector potential of the driver. The optical response of the lattice under study can be understood in terms of the polarization, which is defined as P(t) = t −∞ j(t )dt [38]. As we observe from Fig. 7(a), the time average of the currents in the case of the sin 2 driver would be smaller than the sinc driver. The currents caused by the Bloch oscillations using a sinc driver can lead to a very strong polarization response in the solids; mostly because of the same polarity of the current. The strong current in the case of the sinc driver can be understood in terms of the temporal evolution of the CB2 and CB3 band population, contributing the harmonics till ∼ 50 eV. In Fig. 7(b) and (c), we have compared the temporal evolution of the CB3 and CB2 band population using sin 2 and a sincshaped driver. The strong vector potential associated with the sinc driver plays a crucial role in populating the higher energy bands by ∼ 5 − 35% or so, while using the sin 2 driver with similar peak field amplitude is not observed.
C. HHG cutoff and yield using sinc driver
In Fig. 8, we have further explored the effect of the delay parameter on the HHG cutoff and the harmonic yield. We have presented the HHG spectra for different delay parameters of the sinc driver and the variation of the harmonic cutoff energy [ Fig. 8(b)] and harmonic yield [ Fig. 8(c)] as a function of the delay parameter t d . In Fig. 8(a), two energy ranges are highlighted, for which the harmonic yield is presented in Fig. 8(c). It is observed that there is an optimum delay parameter around t d ∼ 0.7τ which is responsible for the maximum cutoff energy. The maximum harmonic yield in two energy ranges highlighted in Fig. 8(a) are also presented, and t d ∼ 0.7τ is observed to be an optimum delay parameter. This optimum delay parameter can be understood from the Fig. 5(b),(c), and (e), wherein it can be observed that the stronger vector potential can drive the electron past the BZ boundary (i.e., the equivalent of the coming from opposite side because of the periodicity of the band-structure), and as a result, the lower harmonic cutoff is expected. We have seen in Fig. 5(b) that the vector potential amplitude for the t d = 1τ case is large as compared to the t d = 0.7τ case, the corresponding timedependent crystal momentum k(t) in Fig. 5(c) for t d = 1τ case would cause the driving of the electron from k(2.2τ) = 0 to the maximum k(2.64τ) = 0.55, which eventually need to have longer trajectory as compare to the other two cases, to recombine around 3.37τ. So in principle, there will be competing mechanisms; the stronger vector potential will try to push the electrons past the edge of the BZ, but in a process, eventually ends up having large dispersion of electronic wavefunction, resulting in lower harmonic yield and the harmonic cutoff (harmonic cutoff would depend on the extent the electron is moved in a band).
From Fig. 8, we can see that using a sinc driver, we can temporally control the harmonic cutoff. The harmonic cutoff energy can also be tailored by adjusting the delay parameter in a systematic way. The control over harmonic yield in two sample energy ranges also adds to the utility of the sinc drivers.
IV. SUMMARY
In summary, we have studied the interaction of the sincshaped driver with the 1D periodic potential, which is aligned along the polarization direction of the driving laser pulse. The vector potential associated with the proposed field profile [refer Eq. (3)] mimics the sinc function, which for most of the pulse duration does not change the polarity. This typical characteristic of the vector potential drive the electron resulting in a very efficient Zener tunneling to higher conduction bands at the minimum-band gap of the neighboring bands. It is ob- served that by controlling the delay parameter in Eq. (3), the emission of the HHG cutoff energy can be temporally controlled systematically. The harmonic-cutoff energy emission time is found to scale as t peak ∝ t Furthermore, the harmonic yield in a couple of energy ranges and the harmonic cutoffs are also studied by varying t d . An optimum delay parameter corresponding to the maximum harmonic cutoff energy and yield is also observed. As a representative case to test the proof-of-concept, a different lattice parameter, say d = 6 a.u. in the potential V (x) is also used and the results are presented in the Appendix A. The emission time of the cutoff energy with the delay parameter again showed the well-defined variation with different scaling parameter [t peak ∝ t 8/5 d ]. This is expected as the electron dynamics for a given field profile depend solely on the band structure describing the periodic lattice, and hence the scaling parameter will have the imprint of the band structure of the periodic crystal under investigation. Further detailed analysis of the problem we reserve for the future. The detailed analysis of the chirp of the emitted harmonics for different lattice constant would also be an interesting study.
ACKNOWLEDGMENTS
Authors would like to acknowledge the DST-SERB, Government of India, for funding the project CRG/2020/001020.
FIG. 2 .
2The temporal electric field profile of the 3.2 µm sin 2 and sinc laser pulse [t d = 0.7τ] with a peak intensity of 6 × 10 11 W cm −2 are presented (a), along with the respective vector potentials (b). The y−axis scaling parameter is also mentioned (a).
FIG. 4 .
4Temporal dependence of the band population of various bands [as labeled inFig. 1] are presented for the sinc (a) and sin 2 (a) driver. For both cases, the laser parameters are the same as inFig. 3.
FIG. 5 .
5Temporal evolution of band population of CB3 (a) along with the vector potential A(t) (b)
FIG. 6 .
6The Gabor transform of the HHG spectra for the delay parameters t d = 0.5τ (a), 0.7τ (b), and 1τ (c) are presented. All the other laser conditions are the same as inFig. 5. A vertical dashed white line at 3τ (a), 3.1τ (b), and 3.37τ (c) represents the time instants when the maximum energy harmonics are emitted. The variation of the time instant corresponding to the harmonic cutoff t peak with the delay parameter is presented in (d), wherein the red dots denote the observed values from the simulation and the solid line represents the ∝ t
FIG. 7 .
7Time evolution of the total integrated current [Eq.(11)] (a) are presented for sin 2 pulse and for the sinc driver with the time delay 0.5τ, 0.7τ, and 1τ. The temporal evolution of the CB3 (b) and CB2 (c) populations is also illustrated using different laser profiles.
FIG. 8 .FIG
8Harmonic spectra for different delay parameters are presented (a), and the energy ranges 5 -15 eV and 30 -40 eV are highlighted. The variation of the harmonic cutoff energy (b) and harmonic yield (c) with the delay parameter is presented. The harmonic yield in (c) is presented for the two energy ranges highlighted in (a), and the yield for 30 -40 eV is scaled up by 20 times for better representation. . A1. The Gabor transform of the HHG spectra for the delay parameters t d = 0.5τ (a), 0.7τ (b), and 1τ (c) are presented. All the other laser conditions are the same as in Fig. 6. A vertical dashed white line at 3.08τ (a), 3.22τ (b), and 3.49τ (c) represents the time instants when the maximum energy harmonics are emitted. The variation of the time instant corresponding to the harmonic cutoff t peak with the delay parameter is presented in (d), wherein the red dots denote the observed values from the simulation and the solid line represents the ∝ t
t d being the delay parameter [refer Eq. (3)].
arXiv:2305.00752v1 [physics.atm-clus] 1 May 2023−15
0
15
30
45
60
75
90
−1 −0.75 −0.5 −0.25
0
0.25
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0.75
1
VB0
VB1
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CB5
Energy (eV)
k (units of π/d)
FIG. 1. The band structure for periodic potential V (x) = −0.37[1 +
cos(2πx/d)] a.u. is calculated for first Brillouin Zone. The bands
are labeled as 'Valence Band' (VB) and 'Conduction Band' (CB).
Minimum band gap energy between CB1 and VB1 is ∼ 4.2 eV.
(e) we have shown the VB1 and CB3 bands in the extended Brillouin Zone [|k| > π/d] along with the energy difference between the two [
Appendix AIn order to further verify the capability of the sinc-shaped driver to temporally control the cutoff harmonic emission, we have carried out the simulations for the lattice parameter d = 6 a.u. using the same Mathieu-type potential[35]. InFig. A1(a), (b)and (c) we have presented the Gabor transform for HHG spectra emitted with t d = 0.5τ (a), 0.7τ (b) and 1τ (c). The time at which maximum energy is emitted (t peak ) is plotted inFig. A1(d)as function of the delay parameter, and it is observed that for this particular lattice parameter t peak ∝ t 8/5 d .
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| [] |
[
"Instruction-ViT: Multi-Modal Prompts for Instruction Learning in ViT",
"Instruction-ViT: Multi-Modal Prompts for Instruction Learning in ViT"
] | [
"Zhenxiang Xiao ",
"Yuzhong Chen ",
"Lu Zhang ",
"Junjie Yao ",
"Zihao Wu ",
"Xiaowei Yu ",
"Yi Pan ",
"Lin Zhao ",
"Chong Ma ",
"Xinyu Liu ",
"Wei Liu ",
"Xiang Li ",
"Yixuan Yuan ",
"Dinggang Shen ",
"Dajiang Zhu ",
"Tianming Liu ",
"Xi Jiang "
] | [] | [] | Prompts have been proven to play a crucial role in large language models, and in recent years, vision models have also been using prompts to improve scalability for multiple downstream tasks. In this paper, we focus on adapting prompt design based on instruction tuning into a visual transformer model for image classification which we called Instruction-ViT. The key idea is to implement multi-modal prompts (text or image prompt) related to category information to guide the finetuning of the model. Based on the experiments of several image captionining tasks, the performance and domain adaptability were improved. Our work provided an innovative strategy to fuse multi-modal prompts with better performance and faster adaptability for visual classification models. | 10.48550/arxiv.2305.00201 | [
"https://export.arxiv.org/pdf/2305.00201v1.pdf"
] | 258,426,716 | 2305.00201 | a677938545f63ad44c87d09f85dd231980a8476f |
Instruction-ViT: Multi-Modal Prompts for Instruction Learning in ViT
Zhenxiang Xiao
Yuzhong Chen
Lu Zhang
Junjie Yao
Zihao Wu
Xiaowei Yu
Yi Pan
Lin Zhao
Chong Ma
Xinyu Liu
Wei Liu
Xiang Li
Yixuan Yuan
Dinggang Shen
Dajiang Zhu
Tianming Liu
Xi Jiang
Instruction-ViT: Multi-Modal Prompts for Instruction Learning in ViT
Prompts have been proven to play a crucial role in large language models, and in recent years, vision models have also been using prompts to improve scalability for multiple downstream tasks. In this paper, we focus on adapting prompt design based on instruction tuning into a visual transformer model for image classification which we called Instruction-ViT. The key idea is to implement multi-modal prompts (text or image prompt) related to category information to guide the finetuning of the model. Based on the experiments of several image captionining tasks, the performance and domain adaptability were improved. Our work provided an innovative strategy to fuse multi-modal prompts with better performance and faster adaptability for visual classification models.
Introduction
It has been a long standing goal of humanity to develop Artificial General Intelligence (AGI) that exhibits human-level or even surpassing intelligence. An essential characteristic of human intelligence is its ability to process information from multiple modalities, which enables individuals to comprehend their surroundings through multiple sources of in-formation and communicate effectively with others (Zhao et al., 2023). Similarly, artificial intelligence systems are also expected to efficiently handle, integrate, and utilize multimodal data to solve real-world problems. The recent breakthroughs of large language models (LLMs) have provided new insights into realizing this goal. LLMs were initially proposed in the field of Natural Language Processing (NLP) to solve various complex NLP tasks, and have demonstrated remarkable abilities in learning and reasoning. Compared to traditional language models, these LLMs adopt a novel prompt technique which allows the pre-trained LLMs to be adapted to downstream tasks without fine-tuning the models themselves. Through the flexible prompt design, the language model could be pre-trained on massive amounts of raw text and perform few-shot or even zero-shot learning, thus adapting to new scenarios with few or no labeled data (Liu et al., 2023c). For example, the in-context prompt in the GPT series, which is based on auto-regressive pre-training and prompt-based fine-tuning, allowed the model to produce an ideal result for previously unseen tasks without the need to update any parameter .
While large-scale AGI uni-modal (images or texts) models have demonstrated impressive performance in a variety of tasks (Brown et al., 2020;Kirillov et al., 2023), the complexity and diversity of many real-world problems in artificial intelligence often require the integration of information from multiple modalities, such as text, image, and audio. Multimodal models utilize various methods to fuse data from different modalities, and these methods are often categorized as early (feature) fusion, late (decision) fusion, or intermediate (hybrid) fusion, based on the level in the network at which representations are fused (Baltrušaitis et al., 2018). However, the choice of fusion method remains highly dependent on the specific domain, data, and task, and there are currently no universal fusion rules. Multi-modal models have shown their potential in enhancing performance across various tasks, such as speaker diarization (Gebru et al., 2017), text-to-image generation (Rombach et al., 2022), and image description (OpenAI, 2023), but they lack processing methods for some specific tasks like segmentation (Kirillov et al., 2023).
Recently, several studies have attempted to introduce arXiv:2305.00201v1 [cs.CV] 29 Apr 2023 prompts into visual models so that they can cope with multiple tasks. For example, given a pair of input-output images as the visual prompting for a task example, the model could automatically generate an output image which is consistent with the given examples for a new input image (Bar et al., 2022), and multiple tasks could be finished as an image inpainting task. The Stable Diffusion (Rombach et al., 2022) controls the content of the generated images by inserting text prompts to the latent space. The Segment Anything Model (Kirillov et al., 2023) showed impressive segmentation capabilities on numerous segmentation tasks by adding segmentation prompts which could be points, boxes, text, and masks.
In this work, we concentrate on visual instruction tuning for image captions. The instruction tuning method was first proposed for NLP tasks (Wei et al., 2021). By fine-tuning language models on a collection of datasets described via instructions, the instruction tuning method substantially improves zero-shot performance on unseen tasks. For our design, the instruction tuning method is introduced into the vision transformer (ViT) by adding multi-modal prompts (text or image prompt) related to category information to guide the fine-tuning of the model. Specifically, first, some potential categories of images or text prompts are considered as instruction prompts and contact with the image tokens. Second, the image and instruction prompt tokens are jointly fine-tuned by the model, which we called the Instruction Vision Transformer (Instruction-ViT), to update the representation. Last, a similarity score is calculated between the image CLS token and those potential categories' instruction prompt tokens to finish the image captioning task.
Our contributions and the main findings are summarized as follows:
(1) We introduced instruction tuning to the vision transformer, and both image and text could be used as the instruction prompt to guide tuning.
(2) By conducting experiments on several image captioning tasks, our Instruction-ViT showed better performance and faster adaptability.
Related Work
Large-Scale Multi-Modal Models
Large-Scale Multi-Modal Models (LMMs) are trained on large-scale datasets to effectively process multiple modalities, particularly for visual-language downstream tasks. UNITER achieves state-of-the-art performance on various downstream tasks by jointly encoding textual and visual information in a shared representation space (Chen et al., 2020). CLIP utilizes different encoders for images and text, matching them in the latent space to achieve a powerful multi-modal encoder model (Radford et al., 2021). ALIGN also uses a dual-encoder architecture to align visual and language representations by training image-text pairs without manual annotations (Jia et al., 2021). BLIP pre-trains a multi-modal mixture of encoder-decoder model to tackle both understanding-based and generation-based tasks (Li et al., 2022). Flamingo is a family of visual language models trained on large-scale multi-modal web corpora, and can easily adapt to both classification and generation tasks (Alayrac et al., 2022). GPT-4, the latest version of GPT models, is a large-scale multi-modal model which is expected to be able to process multiple types of data, including texts, images, audio, and video (OpenAI, 2023). Our work also focuses on using pre-trained visual-language models to align language and visual features to better perform visual downstream tasks.
Prompt Tuning
Prompt tuning is a technique used in NLP to improve the performance of language models such as GPT and its other variants (Floridi & Chiriatti, 2020). Prompt tuning involves fine-tuning a pre-trained language model on a specific task by providing it with a set of prompts or examples relevant to the task (Gu et al., 2021). Unlike conventional fine-tuning, which involves modifying the pre-trained model weights or parameters, prompt tuning requires no changes in the pretrained model weights. Moreover, prompt tuning becomes increasingly competitive at larger scales, as models with billions of parameters are becoming more common. Existing works show that prompt tuning achieves comparable performance on model tuning, which involves tuning all of the model weights (Liu et al., 2022;Lester et al., 2021). This is significant because altering the underlying model can be a costly process. In prompt tuning, the model is "frozen" and can be used as is. Furthermore, prompt tuning requires less labeled data compared to other methods.
The goal of prompt tuning is to improve the model's ability to generate high-quality outputs for a specific task, such as text classification or language translation. By training the model on specific prompts or examples, it can learn to generate more accurate and relevant outputs for that task (Liu et al., 2023b). Generally, prompt tuning involves providing the model with a set of input-output pairs in the source and target languages, respectively. By fine-tuning the model on these examples, it can learn to generate more accurate translations for new inputs (Liu et al., 2022). Prompt tuning is often used in conjunction with other techniques such as transfer learning and data augmentation to further improve the performance of NLP models (Tu et al., 2023).
Prompt in Visual Space
Prompt tuning is proposed to adapt to different downstream tasks, reducing the amount of parameter storage compared to fine-tuning methods while improving the performance on unknown tasks (Brown et al., 2020;Schick & Schütze, 2020;Wei et al., 2021). Visual Prompt Tuning (Jia et al., 2022) introduces the prompt method into the visual model, which only trains very few parameters to obtain higher classification accuracy than full fine-tuning method. For visionlanguage models, CoOp adds an additional learnable context prompt to the input of the text encoder to enhance the zeroshot learning capability (Zhou et al., 2022b). To further improve the robustness of the class shift of CoOp, CoCoOp embeds the instance-conditional token obtained on image encoder features from the basis of the context token (Zhou et al., 2022a).
Instruction Tuning
Instruction fine-tuning, also known as instruction tuning, is a fine-tuning technique initially introduced for LLMs (Wei et al., 2021). Rather than fine-tuning on a specific downstream task as in BERT-based (Devlin et al., 2018) model tuning, instruction tuning employs data comprising concise instructions and corresponding outputs across a diverse range of tasks and domains. This approach integrates multiple downstream tasks into a single, generic model through a one-time fine-tuning process, considerably reducing total training time and storage space requirements for separate models catering to different tasks. In addition, instruction fine-tuned models not only perform well on the instructions encountered during instruction tuning but also generalize effectively to unseen instructions, thus significantly enhancing zero-shot in-context learning capabilities (Ouyang et al., 2022).
In the field of NLP, based on GPT-3 (Brown et al., 2020), Ouyang et al. utilizes instruction tuning with RLHF (Christiano et al., 2017) to develop InstructGPT, which better aligns model responses with user intent and minimizes untruthful and toxic content in the output. With further instruction tuning, OpenAI introduces ChatGPT and GPT-4 (OpenAI, 2023), which represent significant strides towards AGI models (Zhao et al., 2023). Another recently released instruction-tuned model, Alpaca, employs GPT-3.5 to generate a 52k instruction-following dataset (Wang et al., 2022b) and uses it to fine-tune LLaMA 7b (Touvron et al., 2023). This achieves comparable performance with GPT-3.5 while requiring a smaller scale and fewer computational resources (Taori et al., 2023).
In addition to NLP, recent work has also extended instruction tuning to multi-modal model fine-tuning. Liu et al. leverages GPT-4 to generate instruction-following data (Peng et al., 2023) based on images and the corresponding cap-tions. The resulting LLaVa model demonstrates competitive results with GPT-4 on visual and language understanding tasks.
Methods
We propose Instruction-ViT, a unified framework to align the input of image and prompts. In this section, we will present the method of creating prompt tokens, provide details of the model's backbone, and discuss the completion of downstream tasks, as well as calculation of loss, respectively. Finally, we introduce our training strategy.
Prompt
We construct the prompt as shown in the bottom right of Figure 1. In our work, we use the text of class name, the corresponding image, and the combination of text and image as our prompts, respectively. For the text prompt, we use 30 sentence templates in the same way as OpenCLIP (Schuhmann et al., 2022), of the form like a photo of a {Class N ame}. After that, we use the pretrained CLIP text encoder as our prompt encoder, use the 30 constructed prompts as input, and obtain the average result as the prompt token x pt . CLIP's pre-trained image encoder is used as the Prompt Encoder for creating image prompt tokens x pi . With this encoder, we randomly select an image as the prompt of one class in the training set and finally obtain the prompt tokens. By averaging the text and image prompt tokens together, we obtained the mixed prompt tokens x pm for each class. Therefore, the input prompt tokens can be represented as x p = {x|x ∈ x pt or x ∈ x pi or x ∈ x pm } in this work.
Instruction Prompt in Vision Transformer
As shown in Figure 1, we adopt a ViT as the backbone of our model (Dosovitskiy et al., 2020). For the input of the transformer module, we create a learnable [CLS] token x cls which can represent the global image features and extract prompt features. The other part is the input image, which will be divided into patches and encoded to a sequence of patch embeddings x im by the Embed module. In addition, x cls and x im are added with positional embeddings to retain positional information. The last part is instruction prompt x p , and we can represent the input of our Transformer module as:
x in = [x cls ; x im ; x p ]
where x cls , x im and x p represent the [CLS] token, input image patch embeddings and prompt tokens, respectively.
The input x in is then fed into the Transformer module and uses self-attention mechanism of the Transformer module to make [CLS] token utilize features from both x im and x p . In Algorithm 1 we show the core implementation of our work.
Downstream Task and Loss Construction
For the final downstream task result, we are able to achieve different downstream task modules. In our work, a classification head is added after the CLS token to accomplish the classification task. For the prediction result y pred , we use the cross-entropy loss as the loss function of the classification task, then the loss of classification is:
loss pred = CELoss(y pred , target)
where target is the ground truth, CELoss is the formula to calculate cross-entropy loss.
To improve the alignment of the different modal prompts and the input image, we use the same way as (Dosovitskiy et al., 2020) to measure the similarity between the output [CLS] token and prompt tokens by calculating the cosine similarity. The similarity score can be calculated by the formula: Score = y cls y T p where y cls represents the output of [CLS] token, y text represents the output of prompt tokens, and both y cls and y text are L2 regularized. The class with the largest similarity score with CLS token is the correct target for our expectation, therefore we use the similarity score as the part of loss. The loss can be represented as:
loss score = − log exp (z + ) exp (z i )
where z + represents the similarity score of target sample and z i represents each similarity score. Then, the final loss can be represented by: loss = loss pred + loss score
Training Strategy
To keep the input image as the main body and minimize the computation time, we limit the number of input prompts in validation. We performed an initial filtering of the potential classes, as shown in Figure 2. For an input image, we use zero-shot CLIP image encoder and text encoder to extract the feature from input image and text templates representing the latent class information. Similar to the calculation of the final result, we calculate the similarity score by the formula:
Score = F image F T text
where F image and F text represents the L2-normalized image and text extracted features. We select K prompts with the highest similarity as the input prompts to the next module. For the other N-K prompt tokens, we compute their average results as the input prompts. In this way, we select K+1 prompt tokens, thus reducing the computation time. Figure 1. The overall framework of Instruction-ViT. For each image input, the corresponding latent text or visual features are considered as the prompts, by using Transformer's attention mechanism to combine the features of input image and prompts. CLS token is used to complete the downstream task of classification, and the similarity scores computed by CLS and prompt tokens are used to assist in the fine-tuning of the model. At the training stage, the pink module is fine-tuning and the navy blue module keeps frozen.
Experiments
Datasets
We used 4 well known image classification datasets including Caltech-101 (Fei-Fei et al., 2004), Oxford-III Pets (Parkhi et al., 2012), Stanford Cars (Krause et al., 2013) and Oxford Flowers 102 (Nilsback & Zisserman, 2008).
Implementation Details
In our work, we adopt the network architecture of 12-layer Transformer blocks with 768 hidden sizes and 12 attention heads in the same way as previous work (Dosovitskiy et al., 2020). For creating prompts tokens, the image and text encoder are adopted from the pre-trained parameters of CLIP.
In the training stage, the model was trained for 20 epochs (100 epochs in Stanford Cars due to harder classification) with a batch size of 256. We use Adam (Kingma & Ba, 2014) optimizer with a learning rate 1e-4. We set the foot learning rate to 1e-5, with a linear warm up over the first 5 Figure 2. Running mechanism of prompts selected in validation.
For an input image of the validation set, feature extraction is performed using the zero-shot CLIP model for the potentially possible class and the image, and its similarity score is calculated. The K prompt tokens with the highest similarity and the average of remaining N-K prompt tokens are selected to next module. epochs in the cosine decay strategy. For data augmentation, we adopt RandAugment (Cubuk et al., 2019) and Mixup methods (Zhang et al., 2017).
Evaluation
We evaluated our model with two methods. The first is fine-tuning the full models in the downstream image classification task. The second is prompt tuning where only the project head and the prompt are learnable while the other parameters are frozen in training. We reported the top-1 accuracy on each dataset task.
Result
FINE TUNING RESULT
As shown in Table 1, we compare our model with other models, including ViT (Dosovitskiy et al., 2020), DeiT (Touvron et al., 2021a), CaiT (Touvron et al., 2021b), PiT (Heo et al., 2021) , ResNet (He et al., 2016) and EfficientNet (Tan & Le, 2019). As a general observation, the averaged accuracy of our proposed model outperforms other models in fine tuning performance, both ViT-based models, and CNN-based models. The experimental results show that our proposed method can optimize the current ViT-based approach by introducing additional information in prompts.
VISUAL PROMPT TUNING RESULT
We additionally compare the difference of model performance between our training method and the visual prompt tuning (VPT) method in (Jia et al., 2022). In this comparison, we keep most of the parameters of the model frozen and fine tune only some of the parameters. In our approach, we fine tune the classification heads and the prompt embedding layer, while the VPT method fine tunes the head and visual prompt proposed. As shown in Table 2, our proposed method is superior to VPT. The results of the experiments demonstrate the feasibility of our proposed methodology for creating special prompts and also prove that the other modal prompt, such as text prompt, can help the completion of visual tasks by our method. In addition, we compare the difference of the accuracy between the three proposed modal prompts. In the four datasets of our experiments, the three modal prompts have their own advantages. In Oxford-III Pets and Oxford Flowers 102, text prompt yields the highest accuracy of 84.74% and 63.20%, respectively. In the Caltech-101 dataset, image prompt can achieve the optimal accuracy of 80.85%, while in the Stanford Cars dataset, the mixed prompt of text and image reaches the optimal accuracy of 66.11%. These results suggest the importance of using multiple modalities of prompts in different scenarios.
Conclusion
In this work, we introduced Instruction-ViT, a simple and effective approach to align the input and prompts across varying modalities. It utilizes the pre-trained parameters from ViT-B as the backbone and CLIP encoders along with a flexible head module for completing downstream tasks like image classification. We show that Instruction-ViT can effectively use uni-modal prompts (e.g., images or texts) as well as multi-modal prompts (e.g., combined image and text features). Experimental results show that Instruction-ViT optimizes the ViT-based model by incorporating prompts in different modalities, and the prompts in different modalities can enhance the effect of the model with fewer parameter training.
In the future, we plan to improve Instruction-ViT from the following perspectives:
(1) We would like to explore the influence of different modules in our Instruction-ViT framework by using more powerful backbones and prompt generators. For example, we will use the Swim Transformer as the backbone and the BERT as the text prompt generator.
(2) Due to the flexibility of our proposed prompt approach, we will further explore how to design the prompts to achieve better results with our proposed framework. In addition, we will also test different types of prompts, for example using image descriptions as text prompts or using other modality data as prompts such as audio.
(3) Following the represented work like YOLOS (Wang et al., 2022a) and SAM (Kirillov et al., 2023), we will further experiment with the performance of other downstream tasks such as target detection and image segmentation.
Table 1 .Table 2 .
12Fine tuning performance in 4 datasets. Fine tuning performance using VPT training strategy in 4 datasets.MODEL
CALTECH101
PETS
CARS FLOWERS AVERAGE
VIT-B
97.61
94.19 90.90
99.59
95.57
DEIT-B
96.87
94.71 89.85
96.28
94.43
CAIT-S-24
96.56
94.27 91.06
96.43
94.58
PIT-B
96.73
95.29 91.59
97.25
95.22
RESNET-50
95.55
92.47 87.23
82.93
89.54
RESNET-101
96.72
93.18 87.60
85.76
90.82
RESNET-152
97.08
93.04 87.60
88.64
91.59
EFFICIENTNET-B0
88.74
86.41 83.42
68.98
81.89
EFFICIENTNET-B1
91.59
87.99 85.10
75.28
84.99
EFFICIENTNET-B2
93.48
89.17 84.45
78.11
86.30
EFFICIENTNET-B3
97.59
88.43 84.01
99.59
86.56
OURS(VIT-B)
97.54
94.19 91.08
99.56
95.59
MODEL
PROMPT CALTECH101
PETS
CARS FLOWERS AVERAGE
VIT-B
73.83
81.69 46.93
58.76
65.30
OURS (VIT-B)
TEXT
79.81
84.74 65.75
63.20
73.38
OURS (VIT-B)
IMAGE
80.85
84.58 66.06
60.42
72.98
OURS (VIT-B)
MIX
79.78
84.39 66.11
61.10
72.85
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| [
"https://github.com/tatsu-lab/"
] |
[
"Long-Context Language Decision Transformers and Exponential Tilt for Interactive Text Environments",
"Long-Context Language Decision Transformers and Exponential Tilt for Interactive Text Environments"
] | [
"Nicolas Gontier ",
"Pau Rodriguez ",
"Issam Laradji ",
"David Vazquez ",
"Christopher Pal "
] | [] | [] | Text-based game environments are challenging because agents must deal with long sequences of text, execute compositional actions using text and learn from sparse rewards. We address these challenges by proposing Long-Context Language Decision Transformers (LLDTs), a framework that is based on long transformer language models and decision transformers (DTs). LLDTs extend DTs with 3 components: (1) exponential tilt to guide the agent towards high obtainable goals, (2) novel goal conditioning methods yielding significantly better results than the traditional return-to-go (sum of all future rewards), and (3) a model of future observations. Our ablation results show that predicting future observations improves agent performance. To the best of our knowledge, LLDTs are the first to address offline RL with DTs on these challenging games. Our experiments show that LLDTs achieve the highest scores among many different types of agents on some of the most challenging Jericho games, such as Enchanter. | 10.48550/arxiv.2302.05507 | [
"https://export.arxiv.org/pdf/2302.05507v1.pdf"
] | 256,827,635 | 2302.05507 | 7b47fd0a3ed6b2a63dbcb110aba95e7573a742c4 |
Long-Context Language Decision Transformers and Exponential Tilt for Interactive Text Environments
Nicolas Gontier
Pau Rodriguez
Issam Laradji
David Vazquez
Christopher Pal
Long-Context Language Decision Transformers and Exponential Tilt for Interactive Text Environments
Text-based game environments are challenging because agents must deal with long sequences of text, execute compositional actions using text and learn from sparse rewards. We address these challenges by proposing Long-Context Language Decision Transformers (LLDTs), a framework that is based on long transformer language models and decision transformers (DTs). LLDTs extend DTs with 3 components: (1) exponential tilt to guide the agent towards high obtainable goals, (2) novel goal conditioning methods yielding significantly better results than the traditional return-to-go (sum of all future rewards), and (3) a model of future observations. Our ablation results show that predicting future observations improves agent performance. To the best of our knowledge, LLDTs are the first to address offline RL with DTs on these challenging games. Our experiments show that LLDTs achieve the highest scores among many different types of agents on some of the most challenging Jericho games, such as Enchanter.
Introduction
People spend a significant fraction of their lives performing activities closely linked with natural languages, such as having conversations, writing e-mails, filling out forms, reading and writing documents, and so on. Recently, the excitement around the use of Large Language Models (LLMs) for dialogue has brought the setting of interactive dialogue into the spotlight. Interactive text-based games allow one to explore and test interactive agents, alternative neural architectures, and techniques. However, text environments remain challenging for existing Reinforcement Learning (RL) agents Figure 1. Overview of our approach: Noisy trajectories are generated from a high quality game walkthrough by taking 100 random steps at each 5% of the trajectory. The collection of trajectories on multiple games is used to train our LLDT model offline to predict a goal condition, next action, and next observation. The LLDT is then evaluated in each game environment, initialized with 5 random seeds. since the action space is vast due to the compositional nature of language, making exploration difficult. Fortunately, language has the advantage that knowledge can often be reused across environments, such as the fact that fire burns or that doors open. To solve real-world text-based tasks and play rich text-based games well, RL agents can also benefit from the knowledge about the human world acquired from large offline data sources by leveraging pre-trained LLMs.
In real-world settings, the low-performing behavior exhibited by online RL agents during learning makes them impractical to use with humans in the loop. This situation arises in many other contexts (Levine et al., 2020) and has motivated a lot of research on offline RL. Offline RL methods have a long history, but more recently, several approaches have been proposed that focused on using powerful transformerbased sequence models, including Trajectory Transformers (TTs) (Janner et al., 2021), and Decision Transformers The model encodes the sequence of observations (o), goal conditions (g), and actions (a) up to time step t. The first o1 and last ot observations are fully written, but to shorten the input sequence, the other intermediate observations are replaced by a special "< STATE >" token. The decoder predicts the goal condition gt, action to take at, and next observation ot+1.
(DTs) (Chen et al., 2021). However, these approaches are formulated and examined within continuous control robotics problems. Unlike the methods above, our approach is designed to handle the complexity and richness of human language by leveraging pre-trained LLMs.
Motivated by the analogy of text-games to intelligent text assistants helping people with various tasks, we assume that a few expensive expert demonstrations are available for learning. As such, we use the Jericho text games , which provide a single golden path trajectory per game. To create a large and diverse dataset, we then generate trajectories with perturbations from that golden path as described in Section 4.2 and depicted in Figure 1. The complexity and richness of Jericho games make them a reasonable proxy for the kind of data one might obtain in real-world assistive agent settings.
In this work, we use a pre-trained Transformer language model that we fine-tune on offline game trajectories to predict: trajectory goal conditions, future actions, and observations. To sample high goal conditions from our model, we convert distributions over discrete token representations of numerical values into continuous ones, allowing us to maximize goal conditions that are likely achievable through an exponential tilting technique. In addition, we compare different conditioning methods and introduce an auxiliary loss to predict future observations. We will refer to our approach as Long-Context Language Decision Transformers (LLDTs) with exponential tilt. Our approach is visualized in Figure 2. See Table 1 for a comparison of how our formulation for density estimation and decision-making is situated with respect to prior frameworks. We also note that none of these previous frameworks have been applied to text-based action spaces, so none have leveraged pre-trained LLMs as in our framework.
To conclude, our contributions can be summarized as follows: (1) Our work is the first to address the challenging Jericho text-based games in an offline return conditioned sequence learning setup, wherein we train models on noisy walkthrough trajectories from multiple games simultaneously.
(2) We improve agent behavior with fewer assump-tions by letting the model predict goal conditions in a manner where no knowledge of the maximum score is needed through our use of an exponential tilting technique. (Section 5.1).
(3) We explore and empirically compare 3 novel definitions of goal conditioning that perform better than the return-to-go perspective of Decision Transformers (DTs). (Section 5.2). (4) We propose a novel auxiliary loss to train DTs that draws parallels to model-based RL and empirically shows better performance compared to the traditional model-free loss of DTs (Section 5.3).
Methodology
Problem setup
Text-based games can be formulated as partially observable Markov decision processes (POMDP) described by (S, T , A, O, R, γ). The current game state s t ∈ S is partially observable in o t ∈ O which is often a text description of the current scene (inventory, location, items). The agent can take an action a t ∈ A to interact with the environment and causes a state change based on a transition function T (s t , a t ) leading to a new state s t+1 ∈ S. Some games are stochastic in that the same action for the same state can lead to different states. Once the agent transitions to the new state, a reward r t is given by an unknown reward function R(s t , a t ) that the game designers defined. The reward can either be positive, negative, or neutral.
Offline Reinforcement Learning. The goal of the agent is to learn a policy π(a t |s t ) which maximizes the expected return E[ T t=0 r t ] in the POMDP by observing a series of static trajectories obtained in the same or similar environments. Each trajectory is defined as τ = (o 0 , a 0 , r 0 , o 1 , a 1 , r 1 , ..., o T , a T , r T ), and it is obtained by observing rollouts of arbitrary policies. This setup is similar to supervised learning, where models are trained from a static dataset. It is more difficult than online reinforcement learning since agents cannot interact with the environment to recollect more data.
Reinforcement Learning in text-based games. One of the main differences between traditional RL environments, such as Atari or Mujoco, and text-based environments is that both A and O consist of text. Therefore, due to the compositional nature of language, A is significantly more complex than in common RL scenarios, where the action space is restricted to a few well-defined actions. To deal with such complexity, we model A, O and R with a large pre-trained language model: x i = LLM(x i |x 1:i−1 ), where x i is the i th text token in a text sequence of length L. The goal is that the LLM uses its pre-existing knowledge about the world (e.g., doors can be opened), to propose valid actions given an observation.
Decision Transformers. To perform offline learning on text-based games, we adapt the language model (particularly LongT5 (Guo et al., 2022)) to be a decision transformer (DT) (Chen et al., 2021) which abstracts reinforcement learning as a sequential modeling problem. DTs are trained with the language modeling objective on sequences of {g t , o t , a t } T t=0 triples, where the goal condition g t is defined as the undiscounted sum of future rewards, or returnto-go: g t = T i=t r i . Consequently, we have a model that can be conditioned on a desired goal (or return, in this case). In the following subsections, we discuss the novelties we bring to the original formulation of DTs.
Goal conditioning
One limitation of DTs is that the best final score of a game must be known to condition on it at the first step with g 0 (Chen et al., 2021). Although we have g 0 for the training trajectories, it is impossible to know the best target score when starting a new game. This is especially problematic for Jericho games where maximum scores vary greatly between games .
One solution is to normalize g 0 during training with the maximum game score. This procedure leads to goal conditions between 0 and 1 for the training games and allows to use an initial goal condition of 1 at test time. However, this solution also assumes that we know the maximum score of every game since intermediate rewards returned by the environment r t also need to be normalized to update g t+1 :
g t+1 = g t − r t max score ,(1)
To remove the dependence on manual goal conditioning, we take a similar approach to Lee et al. (2022) and train the model on ordered sequences of {o t , g t , a t } T t=0 triples instead of {g t , o t , a t } T t=0 . Moving the goal condition g t after the observation o t allows us to predict the goal condition based on the current observation rather than manually defining it. This variation allows deciding at inference time if we want to sample or replace the value of g t based on P θ (g t |o t ) (θ being the parameters of our system) by modeling the joint probability of a t and g t as:
P θ (a t , g t |o t ) = P θ (a t |g t , o t ) · P θ (g t |o t ).
(2)
One challenge is that sampling g t can produce low and inaccurate target returns. To alleviate this issue, we perform exponential tilting on the predicted probabilities of g t . In particular we sample g t like so:
g t = argmax g P θ (g t |o t ) · exp(αg t ) ,(3)
with α ≥ 0 being a hyper-parameter that controls the amount of tilting we perform. This allows us to sample high but probable target returns. We compare results with α = {0, 1, 10, 20} in Section 5.1.
Another significant advantage of predicting the goal condition g t based on o t is that we can explore various strategies of goal conditions that cannot be defined manually at inference time. We describe below the original return-to-go used by decision transformers and three novel goal condition strategies.
Return-To-Go (RTG): g t = T i=t r i , is the original strategy of the return-to-go. It is the undiscounted sum of future rewards, which will be high at the beginning of trajectories achieving a high score. These values will decrease as the agent progresses since fewer future rewards will be available in a trajectory with intermediate rewards.
Immediate Reward (ImR): In the setting where g t = r t , each step is conditioned on the reward observed right after the predicted action. We expect that with this goal condition method, the agent will learn what type of actions usually yield higher rewards (opening chest -vs-moving in a direction). We expect this strategy to encourage the model to get high rewards as fast as possible. However, we expect this strategy to work well only for environments with dense reward signals.
Final Score (FinS): g t = T i=0 r i . In this setting, each step is conditioned on the final score achieved by the agent. The final score is defined as the sum of all rewards observed during the entire trajectory. Note that, unlike all the other goal condition definitions, this score will not change over the course of a trajectory. This setting is closer to the traditional RL paradigm in which we often define rewards based on the final performance of an agent: did it win or did it lose. We expect the agent to learn to differentiate successful from unsuccessful trajectories in this setting. Since the model is not conditioned on immediate rewards, we expect it will produce longer trajectories, which can eventually achieve higher final scores.
Density Estimation (L(θ))
Decision Making (π(a|s; η)) DTs Piché et al. (2022)). We compare our approach with Decision Transformers (DTs) (Chen et al., 2021), Reward Weighted Regression (RWR) (Peters & Schaal, 2007;Dayan & Hinton, 1997), Reward-Conditioned Policies (RCP) (Kumar et al., 2019) (also used by Multi-Game Decision Transformers (Lee et al., 2022)), Reweighted Behavior Cloning (RBC) (Piché et al., 2019) (also used by Trajectory Transformer (TT) (Janner et al., 2021)), and Implicit RL via supervised learning (IRvS) (Piché et al., 2022). Where s represents the state as encoded by the model and depends on the architecture and inputs used.
log p θ (a t | o t , G t ) p θ (a | s t , G t ) RWR exp(η −1 G t ) log p θ (a t | o t ) p θ (a | s t ) RCP log p θ (a t | o t , G t )p θ (G t | o t ) p θ (a | s t , G)p θ (G | s t ) exp(η −1 G − κ(η)) RBC log p θ (G t | o t , a t )p θ (a t | o t ) p θ (G | s t , a)p θ (a | s t ) exp(η −1 G − κ(η)) IRvS log p θ (a t , G t | o t ) p θ (a, G | s t ) exp(η −1 G − κ(η)) MB-RCP (ours) log p θ (o t+1 | a t , o t , G t )p θ (a t | o t , G t )p θ (G t | o t ) p θ (a | s t , G)p θ (G | s t ) exp(η −1 G − κ(η))Average Return-To-Go (AvgRTG): g t = T i=t ri (T −t) .
In this setting, each step is conditioned on the average of all future rewards. This is also defined as the return-to-go divided by the number of steps remaining. The motivation for this goal condition is that it will capture the sparsity of rewards in a trajectory, unlike all the others.
To reduce the variance in the numbers observed between different games, all goal condition numbers during training are normalized by the maximum score of the current game, where:
g t = int 100 · g t max score .(4)
At inference time, we can either manually specify goal condition numbers (assuming we know the game maximum score), or we can let the model predict those goal condition numbers with exponential tilt (more flexible).
We experiment with all these goal condition definitions in our experiments and report results in Section 5.2.
Next State Prediction
To give more training signal to the model and make it more robust to stochastic environments, we also experiment with learning to predict the next observation o t+1 . Concretely, we predict o t+1 after taking action a t in state s t . Although the prediction of the next observation is not used to interact with the environment at test time, we believe that the agent will perform better if it can predict how its action will impact the world. Furthermore, predicting the next observation indirectly informs the model about the stochasticity of the environment. This technique draws parallels with the model-based paradigm in Reinforcement Learning, where the agent can predict how the environment will evolve after each action. Formally, the model estimates the following probability:
P θ (o t+1 , a t , g t |o t ) =P θ (o t+1 |a t , g t , o t )· P θ (a t |g t , o t ) · P θ (g t |o t ),(5)
which is a type of Reward Conditioned Policy (RCP) with the additional term P θ (o t+1 |a t , g t , o t ). We call our technique model-based reward conditioned policy (MB-RCP). We compare our formulation to prior work in Table 1. We are interested in using this additional prediction as a form of regularization and therefore treat predicting the next observation as an auxiliary loss, leading to:
L = L CE ([ĝ tât ]; [g t a t ]) + λ · L CE (ô t+1 ; o t+1 ) 1 + λ ,(6)
with L CE being the regular cross entropy loss and λ being a hyper-parameter set to 0.5 in all our experiments. This weighted average prevents the model from spending too much of its representation power on the next observation prediction, as it is not strictly required to be able to interact in an environment. At inference time, only the next goal condition and next action predictions will be used. We perform an ablation study on this aspect of our approach by comparing models trained with (λ = 0.5) and without (λ = 0) this auxiliary loss and report our results in Section 5.3.
Related Work
Upside-down RL (UDRL) (Schmidhuber, 2019;Kumar et al., 2019;Piché et al., 2022) poses the task of learning a policy as a supervised learning problem where an agent is conditioned on an observation and a target reward to produce an action. Instead of generating the next action for a target reward, goal-conditioning methods generate trajectories conditioned on an end-goal (Ghosh et al., 2019;Paster et al., 2020). Most relevant to our work, Chen et al. (2021) recast supervised RL as a sequence modeling problem with decision transformers (DTs), but they did not examine text environments. DTs have been extended to multi-task environments by training them on multiple Atari games (Lee et al., 2022). To address the problem of modelling text-based environments Furman et al. (2022) proposed DT-BERT for question answering in TextWorld environments (Côté et al., 2018). However, the maximum number of steps in their trajectories is 50, and the environments are only differing in their number of rooms and objects. Here we go a step further and propose to learn agents that fully solve Jericho games with diverse game dynamics and scenarios by training on offline trajectories across multiple games.
Jericho is a challenging python framework composed of 33 text-based interactive fiction games . It was initially introduced with a new Template-DQN, and compared with the Deep Reinforcement Relevance Network (DRRN) (He et al., 2016). However, both methods are trained online, which requires an expensive simulator and requires domain-specific knowledge, such as the set of possible actions in order to be trained. Yao et al. (2020) proposed CALM, extending DRRNs to solve the problem of it needing to know the set of possible actions in advance. They use a GPT-2 (Radford et al., 2019) language model to generate a set of possible candidate actions for each game state. Then, they use an RL agent to select the best action among the (top-k=30) generated ones.
One of the main challenges of leveraging language models to solve Jericho games is to encode the full context of the game trajectory. As such, KG-A2C (Ammanabrolu & Hausknecht, 2020) and Q*BERT (Ammanabrolu et al., 2020) use a knowledge graph to represent the environment state at each step and learn a Q-value function. SHA-KG (Xu et al., 2020) uses graph attention network (Veličković et al., 2018) to encode the game history and learn a value function. RC-DQN (Guo et al., 2020) uses a reading comprehension approach by retrieving relevant previous observations, encoding them with GRUs (Cho et al., 2014), and learning a Q-value function. DBERT-DRRN (Singh et al., 2021) leverages a DistilBERT to encode state and action and feed it to an MLP to learn a Qvalue function. XTX (Tuyls et al., 2022) re-visits different frontiers in the state space and performs local exploration to overcome bottleneck states and dead-ends. CBR (Atzeni et al., 2022) stores previous interactions in memory and leverages a graph attention network (Veličković et al., 2018) to encode the similarity between states.
The above previous methods are online-based RL, thus suffering from sample inefficiencies. Here, we take a simpler approach by simply leveraging long context transformers like LongT5 (Guo et al., 2022) to model the sequence of state observations, target goal scores, and actions of past game trajectories as a sequence of tokens. Then, given a state observation, we leverage exponential tilt (Piché et al., 2022;Lee et al., 2022) to produce the action with the best possible target goal score. We find that our LLDT approach is effective enough to outperform all previous methods that we have examined on Jericho games.
Experimental setup 4.1. Jericho Engine
Jericho is a well-known Python framework that consists of 33 text-based interactive fiction games that are challenging learning environments . Developers manually create them, each having its own way of defining the rules and goals for each game, making the games quite diverse.
Text adventure games are challenging on their own because of their combinatorially large action space and sparse rewards. Usually, text adventure games have a large action vocabulary (around 2000 words on average), and each action is made of multiple words (1 to 4 on average). This makes the action space as big as 2000 4 = 1.6 × 10 13 . To alleviate this issue, the Jericho benchmark provides a list of valid actions for each state. However, this makes the environment much slower as the game engine validates all possible actions against the simulator. In addition, the action space becomes dynamic as it changes from state to state. The above challenge in combination with extremely sparse rewards makes text adventure games very challenging for current RL methods.
In this work, we focus on a subset of Jericho games, in particular, the ones belonging to the Zork Universe: enchanter, sorcerer, spellbrkr, spirit, ztuu 1 . We generate trajectories (Section 4.2) for each of these games and train our model on the collection of all trajectories from all games.
Data Collection
Jericho provides one human walkthrough trajectory per game that achieves the maximum score. However, since some games are stochastic, every walkthrough is only valid for a specific default seed when initializing the game. To obtain a more diverse dataset with incorrect or partially correct trajectories, we propose to generate trajectories by following the walkthrough trajectories for some steps and then deviating from them. Concretely, to collect a large number of trajectories with different performances we follow the walkthrough trajectory for X% of its total number of steps and then take 100 additional random steps. We repeat that procedure 10 times for each X ∈ [0, 5, 10, ..., 85, 90, 95]. When X = 0%, this is the same as a fully random trajectory. When X = 95%, the agent follows the walkthrough path for 95% of the steps and then takes 100 random steps. This results in a collection of 201 trajectories, including 1 original walkthrough for each game. Note that we also tried to include TDQN and DRRN trajectories trained on individual games, but these agents did not bring any significant information gain in our collection of trajectories.
To not overfit on the default seed for each game, we ran the same procedure on 5 different seeds. This resulted in 1,005 trajectories of various lengths and qualities for each game. Note that only 1 (or 5 if the game is not stochastic) of those obtain a 100% final score by following the walkthrough actions given by Jericho. We report in Appendix A the normalized scores ( Figure 4) and lengths ( Figure 5) observed in the collection of trajectories collected for each game. The top part of Figure 1 illustrates the data generation procedure.
Sequence Definition
In this section, we describe in detail what defines ordered sequences of {o t , g t , a t } T t=0 triples in the setting of Jericho games collected as described in Section 4.2.
To train an encoder-decoder architecture, trajectories are split between input and output sequences after a random number of steps. After sampling a random index t ∈ [0, T − 1] to split the trajectory, the input sequence is defined as [o 0 , g 0 , a 0 , o 1 , ..., g t−1 , a t−1 , o t ] and the output sequence is defined as [g t , a t , o t+1 ] (also depicted in Figure 2). Each of these {o t , g t , a t } T t=0 elements are represented in natural language text as described below and concatenated together to form two long text sequences: one for the input, one for the output. a t : each intermediate action is written as returned by agents playing the game, with the addition of special token delimiters. Each action is written in text like this: "Action: {a_t} </s></s>", with {a_t} being replaced by the action taken by the agent. g t : each goal condition is computed based on the list of intermediate rewards returned by the environment against the agent playing, depending on the strategy used among the ones described in Section 2.2. With the addition of special token delimiters, each goal condition is written in text like this: "GC: {g_t} </s></s>", with {g_t} being replaced by the goal condition number after being normalized between 0 and 100 (see Equation 4). o t : state observations are defined by multiple state characteristics available to Jericho games. At each step, we use (i) candidate actions available, (ii) the message returned by the game engine, (iii) the description of the current room (if it is not already present in the message returned by the game engine), and (iv) the current inventory of the agent. Each observation, with the addition of special token delimiters, is written in text like this: "Actions: {cand} </s></s> State: {msg} </s></s> Description: {desc} </s></s> Inventory: {inv} </s></s>", with {cand}, {msg}, {desc} and {inv} being the list of candidate actions, the game message after taking the previous action, the description of the current room, and the inventory of the player respectively, all given by the Jericho game engine.
After realizing that game trajectories can be as long as 1000 steps and that the current definition of {o t , g t , a t } T t=0 triples can make input sequences as long as tens of thousands of tokens, we greatly simplified the definition of input sequences. We replaced state observations o t to be a single placeholder token "<STATE>" for all intermediate observations except the first (o 0 ) and current one (o t ) as depicted in Figure 2.
Experimental Results
Since Jericho games have long storylines, we leverage LongT5 (Guo et al., 2022), a text-to-text Transformer with a wide attention span. We use the pre-trained LongT5-base model in all experiments as the base for our encoder-decoder architecture. We then fine-tuned the model for multiple epochs on the generated trajectories from Section 4.2. The hyperparameter settings can be found in Appendix B.
For each game, we initialize its environment with a random seed. We let the model predict the next goal condition and action at each step. The agent performs the predicted action, leading to the next observation in the environment. The model uses this observation as context for the next step. We run these steps in a cycle until we reach the end of the game and compute the final score. The game ends when the agent reaches the final state, the model generates an invalid action, or the model fails to generate an action. We repeat this process on 5 different random seeds and take the average final score. The bottom part of Figure 1 illustrates the training and evaluation process. Next we examine the effect of exponential tilt, the effect of different goal conditioning methods, and the effect of learning to predict the next observation o t+1 as part of the loss function.
The Effect of Exponential Tilt
To analyze the effect of exponential tilt we compare a model that doesn't have access to the maximum score of a game (but uses various amounts of exponential tilt) with one that has that information.
We fine-tuned our model with the loss function described in Equation 6 on all our generated trajectories split into input and output sequence pairs as described in Section 4.3 and depicted in Figure 2. The model was trained with the regular return-to-go goal condition (g t = T i=t r i ) and with λ = 0.5 for the auxiliary loss of predicting o t+1 . We tested the model on all games, normalized the obtained score based on the maximum human score for each game, and recorded the average across games and 5 random seeds for each game.
To measure the effect of exponential tilt, the predicted g t were sampled according to Equation 3 with α = 0, 1, 10, 20 training steps (in thousands) average normalized score 0% 10% 20% 30% 40% 20 40 60 80
Optimal GC
Predicted GC / alpha=0
Predicted GC / alpha=1
Predicted GC / alpha=10
Predicted GC / alpha=20 TDQN DRRN KG-A2C (Ammanabrolu & Hausknecht, 2020) CALM (Yao et al., 2020) SHA-KG (Xu et al., 2020) MPRC-DQN (Guo et al., 2020) RC-DQN (Guo et al., 2020) Bike+CBR (Atzeni et al., Figure 3. Average normalized score across different Jericho games (enchanter, sorcerer, spellbrkr, spirit, ztuu) with various amounts of exponential tilt ("Predicted GC" lines). We also report the performance of a model being conditioned on the optimal goal according to each game's maximum score ("Optimal GC" line).
The average normalized score of various baselines trained on each game is depicted with dotted lines.
("Predicted GC / alpha=α" in Figure 3). In addition, we evaluated the model with the goal-condition being manually given ("Optimal GC" in Figure 3) at each step. In the first step the model is conditioned with g 0 = 100 (the maximum possible according to Equation 4) and at every step, g t is reduced by the amount of observed reward as in Equation 1. This "Optimal GC" evaluation assumes we know the game's maximum score. We aim to achieve similar performance by simply predicting g t instead of manually defining it.
We report in Figure 3 the normalized score averaged across games for each method of predicting g t at different training stages of the model. As we prioritize high numerical return-to-go over their likelihood (α increasing), the model's performance is getting closer to the "Optimal GC" performance. During training the model is exposed to trajectories of various performances (detailed in Figure 4), so without any exponential tilt the model will output the most probable goal-condition based on what it observed during training, which is less than ideal (solid red "Predicted GC / alpha=0" line). If we slightly prioritize higher numerical values (solid yellow "Predicted GC / alpha=1" line), the performance of the model improves slightly but is still very unstable. As α increases to 10 (solid green line) and 20 (solid orange line), the performance is on par with the model that was manually given the "optimal" goal condition based on the game's maximum score. In realistic scenarios, we do not have the optimal goal condition when starting a new game. In addition, predicting the goal condition offers greater flexibility in the design of goal-conditions. We can now explore conditioning methods that would be impossible to define manually during run time. This is exactly what we explore in the next section.
The above results demonstrate two things: (1) the numerical value of the goal-condition has indeed an effect on the quality of the next generated answer, and (2) it is possible to recover the same performance as the "optimal" goalconditioning by increasing the amount of exponential tilt without knowing the game's maximum score.
In addition, we show in Figure 3 the reported average performance of various previous works on the same set of games (dotted lines). Although not directly comparable since all previous methods were trained on each game in an online RL fashion, our offline method beats previous methods with very little training in the case of "Optimal GC" and "Predicted GC / alpha=10 & 20".
Our Goal Conditioning Strategies
To evaluate our goal conditioning strategies, we fine-tune 4 models with the loss function described in Equation 6 on all our generated trajectories split into input and output sequence pairs as described in Section 4.3 and depicted in Figure 2. We train each model with a different goal condition (as described in Section 2.2) and with λ = 0.5 for the auxiliary loss of predicting o t+1 . We test the models on all games after 31.4k training steps and record the average score across 5 random seeds for each game.
In these experiments, we have the model generate goalconditions because at inference time, unlike with return-togo (RTG), we cannot compute the immediate reward (ImR) and the average return-to-go (AvgRTG), even if we know the game maximum score. To be able to manually provide the optimal immediate reward condition, we need to know at each step the maximum achievable reward among all candidate actions, which is infeasible in practice. Similarly in order to provide the optimal average RTG condition, we need to know the number of steps remaining after each state, which is also infeasible in practice. Fortunately, our model can generate these two goal-conditions (as well as any others), while leveraging the exponential tilt for producing better trajectories. Therefore, all models in these experiments are evaluated by sampling g t according to Equation 3 with α = 10. Table 2. Average and the maximum score for each game across 5 random seeds for each goal condition (GC) variation. On the bottom line, scores are normalized according to the maximum human score ('max=#') and averaged across games.
also interesting to note that the best-performing method on average is the "Final Score" conditioning strategy with 50.02% average max score. As mentioned in Section 2.2, this setting is closer to the traditional RL paradigm in which we often define rewards based on the final performance of an agent, which is only based on whether it won or lost a game.
Overall, these results show that the classical return-to-go conditioning method performs poorly in all environments. However, the winning goal conditioning strategy depends on the game which can vary between ImR, FinS, or AvgRTG but not RTG. These results further motivate the advantages of generating goal-conditions that cannot be computed at runtime such as ImR and AvgRTG. Table 3. Average and the maximum score for each game across 5 random seeds and 4 goal conditioning methods, with (λ = 0.5) and without (λ = 0.0) the auxiliary loss on the prediction of the next observation ot+1. Scores are then normalized according to the maximum human score and averaged across games on the bottom line.
Predicting the Next Observation
Here we analyze the effect of predicting the next observation o t+1 as part of the loss function. Therefore, we fine-tuned another 4 models, each with a different goal condition similar to the above section, but with the loss function described in Equation 6 with λ = 0.0 for the auxiliary loss of predicting o t+1 . We tested the models on all games after 31.4k training steps and recorded the average score across 5 random seeds for each game. To compare the effect of the auxiliary loss, we averaged the scores again across all goal-conditioning methods. Table 3 reports the average score, standard deviation, and maximum score obtained on each game over 20 runs (5 random seeds × 4 goal-conditioning methods) for models trained with (λ = 0.5) and without (λ = 0.0) the auxiliary loss on the predicted next observation o t+1 . The bottom line reports the normalized average score based on the maximum human score for each game.
In all games, models trained to predict the next observation o t+1 resulting from the predicted action a t and goalcondition g t perform better than models trained to only predict the goal-condition g t and next action a t . Overall, these results show that our proposed model-based rewardconditioned policy (MB-RCP) learning objective yields stronger performance than the classical reward-conditioned policy (RCP) objective.
Conclusion
In this work, we have proposed Long-Context Language Decision Transformers (LLDTs) as an offline reinforcement learning method for interactive text environments, and we have performed experiments using the challenging textbased games of Jericho. LLDTs are built from pre-trained LLMs followed by training on multiple games simultaneously to predict: the trajectory goal condition, the next action, and the next observation. We have shown that by using exponential tilt, LLDT-based agents get much better performance than otherwise. In fact, the model obtains similar performance as if it was conditioned on the optimal goal, despite the fact that in most realistic scenarios, we do not have access to that optimal goal condition. We have also explored different conditioning methods and observed that the traditional return-to-go was the weakest strategy. Finally we have seen that training the model to predict the next observation as an auxiliary loss improves performance. As future work, we plan on extending this framework to multiple and more diverse games and environments. We hope this work can provide a missing piece to the substantial advances in the application of large language models in the context of real-world interactive task-oriented dialogues.
A. Trajectories Statistics
In this section, we report the normalized scores ( Figure 4) and lengths ( Figure 5) observed in the collection of trajectories collected for each game as described in Section 4.2. Figure 4. Proportion of trajectory normalized scores for a selection of games. In each sub-figure title, n is the number of trajectories and ms is the maximum score. The X-axis is the normalized score the trajectory achieves. The Y-axis is the proportion of trajectories finishing with that score.
Figure 2 .
2Our Long-Context Language Decision Transformer framework. A trajectory of length T is split at a random index t ∈ [0, T − 1].
* Equal contribution 1 Quebec Artificial Intelligence Institute (Mila), Montreal, Canada 2 ServiceNow Research, Montreal, Canada 3 Computer Engineering, Polytechnique Montreal, Canada 4 Canada CIFAR AI Chair. Correspondence to: Nicolas Gontier <[email protected]>.Preliminary work.
100 rnd.
steps
(x10)
100 rnd.
steps
(x10)
100 rnd.
steps
(x10)
Walkthrough (x1)
Game Engine
(seed)
20 break points
(x5)
200+1 trajectories per game
100 rnd.
steps
(x10)
Seeds: 1, 2, 3, 4, 5
Data Generation
Inference
Game Engine
(seed)
Seeds: 6, 7, 8, 9, 10
Observation
Next Action
Score end-of-game
invalid action
1,005
trajectories
per game
Training
Goal Condition
Next Observation
Table 1 .
1Comparison of different policy training and action selection techniques (adapted from
Table 2
2reports the average score, standard deviation, and maximum score obtained on each game across 5 random seeds for all goal-conditioning methods. The bottom line reports the normalized average score based on the maximum human score for each game (max=#). On average, the classical return-to-go goal conditioning method yields weaker performance than all other variants. The average performance of immediate reward (36.36%) and final score (36.62%) conditioning is even higher than the maximum average score of return-to-go conditioning (35.94%). It isGC =
Return-To-Go (RTG)
Immediate Reward (ImR)
Final Score (FinS)
Avg. Return-To-Go (AvgRTG)
avg. stdev.
max.
avg. stdev.
max.
avg. stdev.
max.
avg.
stdev.
max.
enchanter(max=400)
45.00
0.00
45.00
235.00
0.00
235.00
231.00 56.69
280.00
175.00
0.00
175.00
sorcerer (max=400)
124.00 90.63
235.00
112.00 75.93
205.00
124.00 90.63
235.00
132.00 100.43
255.00
spellbrkr (max=280)
31.00
7.35
40.00
31.00
7.35
40.00
25.00
0.00
25.00
40.00
0.00
40.00
spirit (max=250)
18.40
3.88
26.00
22.40
8.69
38.00
26.00 15.35
56.00
5.60
0.80
6.00
ztuu (max=100)
73.00
7.48
85.00
75.00
9.49
90.00
75.00
9.49
90.00
75.00
9.49
90.00
Normalized Average 26.74%
35.94% 36.36%
45.90% 36.62%
50.02% 33.66%
42.84%
we selected games belonging to the same universe to favor transfer of knowledge between games. We plan on also training on all 33 games.
Figure 5. Proportion of trajectory lengths for a selection of games. In each sub-figure title, n is the number of trajectories. The X-axis is the number of steps in a trajectory. The Y-axis is the proportion of trajectories of that length.B. Hyperparametersoptimizer Adafactor learning rate 1e-4 precision 32 batch size 16 max input length 4096 max output length 1024 base model LongT5-base with Transient Global Attention
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Learning to reach goals without reinforcement learning. D Ghosh, A Gupta, J Fu, A Reddy, C Devin, B Eysenbach, S Levine, OpenReviewGhosh, D., Gupta, A., Fu, J., Reddy, A., Devin, C., Eysen- bach, B., and Levine, S. Learning to reach goals without reinforcement learning. OpenReview, 2019.
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Interactive fiction games: A colossal adventure. M Hausknecht, P Ammanabrolu, M.-A Côté, X Yuan, Conference on Artificial Intelligence (AAAI). 2020Hausknecht, M., Ammanabrolu, P., Côté, M.-A., and Yuan, X. Interactive fiction games: A colossal adventure. In Conference on Artificial Intelligence (AAAI), 2020.
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Multi-game decision transformers. K.-H Lee, O Nachum, S Yang, L Lee, C D Freeman, S Guadarrama, I Fischer, W Xu, E Jang, H Michalewski, I Mordatch, Advances in Neural Information Processing Systems (NeurIPS). 2022Lee, K.-H., Nachum, O., Yang, S., Lee, L., Freeman, C. D., Guadarrama, S., Fischer, I., Xu, W., Jang, E., Michalewski, H., and Mordatch, I. Multi-game decision transformers. In Advances in Neural Information Pro- cessing Systems (NeurIPS), 2022.
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| [] |
[
"The diffuse γ-ray background is dominated by star-forming galaxies",
"The diffuse γ-ray background is dominated by star-forming galaxies"
] | [
"Matt A Roth *[email protected] \nResearch School of Astronomy and Astrophysics\nThe Australian National University\nCanberraAustralia\n",
"Mark R Krumholz \nResearch School of Astronomy and Astrophysics\nThe Australian National University\nCanberraAustralia\n\nARC Centre of Excellence for All-Sky Astrophysics in Three Dimensions (ASTRO-3D)\nCanberraAus-tralia\n",
"Roland M Crocker \nResearch School of Astronomy and Astrophysics\nThe Australian National University\nCanberraAustralia\n",
"Silvia Celli \nDipartimento di Fisica dell'Università La Sapienza and INFN\nP. le Aldo Moro 200185RomeItaly\n"
] | [
"Research School of Astronomy and Astrophysics\nThe Australian National University\nCanberraAustralia",
"Research School of Astronomy and Astrophysics\nThe Australian National University\nCanberraAustralia",
"ARC Centre of Excellence for All-Sky Astrophysics in Three Dimensions (ASTRO-3D)\nCanberraAus-tralia",
"Research School of Astronomy and Astrophysics\nThe Australian National University\nCanberraAustralia",
"Dipartimento di Fisica dell'Università La Sapienza and INFN\nP. le Aldo Moro 200185RomeItaly"
] | [] | The Fermi Gamma-ray Space Telescope has revealed a diffuse γ-ray background at energies from 0.1 GeV to 1 TeV, which can be separated into Galactic emission and an isotropic, extragalactic component 1 . Previous efforts to understand the latter have been hampered by the lack of physical models capable of predicting the γ-ray emission produced by the many candidate sources, primarily active galactic nuclei 2-5 and star-forming galaxies 6-10 , leaving their contributions poorly constrained. Here we present a calculation of the contribution of star-forming galaxies to the γ-ray background that does not rely on empirical scalings, and is instead based on a physical model for the γ-ray emission produced when cosmic rays accelerated in supernova remnants interact with the interstellar medium 11 . After validating the model against local observations, we apply it to the observed cosmological star-forming galaxy population and recover an excellent match to both the total intensity and the spectral slope of the γ-ray background, demonstrating that star-forming galaxies alone can explain the full diffuse, isotropic γ-ray background.Many candidate sources have been proposed for the origin of the diffuse, isotropic γ-ray background. These include active galactic nuclei (AGN) (particularly blazars 2-5 ), millisecond pulsars 2 , star-forming galaxies (SFGs) 6-10 , and dark matter annihilation 12 . Previous estimates of their contributions have relied on a highly-uncertain process of empirically scaling the emission from a small sample of local, resolved sources by their estimated cosmological abundances, whereas our approach in this paper is instead to calculate the emission from SFGs directly using a physical model. The cosmic rays (CRs) responsible for γ-ray emission in SFGs (including the Milky Way) are produced by diffusive acceleration at supernova remnant shocks 13 . This process transfers ∼10% of the supernova mechanical energy to relativistic ions, yielding on average ∼ 10 50 erg in CR ions per supernova 14,15 , with another ∼2% (∼ 2 × 10 49 erg) deposited in CR electrons 16 . The resulting CRs follow a power law distribution in particle momentum p of the form dn/dp ∝ p −q 17,18 ; observations of individual supernova remnants, analytical models, and numerical simulations all indicate that the index q is in the range q ≈ 2.0 − 2.6, with a mean value of q ≈ 2.2 − 2.3 19,20 . Some of the CR ions collide inelastically with interstellar medium (ISM) nuclei, producing roughly equal numbers of π 0 , π + and π − mesons that rapidly decay via the channels π 0 → 2γ, π − → µ − +νµ, and π + → µ + + νµ. The decay of π 0 particles is respon-sible for most of the observed Galactic γ-ray foreground, which displays a characteristic spectrum that rises sharply from ∼ 0.1 − 1 GeV as a result of the 135 MeV rest mass of the π 0 particle. As this discussion suggests, a SFG's diffuse γ-ray emission depends primarily on three factors: its total star formation rate (which determines its supernova rate and thus the rate at which CRs are injected), the distribution of γ-ray energies produced when individual CRs collide with ISM nuclei (which depends on the parent CR energy E), and the fraction of CRs (again as a function of E) that undergo inelastic collisions before escaping the galaxy. The first two of these are relatively well-understood, but the third factor, known as the calorimetry fraction f cal (E), is much less certain. It depends on the properties of the galaxy, and the lack of a model for this dependence has previously precluded direct calculation of SFG γray emission. However, Ref. 11 recently introduced a model for f cal (E), based on rates of CR diffusion determined by the balance between the CR streaming instability and ion-neutral damping. In the Methods we describe a new technique to use this model to compute f cal (E), and thus the total γ-ray emission produced by CR ions in SFGs.We supplement the γ-ray production rate from CR ions by adding the contribution from both primary CR electrons, directly injected by supernova remnants, and secondary CR leptons (electrons and positrons), produced in the π ± decay chain; these become important at energies 1 GeV. Our model for these particles includes energy losses due to ionisation, synchrotron emission, bremsstrahlung, and inverse Compton scattering. The model also includes the attenuation of γ-rays produced by both CR ions and leptons due to pair production in collisions with far-infrared photons inside the source galaxy and extragalactic background light photons outside the galaxy, which become important at energies 100 GeV. The radiation that is absorbed by the host galaxy and extragalactic photon fields is reprocessed to lower energies in a pair-production cascade, whereby the initial high energy pair inverse Compton scatters lower energy photons up to γ-ray energies and these, in turn, produce further pairs, and so on. Details of our calculation of all these processes are provided in the Methods.We now have a model that predicts the γ-ray emission of a SFG. The next step in our analysis is to apply this model to a galaxy survey that samples the SFG population out to the epoch of peak cosmological star formation at z ∼ 2. For this purpose, we make use of the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS) 21,22 in the GOODS-S field. We apply our model to the CANDELS sample as arXiv:2109.07598v1 [astro-ph.HE] 15 Sep 2021 | 10.1038/s41586-021-03802-x | [
"https://arxiv.org/pdf/2109.07598v1.pdf"
] | 237,532,419 | 2109.07598 | 485a24f3ddd8f154bad414df75cea5e6d68ed1c3 |
The diffuse γ-ray background is dominated by star-forming galaxies
September 17, 2021
Matt A Roth *[email protected]
Research School of Astronomy and Astrophysics
The Australian National University
CanberraAustralia
Mark R Krumholz
Research School of Astronomy and Astrophysics
The Australian National University
CanberraAustralia
ARC Centre of Excellence for All-Sky Astrophysics in Three Dimensions (ASTRO-3D)
CanberraAus-tralia
Roland M Crocker
Research School of Astronomy and Astrophysics
The Australian National University
CanberraAustralia
Silvia Celli
Dipartimento di Fisica dell'Università La Sapienza and INFN
P. le Aldo Moro 200185RomeItaly
The diffuse γ-ray background is dominated by star-forming galaxies
September 17, 202110.1038/s41586-021-03802-xPublished in Nature
The Fermi Gamma-ray Space Telescope has revealed a diffuse γ-ray background at energies from 0.1 GeV to 1 TeV, which can be separated into Galactic emission and an isotropic, extragalactic component 1 . Previous efforts to understand the latter have been hampered by the lack of physical models capable of predicting the γ-ray emission produced by the many candidate sources, primarily active galactic nuclei 2-5 and star-forming galaxies 6-10 , leaving their contributions poorly constrained. Here we present a calculation of the contribution of star-forming galaxies to the γ-ray background that does not rely on empirical scalings, and is instead based on a physical model for the γ-ray emission produced when cosmic rays accelerated in supernova remnants interact with the interstellar medium 11 . After validating the model against local observations, we apply it to the observed cosmological star-forming galaxy population and recover an excellent match to both the total intensity and the spectral slope of the γ-ray background, demonstrating that star-forming galaxies alone can explain the full diffuse, isotropic γ-ray background.Many candidate sources have been proposed for the origin of the diffuse, isotropic γ-ray background. These include active galactic nuclei (AGN) (particularly blazars 2-5 ), millisecond pulsars 2 , star-forming galaxies (SFGs) 6-10 , and dark matter annihilation 12 . Previous estimates of their contributions have relied on a highly-uncertain process of empirically scaling the emission from a small sample of local, resolved sources by their estimated cosmological abundances, whereas our approach in this paper is instead to calculate the emission from SFGs directly using a physical model. The cosmic rays (CRs) responsible for γ-ray emission in SFGs (including the Milky Way) are produced by diffusive acceleration at supernova remnant shocks 13 . This process transfers ∼10% of the supernova mechanical energy to relativistic ions, yielding on average ∼ 10 50 erg in CR ions per supernova 14,15 , with another ∼2% (∼ 2 × 10 49 erg) deposited in CR electrons 16 . The resulting CRs follow a power law distribution in particle momentum p of the form dn/dp ∝ p −q 17,18 ; observations of individual supernova remnants, analytical models, and numerical simulations all indicate that the index q is in the range q ≈ 2.0 − 2.6, with a mean value of q ≈ 2.2 − 2.3 19,20 . Some of the CR ions collide inelastically with interstellar medium (ISM) nuclei, producing roughly equal numbers of π 0 , π + and π − mesons that rapidly decay via the channels π 0 → 2γ, π − → µ − +νµ, and π + → µ + + νµ. The decay of π 0 particles is respon-sible for most of the observed Galactic γ-ray foreground, which displays a characteristic spectrum that rises sharply from ∼ 0.1 − 1 GeV as a result of the 135 MeV rest mass of the π 0 particle. As this discussion suggests, a SFG's diffuse γ-ray emission depends primarily on three factors: its total star formation rate (which determines its supernova rate and thus the rate at which CRs are injected), the distribution of γ-ray energies produced when individual CRs collide with ISM nuclei (which depends on the parent CR energy E), and the fraction of CRs (again as a function of E) that undergo inelastic collisions before escaping the galaxy. The first two of these are relatively well-understood, but the third factor, known as the calorimetry fraction f cal (E), is much less certain. It depends on the properties of the galaxy, and the lack of a model for this dependence has previously precluded direct calculation of SFG γray emission. However, Ref. 11 recently introduced a model for f cal (E), based on rates of CR diffusion determined by the balance between the CR streaming instability and ion-neutral damping. In the Methods we describe a new technique to use this model to compute f cal (E), and thus the total γ-ray emission produced by CR ions in SFGs.We supplement the γ-ray production rate from CR ions by adding the contribution from both primary CR electrons, directly injected by supernova remnants, and secondary CR leptons (electrons and positrons), produced in the π ± decay chain; these become important at energies 1 GeV. Our model for these particles includes energy losses due to ionisation, synchrotron emission, bremsstrahlung, and inverse Compton scattering. The model also includes the attenuation of γ-rays produced by both CR ions and leptons due to pair production in collisions with far-infrared photons inside the source galaxy and extragalactic background light photons outside the galaxy, which become important at energies 100 GeV. The radiation that is absorbed by the host galaxy and extragalactic photon fields is reprocessed to lower energies in a pair-production cascade, whereby the initial high energy pair inverse Compton scatters lower energy photons up to γ-ray energies and these, in turn, produce further pairs, and so on. Details of our calculation of all these processes are provided in the Methods.We now have a model that predicts the γ-ray emission of a SFG. The next step in our analysis is to apply this model to a galaxy survey that samples the SFG population out to the epoch of peak cosmological star formation at z ∼ 2. For this purpose, we make use of the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS) 21,22 in the GOODS-S field. We apply our model to the CANDELS sample as arXiv:2109.07598v1 [astro-ph.HE] 15 Sep 2021
The Fermi Gamma-ray Space Telescope has revealed a diffuse γ-ray background at energies from 0.1 GeV to 1 TeV, which can be separated into Galactic emission and an isotropic, extragalactic component 1 . Previous efforts to understand the latter have been hampered by the lack of physical models capable of predicting the γ-ray emission produced by the many candidate sources, primarily active galactic nuclei 2-5 and star-forming galaxies [6][7][8][9][10] , leaving their contributions poorly constrained. Here we present a calculation of the contribution of star-forming galaxies to the γ-ray background that does not rely on empirical scalings, and is instead based on a physical model for the γ-ray emission produced when cosmic rays accelerated in supernova remnants interact with the interstellar medium 11 . After validating the model against local observations, we apply it to the observed cosmological star-forming galaxy population and recover an excellent match to both the total intensity and the spectral slope of the γ-ray background, demonstrating that star-forming galaxies alone can explain the full diffuse, isotropic γ-ray background.
Many candidate sources have been proposed for the origin of the diffuse, isotropic γ-ray background. These include active galactic nuclei (AGN) (particularly blazars [2][3][4][5], millisecond pulsars 2 , star-forming galaxies (SFGs) [6][7][8][9][10] , and dark matter annihilation 12 . Previous estimates of their contributions have relied on a highly-uncertain process of empirically scaling the emission from a small sample of local, resolved sources by their estimated cosmological abundances, whereas our approach in this paper is instead to calculate the emission from SFGs directly using a physical model. The cosmic rays (CRs) responsible for γ-ray emission in SFGs (including the Milky Way) are produced by diffusive acceleration at supernova remnant shocks 13 . This process transfers ∼10% of the supernova mechanical energy to relativistic ions, yielding on average ∼ 10 50 erg in CR ions per supernova 14,15 , with another ∼2% (∼ 2 × 10 49 erg) deposited in CR electrons 16 . The resulting CRs follow a power law distribution in particle momentum p of the form dn/dp ∝ p −q 17,18 ; observations of individual supernova remnants, analytical models, and numerical simulations all indicate that the index q is in the range q ≈ 2.0 − 2.6, with a mean value of q ≈ 2.2 − 2.3 19,20 . Some of the CR ions collide inelastically with interstellar medium (ISM) nuclei, producing roughly equal numbers of π 0 , π + and π − mesons that rapidly decay via the channels π 0 → 2γ, π − → µ − +νµ, and π + → µ + + νµ. The decay of π 0 particles is respon-sible for most of the observed Galactic γ-ray foreground, which displays a characteristic spectrum that rises sharply from ∼ 0.1 − 1 GeV as a result of the 135 MeV rest mass of the π 0 particle. As this discussion suggests, a SFG's diffuse γ-ray emission depends primarily on three factors: its total star formation rate (which determines its supernova rate and thus the rate at which CRs are injected), the distribution of γ-ray energies produced when individual CRs collide with ISM nuclei (which depends on the parent CR energy E), and the fraction of CRs (again as a function of E) that undergo inelastic collisions before escaping the galaxy. The first two of these are relatively well-understood, but the third factor, known as the calorimetry fraction f cal (E), is much less certain. It depends on the properties of the galaxy, and the lack of a model for this dependence has previously precluded direct calculation of SFG γray emission. However, Ref. 11 recently introduced a model for f cal (E), based on rates of CR diffusion determined by the balance between the CR streaming instability and ion-neutral damping. In the Methods we describe a new technique to use this model to compute f cal (E), and thus the total γ-ray emission produced by CR ions in SFGs.
We supplement the γ-ray production rate from CR ions by adding the contribution from both primary CR electrons, directly injected by supernova remnants, and secondary CR leptons (electrons and positrons), produced in the π ± decay chain; these become important at energies 1 GeV. Our model for these particles includes energy losses due to ionisation, synchrotron emission, bremsstrahlung, and inverse Compton scattering. The model also includes the attenuation of γ-rays produced by both CR ions and leptons due to pair production in collisions with far-infrared photons inside the source galaxy and extragalactic background light photons outside the galaxy, which become important at energies 100 GeV. The radiation that is absorbed by the host galaxy and extragalactic photon fields is reprocessed to lower energies in a pair-production cascade, whereby the initial high energy pair inverse Compton scatters lower energy photons up to γ-ray energies and these, in turn, produce further pairs, and so on. Details of our calculation of all these processes are provided in the Methods.
We now have a model that predicts the γ-ray emission of a SFG. The next step in our analysis is to apply this model to a galaxy survey that samples the SFG population out to the epoch of peak cosmological star formation at z ∼ 2. For this purpose, we make use of the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS) 21,22 in the GOODS-S field. We apply our model to the CANDELS sample as Figure 1: The γ-ray spectra of nearby SFGs Predicted (lines) and observed (points) spectra for a selection of nearby SFGs detected in γ-rays. The observations shown are taken from a combination of Fermi LAT 23 , HESS 7 , and VERITAS 24 where the horizontal bars show the energy bin and the vertical bars the 1 σ uncertainty limit; in Panel a we show the local starburst galaxies Arp 220 and NGC 253, and Panel b we show the local quiescent galaxies M31 and NGC 4945. The solid lines show model predictions using only stellar data of the type we have available for the CANDELS sample, while the dotted lines shows results predicted if we supplement this with observed gas data. We list the full set of observed quantities used in computing these models in Extended Data Table 1.
described in the Methods. To verify that our approach predicts reasonably accurate γ-ray spectra, we apply it to four local, resolved galaxies with measured γ-ray emission 7,23,24 , chosen to span a wide range of gas and star formation surface densities: Arp 220, NGC 253, M31, and NGC 4945. The input data we use for these calculations are summarised in Extended Data Table 1, and we show the results of the computation in Figure 1, where the solid lines show the spectra derived using only stellar data (as we have for CANDELS) and, for comparison, the dotted lines show the results we obtain if we add directly-measured gas properties (available for these local galaxies). We see that the fits are slightly improved if we make direct use of gas data but, even for the stellar data only, our model reproduces the observed γ-ray spectra to better than a factor of 2 for all galaxies at energies > 1 GeV, and within a factor of ≈ 1.5 for the two more rapidly star-forming galaxies, which, as we show below, are more akin to the population that dominates the γ-ray background.
Having verified that we can obtain accurate predictions of γ-ray spectra from stellar data alone, we carry out two additional validation steps. First, we examine the correlation between galaxies' far-infrared and total γ-ray luminosities, computed as described in the Methods. In Figure 2 we show the resulting distribution of galaxies in the LFIR -Lγ plane, along with a power law fit to the data (blue line), Lγ/erg s −1 = 10 28.26±1.55 (LFIR/L ) 1.14±0.16 . Our model prediction shows good agreement with the observed relation 25 , and we note that both the model and the observed correlation differ noticeably from the calorimetric limit obtained by simply setting f cal (E) = 1 (red line in the Figure). Thus the agreement is non-trivial, and suggests that our model is correctly predicting the variation in galaxies' calorimetry fractions as a function of star formation rate.
Our second validation test is to compare our model with counts of resolved SFGs observed by Fermi LAT. Details of how we perform the comparison are given in the Methods. We show the results in Figure 3, which demonstrates that our model predicts SFG source counts consistent with observations, with the exception that we do not predict sources as bright as the Milky Way's two satellite galaxies, the Large and Small Magellanic Clouds. This is not surprising, since our comparison includes only field galaxies. The FIR-γ correlation The correlation between far-infrared (8 − 1000 µm) and γ-ray (0.1 − 100 GeV) luminosity for the CAN-DELS sample, derived using our model. Points show individual CAN-DELS galaxies, colour-coded by redshift z. The blue line is a power law fit to the CANDELS sample with the shaded band containing 90% of data around the model fit. For comparison, the solid green line shows the empirical relation measured for 14 nearby, resolved SFGs 25 with 2 σ uncertainty in the shaded band. The red line is the calorimetric limit obtained by taking f cal = 1 at all energies in Equation 1, as obtained by Ref. 7 .
Our final step is to compute the contribution of SFGs to the diffuse, isotropic γ-ray background (see Methods for details). We present the results of this calculation in Figure 4, and provide a detailed analysis of the model uncertainties in the Supplementary Information and Extended Data Figure 1. Figure 4 shows that the expected contribution of SFGs to the diffuse isotropic γ-ray background fully reproduces both the intensity and the spectral shape of the observations from ≈ 0.2 GeV to ≈ 1 TeV. We emphasise that we obtain this agreement from our model with no free parameters: our only inputs are the CR injection spectral index (q = 2.2), the energy per supernova (10 51 erg), and the fraction of supernova energy that goes into primary ions and electrons (10% and 2%, respectively) -all quantities that are directly measured in the local Universe -and the distribution of SFGs sampled by CANDELS. The key to the success of the model is the galaxy-by-galaxy calculation of the energy-dependent calorimetry fraction f cal (E), which we demonstrate by also plotting the result (dotted line) we would obtain simply by setting f cal = 1 for all galaxies at all energies. This clearly both overestimates the intensity and yields a spectral slope that is flatter than observed.
We show the relative contributions to the background made by galaxies with differing star formation rates and redshifts in Extended Data Figure 2. The Figure shows that the background at lower energies is dominated by galaxies from just after cosmic noon (z ∼ 1 − 2), while at higher energies, where attenuation by extragalactic background light has a larger effect, the dominant contribution shifts towards lower redshifts, so that at 1 TeV the background is dominated by z ∼ 0.1 sources. At all energies, the dominant contribution comes from galaxies at the upper end of the starforming main sequence, which have high but not extreme star formation rates for their redshift.
It is important to put our finding that SFGs dominate the diffuse, isotropic γ-ray background in the context of recent work, where a number of authors have argued that blazars and other AGN sources contribute substantially or even dominate the background. We provide a more detailed discussion in the Supplementary Information, but here note that we find that, while blazars dominate the resolved component of the extragalactic γ-ray background, as shown in Figure 3, SFGs dominate the unresolved component. This finding is consistent with statistical analyses of angular fluctuations in the isotropic background and cross-correlations between it and galaxies and quasars, which strongly disfavour blazars as a dominant contributor 2,8,26 . Indeed, a straightforward extrapolation of the number counts of observed blazars 27 , illustrated by the orange band in Figure 3, 2 also suggests that blazars do not dominate the unresolved background. Our finding that SFGs alone are able to reproduce the full background is also consistent with the conclusions of Refs. 9 and 28 that, in the absence of either a physical model for the γ-ray emission of SFGs or a much larger sample of resolved galaxies, it is not possible to rule them out as a dominant contributor. We conclude by pointing out that the methodology we have introduced can also be applied to predict luminosity functions and background contributions from SFGs at other wavelengths and in other messengers driven by CRs. Most immediately, observations by the upcoming Cherenkov Telescope Array 29 and Large High Altitude Air Shower Observatory 30 should both extend the population of γ-ray-detected SFGs and push existing detections to substantially higher energies. Our model makes clear predictions for both source counts and spectral shapes that can be tested against these data. In the longer term, application of this model to neutrinos will yield predictions that will be testable by IceCube and other neutrino observatories (see the Supplementary Information and Extended Data Figure 3 for further information), and application to synchrotron emission from CR electrons can be used to make predictions for the radio sky that will be testable with the Square Kilometer Array (SKA) and other next-generation radio telescopes. Moreover, because the basis of these predictions is a coherent physical model, rather than just empirical scalings, these predictions can all be made self-consistently.
Supplementary Information is available for this paper.
Methods
Here we describe our methods to compute γ-ray emission from a single SFG due to both CR ions and leptons, to determine the flux received at Earth from that galaxy, and to apply these models to the CANDELS sample, as well as the details of the Monte Carlo estimation for low redshift source counts.
γ-ray emission model for CR ions
In our model, the total rate of γ-ray emission per unit energy from a SFG is the sum of an ionic component and a leptonic component,
dṄγ/dEγ = dṄγ/dEγ ion + dṄγ/dEγ lepton .
We compute the ionic component as
dṄγ dEγ ion = ∞ mpc 2 1 σpp dσγ dEγ (Eion) f cal (Eion) dṄion dEion dEion. (1)
Here dṄion/dEion is the rate per unit energy at which supernovae injection CR ions of energy Eion into the galaxy, σpp = 40 mbarn is the mean proton-proton inelastic cross-section, dσγ/dEγ(Eion) is the differential cross section for production of γ-rays of energy Eγ by CR ions of energy Eion, and f cal (Eion) is the calorimetry fraction for CR ions of energy Eion. We take dσγ/dEγ(Eion) from the parameterised model of Ref. 31 . We compute dṄion/dEion from the galactic star formation ratė M * by assuming that stars form with a Chabrier initial mass function 32 , which gives the distribution of masses for newly-formed stars, and that stars with initial mass of 8 − 50 M , where M is the mass of the Sun, end their lives as supernovae 33 . Each supernova injects 10 50 erg of energy in CR ions 14,15 , distributed in energy for CR energies Eion > mpc 2 as dṄion/dEion = φṀ * (pion/p0) −q dpion/dEion exp (−Eion/Ecut), where p0 = 1 GeV/c, the cutoff energy Ecut = 10 8 GeV, and the spectral index q = 2.2 19,20 . The exact choice of the cutoff energy above Eion ∼ PeV makes no practical difference because the injection spectral index q > 2, so only a small fraction of the total CR energy is injected at PeV energies regardless of Ecut, and any CRs that are injected at such high energies produce photons that we do not observe due to γγ opacity. The normalisation factor φ that corresponds to our choice of initial mass function, supernova mass range, and CR energy per supernova is
φ = 7.15 × 10 42 s −1 GeV −1 M −1 yr.
The only remaining unknown in Equation 1 is the calorimetry fraction f cal (Eion), which we compute from the recent model of Ref. 11 . The basic premise of the model is that, in the neutral phase that dominates the mass of the ISM and thus the set of available targets for γ-ray production, CR transport is primarily by streaming along magnetic field lines. However, this yields approximately diffusive transport when averaged over scales comparable to or larger than the coherence length of the magnetic field, with a diffusion coefficient
D ≈ Vsthg/M 3 A ,
where Vst is the CR streaming speed, hg is the gas scale height, and MA is the Alfvén Mach number of the turbulence. For diffusive transport with losses in a disc geometry, the calorimetry fraction is given by (using the favoured parameters of Ref. 11 )
f cal (Eion) = 1 − 0F1 1 5 , 16 25 τeff + 3 τeff 4 M 3 A 0F1 9 5 , 16 25 τeff −1 ,(2)
where 0F1 is the generalised hypergeometric function and τ eff is the dimensionless effective optical depth of the ISM, given by
τ eff = σpp ηpp Σg hg c 2 D0 µp mH .(3)
Here ηpp = 0.5 is the elasticity of pp collisions, Σg is the gas surface density of the galactic disc, c is the speed of light, D0 is the diffusion coefficient at the galactic midplane, µp = 1.17 is the number density of nucleons per proton, and mH = 1.67 × 10 −24 g is the mass of a hydrogen atom.
To evaluate the calorimetry fraction for a CR of energy Eion, we must therefore determine the midplane diffusion coefficient D0 for CRs of that energy, which in turn depends on the streaming speed Vst. This speed is dictated by the balance between excitation of the streaming instability and dissipation of the instability by ion-neutral damping, the dominant dissipation mechanism in the weakly-ionised neutral ISM. Balancing these two effects yields a CR proton streaming velocity
Vst ≈ min c, VAi 1 + γ d χ MA c ρ 3/2 4 π 1/2 C e uLA µi γ −q+1 (4)
where VAi is the ion Alfvén speed, γ d = 4.9 × 10 13 cm 3 g −1 is the ionneutral drag coefficient, χ is the ionised mass fraction, ρ = Σg/2hg is the midplane mass density of the ISM, C is the midplane number density of CRs, e is the elementary charge, uLA is the velocity dispersion of Alfvén modes in the ISM at the outer scale of the turbulence, µi is the atomic mass of the dominant ion species, q is the index of the CR energy distribution, and γ = Eion/mpc 2 is the Lorentz factor of the CR. Since γ-ray production in our model is dominated by galaxies with high star formation rates and gas surface densities within which i) the ISM is molecule-dominated, ii) the main ionised species is C + , and iii) the magnetic field is set by a turbulent dynamo, we adopt the fiducial parameters of Ref. 11 appropriate for such galaxies. Specifically, we take χ = 10 −4 , µi = 12, MA = 2, VAi = uLA/χ 1/2 MA, and uLA = σg/ √ 2, where σg is the gas velocity dispersion of the galaxy. We also adopt q = 2.2, consistent with our assumed injection spectrum. We have chosen this set of parameters without any fine tuning, by selecting values that are generally accepted as being the most appropriate for the type of source that dominates the emission. However, we explore the parameter space in the Supplementary Information and show a selection of results in Extended Data Figure 1.
At this point we have specified all the ingredients required to compute f cal (Eion) for a galaxy of known Σg, σg, and hg, save one: C, the CR number density. We estimate this as follows: consistent with our discussion above, for a galaxy with star formation rate per unit areaΣ * , the CR ion injection rate per unit area is
dṄion dA = φΣ * ∞ mpc 2 pion p0 −q Eion pion e −E ion /E cut dEion ≈ 2.61 × 10 43Σ * s −1 M −1 yr.(5)
The CR number density at the midplane is then given by
C ≈ t loss 2hg dṄion dA ,(6)
where t loss is the CR loss time. This is given by t loss = 1/ t −1 col + t −1 diff , where the timescale for losses in inelastic hadronic collisions is tcol = 1/ (ρσppηppc/µpmH) and the diffusive escape time tdiff = h 2 g D −1 0 . For the systems with high gas densities and high star formation rate that dominate γ-ray production, the loss time is generally dominated by collisional losses. Conversely, for systems forming stars more sedately and with lower density environments, it is generally determined by the diffusive escape time.
As a final note, we point out that the model of Ref. 11 applies at CR energies up to tens of TeV in galaxies whose interstellar media are mainly molecular (most galaxies with star formation rates above a few Solar masses per year), but may break down above tens of GeV in galaxies where the gas is mostly atomic 34 . Thus the model might not predict the correct degree of calorimetry at 100 GeV energies in dominantly atomic galaxies. As shown in Extended Data Figure 2, however, low star formation rate galaxies make only a small contribution to the background, and thus a possible error in them will have minimal effects on the final result. 5
We illustrate the behaviour of f cal (Eion) over a range of gas surface densities and redshifts as applied to the CANDELS sample (see below) in Extended Data Figures 4 and 5. γ-ray emission model for CR leptons CR electrons and positrons (since both behave identically, for brevity we just write electrons, but everything that follows should be understood as applying to a mix of both) are either injected directly by diffusive shock acceleration in supernova remnants at the same time as CR ions (primary production), or appear in the decay of charged pions π ± produced by collisions of CR ions with the ISM (secondary production). We assume the former carry a total energy equal to 2% of supernova kinetic energy 16 , and have the same injection spectrum as the CR ions, dṄe/dEe|1 ∝ p −q e (Ee/pe)e −Ee/E cut,e , with q = 2.2, but a lower spectral cut-off energy at Ecut,e = 100 TeV 35 . For the latter we follow Refs. 36,37 : we first compute the rate at which CR ions produce pions of energy Eπ,
dṄπ dEπ = nH c Kπ β σpp(Eion) dṄion dEion f cal (Eion) ,(7)
where nH = Σg/(2µpmHhg) is the ISM number density, c the speed of light, β is the CR velocity divided by c, Eπ = Kπ(E − mpc 2 ), and Kπ = 0.17 is the fraction of energy transferred from the CR to the pion.
Then the rate at which these pions produce secondary electrons is
dṄe dEe 2 = 2 1 Ee/Eπ f ν (2) µ (x) dṄπ dEπ Ee x dx x ,(8)
where f ν (2) µ (x) is a dimensionless fititng function given in Ref. 36 .
Thus the total CR electron injection rate is dṄe/dEe = dṄe/dEe|1 + dṄe/dEe|2. The electrons are subject to four dominant loss processes: collisional ionisation, synchrotron radiation, bremsstrahlung, and inverse Compton scattering; as discussed in the main text, diffusive escape from the galaxy is negligible in comparison. We adopt the following parameterisations from the literature for the total energy loss rates dEe/dt for electrons of energy Ee:
− dEe dt ion = 9 4 σT c mec 2 nH ln γ + 2 3 ln mec 2 15eV (9) − dEe dt sync = 1 6π σT c B 2 γ 2 β 2 (10) − dEe dt brems = 3 π ασT cmcc 2 γnH ln γ + ln 2 − 1/3 γ 15 Φ1,H(1/4αγ)/8 γ 15 (11) − dEe dt IC = 20 π 4 σT cγ 2 u rad Y (4γE peak /mec 2 ).(12)
In these expressions, σT is the Thomson cross section, α is the fine structure constant, me is the electron mass, γ = Ec/mec 2 is the electron Lorentz factor, E peak is the energy where the infrared background peaks (derived from the dust temperature T dust = 98 (1 + z) −0.065 + 6.9 logṀ * /M * 38 and injected as a diluted modified black body spectrum 39 which peaks in photon number at E peak = 2.82 kBT dust where kB is the Boltzmann constant), B = VAi/ √ 4πnHµpmHχ is the magnetic field strength, u rad is the radiation energy density (which based on empirical measurements in nearby galaxies we set equal to the magnetic energy density 40 , u rad = umag = B 2 /8π), and Φ1,H and Y are dimensionless numerical fitting functions; these expressions are taken from Refs. 41 (ionisation), 42 (synchrotron), 41 (bremsstrahlung), and 43 (inverse Compton), and the definitions of the fitting functions are given in those references.
Given these loss rates, the steady-state spectrum in the galaxy is given approximately by dNe dEe
dṄe dEe t loss (Ee) ,(13)
where t loss,i = Ee/ dEe dt i is the loss time due to process i, and t loss =
i t −1 loss,i −1
is the total loss time. Of the four loss processes, only bremsstrahlung and inverse Compton scattering produce γ-rays. We compute the resulting emission using expressions analogous to Equation 1. For bremsstrahlung,
dṄγ dEγ brems = c nH Eγ ∞ Eγ σBS (Eγ, Ee) dNe dEe dEe,(14)
where σBS is the cross section for production of photons of energy Eγ by electrons of energy Ee; we take our expression for σBS from Refs. 41,44,45 .
Similarly, for inverse Compton we have 35,46
dṄγ dEγ IC = 3 4 σTc u rad E 2 peak ∞ E e,min dNe dEe G (a, Γ) γ 2 dEe,(15)
where G(a, Γ) = 2a ln a + (1 + 2a)(1 − a) + (Γa) 2 (1 − a)/2(1 + Γa), Γ = 4E peak Ee/m 2 e c 4 , and a = Eγ/Γ(Ee − Eγ). The total leptonic contribution to γ-ray production is simply dṄγ/dEγ
lepton = dṄγ/dEγ brmes + dṄγ/dEγ IC .
We show the contribution of leptonic emission to the diffuse, isotropic γ-ray background divided by emission mechanism, and by primary versus secondary, in Extended Data Figure 6.
γ-ray flux at Earth
To obtain the total observed γ-ray flux for a galaxy, we must account for the attenuation of γ-ray photons by galactic far-infrared and extragalactic background light (EBL) photons. We compute the optical depth τγγ due to the former using the model of Ref. 47 , and the optical depth from the latter, τEBL, using the model of Ref. 48 . Taking these into account, we can compute the specific photon flux dFγ/dEγ (i.e., the number of photons per unit area, time and energy) received at the Earth from a galaxy at redshift z as
dFγ dEγ = (1 + z) 2 4π d 2 L (z) dṄγ dEγ Eγ (1+z) e −τ EBL (Eγ, z) e −τγγ (Eγ(1+z))(16)
where dṄγ/dEγ
Eγ (1+z)
is the total γ-ray production from both CR ions and electrons evaluated at an energy Eγ(1 + z), and dL (z) is the luminosity distance of the source. The radiation that is absorbed by the host galaxy and extragalactic photon fields is reprocessed to lower energies in the pair-production cascade. We parameterise the photon spectrum from this cascade using the method developed in Ref. 49 . For the purposes of our calculation here, we include the effect of the cascade by adding a component to dFγ/dEγ with a spectral shape as computed by Ref. 49 , and with a normalisation such that its energy is equal to the integrated energy lost to photon-photon scattering.
Application of the model to CANDELS galaxies
We apply our model to each individual galaxy in the CANDELS sample. The full sample contains 34,930 galaxies, but we exclude those whose parameters are uncertain because they contain bright active galactic nuclei, have unreliable redshifts, or lack a good fit to the surface brightness profile. This leaves a sample of 22,279 galaxies.
Our calorimetry model requires, as input, the gas surface density Σg, scale height hg, and velocity dispersion σg, along with the surface density of star formationΣ * . However, the CANDELS data set in Ref. 22 , that we use, provides only the cosmological redshift z, stellar mass M * , halflight or effective radius Re (corrected to 5000 Å according to Ref. 50 ), and total star formation rateṀ * for our sample galaxies. We must therefore estimate the gas properties from observed correlations between gas and stellar properties. We do so as follows.
The half light radius Re at 5000 Å serves as a first order estimate of how the star formation and matter are distributed throughout the galactic disc. We therefore estimate the star formation rate surface density aṡ Σ * =Ṁ * /2πR 2 e and the stellar surface density as Σ * = M * /2πR 2 e . We estimate the gas surface density from the observed correlation between gas, stellar, and star formation surface densities given by Ref. 51 :
Σg M pc −2 = 10 10.28Σ * M yr −1 pc −2 Σ * M pc −2 −0.48(17)
Similarly, there is a strong correlation between galaxy star formation rates and velocity dispersions, which we use to derive σg. For this purpose we fit the relationship using the MaNGA galaxy sample 52 . A powerlaw fit to the data obtained in this survey (Fig. 6 of Ref. 52 Finally, we derive the gas scale height under the assumption that the gas is in vertical hydrostatic equilibrium, in which case the scale height is 53
hg = σ 2 g πG (Σg + Σ * )(19)
With these gas properties in hand, we calculate an observed spectrum for each galaxy in the CANDELS sample by using Equation 16, and we then sum over the sample to predict the γ-ray flux per unit energy per unit solid angle Φ(Eγ) produced by SFGs. In practice, we compute the sum as:
Φ (Eγ) = 1 ΩS n zbin j=1 fcorr,j n S,j i=1 dFγ,i dEγ i,j(20)
Here ΩS = 173 arcmin 2 is the solid angle surveyed by CANDELS, nS,j is the number of surveyed galaxies in the jth redshift bin, (dFγ/dEγ)i,j is the flux from the ith galaxy in this bin predicted using Equation 16, and n zbin the number of redshift bins. The factor fcorr,j is the ratio of the expected total star formation rate in each redshift bin (based on the measured cosmic star formation history 54 and obtained by integrating the star formation rate density over the volume in each redshift bin) to the sum of the star formation rates of CANDELS galaxies in that bin; its purpose is to correct for the fact that, due to its limited field of view and various observational biases, the distribution of star formation with respect to redshift in CANDELS does not precisely match the total star formation history of the Universe. We use redshift bins of size ∆z = 0.1, chosen to ensure that the number of sample galaxies in each bin is large enough that the uncertainty in the mean spectrum due to Poisson sampling of the galaxy population is small. For the purposes of constructing Figure 2, we also require not just the flux, but the total γ-ray luminosity in the Fermi band. We compute this by integrating Eγ dṄγ/dEγ (computed as the sum of Equations 1, 14, and 15) from Eγ = 0.1 − 100 GeV. Since this comparison also requires the far-infrared luminosity, we convert the star formation rate to an farinfrared luminosity in the 8−1000 µm band using the relation in Refs. 7,55 , corrected to a Chabrier IMF 56 ; this conversion is valid for star formation rates 1 M yr −1 , which encompasses almost all of the observed sample to which we wish to compare.
Monte Carlo estimation for the local population
To estimate the observable source count distribution for Fermi LAT, as shown in Figure 3, we cannot use the CANDELS catalogue directly, because CANDELS has a narrow field of view that provides very little sampling of galaxies at z 0.1, whereas these local sources are the only ones Fermi LAT can resolve. We therefore use a Monte Carlo scheme to simulate a nearby, low-redshift (z < 0.1) galaxy population that follows the observed distribution of star formation rates in the local Universe to account for cosmic variance, and where the correlation between galaxy star formation rate and γ-ray luminosity is the same as what our model predicts for the low redshift (z < 1.5) part of the CANDELS sample.
The first step is to produce a sample of SFGs. To do so, we draw galaxies from the observed distribution of star formation rates in the local Universe 57,58 . For each SFG drawn, we also draw an associated redshift in the range z = 0 − 0.1, with probability proportional to the co-moving volume element. We continue drawing galaxies until the total star formation rate of the population we have drawn matches the integrated star formation rate within the volume z = 0 − 0.1 as determined from the cosmic star formation history 54 . The second step is to assign γ-ray luminosities for these galaxies based on our model for the CANDELS galaxies. For this purpose, we apply our model to predict the photon luminosity integrated over the 1 -100 GeV band (i.e., the number of photons per unit time emitted in this energy range) for all CANDELS galaxies with z < 1.5, and fit a power law relationship between this luminosity and the star formation rate; we neglect γγ opacity in this calculation, since this effect is unimportant for the galaxies at z < 0.1 and the energy range < 100 GeV that we are simulating. We then assign each of our SFGs a γ-ray photon luminosity using this powerlaw fit, and in conjunction with the redshift, an observed photon flux S.
At this point we have a sample of γ-ray photon fluxes S for simulated z < 0.1 SFGs, which we can place in bins of S to construct a synthetic prediction for S 2 (dN/dS). We carry out 13,000 Monte Carlo trials of this type, and in each bin of S record the mean and the 68% and 90% probability intervals, which we show as the blue points and bands in Figure 3. Our method for computing the analogous confidence intervals for the observations is described in the Supplementary Information.
7
Extended Data Table 1: Local galaxy data For each entry, we give a value followed by the reference from which that value is taken.
NGC 4945 is observed edge-on, so measurements of the gas scale height and gas surface density are unavailable. We derive them in the usual manner, as described in the Methods, using the measured gas velocity dispersion. † The gas data come exclusively from the nuclear starburst region, so we give two effective radii and SFR estimates: the first is for the entire galaxy, and the second is for the circumnuclear disk / nuclear starburst region only. We use the former for our stellar data spectrum prediction and the latter for our gas prediction. 3; and finally d the conversion fraction of supernova energy to CR electrons for values of 1% and 3%, which is equivalent to 10% and 30% of the total energy injected in all cosmic ray species. Note that varying the total CR energy budget results in a trivial scaling of the result by the same fraction, and thus is not shown.
Extended Data Figure 2: The contribution of SFGs in theṀ *z plane The contribution of SFGs to the total γ-ray spectrum at selected energies in the star formation rate (Ṁ * ), redshift (z) plane. Coloured pixels show the fractional contribution (as indicated in the colourbar) from galaxies in each bin ofṀ * and z to the diffuse isotropic γ-ray background at the indicated energy; a fractional contribution of unity corresponds to that pixel producing all of the background, with no contribution from galaxies outside the pixel. Grey points show individual CANDELS galaxies in regions ofṀ * and z that contribute < 10 −3 of the total. Flanking histograms show the fractional contribution binned in one dimension -Ṁ * (right) and z (top). We see that the background at lower energies is dominated by emission from galaxies on the high side of the star forming main sequence at z ∼ 1 − 2, while at high energies it is dominated by the brightest systems at low redshift. Extended Data Figure 3: The diffuse isotropic γ-ray and neutrino backgrounds The blue line and black points show the model-predicted and observed γ-ray background, and are identical to those shown in Figure 4. The red lines show our model prediction for the neutrino background (single flavour) with Ecut = 100 PeV (solid line) and Ecut = 1 PeV (dashed line), computed as described in the Supplementary Information. We assume a neutrino flavour ratio at the detector of (νe : νµ : ντ ) = (1 : 1 : 1). The red filled band shows a power law fit 73 to the single flavour astrophysical neutrino background with the 90% likelihood limit, as measured by IceCube, which is also shown as grey points, where the horizontal bars show the energy bin and the vertical bars the 1 σ uncertainty limit. Extended Data Figure 4: Cosmic ray calorimetry in the E -Σg plane Mean calorimetry fraction f cal (E) in the surface gas density Σg, cosmic ray energy E plane, binned in redshift intervals. This figure is constructed by deriving the gas surface density and energy dependent calorimetry fraction for each galaxy in the CANDELS sample using our model. The colour of each pixel gives the mean calorimetry fraction of all the galaxies within that particular range of Σg, E, and redshift. The horizontal white stripes correspond to ranges of Σg into which no CANDELS galaxies fall for the corresponding redshift range. Several physical processes contribute to the behaviour visible in the plot. At low Σg, galaxies have low f cal at all energies E because there are few targets for hadronic collisions with CRs. As Σg increases, the increased ISM density results in efficient calorimetry and conversion of CR energy into γ-rays for low CR energies; however, at higher energies the CR number density is low, yielding a high CR streaming velocity and rapid escape, resulting in low f cal . As Σg increases further, the increasing density results in the streaming instability being suppressed efficiently by ion-neutral damping towards lower energies, reducing the calorimetry fraction further. Finally, at the highest Σg, the streaming instability is suppressed completely by ion-neutral damping, but streaming is still limited to the speed of light. Consequently, increasing Σg further only results in increased collisions, and thus a higher calorimetry fraction.
Supplementary Information 1 Confidence intervals for source count distributions
Calculation of confidence intervals on S 2 (dN/dS) (as shown in Figure 3) for the observed sources (both the Fermi-observed SFGs and our modelpredicted CANDELS SFGs) is non-trivial, because both surveys cover a fraction f < 1 of the sky, and the number of sources per bin for at least some bins of S is small, so we cannot compute the uncertainty by assuming that we are in the large N limit. To perform the calculation, we assume that the SFG population follows a Poisson distribution on the sky (i.e., we are in the cosmological isotropic limit), so if the entire sky contains Ntot SFGs within some bin of photon flux, the probability that N obs will be found within the observable region can be written
P (N obs |Ntot) = (f Ntot) N obs e −f N tot N obs ! .(21)
We wish to solve the inverse problem, i.e., given an observed number N obs , what is the probability distribution of Ntot? The answer is given by Bayes's Theorem, which requires P (Ntot|N obs ) = P (N obs |Ntot)
P (Ntot) P (N obs )(22)
where P (Ntot|N obs ) is the posterior probability, P (Ntot) is the prior probability, and P (N obs ) is a normalisation factor. We adopt a flat prior P (Ntot) ∝ 1, so we can then write
P (Ntot|N obs ) = N N N obs tot e −f N tot ,(23)
where N is a normalisation constant. For e −f < 1, which is always the case since 0 < f ≤ 1, the value of N required to guarantee that
N tot P (Ntot|N obs ) = 1 is N = 1 Li−N obs (e −f ) ,(24)
where Lis (z) is the polylogarithm of order s.
To compute the confidence interval we require the cumulative distribution function. In the discrete case this is given by calculating the probability that Ntot < N , which is
P (Ntot < N ) = N N −1 n=0 n N obs e −f n (25) = 1 − N ∞ n=N n N obs e −f n (26) = 1 − N ∞ i=0 (i + N ) N obs e −f (i+N ) (27) = 1 − e −f N Φ e −f , −N obs , N Li−N obs (e −f )(28)
where Φ (z, s, a) is the Lerch Phi function (sometimes also referred to as the Lerch Zeta function). To obtain a particular percentile p in the range 0 to 1, we simply use the continuous forms of the polylogarithm and the Lerch Phi functions, set p = P (Ntot < N ) and invert the problem numerically to find the appropriate value for N ; for the purposes of Figure 3, we are interested in the 90% confidence interval, so we take p = 0.05 and p = 0.95. For the special case N obs = 0, the result simplifies to
p = 1 − e −f N 1 Li0 (e −f ) + 1 ,(29)
which we can invert numerically for p = 0.9 to obtain the 90% confidence upper limit.
In order to use the result we have just derived, we require a value for f . For the CANDELS data points, this is straightforward: our data come from the GOODS-S field, which has an area of 173 arcmin 2 , corresponding to f = 1.16 × 10 −6 . Assigning a value of f to the Fermi data is more complex: Fermi LAT surveys the entire sky, but it cannot detect faint sources, such as SFGs, that are too close to the Galactic plane because they are hidden by the Galactic diffuse foreground. As a result, the effective survey area depends at least somewhat on the flux and spectral shape of the target SFG -brighter and harder sources can be detected closer to the plane than fainter and softer ones. Capturing this effect in detail would require extensive testing of the Fermi reduction pipeline using artificial sources, which is beyond the scope of this work. For the purposes of computing the confidence intervals shown in Figure 3, we ignore this complexity, and roughly estimate that SFGs are undetectable within 15 • degrees of the Galactic plane, which corresponds to approximately f = 0.7. 15
Here we investigate the sensitivity of our results to our choice of model parameters. The set of parameters we have adopted to derive the key result of this study were deliberately chosen to be approximately the consensus value. We purposefully chose not to fine-tune our parameters to obtain a best fit, nor, more importantly, have we had to adopt extremal values of any parameter to force the calculation in the direction of making the contribution of SFGs dominant. However, it is still of interest to explore the parameter space within a reasonable range of variation. The key tunable parameters in our model are: (1) the total energy per unit mass of stars formed that is ultimately injected in CRs, (2) the Alfvén Mach number MA, (3) the ionisation fraction χ, (4) the CR injection spectral index q, and (5) the ratio of CR energy injected into primary leptons to that injected into primary protons. We discuss each of these in turn.
The total CR energy budget is constrained by observations of both individual SN remnants 15 and by observation of the total γ-ray luminosity of local starburst galaxies, which are generally thought to be fully calorimetric 16,25,34 . These observations constrain this energy budget to be within a factor of ≈ 2 of our fiducial choice. Varying this parameter within that range would increase or decrease our predicted background from SFGs by the same factor, while leaving the overall shape of the spectrum unchanged.
The Alfvén Mach number MA cannot be measured directly in external galaxies, but is determined by the ratio of the magnetic uB and kinetic u kin densities in the flow, MA = (3uB/u kin ) −0.5 11 , which in turn is well constrained by dynamo theory and simulations 11,74,75 . Galactic magnetic fields are driven by both turbulent and αΩ dynamos. The growth timescale for the latter is the orbital period, and for the former is the eddy turnover time, which for a galaxy that is marginally stable against gravity is also comparable to the orbital period 76 ; since essentially all galaxies are many orbital periods old, we can assume that galactic dynamos have reached saturation. For the turbulent dynamo driven by supersonic motion, the saturation level of uB/u kin is determined by the Reynolds and magnetic Prandtl numbers. Both of these dimensionless numbers are large in star-forming galaxies (Re 10 6 76 and Pr ∼ hg/lAD 100, where hg is the gas scale height and lAD is the length scale at which magnetic fields decouple from the gas due to ambipolar diffusion 11 ). In this regime, Ref. 75 Figure 1, where we find that lowering MA to 1.6 yields a slightly fainter background, while raising it to MA = 2.3 makes the predicted background somewhat brighter; the overall shape is largely unchanged in either case. For illustrative purposes we also include the extremal values of 1.1 and 3.0, corresponding to near-equipartition and very sub-equipartition field strengths.
The ionisation fraction χ has a larger plausible range: in galaxies with predominantly atomic interstellar media such as the Milky Way, it reaches χ ≈ 10 −2 , while in extreme starbursts with very high densities it might reach as low as 10 −6 , indicating a 2 dex range around our fiducial choice 11,25 . However, this parameter also enters the problem to the 1/4 power. We show the effects of varying χ in Panel b of Extended Data Figure 1; it is clear from this figure that, despite its larger range of variation, varying χ within its plausible range has a smaller effect than varying MA.
The injection spectral index q is constrained by observations of local SN remnants, which require that it lie in the range q ≈ 2.1 − 2.3 19,20 . We explore this range of variation in Panel c of Extended Data Figure 1, which shows that changing the injected spectral index induces, as might be expected, a similar change in the spectral slope of the predicted background. However, given the uncertainties, it is not clear exactly which spectral index in this range would give the best agreement with the data, and we note that all of these models lie far from what would be expected for full calorimetry, as is clear from comparing Extended Data Figure 1 to the line for full calorimetry in Figure 4 of the main text; even for q = 2.1 or 2.3, the overall spectral slope is determined mostly by variation of f cal (E), not by the choice of injection spectrum.
Finally, our fiducial choice of ratio of primary electrons to protons is motivated by what is required to reproduce the FIR-radio correlation 16 . However, given various uncertainties, this could plausibly change at the factor of ≈ 2 level. We show the effects of varying the fraction of SN energy that goes into primary CR electrons (while holding CR ions constant) in Panel d of Extended Data Figure 1. As expected, the effects are minimal except below ≈ 1 GeV in energy, and even there are small, since reducing the energy in primary electrons of course does not affect emission from secondaries, which are equally important.
The overall message of Extended Data Figure 1 is that the conclusion that SFGs contribute at least ≈ 50% of the total diffuse γ-ray background seems inescapable, even if we choose extreme values for model parameters. However, there is also another, somewhat stronger conclusion to draw: within the reasonable parameter space, the spectral shape we derive, which is dictated primarily by our model for f cal (E), matches the spectral shape of data well over a 3 decade range in γ-ray energy. This means that, if we were to adopt extreme values of parameters such that SFGs contribute only ≈ 50% of the background, leaving the remainder to be made up by some other unknown source population, matching the observed spectrum would require implausible fine-tuning: this other source population would have to produce a spectral shape and magnitude nearly identical to that of SFGs, over the entire energy range from ≈ 0.3 GeV to ≈ 1 TeV.
The one place where there is some minor tension between the spectral shape predicted by our model and that observed is near the EBL-induced cutoff of the spectrum between ∼100 GeV and ∼1 TeV, where our model falls slightly below the data (though the data themselves are uncertain in this energy range due to the difficulties of background subtraction). One possible explanation is that the model prediction in this energy range is sensitive to our chosen functional form for the EBL optical depth, and could conceivably be fit better by alternative models. Moreover, due to the EBL, the background at these energies is dominated by low-redshift, hard γ-ray sources (see Extended Data Figure 2), which are poorly sampled by the deep, narrow field of view in CANDELS. We have partly accounted for this by correcting the CANDELS star formation density using the measured star formation history, but a better solution, which we leave for future work, would be to supplement CANDELS by a wide-field, low-redshift galaxy survey. spectrum in different classes of galaxies 78,79 . By contrast, Figure 1 demonstrates that none of the four nearby resolved galaxies shown have spectra that are well described by a γ-ray spectrum in the form of a pure power law over the energy range from Eγ = 1 − 1000 GeV; our model correctly captures this behaviour, but earlier pure power law models did not. Similarly, we calculate f cal as a function of energy directly, rather than relying on an empirical FIR-γ correlation, and our calorimetry fractions are on average larger than those implicitly assumed in earlier works. This is because many of the lower estimates for the contribution from SFGs to the γ-ray background rely on a FIR-γ correlation derived from early Fermi detections of < 10 individually-resolved SFGs 7 that yields somewhat lower γ-ray luminosities than more recent fits using a larger (but still small) sample of SFGs 25 , and with which our model agrees (Figure 2). Thus the reason we find that SFGs can produce the full background, whereas earlier models could not, is that our model predicts γ-ray emission that is both somewhat brighter and has a more complex spectral shape than the values adopted in earlier work.
Likewise, earlier claims that a variety of other source classes dominate the diffuse, isotropic background have also relied on extrapolated empirical correlations with large uncertainties. For instance Ref. 80 estimates the contribution from misaligned active galactic nuclei using a radio-γ correlation derived from a sample of 16 resolved objects, coupled with a radio luminosity function extrapolated to redshifts considerably higher than those wellsampled by observations 81 . By contrast, our assignment of γ-ray luminosities to SFGs is based on a physical model that agrees with local observations, and the CANDELS catalogue from which we draw our distribution of SFG properties has very good completeness over the range of redshift and star formation rate that dominates production of the diffuse, isotropic γ-ray background (see Extended Data Figure 2).
Neutrinos
In addition to electrons and positrons, the decay of π ± also produces neutrinos, which are of particular interest as they propagate largely unhindered from the source to the observer. Our goal here is to compute the all-species neutrino flux due to SFGs, so that we may compare to the astrophysical neutrino background measured by IceCube 73 .
The relationship between the γ-ray and neutrino spectra is approximately given by E 2 ν Fν (Eν = Eγ/2) = (3/2)E 2 γ Fγ (Eγ) 82 . However, we compute the neutrino flux from the charged pion decay in our sample galaxies using the more detailed method in Ref. 35,36,371 .
Charged pion decay produces neutrinos in two steps: the initial decay of the pion creates a muon and a muon neutrino, and then the muon decays, yielding an electron, an electron neutrino, and a second muon neutrino (where we do not distinguish between particles and anti-particles). The all-flavour neutrino spectrum is then a sum over the energy distributions of all three neutrinos produced in this chain, given by
dṄν dEν (Eν ) = 2 1 0 fν e (x) + f ν (2) µ (x) dṄπ dEπ Eν x dx x + 2 λ λ 0 dṄπ dEπ Eν x dx x ,(30)
where λ = 1 − (mµ/mπ) 2 , x = Eν /Eπ, the second integral accounts for the muon neutrinos produced in the initial charged pion decay, and the functions fν e and f ν (2) µ describe the energy distributions for the electron and muon neutrinos produced by decay of the secondary muon, respectively;
we take them from Equations 40 and 36 of Ref. 35,36 . The ratio of neutrino flavours at the source is (νe : νµ : ντ ) = (1 : 2 : 0). However, neutrino oscillations will bring this to an even (νe : νµ : ντ ) = (1 : 1 : 1) for an observer at Earth. Equation 30 is the analogue to Equation 1 of the main text for γ-rays, and we can compute the resulting specific neutrino flux for each galaxy from Equation 2 of the main text simply by replacing dṄγ/dEγ with dṄν /dEν and setting the opacities τγγ = τEBL = 0. We use this to calculate a predicted neutrino flux from each CANDELS galaxy, and we sum to compute the neutrino background due to SFGs using Equation 3 of the main text, exactly as we do for the γ-ray background. We plot the resulting predicted neutrino spectrum in Extended Data Figure 3.
We find that our model predicts that SFGs produce a neutrino flux that is ≈ 15% of the astrophysical neutrino background, as measured by IceCube 73 , for a CR spectral cutoff energy of Ecut = 100 PeV. However, while the choice of Ecut has no significant effect on the γ-ray spectrum (as explained in the main text), it does matter for the neutrino spectrum due to the high energies of the astrophysical neutrinos observable by IceCube. Consequently, we find that SFGs produce 15% of the observed neutrino background if we adopt a smaller value of Ecut 20 . To illustrate this, in Extended Data Figure 3 we show two calculations: one with our fiducial Ecut = 100 PeV, and one with a smaller Ecut = 1 PeV. The cutoffs in the neutrino spectrum shown in Extended Data Figure 3 are a direct result of the adopted value of Ecut. We also note that the normalisation of our predicted neutrino spectrum is sensitive to bright, hard neutrino sources at low redshift, which dominate at high energy but are poorly sampled by the small CANDELS field of view. This suggests that it would be worthwhile in the future to repeat this analysis using a survey of SFGs that is wider but shallower than CANDELS. A further consideration relates to the exact form of the cosmic ray calorimetry at ultra high energies. Here we would expect cosmic rays with sufficiently large gyro radii to interact with and scatter off the external turbulence above the damping scale, significantly decreasing the diffusion coefficients and increasing calorimetry in turn. We leave an exploration of this mechanism and the injection cutoff energy for future work.
Figure 2 :
2Figure 2: The FIR-γ correlation The correlation between far-infrared (8 − 1000 µm) and γ-ray (0.1 − 100 GeV) luminosity for the CAN-DELS sample, derived using our model. Points show individual CAN-DELS galaxies, colour-coded by redshift z. The blue line is a power law fit to the CANDELS sample with the shaded band containing 90% of data around the model fit. For comparison, the solid green line shows the empirical relation measured for 14 nearby, resolved SFGs 25 with 2 σ uncertainty in the shaded band. The red line is the calorimetric limit obtained by taking f cal = 1 at all energies in Equation 1, as obtained by Ref. 7 .
Figure 3 :
3The γ-ray source count distribution Red points show the compensated distribution of SFG luminosities 25 S 2 (dN/dS) on the sky as a function of photon flux S integrated from 1 -100 GeV as seen by Fermi LAT; error bars show 90% confidence intervals or upper limits. The three brightest non-empty bins each contain only a single SFG, which we have labelled. Green points show model-predicted source counts for observed CANDELS galaxies at z > 0.1 with the 90% confidence limit, and blue points show Monte Carlo realisations of the z < 0.1 SFG population, with the light and dark shaded bands indicating 68% and 90% confidence intervals. Black squares show Fermi-detected blazars, and the orange band shows the blazar distribution model of Ref.27 within the 1 σ range. Finally, the red vertical band indicates the flux range over which Fermi observations become incomplete; the left edge of this band is the 4FGL threshold for 98% detection efficiency for sources with spectral index 2.3 27 .
Figure 4 :
4The diffuse isotropic γ-ray background Black data points show Fermi 50-month observations 1 where the horizontal bars show the energy bin and the vertical bars the 1 σ uncertainty limit; the thick blue line shows our prediction for the total background due to SFGs; thin solid lines in other colours show the fractional contribution to this total from π 0 decay (CR ions; green line), leptonic emission processes (CR electrons and positrons; orange line), and γγ scattering and the resulting pair production cascade (red line). Broken thin blue lines show the predicted background if we turn off parts of our model: the blue dotted line shows the spectrum we would obtain if we set f cal = 1 for all galaxies at all energies, while the blue dashed line shows the background we would see in the absence of γγ opacity.
Extended Data Figure 1 :
1The effect of varying model parameters The plots presented here show the result of our calculations when varying the model parameters as discussed in the Supplementary Information. Our fiducial choice is plotted as a solid blue line, with the dashed and dash-dotted lines showing the spectrum for the upper and lower limits respectively of the varied parameter. The black points correspond to the Fermi data as in Figure 4. Plot a shows MA plotted for reasonable values of 1.6 and 2.3, and extremal values of 1.1 and 3.0; b the ionisation fraction χ for values of 10 −2 and 10 −6 ; c the injection index q for values 2.1 and 2.
find uB/u kin = 0.040 − 0.064, which corresponds to Alfvén Mach numbers of 2.9 and 2.3. This provides an absolute upper limit on MA, expected from the turbulent dynamo alone. The αΩ dynamo will further enhance the field strength by creating a coherent component on top of the turbulent one. Simulations in Ref. 74 find that the αΩ dynamo in galactic discs saturates at B coh /B turb = 0.27 − 0.42, corresponding to a factor of 1.27 2 − 1.42 2 increase in uB compared to our estimate for the turbulent dynamo only. If we take the minimum value for the turbulent energy from Ref. 75 (0.04) and the minimum αΩ amplification factor from Ref. 74 (1.27 2 ), this gives MA = 2.27, while the maximal values 0.064 and 1.42 2 give MA = 1.61. Our fiducial choice of MA = 2 is simply the middle of this fairly narrow range of reasonable values. We show the results of varying MA in Panel a of Extended Data
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Comparison to earlier work Our conclusion that emission from SFGs dominates the diffuse, isotropic γ-ray differs from some earlier work. It is therefore important to examine the precise reasons why this is the case. One contributing factor is that earlier models were forced to adopt single power laws for the emitted γ-ray
We caution readers, a number of recent publications calculate the neutrino spectrum using an incorrect formula for the parameterisation function gν e (x) given in Ref.35 . An Erratum has been published in Ref.36 . Use of the incorrect formula leads to overestimation of the neutrino emission by a factor of ∼2.
AcknowledgementsThis research has made use of the NASA/IPAC Infrared Science Archive, which is funded by the National Aeronautics and Space Administration and operated by the California Institute of Technology. Funding for this work was provided by the Australian Government through the Australian Research Council, awards FT180100375 (MRK) and DP190101258 (RMC and MRK), and the Australian National University through a research scholarship (MAR). RMC thanks Oscar Macias and Shin'ichiro Ando for enlightening conversations while a Kavli IPMUfunded guest of the GRAPPA Institute at the University of Amsterdam.Author contributionsAll authors were involved in the design of the study and the interpretation of the results. MAR performed the modelling and data analysis with input from MRK, RMC, and SC. The manuscript was written by MAR, MRK and RMC, and reviewed by all authors.Author informationThe authors declare no competing interests, financial or otherwise. Correspondence and requests for materials should be addressed to the first author. Reprints and permissions information is available at www.nature.com/reprints.
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To construct this figure, for each CANDELS sample galaxy, we apply our model to compute Σg and f cal (E) at the indicated energies. The colour indicates the average f cal (E) value computed over bins of (z, Σg), while contours indicate the density of the CANDELS sample in this plane. Note that the non-monotonic behaviour of f cal (E). Cosmic ray calorimetry in the z -Σg plane Mean calorimetry fraction in the surface gas density (Σg). 5redshift (z) plane at CR energies E = 1 GeV, 10 GeV, 1 TeV and 10 TeV. with Σg that is most prominently visible in the 1 TeV panel is expected, for the reasons explained in the caption of Extended Data Figure 4.Extended Data Figure 5: Cosmic ray calorimetry in the z -Σg plane Mean calorimetry fraction in the surface gas density (Σg), redshift (z) plane at CR energies E = 1 GeV, 10 GeV, 1 TeV and 10 TeV. To construct this figure, for each CANDELS sample galaxy, we apply our model to compute Σg and f cal (E) at the indicated energies. The colour indicates the average f cal (E) value computed over bins of (z, Σg), while contours indicate the density of the CANDELS sample in this plane. Note that the non-monotonic behaviour of f cal (E) with Σg that is most prominently visible in the 1 TeV panel is expected, for the reasons explained in the caption of Extended Data Figure 4.
Extended Data Figure 6: Contributions to the diffuse isotropic γ-ray background The blue line and black points show the model-predicted and observed γ-ray background, and are identical to those shown in Figure 4. The green line shows the contribution from π 0 decay, the olive lines the contribution from bremsstrahlung emission, and the cyan lines the contribution from the inverse Compton emission. In both cases, dashed lines show the spectrum produced by primary CR electrons and the dash-dotted lines the spectrum from secondary electrons and positrons. The red line shows the contributions from the EBL cascadeExtended Data Figure 6: Contributions to the diffuse isotropic γ-ray background The blue line and black points show the model-predicted and observed γ-ray background, and are identical to those shown in Figure 4. The green line shows the contribution from π 0 decay, the olive lines the contribution from bremsstrahlung emission, and the cyan lines the contribution from the inverse Compton emission. In both cases, dashed lines show the spectrum produced by primary CR electrons and the dash-dotted lines the spectrum from secondary electrons and positrons. The red line shows the contributions from the EBL cascade.
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| [] |
[
"Statistics of blocks in k-divisible non-crossing partitions",
"Statistics of blocks in k-divisible non-crossing partitions"
] | [
"Octavio Arizmendi \nUniversität des Saarlandes\nFR 6.1−Mathematik66123SaarbrückenGermany\n"
] | [
"Universität des Saarlandes\nFR 6.1−Mathematik66123SaarbrückenGermany"
] | [] | We derive a formula for the expected number of blocks of a given size from a non-crossing partition chosen uniformly at random. Moreover, we refine this result subject to the restriction of having a number of blocks given. Furthermore, we generalize to k-divisible partitions. In particular, we find that, asymptotically, the expected number of blocks of size t of a k-divisible non-crossing partition of nk elements chosen uniformly at random is kn+1 (k+1) t+1 . Similar results are obtained for type B and type D k-divisible non-crossing partitions of Armstrong. | 10.37236/2431 | [
"https://arxiv.org/pdf/1201.6576v3.pdf"
] | 17,222,201 | 1201.6576 | 9f45b9b86f3adc9531dad0d1a3f481df03dc0db2 |
Statistics of blocks in k-divisible non-crossing partitions
December 21, 2013
Octavio Arizmendi
Universität des Saarlandes
FR 6.1−Mathematik66123SaarbrückenGermany
Statistics of blocks in k-divisible non-crossing partitions
December 21, 2013
We derive a formula for the expected number of blocks of a given size from a non-crossing partition chosen uniformly at random. Moreover, we refine this result subject to the restriction of having a number of blocks given. Furthermore, we generalize to k-divisible partitions. In particular, we find that, asymptotically, the expected number of blocks of size t of a k-divisible non-crossing partition of nk elements chosen uniformly at random is kn+1 (k+1) t+1 . Similar results are obtained for type B and type D k-divisible non-crossing partitions of Armstrong.
Introduction
In this paper we study some statistics of the block structure of non-crossing partitions. A first systematic study of non-crossing partitions was done by G. Kreweras [15]. More recently, much more attention has been paid to non-crossing partitions because, among other reasons, they play a central role in the combinatorial approach of Speicher to Voiculescu's free probability theory [20]. For an introduction to this combinatorial approach, see [16].
A non-crossing partition of {1, ..., kn} is called k-divisible if the size of each block is divisible by k. The poset of k-divisible non-crossing partitions was introduced by Edelman [10] and reduces to the poset of all non-crossing partitions for k = 1. We denote these posets by N C k (n) and N C(n), respectively.
As can be seen in [1] and [2], k-divisible non-crossing partitions play an important role in the calculation of the free cumulants and moments of products of k free random variables. Moreover, in the approach given in [2] for studying asymptotic behavior of the size of the support, when k → ∞, understanding the asymptotic behavior of the sizes of blocks was a crucial step.
In this direction, a recent paper by Ortmann [17] studies the asymptotic behavior of the sizes of the blocks of a uniformly chosen random partition. This lead him to a formula for the right-edge of the support of a measure in terms of the free cumulants, when these are positive. He noticed a very simple picture of this statistic as n → ∞. Roughly speaking, in average, out of the n+1 2 blocks of this random partition, half of them are singletons, one fourth of the blocks are pairings, one eighth of the blocks have size 3, and so on.
Trying to get a better understanding of this asymptotic behavior, the question of the exact calculation of this statistic arose. In this paper, we answer this question and refine these results by considering the number of blocks given. Moreover, we generalize to k-divisible partitions, as follows.
Theorem 1. The sum of the number of blocks of size tk over all the k-divisible non-crossing partitions of {1, 2, .., kn} is given by
n(k + 1) − t − 1 nk − 1 .(1)
In particular, asymptotically, we have a similar phenomena as for the case k = 1; about a k k+1 portion of all the blocks have size k, then a k (k+1) 2 portion have size 2k, then k (k+1) 3 are of size 3k, etc. More generally, for any Coxeter Group W , Bessis [7] and Brady and Watt [8] defined the poset N C(W ) of non-crossing partitions for a reflexion group, making N C(A n−1 ) isomorphic to N C(n). Furthermore, Armstrong [5] defined the poset N C k (W ) of k-divisible non-crossing partitions for a reflexion group.
Many enumerative results are now known for this noncrossing partitions for reflection groups (see e.g. [4,12,13,14,19]).
However, analogous results to Theorem 1 have not been studied. We address this problem for the types B and D.
Theorem 2. (1)
The sum of the number of non-zero pairs of blocks {−V, V } of size tk over all the non-crossing partitions in N C k B (n) is given by
nk n(k + 1) − t − 1 nk − 1 .
(2) The number of the non-crossing partitions in N C(B n ) with a zero-block of size 2t is
n(k + 1) − t − 1 nk − 1 .
While for k-divisible partitions type D we get the following.
(k(n − 1) + 1) (k + 1)(n − 1) − t k(n − 1) − 1 + k(n − 1) (k + 1)(n − 1) − t − 1 k(n − 1) − 2 .
Theorems 2 and 3 show that, asymptotically, the behavior of sizes is the same for type B and type D non-crossing partitions as for the classical non-crossing partitions (or type A). Again, a k k+1 portion of all the blocks have size k, then a k k+1 portion of the remaining blocks are of size 2k,, etc. Another consequence is that the expected number of blocks of a k-divisible non-crossing partition is given by kn+1
k+1 . An equivalent formulation of this result was also observed by Armstrong [5, Theorem 3.9] for any Coxeter group. It is then a natural question if this simple formula can be derived in a bijective way. We end with a bijective proof of this fact, for type A and B k-divisible non-crossing partitions.
Let us finally mention that there exists a type B free probability. Free probability of type B was introduced by Biane, Goodman and Nica [9] and was later developed by Belinschi and Shlyakhtenko [6], Nica and Février [11] and Popa [18].
Apart from this introduction, the paper is organized as follows. In Section 1 we give basic definitions and collect known results on the enumeration of classical non-crossing partitions and non-crossing partitions of type B and D . The main new enumerative results concerning the numbers of blocks of a given size for a non-crossing partition of type A, B and D chosen at random are given in Sections 2, 3 and 4, respectively. Finally, in Section 5 we give a bijective proof of the fact that in average the number of blocks is given by kn+1 k+1 and some further consequences of this bijection.
Preliminaries
Classical non-crossing partitions
Definition 1.1. (1) We call π = {V 1 , ..., V r } a partition of the set [n] := {1, 2, .., n} if and only if V i (1 ≤ i ≤ r) are pairwise disjoint, non-void subsets of S, such that V 1 ∪ V 2 ... ∪ V r = {1, 2, .., n}. We call V 1 , V 2 , .
.., V r the blocks of π. The number of blocks of π is denoted by |π|.
(2) A partition π = {V 1 , ..., V r } is called non-crossing if for all 1 ≤ a < b < c < d ≤ n if a, c ∈ V i then there is no other subset V j with j = i containing b and d.
(3) We say that a partition π is k-divisible if the size of all the blocks is a multiple of k. If all the blocks are exactly of size k we say that π is k-equal.
We denote the set of non-crossing partitions of [n] by N C(n), the set of k-divisible non-crossing partitions of [kn] by N C k (n) and the set of k-equal non-crossing partitions of [kn] by N C k (n). N C k (n) is a poset (see Remark 1.2 below) and was first studied by Edelman [10], who calculated many of its enumerative invariants. Remark 1.2. (1) N C k (n) can be equipped with the partial order of reverse refinement (π σ if and only if every block of π is completely contained in a block of σ).
(2) For a given π ∈ N C(n) ∼ = N C({1, 3, . . . , 2n − 1}) we define its Kreweras complement Kr(π) := max{σ ∈ N C(2, 4, . . . 2n) : π ∪ σ ∈ N C(2n)}.
The map
Kr : N C(n) → N C(n) is an order reversing isomorphism. Furthermore, for all π ∈ N C(n) we have that |π| + |Kr(π)| = n + 1, see [16] for details.
There is a graphical representation of a partition π ∈ N C(n) which makes clear the property of being crossing or non-crossing, usually called the circular representation. We think of [n] as labelling the vertices of a regular n-gon, clockwise. If we identify each block of π with the convex hull of its corresponding vertices, then we see that π is non-crossing precisely when its blocks are pairwise disjoint (that is, they don't cross). [12] in their circular representation. Remark 1.3. The following characterization of non-crossing partitions is sometimes useful: for any π ∈ N C(n), one can always find a block V = {r + 1, . . . , r + s} containing consecutive numbers. If one removes this block from π, the partition π \ V ∈ N C(n − s) remains non-crossing.
We recall the following result which gives a formula for the number of partitions with a given type [15]. Proposition 1.4. Let r 1 , r 2 , ...r n be nonnegative integers such that r 1 + 2r 2 ... + nr n = n. Then the number of partitions of π in N C(n) with r 1 blocks of size 1, r 2 blocks of size 2 ,. . . , r n blocks of size n equals n! p r (n − m + 1)! ,
where p r = r 1 !r 2 ! · · · r n ! and r 1 + r 2 ... + r n = m.
It is well known that the number of non-crossing partition is given by the Catalan numbers 1 n+1 2n n . More generally, for k-divisible non-crossing partitions we have the following [10]. Proposition 1.5. Let N C k (n) be the set of non-crossing partitions of [nk] whose sizes of blocks are multiples of k. Then
#N C k (n) = (k+1)n n kn + 1 .
On the other hand, from Proposition 1.4, we can easily count k-equal partitions.
Corollary 1.6. Let N C k (n) be the set of non-crossing partitions of nk whose blocks are of size of k. Then
#N C k (n) = kn n (k − 1)n + 1 .
The reader may have noticed from Proposition 1.5 and Corollary 1.6 that the number of (k + 1)-equal non-crossing partitions of [n(k + 1)] and the number of k-divisible non-crossing partitions of [nk] coincide. We derive a bijective proof of this fact and study further consequences in Section 5.
Non-crossing partitions for Coxeter groups of classical types
For any Coxeter Group W , Bessis [7] and Brady and Watt [8] defined the poset of non-crossing partition for the reflexion group N C(W ).
For the three infinite families of finite irreducible Coxeter groups, A n−1 (the symmetric group), B n (the hyperoctahedral group) and D n (an index 2 subgroup of B n ) known as classical groups, there exists combinatorial realizations of N C(W ). While N C(A n−1 ) is isomorphic to N C(n), the combinatorial realizations of the lattices N C(B n ) and N C(D n ), that we explain in this section, were done by Reiner [19] and Reiner and Athanasiadis [4], respectively. Furthermore, Armstrong [5] defined the poset N C k (W ) of k-divisible non-crossing partitions for the reflexion group W , making N C k (A n−1 ) isomorphic to N C k (n). The construction of Reiner can be easily be generalized to N C k (B n ). However, a combinatorial realization of N C k (D n ) was not known, until the recent paper by Krattenthaler and Müller [13].
We use these combinatorial realizations to define non-crossing partitions for the Coxeter groups B n and D n .
Let us start with the definition of type B n partitions.
Definition 1.7. Let [±n] := {1, 2, ..., n, −1, −2, ..., −n}. A non-crossing partition of type B n is a non- crossing partition π of [±n] such that for each block V of π the block −V = {−x|x ∈ V }
is also a block of π, and there is at most one block, called the zero-block, which satisfies V = −V . We denote by N C B (n) the set of non-crossing partitions of type B n .
There is a circular representation for non-crossing partitions of type B n obtained as follows. Arrange 2n vertices on a circle and label clockwise with the integers 1, 2, ..., n, −1, −2, ..., −n. For each block V of π, draw the convex hull of the vertices whose labels are the integers in B. Then π is a non-crossing partition of type B n if the convex hulls do not intersect. Notice that the circular representation of the non-crossing partitions of type B n correspond to the circular representation partitions in N C(2n) which are centrally symmetric. The formula for the number of partitions of type B n with the sizes of blocks given was obtained by Athanasiadis [3]. Proposition 1.8. Let s + r 1 , r 2 , ...r n be nonnegative integers such that s + r 1 + 2r 2 ... + nr n = n. Then the number of partitions of π in N C B (n) with r i non-zero pairs of blocks of size i and with a zero-block of size 2s, is given by
n! p r (n − m)!(3)
where p r = r 1 !r 2 ! · · · r n ! and r 1 + r 2 ... + r n = m.
We define Type D partitions via their circular representation. The type D circular representation of a partition π ∈ N C B (n) is the drawing obtained as follows. Arrange 2n − 2 vertices labeled with 1, 2, ..., n − 1, −1, −2, ..., −(n − 1) on a circle and put a vertex labeled with ±n at the center. For each block V of π, draw the convex hull of the vertices whose labels are in B. Definition 1.9. A partition π ∈ N C B (n) is a non-crossing partition of type D n if the convex hulls of its type D n circular representation do not intersect in their interiors and if there is a zero-block V then {n, −n} ∈ V . We denote by N C D (n) the set of non-crossing partitions of type D n .
The formula for the number of partitions of type D with the sizes of the blocks given was obtained by Athanasiadis and Reiner [4]. Proposition 1.10. Let s + r 1 , r 2 , ...r n be nonnegative integers such that s + r 1 + 2r 2 ... + nr n = n. Then the number of partitions of π in N C D (n) with r i non-zero pairs of blocks of size i and with a zero-block of size 2s, is given by
(n − 1)! p r (n − m − 1)! if s ≥ 2 (4) 2 (n − 1))! p r ((n − 1) − m)! + r 1 (n − 1)! p r (n − m)! if s = 0 (5)
where p r = r 1 !r 2 ! · · · r n ! and r 1 + r 2 ... + r n = m.
Some basic combinatorial identities
For the convenience of the reader, we end this section by recalling some combinatorial identities that we use often in Sections 2, 3 and 4.
The following two summation lemmas will enable us to use Propositions 1.4, 1.8 and 1.10 to get the number of blocks of size t subject to the restriction of having a fixed number m of blocks.
Lemma 1.11. The following identity holds
r1+r2+···rn=m r1+2r2+···(n)rn=n m! r 1 ! · · · r n ! = n − 1 m − 1 .(6)
Proof. This is proved easily by counting in two ways the number of paths from (0, 0) to (n − 1, m − 1) using the steps (a, b) → (a, b + 1) or (a, b) → (a + 1, b) by observing that
(m + 1)! r 1 ! · · · r n ! = m + 1 r 1 m + 1 − r 1 r 2 · · · m + 1 − (r 1 + · · · r n−1 ) r n .
Lemma 1.12. The following identity holds r1+r2+···rn=m r1+2r2+···nrn=n
(m − 1)!r t r 1 ! · · · r n ! = n − t − 1 m − 2 .(7)
Proof. We make the change of variabler t = r t − 1 andr i = r i for i = t. Then r1+r2+···rn=m r1+2r2+···nrn=n
(m − 1)!r t r 1 ! · · · r n ! = r1+r2+···rn=m−1 r1+2r2+···(n−t)rn−t=n−t (m − 1)! r 1 ! · · ·r n ! = n − t − 1 m − 2 ,
where we used the Lemma 1.11 in the last equality.
Finally, we remind the so-called Chu-Vandermonde's identity which will enable us to remove the restriction of having a number of blocks given.
n − t − 1 m − 2 .(9)
Proof. First we treat the case k = 1. In order to count the number of blocks of size t of a given partition all partitions π with r 1 blocks with size 1, r 2 blocks of size 2, . . . , r n blocks of size n, we need to multiply by r t . So we want to calculate the following sum r1+r2+···rn=m r1+2r2+···nrn=n
n!r t (n + 1 − m)!p r = n m − 1 r1+r2+···rn=m r1+2r2+···nrn=n (m − 1)!r t p r = n m − 1 n − t − 1 m − 2 .
We used Lemma 1.12 in the last equality. This solves the case k = 1.
For the general case we follow the same strategy. In this case we need (r 1 , . . . , r n ) such that r i = 0 if k does not divide i. So the condition r 1 + r 2 + · · · + r n = m is really r k + r 2k + · · · + r nk = m and the condition r 1 + 2r 2 + · · · + nr n = nk is really kr k + 2kr 2k + · · · + (nk)r nk = nk, or equivalently r k + 2r 2k + · · · + nr nk = n. Making the change of variable r ik = s i we get.
r k +r 2k +···+r nk =m kr k +2kr 2k +···+(nk)r nk =nk (nk)!r tk (nk + 1 − m)!r k !r 2k ! . . . r nk ! = s1+s2+···+sn=m s1+2s2+···+nsn=n (nk)!s t (nk + 1 − m)! n i=0 s i ! .(10)
Now, the last sum can be treated exactly as for the case k = 1, yielding the result. This reduction to the case k = 1 will be obviated for types B and D.
Now we can prove Theorem 1.
Theorem 1. The sum of the number of blocks of size tk over all the k-divisible non-crossing partitions of {1, ..., kn} is given by
n(k + 1) − t − 1 nk − 1 .(11)
Proof. We use Proposition 2.1 and sum over all possible number of blocks. Lettingm = m − 1, we get nk m=1 nk m − 1
n − t − 1 m − 2 = nk−1 m=0 nk nk −m n − t − 1 m − 1 .
Now, using the Chu-Vandermonde's identity for s = nk − 1, x = n(k + 1) − t − 1 and y = nk − 1 we obtain the result.
Corollary 2.2. The expected number of blocks of size tk of a non-crossing partition chosen uniformly at random in N C k (n) is given by
(nk + 1) n(k+1)−t−1 nk−1 (k+1)n n .(12)
Moreover, similar to the case k = 1, asymptotically the picture is very simple, about a k k+1 portion of all the blocks have size k, then k k+1 of the remaining blocks are of size 2k, and so on. This is easily seen using (12). Proof. Summing over t, in (11), we easily get the result.
Finally, from the previous corollary one can calculate the expected number of block of k-divisible non-crossing partition.
Corollary 2.5. The expected number of blocks of a k-divisible partition of [kn] chosen uniformly at random is given by kn+1 k+1 . Remark 2.6. 1) Corollary 2.5 was proved by Armstrong [5] for any Coxeter group.
2) When k = 1 there is a nice proof of Corollary 2.5. Recall from Remark 1.2 that the Kreweras complement Kr : N C(n) → N C(n) is a bijection such that |Kr(π)| + |π| = n + 1 . Then summing over all π ∈ N C(n) and dividing by 2 we obtain that the expected value is just n+1 2 . This suggests that there should be a bijective proof of Corollary 2.5. This is done in Section 5.
Number of blocks in k-divisible partitions of type B
In this section we give analogous results for k-divisible non-crossing partitions of type B defined in [5] Definition 3.1. A partition π in N C B (n) is k-divisible if the size of all blocks in π are multiples of k. We denote by N C k B (n) the set of k-divisible non-crossing partitions of type B kn . The number of k-divisible non-crossing partitions of type B was given in [4]. Remark 3.3. The definition of a k-divisible partition of type B n seems quite natural. However, there is small detail on what the size of the zero-block shall be. Notice in the previous definition that for k even and n an integer, in order for a partition to be in N C k B (n), the size of the zero-block is a multiple of 4. Moreover, there exist 2-divisible non-crossing partitions of type B for n odd, and for instance it makes sense to talk about N C 2 B (3/2). We believe that the size of the zero-block of a partition in N C B (n) should be defined as half of the size of this block seen as a partition in N C(n). With this definition, the k-divisible partitions must belong to N C B (kn) for some n, and for instance, N C 2 B (3/2) would be empty. We will not continue with this discussion but rather refer the reader to Section 4.5 of [5].
Of course, our counting results depend on the definition of the size of the zero-block. To avoid this problem we count the number of non-zero block and the number of zero-blocks separately. Also, to avoid confusion we talk about the size of a pair instead of talking about the size of a block. We proceed as for the case of classical non-crossing partitions. We start by calculating the expected number of non-zero blocks of a given size t, subject to the restriction of having m non-zero blocks. Before this, we need also to fix the size of the zero-blocks as we state in the following lemma.
Lemma 3.5. The number of pairs of non-zero blocks of size t of all non-crossing partitions in N C(B n ) with m pairs of non-zero blocks and a zero-block of size 2s is given by
n n − 1 m − 1 n − t − s − 1 m − 2 .(13)
Proof. We use (3) and Lemma 1.12 in the same way as for the classical non-crossing partitions.
n!r t (n − m)!p r = n n − 1 m − 1 r1+r2+···rn=m r1+2r2+···nrn=n−s (m − 1)!r t p r = n n − 1 m − 1 n − t − s − 1 m − 2 .
Proposition 3.6. The sum of non-zero pairs of blocks{−V, V } of size t over all the non-crossing partitions in N C B (n) is given by
n 2n − t − 1 n − 1 .(14)
Proof. First, we use Lemma 3.5 and sum over all possible sizes s of the zero-block in (13). Thus, the number of non-zero blocks of size t of all the non-crossing partitions in N C B (n) with m pairs of non-zero blocks is given by
n n − 1 m − 1 n − t m − 1 .(15)
Next, we sum over m in 15, using Chu-Vandermonde's identity for x = n − 1, y = n − t and s = n − 1.
Now, we count the number of zero-blocks of size 2t over all the partitions.
Proposition 3.7. The number of non-crossing partitions in N C B (n) with a zero-block of size 2t is given by
2n − t − 1 n − 1 .(16)
Proof. For counting the number of zero-blocks of a certain size we first we calculate the number of partitions with a zero-block of size 2t and m non-zero blocks using 3 and Lemma 1.12, yielding n m
n − t − 1 m − 1 .
Summing over m by means of Chu-Vandermonde's identity we get the desired result.
It is straightforward to pass to k-divisible partitions, and obtain Theorem 2.
(k + 1) − t − 1 nk − 1 .(17)
2) The number of the non-crossing partitions in N C k B (n) with a zero-block of size 2t then
n(k + 1) − t − 1 nk − 1 .(18)
Corollary 3.8. The expected number of unordered pairs {−V, V } of size tk of a non-crossing partition chosen uniformly at random in N C k B (n) is given by
(nk + 1) n(k+1)−t−1 nk−1 (k+1)n n .(19)
The fact that formulas (12) and (19) coincide may be explained by the remarkable observation by Biane, Goodman and Nica [9] that there is an (n + 1) − to − 1 map from N C B (n) to N C(n). More precisely, for any subset X ⊂ Z, let Abs(X) := {|x| : x ∈ X} and for a partition π ∈ N C(n), denote by Abs(π) the set {Abs(V ) : V ∈ π}. Proposition 3.9. Let n be a positive integer. Then the map Abs :
N C B (n) → N C(n) is a (n+1)−to−1 map from N C B (n) onto N C(n).
Finally, we consider the asymptotics when n → ∞. Since when n is large the number of zero-blocks is negligible, independent of the choice of how to count them, we obtain the same behavior as for the classical partitions. ..e i are arranged on the outer circle in the clockwise order and e s+1 , e s+2 , ..., e t arranged on the inner circle in counterclockwise order for some i. Then we draw curves, which lie inside of the annulus, connecting e i and e i+1 for all i = 1, 2, ..t, where e t+1 = e 1 . If we can draw the curves for all the blocks of π such that they do not intersect, then we call π non-crossing partition on the (2k(n − 1), 2k) annulus. We say that a block on the outer circle is visible if there is no block between this block and the inner circle.
Following Kim [14] the combinatorial realization of k-divisible non-crossing partitions of type D is defined as follows.
Definition 4.1. A k-divisible non-crossing partition π of type D n is a non-crossing partition π on the (2k(n − 1), 2k) annulus satisfying the following conditions:
1. If V ∈ π, then −V ∈ π.
2. For each block V ∈ π with t elements, if e 1 , e 2 , ..., e t are the elements of V such that e 1 , e 2 , ..., e s are arranged on the outer circle in clockwise order and e s+1 , e s+2 , ..., e t are arranged on the inner circle in counterclockwise order, then |e i+1 | ≡ |e i | + 1 mod k for all i = 1, 2, ..t, where e t+1 = e 1 .
3. If there is a zero-block V of π, then V contains all the integers on the inner circle and at least two integers on the outer circle.
4. If π has no block with elements in both the inner circle and the outer circle, then for any inner block V i and any visible block V j and the partitionπ with V i ∪ V j instead of V i and V j satisfies Condition 2.
We denote by N C k D (n) the set of k-divisible non-crossing partitions of type D n Remark 4.2. As pointed out in [14], Condition 4 was overlooked in [13]. Also, Condition 4 is stated differently in [14]. However, for our purposes they are equivalent.
The number of k-divisible non-crossing partitions of type D was given in [5]. n + (k + 1)(n − 1) + 1 n .
We consider two cases which we call type D1 and type D2 depending on whether there is a zero-block or not. Type D1 Definition 4.4. A k-divisible non-crossing partition of type D n is of type D1 n if it has a zero-block We denote by N C k D1 (n) the set of k-divisible non-crossing partitions of type D1 n . The number of k-divisible non-crossing partitions of type D1 was calculated in [13].
Proposition 4.5. Let n be a positive integer, s ≥ 2 and r 1 , r 2 , ...r n ∈ N ∪ {0} be such that s + r 1 + 2r 2 ... + nr n = n. Then the number of partitions of π in N C k D1 (n) with r i pairs {−V, V } of non-zero blocks size i a zero-block of size 2s is given by
(k(n − 1))! (k(n − 1) − m)!p r .(20)
Lemma 4.6. The number of pairs of non-zero blocks of size tk of all non-crossing partitions in N C k D1 (n) with 2m non-zero blocks and a zero-block of size 2s is
(k(n − 1)) k(n − 1) − 1 m − 1 n − t − s − 1 m − 2 .(21)
Proof. In the same way as for the non-crossing partitions of type B, we use (20) and Lemma 1.12.
r1+r2+···rn=m r1+2r2+···nrn=n−s (k(n − 1))!r t (n − m)!p r = (k(n − 1)) k(n − 1) − 1 m − 1 r1+r2+···rn=m r1+2r2+···nrn=n−s (m − 1)!r t p r = n k(n − 1) − 1 m − 1 n − t − s − 1 m − 2 .
Proposition 4.7. The number of non-zero blocks of size tk of all the non-crossing partitions in N C k D1 (n) then
k(n − 1) (k + 1)(n − 1) − t − 2 k(n − 1) − 1 .(22)
Proof. We use Lemma 4.6 and sum over all possible sizes s of the zero-block in (21). The only difference with type B is that now s ≥ 2. Thus, the number of non-zero blocks of size tk of all the non-crossing partitions in N C k D1 (n) with m non-zero block is given by
k(n − 1) k(n − 1) − 1 m − 1 n − t − 2 m − 1 .
Summing over m, using Chu-Vandermonde's identity for x = k(n−1)−1, y = n−t−2 and s = k(n−1)−1, we get the result.
In a similar way as for type B, we calculate the number of zero-blocks of a given size over all the non-crossing partitions of type D.
Proposition 4.8. The number of non-crossing partitions in N C k D (n) with a zero-block of size 2tk is given by
(k + 1)(n − 1) − t k(n − 1) − 1 . (23)
Proof. This is analogous to the case of N C k B (n).
Type D2
Definition 4.9. A k-divisible non-crossing partition of type D n is of type D2 n if it has no zero-block. We denote by N C k D2 (n) the set of k-divisible non-crossing partitions of type D2 n .
The number of k-divisible non-crossing partitions of type D2 was calculated in [13].
Proposition 4.10. Let n be a positive intege, r 1 , r 2 , ...r n ∈ N ∪ {0} be such that r 1 + 2r 2 ... + nr n = n. Then the number of partitions of π in N C k (D2) with r i non-zero pairs of size ik block is given by
2 (k(n − 1))! (k(n − 1) − m)!p r + r 1 (k(n − 1)! (k(n − 1) − m + 1)!p r .(24)
where p r = r 1 !r 2 ! · · · r n ! and r 1 + r 2 ... + r n = m.
Proposition 4.11. The number of blocks of size tk of all non-crossing partitions in N C k D2 (n) is given by
k(n − 1) (k + 1)(n − 1) − t − 2 k(n − 1) − 2 + 2 (k + 1)(n − 1) − t − 1 k(n − 1) − 2 (25)
for t > 1 and by
k(n − 1) (k + 1)(n − 1) − 3 k(n − 1) − 2 + 2 (k + 1)(n − 1) − 2 k(n − 1) − 2 + (k + 1)(n − 1) − 1 k(n − 1) − 1(26)
for t = 1.
Proof. Using (24) we count the number of blocks of size tk over all non-crossing partitions in N C k D (n) with m non-zero pairs and then sum over m. This is 2 r1+···+rn=m r1+···+nrn=n r t (k(n − 1))! (k(n − 1) − m)!p r + r1+···+rn=m r1+···+nrn=n r t r 1 (k(n − 1))! (k(n − 1) − m + 1)!p r .
The first part of the sum can be treated in the same way as the classical case using Lemma 1.12.
2 r1+···+rn=m r1+···+nrn=n
r t (k(n − 1))! p r (k(n − 1) − m)! = 2(k(n − 1)) k(n − 1) − 1 m − 1 r1+···+rn=m r1+···+nrn=n r t (m − 1)! p r = 2(k(n − 1)) k(n − 1) − 1 m − 1 n − t − 1 m − 2 .
Summing over m, by means of the Chu-Vandermonde's formula, we get
2k(n − 1) (k + 1)(n − 1) − t − 1 k(n − 1) − 2 .(27)
For the second part of the sum we divide in two cases, t = 1 and t > 1. If t > 1, then making the change of variabler 1 = r 1 − 1, andr i = r i , for i = 1, we get r1+···+rn=m r1+···+nrn=n
r t r 1 (k(n − 1))! (k(n − 1) − m + 1)!p r = r1+···+rn=m−1 r1+···+nrn=n−1r t (k(n − 1))! (k(n − 1) − m + 1)! n i=1r i ! = (k(n − 1)) k(n − 1) − 1 m − 2 n − t − 2 m − 3 .
Summing over m we get
(k(n − 1)) (k + 1)(n − 1) − t − 2 k(n − 1) − 2 .(28)
Thus, combining (27) and (28) we get the result for t > 1. Now, for t = 1, the sum r1+···+rn=m r1+···+nrn=n r 1 r 1 (k(n − 1))! (k(n − 1) − m + 1)!p r can be split as r1+···+rn=m r1+···+nrn=n
r 1 (r 1 − 1)(k(n − 1))! (k(n − 1) − m + 1)!p r + r1+···+rn=m r1+···+nrn=n r 1 (k(n − 1))! (k(n − 1) − m + 1)!p r .
Each of the terms can be treated as before, yielding
k(n − 1) k(n − 1) − 1 m − 2 n − 3 m − 3 + k(n − 1) m − 1 n − 2 m − 2 .
Again, summing over m we get
k(n − 1) (k + 1)(n − 1) − 3 k(n − 1) − 2 + (k + 1)(n − 1) − 1 k(n − 1) − 2 .(29)
Combining (27) and (29) we get the result for t = 1.
Now putting together Propositions 4.8, 4.7 and 4.11 we get, after some small simplifications, Theorem 3.
N C k D (n) is (k(n − 1) + 1) (k + 1)(n − 1) − t k(n − 1) − 1 + k(n − 1) (k + 1)(n − 1) − t − 1 k(n − 1) − 2 .
Notice that even though the formulas of Proposition 4.11 look very different, for t = 1 and for t > 1, Theorem 3 gives the same formula for all t.
Finally, using the Theorem 3 and Proposition 4.3 we get the asymptotics when n → ∞. Surprisingly, we get the same behavior for the cases of N C k (n) and N C k B (n). Corollary 4.12. When n → ∞ the expected number of pairs of size tk of a non-crossing partition chosen uniformly at random in N C k (D n ) is asymptotically nk (k+1) t+1 .
The bijection
In this section we give a bijective proof of the fact that N C k (n) = N C k+1 (n). From this bijection we derive Corollary 2.5.
Lemma 5.1. For each n and each k let f : N C k+1 (n) → N C k (n) be the map induced by the identification of the pairs {k + 1, k + 2}, {2(k + 1), 2(k + 1) + 1}, . . . , {n(k + 1), 1}. Then f is a bijection.
Proof. First, we see that the image of this map is in N C k (n). So, let π be a (k + 1)-equal partition.
(i) Every block has one element on each congruence mod k +1. Indeed, because of the characterization of non-crossing partitions on Remark 1.3, there is at least one interval, which has of course this property. Removing this interval does not affect the congruence in the elements of other blocks. So by induction on n every block has one element of each congruence mod k + 1.
(ii) Note that for each two elements identified we reduce 1 point. So suppose that m blocks (of size k) are identified in this bijection to form a big block V . Then the number of vertices in this big block equals m(k + 1) − #( identified vertices)/2. Now, by (i), there are exactly two elements in each block to be identified with another element, that is 2m . So |V | = m(k + 1) − #(identified vertices)/2 = m(k + 1) − (2m)/2 = mk.
this proves that f (π) ∈ N C k (n). Now, it is not very hard to see that by splitting the points of π ∈ N C k (n) we get a unique inverse f −1 (π) ∈ N C k+1 (n).
Theorem 3 .
3The sum of the number of pairs of blocks {−V, V } of size tk over all the non-crossing partitions in N C k D (n) is given by
Figure 1 :
1Crossing and Non-Crossing Partitions Fig. 1 shows the non-crossing partition {{1, 2, 5, 9}, {3, 4}, {6}, {7, 8}, {10, 11, 12}} of the set [12], and the crossing partition {{1, 4, 7}, {2, 9}, {3, 11, 12}, {5, 6, 8, 10}} of
Figure 2 :
2Type B partitions
Figure 3 :
3Type D partitions
of blocks in k-divisible non-crossing partitions.First we calculate the expected number of blocks of a given size t, subject to the restriction of having m blocks from which the main result will follow.Proposition 2.1. The sum of the number of blocks of size tk of all non-crossing partitions in N C k (n) with m blocks nk m − 1
Corollary 2 . 3 .
23When n → ∞ the expected number of blocks of size tk of a non-crossing partition chosen uniformly at random in N C k (n) is asymptoticallynk (k+1) t+1 .The following is a direct consequence of the last proposition.
Corollary 2 . 4 .
24The sum of the number of blocks of all the k-divisible non-crossing partitions in N C k (n) is n(k + 1) − 1 nk .
Proposition 3. 2 .
2Let N C k B (n) be the set of non-crossing partitions of k-divisible non-crossing partitions of type B kn Then
Definition 3 . 4 .
34Let π ∈ N C B (n) and V a block of π. The size of the unordered pair {V, −V } is 1 2 |V ∪ (−V )|, where for a subset A ⊂ Z, |A| denotes its cardinality.
Theorem 2. 1 )
1The sum of the non-zero pairs of blocks {−V, V } of size tk of over all non-crossing partitions in N C k B (n) is given nk n
Corollary 3 . 10 .
310When n → ∞ the expected number of pairs of blocks of size t of a non-crossing partition chosen uniformly at random in N C k (B n ) is asymptotically nk (k+1) t+1 . 4 Number of blocks in k-divisible partitions of type D Krattenthaler and Müller [13] solved the problem of finding a combinatorial realization of k-divisible non-crossing partitions of type D. Let π be a partitions of {1, 2, ...kn, −1, −2, ..., −kn}. We represent π in an annulus with the integers 1, 2, ..., k(n − 1), −1, ..., −(kn − k) on the outer circle in clockwise order and the integers k(n − 1) + 1, ...kn, −(k(n − 1) + 1), ..., −kn in the inner circle in counterclockwise order. A block V = {e 1 , e 2 , ..., e s } with e 1 , e 2 , .
Proposition 4 . 3 .
43The number of k-divisible non-crossing partitions of type D n is given by (k + 1)(n − 1)
Figure 4 :
42-divisible partitions of type D1 and type D2
Theorem 3 .
3The sum of the number of pairs of blocks {V, −V } of size tk over all the partitions in
Figure 5 :
5Bijections between 3-equal and 2-divisible non-crossing partitionsNow we give a proof of Corollary 2.5.
Figure 6 :
6A 3-equal and its Kreweras complement divided mod 3.
AcknowledgementsThe author is grateful to Janosch Ortmann for asking some of the questions treated in this paper and would like to thank Pablo Soberón for comments that helped improving this paper. He is indebted to Professor Christian Krattenthaler for bringing to his attention non-crossing partitions of type B and D and for making him available the preprint[12].Proof. For each n and each k and each 0 < i ≤ k+1 let f i : N C k+1 (n) → N C k (n) the map induced by the identification of the pairs {k+1+i, k+1+i+1}, . . . {2(k+1)+i, 2(k+1)+i+1}, ...{n(k+1)+i, n(k+1)+i+1} (we consider elements mod nk). Then by the proof of the previous lemma, each f i is a bijection. So, let π be a fixed (k + 1)-equal partition. Considering all the bijections f i on this fixed partition, we see that every point j is identified twice (one with f j−1 and one with f j ). So for each partition π in N C k+1 (n), the collection (f i (π)) k i=1 consists of k + 1 partitions in N C k (n) whose number of blocks add kn + 1. Remark 5.2. Note that for a k-divisible partition on [2kn] points, the property of being centrally symmetric is preserved under the bijections f i (see e. g.Fig.5), and then the arguments given here also work for the partitions of type B. We expect that a similar argument works for type D.In the following example we want to illustrate how the bijection given by Lemma 5.1 allows us to count k-divisible partitions with some restrictions by counting the preimage under f . Example 5.3. Let N C k 1→2 (n) be the set of k-divisible non-crossing partitions of [kn] such that 1 and 2 are in the same block. It is clear that π ∈ N C k 1→2 (n) if and only if f −1 (π) satisfies that 1 and 2 are in the same block. Now, counting the (k + 1)-equal non-crossing partitions of [(k + 1)n] such that 1 and 2 are in the same blocks is the same as counting non-crossing partitions of [(k + 1)n − 1] with n − 1 blocks of size k + 1 and 1 block of size k containing the element 1, since 1 and 2 can be identified. From Proposition 1.4, the size of this set is easily seen to bewhere the first factor of the LHS is the probability that the block of size k contains the element 1.Let us finally mention that the bijections f i are closely related to the Kreweras complement of a (k + 1)-equal non-crossing partitions, which was considered in[2]. Indeed Kr(π) can be divided in a canonical way into k + 1 partitions of [n], π 1 , ..., π k+1 , such that |π i | = |f i (π)|.Fig. 5shows the bijections f 1 , f 2 and f 3 for k = 3, n = 4 and π = {{1, 8, 9}, {2, 6, 7}, {3, 4, 5}, {9, 10, 11}}, whileFig. 6shows the same partition asFig. 5with its Kreweras complement divided into the partitions π 1 , π 2 and π 3 .
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| [] |
[
"Federated Learning with Domain Generalization",
"Federated Learning with Domain Generalization"
] | [
"Liling Zhang [email protected] \nCollege of Computer Science\nChongqing University\n\n",
"Xinyu Lei \nDept. of Computer Science and Engineering\nMichigan State University\n\n",
"Yichun Shi [email protected] \nDept. of Computer Science and Engineering\nMichigan State University\n\n",
"Hongyu Huang [email protected] \nCollege of Computer Science\nChongqing University\n\n",
"Chao Chen [email protected] \nCollege of Computer Science\nChongqing University\n\n"
] | [
"College of Computer Science\nChongqing University\n",
"Dept. of Computer Science and Engineering\nMichigan State University\n",
"Dept. of Computer Science and Engineering\nMichigan State University\n",
"College of Computer Science\nChongqing University\n",
"College of Computer Science\nChongqing University\n"
] | [] | Federated Learning (FL) enables a group of clients to jointly train a machine learning model with the help of a centralized server. Clients do not need to submit their local data to the server during training, and hence the local training data of clients is protected. In FL, distributed clients collect their local data independently, so the dataset of each client may naturally form a distinct source domain. In practice, the model trained over multiple source domains may have poor generalization performance on unseen target domains. To address this issue, we propose FedADG to equip federated learning with domain generalization capability. FedADG employs the federated adversarial learning approach to measure and align the distributions among different source domains via matching each distribution to a reference distribution. The reference distribution is adaptively generated (by accommodating all source domains) to minimize the domain shift distance during alignment. In FedADG, the alignment is fine-grained since each class is aligned independently. In this way, the learned feature representation is supposed to be universal, so it can generalize well on the unseen domains. Intensive experiments on various datasets demonstrate that FedADG has comparable performance with the state-of-the-art. | null | [
"https://export.arxiv.org/pdf/2111.10487v2.pdf"
] | 244,478,326 | 2111.10487 | 45cc5f6167cc8b517ec1f8d58bd2cb36eccc46af |
Federated Learning with Domain Generalization
Liling Zhang [email protected]
College of Computer Science
Chongqing University
Xinyu Lei
Dept. of Computer Science and Engineering
Michigan State University
Yichun Shi [email protected]
Dept. of Computer Science and Engineering
Michigan State University
Hongyu Huang [email protected]
College of Computer Science
Chongqing University
Chao Chen [email protected]
College of Computer Science
Chongqing University
Federated Learning with Domain Generalization
Federated Learning (FL) enables a group of clients to jointly train a machine learning model with the help of a centralized server. Clients do not need to submit their local data to the server during training, and hence the local training data of clients is protected. In FL, distributed clients collect their local data independently, so the dataset of each client may naturally form a distinct source domain. In practice, the model trained over multiple source domains may have poor generalization performance on unseen target domains. To address this issue, we propose FedADG to equip federated learning with domain generalization capability. FedADG employs the federated adversarial learning approach to measure and align the distributions among different source domains via matching each distribution to a reference distribution. The reference distribution is adaptively generated (by accommodating all source domains) to minimize the domain shift distance during alignment. In FedADG, the alignment is fine-grained since each class is aligned independently. In this way, the learned feature representation is supposed to be universal, so it can generalize well on the unseen domains. Intensive experiments on various datasets demonstrate that FedADG has comparable performance with the state-of-the-art.
Introduction
In deep learning, a good model should be trained over large-scale datasets to ensure its high performance. These large-scale datasets are often collected by multiple distributed clients. To train the model, a straightforward way is to let the clients upload their local data to a centralized server for training. However, some clients' local data (e.g., biometric health records, financial records, location information) maybe highly privacy-sensitive and they are reluctant to share with any other entities. Fortunately, the proposal Federated Learning (FL) [25] provides a privacy-preserving mechanism that enables a centralized server to train the model without requiring clients Each client's data forms a distinct source domain. We aim to develop a solution to learn a classifier (on multiple source domains) that can be used for "unseen domain" with good performance.
to share their private data. In one iteration of FL, a server sends the global model to all clients. Then, each client trains the global model using the local data. Next, each client sends the model update to the server, which is possible for model update aggregation and new global model generation. After multiple rounds of iteration, the model can be well trained.
In FL, since distributed clients collect their local data independently, each client's dataset may naturally form a distinct domain (a domain is defined as a set of labeled training data that are sampled from a specific distribution [4,37]). For example, Fig. 1 shows a FL task in which clients need to use their collected bird images in training. Each client usually collects different bird species (using different cameras and different shot angles), so each client's collected dataset forms a distinct domain. Here, the domain formed by one client's dataset is called a source domain, so there are multiple source domains in FL.
Most previous FL studies assume that the test dataset is a subset of client dataset. There is a lack of studies for another common usage scenario in which the data of the target dataset (i.e., test dataset) is absent from FL train-ing process. It is required to build a model that has high performance when testing over the related but unseen target dataset (note that the target dataset forms the target domain). However, the FL-trained model may have poor performance on target domains due to the discrepancies between source domains and target domains.
The above issue can be addressed by Domain Generalization (DG) [4,13,27] technique, but the previous techniques of domain generalization cannot be directly applied to FL setting. Domain generalization aims to train a machine learning model from one or several different source domains while ensuring the trained model can be generalized well on target domains. Most conventional solutions finish the domain generalization task in a centralized manner. That is, a centralized server (with access to all source domain data) is responsible for the domain generalization task. For example, Jigsaw puzzle based Generalization (JiGen) [5] requires data decomposed from multi-source domains to be mixed to train a classifier. Besides, MixStyle [43] needs to mix features from different source instances to synthesize new domains. However, accessing to sources domains by the centralized server is prohibitive in FL to meet the security requirements.
Therefore, these conventional techniques cannot be easily applied to domain generalization in FL. There are two proposed schemes (i.e., COPA [40] and FedDG [23]) that study domain generalization problems in FL. COPA is the abbreviation of Collaborative OPtimization and Aggregation, while FedDG is the abbreviation of Federated Domain Generalization. Both of them suffer from some limitations. For COPA, it requires each IoT device to share its local data size. Moreover, it leaks the global information (i.e., domain variation) to each device for batch normalization (BN) layer parameters tuning. In a nutshell, COPA sacrifices security for domain generalization. For FedDG, it allows each device's local data information (i.e., image amplitude spectrum) to be shared with other entities. However, the shared image amplitude spectrum contains class-relevant information, which can be used for training a classifier [33]. It leaks sensitive information about the device's local data. In summary, both COPA and FedDG sacrifice security for domain generalization. Different from the two schemes, our solution aims to achieve domain generalization without the above information leakage.
In this paper, we propose Federated Adversarial Domain Generalization (FedADG) scheme to address the domain generalization problem in FL. FedADG design has two key insights as described below. First, FedADG exploits the idea to learn the domain-invariant feature representation by aligning each distribution of source domain data to a reference distribution in a distributed manner. In the alignment, we employ Adversarial Learning Network (ALN) to measure the distance between distributions in FL setting. Furthermore, we propose the Federated ALN (FedALN) technique to train ALN in FL setting. In this way, FedADG can learn the domain-invariant features while eliminating the requirement for a centralized server to access clients' local data. Second, FedADG uses the idea to adaptively learn a dynamic distribution (by accommodating to all source domains) as the reference distribution. This approach can minimize the domain shift distance during alignment.
Compared with using a pre-selected fix reference distribution, our approach reduces the distortion of extracted feature representation. Therefore, the key information of the original source domain data can be largely preserved, resulting in the high generalization performance of FedADG. Besides, FedADG takes the label information (coded as a one-hot vector) as input during the alignment process. Hence, FedADG supports class-wise alignment, which can further improve its performance on target domains. Furthermore, compared with using the fixed reference distribution, using the dynamically generated reference distribution approach can get more discriminative features after alignment. The discriminative features are helpful to improve the performance of FedADG.
The high performance of FedADG can be explained via visualization, so FedADG gains some explainability to some extent. The more explainability a FedADG scheme has, the deeper understanding that users achieve. An explainable machine learning model can help users in two folds. First, it can help users to tune model parameters efficiently, making it easier for further model optimization. Second, it is more trustworthy to be used in sensitive and critical areas, where its value can be enormous. Note that most previous domain generalization solutions lack explainability.
We summarize our contributions as follows:
• We propose FedALN to learn the domain-invariant features in FL while eliminating the requirement for a centralized server to access clients' local data.
• We propose FedADG which employs the adaptively generated reference distribution and class-wise alignment technique in FedADG to ensure its high performance.
• The explainability of FedADG's high performance brings in two immediate benefits. First, it is easier for users to tune parameters and have further model optimization. Second, it is more trustworthy to be used in practice.
The remainder of the paper is organized as follows. Sec. 2 introduces some preliminary knowledge. Sec. 3 introduces some related works of this paper. Sec. 4 presents the FedADG scheme and its training process in detail. Sec. 5 analyzes the principle of FedADG. Sec. 6 demonstrates the experimental results, and the efficiency and effects of FedADG scheme are analyzed. Sec. 8 concludes this paper.
Preliminaries
Federated Learning
Federated learning [25,26] is a distributed machine learning method that learns a global model across multiple clients without revealing the device's local dataset. Fig.1 illustrates the framework of FL, which includes K clients and a centralized server. Learning a global model on FL requires multiple iterations of training on both the server and the client. In one iteration of FL, the server sends the initialized global model to all clients. Then, each client trains the global model on their local dataset. Next, each client's model updates are sent to the centralized server and used for aggregation to generate a new global model. After multiple rounds of iterations, the global deep learning model can be well trained.
Generative Adversarial Network
Generative Adversarial Network (GAN) is first proposed in [14]. GAN endows the generative model with the ability to generate given distribution outputs via an adversarial procedure. GAN has two components: a generator (G) and a discriminator (D). For generator model, the generator takes random noise samples z from a given prior distribution as input. Then, the generator model is trained to output fake samples that are similar to the real training samples. For discriminator model, the discriminator takes the samples output by the generator model and the real training samples as input. Next, the discriminator learns to distinguish whether an input is fake (generated) or true (from real training samples). The generator and discriminator perform multiple rounds of adversarial training. The training objective can be expressed as
arg min G max D V (D, G) =E x∼pr(x) [log D(x)]+ E z∼pg(z) [log (1 − D(G(z)))],(1)
where p r (x) and p g (z) denote the distribution of the real training sample and the prior distribution used in generator, respectively. Compared with the classic GAN, FedADG introduces several new components to achieve our purposes in the FL setting.
Related Work
Federated Learning. Federated learning [25,26] is a decentralized approach that leaves training data distributed on multiple clients and learns a global model by aggregating the locally-uploaded parameters on server. In FL, clients do not need to share their local data to any other entities, so the local data is protected. To improve the performance of the FL-trained model, researchers have proposed many optimized schemes, such as Federated learning with the Proximal term (FedProx) [22], Federated Normalized averaging algorithm (FedNova) [38], and MOdel-cONtrastive learning (MOON) [21]. Most previous FL studies assume that the test dataset is a subset of client dataset. Different from the previous papers, this paper mainly focuses on enabling FL to train a model that has good performance on unseen target domains. Domain Generalization. The requirement of learning a model from multiple seen source domains for unseen domains motivates the research of domain generalization. Most previous solutions [1,5,19] consider the domain generalization problem in a centralized setting. In these papers, a centralized server has access to data from all source domains and it is responsible for training a machine learning model that has domain generalization capability. However, these solutions expose the source domain data to the server. This is not allowed in FL, so these solutions cannot be directly used in FL setting. To sum up, we summarize the comparison between the previous solutions and FedADG, as shown in Tab. 1. Domain Adaptation. A similar concept is Unsupervised Domain Adaptation (UDA), which aims to learn an ML model from one or multi-source domain(s) that performs well on a different (but related) target domain [3]. UDA techniques assume the availability of unlabeled target domain data. Even if Peng et al. [28] propose a privacypreserving approach, but its test dataset participates in the training process, which is prohibited in domain generalization. Therefore, UDA techniques cannot be directly used in this paper.
Problem Statement and FedADG Scheme
In this section, we first have the problem statement. Then, we introduce the FedADG scheme. For ease of reading, we summarize the frequently used notations in Tab. 2.
Problem Statement
In this paper, we aim to develop a solution to learning a ML model with non-shared data from multi-source domains. Suppose that there are K source domains S = {S k } K k=1 , and a sample-label pair from source domain k is denoted by (x ki , y ki ), where x ki ∈ R d×1 and y ki ∈ R m×1 . The ML model trained over the K source domains should have high performance on the unseen target domains. Besides, the proposed solution should follow the same security principle as the traditional FL: only model parameters (e.g., updated gradients) can be sent to the server, and no information about local data can be shared directly. components, which are described as follows. Feature Extractor. Feature extractor can extract latent features from the raw data for each client. Besides, the extracted features can be applied to the classification task. Discriminator. Given features extracted from raw data (from a source domain) and features generated by distribution generator, the discriminator is used to distinguish the extracted features and the generated features. During training, the discriminator gains its ability to distinguish the above two types of features. Besides, a Random Projection (RP) layer is pre-pended to the discriminator. The RP layer is used to stabilize the training of ALN. Distribution Generator. On input random noise samples and one-hot vector (used for label encoding), distribution generator generates features, which follow a certain distribution (i.e., the reference distribution). Note that the above three components constitute the Adversarial Learning Network (ALN). Classifier. Given features as the input, the classifier outputs the predicted label.
FedADG Components
FedADG Loss Function
FedADG loss function consists of adversarial loss function and classification loss function. Adversarial Loss Function. The adversarial loss function includes three loss functions: L adv d , L adv f , and L adv g . They are elaborated below. L adv d . The loss function L adv d is used to update the parameters in discriminator. During adversarial learning, features extracted by feature extractor F (·) are regarded as negative samples, while features generated by distribution generator G(·) are regarded as positive samples. Given the two types of features with the same one-hot vector (encoding a label y), the discriminator D(·) outputs the probability that they are positive samples. Besides, the output of the D(·) is used to calculate L adv d to measure the difference between the two types of samples. L adv d is defined as
L adv d = − (E x∼p(h) [(1 − D(h|y)) 2 ] + E z∼p(h ) [D(h |y) 2 ]),(2)
where h = F (x) and p(h) is the F (·) generated distribution over input data x. Likewise, h = G(z) and p(h ) is the G(·) generated distribution over input data z. The random noise z is drawn from [0, 1) uniformly.
p(h) feature distribution F (·) feature extractor G(·) distribution generator D(·) discriminator C(·) classifier
Note that a random projection layer is pre-pended to the discriminator (as shown in Fig. 2). The random projection function is used to linearly transform data from d 1 dimensions to d 2 dimensions [11], where d 1 > d 2 . It can be represented as d 1 ×d 2 matrix R. Let an n×d 1 matrix h represent a d 1 -dimensional data set. Each row in h represents d 1 -dimensional data and n is the number of data. Let h denote the projected data set and we haveh = h × R. In this work, the random projection layer helps stabilize the ALN training as well as reduce computation.
L adv f . For L adv f , it is used by discriminator to evaluate the possibility that h is the positive sample. In adversarial learning, given a fixed D(·), L adv f is used to update the parameters in the feature extractor. In the process of training feature extractor, the negative samples h extracted by F (·) are used to deceive the discriminator (in a successful deception, discriminator treats h as positive samples).
Thus, L adv f is given by
L adv f = E x∼p(h) [(1 − D(h|y)) 2 ].(3)
L adv g . For L adv g , it is used by discriminator to evaluate the possibility that h is the positive sample. In adversarial learning, given a fixed D(·), L adv g is used to update the parameters in the distribution generator. Specifically, L adv g is given by
L adv g = E z∼p(h ) [(1 − D(h |y)) 2 ].(4)
In the definitions of L adv d , L adv f , and L adv g , we borrow the idea from [24] to use the least-squared term instead of the log-likelihood term. This approach helps to address the non-convergence problem during training. Classification Loss Function. Let L err be the loss on the classifier's predictions. It is used to measure the error between the label C(h) (h = F (x)) predicted by the classifier C(·) and the real label y of the data. L err is the standard cross-entropy loss [30] in FedADG. During training, L err controls the update of both feature extractor and classifier. In order to prevent overfitting, label smoothing regularization [34] is adopted in computing L err to reduce the weight of the positive samples in L err . Complete Loss Function. The complete loss function of FedADG is
L F edADG = L adv d + L adv g + λ 0 L adv f + λ 1 L err , (5)
where L adv d , L adv g , and L adv f are given in Eq. (2)-(4), respectively. Both λ 0 and λ 1 are adjustable weight hyperparameters. During training, the objective of FedADG is to minimize L F edADG . Figure 2: Illustration of the proposed FedADG scheme. FedADG first aligns each distribution of source domain data to the generated reference distribution through ALN on each client. Meanwhile, via minimizing the loss function of distribution generator, the generated reference distribution is moving close to the "center" of all source domain distributions. Note that the above alignment process is performed in a class-wise manner by using a one-hot vector (encoding the data label). Besides, FedADG uses the loss function of a classifier to assist the learning of the feature extractor. After training, in FedALN, the reference distribution and all the distributions of source domains data are aligned to learn a domain-invariant representation for domain generalization.
FedADG Training Process
The detailed FedADG training is presented in Algorithm 1. The FedADG training process includes two phases: server execution and client update. Server Execution Phase. The server is used to aggregate the model parameters uploaded by the clients. To begin the training, in Step s1, the server initializes the parameters w = {w f , w c , w g } of three network components (i.e., feature extractor F (·), classifier C(·), and distribution generator G(·)) and distributes them to all clients. During the training process, in Step s2 and Step s3, the server receives and aggregates model parameters from all clients to obtain new parameters. Then, the server sends the aggregated parameters to the clients. After multiple rounds of server-client interaction, the model can be welltrained. Note that the ML model (that is constructed as the series connection of feature extractor and classifier) is applied to target domains. Client Update Phase. In the training process, the client uses the local discriminator and receives the parameters w of other components from the server to train on the local data. Specifically, as shown in Step c2 and Step c3, L err is used to control the training of classifier C(·) and feature extractor F (·). Then, the parameters of F (·) and C(·) are updated to minimize the loss λ 0 L adv f +λ 1 L err . The parameters of the discriminator D(·) are updated to minimize the loss L adv d . In Step c8, the output of D(·) for the given positive samples with y is used to update the parameters of G(·) to minimize the loss L adv g . After the local training is completed, the client uploads the parameters w of F (·), C(·), and G(·) to the server.
FedADG Analysis
In this section, we first analyze how to learn domaininvariant features in FedADG. Then, we explain how FedADG achieves high performance on target domains.
Algorithm 1 FedADG Training Algorithm
Input: source domains S = {S k |k = 1, ..., K}, onehot vector y, model parameters of F (·), C(·), and G(·) w = {w f , w c , w g }, parameter of D(·) w d , etc.
Output: Feature extractor F (·) and Classifier C(·)
Server executes:
Step s1: Initialize w 1 for round t = 1, 2, . . . , T do for each client k = 1, 2, . . . , K in parallel do
Step s2: w k t+1 ← ClientUpdate(k, w t ) end for
Step s3:
w t+1 ← 1 K K k=1 w k t+1 end for ClientUpdate(k, w): // Execute on client k Receive w = {w f , w c , w g } from server for epoch i = 1, 2, . . . , E 0 do
Step c2: Sample one mini-batch S x from S k
Step c3: Update w f and w c on S x to minimize L err end for for epoch j = 1, 2, . . . , E 1 do
Step c4: Sample one mini-batch S x from S k
Step c5: Update w f and w c on S x to minimize λ 0 L adv f + λ 1 L err
Step c6: Use random number generator to generate one mini-batch random numbers S z
Step c7: Update w d on S x and S z to minimize L adv d
Step c8: Update w g on S z and y to minimize L adv g end for
Step c9: Upload the trained w to server
How to Learn Domain-Invariant Features
Under the FL settings, FedADG aligns the distributions of all source domains data to learn the domain-invariant features. In the previous domain generalization techniques, the centralized server can access each client's local data. Thus, it can learn a domain-invariant feature via directly minimizing the discrepancy between the source domains using Maximum Mean Discrepancy (MMD) distance metric [20]. However, in FL, the server can not access each clients' local data, making it hard to learn the domain-invariant features. In our proposed Adversarial Learning Network (ALN), the distribution generator is shared among clients, indicating that the reference distribution is identical for all clients. Thus, once the discriminator is hard to distinguish between the feature extracted from feature extractor and the feature generated from distribution generator, the generated features are considered to be invariant across multi-source domains. Note that ALN can be trained in a federated manner (i.e., FedALN), which eliminates the requirement for centralized training.
How to Achieve High Performance
There are two candidate approaches to obtain the reference distribution in FedADG: pre-selected fixed distribution and adaptively generated distribution. Using the adaptively generated distribution can increase the performance of FedADG due to the following three reasons. Less Distortion During Alignment. As shown in Fig. 3, we employ t-SNE [35] to visualize the source domain features and the reference distribution features before training the model. Gaussian distribution is used as the fixed distribution. It can be observed that the adaptively generated distribution would locate close to the "center" of the distributions from all the source domain features. Hence, the distances between the adaptively generated distribution and the distributions (of source domain data) are smaller than the distances between the fixed reference distribution and the source domain distributions. Thus, using the adaptively generated distribution can reduce the distortion of extracted feature representation during alignment. Less distortion means that the key information of the original source domain data can be largely preserved, resulting in the high generalization performance of FedADG. Class-Wise Alignment. FedADG uses the label information (encoded in a one-hot vector) in the adversarial training. Thus, the distribution generator generates features for each class in training. It means that the distributions of source domains data are aligned in a class-wise manner. This fine-grained class-wise alignment approach can further improve the performance of FedADG. More Discriminative Features. Fig. 4 shows the source domain features and the reference distribution features after training the model. The distances between different class clusters in Fig. 4b are more evident than that in [29,41], we simulate the computation of clients on the Linux server and then measure FedADG performance. Since the learning process is exactly the same, the performance metrics measured are accurate in our experiments.
Datasets. All experiments are based on three widely used datasets in DG, i.e., VLCS (Pascal [9], LabelMe [31], Caltech-101 [10], and SUN [6]), PACS [18] (Photo, Art painting, Cartoon, and Sketch), and Office-Home [36] (Real-World, Clipart, Product, and Art). All of them have four sub-datasets, which form distinct domains. For each dataset, we utilize the leave-one-domain-out validation strategy. That is, we let one dataset serve as the target domain and use the rest three datasets as source domains. Like [13], each domain is divided into a training set (70%) and a validation set (30%) randomly. The welltrained model is tested on the target domain data. Besides, we follow the protocol of [18] to perform experiments on PACS. For Office-Home, we use the same protocol as [8]. Parameter Settings. In all experiments, the parameters of feature extractor are initialized with pre-trained weights using ImageNet [7]. The hyper-parameters of the feature extractor and the initial learning rates of different components on different datasets are detailed in Table 3.
Notice that the hyper-parameters λ 0 (0 < λ 0 < 1) and λ 0 (0 < λ 1 < 1) train the feature extractor together, and λ 0 + λ 1 = 1. In particular, lr f is the learning rate of the feature extractor and classifier, the learning rate of distribution generator and discriminator are lr g and lr d , respectively. In our experiments, unless otherwise stated, the hyper-parameters and learning rates are set as the above default configuration.
Performance Evaluation
In this section, we compare FedADG with several recent domain generalization solutions on VLCS, PACS, and
Office-Home datasets. These solutions are briefly introduced as follows.
1. DANN [12], a neural network that can both accurately classify source data and have features that are invariant across multiple source domains. DANN is the abbreviation of Domain-Adversarial Neural Network.
2. JiGen [5], a supervised framework for learning to generalize across visual domains by solving jigsaw puzzles.
3. Epi-FCR [19], a scheme to learn domain shift using episodic training. Epi-FCR is the abbreviation of Episodic-Feature and Classifier Regularisation.
MTSSL [2]
, a method for enabling models to learn transferable features through a self-supervised task of Gabor filter bank response prediction. MTSSL is the abbreviation of Multi-Task Self-Supervised Learning. [39], a network that uses self-supervised learning and metric learning to improve classifier performance on target domains. EISNet is the abbreviation of Extrinsic and Intrinsic Supervision Network.
EISNet
6. L2A-OT [42], a method to learn domain-invariant features by augmenting the source domain with synthetic data. L2A-OT is the abbreviation of Learning to Augment by Optimal Transport.
7. DSON [32], a scheme that combines batch normalization and instance normalization to enhance generalization performance on target domains. DSON is the abbreviation of Domain Specific Optimized Normalization.
8. Mixstyle [43], a method for mixing features across source domains to synthesize new source domains to optimize model generalization.
9. RSC [16], a method to discard dominant features of training data to optimize the generalization ability of a model. RSC is the abbreviation of Representation Self-Challenging.
All prior solutions require centralized data access, whereas FedADG is used for domain generalization in a distributed way. We also compare it with a recent stateof-the-art DG method: FedDG [23], which does not centralize the dataset and is also trained in the FL setting. Besides, FedAvg [25] is used as a baseline. We do not compare COPA [40] because of the following reasons. First, it sacrifices security (refer to Section 1 for more details). Second, the project codes are not publicly available. We also do not compare FL optimization methods (e.g., Fed-Prox, FedNova, and MOON) since these papers do not focus on the domain generation problem. These optimization methods differ from FedADG in the following aspects. First, the source domain data in FedADG are from different clients with domain discrepancy, instead of different subsets from the same dataset [40]. Second, the discrepancy between the test datasets and training datasets in FedADG also makes it more complex than those in federated optimization methods. Moreover, FedADG requires building a model that has high performance when testing over the related but unseen target dataset rather than seen dataset. Note that all solutions used in the comparison are constructed using the same pre-trained network as FedADG.
For each test, we run 5 trails and report the average results which are shown in Table 4, Table 5, and Table 6. In the three tables, each column containing experimental results (except Avg. column) shows the results when one domain is chosen as the target domain. We highlight the best results in bold font. VLCS. Table 4 shows the domain generalization accuracy on VLCS. We use two pre-trained networks, AlexNet and ResNet18, as the backbone to compare FedADG with some recent domain generalization solutions. Table 4 shows that FedADG outperforms most of the compared centralized solutions. The performance is comparable to the recent RSC solution. Besides, FedADG has good performance in both small and large backbone networks. PACS. Table 5 shows the domain generalization accuracy on PACS. We use the same backbone network as in VLCS. In Table 5, we find that the performance of FedADG is better than most of the compared centralized solutions. Furthermore, the performance of FedADG is obviously improved compared to FedDG, which is also a distributed DG method. Besides, we observe that FedDG does not improve performance like FedADG (as the back- bone size increases from AlexNet to ResNet18). The accuracy of FedADG is slightly worse than the recent solution L2A-OT. Specifically, FedADG significantly improves the performance in the Sketch domain. Office-Home. We also evaluate FedADG on the Office-Home dataset and the results are shown in Table 6. ResNet18 and ResNet50 are applied as the backbone. In Table 6, we observe that FedADG is better than other solutions. Compared with Traditional Centralized Method. Table 4, Table 5, and Table 6 present the domain generalization accuracy of FedADG and prior traditional centralized machine learning solutions without FL. These traditional ML approaches are described in Section VI-B. In these tables, the paradigm of these traditional ML solutions is represented as "centralized w/o privacy concerns". That is, these solutions require the centralized server to access source domain data and expose sensitive local information. In Table 4 and Table 5, we find that the performance of FedADG is comparable to the traditional ML methods. In particular, the generalization accuracy of the AlexNetbased FedADG on PACS is over 1% higher than the centralized approaches (e.g., MTSSL, Epi-FCR, JiGen). Besides, Table 6 shows that FedADG performs significantly better than other traditional ML methods on Office-Home dataset. In summary, our proposed FedADG can achieve good domain generalization capability while still protecting data privacy.
Ablation Study
An ablation study investigates the performance of FedADG by removing a certain component to understand the contribution of the component to the overall FedADG scheme. We perform ablation experiments on VLCS and PACS datasets using AlexNet. Specifically, we focus on the distribution generator and discriminator, along with the data label (encoding as a one-hot vector) in these two components. When we remove the one-hot vectors from both distribution generator and discriminator, the remained FedADG is denoted as "FedADG w/o one-hot". We use "FedADG w/o RP" to denote FedADG without the Random Projection (RP) layer. "FedADG w/o G&D" represents FedADG without distribution generator and discriminator. Fig. 5 shows the ablation study results. The results analysis is performed as follows. FedADG w/o one-hot. As shown in Fig. 5, FedADG has higher accuracy in each target domain than FedADG w/o one-hot. These results demonstrate that the class-wise alignment can increase the generalization performance of FedADG. FedADG w/o RP. Fig. 5 shows that the accuracy of FedADG w/o RP is less than FedADG. The function of random projection is to decrease the dimension of features. The low-dimension features stabilize the training of ALN and help to do feature alignment. Thus, the random projection layer can improve the performance of FedADG on target domains. FedADG w/o G&D. Fig. 5 shows that the accuracy of FedADG w/o G&D is less than FedADG. In terms of av-erage accuracy, FedADG is over 3% higher than FedADG w/o G&D because the latter lacks of domain generaliza- Figure 5: Ablation study on VLCS and PACS datasets. "FedADG w/o one-hot" means to remove the one-hot vector from FedADG. "FedADG w/o RP" represents the model by removing the random projection layer from FedADG. Besides, "FedADG w/o G&D" represents the model by removing distribution generator and discriminator from FedADG. tion design.
In-domain Performance Evaluation
The previous experimental results are tested on out-ofdomain data. We also measure the performance of FedADG on in-domain data. We consider the commonly used experimental setting: both the training and testing data come from the same domain. AlexNet and ResNet18 are applied as the backbone and PACS dataset is used in experiments. Table 7 compares the in-domain performance of FedADG and FedAvg. In Table 7, "FedAvg (in)" and "FedADG (in)" represent the in-domain performances of FedAvg and FedADG, respectively. The results show that the in-domain performance of FedADG is comparable to that of FedAvg. It indicates that FedADG can be used on both source domains and the unseen target domain. In practice, the clients can train both FedAvg (used for in-domain data) and FedADG (used for out-domain data).
Impact of Different Reference Distributions
As we have discussed in Section 4.2, the reference distribution helps to align the feature distributions of all source domain data, which in turn improves the accuracy of classification on target domains. In general, most existing works adopt fixed reference distribution without considering the distortion it may cause to the source domain distribution. In this part, we investigate how the adaptively generated distribution can outperform the fixed settings such as Gaussian distribution (N ), Uniform distribution (U), and Laplace distribution [20]. The experimental results of different reference distributions using AlexNet on PACS are shown in Table 8.
In Table 8, the parameter of the Laplace distribution we compared in the experiment is 1/ √ 2, which is proved by Li et al. [20] to have the best effect on target domains. Moreover, we find that the accuracy of the Laplace distribution with the parameter of 1/ √ 2 is higher than the other two fixed distributions in the table. By observing all experimental results in Table 8, we notice that the average accuracy of the adaptively generated reference distribution can be 10% higher than the accuracy of the fixed reference distributions. Especially in the target domains of Cartoon and Sketch, the accuracy of using the adaptively generated distribution in FedADG is about 20% higher than the accuracy of the fixed reference distribution. The remarkable result of FedADG supports the effectiveness of using adaptively generated distribution. It proves that the generated adaptive reference distribution can promote the performance of the model for target domains.
Limitation
The major limitation of FedADG is that it requires extra resources consumption in training compared with the traditional FL schemes (e.g., FedAvg). On the one hand, clients need more computing resources to perform local adversarial training of ALN. On the other hand, clients have a larger communication overhead since they are required to submit the updates of ALN to the server as well as receive a new global ALN model in federated training. In a nutshell, FedADG achieves DG at the cost of extra resources consumption. Note that the extra resources consumption is not very heavy and can be easily handled by the majority of smart devices on the market.
Conclusion
In this paper, we propose FedADG scheme under the federated learning setting for domain generalization. The main idea of FedADG is to learn the domain-invariant feature representation in FL while eliminating the requirement for a centralized server to access clients' local data. First, we propose the federated adversarial learning approach to measure and align the distributions among different source domains via matching each distribution to the reference distribution. Specifically, we use the federated adversarial learning technique to adaptively learn a dynamic distribution (by accommodating all source domains) as the reference distribution. Therefore, the learned feature representation tends to be universal. Then, our proposed FedADG uses the adaptively generated reference distributions and class-wise alignment technique. It ensures that FedADG has good generalization performance over the unseen target domains while protecting local data privacy. Furthermore, we analyze the explainability of FedADG, which helps researchers to optimize the model and make the model more trustworthy. Finally, the effectiveness of FedADG has been demonstrated by intensive simulations. Thus, FedADG significantly boosts FL performance. There are two directions to launch further research. First, we aim to handle the scenario in which the unseen target domain contains more classes than the seen source domain. Second, we plan to find an optimization method to automatically balance the classification training epoch and the alignment training epoch to obtain a better federated generalization performance.
Figure 1 :
1In FL, K different clients collaboratively train a machine learning model for object classification task.
Fig. 2 shows
2FedADG scheme. It can be seen from the figure that each client's local model mainly consists of four
Figure 3 :Figure 4 :
34T-SNE visualization of FedADG features before training (fixed distribution v.s. adaptively generated distribution). The red asterisk marker represents the features of reference distribution. Each marker (except the asterisk) represents a distinct source domain. Each color (except red) represents a distinct class label. Each feature is projected into two-dimensional space.(a) Source domain features and fixed distribution features. (b) Source domain features and generated distribution features. T-SNE visualization of FedADG features after training (fixed distribution v.s. adaptively generated distribution). There are three source domains (A, C, S) data of the PACS dataset. In the legend, Ai represents the class label i of source domain A. The meanings of different colors and markers are the same as Fig. 3. (a) Source domain features and fixed distribution features. (b) Source domain features and generated distribution features.
Fig
Fig. 4a. It indicates that FedADG is capable of learning more discriminative features among different classes for different source domains. Therefore, FedADG has good domain generalization performance.
Table 1 :
1The comparison between the previous solutions and FedADG.Notation Description
K
number of clients
S
source domain
n
number of source domain data
(x, y)
data and its label
z
random noise
h
feature of source domain data
Table 2 :
2Notations.
Table 3 :
3Implementation. We conduct our experiments using Pytorch 1.7.1 deep learning framework and Python 3.6.5 on Ubuntu 16.04. We use four Linux terminals to simulate the deployment of FedADG. Our server uses Geforce RTX 2080ti GPU with 24G RAM for computing. Following most of the previous studies on FLParameters setting in experiments on different
datasets.
on three different datasets. Then, we have an ablation
study of the FedADG scheme. Afterward, we investigate
the in-domain performance of FedADG. Last, we study
the impact of the different reference distributions.
6.1 Experimental Settings
Table 4 :
4The average classification accuracy using leave-one-domain-out validation on VLCS dataset.
Table 5 :
5The average classification accuracy using leave-one-domain-out validation on PACS dataset.Paradigm
Backbone
Method
Real
Clipart Product
Art
Avg.
Centralized w/o
privacy concern
ResNet18
JiGen [5]
72.79
47.51
71.47
53.04
61.20
ResNet18
DSON [32]
74.68
45.70
71.84
59.37
62.90
ResNet18
RSC [16]
74.54
47.90
71.63
58.42
63.12
Distributed
ResNet18
FedAvg [25]
71.31
52.11
67.60
48.00
59.76
ResNet18
FedADG (ours)
74.98
53.98
70.83
58.13
64.48
Centralized w/o
privacy concern
ResNet50
Mixstyle [43]
69.20
53.20
68.20
51.10
60.43
ResNet50
RSC [16]
75.10
51.40
74.80
60.70
65.50
Distributed
ResNet50
FedAvg [25]
71.83
54.06
69.14
53.07
62.03
ResNet50
FedADG (ours)
76.48
56.09
74.87
60.27
66.93
Table 6 :
6The average classification accuracy using leave-one-domain-out validation on Office-Home dataset.Paradigm
PACS
Backbone
Sketch
Artpaint Cartoon
Photo
Avg.
Distributed
FedAvg (in)
AlexNet
99.98
99.98
99.97
99.98
99.98
FedADG (in)
AlexNet
98.13
98.06
97.96
98.03
98.05
FedAvg (in)
ResNet18
99.12
98.08
98.38
95.83
97.85
FedADG (in)
ResNet18
99.83
98.47
94.08
99.73
98.03
Table 7 :
7The average in-domain classification accuracy using leave-one-domain-out validation on PACS dataset.
Unseen domain (→) Sketch Artpaint Cartoon Photo Avg.fixed reference distribution
N ∼ (0, I)
49.45
53.32
53.54
75.69 58.00
U ∼ [−1, 1]
38.23
57.71
55.33
81.26 58.13
Laplace(1/
√
2)
44.29
54.69
55.84
84.31 59.78
adaptively generated distribution
FedADG (ours)
69.15
68.99
70.14
87.01 73.82
Table 8 :
8Experimental results when using different reference distributions in FedADG on PACS dataset.
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| [] |
[
"SUSPENSION HOMOTOPY OF 6-MANIFOLDS",
"SUSPENSION HOMOTOPY OF 6-MANIFOLDS",
"SUSPENSION HOMOTOPY OF 6-MANIFOLDS",
"SUSPENSION HOMOTOPY OF 6-MANIFOLDS"
] | [
"Ruizhi Huang ",
"Ruizhi Huang "
] | [] | [] | For a simply connected closed orientable manifold of dimension 6, we show its homotopy decomposition after double suspension. This allows us to determine its K-and KO-groups easily. Moreover, for a special case we refine the decomposition to show the rigidity property of the manifold after double suspension. dimension n, Bardon [Bar] classified the simply connected 5-manifolds, and Wall [Wal2], Jupp [Jup] and Zhubr [Zhu1, Zhu2] classified the simply connected 6-manifolds. More recently, Kreck and Su [KS] classified certain non-simply connected 5-manifolds, while Crowley and Nordström [CN] and Kreck [Kre] studied the classification problem of various kinds of 7-manifolds. | 10.2140/agt.2023.23.439 | [
"https://export.arxiv.org/pdf/2104.04994v1.pdf"
] | 233,210,586 | 2104.04994 | 81476920d3bfd41fc4234cacd9767c3c6a90b3d9 |
SUSPENSION HOMOTOPY OF 6-MANIFOLDS
11 Apr 2021
Ruizhi Huang
SUSPENSION HOMOTOPY OF 6-MANIFOLDS
11 Apr 2021arXiv:2104.04994v1 [math.AT]
For a simply connected closed orientable manifold of dimension 6, we show its homotopy decomposition after double suspension. This allows us to determine its K-and KO-groups easily. Moreover, for a special case we refine the decomposition to show the rigidity property of the manifold after double suspension. dimension n, Bardon [Bar] classified the simply connected 5-manifolds, and Wall [Wal2], Jupp [Jup] and Zhubr [Zhu1, Zhu2] classified the simply connected 6-manifolds. More recently, Kreck and Su [KS] classified certain non-simply connected 5-manifolds, while Crowley and Nordström [CN] and Kreck [Kre] studied the classification problem of various kinds of 7-manifolds.
In the mentioned literature, the homotopy classification of M was usually carried out as a byproduct in terms of a system of invariants. However, it is almost impossible to extract nontrivial homotopy information of M directly from the classification. On the other way around, the unstable homotopy theory is a powerful tool to study the homotopy properties of manifolds. There are several interesting attempts in recent years along this way. For instance, Beben and Theriualt [BT1] studied the loop decompositions of (s − 1)-connected 2s-manifolds, while Beben and Wu [BW] and Huang and Theriault [HT] studied the loop decompositions of the (s − 1)-connected (2s + 1)-manifolds. The homotopy groups of these manifolds were also investigated by Sa. Basu and So. Basu [BB, Bas] from different point of view. Moreover, a theoretical method of loop decomposition was developed by Beben and Theriault [BT2], which is quite useful for studying the homotopy of manifolds. Additionally, the homotopy type of the suspension of a connected 4-manifold was determined by So and Theriault [ST].
In this paper, we study the homotopy aspect of simply connected 6-manifolds. Let M be a simply connected closed orientable 6-manifold. By Poincaré duality and the universal coefficient theorem
we have (1) H * (M ; Z) = Z ⊕d ⊕ T * = 2, Z ⊕2m ⊕ T * = 3, Z ⊕d * = 4, Z * = 0, 6 0 otherwise,
where m, d ≥ 0, and T is a finitely generated abelian torsion group. Our first main theorem concerns the double suspension splitting of M . Denote ΣX be the suspension of any CW -complex X. Denote P n (T ) be the Moore space such that the reduced cohomology H * (P n (T ); Z) ∼ = T if * = n and 0 otherwise [N2].
Theorem 1.1. Let M be a simply connected closed orientable 6-manifold with homology of the form (1). Suppose that T has no 2 or 3-torsion. Then there is an integer c with 0 ≤ c ≤ d determined by the cohomology ring of M such that
• if c = 0, Σ 2 M ≃ ΣW 0 ∨ d−1 j=1 (S 4 ∨ S 6 ) ∨ 2m i=1
S 5 ∨ P 6 (T ) ∨ P 5 (T ),
where W 0 ≃ (S 3 ∨ S 5 ) ∪ e 7 ;
• if c = d,
Σ 2 M ≃ ΣW d ∨ d−1 i=1 Σ 2 CP 2 ∨ 2m i=1
S 5 ∨ P 6 (T ) ∨ P 5 (T ),
where W d ≃ ΣCP 2 ∪ e 7 ;
• if 1 ≤ c ≤ d − 1,
Σ 2 M ≃ ΣW c ∨ c−1 i=1 Σ 2 CP 2 ∨ d−c−1 j=1 (S 4 ∨ S 6 ) ∨ 2m i=1
S 5 ∨ P 6 (T ) ∨ P 5 (T ),
where W c ≃ (ΣCP 2 ∨ S 3 ∨ S 5 ) ∪ e 7 .
Notice the number c is indeed determined by the Steenrod square Sq 2 : H 2 (M ; Z/2Z) → H 4 (M ; Z/2Z), while there is an ambiguous term W c (0 ≤ c ≤ d). Since we only need the suspension of W c and the Hopf element η i ∈ π i+1 (S i ) is detected by Sq 2 , the ambiguity reduces to the components of the attaching map of the top cell of W c to the wedge summand ΣCP 2 and S 3 .
groups of M K(M ) ∼ = Z ⊕2d+1 ⊕ T, KO(M ) ∼ = d (Z ⊕ Z/2Z).
If we specify to the case when d = 1, we can obtain the complete description of M after double suspension, based on the work of Yamaguchi [Yam] (also summarised and corrected by Baues [Bau]). [Tod], where η i ∈ π i+1 (S i ) is the Hopf element. Let V 3 be the manifold as the total space of the sphere bundle of the oriented R 3 -bundle over S 4 determined by its first Pontryagin class p 1 = 12s 4 , where s 4 ∈ H 4 (S 4 ; Z) is a generator. Theorem 1.3. Let M be a simply connected closed orientable 6-manifold with homology of the form
Denote η 3 i = η i+2 • η i+1 • η i ∈ π i+3 (S i )
(1) such that d = 1. Let x, y ∈ H * (M ; Z) be two generators such that deg(x) = 2 and deg(y) = 4. Denote x 2 = ky for some k ∈ Z. Suppose T has no 2 or 3-torsion. Then
• if k is odd, then M is Spin, moreover Σ 2 M ≃ Σ 2 CP 3 ∨ 2m i=1 S 5 ∨ P 6 (T ) ∨ P 5 (T ),
when k ≡ 1 mod 6, while
Σ 2 M ≃ Σ 2 V 3 ∨ 2m i=1
S 5 ∨ P 6 (T ) ∨ P 5 (T ), when k ≡ 3 mod 6;
• if k is even and V is non-Spin
Σ 2 M ≃ S 4 ∨ Σ 4 CP 2 ∨ 2m i=1
S 5 ∨ P 6 (T ) ∨ P 5 (T );
• if k is even and V is Spin
Σ 2 M ≃ (S 4 ∪ λη 3 4 e 8 ) ∨ S 6 ∨ 2m i=1 S 5 ∨ P 6 (T ) ∨ P 5 (T ),
where λ ∈ Z/2 is determined by M .
It should be remarked that there is no ambiguity in the term (S 4 ∪ λη 3 4 e 8 ) in the last decomposition of Σ 2 M . Indeed, the stable cube element η 3 n ∈ π n+3 (S n ) (n ≥ 2) is detected by the secondary operation T [Har,Exercise 4.2.5], and in our case the homotopy decomposition has to preserve the module structure induced by the cohomology operations. Moreover, it is clear that the number k mod 2 and the spin condition of M are determined by the Steenrod square Sq 2 . And we will also see that ΣCP 3 and ΣV 3 can be distinguished by the Steenrod power P 1 : H 3 (ΣM ; Z/3Z) → H 7 (ΣM ; Z/3Z). Hence, we obtain the following rigidity result for manifolds in Theorem 1.3 after double suspension. The paper is organized as follows. In Section 2 we reduce the decomposition problem of 6manifolds to that of ones whose third Betti numbers are zero. In Section 3, we give a detailed procedure to decompose 6-manifolds after double suspension by homology decomposition method. Section 4 and Section 5 are devoted to prove Theorem 1.1 and Theorem 1.3 respectively. In Section 6, we compute some homotopy groups of odd primary Moore spaces used in Section 3. He would like to thank Professor Stephen Theriault for his international online lecture series "Loop Space Decomposition", which stimulated his research interest in the homotopy of 6-manifolds.
2. Reducing to the case when b 3 (M ) = 0
The following well known splitting theorem of 6-manifolds was proved by Wall [Wal2] in smooth category, while Jupp [Jup] pointed out that the theorem holds in topological category by the same argument.
Theorem 2.1. [Wal2, Theorem 1] Let M be a simply connected closed orientable 6-manifold with third Betti number b 3 (M ) = 2m. Then there exists a 6-manifold M 1 such that
M ∼ = M 1 # # m S 3 ×S 3 .
Corollary 2.2. Let M and M 1 be the manifolds in Theorem 2.1. Then
ΣM ≃ ΣM 1 ∨ m i=1 (S 4 ∨ S 4 ).
Proof. Let M ′ 1 and M ′ be the 5-skeletons of M 1 and M respectively. It is known that S 3 ∨ S 3 is the 5-skeleton of S 3 × S 3 and Σ(S 3 × S 3 ) ≃ Σ(S 3 ∨ S 3 ) ∨ S 7 . In particular, there is a homotopy
retraction r : Σ(S 3 × S 3 ) → Σ(S 3 ∨S 3 ). For the connected sum M 1 #(S 3 × S 3 ), there are the obvious pinch maps q 1 : M 1 #(S 3 × S 3 ) → M 1 and q 2 : M 1 #(S 3 × S 3 ) → S 3 × S 3 . Consider the composition φ : Σ(M 1 #(S 3 × S 3 )) µ ′ −→ Σ(M 1 #(S 3 × S 3 )) ∨ Σ(M 1 #(S 3 × S 3 )) Eq1∨(r•Eq2) −→ ΣM 1 ∨ Σ(S 3 ∨ S 3 ),
where µ ′ is the standard co-multiplication of suspension complex, and E denotes the suspension of a map. It is easy to see that φ induces an isomorphism on homology and then is a homotopy equivalence by the Whitehead Theorem. Since
M ′ ≃ M ′ 1 ∨ m i=1 (S 3 ∨ S 3 ),
by repeating the above argument, we can show the decomposition in the corollary.
In Theorem 2.1, the connected summand M 1 satisfies b 3 (M 1 ) = 0. Hence by Corollary 2.2 it suffices to consider such 6-manifolds in the sequel.
Homology decomposition of M after suitable suspensions
Let M be a simply connected closed orientable 6-manifold with b 3 (M ) = 0. By Poincaré duality and the universal coefficient theorem we have
(2) H * (M ; Z) = Z ⊕d ⊕ T * = 2, T * = 3, Z ⊕d * = 4, Z * = 0, 6 0 otherwise,
where d ≥ 0, and T is a finitely generated abelian torsion group. We may denote
(3) T = ℓ ⊕ k=1 Z/p r k k Z,
where each p k is a prime and r k ≥ 1.
Instead of using skeleton decomposition, we may apply homology decomposition to study the cell structure of M . For any finitely generated abelian group A, let P n (A) be the Moore space such that the reduced cohomology H * (P n (A); Z) ∼ = A if * = n and 0 otherwise [N2]. The information on the homotopy groups of P n (T ) used in this section will be proved in Section 6.
Theorem 3.1. [Hat,Theorem 4H.3] Let X be a simply connected CW -complex.
Denote H i = H i (X; Z). Then there is a sequence of complex X (i) (i ≥ 2) such that • H j (X (i) ; Z) ∼ = H j (X; Z) for j ≤ i and H j (X (i) ; Z) = 0 for j > i; • X (2) = P 3 (H 2 )
, and X (i) is defined by a homotopy cofibration
P i (H i ) fi−1 −→ X (i−1) ιi−1 −→ X (i) , where f i−1 induces a trivial homomorphism f i−1 * : H i−1 (P i (H i ); Z) → H i−1 (X (i−1) ; Z); • X ≃ hocolim{X (2) ι2 −→ · · · ιi−2 −→ X (i−1) ιi−1 −→ X (i) ιi −→ · · · }.
From this theorem, it is clear that the homology decomposition is compatible with the suspension functor. That is, for X in Theorem 3.1 the sequence of the triples (ΣX (i) , Ef i , Eι i ) contributes to a homology decomposition of ΣX.
3.1. Structure of M (3) . By Theorem 3.1 and (2),
M (2) ≃ d i=1 S 2 ∨ P 3 (T ), P 3 (T ) f2 −→ M (2) ι2 −→ M (3) .(4)
Notice that from (3), we have P n (T ) ≃ ℓ k=1 P n (p r k k ) by [N3] or [N2].
Lemma 3.2. The map f 2 in (4) is null-homotopic, and hence
M (3) ≃ d i=1 S 2 ∨ P 3 (T ) ∨ P 4 (T ). Proof. Since P 3 (T ) ≃ ℓ k=1 P 3 (p r k k ), there is the embedding j : ℓ i=1 S 2 → P 3 (T ) of the bottom cells.
Consider the following commutative diagram
π 2 ( ℓ i=1 S 2 ) hur ∼ = j * / / / / π 2 (P 3 (T )) f2 * / / hur ∼ = π 2 (M (2) ) hur ∼ = H 2 ( ℓ i=1 S 2 ; Z) j * / / / / H 2 (P 3 (T ); Z) f2 * =0 / / H 2 (M (2) ; Z),
where the Hurewicz homomorphisms hur are isomorphisms by the Hurewicz theorem, f 2 * = 0 on homology by Theorem 3.1, and both j * are epimorphisms. In particular, f 2 * • j * is trivial on homotopy groups, and hence f 2 • j is null homotopic. Then with (4) we have the diagram of homotopy cofibrations
ℓ i=1 S 2 ℓ i=1 p r k k / / * / / ℓ i=1 S 3 i2•( ℓ i=1 p r k k ) ℓ i=1 S 2 0 / / j M (2) / / M (2) ∨ ℓ i=1 S 3 P 3 (T ) f2 / / M (2) ι2 / / M (3) ,
where p r k k : S n → S n is a map of degree p r k k , and i 2 :
ℓ i=1 S 3 → M (2) ∨ ℓ i=1
S 3 is the injection onto the sphere summands. It follows that
M (3) ≃ M (2) ∨ P 4 (T ) ≃ d i=1 S 2 ∨ P 3 (T ) ∨ P 4 (T ),
and the proof of the lemma is completed.
The following corollary follows from Lemma 3.2 and will be used in Lemma 3.6.
Corollary 3.3. The homotopy cofiber of the obvious inclusion j :
P 3 (T ) ∨ P 4 (T ) → M (3) → M is a Poincaré duality complex V with cell structure V = d i=1 S 2 ∪e 4 (1) ∪e 4 (2) . . .∪e 4 (d) ∪e 6 .
Moreover, by [Wal2,Theorem 8] V is homotopy equivalent to a closed smooth manifold.
3.2. Structure of M (5) . By Theorem 3.1 and Lemma 3.2,
M (3) ≃ d i=1 S 2 ∨ P 3 (T ) ∨ P 4 (T ), d i=1 S 3 f3 −→ M (3) ι3 −→ M (4) = M (5) .(5)
We need to study the map
f 3 : d i=1 S 3 → d i=1 S 2 ∨ P 3 (T ) ∨ P 4 (T ). Let i 3 : P 4 (T ) → d i=1 S 2 ∨ P 3 (T ) ∨ P 4 (T ) be the inclusion. Define the complex Y by the homotopy cofibration P 4 (T ) ι3•i3 −→ M (4) = M (5) −→ Y.
Lemma 3.4. The map f 3 in (5) factors as
f 3 : d i=1 S 3 f ′ 3 −→ d i=1 S 2 ∨ P 3 (T ) i1∨i2 −→ d i=1 S 2 ∨ P 3 (T ) ∨ P 4 (T ),
for some f ′ 3 , where i 1 and i 2 are inclusions. Moreover, there is the homotopy cofibration
(6) d i=1 S 3 f ′ 3 −→ d i=1 S 2 ∨ P 3 (T ) ι ′ 3 −→ Y, and M (5) ≃ Y ∨ P 4 (T ).
Proof. First there is the diagram of homotopy cofibrations * / /
P 4 (T ) i3 P 4 (T ) ι3•i3 d i=1 S 3 f3 / / M (3) q1,2 ι3 / / M (5) d i=1 S 3 f ′ 3 / / d i=1 S 2 ∨ P 3 (T ) ι ′ 3 / / Y,
where q 1,2 is the obvious projection, ι ′ 3 is induced from ι 3 , and f ′ 3 := q 1,2 • f 3 . The diagram immediately implies that (6) is a homotopy cofibration.
Denote q 3 : d i=1 S 2 ∨ P 3 (T ) ∨ P 4 (T ) → P 4 (T ) be the canonical projection. By Theorem 3.1,
f 3 * : H 3 ( d i=1 S 3 ; Z) → H 3 ( d i=1
S 2 ∨ P 3 (T ) ∨ P 4 (T ); Z) is trivial. In particular, q 3 * • f 3 * = 0. Then by the Hurewicz Theorem q 3 • f 3 is null homotopic. Further, by the Hilton-Milnor Theorem (see §XI.6 of [Whi] for instance), π 3 (S 2 ∨ P 3 (T ) ∨ P 4 (T )) ∼ = π 3 (S 2 ∨ P 3 (T )) ⊕ π 3 (P 4 (T )), and hence
[ d i=1 S 3 , S 2 ∨ P 3 (T ) ∨ P 4 (T )] ∼ = [ d i=1 S 3 , S 2 ∨ P 3 (T )] ⊕ [ d i=1 S 3 , P 4 (T )].
Under this isomorphism, the homotopy class of f 3 corresponds to
[f ′ 3 ] + [q 3 • f 3 ]. However since we already show that [q 3 • f 3 ] = 0, we have f 3 ≃ (i 1 ∨ i 2 ) • f ′ 3 , and M (5) ≃ Y ∨ P 4 (T ) as required.
3.3. Structure of ΣM (5) . From this point, we may need extra conditions on the torsion group T .
First recall that we already showed that by Lemma 3.4
M (5) ≃ Y ∨ P 4 (T ), d i=1 S 3 f ′ 3 −→ d i=1 S 2 ∨ P 3 (T ) ι ′ 3 −→ Y.(7)
Let q 1 :
d i=1 S 2 ∨ P 3 (T ) → d i=1 S 2 be the canonical projection. Denote f ′′ 3 := q 1 • f ′ 3 = q 1 • q 1,2 • f 3 .
Also recall P n (T ) ≃ ℓ k=1 P n (p r k k ). Let us suppose each p ≥ 3 from now on.
Lemma 3.5. Suppose T has no 2-torsion.
ΣM (5) ≃ ΣX ∨ P 5 (T ) ∨ P 4 (T ),
where X is the homotopy cofiber of the map f ′′ 3 :
d i=1 S 3 → d i=1 S 2 .
Proof. There is the diagram of homotopy cofibrations * / /
P 3 (T ) i2 P 3 (T ) ι ′ 3 •i2 d i=1 S 3 f ′ 3 / / d i=1 S 2 ∨ P 3 (T ) q1 ι ′ 3 / / Y d i=1 S 3 f ′′ 3 / / d i=1 S 2 / / X,
where i 2 is the canonical inclusion. Since π 4 (P 4 (p r )) = 0 for odd p by Lemma 6.3, we have ΣY ≃ ΣX ∨ P 4 (T ). The lemma then follows from (7).
3.4. Structure of Σ 2 M. Recall we have when T has no 2 torsion by Lemma 3.5 ΣM (5) ≃ ΣX ∨ P 5 (T ) ∨ P 4 (T ),
S 5 f5 −→ M (5) ι5 −→ M.(8)
Further by Corollary 3.3 we have the homotopy cofibration
S 5 → X → V,
where X is defined in Lemma 3.5 without restriction on T , and V is a closed smooth manifold. We may further suppose T has no 3-torsion.
Lemma 3.6. Suppose T has no 2 or 3-torsion. Then
Σ 2 M ≃ Σ 2 V ∨ P 6 (T ) ∨ P 5 (T ).
Proof. By the Hilton-Milnor theorem, we may write the suspension of f 5 as
Ef 5 = g(1)
5 + g
5 + g
5 + θ : S 6 → ΣM (5) ≃ ΣX ∨ P 5 (T ) ∨ P 4 (T ), for some θ, where Eθ = 0, g (i) 5 = q i • Ef 5 , and q i is the canonical projection of ΣX ∨ P 5 (T ) ∨ P 4 (T ) onto its i-th summand. Then g (2) 5 = 0 by Lemma 6.4, and Eg
(3) 5 = 0 by Lemma 6.6. It follows that E 2 f 5 = Eg
(1) 5 . Furthermore, there is the diagram of homotopy cofibrations * / / P 4 (T ) ∨ P 3 (T ) j5 P 4 (T ) ∨ P 3 (T )
j S 5 f5 / / M (5) ι5 / / π5 M π S 5 / / X / / V,
where the homotopy cofibration in the last column is defined in Corollary 3.3 by using Lemma 3.2, and similarly the homotopy cofibration in the middle column can be also defined by using Lemma
Then it is clear g
5 ≃ E(π 5 • f 5 ) and the lemma follows.
Proof of Theorem 1.1 and Corollary 1.2
In Lemma 3.6 we have established the double suspension splitting of M when b 3 (M ) = 0, and are left to consider the homotopy of V after suspension. Recall that V is a Poincaré Duality complex of dimension 6, and its 5-skeleton V 5 = X is the homotopy cofiber of the map f ′′ 3 :
d i=1 S 3 → d i=1 S 2 by
Lemma 3.5. The following lemma, as a special case of [H,Lemma 6.1], determines the suspension homotopy type of X.
ΣX ≃ c i=1 ΣCP 2 ∨ d−c j=1 (S 3 ∨ S 5 ),
for some 0 ≤ c ≤ d.
We may apply the method in [H, Section 3] to decompose Σ 2 V , in the same way that we used it to prove [H, Lemma 6.4 and Lemma 6.6].
Lemma 4.2. Suppose ΣX decomposes as in Lemma 4.1.
• If c = 0,
Σ 2 V ≃ ΣW 0 ∨ d−1 j=1 (S 4 ∨ S 6 ), where W 0 ≃ (S 3 ∨ S 5 ) ∪ e 7 ;
• if c = d,
Σ 2 V ≃ ΣW d ∨ d−1 i=1 Σ 2 CP 2 , where W d ≃ ΣCP 2 ∪ e 7 ; • if 1 ≤ c ≤ d − 1, Σ 2 V ≃ ΣW c ∨ c−1 i=1 Σ 2 CP 2 ∨ d−c−1 j=1 (S 4 ∨ S 6 ), where W c ≃ (ΣCP 2 ∨ S 3 ∨ S 5 ) ∪ e 7 .
Proof. As we pointed out that the proof is similar to that of [H,Lemma 6.4 and Lemma 6.6], we may only sketch it. The interested reader can find the details of the method in [H,Section 3]. With Lemma 4.1 let g :
S 6 → ΣX ≃ c i=1 ΣCP 2 ∨ d−c j=1 (S 3 ∨ S 5 )
be the attaching map of the top cell of ΣV . To apply the method in [H,Section 3], we only need the information of homotopy groups π 6 (ΣCP 2 ) ∼ = Z/6Z by [Muk,Proposition 8.2(i)], π 6 (S 3 ) ∼ = Z/12Z and π 6 (S 5 ) ∼ = Z/2, which are all finite cyclic groups. Then we can represent the attaching map Eg of the top cell of Σ 2 V by a matrix B, and apply [H,Lemma 3.1] to transform B to a simpler one C. The new matrix representation C of the attaching map, corresponding to a base change of ΣX through a self homotopy equivalence, will give the desired decomposition. Now we can prove Theorem 1.1.
Proof of Theorem 1.1. First by Theorem 2.1 and Corollary 2.2, we have
ΣM ≃ ΣM 1 ∨ m i=1 (S 4 ∨ S 4 ),
where M 1 is a closed 6-manifold with homology of the form (2). In particular b 3 (M 1 ) = 0. Hence by Lemma 3.6 and Lemma 4.2, we have that if 1 ≤ c ≤ d − 1
Σ 2 M ≃ ΣW c ∨ c−1 i=1 Σ 2 CP 2 ∨ d−c−1 j=1 (S 4 ∨ S 6 ) ∨ P 6 (T ) ∨ P 5 (T ) ∨ 2m i=1 S 5 ,
where W c ≃ (ΣCP 2 ∨ S 3 ∨ S 5 ) ∪ e 7 . The decompositions for the other two cases when c = 0 or d can be obtained similarly. Finally, notice that c records the number of the non-trivial Steenrod square Sq 2 : H 2 (Σ 2 M ; Z/2Z) → H 4 (Σ 2 M ; Z/2Z), which is preserved by the decomposition and the suspension operator. Since Sq 2 is the cup square on the elements of H 2 (M ; Z/2Z), this completes the proof of Theorem 1.1.
To prove Corollary 1.2, we need the Bott periodicity showed in the following table: (P 5 (T )) = 0 requires that T has no 2-torsion.
Table 1. K −i (S 0 ) and KO −j (S 0 ) i mod 2 0 1 K −i (S 0 ) Z 0 j mod 8 0 1 2 3 4 5 6 7 KO −j (S 0 ) Z Z/2Z Z/2Z 0 Z 0 0 0 From
Lemma 4.3. Let W c be the complex defined in Lemma 4.2 for 0 ≤ c ≤ d.
• K(P 5 (T )) ∼ = T, K(P 6 (T )) = 0,
• KO 2 (P 5 (T )) = KO 2 (P 6 (T )) = 0, spectively. Then the top attaching map b is determined by a generator b W ∈ π 5 (S 2 ∪ kη2 e 4 ) of infinite order, the second Stiefel-Whitney class ω 2 (V ) ∈ H 2 (V ; Z/2) of V , and an indeterminacy term b ′ ∈ Z/2 which depends on the following three cases.
• KO 1 (ΣCP 2 ) ∼ = KO 1 (S 3 ∨ S 5 ) ∼ = Z ⊕ Z/2Z, KO 1 (Σ 2 CP 2 ) ∼ = KO 1 (S 4 ∨ S 6 ) = 0, • KO 1 (W 0 ) ∼ = KO 1 (W d ) ∼ = Z ⊕ Z/2Z, KO 1 (W c ) ∼ = 2 (Z ⊕ Z/2Z).KO(M ) ∼ = KO 2 (Σ 2 M ) ∼ = KO 2 (ΣW c ) ⊕ c−1 i=1 KO 2 (Σ 2 CP 2 ) ⊕ d−c−1 j=1 KO 2 (S 4 ∨ S 6 ) ⊕ 2m j=1 KO 2 (S 5 ) ⊕ KO 2 (P 6 (T )) ⊕ KO 2 (P 5 (T )) ∼ = 2 (Z ⊕ Z/2Z) ⊕ c−1 (Z ⊕ Z/2Z) ⊕ d−c−1 (Z ⊕ Z/2Z) = d (Z ⊕ Z/2Z).
The case when
• If k is odd, then V is Spin and the homotopy type of V is unique determined by k, and b = b W ;
• If k is even and V is non-Spin, then the homotopy type of V is unique determined by k, and b = b W + η 4 with η 4 representing the generator of a Z/2 summand determined by ω 2 (V );
• If k is even and V is Spin, then V has precisely two homotopy types depending on the value
of b ′ ∈ Z/2, and b = b W + b ′ .
Remark 5.2. In Theorem 5.1, b W , as a generator of the Z-summand of π 5 (S 2 ∪ kη2 e 4 ), is indeed a relative Whitehead product when k = 0 by [Yam, Lemma 2.6]. It is possible that the suspension map Eb W is not null-homotopic. η 4 is derived from the homotopy class of
S 5 b −→ S 2 ∪ kη2 e 4 q −→ S 4 ,
where q is the quotient map onto the 4-cell of S 2 ∪ kη2 e 4 (c.f. [Bau,Section 1]). b ′ is from a class of π 5 (S 2 ) ∼ = Z/2{η 3 2 } by [Yam, Lemma 2.6] or [Bau,Section 1]. Also, as pointed out in Mathematical Reviews [MR], the original theorem of [Yam] was misstated which is corrected here and in [Bau, Section 1] as well.
Thanks to Theorem 5.1 and Remark 5.2, we can describe the suspension homotopy type of V .
Recall that π 6 (ΣCP 2 ) ∼ = Z/6Z{Eπ 2 } [Muk, Theorem 8.2(i)], where π 2 : S 5 → CP 2 is the Hopf map, and E is the suspension of a map.
Proposition 5.3. Let V ≃ S 2 ∪ kη2 e 4 ∪ b e 6 be a closed smooth manifold.
• If k is odd, then V is Spin and
ΣV ≃ ΣCP 2 ∪ k ′ Eπ2 e 7 ,
where k ′ = 1 or 3 such that k ′ ≡ ±k mod 6;
• If k is even and V is non-Spin
ΣV ≃ S 3 ∨ Σ 3 CP 2 ;
• If k is even and V is Spin
ΣV ≃ (S 3 ∪ b ′ η 3 3 e 7 ) ∨ S 5 ,
where b ′ ∈ Z/2 is from Theorem 5.1.
Proof. It is clear that the decompositions for the two cases when k is even follows immediately from Theorem 5.1 and Remark 5.2. When k is odd, V is spin and ΣV ≃ ΣCP 2 ∪ EbW e 7 by Theorem 5.1.
Also notice that ΣCP 2 ∪ EbW e 7 ≃ ΣCP 2 ∪ −EbW e 7 . Hence, to prove the statement in the proposition it suffices to show that the suspension map E : π 5 (S 2 ∪ kη2 e 4 ) → π 6 (ΣCP 2 ) sends the generator b W to kEπ 2 up to sign.
For this purpose, start with the diagram of homotopy cofibrations (9)
S 3 k kη2 / / S 2 / / S 2 ∪ kη2 e 4
which defines the map r. Then there is the diagram of homotopy fibrations (10)
S 1 / / Z / / r S 2 ∪ kη2 e 4 fx / / r K(Z, 2) S 1 / / S 5 π2 / / CP 2 fc / / K(Z, 2),
where f c and f x represent the generators c ∈ H 2 (CP 2 ; Z), and x ∈ H 2 (S 2 ∪ kη2 e 4 ; Z) respectively, and Z is the homotopy fibre of f x mapping to S 5 by the induced mapr. By analyzing the Serre spectral sequences of the homotopy fibrations in Diagram (10), it can be showed that Z ≃ P 4 (k) ∪ e 5 andr * : H 5 (S 5 ; Z) → H 5 (Z; Z) is of degree k. Since by Lemma 6.3 π 4 (P 4 (k)) = 0 when k is odd, we see that Z ≃ P 4 (k) ∨ S 5 , and thenr * is of degree k on homology. Moreover, by the naturality of the Hurewicz homomorphism and Lemma 6.4, it is easy to see thatr * : π 5 (Z) ∼ = Z → π 5 (S 5 ) is of degree k. It follows that r * : π 5 (S 2 ∪ kη2 e 4 ) ∼ = Z → π 5 (CP 2 ) ∼ = Z is of degree k by Diagram (10).
Now the naturality of suspension map induces the following commutative digram (11)
π 5 (S 2 ∪ kη2 e 4 ) r * / / E π 5 (CP 2 ) E π 6 (ΣCP 2 ) Er * / / π 6 (ΣCP 2 ),
where E : π 5 (CP 2 ) ∼ = Z → π 6 (ΣCP 2 ) ∼ = Z/6Z is surjective by [Muk,Theorem 8.2(i)]. We have showed that r * in Diagram (11) is of degree k. On the other hand, from the last column of Diagram (9) we have the homotopy cofibraiton ΣCP 2 Er −→ ΣCP 2 −→ P 6 (k).
Applying the Blakers-Massey Theorem [BM], we obtain the exact sequence (12) π 6 (ΣCP 2 ) Er * −→ π 6 (ΣCP 2 ) −→ π 6 (P 6 (k)).
Since π 6 (P 6 (k)) = 0 by Lemma 6.3 and π 6 (ΣCP 2 ) ∼ = Z/6Z{Eπ 2 }, we see that Er * is an isomorphism from (12). Then by Diagram (11), it follows that E : π 5 (S 2 ∪ kη2 e 4 ) → π 6 (ΣCP 2 ) sends the generator b W to kEπ 2 up to sign. This proves the statement in the case when k is odd, and we have completed the proof of the proposition.
Σ 2 M ≃ Σ 2 M 1 ∨ m i=1 (S 5 ∨ S 5 ) ≃ Σ 2 V ∨ P 6 (T ) ∨ P 5 (T ) ∨ m i=1 (S 5 ∨ S 5 ),
where M 1 is a closed 6-manifold with homology of the form (2) such that b 3 (M 1 ) = 0 and d = 1.
Moreover, By Corollary 3.3 and the assumption on the ring structure of H * (M ; Z), V ≃ S 2 ∪ kη2 e 4 ∪ b e 6 for some attaching map b. Denote λ = b ′ in Theorem 5.1. The theorem for the two cases when k is even then follows immediately from Proposition 5.3. For the case when k is odd, recall that there is the fibre bundle [HBJ, Section 1.1]
S 2 −→ CP 3 σ −→ S 4 ,
with its first Pontryagin class p 1 = 4s 4 where s 4 ∈ H 4 (S 4 ; Z) is a generator. Then pullback this bundle along the self-map of S 4 of degree 3, we obtain the 6-manifold V 3 with bundle projection σ 3 onto S 4 in the following diagram of S 2 -bundles
S 2 / / V 3 σ3 / / S 4 3 S 2 / / CP 3 σ / / S 4 .
From this diagram, it is easy to see that the first Pontryagin class of σ 3 is 12s 4 as required and
x 2 = 3y, where by abuse of notation x, y ∈ H * (V 3 ; Z) are two generators such that deg(x) = 2 and deg(y) = 4. Hence by Proposition 5.3, ΣV ≃ ΣCP 3 when k ≡ ±1 mod 6 and ΣV ≃ ΣV 3 when k ≡ 3 mod 6, and then the two decompositions when k is odd in the theorem follows. This completes the proof of the theorem.
Proof of Corollary 1.4. As the discussions before Corollary 1.4, the number k mod 2 and the spin condition of M are determined by the Steenrod square Sq 2 . Since the attaching maps of the top cells of ΣCP 3 and ΣV 3 are Eπ 2 of order 6 and 3Eπ 2 of order 2 respectively by Proposition 5.3, after localization at 3 we can consider the Steenrod power P 1 : H 3 (ΣM ; Z/3Z) → H 7 (ΣM ; Z/3Z). Then since ΣV 3 ≃ (3) S 3 ∨S 5 ∨S 7 , P 1 acts trivially on its cohomology. In contrast, ΣCP 3 ≃ (3) S 3 ∪ α1 e 7 ∨S 5 withe α 1 an element detected by P 1 [Har, Section 1.5.5]. Hence, ΣCP 3 and ΣV 3 can be distinguished by P 1 . Moreover, the stable cube element η 3 n ∈ π n+3 (S n ) (n ≥ 2) is detected by the secondary operation T [Har,Exercise 4.2.5]. And there is no indeterminacy since in the either case S 4 ∪ λη 3 4 e 8 splits off as a wedge summand of the double suspension of the manifold. From the above discussions on cohomology operations, we can prove the corollary easily by the decompositions in Theorem 1.3.
Some computations on homotopy groups of odd primary Moore spaces
In this section, we work out the homotopy groups of Moore spaces used in Section 3. Consider the Moore space P 2n+1 (p r ) with n ≥ 1, p ≥ 3 and r ≥ 1. We have the homotopy fibration
(13) F 2n+1 {p r } −→ P 2n+1 (p r ) q −→ S 2n+1 ,
where q is the pinch map of the bottom cell. Cohen-Moore-Neisendorfer proved the following the famous decomposition theorem.
Theorem 6.1. [CMN, N] Let p be an odd prime. There is a p-local homotopy equivalence
ΩF 2n+1 {p r } ≃ (p) S 2n−1 × ∞ k=1 S 2p k n−1 {p r } × ΩΣ α P nα (p r ),
where S i {p r } is the homotopy fibre of the degree map p r : S i → S i , and α P nα (p r ) is an infinite bouquet of mod p r Moore spaces with only finitely many in each dimension and the least value of n α is 4n − 1.
We also need the following classical result.
Lemma 6.2. [N2, Proposition 6.2.2] Let p be an odd prime.
P m (p r ) ∧ P n (p r ) ≃ P m+n (p r ) ∨ P m+n−1 (p r ).
Lemma 6.3. [ST, So] Let p be an odd prime.
π 3 (P 3 (p r )) = Z/p r Z, π n (P n (p r )) = 0, for n ≥ 4.
Proof. The cases when n = 3 and 4 were already proved in [ST,Lemma 2.1] and [So,Lemma 3.3] respectively, while the remaining cases follow immediately from the Freudenthal Suspension Theorem.
Lemma 6.4. Let p be an odd prime.
π n+1 (P n (p r )) = 0, for n ≥ 3.
Proof. π 4 (P 3 (p r )) = 0 was showed in [So,Lemma 3.3]. Let us consider π 5 (P 4 (p r )). By the classical EHP-sequence (Chapter XII, Theorem 2.2 of [Whi]), there is the exact sequence 0 = π 4 (P 3 (p r )) → π 5 (P 4 (p r )) H → π 5 (P 4 (p r ) ∧ P 3 (p r )) P → π 3 (P 3 (p r )) → π 4 (P 4 (p r )) = 0.
By Lemma 6.2, π 5 (P 4 (p r ) ∧ P 3 (p r )) ∼ = π 5 (P 6 (p r ) ∨ P 7 (p r )) ∼ = Z/p r Z.
Hence, by Lemma 6.3 and the above exact sequence, P is an isomorphism and then π 5 (P 4 (p r )) = 0. The remaining cases follow immediately from the Freudenthal Suspension Theorem, and this completes the proof of the lemma.
In the remaining two lemmas, we exclude the case when p = 3.
Lemma 6.5. Let p ≥ 5.
π n+2 (P n (p r )) = 0, for n ≥ 6.
Proof. By the Freudenthal Suspension Theorem, it suffices to show π 9 (P 7 (p r )) = 0. For that let us compute π 9 (F 7 {p r }) first. By Theorem 6.1, π 9 (F 7 {p r }) ∼ = π 8 (ΩF 7 {p r }) ∼ = π 8 (S 5 ) (p) .
Since π 8 (S 5 ) ∼ = Z/24Z and p ≥ 5, π 9 (F 7 {p r }) = 0. Now from the exact sequence of homotopy groups of the homotopy fibration (13) (n = 3) 0 = π 9 (F 7 {p r }) → π 9 (P 7 (p r )) → π 9 (S 7 ) (p) = 0, we see that π 9 (P 7 (p r )) = 0.
Lemma 6.6. Let p ≥ 5. The suspension morphism E : π 6 (P 4 (p r )) → π 7 (P 5 (p r )) ∼ = Z/p r Z is trivial.
Proof. On the one hand there is the EHP-sequence of P 4 (p r ) π 6 (P 4 (p r )) E → π 7 (P 5 (p r )) H → π 7 (P 5 (P r ) ∧ P 4 (p r )) → π 5 (P 4 (p r )) = 0,
where π 5 (P 4 (p r )) = 0 by Lemma 6.4, and π 7 (P 5 (P r ) ∧ P 4 (p r )) ∼ = π 7 (P 8 (P r ) ∧ P 9 (p r )) ∼ = Z/p r Z by Lemma 6.2. It follows that (14) π 7 (P 5 (p r ))/Im(E) ∼ = Z/p r Z.
On the other hand there is the EHP-sequence of P 5 (p r ) π 9 (P 6 (P r ) ∧ P 5 (p r )) P → π 7 (P 5 (p r )) → π 8 (P 6 (p r )) = 0,
where π 8 (P 6 (p r )) = 0 by Lemma 6.5, and π 9 (P 6 (P r ) ∧ P 5 (p r )) ∼ = π 9 (P 10 (P r ) ∧ P 11 (p r )) ∼ = Z/p r Z by Lemma 6.2. It follows that (15) π 7 (P 5 (p r )) ∼ = Z/p r Z/Ker(P ).
Combining (14) and (15), we see that π 7 (P 5 (p r )) ∼ = Z/p r Z, and Im(E) = Ker(P ) = 0. The proof of the lemma is completed.
Corollary 1 . 4 .
14Let M and M ′ be two manifolds in Theorem 1.3. Then Σ 2 M ≃ Σ 2 M ′ if and only if H * (Σ 2 M ; Z) ∼ = H * (Σ 2 M ′ ; Z) as abelian groups, and H * (Σ 2 M ; Z/pZ) ∼ = H * (Σ 2 M ′ ; Z/pZ) as Z/2Z{Sq 2 , T}-modules when p = 2, and as Z/3Z{P 1 }-modules when p = 3.
Acknowledgements. Ruizhi Huang was supported by National Natural Science Foundation of China (Grant nos. 11801544 and 11688101), and "Chen Jingrun" Future Star Program of AMSS.
H 2 (M ; Z) ∼ = Z By Lemma 3.6, we may consider the torsion free case first. In [Yam], Yamaguchi classified the homotopy types of CW -complexes of the form V ≃ S 2 ∪ e 4 ∪ e 6 . Specifying to the case when V is a manifold, we can summarized the necessary result in the following theorem (c.f. [Bau, Section 1]). Theorem 5.1. [Yam, Corollary 4.6, Lemma 2.6, Lemma 4.3] Let V ≃ S 2 ∪ kη2 e 4 ∪ b e 6 be a closed smooth manifold, where kη 2 with k ∈ Z and b are the attaching maps of the cells e 4 and e 6 re-
Nevertheless, Theorem 1.1 is still useful, for instance, to calculate the K-group or the KO-groupof M in Corollary 1.2. In particular, when M is a Calabi-Yau threefold it partially reproduces the result of Doran and Morgan [DM, Corollary 1.10] on its K-group by different method, and provides new computation on its KO-group. Moreover, there are many examples of simply connected Calabi-Corollary 1.2. Let M be the manifold in Theorem 1.1. Then for the reduced K-group and KO-Yau threefolds. For instance, based on Kreuzer and Skarke [KSk], Batyrev and Kreuzer [BK] showed
that there are exactly 473 800 760 families of simply connected Calabi-Yau 3-folds corresponding to
4-dimensional reflexive polytopes.
Table ( 1
() we can easily calculate the following, where only KO2
Proof of Theorem 1.2. Let us only compute the KO-group of M when 1 ≤ c ≤ d − 1, while theother cases can be computed similarly. By Theorem 1.1 and Lemma 4.3, we have
Simply connected five-manifolds. D Barden, Ann. Math. 821D. Barden, Simply connected five-manifolds, Ann. Math. 82 (1965), 365-385. 1
The homotopy type of the loops on (n − 1)-connected (2n + 1)-dimensional manifolds. Sa, Basu, Algebraic topology and related topics. SingaporeBirkhauser/SpringerSa. Basu, The homotopy type of the loops on (n − 1)-connected (2n + 1)-dimensional manifolds, Algebraic topology and related topics, pp. 1-25, Trends. Math., Birkhauser/Springer, Singapore, 2019. 1
Homotopy groups of highly connected manifolds. Sa, So Basu, Basu, Adv. Math. 3371Sa. Basu and So. Basu, Homotopy groups of highly connected manifolds, Adv. Math 337 (2018), 363-416. 1
Integral cohomology and mirror symmetry for Calabi-Yau 3-folds, from. V V Batyrev, M M Kreuzer, Mirror Symmetry V: Proceedings of the BIRS Workshop on Calabi-Yau Varieties and Mirror Symmetry. 38V. V. Batyrev and M. M. Kreuzer, Integral cohomology and mirror symmetry for Calabi-Yau 3-folds, from: "Mirror Symmetry V: Proceedings of the BIRS Workshop on Calabi-Yau Varieties and Mirror Symmetry (December 2003)", AMS/IP Stud. Adv. Math. 38 (2006), 255-270. 1
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| [] |
[
"Domain walls and perturbation theory in high temperature gauge theory: SU(2) in 2+1 dimensions",
"Domain walls and perturbation theory in high temperature gauge theory: SU(2) in 2+1 dimensions"
] | [
"C Korthals Altes \nCentre Physique Theorique au CNRS\nB.P. 90713288Luminy, MarseilleFrance\n",
"A Michels \nTheoretical Physics\nUniversity of Oxford\n1 Keble RoadOX1 3NPOxfordUK\n",
"M Stephanov \nDepartment of Physics\nUniversity of Illinois at Urbana-Champaign\n1110 West Green Street61801UrbanaUSA\n",
"M Teper \nTheoretical Physics\nUniversity of Oxford\n1 Keble RoadOX1 3NPOxfordUK\n"
] | [
"Centre Physique Theorique au CNRS\nB.P. 90713288Luminy, MarseilleFrance",
"Theoretical Physics\nUniversity of Oxford\n1 Keble RoadOX1 3NPOxfordUK",
"Department of Physics\nUniversity of Illinois at Urbana-Champaign\n1110 West Green Street61801UrbanaUSA",
"Theoretical Physics\nUniversity of Oxford\n1 Keble RoadOX1 3NPOxfordUK"
] | [] | We study the detailed properties of Z 2 domain walls in the deconfined high temperature phase of the d = 2 + 1 SU(2) gauge theory. These walls are studied both by computer simulations of the lattice theory and by one-loop perturbative calculations. The latter are carried out both in the continuum and on the lattice. We find that leading order perturbation theory reproduces the detailed properties of these domain walls remarkably accurately even at temperatures where the effective dimensionless expansion parameter, g 2 /T , is close to unity. The quantities studied include the surface tension, the action density profiles, roughening and the electric screening mass. It is only for the last quantity that we find an exception to the precocious success of perturbation theory. All this shows that, despite the presence of infrared divergences at higher orders, high-T perturbation theory can be an accurate calculational tool. | 10.1103/physrevd.55.1047 | [
"https://export.arxiv.org/pdf/hep-lat/9606021v1.pdf"
] | 15,362,351 | hep-lat/9606021 | 4e5074d90abc9046e30aa161b69997437edee62e |
Domain walls and perturbation theory in high temperature gauge theory: SU(2) in 2+1 dimensions
Jun 1996
C Korthals Altes
Centre Physique Theorique au CNRS
B.P. 90713288Luminy, MarseilleFrance
A Michels
Theoretical Physics
University of Oxford
1 Keble RoadOX1 3NPOxfordUK
M Stephanov
Department of Physics
University of Illinois at Urbana-Champaign
1110 West Green Street61801UrbanaUSA
M Teper
Theoretical Physics
University of Oxford
1 Keble RoadOX1 3NPOxfordUK
Domain walls and perturbation theory in high temperature gauge theory: SU(2) in 2+1 dimensions
Jun 1996arXiv:hep-lat/9606021v1 27
We study the detailed properties of Z 2 domain walls in the deconfined high temperature phase of the d = 2 + 1 SU(2) gauge theory. These walls are studied both by computer simulations of the lattice theory and by one-loop perturbative calculations. The latter are carried out both in the continuum and on the lattice. We find that leading order perturbation theory reproduces the detailed properties of these domain walls remarkably accurately even at temperatures where the effective dimensionless expansion parameter, g 2 /T , is close to unity. The quantities studied include the surface tension, the action density profiles, roughening and the electric screening mass. It is only for the last quantity that we find an exception to the precocious success of perturbation theory. All this shows that, despite the presence of infrared divergences at higher orders, high-T perturbation theory can be an accurate calculational tool.
Introduction
Non-Abelian gauge theories possess many surprising aspects. An example is the linear confining potential present in both the three and four dimensional SU(N) theories at low temperature, T . At some finite T there is a phase transition and confinement is then lost [1,2]; but this is not unexpected because simple energy versus entropy arguments tell us that a confining 'flux tube' will condense into the vacuum at some finite value of the temperature. These phenomena are non-perturbative and have so far defied analytic, as opposed to numerical, approaches. However, at sufficiently high T , there would appear to be an important theoretical simplification. All these theories are asymptotically free so that the effective interaction on the relevant energy scale, T , should become small at high T and the physics of the gluon plasma should become accurately calculable in perturbation theory. Unfortunately there are infrared divergences in higher orders of perturbation theory, which are associated with the perturbative masslessness of the magnetic gluon. Although we expect this gluon to acquire a mass through the non-perturbative physics of the dimensionally reduced theory, this leaves room for uncertainty about how reliable high-T perturbation theory really is.
However even this naive picture of gauge theories at high T -as a weakly interacting plasma of gluons -contains surprises. There turns out to be a symmetry associated with the centre of the group, Z(N), which is spontaneously broken at high T . Separating two different Z(N) vacua will be a domain wall whose properties have been calculated in perturbation theory for T → ∞ [3]. However the reality of these domain walls is controversial for a variety of reasons. Firstly, there are the general doubts about high-T perturbation theory that we alluded to above. Secondly, all this is in the usual Euclidean formulation of finite temperature field theories and there is a question of what if anything these domain walls might correspond to in Minkowski space-time. Finally the walls have peculiar thermodynamic properties, which become more acute when one includes quarks into the theory [4]. It is important to resolve these uncertainties not only because of the theoretical interest of these domain walls, but also because related structures can be associated with important physical phenomena when one considers the Standard Model in the early universe [5].
In this paper we address the particular problem associated with the uncertain status of perturbation theory at high T . We shall do so by calculating the properties of the domain walls both in perturbation theory and by a fully non-perturbative Monte Carlo computer simulation. We shall work with the SU(2) gauge group because the problems should not be any different for larger groups. Moreover we shall work in 2+1 dimensions rather than in the more physical case of 3+1 dimensions. The reason is that the computational resources needed are much less in the former case, and only there can we perform calculations with enough precision and control to be really useful. At the same time, the origin of the infrared problems is similar in d = 2 + 1 and in d = 3 + 1, and their severity is, if anything, greater in the lower dimensional case.
We shall find in the calculations described below that one-loop perturbation theory does indeed work remarkably -even precociously -well for our Z 2 domain walls. By implication, this provides evidence for the general applicability of perturbation theory at high T .
At the same time we emphasise that we make no attempt, in this paper, to address the other controversial aspects of domain walls and so do not attempt to settle the interesting question of their potential role in, for example, separating bubbles of different vacua in the early universe.
We shall now outline the contents of this paper. In Section 2 we give a heuristic introduction to domain walls and the thermodynamics of gauge theories at high temperatures. This will provide the general background for the more detailed and specific calculations of the later sections. We then turn to the perturbative calculation, at one-loop, of the properties of domain walls. In Section 3 we perform the calculation of the surface tension for the continuum theory. The calculation is by a method that easily extends to other dimensions and so, as well as obtaining expressions for d = 2 + 1, we can compare with previous d = 3 + 1 calculations. In order to compare with our later numerical work it is useful to have similar results for a finite value of the lattice spacing a. This calculation is carried out in Section 4. In Section 5 we calculate the action density in the domain wall, as a function of the distance from the centre of the wall, since this is one of the quantities we shall later calculate numerically. In Section 6.1 we calculate the effects of oscillations of the wall once its length is large -the 'roughening' of the wall, which has been neglected in most previous studies. In Section 6.2 we show how the finite size of the volume, in the direction orthogonal to the wall, affects the profile and surface tension of the wall. This is important to understand since our numerical work will necessarily be on lattices of a finite size. We then turn, in Section 7, to a description of our simulations and the results we obtain thereby. We begin, in Section 7.1, by describing how we can simulate domain walls through the use of twisted boundary conditions. In Section 7.2 we point out that these domain walls can be viewed as 't Hooft disorder loops that have been squashed by the short Euclidean time direction. Section 7.3 describes how we calculate the properties of the domain wall and the electric screening mass. Section 7.4 lists the large-volume raw 'data' from which we will eventually extract physical quantities. Section 7.5 describes in detail how we control finite volume effects. This is crucial since the potential problems with perturbation theory are infrared ones. Finally, in Section 7.6 we compare our numerical results with those of perturbation theory. Section 8 contains our conclusions.
Our theoretical analysis, in Sections 3 to 6, is performed for the general case of SU(N) gauge fields in 2 < d ≤ 4 dimensions. Our numerical results, on the other hand, are for the particular case of SU(2) gauge fields in d = 2 + 1. The preliminary results of this study appeared in the Proceedings of the 1994 Lattice Conference [6]. A study of the case of SU(3) gauge fields in d = 2 + 1 has recently been reported [7]. Both the SU(2) and SU (3) results are in agreement with our theoretical analysis. In addition both show the same large deviation from D'Hoker's self-consistent formula for the Debye mass.
General considerations
In 2+1 dimensions the gauge coupling, g 2 , is dimensionful and its value sets the mass scale for the theory. If we perform a perturbative calculation of a quantity in which there is a dominant momentum scale, Q, then the effective expansion parameter will clearly be g 2 /Q so that the theory rapidly becomes free at short distances. So at high T the effective expansion parameter, g 2 /T , will be small and we can expect that we should be able to apply perturbation theory. All this is very similar to the case in 3+1 dimensions. There the coupling is dimensionless but this difference is only apparent: the scale-invariance is anomalous, the coupling runs and its value only serves to set the overall mass scale (dimensional transmutation). The coupling becomes small at short distances, so that its value at high temperature, g 2 (T ), should be small enough for us to apply perturbation theory. In other ways the two theories are also similar: numerical simulations [25] show that the d = 2 + 1 theory has linear confinement, a deconfining temperature, T c , and a glueball spectrum that are similar in many ways to that of the theory in 3+1 dimensions.
So we expect a hot gauge theory to be, to a very good approximation, a plasma of free gluons, with interactions given in terms of the small effective coupling at the ambient temperature T . These gluons are screened just as are photons in a plasma of charged particles. The Debye screening mass, m D , that they acquire grows with T ; to lowest order m 2 D is O(g 2 ) so we expect, on purely dimensional grounds, that m D ∼ g(T )T in d = 3 + 1 and m D ∼ gT 1/2 in d = 2 + 1.
However this simple picture is not the whole story. An additional and important role is played by the center, Z(N), of the gauge symmetry group SU(N). This sub-group has a special status because the gluons, which transform according to the adjoint representation, are invariant under gauge transformations that belong to the centre. Since the gluons only feel SU(N) gauge transformations modulo any Z(N) transformation, their symmetry group is really SU(N)/Z(N). Sources in the fundamental representation, on the other hand, are not invariant under transformations in Z(N). Consider such a heavy source in the usual Euclidean space-time formulation of the high-T field theory where the Euclidean time is periodic with period 1/T . As is well known the presence of such a static heavy source leads to an extra factor of P ≡ (1/N) tr P exp{i A 0 dt} -a Polyakov loop -in the partition function. In the low T confining phase P = 0 while at high T | P | ≃ 1. Since the gluons are screened, the correlations are short range at high T and so if, for example, P is close to 1 at one point, the vacuum will have P ≃ 1 everywhere. But the Z(N) symmetry tells us that there must be N such vacua, in each of which the physics is identical, and which are differentiated by P being close to one of the N complex N'th roots of unity. The picking of one of these vacua corresponds to the spontaneous breaking of the symmetry. That the spontaneous breaking takes place at high T rather than at low T is remarkable but not impossible; it is, for example, a commonplace in (self-)dual theories.
As soon as we have spontaneous symmetry breaking we have the possibility of domain walls which will occur at the interface between two of the vacua. (Throughout this paper we shall call these objects 'walls', and will speak of their 'surface' tension, even though the interface is really a string when we are in 2 spatial dimensions. This is to avoid confusion with the confining string.) In the case of interest to us, SU(2), such a domain wall would separate regions of our Euclidean space-time volume that are characterised by P ≃ +1 on the one side and P ≃ −1 on the other. One can compute the free energy density of the wall for small coupling and one indeed finds a positive excess over the free energy of the gluon plasma.
Thus we have a picture of the gluon plasma that parallels that of a ferro-magnetic substance below the Curie temperature: the average of the Polyakov loop, which arises from the heavy fundamental source, being the order parameter. But what is the analogue of the external field? Such a field is needed in order to make the system choose a particular direction of magnetisation on a macroscopic scale. Without such an analogue one cannot trigger a thermodynamical state where our order parameter, P , takes the value +1 or one where it takes the value −1.
As we remarked in the Introduction, the 'reality' of these domain walls is controversial. Of course, the walls have been observed [8] in (d = 3 + 1) Monte Carlo simulations of the theory, but this has only been for large lattice spacings, and there have been speculations that the walls would not survive into the continuum limit [9]. In this paper we shall address the question of whether these domain walls do indeed exist in the continuum limit and, more specifically, whether high-T perturbation theory is reliable. But first, in this section, we introduce the basic framework within which we work.
Yang-Mills fields: the basic parameters
We consider a pure SU(N) gauge theory in 2 or 3 space dimensions. We will concentrate in this subsection on the thermodynamic quantities that one can define by enclosing the system in a box of size L y L z . (For the threedimensional case we add an x-direction.) The Hamiltonian for this system reads:
Ĥ ≡ 1 2 x tr g 2Ê 2 + 1 g 2B 2 ,(1)
whereÊ is the canonical momentum for andB mn ≡ ∂ mÂn − ∂ nÂm + i[ m , n ]. We use the standard notation for the fields as N × N matrices in the Lie algebra of the defining representation of SU(N). We introduce the free energy F of the system in a heat bath at temperature T through the Gibbs trace over the physical states of the system. By 'physical' we mean that the states obey the Gauss constraint
∇Ê + i[Â,Ê] |ψ = 0.(2)
The free energy is defined by
exp − F T = Tr phys exp −Ĥ T .(3)
As is well known, the Gibbs trace can be related to the Feynman path integral by
Tr phys exp −Ĥ T = DA 0 DA exp − 1 g 2 S(A) .(4)
Here the integration over A 0 implements the Gauss law, the action S(A) is equal to (1/4) dxdt trF µν F µν , and the temperature enters the formalism through the fact that we make the potentials periodic in the Euclidean time t with period 1/T . The relationship (4) is fundamental in that it allows us to calculate the free energy using the whole panoply of methods available for calculating path integrals; such as perturbation theory(see Sections 3 and 4) and Monte Carlo methods (see Section 7). Throughout this paper we suppose the potentials to be periodic in the spatial directions. However, this does not mean that any gauge transformation Λ has to be periodic, only that it should be periodic modulo an element, z k = exp{ik 2π N }, of the centre Z(N) of SU(N). That this is so is easily seen from the transformation properties of the potential:
A Λ µ = Λ † A µ Λ − iΛ † ∂ µ Λ.(5)
If say Λ(L z ) = Λ(0)z k , then the z k will commute with all the matrices in (5) and so will disappear from the right hand side, leaving the transformed potential still periodic [10]. We shall denote such gauge transformations bŷ Λ k . If k = (k y , k z ) then this is a gauge transformation which is periodic up to z ky in the y-direction and up to z kz in the z-direction. The interest of these extended gauge transformations lies in two facts:
• they leave the Hamiltonian (1) invariant;
• they serve to distinguish subspaces, in the space of physical states, which possess a given number, e y and e z , of electric fluxes in the y and z directions respectively. Clearly this distinction is modulo N.
The above notion of electric flux is developed in detail in [10]. The following remarks represent no more than a heuristic outline. Suppose first that we have opposite fundamental sources at x 1 and x 2 . To make this system gauge invariant we need to join the sources by a finite string P exp{i x 2 x 1 dx} running between them. Such a string operator creates the unit fundamental (electric) flux that must flow between the sources. Suppose we are now in the purely gluonic system with no sources and suppose we wish to add a unit of electric flux across the whole volume, running in the z-direction. From the above we expect that we can do so by applying to our state the periodic string operator tr P exp{i Lz 0 dz z }. Unlike a contractible string operator loop, which would represent some local excitation, this operator will clearly feel the centre element of the gauge transformationΛ k if k z = 0. Indeed it is easy to see that it will acquire a factor of z kz . If we create a state with n such units of electric flux in the z-direction, then it will acquire a factor of (z kz ) n . So if we label a state with electric flux e = (e y , e z ) by |e , it will be an eigenstate ofΛ k with eigenvalue exp{ik·e 2π N }, (assuming trivial transformation properties for the fluxless state |0 ). Clearly the state with N + 1 fluxes has the same transformation property as that with 1 unit of flux; as one would expect in a non-Abelian theory.
Since the Hamiltonian is invariant underΛ k , the energy eigenstates can be simultaneously labeled by e and we can define a free energy F e by restricting the Gibbs trace in (3) to a given electric flux sector:
exp − F e T = Tr e exp −Ĥ T(6)
With this definition relation (3) can be rewritten as [10]:
exp − F e T = 1 N d−1 k exp −ik·e 2π N Z k ,(7)
where
Z k ≡ Tr phys exp −Ĥ TΛ k .(8)
The Gibbs traces Z k in (8) can be expressed as "twisted" path integrals and can be computed using Monte Carlo methods. These path integrals have the following defining property. Suppose, for example, that k = (0, k z ). Then due to the occurrence of the transformationΛ k in (8) one picks up a factor of exp{ik z 2π/N} in the gauge transform relating A(t, y, 0) to A(t, y, L z ) and A(0, y, z) to A(L t , y, z), after going around the boundary of the box in the z-t directions. This multivaluedness does not affect the gluon field A µ . In Section 7 we will perform Monte Carlo simulations of such a twisted partition function.
The two important parameters: string and surface tension
Now we are ready to give a thermodynamic characterisation of the various phases of the gauge theory, through the behaviour of the flux free energies F e as the temperature is varied. To obtain simple formulas we will restrict ourselves to SU(2), but the end results will be valid for any SU(N). The interesting quantity is the flux free energy F 01 in the elongated direction. We want to compare it to the flux free energy F 00 and see how the difference behaves for low and high temperatures. This is straightforward using (7) for the flux free energies. But we still need some theoretical input on how the twisted functionals behave. What one finds for low T is that
1 − Z 01 Z 00 = C exp − ρ(T ) T L z ,(9)
while 1 − (Z ky1 /Z 00 ) is exponentially smaller for any k y = 0. On the other hand for high enough T one finds:
Z 01 = D exp − σ(T ) T L y Z 00(10)
and Z ky1 is exponentially smaller for any k y = 0. The C and D are some pre-exponential factors. The evidence for (9) and (10) comes from Monte Carlo simulations, as well as analytic Hamiltonian analyses of gauge Potts models [11]. So one gets for the free energy difference:
F 01 − F 00 ∼ ρ(T )L z if T is small (11) F 01 − F 00 ∼ L z exp − σ(T ) T L y if T is large(12)
Clearly what (11) is telling us is that at low temperatures we are in a confining regime, where imposing unit electric flux across the lattice costs us an energy that is proportional to the length traversed by the flux; and the tension of this flux 'string' is ρ. So the free energy difference becomes very large at large L z , but is insensitive to the transverse spatial dimensions. At some critical temperature T c this behaviour changes into that of (12). The free energy difference now becomes exponentially small with the transverse size. This behaviour suggests that there is a wall, with an energy density independent of the transverse direction, and a total energy proportional to σ. This quantity has been computed in a semiclassical approximation [3,12] at very high temperatures, where perturbation theory should apply. The purpose of the present study is to check the validity of this approximation using Monte Carlo methods.
The way the surface tension enters the free energy difference is through the exponential. This was first noticed by Bhattacharya et al. [13] and was recently discussed in a Z(2) gauge model [14]. It is reminiscent of an energy difference induced by tunneling. As we will see in the next section this is indeed a tunneling through a potential that arise from quantum one-loop effects.
Effective Action and Polyakov Loops
As we have seen above, the fundamental quantity of interest is the ratio of twisted path integrals Z k . These are computed by converting from the vector potentials, as integration variables, to Polyakov loops:
Ω(x) ≡ P exp i 1/T 0 dtA 0 (t, x)(13)
and by integrating out the remaining variables to get an effective action for Ω(x). In a suggestive notation:
exp{−S eff (Ω)} = DA δ Ω − P exp i 1/T 0 dtA 0 exp − 1 g 2 S(A)(14)
The reader, in looking at this equation, should keep in mind that only the eigenvalues λ i of the loop Ω are gauge invariant. So it is only these that should appear in the delta function constraint and in S eff . Note also the relation between Ω and P :
P = 1 N trΩ = 1 N i λ i .(15)
The S eff has been worked out [13,12,15] to two loop order in the d = 3 + 1 case. In Section 3 we will derive the one loop result for any d, and in particular for the case of interest in this paper, d = 2 + 1.
It is important to note that the effective action does not depend on the boundary conditions if the volume is large enough. It is also easy to see that the twist k = (0, k z ) of the previous section corresponds to the following boundary conditions in the effective theory for Ω(y, z):
Ω(y, L z ) = exp{ik z 2π N }Ω(y, 0).(16)
With such boundary conditions Z k is given by:
Z k = (k) DΩ exp (−S eff (Ω)) .(17)
We also note that the path integral has a formal resemblance to that of a spin model partition function. The effective action, when evaluated in perturbation theory, will start with a classical kinetic term. The first nonzero contribution for constant Ω(x) appears at one-loop. So to this order we can write:
S eff (Ω) = x tr T 2g 2 |∇Ω| 2 + V eff (Ω) ,(18)
At high T the coefficient of the gradient term is large and we can expect that the path integral will be saturated by smooth configurations of Polyakov loops. We shall see in the next section that V eff reaches its minima when all N eigenvalues of Ω coincide. Due to a condition det Ω = 1 there are N such minima: Ω ∈ Z(N). The order parameter distinguishing between these N degenerate phases is the value of the Polyakov loop P (15). The Z(N) symmetry in the effective theory of Polyakov loops is due to the existence of gauge transformations which are periodic in the Euclidean t direction up to an element of Z(N): Λ(1/T ) = Λ(0)z k . These transformations leave S(A) invariant but multiply Polyakov loops by z k .
In spin model language, we are in an ordered phase when T is large. This is a phase in which the entropy of the spin system is low. However the spin system is, at the same time, supposed to be describing a very hot gauge system with a large entropy in terms of quantum states. It is this complementary feature of the spin and gauge systems, that has given rise to a lot of confusion in the past few years. For example, one can ask the following question: what is the meaning of a localised surface in the spin model in terms of the gauge model? In this paper we will not go into this question, but take the pragmatic point of view that we just want to calculate the exponent in the decay law (12) for the hot fluxes. Nonetheless, one should bear in mind, that if one does assume the existence of a well localised interface, this leads to unusual thermodynamic properties; as we shall see below.
Some Thermodynamic Properties of the Wall in Gauge Theory
The study of surface effects requires a careful specification of boundary conditions. Only when that is done can we separate the free energy of the system into well defined bulk and surface free energies. Let us suppose we have in our two dimensional 'box' a domain wall, separating two domains. Then the free energy F will be for large L y and L z :
F = f L y L z + σL y(19)
Here we assume that the size of the box is much larger than any of the microscopic quantities in the system. What this equation describes is simply a wall with a constant free energy density in the y-direction, and a non-trivial free energy profile in the zdirection. When integrated over z, this profile gives the interface tension σ. This profile is like a soliton.
In our three dimensional SU(2) gauge theory we have a dimensionful coupling constant g, with dimension √ mass. The only other dimensionful quantity in our problem is the temperature T . We will typically work in the regime where g 2 /T is a small number. So on the basis of dimensions alone the interface tension has a simple form:
σ = a(g 2 /T )T 2(20)
with function a positive and dimensionless. We can apply semi-classical methods to calculate a. These methods typically give, when applied to solitons such as monopoles and sphalerons, E sol ∼ scale coupling (21) for the energy E sol in terms of the scale (Higgs expectation value). So it is not surprising that we obtain, as described in detail in the next section, a similar result for our profile:
σ = α 0 T 2 g √ T = α 0 T 5/2 g .(22)
where α is a numerical factor of geometrical origin. The expression (22) can be easily understood if one realizes that the domain wall has a width w of the order of the screening length w ∼ 1/(gT 1/2 ), carries an excess of free energy density ∆f ∼ T 3 and that σ ∼ w∆f .
We are now ready for the following thermodynamical observations. First consider a droplet of radius R of 'minus' phase in a sea of 'plus' phase. The free energy excess due to the presence of this droplet equals ∆F = 2πRσ.
Now, the probability for the appearance of such a droplet is exp{−∆F/T }. From the explicit dependence of σ on T we learn that this probability becomes exponentially small for large T . So we find that an ordered phase prevails at high temperature 1 . The second observation is a simple consequence of the positivity and the explicit temperature dependence of the interface tension. The interface entropy equals
entropy = − ∂ ∂T σ = − 5 2 α 0 T 3/2 g(24)
So the interface entropy is negative! As is the interface internal energy ǫ, although the free energy given by the difference −T σ is positive. Such a negative interface entropy is to be expected quite generally for models that order at high temperature: the free energy starts to grow at the critical temperature, and consequently the entropy is negative. So the presence of the wall diminishes the entropy of the system.
Continuum calculation
In this section we calculate S eff , the surface tension and the profile of the order parameter (Polyakov loop) of the Z(N) wall. When the temperature is high the effective gauge coupling becomes small and one can use perturbation theory to calculate the effective action (18). This has been done by several authors [16,17] for d = 4. Here we derive similar results for d = 3. Our method is simpler and is trivially generalized to arbitrary d. The lattice version of these results is presented in Section 4. We start from the partition function of the pure gauge theory at finite temperature:
Z = DA µ exp − 1 4g 2 1/T 0 dt x tr F µν F µν = DA 0 DA exp − 1 2g 2 1/T 0 dt x tr (∂ 0 A + DA 0 ) 2 + B 2(25)
We use some obvious short hand notations. The gauge potentials: A µ ≡ iA a µ T a and the hermitian generators of the SU(N) group are normalized as: tr T a T b = δ ab . The covariant derivative is:
DA 0 ≡ ∇A 0 + i[A, A 0 ].(26)
Bold symbols denote vectors (d − 1 components). The exception is B which is an antisymmetric tensor ((d − 1)(d − 2)/2 components):
B ik = ∂ i A k − ∂ k A i + i[A i , A k ](27)
In contrast to [16,17] we do not begin by fixing the gauge. First we rewrite the partition function in the form of an integral over physical fluctuations at very high T . These are the transverse components of the vector potential A. To achieve this we integrate over A 0 which is clearly an auxiliary field.
For every given configuration A(x, t) we can do this easily: the integral is Gaussian. We obtain:
Z = DA det(−D 2 ) −1/2 × exp − 1 2g 2 1/T 0 dt x tr (∂ 0 A − DD −2 D∂ 0 A) 2 + B 2(28)
Thus far our manipulations have been exact. Now we use the smallness of g. In the leading order in g (saddle point approximation) we need to keep only terms quadratic in A in the exponent and can neglect the pre-exponent. As expected the effect of A 0 is to project out the longitudinal fluctuations of A from the kinetic term in (28). The B 2 term does not contain them already. We see that the integral over A factorises into integrals over d − 2 (times N 2 − 1) transversal fluctuations and a trivial infinite factor.
Each of these integrals represents a very well known partition function of the photon gas, the logarithm of which is:
ln Z 0 = (d − 2) ln det(−∂ 2 0 − ∇ 2 ) −1/2 = −(d − 2) Vd d−1 k (2π) d−1 ln(1 − exp{−|k|/T }),(29)
where V is the volume of space. Thus we obtain the Stefan-Boltzmann law for the free energy density of the hot gluon gas (with a multiplicity of 'photons' N 2 − 1):
F = −(N 2 − 1) T V ln Z 0 = −cT d ,(30)
where c is:
c = (N 2 − 1)(d − 2) Γ(d/2) π d/2 ζ(d).(31)
Next, we want to find the dependence of the free energy density on the value of the Polyakov loop. To this end we calculate the partition function (25) about a constant background field A 0 . We make a shift:
A 0 = A ′ 0 + A 0 and integrate over A ′ 0 . This results in simply replacing ∂ 0 A with ∂ 0 A + i[A 0 , A](32)
in equation (28). For a given matrix A 0 there will be at least N − 1 generators that commute with it ('neutral') and at most N(N − 1) that do not ('charged'). The 'charges' are given by: q i −q j , where q i are the eigenvalues of the matrix A 0 /(2πT ). These are related to the eigenvalues λ i of the Polyakov loop Ω (13) by: λ i = exp{i2πq i }. The contribution of 'neutral' bosons is unchanged by (32) and is still given by (29). The partition function of the 'charged' gluons is given by:
ln Z q = (d − 2) ln det(−(∂ 0 + i2πqT ) 2 − ∇ 2 ) −1/2 = −(d − 2) Vd d−1 k (2π) d−1 ln(1 − exp{−|k|/T + i2πq}).(33)
where q = q i − q j . The periodicity in q reflects the Z(N) symmetry.
Following [13] we choose q 1 = ... = q N −1 = q/N. Only N − 1 eigenvalues are independent: i q i = 0 and thus q N = q/N − q. As q varies from 0 to 1 the Polyakov loop:
P = 1 N tr exp i T A 0(34)
changes from 1 to exp(i2π/N). In the SU(2) case q parameterises the only path between the two minima of the effective potential for the Polyakov loop. In SU(3) it parametrises the lowest action path [12]. It is possible that it does the same for N > 3. With our choice of A 0 there are (N − 1) 2 'neutral' gluons and 2(N − 1) gluons with 'charges' equal to ±q. The free energy density as a function of the Polyakov loop order parameter (parameterised by q) is now given by:
F (q) = −cT d + V (q)T d(35)
All the dependence on q comes from the 'charged' gluons and is given by the universal function:
V (q) = −2(N − 1)(d − 2) d d−1 k (2πT ) d−1 ln(1 − exp{−|k|/T }(1 − cos 2πq)) = 2(N − 1)(d − 2) Γ(d/2) π d/2 ∞ ν=1 1 ν d (1 − cos 2πqν).(36)
For d = 4 this function is expressed via the Bernoulli polynomial B 4 (q). For d = 3 only a numerical evaluation is possible.
For q varying slowly on the scale of 1/T one can use the tree expression for the gradient term [12]. Then the free energy density reads:
F (q) = N − 1 N 2πT g ∇q 2 + (V (q) − c)T d ≡ T d γ 2 2 (∇q) 2 + V (q) − c (37) where we defined a dimensionful parameter γ: γ = N − 1 N 8π 2 g 2 T d−2 .(38)
This completes the calculation of S eff of Section 2.3:
S eff = x F /T .
The profile of the wall between phases with q(−∞) = 0 and q(∞) = 1 is given by the function q 0 (z) which minimizes (37) under corresponding boundary conditions. It satisfies:
γ dq(z) dz = 2V (q).(39)
The width of the wall is controlled by γ: w ∼ γ ∼ 1/(gT d/2−1 ). The exact solution q 0 (z) for d = 4 is known [12]. For d = 3 we solve (39) numerically. The surface tension σ is given by the integral of the excess of the free energy density inside the wall and is proportional to the action on the trajectory q 0 (z):
σ = T d dz 2V (q 0 (z)) = γT d 1 0 dq 2V (q) = α 0 T d/2+1 g ,(40)
where we defined:
α 0 ≡ 4π N − 1 N 1 0 dq V (q).(41)
For d = 3 and N = 2 we find using (36): α 0 = 5.104 . An interesting property of d = 3 is that the second derivative of V (q) diverges at q = 0 mod 1 (see (36)). In fact, V (q) ∝ q 2 ln(1/q) rather than q 2 for small q. One can see from (39) that this results in a Gaussian rather than exponential fall-off in the tails of the solution. This is related to the fact that the Debye mass in the Thomas-Fermi approximation is divergent in d = 3 if the charged particles are massless: the well-known dk 2 /k 2 . On the other hand, D'Hoker argued [18] that all infrared divergences in QCD T 3 are cut off by a Debye mass m D of order g T ln(T /g 2 ) (see eq. (117)). This means that the fast Gaussian fall-off should saturate at the exponential when q gets small enough (2πqT ≪ m D ) , i.e., sufficiently far from the center of the wall. 2 This behavior of the tail does indeed occur in our Monte Carlo study (see Section 7.5).
Lattice version of the interface tension calculation
To compare numerical lattice data with theory requires a lattice version of the calculation of the previous section.
In this section we develop a one loop expression on a lattice with L t sites in the temperature direction and an infinite number in all space directions. Quantities computed in this section are the effective potential, the profile of the wall, and the surface tension. In Section 5 we calculate the expectation values of electric and magnetic plaquettes. In Section 6 the reader will find estimates of finite size corrections. In the dimensionality we are interested in we have to go to higher loops to get a finite Debye mass. We follow to this end the ideas of D'Hoker [18], which amount to a very simple prescription.
Let us first fix our notation. The continuum action of Section 3 becomes on the lattice [19]:
1 g 2 S cont → βS lat = β p Re[1 − 1 N trU p ](42)
Here every plaquette p is summed over only once; in contrast to the continuum action where the sum over µ, ν included both orders. The lattice spacing a is related to the temperature T by
aL t = 1 T (43)
which follows immediately from the fact that the length of the system in the fourth direction is 1/T .
The lattice action should become the continuum action in the limit where the dimensionless quantity aA µ becomes very small:
U P ∼ exp{ia 2 F µ,ν }.(44)
The lattice coupling becomes in that limit:
β 2N = a d−4 g 2 (45) or β 2N L 4−d t = T 4−d g 2(46)
This is the relation between lattice and continuum parameters in any number of dimensions d. For d = 4 all reference to dimensionful properties drops out. Perturbation theory is defined by taking the dimensionless parameter g 2 T 4−d small, or β large at fixed L t .
First we need a definition of the surface tension on the lattice. This can be done by taking the lattice analogues of twisted and untwisted boxes and their corresponding partition functions. This corresponds precisely to the way we measure the surface tension on the lattice (see section 7):
exp − σ T L d−2 ≡ Z(twisted) Z(untwisted) .(47)
Although our aim is to study a continuum theory, all our Monte Carlo results come from a lattice with finite lattice spacing a. The value of a has a meaning only in comparison with physical length parameters. For our system there are two such parameters: the inverse temperature 1/T and the Debye length, or the width of the wall w. The width w ∼ 1/(gT d/2−1 ) is large compared to 1/T for small g. To the order we are interested in we can neglect corrections due to finiteness of the ratio of w over 1/T or a. We choose large β so that the ratio w/a is big: 10 − 20, and we neglect corrections due to its finiteness. 3 However, to save computer time we choose to keep the ratio of 1/T to a relatively small. This ratio is the number of lattice sites in the temporal direction: L t . To compare with our MC data we calculate the quantities discussed in Section 3 for any L t to the leading order in g.
To achieve this we notice, that to the order we are working in we have a theory of free gluons interacting only with the background. All path integrals are Gaussian and factorise into integrals over momentum modes. We need only to substitute the lattice periodic momenta for continuum ones:
k i −→ 2 a sin ak i 2 ≡k i ; (48) k 0 + 2πqT −→ 2 a sin a k 0 + 2πqT 2 ≡k 0 (q).(49)
The lattice momentum varies inside the Brillouin zone: −π/a < k i < π/a. The component k 0 is discrete and takes values: 2πT n, n = 0, 1, . . . , L t − 1.
The partition function for a 'charged' gluon normalized by the 'neutral' one is given by:
ln Z q Z 0 = − d − 2 2 ln det(−k 0 (q) 2 − ik 2 i ) − ln det(−k 2 0 − ik 2 i ) . (50)
We use the proper time trick to calculate (50). To simplify formulas we work in lattice units a = 1. The r.h.s. of (50) becomes:
d − 2 2 V T k 0 π −π i dk i 2π ∞ 0 dt t exp{−(k 0 (q) 2 + ik 2 i )t} − exp{−(k 2 0 + ik 2 i )t}(51)
To rewrite the sum over k 0 = 0, 2π/L t , . . . , 2π(1 − 1/L t ) (remember that T = 1/L t in our units) we use the Poisson summation formula, which for a function f (k 0 ) periodic with a period 2π has the form:
2π(1−1/Lt) k 0 =0 f (k 0 ) = ∞ ν=−∞ π −π dk 0 2π e iLtνk 0 f (k 0 ).(52)
We substitute (52) into (51), shift the variable k 0 and obtain for ln Z q /Z 0 :
d − 2 2 V T ∞ ν=−∞ π −π µ dk µ 2π ∞ 0 dt t exp{−(k 2 0 + iL t νk 0 − i2πqν + ik 2 i )t} − exp{−(k 2 0 + iL t νk 0 + ik 2 i )} . (53)
Integration over k µ produces modified Bessel functions:
I n (2t) = π −π dk 2π exp{2t cos k + ink},(54)
and t can be rescaled after that. Finally, we obtain the expression which replaces V (q), eq. (36), at finite L t :
V lat (q) = 2(N −1)(d−2)L d t ∞ 0 dt t ∞ ν=1 e −dt I νLt (t)[I 0 (t)] d−1 (1−cos 2πqν) (55)
It is instructive to see how the continuum limit L t → ∞ (i.e., aT → 0) is recovered. For large n and large t > ∼ n the Bessel function behaves as:
e −t I n (t) ≈ 1 √ 2πt exp{− n 2 2t } and e −t I 0 (t) ≈ 1 √ 2πt .(56)
Only large t of order (νL t ) 2 will contribute to the integral in (55). After rescaling t by (νL t ) 2 and integrating we get the same expression as in (36). The variation of q with z can be also taken into account. The gradient term on the lattice becomes:
γ 2 2 (q(z + 1) − q(z)) 2 ,(57)
so that the profile equation (39) is replaced with:
q(z + 1) = q(z) + 2 γ 2 V lat (q),(58)
where similarly to the continuum case
γ 2 = 8π 2 N − 1 N T 2−d g 2 = 4π 2 N − 1 N 2 βL d−2 t .(59)
The interface tension is given by an expression similar to (40) except for α 0 being replaced by its lattice version:
α = 4π N − 1 N 1 0 dq V lat (q) .(60)
Action density profiles
An important quantity that we measure on the lattice is the expectation value of the plaquette action. In the continuum limit this corresponds to E 2 + B 2 , where the Euclidean E is:
E = −∂ 0 A − DA 0 .(61)
We also measure separately the expectation values of the plaquettes of each orientation, which correspond to E 2 y , E 2 z , B 2 in our d = 3 case. An interesting result that we find in our Monte Carlo study is that these expectation values display nontrivial profiles, correlated with the position of the domain wall. One expects that at high T a perturbative calculation of these quantities is possible. In this section we perform such a calculation. 4 We start again with the partition function (25) on a constant background A 0 . At high T the effective interaction is weak and we apply a saddle point approximation. We write:
Z(A) = DA µ exp − 1 2g 2 1/T 0 dt x (E 2 + B 2 ) ,(62)
where we linearize E and B:
E = −∂ 0 A − i[A 0 , A] − ∇A 0 ; B = ∇ × A.(63)
We have already calculated the integral (62) in Section 3:
Z(A) ≡ Z(q) = exp −T d−1 x (V (q) − c) .(64)
Now, to find the action density profiles we want to calculate things like E 2 and B 2 , where the average is understood in terms of the probability distribution given by the integrand in (62). These averages depend on A, or q. The easiest way to calculate E 2 is to introduce a parameter, say ǫ, in front of this term in the exponent, do the integral and differentiate the logarithm of the result over ǫ. We shall do this shortly, but before that let us improve a bit on the formula (64). What we need to include is the contribution of zero point energies of the modes of the fields A. This contribution does not depend on the temperature and is not present in (64), but it gives an overwhelmingly dominant contribution to E 2 and B 2 . Indeed, this contribution is of order (1/a) d , where a is the lattice spacing (the UV cutoff), and is much larger than T d -the thermal contribution. To see what we are missing consider calculating E 2 + B 2 from (64). For that one needs only to differentiate (the logarithm of) the right hand side over (1/g 2 ). But the right hand side does not depend on g! This means that the quantity E 2 + B 2 (which is obviously not zero) gives the sum of zero point energies and has no thermal contribution at that order. To make this fact explicit imagine rescaling g 2 in (64) by a factor 1/ǫ. One can absorb this factor by rescaling the fields A by √ ǫ. This will change the measure by
a factor ( √ ǫ) (N 2 −1)(d−1)N , where N = T −1 V/a d is the total number of the lattice sites. The number (N 2 − 1)(d − 1)
N is simply the number of non-zero modes, or non-zero eigenvalues of the matrix of the quadratic form in the exponent (62). Eventually we get instead of (64):
Z(q) = g (N 2 −1)(d−1)N exp −T d−1 x (V (q) − c) .(65)
Using (65) we get for any T :
1 2g 2 E 2 + B 2 = (N 2 − 1)(d − 1) 2 a −d .(66)
It means that each space component of the vector E or the tensor B has the average at T = 0 (due to Euclidean invariance):
1 2g 2 E 2 x = N 2 − 1 d a −d .(67)
This is the dominant contribution and should be compared to β(1 − trU P /N) per plaquette which we measure (in our case N = 2, d = 3). It is indeed equal to 1 in lattice units up to a small correction. Part of this correction is the thermal effect.
To calculate the thermal part of the action density, multiply only the term E 2 in (62) by a factor ǫ. Then consider the following transformation:
t ′ = t/ √ ǫ, A ′ 0 = A 0 √ ǫ, T ′ = T √ ǫ and g ′2 = g 2 / √ ǫ.
In terms of new variables the integral is the same up to a Jacobian factor and thus:
Z(q) = DA µ exp − 1 2g 2 1/T 0 dt x (ǫE 2 + B 2 ) = g ǫ 1/d (N 2 −1)(d−1)N exp −( √ ǫT ) d−1 x (V (q) − c)(68)
Therefore,
1 2g 2 E 2 = (d − 1) N 2 − 1 d a −d − d − 1 2 (c − V (q))T d .(69)
The E 2 is a sum of d − 1 components each contributing equally. Using (66) and (69) we get also:
1 2g 2 B 2 = (d − 1)(d − 2) 2 N 2 − 1 d a −d + d − 1 2 (c − V (q))T d .(70)
Note that while (1/2g 2 ) E 2 + B 2 is totally due to the vacuum zero point energy, it is (1/2g 2 ) B 2 −E 2 that contains the thermal energy. The vacuum term in this quantity cancels in d = 4, because there is the same number of E and B components. So far we have neglected the variation of q with z. This can be easily corrected for. It contributes only to E 2 z the amount (γ 2 /2)(∂ z q) 2 T d , which is equal to V (q)T d due to (39).
Finally, we write down expressions for the plaquette action densities in our case (N = 2, d = 3). We use lattice units: a = 1, in which T = 1/L t , and β is given by (45):
1 2g 2 B 2 = 1 + (c − V (q)) 1 L 3 t + O(1/β); 1 2g 2 E 2 y = 1 − 1 2 (c − V (q)) 1 L 3 t + O(1/β); 1 2g 2 E 2 z = 1 − 1 2 (c − 3V (q)) 1 L 3 t + O(1/β).(71)
The corrections of order 1/β are due to non-quadratic terms and are beyond our approximation. However, they should be the same for all plaquettes at that order. They cancel in, e.g.:
B 2 − E 2 y or B 2 (q) − B 2 (0)
. The formulas (71) are in good agreement with our MC data (see Section 7).
What do we learn from all this? One can see, for example, that the thermal part in the fluctuations of B 2 becomes negative at some q near 1/2, i.e., inside the wall. This means that the B 2 becomes smaller than the contribution of the vacuum fluctuations to that quantity! 5 This is just another side of the old puzzle with negative entropy and thermal energy density [4,9].
6 Finite size corrections 6
.1 Roughening
Due to the long wavelength thermal fluctuations of its shape, an interface in 3 spatial dimensions oscillates from its central position by a distance which grows as the logarithm of its area. This roughening also occurs in 2 dimensions, where the effect is proportional to √ L y . Here we calculate the effect of the roughening on our measurements of the profile of the wall and the interface tension and show that this effect is rather small.
The roughening is due to long wavelength fluctuations of the shape of the interface. These fluctuations are therefore essentially classical: ω k ≪ T . We can consider our interface in 2 spatial dimensions as a classical string of length L y with tension and mass per unit length equal to σ. The string is a set of free oscillators, the normal modes. The amplitudes f k of these are the Fourier components of f (y), the shape of the string at a given instance. Each oscillator (mode) has energy T in the heat bath (equipartition). On the other hand, the mean energy of such an oscillator is σω 2 k f 2 k . The mean square of the fluctuation of the string is then:
Ly 0 dyf 2 (y) = k f 2 k = k T σω 2 k(72)
The dispersion law is: ω k = k. If we take L y = ∞ and replace the sum with an integral it will diverge linearly: the soft modes get out of hand if L y does not cut them off. For finite L y the values of momenta q are given by the periodic b.c.:
k = 2π L n, n = 1, 2, 3, . . . ,(73)
where k = 0 (translational mode) is removed by our procedure of shifting the center of the interface. In principle, our string approximation will break down at some large k max , but the value of this UV cutoff is not essential for the long wavelength effect we are interested in. We put k max = ∞. Thus we have:
f 2 (y) = 2 L y ∞ n=1 T σ L 2 y (2πn) 2 .(74)
The factor of 2 is because for each n there is a cosine and a sine mode. The sum over n can be evaluated and we get for the mean square deviation of the wall from a straight line:
∆w 2 = f 2 (y) = T σ L y 12 .(75)
Let us estimate this effect in our case. Take, for example, L t = 3 and β = 75:
∆w = 2 α L 3 t β L y 12 1/2 ≈ L y /50,(76)
where we used the formula (40) for σ and the relation (45). For L y = 12 − 60 this varies from 0.5 to 1 or so, as compared to w ∼ 17. We observe a slight variation of the width of the wall of roughly this size in our MC data, although it is no doubt optimistic to be applying a string formalism in a situation where the length of the wall is comparable to its width! We conclude that roughening does not affect our estimates of the profile of the wall. This effect is small, because the wall is stiff, and L y is not too large. So what we measure numerically is really the intrinsic profile of the interface.
One can also calculate the correction to the interface tension from the string-like fluctuations of the interface using the same idea that the string is a set of oscillators in a heat bath. A difficulty lies in the fact that unlike f 2 which is a convergent sum of f 2 k , the amplitudes of the oscillators, the sum of their free energies, −T ln(T /ω k ) + const, is divergent. This divergence is ultraviolet, however, and can be subtracted when computing the finite size dependence, similarly to the Casimir effect.
Another, more illuminating way of deriving this correction is to consider the F (q) in (37) as an effective potential energy for the long wavelength classical thermal fluctuations of the wall. The profile q 0 (z) satisfying (39) is a minimum of F (q) for the corresponding boundary conditions. The leading exponential behavior of the partition function is then:
Z wall ∼ exp − 1 T x [ F (q 0 (x)) − F (0) ] = exp − σL y T ,(77)
where we subtracted the bulk free energy. The pre-exponential correction to (77) is due to fluctuations of q(y, z) around q 0 (z) and is given by the determinant:
det ′ (−γ 2 ∇ 2 + V ′′ (q 0 (z))) −1/2 ,(78)
where the prime on 'det' denotes the fact that we omitted the translational mode. This mode is proportional to ∂ z q 0 . Properly normalized it produces a factor σL y /γ 2 T and an integration over the position of the center. The spectrum of the operator in (78) can be written as
λ = γ 2 k 2 + λ m(79)
where k = 2πn/L y and λ m are the eigenvalues of −γ 2 ∂ 2 z + V ′′ (q 0 (z)). The λ = γ 2 k 2 band attached to the zero eigenvalue λ m = 0, corresponds to fluctuations of the wall as a whole. The roughening effects are due to these gapless fluctuations. The corresponding determinant is det ′ (−γ 2 ∂ 2 y ). We can regularize it in the UV by dividing it by a similar determinant with L y = ∞. On dimensional grounds:
det ′ (−γ 2 ∂ 2 y ) Ly det(−γ 2 ∂ 2 y ) ∞ = L 2 y γ 2 · const,(80)
Other eigenvalues are not related to the roughening and we neglect their contribution in our estimate. Collecting all the factors we obtain:
Z wall ≈ const dz σ T L y exp − σL y T = const σL 2 z T L y exp − σL y T .(81)
From (81) we can read off the correction to the interface tension 6 :
∆σ = T 2L y ln T L y σL 2 z + O(T /L y )(82)
This means that the correction to α, in the way we measure it in our simulations (see Section 7), is given by:
∆α = 4L 3/2 t L y β ∂ ∂β ∆σL y T ≈ − L 3/2 t L y √ β .(83)
For example, at β = 75, L t = 3 and L y ∼ 50 this correction is only about 0.2%.
Finite L z corrections
In the previous section we discussed the roughening effect which introduces corrections of the type (ln L y )/L y to σ. In this section we discuss another source of finite size effects: the finiteness of L z . There are two ways the finiteness of L z affects the free energy of the wall. First, there is a correction to V (q) itself because it is given by one-loop integrals which depend on L z through the quantization of momenta running in the loop. 7 This correction should be of order (T /L z ) 2 and is relatively small. For L z > ∼ w this correction is beyond our leading order in g approximation, because (T /w) 2 is of order g 2 .
The second source of corrections is due to the finiteness of L z /w. It is obtained by calculating the action of a particle with a 'mass' γ 2 which, moving in the potential −V (q), returns to its starting value of q in a period of 'time' 2L z This finite size effect we estimate here and show that, for large enough L z , it is exponential in d > 3, i.e., exp{−L z /w}, and Gaussian in d = 3, i.e., exp{−L 2 z /w 2 }. So we consider a trajectory which starts at rest at q = ε and arrives at q = 1 − ε precisely after a given 'time' L z so as to satisfy the boundary 6 This correction is analogous to the Luscher's correction [21], but for a classical string in a thermal bath, rather than a strip (quantum string at T = 0). The universal coefficient of the Coulomb correction 1/L to the free energy in Luscher's case becomes in our case the coefficient of the ln L correction. It is especially obvious in the interpretation given by Stack and Stone [22]. 7 There is, of course, a similar dependence on L y .
condition on the Polyakov loop. The trajectory satisfies a Lagrange-Euler equation which can be integrated to give ('energy' conservation):
γ 2 2 (q ′ ) 2 − V (q) = −V (ε) ≡ E.(84)
We can use it to relate L z to ε:
L z (ε) = 1−ε ε dq 2 γ 2 (V (q) − V (ε))(85)
The action can be cast into the form:
S(ε) = Lz 0 dz [ γ 2 2 (q ′ ) 2 + V (q)] = 1−ε ε dq 2γ 2 (V (q) − V (ε)) + V (ε)L z (ε),(86)
which is convenient for the numerical evaluation of S(ε). Also note, that this form is familiar in theoretical mechanics as dS = pdq − Edt. The equations (85) and (86) give a parametric representation of S as a function of L z which we use to evaluate the correction numerically.
To find the asymptotic dependence of S on L z one can use the relation:
dS dL z = V (ε),(87)
which is a consequence of dS = pdq − Edt, and can be also derived explicitly from (85), (86). Now we use the asymptotic form of V (q) at small q. For d > 3 it is:
V (q) = bq 2 + O(q 4 ) (d > 3),(88)
where b is a constant which depends on N and d (96). Using this form we get from (85):
L z (ε) = 2 γ 2 2b ln 1 ε + O(1), (d > 3).(89)
Integrating the equation:
dS dL z = bε 2 + O(ε 4 ) ≈ const b exp − 2b γ 2 L z , (d > 3),(90)
we obtain the asymptotic form of the large L z correction to S:
δS ∼ − const bγ 2 2 exp − 2b γ 2 L z , (d > 3).(91)
For d = 3, however, we have:
V (q) = bq 2 ln 1 q + O(q 2 ), (d = 3).(92)
where b is given by (97). This leads to
L z (ε) = 8γ 2 b ln 1 ε + O(1), (d = 3);(93)
and
dS dL z ≈ const b 2 L 2 z 8γ 2 exp{− bL 2 z 4γ 2 }, (d = 3);(94)
Integrating we obtain:
δS ∼ − const bL z 4 exp{− b 4γ 2 L 2 z }, (d = 3).(95)
We see that the asymptotic form of the correction is related to the way the tail of the wall decays: exponential in d > 3 and Gaussian in d = 3. 8 From (36) we find for b in (88):
b = (2π) 2 (N − 1)(d − 2) Γ(d/2) π d/2 ζ(d − 2), (d > 3);(96)
To get an idea of the size of this correction in our case let us estimate it for the case of β = 100, L t = 4, L z = 120. We get γ 2 = π 2 βL t = 400π 2 ≈ 60 2 . The exponent is bL 2 z /4γ 2 ≈ (L z /50) 2 ≈ 5.8 and e −5.8 ≈ 1/300. The preexponent bL z /4 ≈ 200. Thus δS ∼ 1. This should be compared to:
S 0 = 1 0 dq 2γ 2 V (q) ≈ 50.(98)
Thus the correction, δS/S 0 , is of the order of a (few) percent. In the following section we shall evaluate these corrections, numerically, for all L z and not just large L z as herein.
Numerical Simulations of Domain Walls
As we have seen, at high temperatures the theory appears to have degenerate vacua which are separated by domain walls. At asymptotic temperatures many properties of these domain walls can be calculated in perturbation theory; indeed the existence of these interfaces can only be seen when one goes beyond tree level. However, as we remarked earlier, the presence of infrared divergences in higher-orders has raised doubts about the applicability of perturbation theory and, indeed, about the actual existence of the interface.
To address these doubts we have performed accurate computer simulations of the domain walls, and have compared what we find with the results of the perturbative calculations. These computer simulations will be described in this section. If we simulate the high temperature SU(2) gauge theory in a finite but large spatial volume, with periodic boundary conditions, then we expect some fraction of the field configurations to contain both Z(2) phases in different portions of the torus. Such a configuration will contain domain walls separating the two phases and in principle one could study the domain walls by focusing on these particular field configurations. However the relative probability of such configurations is very small for the temperatures of interest and they would not be encountered in a typical Monte Carlo calculation. So we have to use an alternative less direct method. What we do is to impose twisted boundary conditions on our system, so enforcing the existence of at least one domain wall. This will be described in Section 7.1. In Section 7.2 we show how the domain wall can be interpreted as a 't Hooft disorder loop. We then specify the physical quantities that we plan to calculate and describe the methods by which we do so in Section 7.3. Section 7.4 summarises our Monte Carlo results. Of course, it is crucial to demonstrate that we have all finite-volume effects under control -after all, it is infrared effects that are the potential problem here -and this we do in Section 7.5. Finally, in Section 7.6, we will take our raw 'data' and use it to extract quantities that are of direct physical interest in the present context and compare them to the perturbative predictions.
Twisted Boundary Conditions
We work on lattices of size L y × L z × L t in lattice units. The Euclidean time extent determines the temperature, aT = 1/L t , of the field theory. The partition function contains the factor exp(−βS) where β = 4/(ag 2 ) and the lattice action is as in (42):
S = p [1 − 1 2 trU p ],(99)
where U p is the path ordered product of the SU(2) matrices, U l , on the links, l, that form the boundary of the plaquette p.
The simplest and most usual way to introduce twisted boundary conditions is as follows [20]. We change the above action to a twisted action, S tw , by replacing trU p with −trU p for those plaquettes in the zt plane that emanate from the sites (y, z, t) where z and t are fixed to some particular values, say z = j and t = k, while y takes all values from 1 to L y (see Fig. 1).
The system with this altered action and with periodic boundary conditions is equivalent to the system with the original action but with twisted boundary conditions [23]. This we see from the following argument. Firstly, let us choose labeling z of the sites so that j = L z , i.e. the twist is between z = L z and z = 1. Secondly, to include the possibility of boundary conditions that are not periodic it is convenient to extend our labeling to include z = 0, as well as z = 1, . . . , L z . If the system is periodic then corresponding sites and links with z = 0 and z = L z are identified (and similarly for other directions). Then the system with a twisted action can be viewed as a system with the original action but with fields which are not periodic. To be more specific they are periodic except that for y = 1, . . . , L y and t = k the time-like link at z = 0 is mapped into the negative of itself at z = L z . This is the lattice version of the twisted boundary conditions described in Section 2.1.
One can move the zt position of the line of twisted plaquettes by flipping the sign of all U l which bound these plaquettes from one of the sides; but one cannot undo the twist completely. It should be clear that the position of the twist does not carry any physical significance since it can be moved by such a redefinition of the variables U l .
How does the twist lead to the presence of a domain wall? To see this consider the same labeling of sites as we have just used. With free boundary conditions the system would spend most of the time in one of the two phases where Polyakov loops are all near +1 or all near −1. With the twisted boundary condition a Polyakov loop at z = 0 is mapped onto negative of itself at z = L z . Therefore a homogeneous configuration is frustrated and the Polyakov loops must create a nontrivial profile in the z direction to interpolate between z = 0 and z = L z .
So to study the high T properties of domain walls we perform Monte Carlo calculations on lattices with periodic boundary conditions but with a twisted action. How well defined is the domain wall in practice? To answer this question we show in Fig. 2a the distribution of Polyakov loops on a typical field configuration taken from a 30 × 80 × 2 lattice at β = 100. In physical units this corresponds to a temperature T ∼ 30T c where T c is the deconfining temperature. We see that the domain wall is very well defined, with relatively small fluctuations around a smooth background distribution. This is in fact the highest value of T at which we work. The lowest value is on a 12 × 30 × 2 lattice at β = 7 corresponding to T ∼ 2T c . There a typical field configuration looks as in Fig. 2b. The fluctuations are now much larger, but the domain wall can still be unambiguously located. So it is clear that, for the range of T we study, there is no ambiguity in identifying the domain wall.
Domain Walls as Disorder Loops
Before going on to the details of the calculations, we address the following natural question. Since the twist is entirely symmetric in z and t why should the 'domain wall' separate regions in z rather than regions in t? This question can be plausibly answered, in a way that highlights the physics, by first considering the twisted system at very low T where the system is manifestly rotationally invariant. Here introducing a twist introduces into the system a 't Hooft disorder loop [24] which is closed through the boundary in the y-direction. This loop will presumably be a flux tube whose width will be on the order of the characteristic length scale of the theory, which here is 1/g 2 . Its special property is that it if one considers the gauge potential on a closed path that encircles the disorder loop far from its centre, then the presence of this loop leads to the potential acquiring a gauge transformation that goes from 1 to a non-trivial element of the centre as we go once around this closed path. So if we take a large Wilson loop and pierce it once (or an odd number of times) by this disorder loop, then the value of the Wilson loop is changed by a factor of −1 as compared to the value it would possess in the absence of the disorder loop. Hence the name 'disorder loop'. It is clear that if the vacuum contained a condensate of such loops, then these would be sufficient to ensure that large Wilson loops varied as the exponential of their area, so that we had linear confinement. Suppose we now increase T by reducing the extent of the system in the t-direction. Clearly at some point the time extent will become smaller than the width of the flux tube, the tube will become squeezed so that it extends right across the time direction while still extending over a finite region in the z-direction. This will occur once T is sufficiently large compared to g 2 ; presumably around the deconfining transition. So at high temperatures our 'domain wall' is actually a squeezed disorder loop that closes upon itself through the y-direction. It is indeed symmetric in z and t except for the deformation induced by the limited t extent. The fact that the Polyakov loops on either side of the wall have opposite signs is what one might expect from such a squeezed 't Hooft disorder loop.
We have simplified the above argument by assuming that the disorder loops exist as definite field fluctuations in the low-T theory. This is assuming a great deal of course. Whether they do so exist is one of the central questions in the still unresolved problem of colour confinement. This makes the connection between these loops and the domain walls at high-T of added interest.
Quantities Calculated
We perform calculations on lattices with and without a twist. The simplest and most interesting quantity we extract is the extra action, S w , associated with the presence of a domain wall. If both twisted and untwisted lattices are of the same size, then
S w = S tw − S nt(100)
where S tw , S nt are the average values of the twisted and untwisted actions, as defined in Section 7.1. The extra action is related to the free energy of
the wall, F w ≡ F tw − F nt , by ∂ ∂β F w T = S w(101)
where the derivative is taken at constant values of L y , L z , L t . Using this relationship we shall test the one-loop prediction for F w . It might appear that changing the action for a line of L y parallel plaquettes could introduce some additional local contribution to S tw − S nt which is not related to the free energy of the wall. That this is not so one can see by considering a system with 2 parallel twists. This system is equivalent to a system without a twist after a redefinition of variables U l which move the twists to a single position where they cancel each other.
In addition to making predictions for the domain wall free energy, perturbation theory can also be used to predict the detailed shape of the domain wall as it interpolates between the two Z(N) vacua. In the Monte Carlo calculations the domain wall is free to move and so if we are to obtain an average profile, we need to shift our origin, in each Monte Carlo generated configuration, to the centre of the domain wall. We also need to take into account the presence of the twist, since the Polyakov loops change sign as one moves through it. Our algorithm is as follows. Consider a single Monte Carlo generated field configuration. First we average Polyakov loops over y. We write this average as p(z). We now want to identify the location, z = z c , of the centre of the domain wall. This is defined operationally as follows. We first identify the values of z where p(z) changes from positive to negative values (factoring out, of course, the trivial change at the twist itself). Clearly the number of such changes must be odd. In practice the domain wall is very smooth at high T -as one can see in Fig. 2 -and it is almost always the case that there is only one place where there is a sign change. This occurs between sites and we shift our origin in z so that the sites where the sign changes are labeled by z = 0 and z = 1. We now ensure that p(z = 1) > 0 by multiplying the whole profile by -1, if necessary. Our range of z is now from −L z /2 + 1 to +L z /2. Somewhere in this range there is the twist and the value of p(z) will flip sign there. If this occurs for z ≥ 1 then we flip the signs of p(z) for values of z beyond the twist; if it occurs for z ≤ 0 then we flip the signs for z before the twist. In this way we obtain a wall profile with p(z ≥ 1) > 0 and p(z ≤ 0) < 0. We can now average this profile over many configurations to obtain an average profile. This will be symmetric about z = 1/2 so we can fold the profile over (with a sign flip) so that it is defined for 1 ≤ z ≤ L z /2 and is positive. This is our final averaged profile. Note that in any individual configuration the centre of the domain wall may be closer to z = 0 than to z = 1. That is to say, our profile is 'smeared' over distances δz ∼ 1/2.
In rare cases a given configuration contains more than one sign change in p(z) (always factoring out the trivial sign change at the twist). The number of these sign changes is clearly odd. There are two possibilities. One is that we have a configuration with more than one domain wall, i.e. the one enforced by the twist plus pairs that are genuine quantum fluctuations. In this case we would typically expect at least one large gap in z between the walls. The more trivial possibility is that we might be simply seeing a large fluctuation of the values of p(z) near the centre of the wall (where the values are small on the average). This would be characterised by very small gaps between the locations of the sign changes. The first type of configuration, which we should not include in our average, did not occur in any of the calculations that we include below. (It does occur if we approach the deconfining transition or if we make the extent in y of the lattice, and hence of the wall, sufficiently small.) The second type of configuration we should include and we do so by taking its centre to be located in the middle sign change. In practice these configurations are so rare that there is no visible change in any extracted quantities whether we include them or not.
Having obtained a centre for the wall from the Polyakov loop distribution, we can also define an action profile for the wall, and we can clearly do this separately for the different µν components of the action.
A quite different but equally interesting quantity is the electric screening mass, m D . This can be obtained from the lightest mass, m P , that couples to Polyakov loops, and hence from the tail of the wall profile. We expect that for large enough z,
p(∞) − p(z) ∝ e −am P z(102)
where p(∞) can be obtained either by working with very large lattices, or by performing simulations on a lattice without a twist and using the average value of the Polyakov loop obtained therein. So if we define an effective mass by
am eff (z) = ln p(∞) − p(z − 1) p(∞) − p(z)(103)
then
am P = lim z→∞ am eff (z)(104)
In practice we would extract m P from m eff once we were at large enough z that the latter had become independent of z. The electric screening mass, m D , should then be given by m P = 2m D (see below). On a finite lattice the above needs to be altered because we expect contributions going both ways around the z-torus. So instead of (102) we use
p(∞) − p(z) ∝ e −am P z + e −am P (Lz−z)(105)
and alter (103) correspondingly.
Since we need to calculate the average action without a twist, we can also calculate the screening mass on these untwisted field configurations. Here we follow standard techniques for such mass calculations [25]. We construct p y = 0 sums of Polyakov loops at each value of z and then obtain the vacuumsubtracted correlation function, C(z 1 − z 2 ), as a function of their separation, z = z 1 − z 2 . For large separation z we have C(z) ∝ exp(−am P z). We define an effective mass am eff (z) = ln[C(z − 1)/C(z)], and we increase z until m eff becomes independent of z. At this point we can estimate m P = m eff . This calculation has the advantage that we can prove that m eff (z) ≥ m eff (∞). In practice we modify this formalism for the periodicity in z as described above. In addition we calculate with a range of smeared Polyakov loops and use the correlation function that minimizes m eff (z = 1); in the spirit of a variational calculation. However, this turns out not to be really necessary here; unlike the situation at T = 0. As we shall see below, this method turns out to be much more efficient for the calculation of screening masses than using the tails of domain walls.
Monte Carlo simulations
Our Monte Carlo simulations were performed on a variety of periodic lattices with and without a twist. We used a standard heat-bath update algorithm mixed with over-relaxation steps.
The control of finite volume effects is particularly important in these calculations since it is infra-red effects that are usually seen as being at the root of any possible breakdown of perturbation theory at high temperatures. We have therefore performed extensive numerical checks of finite volume effects and these will be described in detail in Section 7.5. In this section we shall confine ourselves to a presentation of those results that have been obtained on lattices which are sufficiently large that any finite-volume corrections are much smaller than our (very small) statistical errors. This will, of course, need to be demonstrated and we shall do so later. Now, let us estimate how large the required volumes must be in lattice units. If we use a periodic L y × L z × L t lattice, this corresponds to the fields in the spatial L y × L z volume being at a temperature aT = 1/L t . The dimensionless inverse coupling β, as we have seen, is related to the dimensionful coupling, g 2 by β = 4/ag 2 . Thus in physical units the temperature is T /g 2 = β/4L t , and, at a fixed value of L t , β ∝ T . Perturbation theory is expected to be most reliable at very high T , so we want to study the theory for very large β. The characteristic length scale at high T is of the order of 1/gT 1/2 : the inverse of the Debye mass 1/m D and the width of the wall w are of that order. Therefore the spatial sizes in units of a must satisfy:
L y , L z ≫ 1 ag √ T = L t T g 2(106)
From (106) we see that for a fixed value of the temperature in physical units, T /g 2 , the required volumes will be smallest, in lattice units, for L t = 2 (since L t = 1 is not sensible). Since the perturbative properties of the domain wall can be calculated on the lattice, the minimal calculation one might perform is to do everything at L t = 2. However in this case a is as large as possible in units of 1/T (aT = 1/L t = 1/2) and since there have been suggestions [9] that high-T perturbation theory might break down as a → 0 we choose to perform calculations for several values of L t . (This will have other advantages that will become apparent below.) We shall cover a range of temperatures for L t = 2, 3, 4 and we shall perform a calculation at one reasonably high value of T for the case L t = 6. In this latter case aT = 1/6 which is surely small enough that any breakdown of perturbation theory, as a → 0, should have become prominent.
In Table 1 we list the average values of the plaquette, 1 − s nt ≡ 1 2 trU p , for the calculations without a twist. In Table 2 we do the same for the corresponding quantity, 1 − s tw , with a twist. We show the values of β, the lattice sizes, the number of Monte Carlo sweeps and the average plaquette action. In the twisted case we perform 'measurements' every Monte Carlo sweep; in the untwisted case every four sweeps. The typical number of thermalisation sweeps prior to taking any measurements is between 25000 and 50000. The errors, given in brackets, are typically based on 40 or 50 bins. In a few of the lower statistics cases we use as few as 25 bins. The reader will note that at some values of the parameters we have several different lattices. These arose during the finite volume studies that will be described in detail later on. The measurements that we list here are those that do not suffer significant finitesize corrections (and are statistically accurate enough to be useful). At the different values of L t we have chosen values of β such that the temperatures, in units of g 2 , are roughly the same, although for higher L t we are forced to cover more limited ranges of T . (The reader may be puzzled that in some cases β has not been chosen exactly proportional to L t ; for the purposes of the work in this paper no particular significance should be attached to these choices.) If we now multiply s tw − s nt by the number of plaquettes in the twisted lattice, which is the one containing the domain wall, we obtain a value for S w and hence, from (101), information on F w . We shall see later on what this comparison tells us about the accuracy of perturbation theory.
As described in the previous section, the lightest mass that couples to Polyakov loops, is of particular interest because it is related to the Debye screening mass. It can be calculated either from correlations of Polyakov loops in the system without a twist, or from the way the tail of the domain wall merges into the vacuum once we are far enough away from the centre of the wall. In Fig. 3 we show the effective masses as obtained by the two methods. In Fig. 3a we have chosen our highest value of T for L t = 3 while in Fig. 3b we show what one obtains for a medium value of T with L t = 2. We see that in both cases the values of am eff (z) as obtained from Polyakov loop correlations do become independent of z at larger z, and that these 'plateaux' occur early enough for the errors to be very small. Since in this case m eff (z) is always an upper bound on m P , we can extract an accurate estimate of m P using the first value of the effective mass that is, within errors, on the plateau. The effective masses obtained from the domain walls are clearly consistent with being asymptotic to these mass values. However it is equally clear that they would give us much less accurate estimates of m P . (We would need to do fits with at least two masses, since there are no convenient plateaux, and so the errors on m P would be perhaps an order of magnitude greater. Moreover the assumption that the effective masses asymptote from below, while reasonable, introduces a difficult to quantify extra systematic error.) So from now on we shall only use the values of m P as extracted from Polyakov loop correlations. These are listed in Table 3, for those lattice volumes which do not suffer significant finite-volume corrections.
To obtain the extra action of the domain wall, S w , at a particular value of β, we take the difference s tw − s nt at that β and multiply by the number of plaquettes on the twisted lattice, which contains the domain wall. We expect that this extra action will be proportional to the length of the domain wall, i.e. to L y , as long as L y is not very small. (This and the related question of roughening will be addressed when we discuss finite volume corrections.) So we form the quantity S w /L y , which is the extra action of the wall per unit length (in units of the lattice spacing). If we have values of this quantity for several values of L y at a given value of β and L t , we can average them to obtain our best overall estimate. In Table 4 we present our final averages for this quantity and for the mass am P , as obtained by averaging the values given in Tables 1-3. These will form the basic raw material for our later comparisons with perturbation theory.
Finite Volume Corrections
We shall be using our values of S w to test perturbation theory. The details of the T and L t dependence will be important in this comparison. Since the size of the domain wall varies with T , it is important that we control any finite volume corrections at all values of our parameters. Otherwise part of the T dependence we observe might be due to such corrections. In this section we describe in detail how we control finite size effects. We begin with effects of finite L z and then consider finite L y .
To establish how the finite periodicity in the z-direction affects the action of the domain wall, we perform numerical calculations for a large range of values of L z . Since these effects may well vary with the lattice spacing, i.e. with aT = 1/L t , we perform such calculations for two different values of L t , but at the same value of physical temperature T /g 2 . Since the finite-size corrections may differ for the contributions that are leading and non-leading in g 2 /T , we also perform the calculations for two different values of T /g 2 at the same value of L t . The parameter values and the corresponding values of S w /L y are displayed in Table 5. As we discussed previously, see (106), the natural scale for the domain wall should be of the order of 1/agT 1/2 = √ βL t /2. This is the scale that appears in perturbation theory (38,59): γ = π √ βL t . We therefore plot in Fig. 4 the values shown in Table 5 against the scaled lattice length L z /γ. We see that to a good approximation the finite size effects are indeed just functions of this scaled length. We also see that the finite size effects vary from being very large to being very small over a narrow range of values of L z /γ. Indeed, the domain wall effectively disappears for L z /γ ≤ 0.8, and while the corrections are still large for L z /γ ∼ 0.9 to 1.1, they have become invisible, within our statistical errors by the time L z /γ ∼ 1.35. We therefore see that the values in Tables 2-4, which all correspond to lattices satisfying L z /γ ≥ 1.65, are effectively for L z = ∞. As a final precaution against the unexpected, we show in Table 6 some further calculations obtained for a wide range of values of β and T . Taking these together with the values in Tables 1-3 confirms that the pattern of finite size effects we see in Fig. 4 is indeed characteristic of the range of T and a covered by the calculations in this paper.
Since the finite size effects appear to be insensitive to the value of T , it is interesting to ask whether they can be reproduced in leading order perturbation theory, which, after all, is supposedly exact in the T = ∞ limit. The appropriate formalism is that of the 'ball rolling in the inverse potential' as described in Section 6. To that order the finite size effects are functions of the scaled length L z /γ, as can be seen from eqs. (85,86). In Section 6.2 we solved the equations analytically in the limit of large L z /γ. In this section we solve them numerically for all L z /γ, and different L t , using V lat (q). We note that solution does not exist if the 'time' 2L z after which the particle has to come back oscillating around q = 1/2 is smaller than the period of small harmonic oscillations around this point. This gives for the minimal value of L z : L z /γ = π/4 ln 2 = 1.06 (for L t = ∞). This fits in well with what we observe in Fig. 4. For large L z the correction is exponential in L z 2 in this order. This would be difficult to see given our finite statistical errors. Moreover we know that in the full theory the correction must ultimately be exponential in L z . From Fig. 4 we see that leading order perturbation theory describes the observed finite size effects reasonably well. We also see that our criterion, L z /γ ≥ 1.65, should be a safe one to use for all values of L t .
We now turn to the finite-size effects associated with the transverse length of the domain wall, L y . If L y is small enough then extra domain walls will be produced as quantum fluctuations since the main factor in the suppression of domain wall excitation is ∼ exp(−βS w ) and S w ∝ L y . When L y becomes sufficiently large we expect the leading correction to be that due to roughening, as discussed in Section 6.1. As we see in (83) these corrections should be very small for the parameters we use. Of course at asymptotically large L y the profile of the domain wall, defined by averaging the Polyakov loops over y, will broaden as √ L y . For our values of L y the broadening of the profile is small (see Section 6.1).
To find out what are the corrections at finite L y , we have performed calculations for a range of values of L y . As in our study of the L z dependence we do so for two values of L t at the same value of T /g 2 and for two values of T at the same value of L t . These are presented in Table 7. We see that the surface tension shows no variation with L y at, say, the 2σ level except for L y = 4 at β = 25. Here there appears to be a ∼ 4 − 8% reduction in the tension. This compares well with (83) -recalling that α ∼ 6 for L t = 2, although we should certainly not expect (83) to be accurate for such small L y .
However, while we see that there are no significant L y corrections to the surface tension, this is certainly not the case for the Polyakov loop mass, am P . We see that not only does this mass gap show large finite size corrections for the smallest values of L y , but that these corrections become noticeable for values of L y that are not so small. In fact the pattern we see is consistent with a relative correction of the form ∼ exp(−am P L y ).
The extensive finite-size studies we have carried out in this section show that the potentially dangerous infra-red effects are in fact under control and that the values of the surface tension and mass gap that we shall be using in the next Section, may be regarded as having been obtained on an infinite system.
Surface Tension and Debye Screening Mass
Even without looking at the detailed numbers in our Tables, there are two properties of the domain walls that are immediately apparent. One is that the probability of such a wall being produced at high T is very small. The other is that the walls have a finite width. Do these qualitative features already teach us something?
Consider the finite width. This is significant because at tree level a wall would have infinite width. That is to say, the width of the wall would be ∝ L z however large we made L z . One can easily see this by considering the minimum action configuration that interpolates between the vacuum with all Polyakov loops +1 and the vacuum with all loops −1. The fact that the walls we generate are of a finite width, i.e. independent of L z once L z is sufficiently large, is implicit in the finite volume studies of the previous section. However it is worth showing this explicitly. We define the width of the wall, w(90%), as the distance, in lattice units, from the value of z where the Polyakov loop is 90% of its asymptotic value to the point where it is of the same magnitude but of opposite sign. As a check that there is nothing special about the choice of 90%, we shall also define a width, w(2/3), on the basis of 2/3 of the asymptotic value. In Fig. 5 we show how w varies with L z in two of the cases where we have made measurements for a wide range of lattice lengths. What we see is that the width of the wall does not change with increasing L z once L z > 2w(90%): the wall does indeed have a finite fixed width.
How does this width depend on T ? We extract w from all our large volume calculations (essentially those listed in Table 2) and plot the results against γ = π(βL t ) 1/2 in Fig. 6. This variable, as we have seen, determines the width of the wall in leading-order perturbation theory. As a matter of fact, γ ≈ w(0.97) to that order in the continuum limit,i.e. for L t = ∞. We see from Fig. 6 that for each value of L t the width varies linearly with γ, with significant deviations only at the very lowest values of T . The lines for different L t are close to each other, but do not coincide. This is what one expects in perturbation theory; the width has to be proportional to the one scale, γ = 2π/g √ T , but the constant of proportionality will suffer lattice spacing corrections, i.e. will depend on aT ≡ 1/L t . We show in Fig. 6 the lines one gets in perturbation theory. Clearly these are consistent.
The corrections to perturbation theory are governed by g 2 /T = 4L t /β. We note that for β = 7 at L t = 2, and for the corresponding values of β at other values of L t , this is ≥ 1. It is therefore remarkable that the calculated widths deviate by no more than ∼ 10% from the leading-order perturbative high T expectations. It would seem that not only does perturbation theory work well where we might expect it to, but it even works well where we have little reason to hope it might. We shall see other instances of this later on.
The fact that wall-like quantum fluctuations are very rare tells us that the action a wall costs is positive: S w > 0. Using (101) and noting that, for fixed L t and L y , ∂/∂β ∼ ∂/∂T because β ≡ 4/ag 2 = 4L t T /g 2 , this tells us that ∂ ∂T
F w T = ∂ ∂T L y L t σ(T ) T 2 ≥ 0.(107)
Here we have used the definition of the domain wall surface tension:
F w = aL y σ.(108)
From (107) we immediately deduce that σ(T ) ∝ T δ , where δ ≥ 2, ignoring possible logs. At the same time we expect that it cannot increase faster than T 3 . The perturbative value for the exponent is, as we have seen, δ = 2.5. So we see that our qualitative observations already constrain the temperature variation of the surface tension to lie in the interval δ = 2.5 ± 0.5. In the quantitative comparisons below we shall attempt to make the comparison with perturbation theory much more precise. Before doing so it is interesting to ask what the above T 2 bound means physically. The following is a simple heuristic interpretation. Let m D be the screening mass; then the wall will have a thickness O(1/m D ). A very crude expectation is that σ ∝ T 3 /m D . Now if m D grows faster than T then it cannot be thermally excited and the whole high-T picture of a screened plasma breaks down. The statement that m D grow no faster than T is equivalent to our bound that σ grows at least as fast as T 2 .
We now turn to a quantitative analysis of the results displayed in Table 4. We recall that the leading order perturbative result is
σ = α g T 2.5(109)
The important energy scale is T and the dimensionless expansion parameter on this scale is g 2 /T . Thus the above leading order perturbative result should become exact in the T → ∞ limit. Naively one might expect finite temperature corrections to (109) to be O(g 2 /T ) -as in 4 dimensions -but here in d=2+1 there might well be logarithms and the power itself might be different. We shall see below that this uncertainty about the functional form of the leading corrections will limit the precision with which we can test perturbation theory. As we have seen, (109) is valid both in the continuum and on the lattice, except that in the latter case the value of the constant α will receive calculable lattice spacing corrections. Since T is the (largest) important physical energy scale, we expect that the lattice spacing corrections should depend only on aT ≡ 1/L t . Moreover, since α is a dimensionless physical quantity we expect, on quite general grounds for a pure gauge theory, that the corrections should be O(a 2 T 2 ) ∼ O(1/L t 2 ) for small enough a. The detailed perturbative calculations do in fact bear out this expectation.
From (101,108,109) we obtain
S w = ∂ ∂β F w T = αL y 8L t 2 g 2 T 1/2(110)
to leading order in perturbation theory. We have calculated the value of α as a function of aT ≡ 1/L t , using lattice perturbation theory (60) and show a selection of these values in Table 8. Since the lattice spacing corrections depend on L t , we shall mostly examine the T dependence at fixed L t . Indeed, we have chosen our parameters with this in mind. However before doing so we briefly take the alternative approach of varying L t at fixed β. For this purpose it is useful to rewrite (110) in the form
S w L y = 2 α(L t ) β 2 T g 2 3/2(111)
Now, as we see in Table 4, it is only for β = 75.0 that we have a usefully large range of L t values. So we plot these values of S w /L y against T /g 2 ≡ β/4L t , in Fig. 7. We use logarithmic scales so that a power dependence in T will appear as a straight line. And, indeed, the calculated values do fall on a straight line to a good approximation. The slope suggests a variation ∝ T 1.6 which is close to the perturbative variation of ∝ T 1.5 . To carry the comparison further we need to take into account the fact that in addition to the predicted T 1.5 behaviour, there are different aT lattice corrections at the different values of L t . That is to say, the behaviour predicted by perturbation theory is ∝ α(L t )T 1.5 and not just the power of T. In Fig. 7 we show the complete leading order perturbative prediction and we see that its variation fits that of our data very well. The normalisation is not exactly right -but the difference is small and is decreasing with increasing T just as one would expect from a higher order correction in g 2 /T . If we do indeed try to fit the data with a higher order correction we find that an O(g 2 /T ) correction will not work; but
σ = α g T 2.5 1 + 0.13 g 2 T 0.5 (112)
fits perfectly well. The precise power of the correction is not to be taken too seriously of course; it may be an effective power that partially simulates the effects of logarithms in our limited range of T /g 2 (this range being roughly 3 to 10). We conclude from the above comparison that at temperatures T /g 2 ≥ 3 the T dependence of σ is very close to the perturbative expectation of T 2.5 ; indeed what we find is ∼ T 2.6 . Moreover this slight difference almost entirely disappears when we include the perturbatively calculated aT corrections to α. The remnant discrepancy, a few percent, decreases as T increases and so is consistent with being a higher order perturbative correction, as, for example, in (112). Unfortunately the fact that we do not know the precise functional form of this correction, prevents us from carrying out the quantitative comparison any further than this.
We turn now to consider the bulk of our calculations. We shall consider the T dependence at fixed values of L t ≡ 1/aT , so that the lattice spacing corrections do not vary with T . At the same time, by performing calculations for several values of aT we can see whether the perturbative predictions show any sign of failing as one approaches the continuum limit. To compare our results to the perturbative prediction we define a quantity α eff by
α eff = β 2 2 g 2 T 3/2 S w L y(113)
As we see from (111), to leading order in perturbation theory α eff (L t ) = α(L t ). In Fig. 8 we display our Monte Carlo results for α eff as a function of g 2 /T . The perturbative value of α is also shown, as a horizontal broken line, in each case. We observe that the calculated surface tension indeed approaches the perturbative value as T increases. At the highest values of T the discrepancy is no more than a few percent. Our data is clearly compatible with the leading perturbative result being exact in the T → ∞ limit. In Fig. 9 we plot the ratio α eff /α for all our data. We see that to a good approximation it is a function only of g 2 /T . This supports the idea that the small differences we see between the full and perturbative surface tensions are in fact due to higher order corrections in g 2 /T . We note that these higher order corrections are small over our whole range of T . Indeed, even for g 2 /T ∼ 1.1 the correction is only about 25% of the leading term. Recall that this temperature corresponds to only about twice the deconfining temperature. It is quite extraordinary that lowest order perturbation theory should still be so accurate at such low temperatures.
If we knew the functional form of the leading correction, we would attempt to extrapolate our 'measured' values to T = ∞. Unfortunately we do not. In d = 3 + 1 we would expect the correction to be simply ∝ g 2 /T . However in d = 2 + 1 there are infrared logarithms which may also resum into a power of g 2 /T . So it seems reasonable that the correction should be some effective power of g 2 /T that lies between 0.5 and 1 in our range of T . If we fit our data with a form
σ = α g T 2.5 1 + c g 2 T ε(114)
then we find that while ε = 1 is excluded, powers near ε = 0.65 work perfectly well, as we see in Fig. 10. The intercepts of these fits are compatible with the leading order perturbative predictions. The final question in this context is whether there is any sign that this agreement with perturbation theory breaks down as we approach the continuum limit. The above detailed comparisons have involved reducing the lattice spacing by a factor of 2, i.e. from aT = 1/2 to aT = 1/4. As we see in Figs 8,9 there is no sign of any lattice spacing dependence other than that calculable in perturbation theory. To go further we also calculated the surface tension with aT = 1/L t = 1/6 at β = 75. In Fig. 11 we show the value of α eff /α(L t ) for this L t = 6 point as well as for other L t values at approximately the same temperature (using T /g 2 = β/4L t ). Note that since the T dependence of this quantity is very weak, as we have seen in Fig. 8, it is not important to get T exactly the same at different values of L t . Note also that this is a reasonably high value of T at which to perform such a test: the deviations from leading order perturbation theory are only at the 7% level or so. We see from Fig. 11 that there is no significant deviation from perturbation theory with decreasing a even down to aT = 1/6. It therefore seems extremely unlikely that the continuum surface tension will not be equally well described by perturbation theory.
So far we have focused on the surface tension of the domain wall. However perturbation theory also makes predictions for more detailed aspects of the domain wall, such as its profile. In Fig. 12 we show how these predictions compare with our Monte Carlo calculated Polyakov loop profiles for several parameter values. We see very good agreement with the main discrepancy arising from the fact that the vacuum values of Polyakov loops are ±1 in leading order perturbation theory. This is primarily an artifact arising from the fact that θ = 0 does not mean that cos θ = 1. A less trivial, but almost invisible difference is that the approach to the vacuum at large distances is Gaussian for perturbation theory, but exponential in the full theory.
In Fig. 13 we show some typical examples of profiles of different components of the action density. Again there is very good agreement with perturbation theory.
We turn now to the Debye screening mass, m D . When we expand the trace of the Polyakov loop the first non-trivial term is ∼ A 0 2 so that we expect correlations of Polyakov loops to receive contributions from the exchange of pairs of screened electric gluons. However, as emphasised by [26], at higher order in the coupling, g 2 /T , they also receive contributions from the magnetic gluon and larger numbers of gluons of both kinds. The question naturally arises: which kind of contribution are we seeing when we extract the masses displayed in Table 3? If we expand the normalised correlation function, C(z), of Polyakov loop operators, P , in energy eigenstates
C(z) C(0) = n c n exp (−E n z)(115)
then the coefficients, c n , are just the amplitudes squared
c n = | vac|P |n | 2(116)
with normalisation c n = 1. From our above discussion we expect c n to be largest for the state with two screened electric gluons. Of course, for large enough z, the correlation function will be dominated by the lightest energy, E min , irrespective of the value of c n (as long as it is non-zero), and so we need to check whether the masses we have listed in Table 3 do indeed correspond to states for which c n is large. In Table 9 we list the overlaps, c n , for the states whose masses are listed in Table 3. We provide both the overlap onto the best smeared Polyakov loop operator and the overlap onto the simple, unsmeared Polyakov loop operator. We do so for the extreme values of T at each value of L t . We see that in all cases the normalised matrix element squared is ≥ 79%. Given that we expect the magnetic gluon etc contributions to receive relative suppressions that are powers of g 2 /T , which is ∼ 0.1 at our highest β values, it is clear that the masses we have obtained belong to states that have no such suppression. We therefore claim that we can read off m D from Table 3 using am D = 0.5am P . As we have already seen, the Debye screening mass is infinite at 1-loop, due to an infrared divergence. These divergences go away if we use nonzero gluon masses in the diagrams and so one can try to do a self-consistent calculation for the mass, m D . This has been done by D'Hoker [18] who obtains
m D 2 = g 2 T π ln T m D − 1 + O 1 (ln(T /m D )) ζ(117)
where ζ > 0. We see from (117) that m D /g 2 is a function only of T /g 2 . Naively we would, of course, have expected m D ∼ g √ T since, as we have seen, that is the scale for the domain wall. And indeed this is the leading T dependence in (117), up to a weakly varying logarithm. However the correction term is down only by logarithms and so we might not be surprised to find the comparison with perturbation theory not as good as for the properties of the domain wall, where the corrections are powers of g 2 /T .
In Fig. 14 we plot our masses against T /g 2 . What we actually choose to plot is m D 2 /g 2 T since that way we factor out the supposedly dominant g 2 T factor, and so expose the remaining variation more clearly. We also show the leading perturbative prediction as obtained from the first term of (117). The first observation is that the dominant variation of m D is indeed ∼ gT 1/2 . However there is a substantial additional variation which is too strong to be due to corrections that are higher order in g 2 /T . Indeed if we try a fit of the form
m D 2 = g 2 T c 0 + c 1 g 2 T ε(118)
we find that it simply does not work, even if we remove the lowest T point from the fit. Indeed, as we see, this additional T variation is quite similar to that obtained from (117). However the normalisation is completely off. Even allowing for the L t dependence in our values of m D , there is a discrepancy of about a factor of 3 with perturbation theory. Since the corrections in (117) are only logarithmic we cannot, of course, claim a contradiction with perturbation theory. However it is worrying that there is no trend towards a reduction of the discrepancy even at our highest values of T where 1/ ln(T /g 2 ) ∼ 0.4. This is in stark contrast to other properties of the domain wall where we found the corrections to leading order perturbation theory to be small even for g 2 /T ∼ 1.
Conclusions
In this paper we have carried out extensive perturbative and Monte Carlo calculations of the high-T domain walls which are associated with the spontaneous breaking of a Z(N) symmetry in SU(N) gauge theories. As we argued, these walls can be viewed as 't Hooft disorder loops which become squeezed once the extent in Euclidean time becomes small enough, as it does at high enough T . Our purpose has been to test high-T perturbation theory and to establish whether these unusual objects do really exist in the Euclidean continuum theory. In order to be able to obtain numerical results of sufficient accuracy to be convincing, we have worked with the simplest theory that one may consider as realistic in this context: the SU(2) gauge theory in 2+1 dimensions. This kind of calculation is difficult for several reasons. Firstly the potential problems are infrared and it is therefore crucial to make sure that the volumes used are large enough. This means not only doing detailed numerical finite-size studies, but also calculating the appropriate finite-volume corrections in perturbation theory. For example, in this paper we have shown how one can calculate the effects of roughening on these domain walls. Secondly one-loop perturbation theory becomes exact, at best, only in the limit T → ∞. Now at fixed aT the size of the lattice will grow as T 1/2 , in lattice units, simply because the width of the wall is O(1/T 1/2 ) in d = 2 + 1. This makes it difficult to simultaneously get close to the continuum limit, where aT is small, and to reach very high values of T . For this reason we have performed the perturbative calculations not only for the continuum theory but also for the lattice theory. In this way we can directly compare perturbation theory to the full non-perturbative results one gets from Monte Carlo simulations.
In practice we have carried out simulations for temperatures as high as ∼ 30T c , where T c is the deconfining temperature, and for lattice spacings as small as 1/6T . (This comparison with 1/T is appropriate, since T is the largest important physical energy scale in the problem). At the highest values of T our numerically obtained values of the surface tension agree with the perturbative predictions at the percent level and this is so at all our values of a. Moreover there is agreement at the 25% level or so, even at temperatures as low as g 2 /T ∼ 1. When we look at the variation with a, over the range a = 1/2T to 1/6T , we again find excellent agreement with perturbation theory, with not the slightest hint of any anomaly developing as a → 0.
At the same time we have obtained perturbative predictions for the more detailed properties of the wall, such as the action density profile. These calculations agree very well with our simulations. These profiles are interesting in themselves and show, for example, that B 2 − E 2 -the thermal energy (in Euclidean space) -becomes negative inside the wall (see Section 5).
The only quantity where we fail to find agreement with perturbation theory is for the Debye screening mass. However here the perturbative calculations are not straightforward and the corrections are expected to be a power of 1/ log(T /g 2 ) and not of g 2 /T . So there is no good reason to read too much significance into this particular discrepancy.
Our conclusion is that these high-T domain walls are present in the Euclidean theory, exactly as predicted by perturbation theory, both on the lattice and in the continuum. Since these walls are quantum rather than semiclassical objects, they provide a severe testing ground for high-T perturbation theory. Its success here lends strong support to the usual pragmatic assumption that perturbation theory reliably describes gauge theories at high temperatures.
while in d = 3 the value of b defined as in (92) equals: b = 2π(N − 1), (d = 3).
Figure 1 :Figure 2 :
12Location of the twist. The plaquettes that are indicated will appear with a factor of −1 in the twisted action. Values of Polyakov loops on typical field configurations with a domain wall: for L t = 2 at: (a) β = 100, (b) β = 7.
Figure 3 :Figure 4 :
34Effective masses from Polyakov loop correlations (+) and from the tails of domain walls (⋄) for (a) β = 112.5 with L t = 3, (b) Dependence of the domain wall action density on the (scaled) length of the lattice. Plotted are the values inTable 5and the lines are the leading order perturbation theory expectation (see text).
Figure 5 :Figure 6 :Figure 7 :Figure 8 :Figure 9 :
56789The width of the domain wall as a function of the length of the lattice for (a) β = 75 and L t = 3, (b) β = 25 and L t = 2. Widths are calculated when the Polyakov loop attains 2Widths of the domain wall (defined as in Fig. 5) plotted againstγ = π(βL t ) 1/2 for (a) L t = 2, (b) L t = 3, (c) L t = 4.Lines are corresponding perturbative predictions; the dotted lines in (c) are for the continuum, L t = ∞. Action density of the domain wall (⋄) for L t = 2, 3, 6 at β = 75. The line is a fit (see text). Also shown are the perturbative values (+) for L t = 2The interface tension (with the factor T 2.5 /g removed): numerical values (⋄) compared to leading order perturbation theory (dashed lines). Ratio of numerical and perturbative interface tensions versus the high-T expansion parameter, g 2 /T .
Figure 10 :Figure 11 :Figure 12 :Figure 13 :Figure 14 :
1011121314Trial extrapolations to the T = ∞ limit of the calculated interface tensions. Ratio, at fixed T , of numerical and (leading-order) perturbative interface tensions for decreasing lattice spacing (a ≡ 1/L t T ). Profile of the Polyakov loop within the domain wall (centred at z = 0): numerical values (solid lines) and perturbation theory (dashed lines). Vacuum-subtracted local action densities within the domain wall: numerical calculations at β = 99.97 and L t = 4, compared with perturbation theory (lines). Numerically calculated values of the Debye mass, compared to D'Hoker's self-consistent perturbative prediction.
Table 1 :
1Average plaquette on lattices without a twist.L t
β
lattice no. sweeps
1 − s nt
2
100.0 30×60
820,000
0.98998945(28)
75.0
26×50
400,000
0.98664608(71)
50.0
50×60
400,000
0.97995063(66)
40×60
240,000
0.97994775(87)
30×60
200,000
0.979994952(117)
20×40
440,000
0.97995159(130)
16×60
400,000
0.97994859(101)
25.0
30×48
400,000
0.9597749(15)
20×48
400,000
0.9597747(21)
12×60
400,000
0.9597749(23)
12×40
400,000
0.9597752(23)
12×30
400,000
0.9597734(34)
12×26
400,000
0.9597801(38)
12×20
400,000
0.9597755(47)
15.0
12×24
400,000
0.9326535(71)
7.0
12×20
400,000
0.8533597(159)
3
112.5 40×80
800,000
0.99109673(14)
75.00 30×80
360,000
0.98663259(35)
30×60
800,000
0.98663236(33)
30×50
196,000
0.98663197(77)
24×80
200,000
0.98663262(48)
18×80
200,000
0.98663250(37)
37.47 18×60
400,000
0.97317001(126)
18×46
400,000
0.97316810(130)
22.45 18×46
400,000
0.95504162(204)
10.27 18×32
400,000
0.9004471(62)
4
99.97 40×80
800,000
0.98997622(15)
49.95 24×60
800,000
0.97989112(39)
29.91 24×48
800,000
0.96631709(87)
13.81 24×40
800,000
0.92634313(179)
5
62.40 30×100
400,000
0.98391671(38)
37.33 30×80
400,000
0.97304743(54)
6
75.00 36×90
840,000
0.98662597(12)
Table 2 :
2Average plaquette on lattices with a twist.L t
β
lattice no. sweeps
1 − s tw
2
100.0 30×80
800,000
0.98987388(26)
75.0
26×70
400,000
0.98649347(50)
50.0
50×60
400,000
0.97972630(65)
30×60
200,000
0.97972542(91)
20×60
400,000
0.97972801(91)
16×60
400,000
0.97972622(124)
25.0
30×48
400,000
0.9593664(15)
20×48
400,000
0.9593643(21)
12×60
400,000
0.9594461(23)
12×48
400,000
0.9593680(29)
12×40
400,000
0.9592867(25)
15.0
12×36
400,000
0.9319060(58)
7.0
12×30
400,000
0.8519115(137)
12×20
400,000
0.8512002(174)
3
112.5 40×100
650,000
0.99106642(14)
75.00 30×100
350,000
0.98659536(35)
30×80
350,000
0.98658592(37)
37.47 18×60
400,000
0.97307998(106)
22.45 18×46
400,000
0.95488588(203)
10.27 18×32
400,000
0.9000855(59)
4
99.97 40×120
800,000
0.98996380(18)
49.95 24×84
800,000
0.97986529(39)
29.91 24×64
800,000
0.96627028(56)
13.81 24×48
800,000
0.92624494(158)
24×40
400,000
0.92622498(271)
6
75.00 36×130
600,000
0.98662125(13)
Table 3 :
3Masses obtained from correlations of Polyakov loops.L t
β
lattice
a m p
2
100.0 30×60
0.315(7)
75.0
26×50
0.352(4)
50.0
50×60
0.414(5)
40×60
0.418(6)
30×60
0.413(6)
20×40
0.409(6)
16×60
0.410(5)
25.0
30×60
0.526(6)
20×48
0.529(6)
12×60
0.529(6)
12×40
0.525(7)
12×30
0.520(9)
12×26
0.533(9)
12×20
0.522(10)
15.0
12×24
0.632(8)
7.0
12×20
0.771(7)
3
112.5 40×80
0.2188(44)
75.00 30×80
0.2586(53)
30×60
0.2588(50)
30×50 0.2561(109)
24×80 0.2508(126)
18×80
0.2511(97)
37.47 18×60
0.3351(59)
18×46
0.3399(66)
22.45 18×46
0.395(6)
10.27 18×32
0.489(4)
4
99.97 40×80
0.1897(18)
49.95 24×60
0.2420(19)
29.91 24×48
0.2802(23)
13.81 24×40
0.3411(40)
5
62.40 30×100
0.177(7)
37.33 30×80
0.2226(42)
6
75.00 36×90
0.1541(23)
Table 4 :
4The action density of the domain wall per unit length and averaged Polyakov loop masses.L t
β
S w /L y
a m p
2
100.0 0.055474(183)
0.315(7)
75.0 0.064096(365)
0.352(4)
50.0
0.08048(22)
0.4123(25)
25.0
0.11773(38)
0.5269(27)
15.0
0.1615(20)
0.632(8)
7.0
0.2598(29)
0.771(7)
3
112.5
0.02728(18)
0.2188(44)
75.00
0.03342(30)
0.2572(32)
37.47
0.04811(76)
0.3372(44)
22.45 0.06448(119)
0.395(6)
10.27
0.1041(25)
0.489(4)
4
99.97
0.01788(34)
0.1897(18)
49.95
0.02604(56)
0.2420(19)
29.91
0.03595(80)
0.2802(23)
13.81 0.05661(119) 0.3411(40)
5
62.40
-
0.177(7)
37.33
-
0.223(4)
6
75.00
0.01104(41)
0.1541(23)
Table 5 :
5The action density of the domain wall as a function of the lattice length, L z , for selected values of β and L t .β
L t L z
S w /L y
25 2
60 0.11840(101)
48
0.11722(96)
40
0.11719(71)
30
0.11792(84)
26
0.11497(77)
20
0.06947(84)
50 2
60
0.07979(36)
40
0.07877(36)
34
0.07169(35)
30
0.04981(52)
26
0.01735(44)
75 3 100 0.03340(39)
80
0.03352(32)
60
0.03239(22)
54
0.02940(35)
50
0.02354(44)
46
0.01577(44)
40
0.00640(60)
Table 6 :
6Some additional values of the action density of the domain wall, for values of L z smaller than inTable 2.L t
β
L z
S w /L y
2
100.0 60 0.05498(18)
75.0 50 0.06391(28)
15.0 24 0.1603(14)
3
112.5 60 0.02701(16)
37.47 40 0.04863(66)
4
99.97 80 0.01688(26)
49.95 60 0.02054(58)
29.91 48 0.03557(82)
Table 7 :
7Variation of the action density of the domain wall and Polyakov loop masses with the length, L y , of the wall.β
L t L y
S w /L y
a m p
25 2
30
0.11765(59)
0.529(3)
20
0.11820(86)
0.529(5)
12
0.1184(10)
0.529(6)
8
0.1162(15)
0.492(11)
4
0.1091(21)
≤ 0.384(9)
50 2
50
0.08076(34)
0.396(12)
40
-
0.416(4)
30
0.08068(53)
0.410(10)
20
0.07991(61)
0.409(6)
16
0.08005(58)
0.386(15)
12
0.07937(64)
0.375(9)
8
0.07860(65)
0.335(8)
4
0.07870(179) 0.272(12)
75 3 120 0.03271(54)
-
60
0.03409(47)
-
30
0.03347(28)
-
24
-
0.251(13)
18
-
0.251(7)
12
0.03303(65)
0.218(7)
6
-
≤ 0.183(6)
Table 8 :
8Perturbative values of the constant, α, in the interface tension.L t α
2
6.024
3
5.655
4
5.409
5
5.284
6
5.221
8
5.165
10 5.142
20 5.113
∞ 5.104
Table 9 :
9Overlaps of Polyakov loop operators onto the states coresponding to our values of m p .L t
β
O best O Bl=1
2
100.0 0.88
0.81
25.0
0.93
0.82
7.0
0.98
0.90
3
112.5 0.84
0.79
10.27 0.98
0.89
4
99.97 0.92
0.81
13.81 0.97
0.86
6
75.00 0.89
0.80
A study of discrete gauge models reveals the same phenomenon: they can be mapped onto a two dimensional Ising model with a coupling ∼ log T for T large. That is, for T large both models start to order.
This happens, however, beyond the applicability of our formulas for q(z).
We study the corrections due to the finiteness of L z /w in Section 6.2.
Such profiles were also measured in[20]. Here, we show that at high T one can actually calculate them analytically.
In other words, B 2 − E 2 y becomes negative.
See, however, footnote 2 on page 18 and the related discussion.
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| [] |
[
"Field Tuning the g Factor in InAs Nanowire Double Quantum Dots",
"Field Tuning the g Factor in InAs Nanowire Double Quantum Dots"
] | [
"M D Schroer \nDepartment of Physics\nPrinceton University\n08544PrincetonNJUSA\n",
"K D Petersson \nDepartment of Physics\nPrinceton University\n08544PrincetonNJUSA\n",
"M Jung \nDepartment of Physics\nPrinceton University\n08544PrincetonNJUSA\n",
"J R Petta \nDepartment of Physics\nPrinceton University\n08544PrincetonNJUSA\n"
] | [
"Department of Physics\nPrinceton University\n08544PrincetonNJUSA",
"Department of Physics\nPrinceton University\n08544PrincetonNJUSA",
"Department of Physics\nPrinceton University\n08544PrincetonNJUSA",
"Department of Physics\nPrinceton University\n08544PrincetonNJUSA"
] | [] | We study the effects of magnetic and electric fields on the g factors of spins confined in a twoelectron InAs nanowire double quantum dot. Spin sensitive measurements are performed by monitoring the leakage current in the Pauli blockade regime. Rotations of single spins are driven using electric-dipole spin resonance. The g factors are extracted from the spin resonance condition as a function of the magnetic field direction, allowing determination of the full g-tensor. Electric and magnetic field tuning can be used to maximize the g-factor difference and in some cases altogether quench the EDSR response, allowing selective single spin control. | 10.1103/physrevlett.107.176811 | [
"https://arxiv.org/pdf/1105.1462v2.pdf"
] | 6,257,421 | 1105.1462 | f1a7889ca47dbf43b10a1188faeb95509899861a |
Field Tuning the g Factor in InAs Nanowire Double Quantum Dots
21 Mar 2012
M D Schroer
Department of Physics
Princeton University
08544PrincetonNJUSA
K D Petersson
Department of Physics
Princeton University
08544PrincetonNJUSA
M Jung
Department of Physics
Princeton University
08544PrincetonNJUSA
J R Petta
Department of Physics
Princeton University
08544PrincetonNJUSA
Field Tuning the g Factor in InAs Nanowire Double Quantum Dots
21 Mar 2012(Dated: January 20, 2013)numbers: 8535Gv7170Ej7321La
We study the effects of magnetic and electric fields on the g factors of spins confined in a twoelectron InAs nanowire double quantum dot. Spin sensitive measurements are performed by monitoring the leakage current in the Pauli blockade regime. Rotations of single spins are driven using electric-dipole spin resonance. The g factors are extracted from the spin resonance condition as a function of the magnetic field direction, allowing determination of the full g-tensor. Electric and magnetic field tuning can be used to maximize the g-factor difference and in some cases altogether quench the EDSR response, allowing selective single spin control.
The desire for a fully controllable and scalable quantum computer places several stringent conditions on its constituent qubits [1]. Recent implementations of spinqubits in GaAs heterostructures have made impressive advances, resulting in a well characterized system in which spin initialization, control, and readout have all been achieved [2][3][4]. For GaAs-based qubits, rapid and selective control of single spins remains a challenge. Single spin rotations have been performed using traditional electron spin resonance (ESR) as well as electric-dipole spin resonance (EDSR), but the highest achieved manipulation rate in GaAs is still 2 orders of magnitude slower than the exchange gate [2,3,5,6]. Encoding the qubit in two-or three-electron spin states can eliminate the need for single spin control, but requires local magnetic field gradients or exquisite control of the exchange interaction [7,8].
"Spin-orbit qubits" based on InAs nanowires have recently been proposed as an alternative to the GaAs lateral quantum dot system [9,10]. In the spin-orbit qubit, the qubits are dressed states of spin and orbital degrees of freedom, due to the strong spin-orbit coupling of InAs. Electrical control of the qubit's orbital component allows Rabi frequencies on the order of 60 MHz to be achieved [5,10]. While strong spin-orbit coupling enables fast spin rotations, it has the potential to introduce several complications. In particular, the electronic g factors, spin relaxation time, and EDSR rotation rates are expected to vary with magnetic field direction [11]. Moreover, the spin-orbit interaction couples the spin to the orbital component of the wavefunction, leading to a g-tensor that is sensitive to the gate-tunable confinement potential.
In this Letter, we explore the effects of strong spinorbit coupling on spins confined to an InAs nanowire double quantum dot. We demonstrate magnetic and electric field control of the g factors for each quantum dot. The EDSR spin manipulation rate is a sensitive function of magnetic field direction, allowing the Rabi frequency to be maximized. For specific magnetic field directions the EDSR response can be dramatically reduced. Our results show that electric and magnetic field tuning of the g fac-tor can be used to optimize the EDSR driving rate and allow for selective single spin control.
Our device, shown in Fig. 1(a), is fabricated on a high resistivity, oxidized silicon substrate. Using electron beam lithography, we define an array of gate patterns consisting of a series of 20 nm thick Ti/Au gates, spaced at a 60 nm pitch. Two large side-gates allow the transparency of the nanowire leads to be tuned. A 20 nm layer of SiN x is then deposited as a gate dielectric using plasma enhanced chemical vapor deposition [10]. Single crystal InAs nanowires with a zinc blende structure are grown using a gold catalyzed vapor-liquid-solid process in a metal-organic chemical vapor deposition reactor [13]. Nanowires are removed from the growth substrate with ultrasonication in ethanol and then dispersed on the gate array. The final fabrication step involves defining low resistance ohmic contacts to nanowires with diameters of ∼50 nm that have fallen across a gate pattern. The sample was measured in a dilution refrigerator equipped with a vector magnet system and high frequency coaxial wiring.
We first determine the charge stability diagram by measuring the current at finite applied bias, as shown in Fig. 1(b). The absence of finite bias triangles in the lower left hand corner of the plot indicates the double quantum dot has been completely emptied of free electrons. In this region (N L , N R ) = (0, 0), where N L (N R ) is the number of electrons in the left(right) dot. Measurements at high bias and with relatively transparent tunnel barries have confirmed our identification of the (0,0) charge state. We focus on the two-electron regime, where Pauli blockade results in current rectification at the (1,1)↔(2,0) charge transition, as first observed in vertical double quantum dots by Ono et al. [14]. At positive bias, charge transport at the (1,1)↔(2,0) transition occurs freely through a cycle of (1,0)→(2,0)→(1,1)→(1,0), where steps with double occupancy are limited to spin singlet states due to the ∼9 meV exchange splitting of the (2,0) state. Under negative bias, the cycle is reversed and the system loads the second electron into the (1,1) charge state, where the singlet and triplet states are nearly degenerate. If a (1,1) arXiv:1105.1462v2 [cond-mat.mes-hall] 21 Mar 2012 singlet and triplet states are nearly degenerate. If a (1,1) triplet state is loaded, the charge transition to (2,0) will be blocked due to Pauli exclusion. The left and right insets to Fig. 1(b) show the current through the double dot at positive and negative bias, exhibiting the voltage bias dependence characteristic of Pauli blockade.
In the absence of spin flips or cotunneling, the leakage current in the Pauli blockade regime will be zero. Processes which drive rotations from the (1,1) triplet states to the (1,1) singlet state will lift Pauli blockade, resulting in a measurable leakage current. Pauli blockade enables spindependent readout and has been used to detect the mixing of spin states due to hyperfine fields, ESR, and EDSR in GaAs and InAs quantum dots [3,6,10,15].
We measure the leakage current in the Pauli blockade regime as a function of magnetic field B and the frequency f of the ac excitation applied to a depletion gate (see Fig. 2). A zero field peak is evident, corresponding to rapid singlettriplet mixing due to the hyperfine fields [15,16]. We extract a hyperfine field B N ¼ 3:3 mT from these data using the theory of Jouravlev et al. [17]. Previous reports indicate B N ¼ 0:66 and 1.5 mT in InAs quantum dots, with a value that depends on the number of nuclei the electron spin has appreciable overlap with [10,18]. Away from zero field, the leakage current is nonzero only when the EDSR resonance condition is satisfied. The data in Fig. 2(a) are measured for B kẑ, while the data in Fig. 2(b) are measured for B k ðx þẑÞ. Vertical cuts through the data, shown in Fig. 2(d), clearly indicate that the g factor of each electron is sensitive to the magnetic field direction, with well-resolved EDSR peaks visible for each spin in the lower trace.
We quantitatively measure the directional dependence of the g factors by fixing the magnetic field magnitude at jBj ¼ 80 mT and rotating the field direction. Data are shown in Figs. 3(a)-3(c) for three different field rotations. In each case the position and intensity of the EDSR resonances evolve with the field direction. For each magnetic field direction we fit each resonance peak to Gaussian and extract the g factor. Fitting is only performed for regions of the data where the resonance is clearly visible. After the g factors have been extracted, we perform a simultaneous curve fit of the data sets in Figs. 3(a)-3(c) to a general model of a 3D anisotropic g factor:
gðBÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi g 2 1 B 2 1 þ g 2 2 B 2 2 þ g 2 3 B 2 3 q jBj :(1)
Here g 1 , g 2 , and g 3 are the values of the g factors for the three principal axes, and B 1 , B 2 and B 3 are the magnetic field components along these axes [19,20]. Our fitting also takes into account field offsets of 2-3 mT due to trapped flux in the superconducting magnet coils.
The best fit values for the g tensor are listed in Table I. The principal axes' g factors vary from approximately 7 to 9, suppressed from the bulk value of jgj ¼ 14:7 through a combination of spin-orbit coupling and confinement [21][22][23]. In each case, the g tensor contains one large
V sd ¼ þ4 mV.
The apparent excited state is a resonance associated with the one-dimensional leads that contact the double quantum dot [12]. Right inset: ð1; 1Þ $ ð2; 0Þ transition at V sd ¼ À4 mV. The current in the insets has been multiplied by a factor of 2. Left inset: (1, 1) ↔ (2, 0) transition at V sd = +4 mV. The apparent excited state is a resonance associated with the onedimensional leads that contact the double quantum dot [12]. Right inset: (1, 1) ↔ (2, 0) transition at V sd = -4 mV. The current in the insets has been multiplied by a factor of two.
triplet state is loaded, the charge transition to (2,0) will be blocked due to Pauli exclusion. The left and right insets to Fig. 1(b) show the current through the double dot at positive and negative bias, exhibiting the voltage bias dependence characteristic of Pauli blockade.
In the absence of spin-flips or cotunneling, the leakage current in the Pauli blockade regime will be zero. Processes which drive rotations from the (1,1) triplet states to the (1,1) singlet state will lift Pauli blockade, resulting in a measurable leakage current. Pauli blockade enables spin-dependent readout and has been used to detect the mixing of spin states due to hyperfine fields, ESR, and EDSR in GaAs and InAs quantum dots [3,6,10,15].
We measure the leakage current in the Pauli blockade regime as a function of magnetic field B and the frequency f of the ac excitation applied to a depletion gate (see Fig. 2). A zero field peak is evident, corresponding to rapid singlet-triplet mixing due to the hyperfine fields [15,16]. We extract a hyperfine field B N = 3.3 mT from these data using the theory of Jouravlev et al. [17]. Previous reports indicate B N = 0.66 and 1.5 mT in InAs quantum dots, with a value that depends on the number of nuclei the electron spin has appreciable overlap with [10,18]. Away from zero field, the leakage current is nonzero only when the EDSR resonance condition is satisfied. The data in Fig. 2(a) are measured for B ||ẑ, while the data in Fig. 2(b) are measured for B || (x +ẑ). Vertical cuts through the data, shown in Fig. 2(d), clearly indicate that the g factor of each electron is sensitive to the singlet and triplet states are nearly degenerate. If a (1,1) triplet state is loaded, the charge transition to (2,0) will be blocked due to Pauli exclusion. The left and right insets to Fig. 1(b) show the current through the double dot at positive and negative bias, exhibiting the voltage bias dependence characteristic of Pauli blockade.
In the absence of spin flips or cotunneling, the leakage current in the Pauli blockade regime will be zero. Processes which drive rotations from the (1,1) triplet states to the (1,1) singlet state will lift Pauli blockade, resulting in a measurable leakage current. Pauli blockade enables spindependent readout and has been used to detect the mixing of spin states due to hyperfine fields, ESR, and EDSR in GaAs and InAs quantum dots [3,6,10,15].
We measure the leakage current in the Pauli blockade regime as a function of magnetic field B and the frequency f of the ac excitation applied to a depletion gate (see Fig. 2). A zero field peak is evident, corresponding to rapid singlettriplet mixing due to the hyperfine fields [15,16]. We extract a hyperfine field B N ¼ 3:3 mT from these data using the theory of Jouravlev et al. [17]. Previous reports indicate B N ¼ 0:66 and 1.5 mT in InAs quantum dots, with a value that depends on the number of nuclei the electron spin has appreciable overlap with [10,18]. Away from zero field, the leakage current is nonzero only when the EDSR resonance condition is satisfied. The data in Fig. 2(a) are measured for B kẑ, while the data in Fig. 2(b) are measured for B k ðx þẑÞ. Vertical cuts through the data, shown in Fig. 2(d), clearly indicate that the g factor of each electron is sensitive to the magnetic field direction, with well-resolved EDSR peaks visible for each spin in the lower trace.
We quantitatively measure the directional dependence of the g factors by fixing the magnetic field magnitude at jBj ¼ 80 mT and rotating the field direction. Data are shown in Figs. 3(a)-3(c) for three different field rotations. In each case the position and intensity of the EDSR resonances evolve with the field direction. For each magnetic field direction we fit each resonance peak to Gaussian and extract the g factor. Fitting is only performed for regions of the data where the resonance is clearly visible. After the g factors have been extracted, we perform a simultaneous curve fit of the data sets in Figs. 3(a)-3(c) to a general model of a 3D anisotropic g factor:
gðBÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi g 2 1 B 2 1 þ g 2 2 B 2 2 þ g 2 3 B 2 3 q jBj :(1)
Here g 1 , g 2 , and g 3 are the values of the g factors for the three principal axes, and B 1 , B 2 and B 3 are the magnetic field components along these axes [19,20]. Our fitting also takes into account field offsets of 2-3 mT due to trapped flux in the superconducting magnet coils.
The best fit values for the g tensor are listed in Table I. The principal axes' g factors vary from approximately 7 to 9, suppressed from the bulk value of jgj ¼ 14:7 through a combination of spin-orbit coupling and confinement [21][22][23]. In each case, the g tensor contains one large 1 (color online). (a) Scanning electron microscope image illustrating the device geometry. Four gates are sufficient to fully define the double quantum dot with this device, resulting in a double well confinement potential. The double dot's coupling to the right and left leads are controlled with voltages V rw and V lw , respectively, while V rp and V lp control the interdot coupling and occupation. (b) Current measured through the device as a function of gate voltages V rp and V lp , with source drain bias V sd ¼ À4 mV. Dashed lines are superimposed to illustrate the double dot charge stability diagram. Left inset: ð1; 1Þ $ ð2; 0Þ transition at V sd ¼ þ4 mV. The apparent excited state is a resonance associated with the one-dimensional leads that contact the double quantum dot [12]. Right inset: ð1; 1Þ $ ð2; 0Þ transition at V sd ¼ À4 mV. The current in the insets has been multiplied by a factor of 2. magnetic field direction, with well-resolved EDSR peaks visible for each spin in the lower trace.
We quantitatively measure the directional dependence of the g factors by fixing the magnetic field magnitude at |B| = 80 mT and rotating the field direction. Data are shown in Figs. 3(a)-(c) for three different field rotations. In each case the position and intensity of the EDSR resonances evolve with the field direction. For each magnetic field direction we fit each resonance peak to a Gaussian and extract the g factor. Fitting is only performed for regions of the data where the resonance is clearly visible. After the g factors have been extracted, we perform a simultaneous curve fit of the data sets in Fig. 3(a-c) to a general model of a 3D anisotropic g factor:
g (B) = g 2 1 B 2 1 + g 2 2 B 2 2 + g 2 3 B 2 3 |B|(1)
Here g 1 , g 2 , and g 3 are the values of the g factors for the three principal axes, and B 1 , B 2 and B 3 are the magnetic field components along these axes [19,20]. Our fitting also takes into account field offsets of 2-3 mT due to trapped flux in the superconducting magnet coils.
The best fit values for the g tensor are listed in Table I. The principal axes' g factors vary from approximately 7 to 9, suppressed from the bulk value of |g| = 14.7 through a combination of spin-orbit coupling and confinement [21][22][23]. In each case, the g tensor contains one large axis, and two approximately equal and smaller axis, and two approximately equal and smaller axes, appearing to mimic the cylindrical symmetry of the nanowire itself. However, we find that the large axis is oriented in different directions for each electron, and in general is not aligned with the nanowire axis. These results suggest that the orientation of the g tensor is largely determined by the anisotropy of the confinement potential, not the underlying one-dimensional symmetry of the nanowire.
We performed a second set of measurements with the device tuned to a very different gate voltage configuration to measure the sensitivity of the g tensor to changes in the confinement potential. The data in Fig. 3 were acquired with the double quantum dot tuned such that the tunnel rates to the left and the right leads were balanced. To explore a different confinement potential, we unbalanced the tunnel rates by varying the voltages on the local electrodes, thereby tuning the double quantum dot into a regime where transport shows a large, asymmetric cotunneling peak [see Fig. 4(a)]. Figures 4(b)-4(d) map out the g factor as a function of field orientation for the unbalanced case. The g tensor has been dramatically altered, with extracted values for the principal axes directions and g factors summarized in Table I. For this gate voltage configuration, the upper resonance is highly anisotropic and shifted to higher frequencies. The lower resonance is broadened significantly, which we attribute to enhanced dynamic nuclear polarization [24].
We now focus on the field dependence of the EDSR amplitude, which is expected to depend on both the magnitude and direction of the magnetic field [5]. Figures 3(d)-3(f) show the current extracted from the upper resonance of Figs. 3(a)-3(c), respectively. We find that the on resonance current is strongly modulated by field orientation. Golovach et al. [5] have shown that the effective magnetic field for spin-orbit driven EDSR in a two-dimensional system is given by
B so ðtÞ ¼ 2B Â ð 0 sin! ac tÞ;(2)
where 0 is a dimensionless ''spin-orbit vector'' determined by the applied electric field and the local spin-orbit parameters. In the absence of an applicable theory, we generalize the concept of the spin-orbit vector to three axes, appearing to mimic the cylindrical symmetry of the nanowire itself. However, we find that the large axis is oriented in different directions for each electron, and in general is not aligned with the nanowire axis. These results suggest that the orientation of the g tensor is largely determined by the anisotropy of the confinement potential, not the underlying one-dimensional symmetry of the nanowire.
We performed a second set of measurements with the device tuned to a very different gate voltage configuration in order to measure the sensitivity of the g tensor to changes in the confinement potential. The data in Fig. 3 were acquired with the double quantum dot tuned such that the tunnel rates to the left and the right leads were balanced. To explore a different confinement potential, we unbalanced the tunnel rates by varying the voltages on the local electrodes, thereby tuning the double quantum dot into a regime where transport shows a large, asymmetric cotunneling peak [see Fig. 4(a)]. Figure 4(b)-4(d) map out the g factor as a function of field orientation for the 'unbalanced' case. The g tensor has been dramatically altered, with extracted values for the principal axes directions and g factors summarized in Table I. For this gate voltage configuration, the upper resonance is highly anisotropic and shifted to higher frequencies. The lower resonance is broadened significantly, which we attribute to enhanced dynamic nuclear polarization [24].
We now focus on the field dependence of the EDSR amplitude, which is expected to depend on both the magnitude and direction of the magnetic field [5]. Figure 3(d)-3(f) show the current extracted from the upper resonance of Fig. 3(a)-3(c) respectively. We find that the on resonance current is strongly modulated by field orientation. Golovach et al. [5] have shown that the ef-fective magnetic field for spin-orbit driven EDSR in a two-dimensional system is given by
B so (t) = 2B × (Ω 0 sin ω ac t)(2)
where Ω 0 is a dimensionless "spin-orbit vector" determined by the applied electric field and the local spin-axis, and two approximately equal and smaller axes, appearing to mimic the cylindrical symmetry of the nanowire itself. However, we find that the large axis is oriented in different directions for each electron, and in general is not aligned with the nanowire axis. These results suggest that the orientation of the g tensor is largely determined by the anisotropy of the confinement potential, not the underlying one-dimensional symmetry of the nanowire. We performed a second set of measurements with the device tuned to a very different gate voltage configuration to measure the sensitivity of the g tensor to changes in the confinement potential. The data in Fig. 3 were acquired with the double quantum dot tuned such that the tunnel rates to the left and the right leads were balanced. To explore a different confinement potential, we unbalanced the tunnel rates by varying the voltages on the local electrodes, thereby tuning the double quantum dot into a regime where transport shows a large, asymmetric cotunneling peak [see Fig. 4(a)]. Figures 4(b)-4(d) map out the g factor as a function of field orientation for the unbalanced case. The g tensor has been dramatically altered, with extracted values for the principal axes directions and g factors summarized in Table I. For this gate voltage configuration, the upper resonance is highly anisotropic and shifted to higher frequencies. The lower resonance is broadened significantly, which we attribute to enhanced dynamic nuclear polarization [24].
We now focus on the field dependence of the EDSR amplitude, which is expected to depend on both the magnitude and direction of the magnetic field [5]. Figures 3(d)-3(f) show the current extracted from the upper resonance of Figs. 3(a)-3(c), respectively. We find that the on resonance current is strongly modulated by field orientation. Golovach et al. [5] have shown that the effective magnetic field for spin-orbit driven EDSR in a two-dimensional system is given by
B so ðtÞ ¼ 2B Â ð 0 sin! ac tÞ;(2)
where 0 is a dimensionless ''spin-orbit vector'' determined by the applied electric field and the local spin-orbit parameters. In the absence of an applicable theory, we generalize the concept of the spin-orbit vector to three Fig. 3, and the 'unbalanced' case of Fig. 4. α, β and γ are Euler angles (in radians) which define the transformation from the measurement coordinate system to the g tensor coordinate system, using the ZX Z convention. orbit parameters. In the absence of an applicable theory, we generalize the concept of the spin-orbit vector to three dimensions. While Eqn. 2 gives the Rabi frequency, f r = gµ B B so /2h for on resonance driving, the dependence of the pumped current is less clear. In general, the current will not be proportional to f r as the fluctuating nuclear field shifts the applied excitation on and off of resonance. We have studied the induced current using a simple rate equation model, similar to the approach used by Koppens et al. [25]. For B so < B N we find
I EDSR = eΓ i |B so | 2 2B N 2(3)
where Γ i is the interdot tunneling rate. Fits to Eqn. 3 are displayed in Fig. 3
(d)-3(f). Equation 3
reproduces the periodic modulation of the pumped current; however we are not able to simultaneously fit the three individual rotation sweeps. The discrepancy from the behavior predicted by Eqs. 2 and 3 is not well understood but may involve corrections due to the magnetic field dependence of spin-orbit matrix elements [11]. Equation 3 may also be used to estimate the magnitude of the spin-orbit field. With a slightly different tuning, we have observed EDSR currents exceeding 1.5 pA at |B| = 150 mT. For this tuning, we measure positivebias (nonblockaded) currents of ∼35 pA. Assuming the interdot tunnel coupling is rate limiting, this results in a lower bound estimate of Γ i ∼ 220 MHz, leading to a spin-orbit field |B so | = 1.0 mT. Finally, we may estimate the spin-orbit length l so as
l so = B B so 2 2 e|E| m e ∆ 2(4)
where |E| is the applied electric field, m e is the effective mass, and ∆ is the level spacing [5]. We estimate |E| as 1 mV/60 nm, and find ∆ = 9 meV from finite bias measurements. With these parameters, we find l so ∼ 170 nm. Previous studies have reported values for the spinorbit length in InAs nanowires of 100-200 nm [26,27]. The large variation of the g factors with electric field tuning suggests that g tensor modulation might play a significant role in electrically driven spin manipulation [28]. Assuming the change in the g factor is purely linear in electric field, we find a maximum ∂g/∂V g ∼ 0.03/mV. Taking g = 8 and assuming an applied magnetic field of 80 mT, this yields an effective ac magnetic field of 0.3 mT, which is of the same order as the calculated spinorbit field. Our results suggest that g tensor modulation may be significant for specific electric and magnetic field configurations.
We have demonstrated magnetic and electric field tuning of the electron g factors in an InAs nanowire double quantum dot, which have important implications for spin-orbit qubits. The ability to selectively control single spins may be obtained by tuning magnetic and electric fields to (1) maximize the difference between the g factors of neighboring spins and (2) maximize the difference in the response of neighboring spins. A linear array of quantum dots in an InAs nanowire could be used to implement a Loss-DiVincenzo style spin register, where neighboring spins are tuned out of resonance with electric field tuning of the g tensor. The combination of spectral selectivity with the localized nature of the on-chip generated microwave fields may allow the development of scalable quantum dot arrays in InAs nanowires.
PACS numbers: 85.35.Gv, 71.70.Ej, 73.21.La
FIG. 2
2(color online). Leakage current measured as a function of the magnetic field and applied frequency for (a) B kẑ and (b) B k ðx þẑÞ. The zero field peak, shown in (c), is due to hyperfine field mixing of the spin states, with a fit to the data yielding B N ¼ 3:3 mT. The spin resonance condition is satisfied when hf ¼ g B jBj, resulting in a peak in the leakage current. Cuts through the data in (a) and (b) are shown in (d), demonstrating the dependence of the g factors on magnetic field direction. The upper trace is offset by 0.3 pA for clarity.FIG. 1 (color online). (a) Scanning electron microscope image illustrating the device geometry. Four gates are sufficient to fully define the double quantum dot with this device, resulting in a double well confinement potential. The double dot's coupling to the right and left leads are controlled with voltages V rw and V lw , respectively, while V rp and V lp control the interdot coupling and occupation. (b) Current measured through the device as a function of gate voltages V rp and V lp , with source drain bias V sd ¼ À4 mV. Dashed lines are superimposed to illustrate the double dot charge stability diagram. Left inset: ð1; 1Þ $ ð2; 0Þ transition at
online) (a) Scanning electron microscope image illustrating the device geometry. Four gates are sufficient to fully define the double quantum dot with this device, resulting in a double well confinement potential. The double dot's coupling to the right and left leads are controlled with voltages Vrw and V lw respectively, while Vrp and V lp control the interdot coupling and occupation. (b) Current measured through the device as a function of gate voltages Vrp and V lp , with source drain bias V sd = -4 mV. Dashed lines are superimposed to illustrate the double dot charge stability diagram.
FIG. 2
2(color online). Leakage current measured as a function of the magnetic field and applied frequency for (a) B kẑ and (b) B k ðx þẑÞ. The zero field peak, shown in (c), is due to hyperfine field mixing of the spin states, with a fit to the data yielding B N ¼ 3:3 mT. The spin resonance condition is satisfied when hf ¼ g B jBj, resulting in a peak in the leakage current. Cuts through the data in (a) and (b) are shown in (d), demonstrating the dependence of the g factors on magnetic field direction. The upper trace is offset by 0.3 pA for clarity.FIG.
online) Leakage current measured as a function of magnetic field and applied frequency for (a) B ||ẑ and (b) B || (x +ẑ). The zero field peak, shown in (c), is due to hyperfine field mixing of the spin states, with a fit to the data yielding BN = 3.3 mT. The spin resonance condition is satisfied when hf = gµB|B|, resulting in a peak in the leakage current. Cuts through the data in (a) and (b) are shown in (d), demonstrating the dependence of the g factors on magnetic field direction. The upper trace is offset by 0.3 pA for clarity.
FIG. 3 (
3color online). (a)-(c) Leakage current measured as a function of magnetic field direction and frequency for the rotations depicted in the insets below. The y axes have been rescaled from frequency to units of g, taking a fixed magnetic field amplitude of 80 mT; all angles are radians. In this coordinate system, the unit vector describing the nanowire axis is approximately ð; Þ ¼ ð1:2; 3:1Þ. A fit to Eq. (1) has been superimposed on the data. (d)-(f) Pumped current extracted from the upper resonances in (a)-(c), along with a fit to Eq. (3). FIG. 4 (color online). (a) Measured current through the double dot as a function of V lp and V rp for V sd ¼ À4 mV in the case of unbalanced tuning. (b)-(d) Evolution of the resonance condition for magnetic field rotations about the x, y, and z axes, respectively. In (b)-(d) the y axes have been rescaled into units of g, taking a fixed magnetic field amplitude of 80 mT. online) (a-c) Leakage current measured as a function of magnetic field direction and frequency for the rotations depicted in the insets below. The y-axes have been rescaled from frequency to units of g, taking a fixed magnetic field amplitude of 80 mT; all angles are radians. In this coordinate system, the unit vector describing the nanowire axis is approximately (θ, φ) = (1.2, 3.1). A fit to Eqn. 1 has been superimposed on the data. (d-e) Pumped current extracted from the upper resonances in (a-c), along with a fit to Eq. (3).
FIG. 3 (
3color online). (a)-(c) Leakage current measured as a function of magnetic field direction and frequency for the rotations depicted in the insets below. The y axes have been rescaled from frequency to units of g, taking a fixed magnetic field amplitude of 80 mT; all angles are radians. In this coordinate system, the unit vector describing the nanowire axis is approximately ð; Þ ¼ ð1:2; 3:1Þ. A fit to Eq. (1) has been superimposed on the data. (d)-(f) Pumped current extracted from the upper resonances in (a)-(c), along with a fit to Eq. (3). FIG. 4 (color online). (a) Measured current through the double dot as a function of V lp and V rp for V sd ¼ À4 mV in the case of unbalanced tuning. (b)-(d) Evolution of the resonance condition for magnetic field rotations about the x, y, and z axes, respectively. In (b)-(d) the y axes have been rescaled into units of g, taking a fixed magnetic field amplitude of 80 mT. online) (a) Measured current through the double dot as a function of V lp and Vrp for V sd = -4 mV in the case of unbalanced tuning. (b)-(d) Evolution of the resonance condition for magnetic field rotations about the x, y and z axes respectively. In (b)-(d) the y axes have been rescaled into units of g, taking a fixed magnetic field amplitude of 80 mT.
TABLE I :
IG-tensor Parameters. Values are given for both the balanced case of
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"RAMS-Trans: Recurrent Attention Multi-scale Transformer for Fine-grained Image Recognition"
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"Xuan Jin [email protected] \nAlibaba Group\nHangzhouChina\n",
"Yin Zhang [email protected] \nCollege of Computer Science and Technology\nZhejiang University\nHangzhouChina\n",
"Haiwen Hong [email protected] \nCollege of Computer Science and Technology\nZhejiang University\nHangzhouChina\n\nAlibaba Group\nHangzhouChina\n",
"Jingfeng Zhang \nAlibaba Group\nHangzhouChina\n",
"Yuan He \nCollege of Computer Science and Technology\nZhejiang University\nHangzhouChina\n\nAlibaba Group\nHangzhouChina\n",
"Hui Xue [email protected] \nAlibaba Group\nHangzhouChina\n",
"Yunqing Hu ",
"Xuan Jin ",
"Yin Zhang ",
"Haiwen Hong ",
"Jingfeng Zhang ",
"Yuan He ",
"Hui Xue "
] | [
"College of Computer Science and Technology\nZhejiang University\nHangzhouChina",
"Alibaba Group\nHangzhouChina",
"Alibaba Group\nHangzhouChina",
"College of Computer Science and Technology\nZhejiang University\nHangzhouChina",
"College of Computer Science and Technology\nZhejiang University\nHangzhouChina",
"Alibaba Group\nHangzhouChina",
"Alibaba Group\nHangzhouChina",
"College of Computer Science and Technology\nZhejiang University\nHangzhouChina",
"Alibaba Group\nHangzhouChina",
"Alibaba Group\nHangzhouChina"
] | [] | In fine-grained image recognition (FGIR), the localization and amplification of region attention is an important factor, which has been explored a lot by convolutional neural networks (CNNs) based approaches. The recently developed vision transformer (ViT) has achieved promising results on computer vision tasks. Compared with CNNs, Image sequentialization is a brand new manner. However, ViT is limited in its receptive field size and thus lacks local attention like CNNs due to the fixed size of its patches, and is unable to generate multi-scale features to learn discriminative region attention. To facilitate the learning of discriminative region attention without box/part annotations, we use the strength of the attention weights to measure the importance of the patch tokens corresponding to the raw images. We propose the recurrent attention multi-scale transformer (RAMS-Trans), which uses the transformer's self-attention to recursively learn discriminative region attention in a multi-scale manner. Specifically, at the core of our approach lies the dynamic patch proposal module (DPPM) guided region amplification to complete the integration of multiscale image patches. The DPPM starts with the full-size image patches and iteratively scales up the region attention to generate new patches from global to local by the intensity of the attention weights generated at each scale as an indicator. Our approach requires only the attention weights that come with ViT itself and can be easily trained end-to-end. Extensive experiments demonstrate that RAMS-Trans performs better than concurrent works, in addition to efficient CNN models, achieving state-of-the-art results on three benchmark datasets. | 10.1145/3474085.3475561 | [
"https://arxiv.org/pdf/2107.08192v1.pdf"
] | 236,087,893 | 2107.08192 | 21288897fd9e8ce56104743138caffa00470ca13 |
RAMS-Trans: Recurrent Attention Multi-scale Transformer for Fine-grained Image Recognition
October 20-24, 2021
Yunqing Hu [email protected]
College of Computer Science and Technology
Zhejiang University
HangzhouChina
Alibaba Group
HangzhouChina
Xuan Jin [email protected]
Alibaba Group
HangzhouChina
Yin Zhang [email protected]
College of Computer Science and Technology
Zhejiang University
HangzhouChina
Haiwen Hong [email protected]
College of Computer Science and Technology
Zhejiang University
HangzhouChina
Alibaba Group
HangzhouChina
Jingfeng Zhang
Alibaba Group
HangzhouChina
Yuan He
College of Computer Science and Technology
Zhejiang University
HangzhouChina
Alibaba Group
HangzhouChina
Hui Xue [email protected]
Alibaba Group
HangzhouChina
Yunqing Hu
Xuan Jin
Yin Zhang
Haiwen Hong
Jingfeng Zhang
Yuan He
Hui Xue
RAMS-Trans: Recurrent Attention Multi-scale Transformer for Fine-grained Image Recognition
Chengdu, SichuanOctober 20-24, 202110.1145/1122445.1122456ACM Reference Format: 2021. RAMS-Trans: Recurrent Attention Multi-scale Transformer for Fine-grained Image Recognition. In Chengdu '21: ACM Symposium on Neural Gaze Detection, October 20-24, 2021, Chengdu, Sichuan. ACM, New York, NY, USA, 10 pages. https://doi.org/10.1145/1122445.1122456 * Corresponding author. ACM ISBN 978-1-4503-XXXX-X/18/06. . . $15.00
In fine-grained image recognition (FGIR), the localization and amplification of region attention is an important factor, which has been explored a lot by convolutional neural networks (CNNs) based approaches. The recently developed vision transformer (ViT) has achieved promising results on computer vision tasks. Compared with CNNs, Image sequentialization is a brand new manner. However, ViT is limited in its receptive field size and thus lacks local attention like CNNs due to the fixed size of its patches, and is unable to generate multi-scale features to learn discriminative region attention. To facilitate the learning of discriminative region attention without box/part annotations, we use the strength of the attention weights to measure the importance of the patch tokens corresponding to the raw images. We propose the recurrent attention multi-scale transformer (RAMS-Trans), which uses the transformer's self-attention to recursively learn discriminative region attention in a multi-scale manner. Specifically, at the core of our approach lies the dynamic patch proposal module (DPPM) guided region amplification to complete the integration of multiscale image patches. The DPPM starts with the full-size image patches and iteratively scales up the region attention to generate new patches from global to local by the intensity of the attention weights generated at each scale as an indicator. Our approach requires only the attention weights that come with ViT itself and can be easily trained end-to-end. Extensive experiments demonstrate that RAMS-Trans performs better than concurrent works, in addition to efficient CNN models, achieving state-of-the-art results on three benchmark datasets.
INTRODUCTION
Fine-grained image recognition (FGIR) has been a challenging problem. Most of the current methods are dominated by convolutional neural networks (CNNs). Unlike conventional image classification problems, FGIR has the problem of large intra-class variance and small inter-class variance. Therefore, FGIR methods need to be able to identify and localize region attention in an image that is critical for classification. There is a class of methods called part-based methods [2,36] for FGIR, and some of them use additional supervised information such as bounding box/part annotations to locate key regions. However, labeling bounding boxes/part annotations is a labor-intensive task that requires a lot of resources. How to be able to use the effective information generated by the model itself for region attention localization and amplification is one of the research directions that FGIR has to face.
The effectiveness of CNNs needs no further explanation here. However, we need to emphasize again that one of the key aspects of CNNs that make them effective is their translation invariance and local feature representation capability. CNNs are continuously downsampled as their depth increases, yet the receptive field of the model increases, so that both global and local information of the feature map can be utilized. For example, in networks such as VggNet [29] and ResNet [17], the receptive field of underlying convolutional is smaller and has more local information, while the receptive field of higher convolutional is larger and has more global information. The works [41] and [20] use this characteristic for FGIR. Some works exploit the attention properties of the feature maps of CNN itself, such as [40] and [35] that exploit the attention maps of image features to select region attentions. Transformer [32] has gradually transformed from a research hotspot in NLP [6,7,38] to CV tasks [9,46] in recent years. The proposal of vision transformer (ViT) has brought a new shock to computer vision and aroused the research interest in image sequentialization within the community. ViT flattens the segmented image patches and transforms them into patch tokens. Similar to character sequences in NLP, those tokens will be sent to the multi-head self-attention mechanism for training. Since patch tokens are position-agnostic, position embedding will be added to serve the purpose of adding spatial information. However, when ViT encounters FGIR, there are two main problems need to be solved. First, The model processes all the patch tokens at once, and when the complexity of the dataset increases, such as when the image resolution is high or when an image has a cluttered background, the model may not be able to effectively capture the region attention carried in the patch tokens.
Conversely, when the image resolution is low, this fixed patch size is more likely to make the model lose local information. Second, ViT differs from CNNs in that the length of patch tokens does not change as its encoder blocks increases, thus the receptive field of the model cannot be effectively extended. Therefore, for FGIR, we can do much more than just feed the flattened raw image patches into the Transformer. If we refer to the characteristics of CNNs and introduce attention to local regions in the model, that is, extend the effective receptive field, the recognition performance of the model is likely to be further improved. So then we encounter a very important question, how to explore and discover local information in ViT which focuses on using global information? The latest TransFG [16] gives us a pretty good answer, which is to take advantage of ViT's inherent attention weights. Both in the NLP domain of transformer and ViT training, most of the works simply require the use of the last layer of classification token information, discarding the seemingly accessory attention weights. TransFG multiplies all the attention weights before the last transformer layer to get the importance ranking of patch tokens, and then concatenate the selected tokens along with the global classification token as input sequence to the last transformer layer. However, this kind of hard attention filtering is easy to fail in two cases, one is in the case of small image resolution, and the other is in the case of the high complexity of the dataset. In the former case, a lot of important local information is not easily available, and if most of the tokens information has to be filtered out at this time, it is likely to lose classification performance. In the latter case, a model can easily make wrong judgments based on improper token information when the attention mechanism fails.
Through preliminary visualization experiments, We find that the strength of the attention weights can be intuitively correlated with the extent to which the patches contain the target object. To this end, we propose the recurrent attention multi-scale transformer (RAMS-Trans), which uses the transformer's self-attention mechanism to recursively learn discriminative region attention in a multi-scale manner. Specifically, at the core of our approach lies the proposed dynamic patch proposal module (DPPM) which aims to adaptively select the most discriminative region for each image. The DPPM starts with the complete image patches and scales up the region attention to generate new patches from global to local by the intensity of the attention weights generated at each scale as an indicator. The finer scale network takes as input the tendency regions scaled up from the previous scale in a cyclic manner. The following contributions are made in this paper:
• We reformulate the FGIR problem from a sequence to sequence learning perspective and design a new visual Transformer architecture namely recurrent attention multi-scale transformer (RAMS-Trans). It combines the advantages of CNNs in expanding receptive field, strengthening locality, and the advantages of Transformers in utilizing global information. • As an instantiation, we exploit the transformer framework, specifically, to the use of multi-head self-attention weights to locate and zoom in on regions of interest, to implement our fully attentive feature representation by sequentializing images.
• Extensive experiments show that our RAMS-Trans model can learn superior feature representations as compared to traditional CNNs and concurrent work on three popular FGIR benchmarks (CUB-200-2011, Stanford Dogs, and iNatu-ralist2017).
2 RELATED WORK 2.1 CNN based Fine-grained image recognition FGIR can be divided into the following three directions, localizationclassification sub-networks, end-to-end feature encoding, and external information, of which the first two directions are the main content of this section. The first method is classified as strongly [2,23,36] or weakly supervised [15] according to whether it utilizes bounding box/part annotations information. This class of methods locates key component regions by training supervised or weakly supervised localization sub-networks. Then, the classification subnetwork uses the fine-grained region information captured by the localization sub-network to further improve the classification capability. Mask-CNN [36] is based on part annotations and uses FCN to localize key parts (head, torso) to generate a mask with weighted object/part. However, the acquisition of part annotation can add additional and high markup costs. Many methods use attention mechanisms to set specific sub-network structures so that classification can be done using only image-level annotations. The second type of approach usually designs end-to-end models that encode discriminative features as higher-order information. From Bilinear-Pooling [24] to compact kernel pooling [13], many works use different methods such as designing kernel modules [5] or special loss functions [42] to reduce the dimensionality of higherorder features. However, these methods have difficulty in obtaining fine variance from the global feature view and hardly surpass the previous method. An approach very close to our work is RA-CNN [12], and the common denominator is the learning of regional features under the action of two scales. However, we have the following two key differences from RA-CNN. First, we don't need additional parameters to learn the coordinates of the regions, and we only need to rely on the attention weights attached to the transformer training for the region attention learning. Second, we do not need to force the accuracy of scale 2 to be higher than scale 1, and we are letting the two scales learn from each other and jointly improve the accuracy.
Transformer in Vision
Inspired by the Transformer [32] for NLP tasks [6,7,38], a large number of models have recently emerged that rely heavily on the Transformer for computer vision [9,46]. [3] and [46] are the earlier works to apply transformer to object detection. ViT [9] is the first work to transform a 2D image into an 1D patch tokens, which will be fed into the subsequent transformer layers for training, achieving an accuracy rate comparable to that of CNNs for image recognition. DeiT [31] enhances ViT [9] by introducing a simulation token and employs knowledge distillation to simulate the output of CNN teachers to obtain satisfactory results without training on large-scale datasets. SETR [44] proposes a pure self-attention-based encoder to perform semantic segmentation.
The most related work is TransFG [16] which also leverages attention for FGIR. However, there is a key difference. We take the use of attention weights for amplification and reuse of region attentions, while TransFG only filters the patch tokens in the last layer of the Transformer. Second, we propose a recurrent structure to extract and learn multi-scale features for better visual representation. Our model is superior in various image resolutions and large-scale FGIR datasets (see Section 4).
APPROACH
Our approach (RAMS-Trans) is built on top of the vision transformer (ViT [9]), so we first present a brief overview of ViT and then describe our approach for learning multi-scale features for FGIR.
Preliminaries: Vision Transformer
Image Tokenization. The innovation and key to ViT are that it processes a 2D image into a string-like 1D sequence and then feeds it into blocks stacked by the standard Transformer's encoder. Specifically, ViT reshapes the image , ∈ R × ×3 , with certain patch size, into a 2D sequence of patches , ∈ R ×( × ×3) , where H, W are the height and width of the raw image respectively, 3 is the number of channels of the raw RGB image, and P is the artificially set patch size used to split the image. In ViT, the size of is usually 16 or 32. is the total number of patches split into, = × / 2 . Then ViT maps the vectorized patches into a latent -dimensional embedding space using a trainable linear projection, to obtain the patch tokens ℎ , where ℎ ∈ R × . Similar to BERT, ViT also initializes the class tokens (CLS) for final classification in the tokenization phase, which will be concatenated with the patch tokens and then sent to the subsequent transformer layers. In addition, since the patch tokens input to the subsequent transformer are position-agnostic, and the image processing depends on the spatial information of each pixel, ViT adds the position embedding to each patch, which can be continuously learned in the subsequent training process:
0 = [ || ℎ ] +(1)
where ∈ R 1× and ∈ R (1+ )× are the CLS and the position embedding respectively. However, image tokenization in FGIR with a fixed patch size may have two problems: (1) The model processes all the patch tokens at once, and when the complexity of the dataset increases, e.g., with a cluttered background, the model may not be able to effectively capture the region attention carried in the patch tokens. (2) This kind of fixed patch size makes it easier for the model to lose local information when the image resolution is low.
Encoder Blocks. The main structure of ViT are Blocks, consists of a stack of Transformer's standard encoder. Each block consists of a multi-head self-attention (MSA) and a feed-forward network (FFN), which consists of two fully connected layers. The output of the ℎ layer can be expressed as:
= − 1 + ( ( − 1)) (2) = + ( ( ))(3)
where (·) indicates the Layer Normalization operation [1]. The uniqueness of CNN's image processing lies in the fact that as the depth of the model increases, the raw images are continuously downsampled, while the receptive field of the model keeps getting larger so that both global and local information of the images can be utilized. What makes ViT different from CNNs is with the increasing number of encoder blocks, the length of patch tokens does not change, and the receptive field of the model cannot be effectively extended, which may affect the accuracy of the model on the FGIR.
Proposed Network
Inspired by SCDA [35] and RA-CNN [12], we propose the recurrent attention multi-scale transformer (RAMS-Trans) to solve the above problems. We take two scales in Figure 2 as an example. First, the model accepts the raw input image 1 and then outputs cross-entropy loss1 and the multi-head self-attention weights of each transformer layer after the computation. Then is fed into DPPM, which firstly outputs the corresponding patches mask matrix on the raw image 1 , that is, the binary patch mask matrix, and then gets the coordinate value of the region attention on 1 by the maximum connected region search algorithm according to the matrix. In the second stage, we get the local image input 2 by bilinear interpolation algorithm to zoom in from 1 , which will be recomputed through L-layer encoder blocks to get cross-entropy loss2 and multi-head self-attention weights 2 .
It is important to emphasize that the core of our approach is how to use the characteristics of ViT itself to find the region attention in the raw image, to break the receptive field limitation of ViT due to the fixed size of patches, and then use the multi-scale image representation to achieve more effective recognition of objects. In CNNs, SCDA uses the fusion of multiple channel feature maps to achieve the localization of objects, from which we are inspired that since ViT processes the raw images into patch tokens for subsequent MSA and FFN calculations, can we thus obtain the importance of each patch corresponding to the raw image 1 ? Since one of the cores of the Transformer is MSA, then it is natural to think of using self-attention weights to try to accomplish this. We first perform a visual evaluation:
Visual Evaluation. Relying only on the attention weights incidental to the pre-training of ViT, it is possible to accurately locate the object region in the raw images and discard the irrelevant and noisy regions. In Figure 1., we show some images from three datasets CUB-200-2011, Stanford Dogs, and iNaturalist2017. We extract their attention weights using a ViT model pre-trained on ImageNet21k without fine-tuning the target dataset at all, and then visualize them employing CAM [45]. In Figure 1 we can see that using only the raw attention weights we can well localize the objects in the raw images and mask the background and noise regions. The above visualization process illustrates that the self-attention weights generated by ViT in the calculation of its MSA mechanism can be correlated to some extent with the positions of the target objects in the raw images.
Dynamic Patch Proposal Module. Our goal is to adaptively select a varying number of patches from 1 with 1/2 × 1/2 patches to recompose 2 . We first take out the attention weights of each transformer layer as:
= ( 1/2 ) = [ 0 , 1 , ..., ](4)
= [ 1 , 2 , ..., ] ∈ 1, 2, ...,
= [ 1 , 2 , ..., ] ∈ 1, 2, ...,(5)
where Q, K are Query and Key vectors respectively. Then we regularize the
= 1 =1 +(7)
where is the regularization factor and is the diagonal matrix.
= 1 ∑︁ =1 ( 1 ∑︁ =1 + )(8)
Then we propose to integrate attention weights of all previous layers and recursively apply a matrix multiplication to the modified attention weights in all the layers as:
= =1(9)
We calculate the mean value of all the positions in as the threshold to determine the localization position of the object. In particular, to improve the localization ability and further determine the region attention, we design a magnification factor as a hyperparameter to increase the threshold:
( , ) = 1 ( , ) > 0 ℎ(10)
where˜( , ) is the patch mask matrix and (x, y) is a particular position in these 1/2 × 1/2 positions. Finally, we employ Algorithm 1 to extract the largest connected component of˜to localize and zoom in the region attention in the raw image 1 .
Implementation
We present our implementation details on loss function as well as scale-wise class token. Loss Function In the training phase, our loss function is represented as a multi-task loss consisting of classification 1 and guided 2 :
= 1 + 2(11)
which are complementary to each other. is the coefficient to balance the weight between the two losses, which we take as 1.0 in the experiments. 1 represents the fine-grained classification loss of scale1 and 2 is the guided loss which is designed to guide the mode to select the more discriminative regions. These two losses work together in the backpropagation process to optimize the performance of the model. It enables the final convergent model to make classification predictions based on the overall structural characteristics of the object or the characteristics of region attention. During the testing phase, we removed the scale 2 to reduce a large number of calculations, so our approach will not take too long to predict in practical applications.
Scale-wise Class Token In Sec 3.1 we have described how the class token is generated and its role, which is mainly to exchange information with the patch tokens and finally to feed the class information to the classification layer. However, in our framework, the region attention of the raw image will be positioned and enlarged, and thus the patch tokens will be different between scales, which may affect the final classification performance if the class token is shared between scales. We, therefore, propose the scale-wise class token, i.e., different class-tokens are used to adapt patch tokens of different scales:
0 = [ 1 || 2 || ℎ ] +(12)
We demonstrate the effectiveness of this design in subsequent experiments with different resolutions.
EXPERIMENTS
In this section, we describe our experiments and discuss the results. We first present three datasets and then present our specific experimental details and results for each dataset respectively. Finally, we conduct detailed ablation experiments on our approach to investigate the impact of the components on FGIR in more depth. Note that all our results are reported as accuracies and are compared with the latest methods.
Algorithm 1 Finding Connected Components in Binary Patch Mask Marix
Require: A binary matrix: ; 1: Select a patch as the starting point; 2: while True do 3:
Use a flood-fill algorithm to label all the patches in the connected component containing ; 4: if All the patches are labeld then Search for the next unlabeled patch as ; 8: end while 9: return Connectivity of the connected components, and their corresponding size (patches numbers)
Datasets. A total of three benchmark datasets are presented in our experiments, namely CUB-200-2011 [33], Stanford Dogs [22] and iNaturalist2017 [18]. CUB-200-2011 is a fine-grained dataset on bird classification. In addition to labels, it also contains bounding box/part annotations which are useful for classification. Stanford Dogs contains images of 120 breeds of dogs from around the world. iNaturalist2017 is a large-scale FGIR dataset containing over 5,000 species of animals and plants.
Results on CUB-200-2011
Implementation Details. We load the model weights from the official ViT-B_16 model pre-trained on ImageNet21k. In all experiments, we used the SGD optimizer to optimize with an initial learning rate of 0.03 and a momentum of 0.9. We use weight decay 0. We use cosine annealing to adjust the learning rate with batch size 16. The model is trained for a total of 10,000 steps, of which the first 500 steps are warm-up. We resize the input images by scaling the shortest side to 600 and randomly crop a region of 448 × 448 for training. In the test, we use center crop to change the image size to 448 × 448. We split the image into patches as in the ViT, with the patch size is 16 × 16. The hyperparameter is chosen to be 1.3. We complete the construction of the whole model using Pytorch and run all experiments on the Tesla V-100 GPUs.
Comparison with state-of-the-art methods. The classification accuracies of CUB-200-2011 are summarized in Table 1. All previous FGIR methods test their performance on this dataset. As can be seen in Table 1, our approach outperforms all CNN-based methods and TransFG's PSM module and achieves state-of-the-art performance. Although ViT itself achieves good performance on this dataset, with the addition of our DPPM module, it achieves a further 0.7% improvement, which is rare for the Vit-B16 model which has been adequately pre-trained on a large-scale dataset.
Visualization. In order to visually analyze the selected regions from raw images for our DPPM, we present the amplified regions in the left part of Figure 3. The first, third, and fifth rows are the raw images, and the second, fourth and sixth rows are the visualization of the local images after the proposed patches have been amplified. From Figure 3 we can clearly see that DPPM has amplified the most discriminative regions of birds in each category, such as the head and the bill.
Results on iNaturalist2017
Implementation Details. To fully validate the effectiveness of our RAMS-Trans, we load the officially provided ViT-B16 model pre-trained on ImageNet21k. In all experiments, we use the SGD optimizer to optimize with an initial learning rate of 0.005 and a momentum of 0.9. We use weight decay 0. We use cosine annealing to adjust the learning rate with batch size 16. The model is trained for a total of 1e6 steps, of which the first 500 steps are warm-up. To align with TransFG, we resize the input images by scaling the shortest side to 448 and randomly crop a region of 304 × 304 for training. In the test, we use center crop to change the image size to 304 × 304. We still split the image into 16 × 16 patches with non-overlap and use a hyperparameter alpha of 1.2.
Comparison with state-of-the-art methods. Table 2 summarizes our results and compares them with the CNN-based state-ofthe-art methods and TransFG. Our approach outperforms ResNet152 by 9.5% and outperforms all CNN-based methods. With the pretrained model loaded, our method can achieve an improvement of 1.5% higher than the baseline. It is worth note that for a fair comparison, we report in Table 2 both the result obtained when we run the PSM module of TransFG in our code environment under the non-overlap setting. It can be seen that the DPPM module of our approach outperforms the PSM module of TransFG by 2.4% with the same loading of the ViT-B_16 pre-trained model.
Visualization. In order to visually analyze the selected regions from raw images for our DPPM, we present the amplified regions in the right part of Figure 3. The first, third, and fifth rows are the raw images, and the second, fourth, and sixth rows are the visualization of the local images after the proposed patches have been amplified. From Figure 3 we can clearly see that our RAMS-Trans successfully captures the most discriminative regions for an object, i.e., head, eyes for Amphibia; fins for Actinopterygii; thallus for Protozoa.
Results on Stanford Dogs
Implementation Details. We load the model weights from the official ViT-B_16 model pre-trained on ImageNet21k. In all experiments, we use the SGD optimizer to optimize with an initial learning rate of 0.003 and a momentum of 0.9, aligning with TransFG. We use weight decay 0. We use cosine annealing to adjust the learning rate with batch size 16. The model is trained for a total of 20,000 steps, of which the first 500 steps are warm-up. We resize the input images by scaling the shortest side to 448 and randomly crop a region of 224 × 224 for training. In the test, we use center crop to change the image size to 224 × 224. We split the image into patches as in the ViT, with the patch size is 16 × 16. The hyperparameter is chosen to be 1.0.
Comparison with state-of-the-art methods. The classification results of Stanford Dogs are summarized in Table 3. As can be seen, our approach outperforms all CNN-based methods and TransFG and achieves state-of-the-art results. It is important to note that for TransFG, the DPPM module of ours is aligned with its PSM module. Although they are both hard attention mechanisms, our approach is softer than its simple token filtering way because our approach extends the effective receptive field of the raw images, thus the classification performance is also better.
Method
Backbone Acc.(%) ResNet-50 [17] ResNet-50 84.5 RA-CNN [12] VGG-19 85.3 GP-256 [37] VGG-16 85.8 MaxExt [11] DenseNet-161 86.6 DFL-CNN [34] ResNet-50 87.4 NTS-Net [39] ResNet-50 87.5 Cross-X [26] ResNet-50 87.7 DCL [4] ResNet-50 87.8 CIN [14] ResNet-101 88.1 DBTNet [42] ResNet-101 88.1 ACNet [21] ResNet-50 88.1 S3N [8] ResNet-50 88.5 FDL [25] DenseNet-161 89.1 PMG [10] ResNet-50 89.6 API-Net [47] DenseNet-161 90.0 StackedLSTM [15] GoogleNet 90.4 MMAL-Net [40] ResNet-50 89.6 ViT [9] ViT-B_16 90.6 TransFG & PSM [16] ViT-B_16 90.9
RAMS-Trans
ViT-B_16 91.3 ViT-B_16 67.0 TransFG &PSM [16] ViT-B_16 66.6
RAMS-Trans
ViT-B_16 68.5 Table 3: Comparison of our RAMS-Trans with existing state of the arts methods on Stanford Dogs.
Method
Backbone Stanford Dogs MaxEnt [11] DenseNet-161 84.9 FDL [25] DenseNet-161 84.9 RA-CNN [12] VGG-19 873 SEF [27] ResNet-50 88.8 Cross-X [26] ResNet-50 88.9 API-Net [47] ResNet-101 90.3 ViT [9] ViT-B_16 92.2 TransFG & PSM [16] ViT-B_16 90.0
RAMS-Trans
ViT-B_16 92.4
Visualization. In order to visually analyze the selected regions from raw images for our DPPM, we present the amplified regions in the center part of Figure 3. The first, third and fifth rows are the raw images, and the second, fourth and sixth rows are the visualization of the local images after the proposed patches have been amplified. Figure 3 conveys that the proposed regions do contain more finegrained information of dogs in each category such as the ears, eyes, and fur.
Ablation Experiments
We conduct ablation experiments on our RAMS-Trans framework to analyze how its variants affect the FGIR results. All ablation studies are done on CUB-200-2011 dataset while the same phenomenon can be observed on other datasets as well.
Hyperparameters. Since DPPM in our approach requires the choice of hyperparameter , we experimentally investigate the effect of the threshold on the classification performance. We set 4 sets of alpha values between 1.1 and 1.4 with 0.1 intervals, and the results of all experiments are summarized in Table 5. It can be seen that as the value increases from 1.1 to 1.4, the recognition accuracy first increases and then decreases. The best performance is obtained when is 1.3. Based on this, we can get the following analysis: when the is small, the DPPM will crop as many patches in the raw images as possible, resulting in many non-critical regions being fed into the model again, while when the is large, the DPPM will crop as few patches in the raw images as possible, losing many critical regions to some extent. Both of these situations will lead to a decrease in classification accuracy, thus it is important to choose a suitable threshold value.
Compared to bounding box annotations. In order to further investigate the effectiveness of RAMS-Trans in selecting region attention, we replace our DPPM with the bounding box coordinates that come with the CUB-200-2011 dataset, keeping other settings unchanged, and conduct a comparison experiment. In Table 7, it can be seen that the experiments with a bounding box that utilized the supervised information are instead lower than the baseline. We also compare and analyze these two sets of comparative experiments by zooming in on the local images obtained from the raw images. From Table 7, we can analyze and obtain that using object annotations limits the performance since human annotations only give the coordinates of important parts rather than the accurate discriminative region location. Figure 4 visualizes the change of the feature maps before and after training. It can be seen that the activation region tends to be localized from the original global.
Influence of image resolution. Image resolution is a critical factor for image classification, and there are very many applications in real industrial scenarios. In order to investigate the performance improvement of RAMS-Trans for different image resolutions, we set five different image resolutions for comparison experiments, which are 224 × 224, 256 × 256, 288 × 288 and 320 × 320. Meanwhile, we conduct experiments on the PSM of TransFG with the same settings. As shown in Table 4, our method exceeds the baseline at each set of resolutions, while the PSM module loses performance.
Compared to random proposal method. In order to prove the effectiveness of our proposed DPPM module, we add the experiment of random sampling. Instead of the DPPM module, we use randomly generated coordinate points. As shown in Table 7, the DPPM can improve 0.4% over the random sampling method.
Influence of scale-wise class token. To verify the validity of the scale-wise class token, we add scale-shared class token experiments to experiments 4. As can be seen from Table 6, the experimental results with scale-wise class token are better than those scale-wise class token in most resolution cases.
Influence of patch size To analyze the effect of different patch sizes on FGIR and the effectiveness of our approach at different patch sizes, we load the official ViT-B_32 model provided for pretraining on ImageNet21k. The experimental results are summarized in Table 8. For a fair comparison, we report in Table 8 both the results obtained when we run the PSM module of TransFG in our code environment under the non-overlap setting. It can be seen that : An illustration of learning discriminative details by RAMS-Trans. The first row is the original images, the second raw shows the attention maps generated from raw attention weights, and the third raw shows the attention maps generated from trained attention weights.
when the patch size is 32, the performance of ViT as a baseline drops by 2.1% compared to when the patch size is 16. We can analyze that the larger the patch size is, the more sparsely the raw image is split, and the local information is not well utilized, so the performance of FGIR is degraded, yet our approach still exceeds the baseline by 0.7%.
CONCLUSION
In this paper, we propose a new recurrent attention multi-scale transformer (RAMS-Trans) architecture that combines the advantages of CNNs in expanding the effective receptive field, strengthening locality, and the advantages of Transformers in utilizing global information. Without bounding box/part annotations and additional parameters, RAMS-Trans uses the transformer's selfattention weights to measure the importance of the patch tokens corresponding to the raw images and recursively learns discriminative region attention in a multi-scale manner. Last but not least, our approach can be easily trained end-to-end and achieves state-of-theart in CUB-200-2011, Stanford Dogs, and the large-scale iNatural-ist2017 datasets. The future work is how to locate region attention more precisely to further improve the classification accuracy.
Figure 1 :
1Visualization results of Attention weights on CUB-200-2011, iNaturalist2017 and Stanford dogs datasets. The first, the third and the fifth rows are original images, while the second, the fourth and the sixth raws show the raw attention maps. Best viewed in color.
Figure 2 :
2The framework of recurrent attention multi-scale transformer (RAMS-Trans). The inputs are from global full-size images to local region attention (from left to right). The attention weights of all transformer layers are aggregated to generate the patch mask matrix, of which red 1 indicates an activated patch. The red box indicates the selected patch. Note that the linear projection, the transformer layers, and Fc (Fully Connection) layers are parameter-sharing, while CLS tokens do not.
Figure 3 :
3Attention localization at the local scale for CUB-200-2011, Stanford Dogs and iNaturalist2017. The regions (in second row of each category) learned from multiple image samples, represent consistent region attention for a specific fine-grained category, which are discriminative to classify this category from others.
Figure 4
4Figure 4: An illustration of learning discriminative details by RAMS-Trans. The first row is the original images, the second raw shows the attention maps generated from raw attention weights, and the third raw shows the attention maps generated from trained attention weights.
Table 1 :
1Comparison of our RAMS-Trans with existing state of the arts methods on CUB-200-2011.
Table 2 :
2Comparison of our RAMS-Trans with existing state of the arts methods on iNaturalist2017.Method
Backbone
iNaturalist2017
Resnet152 [17]
ResNet152
59.0
SSN [28]
ResNet101
65.2
Huang et al. [19]
ResNet101
66.8
IncResNetV2 [30]
InResNetV2
67.3
TASN [43]
ResNet101
68.2
ViT [9]
Table 4 :
4Ablation experiments on DPPM with different input resolutions.Resolution
192
224
256
288
320
ViT
81.5 85.3 87.4 88.3 88.9
TransFG & PSM [16] 81.6 84.9 86.8 88.0 89.1
RAMS-Trans
82.2 86.3 88.1 88.9 89.7
Table 5 :
5Impact of Threshold on CUB-200-2011 dataset.Approach
Value of
Accuracy(%)
RAMS-Trans
1.1
90.9
RAMS-Trans
1.2
91.2
RAMS-Trans
1.3
91.3
RAMS-Trans
1.4
91.0
Table 6 :
6Ablation experiments on CLS token with different input resolutionsResolution
192
224
256
288
320
448
scale-sharing 82.2 85.9 87.8 88.5 89.8 91.2
scale-wise
82.2 86.3 88.1 88.9 89.7 91.3
Table 7 :
7Ablation experiments on patch proposal way.Method
Patch Proposal Accuracy(%)
ViT [9]
Bounding Box
89.2
RAMS-Trans Bounding Box
90.5
RAMS-Trans
Random
90.9
RAMS-Trans
DPPM
91.3
Table 8 :
8Ablation experiments on patch size.Method
Patch Size Accuracy(%)
ViT [9]
16 × 16
90.6
TransFG & PSM [16]
16 × 16
90.9
RAMS-Trans
16 × 16
91.3
ViT
32 × 32
88.4
TransFG & PSM [16]
32 × 32
88.9
RAMS-Trans
32 × 32
89.1
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| [] |
[
"FILTER CLASSES OF UPSETS OF DISTRIBUTIVE LATTICES",
"FILTER CLASSES OF UPSETS OF DISTRIBUTIVE LATTICES"
] | [
"Adam Přenosil \nUniversità degli Studi di Cagliari\nCagliariItaly\n"
] | [
"Università degli Studi di Cagliari\nCagliariItaly"
] | [] | Let us say that a class of upward closed sets (upsets) of distributive lattices is a finitary filter class if it is closed under homomorphic preimages, intersections, and directed unions. We show that the only finitary filter classes of upsets of distributive lattices are formed by what we call n-filters. These are related to the finite Boolean lattice with n atoms in the same way that filters are related to the two-element Boolean lattice: n-filters are precisely the intersections of prime n-filters and prime n-filters are precisely the homomorphic preimages of the prime n-filter of non-zero elements of the finite Boolean lattice with n atoms. Moreover, n-filters on Boolean algebras are the only finitary filter classes of upsets of Boolean algebras generated by prime upsets.ifEach m-filter is an n-filter if m < n, but the n-filter P n is not an m-filter for any m < n. Clearly 1-filters in this sense are just ordinary lattice filters. An n-filter is said to be prime if a ∨ b ∈ F implies that a ∈ F or b ∈ F . | null | [
"https://export.arxiv.org/pdf/2111.09806v2.pdf"
] | 257,804,490 | 2111.09806 | 78d99db0545315227c5f37133dc18ca2634221ed |
FILTER CLASSES OF UPSETS OF DISTRIBUTIVE LATTICES
29 Mar 2023
Adam Přenosil
Università degli Studi di Cagliari
CagliariItaly
FILTER CLASSES OF UPSETS OF DISTRIBUTIVE LATTICES
29 Mar 2023arXiv:2111.09806v2 [math.LO]
Let us say that a class of upward closed sets (upsets) of distributive lattices is a finitary filter class if it is closed under homomorphic preimages, intersections, and directed unions. We show that the only finitary filter classes of upsets of distributive lattices are formed by what we call n-filters. These are related to the finite Boolean lattice with n atoms in the same way that filters are related to the two-element Boolean lattice: n-filters are precisely the intersections of prime n-filters and prime n-filters are precisely the homomorphic preimages of the prime n-filter of non-zero elements of the finite Boolean lattice with n atoms. Moreover, n-filters on Boolean algebras are the only finitary filter classes of upsets of Boolean algebras generated by prime upsets.ifEach m-filter is an n-filter if m < n, but the n-filter P n is not an m-filter for any m < n. Clearly 1-filters in this sense are just ordinary lattice filters. An n-filter is said to be prime if a ∨ b ∈ F implies that a ∈ F or b ∈ F .
Introduction
The structure theory of distributive lattices rests on two facts: the prime filter separation property (which states that each filter is an intersection of prime filters) and the correspondence between prime filters and homomorphisms into the two-element distributive lattice B 1 consisting of the elements 0 < 1 (which states that each prime filter is a homomorphic preimage of the prime filter {1} on B 1 ). Together, these facts show that distributive lattices are subdirect powers of B 1 .
Are there other kinds of upward closed subsets (upsets) of distributive lattices besides filters which admit an analogous representation in terms of intersections of homomorphic preimages of a certain canonical upset? We show that if the role of the canonical prime filter {1} on B 1 is taken over by the upset P n of non-zero (x > 0) elements of the finite Boolean lattice with n atoms B n := (B 1 ) n , we obtain precisely what we call n-filters. An n-filter is an upset F of a meet semilattice S which enjoys the following property for each non-empty finite set X ⊆ F :
The relationship between prime n-filters on distributive lattices and the canonical prime n-filter P n on B n extends the relationship between prime filters and the canonical prime filter {1} on B 1 : prime n-filters are precisely the homomorphic preimages of P n . This relationship follows from the observation that prime n-filters on distributive lattices are precisely the unions of at most n ordinary prime filters (which is far from true for n-filters in general).
The relationship between n-filters and prime n-filters on distributive lattices then extends the relationship between filters and prime filters:
n-filters are precisely the intersections of prime n-filters.
The key part of the proof is describing the n-filter generated by an upset of a distributive lattice. Observe that if we apply this description of n-filters to ideal lattices of distributive lattices, we obtain the filter-ideal separation property for distributive lattices:
n-filters can be separated from ideals by prime n-filters.
In other words, the fundamental properties of (prime) filters on distributive lattices have direct analogues for (prime) n-filters.
Are there other kinds of upsets of distributive lattices besides n-filters which share the basic properties of lattice filters? We encapsulate these properties in the definition of a finitary filter class: a class of upsets closed under homomorphic preimages, intersections, and directed unions. The only finitary filter classes of upsets of distributive lattices are indeed the families of n-filters for some n. This is not true for Boolean algebras, where many other finitary filter classes exist. Nonetheless, n-filters are the only finitary filter classes of upsets of Boolean algebras generated by some family of prime upsets. There are also other finitary filter classes of upsets of unital meet semilattices. However, n-filters are the only finitary filter classes of non-empty upsets of unital meet semilattices which are closed under socalled strict homomorphic images. This is not true for general meet semilattices.
To the best of our knowledge, the problem of describing which families of upsets of distributive lattices (and other kinds of ordered algebras) behave like the family of all lattice filters has not been considered in the literature. Equivalently, as explained in more detail in Section 4, this problem asks for a full classification of upsets according to which first-order properties of a particular type they satisfy (namely properties expressible by what we call filter implications and logical implications in Theorem 4.4). In the case of distributive lattices and unital meet semilattices, we provide a full classification of their upsets according to which logical implications they satisfy (Theorems 5.8 and 6.4). The analogous problem for Boolean algebras is substantially more complicated. Indeed, the major questions left open by the present paper are whether there are tractable classifications of upsets of Boolean algebras and meet semilattices. (We conjecture that the answer is negative for Boolean algebras and positive for meet semilattices.)
The conceptual machinery used in the later sections (from Section 4 onwards) to solve these problems will largely be familiar to practitioners of so-called abstract algebraic logic [4], but perhaps not to lattice theorists. This might require the reader to first get to grips with some possibly unfamiliar jargon and notation. We have tried to keep this to the bare minimum required to prove the desired results.
The concept of an n-filter may remind the reader of the so-called n-Helly property for families of sets. A family of sets enjoys this property if for each non-empty finite subfamily S i with i ∈ I i∈I S i = ∅ =⇒ j∈J S j = ∅ for some J ⊆ I with 1 ≤ |J| ≤ n.
This property is named after Helly's theorem in convex geometry, which states that the family of all convex subsets of R n enjoys the (n + 1)-Helly property [6]. In the terminology of the present paper we would say that the non-empty convex subsets of R n form an (n + 1)-filter in the lattice of all convex subsets of R n . The characteristic equational property of this lattice is the n-distributive property introduced by Huhn [7]:
x ∧ (y 1 ∨ · · · ∨ y n+1 ) = (x ∧ z 1 ) ∨ · · · ∨ (x ∧ z n+1 ), where z i := {y j | 1 ≤ j ≤ n + 1 and j = i}.
While n-distributivity will not make any further appearance in the present paper, it is worth observing that this property is intimately related to n-filters: a lattice is n-distributive if and only if its filters can be separated from its ideals by n-prime filters (i.e. the complements of prime n-ideals). This separation property is, in a way, dual to the one considered in this paper, which states that n-filters can be separated from ideals by prime n-filters.
Basic properties of n-filters
We first introduce n-filters, prime n-filters, and m-prime filters in more detail. These may be defined on any meet semilattice (in fact on any poset), but restricting to distributive meet semilattices gives us a better grip on the generation of n-filters. 1 2.1. Defining n-filters. It will be convenient to introduce the notation Y ⊆ n X to abbreviate the claim that Y is a non-empty subset of X of cardinality at most n. The set {1, . . . , n} will be denoted [n]. In other words,
Y ⊆ n X ⇐⇒ Y ⊆ X and |Y | ∈ [n].
Throughout the paper, S will denote a meet semilattice and L will denote a lattice. By a semilattice we shall mean a meet semilattice by default. The top (bottom) element of a meet semilattice, if it exists, will be denoted by 1 (0), but in general we do not require that these elements exist. An upset F of S is called total if F = S. An upset which is not total will be called proper.
An n-filter on a semilattice S for n ≥ 1 is an upset F ⊆ S such that for each non-empty finite X ⊆ S
Y ∈ F for each Y ⊆ n X =⇒ X ∈ F.
We further define a 0-filter to be an upset which is either empty or total. An n-ideal on a join semilattice is an n-filter on the order dual of the join semilattice. A 1-filter (1-ideal) will simply be called a filter (ideal ).
We may in fact restrict to |X| = n + 1 and |Y | = n in the definition of n-filters. For example, the 2-filters on S are precisely the upsets F ⊆ S such that
x ∧ y, y ∧ z, z ∧ x ∈ F =⇒ x ∧ y ∧ z ∈ F.x j ∈ F for each i ∈ [n + 1] =⇒ x 1 ∧ · · · ∧ x n+1 ∈ F.
Proof. The left-to-right implication is trivial. Conversely, suppose that the above implication holds for F . We prove by induction over the cardinality of the set X ⊆ S that if Y ∈ F for each Y ⊆ n X, then X ∈ F . If |X| ≤ n, we may take Y := X. If |X| = n + 1, the claim holds by assumption. Now consider X = {x 1 , . . . , x k+1 } with |X| = k + 1 > n + 1 and suppose that the claim holds for all Y ⊆ k X. If X / ∈ F , then by the inductive hypothesis applied to {x 1 , . . . ,
x k−1 , x k ∧ x k+1 } there is either some Y ⊆ n {x 1 , . . . , x k−1 } such that Y / ∈ F or there is some Y ⊆ n−1 {x 1 , . . . , x k−1 } such that Y ∧ x k ∧ x k+1 / ∈ F .
This is a meet of n + 1 elements and n+1 ≤ k, therefore the inductive hypothesis applies to Y ∪{x k , x k+1 }. In either case we obtain some Z ⊆ n X such that Z / ∈ F .
The family of all n-filters on a semilattice S ordered by inclusion forms a complete lattice Fi n S where meets are intersections of n-filters and directed joins are directed unions of n-filters. We can thus define the n-filter generated by X ⊆ S as the smallest n-filter on S which contains X. An n-filter generated by a finite set will be called finitely generated. Because each n-filter is a directed union of its finitely generated n-subfilters, Fi n S is an algebraic lattice and its compact elements are precisely the finitely generated n-filters.
An element x of a meet semilattice S will be called (meet) m-prime, where 1 ≤ m, if for each non-empty finite family Y ⊆ S
Y ≤ x =⇒ Z ≤ x for some Z ⊆ m Y.
We may again restrict to |I| = m + 1 and |J| = m. We now define an m-prime n-filter on a semilattice S as a meet m-prime element of the lattice Fi n S of n-filters on S. An m-prime n-ideal on a join semilattice is then an m-prime n-filter on its order dual. A 1-prime n-filter will simply be called a prime n-filter. This agrees with the existing definition of prime filters on semilattices. Note that this is a second-order definition, in the sense that it quantifies over subsets of S. However, if either n = 1 or m = 1, it may be reduced to a first-order definition, which quantifies only over elements of S. 2
Fact 2.2. An n-filter on a lattice is prime if and only if its complement is an ideal.
A filter on a lattice is m-prime if and only if its complement is an m-ideal. 2 One can more generally define an n-filter on a poset as an upset F such that for each nonempty finite X ⊆ F the set X has a lower bound in F whenever each Y ⊆n X does. Although the family of all n-filters on a poset need not be closed under intersections, we can still call an n-filter F on a poset m-prime if for each non-empty finite family F i of n-filters for i ∈ I i∈I
F i ⊆ F =⇒ j∈J F j ⊆ F for some J ⊆m I.
With this definition in hand, Fact 2.2 in fact holds for arbitrary posets.
Proof. Let F be a prime n-filter with x, y / ∈ F , and let G and H be the principal filters generated by x and y. Then G F and H F , therefore G ∩ H F as witnessed by some z ∈ G ∩ H such that z / ∈ F . In other words, x, y ≤ z / ∈ F , so x ∨ y / ∈ F and the complement of F is an ideal. Conversely, suppose that the complement of F is an ideal and G and H are n-filters such that G F and H F as witnessed by x ∈ G \ F and y ∈ H \ F . Because the complement of F is an ideal,
x ∨ y / ∈ F . But x ∨ y ∈ G ∩ H, so G ∩ H F .
To prove the left-to-right direction of the second claim, let F be an m-prime filter and x i / ∈ F for i ∈ I be a finite family such that x J := {x j | j ∈ J} / ∈ F for each J ⊆ n I. Let G i be the principal filter generated by x i for i ∈ I. Then x J witnesses that j∈J G j F for each J ⊆ n I. Because F is an m-prime filter, it follows that i∈I G i F , so there is some z / ∈ F such that x i ≤ z for each i ∈ I.
Thus {x i | i ∈ I} / ∈ F and the complement of F is an m-ideal. Conversely, consider filters G i for i ∈ [m+1] with j =i G j F for i ∈ [m+1], as witnessed by the elements x i ∈ j =i G j \ F . Because G i is a filter, y i := {x j | j = i} ∈ G i . Then y j ≤ x i / ∈ F for each j = i. Because the complement of F is an m- ideal, z := {y i | i ∈ [m + 1]} / ∈ F . But z ∈ i∈[m+1] G i , so i∈[m+1] G i F .
The following equivalences thus provide a working definition of prime n-filters and n-prime filters on lattices.
Fact 2.3. An n-filter F on a lattice is prime if and only if
x ∨ y ∈ F =⇒ x ∈ F or y ∈ F.
A filter F on a lattice is m-prime if and only if
x 1 ∨ · · · ∨ x m+1 ∈ F =⇒ j =i x j ∈ F for some i ∈ [m + 1].
While the complement of an m-prime n-filter is always an m-ideal, we shall see that the converse implication fails (even in Boolean algebras): a 2-filter whose complement is a 2-ideal need not be a 2-prime 2-filter.
The following observation shows how ubiquitous (m-prime) n-filters are. Proof. If an upset F ⊆ S is not an n-filter, then this must be witnessed by at least n + 1 distinct elements of S. Moreover, the failure of an n-filter to be m-prime must be witnessed by at least m + 1 distinct subsets, and similarly the failure of an element of a semilattice S to be m-prime must be witnessed by at least m + 1 distinct elements of S.
Let us stress that we count the empty filter as well as the total filter as m-prime n-filters. This is because we wish to identify prime n-filters on distributive lattices (and m-prime n-filters on finite distributive lattices) as the homomorphic preimages of a certain canonical prime (m-prime) n-filter.
2.2.
Constructing n-filters. Homomorphisms, subalgebras, and products interact with (prime) n-filters in the expected ways. Fact 2.7. Let S be a subsemilattice of T and F be an n-filter on T. Then the restriction of F to S is an n-filter on S. Fact 2.8. Let L be a sublattice of M and F be a prime n-filter (m-prime filter) on M. Then the restriction of F to L is a prime n-filter (m-prime filter) on L. Fact 2.9. Let S i for i ∈ I be a family of semilattices and F i be an n-filter on S i for each i ∈ I. Then F := i∈I F i is an n-filter on S := i∈I S i .
To show that restricting an m-prime n-filter to a subsemilattice yields an mprime n-filter, we shall need to make a certain assumption about how the smaller semilattice sits inside the full semilattice.
We say that a subposet P of a poset Q is an ideal subposet of Q if the restriction to P of each ideal on Q is an ideal on P . Equivalently, P is an ideal subposet of Q if for each x, y ∈ P and each u ∈ Q such that x, y ≤ u in Q there is some z ∈ P such that x, y ≤ z ≤ u. In particular, each subsemilattice of a join semilattice is an ideal subposet. A subsemilattice of a meet semilattice which is an ideal subposet will be called an ideal subsemilattice. For example, S is isomorphic to an ideal subsemilattice of the lattice Fi S of all filters on S with respect to the embedding assigning to each a ∈ A the principal filter generated by a. Proof. We already know that the restriction of F to S is an n-filter. Let us call it G. To prove that G is an m-prime n-filter, suppose that i∈I G i ⊆ G for some non-empty finite family of n-filters G i with i ∈ I. Let F i be the upset generated by G i in T. Then each F i is an n-filter. Now if a ∈ i∈I F i , then for each i ∈ I there is some g i ∈ G i such that g i ≤ a. Because S is an ideal subposet of T, there is some g ∈ S such that g i ≤ g for each i ∈ I and g ≤ a. Thus a ≥ g ∈ i∈I G i ⊆ G ⊆ F . We have therefore proved that i∈I F i ⊆ F . Because F is m-prime, there is some J ⊆ m I such that j∈J F j ⊆ F . Restricting both sides to S now yields j∈J G j ⊆ G, therefore G is an m-prime n-filter. Conversely, we may wish to extend an n-filter on a subsemilattice (or more generally, a subposet which is a semilattice) to the full semilattice.
Fact 2.11. Let F be an n-filter on a semilattice S and let S be an ideal subposet of a semilattice T. Then the upset generated by F in T is an n-filter on T.
Proof. Let G be the upset generated by F in T. To show that G is an n-filter, suppose that there are x 1 , . . . , x n+1 ∈ T with j =i x j ∈ G for each i ∈ [n + 1]. That is, there are y 1 , . . . , y n+1 ∈ F such that y i ≤ j =i x i . We need to find z ∈ F such that z ≤ x 1 ∧· · ·∧x n+1 . Because S is an ideal subposet of T, for each i ∈ [n+1] there is some z i such that y j ≤ z i ≤ x i for each j = i. Then j =i z j ≥ y i ∈ F for each i ∈ [n + 1], therefore z := i∈[n+1] z i ∈ F , where the meets take place in S. Moreover, z ≤ z i ≤ x i for each i ∈ [n + 1].
Fact 2.12. Let F be an n-filter on a sublattice L of a lattice M. Then the upset generated by F in M is an n-filter on M.
The simplest way to construct n-filters is to take unions of filters.
Fact 2.13. Let F i for i ∈ I be a finite family of upsets of S such that F i is an n i -filter for each i ∈ I. Then F := i∈I F i is an n-filter for n := i∈I n i .
Proof. We may assume that I is non-empty and 0 < n i for each i ∈ I (in particular, n i < n for each n i ): if n i = 0, then either F i = S, in which case F = S is a 0-filter, or F i = ∅, in which case we may remove F i from the family.
Now consider x 1 , . . . , x n+1 ∈ S such that y i := j =i x j ∈ F for i ∈ [n + 1]. Because n = i∈I n i , there is some k ∈ [n + 1] such that at least n k + 1 ≥ 2 of the meets y i belong to F k , say y j and y k for j = k. But then the meet of each subset of {x 1 , . . . , x n+1 } of cardinality at most n k < n is a submeet of y j or y k , therefore it belongs to F k . Because F k is an n k -filter,
x 1 ∧ · · · ∧ x n+1 ∈ F k ⊆ F .
Fact 2.14. The union of a family of n filters is an n-filter.
We shall see that the converse is true for prime n-filters: each of them decomposes into a union of n filters. However, there is no bound on how many filters are required to obtain a general n-filter: the upset generated by the κ coatoms of the Boolean lattice B κ := (B 1 ) κ is a 2-filter which is not a union of fewer than κ filters.
2.3.
Generating n-filters. We now wish to explicitly describe the n-filter on S generated by some U ⊆ S. We denote this n-filter [U ] n . Because [U ] n always extends the upset generated by U , it suffices to restrict our discussion to cases where U is an upset.
The 1-filter generated by an upset U ⊆ S admits a simple description:
a ∈ [U ] 1 ⇐⇒ there is some non-empty finite X ⊆ U such that X ≤ a.
To extend this description to n-filters, we impose distributivity on S. Recall that a meet semilattice is distributive if
x ∧ y ≤ z =⇒ x ′ ∧ y ′ = z for some x ′ ≥ x and y ′ ≥ y. We say that Y ⊆ S refines X ⊆ S if for all y ∈ Y there is x ∈ X such that x ≤ y.
Lemma 2.15 (Common refinement). Let S be a distributive meet semilattice and let Y i ⊆ S for i ∈ [n] be a family of non-empty finite sets.
If Y i ≤ x for each Y i , then there is some Y ⊆ S such that Y = x and moreover Y refines each Y i .
Proof. This holds by a straightforward induction over n.
Lemma 2.16 (Generating n-filters). Let S be a distributive semilattice and U be an upset of S. Then
a ∈ [U ] n ⇐⇒ there is some non-empty finite X ⊆ U such that Y ∈ U for each Y ⊆ n X and X ≤ a.
We may equivalently replace X ≤ a by X = a in the above.
Proof. Let F ⊆ S be the upset defined by this condition, and let us call a non-empty finite set X ⊆ S admissible if Y ∈ U for each Y ⊆ n X. Because F is distributive and U is an upset, F consists precisely of the meets of admissible subsets of S. Clearly F ⊆ [U ] n . Conversely, we prove that F ⊇ U is an n-filter. Consider x 1 , . . . , x n+1 ∈ F such that y i := j =i x j ∈ F for each i ∈ [n + 1]. We need to prove that i x i ∈ F . That is, we need to express i x i as the meet of an
i ∈ [n + 1] there is an admissible set Y i with Y i = y i . In particular, Y i ≤ x j for each j = i.
Applying the previous lemma to the family Y j for j = i and to x :
= x i yields a set Z i which refines each Y j with j = i such that Z i = x i . It follows that {Z i | i ∈ [n + 1]} = i∈[n+1] x i . It thus suffices to show that {Z i | i ∈ [n + 1]} is an admissible set. If Z ⊆ n {Z i | i ∈ [n + 1]}, then for each z ∈ Z there is i ∈ [n + 1] such that z ∈ Z i . Thus for each j = i there is an element of Y j which lies below z.
Crucially, since Z has at most n elements but j varies over [n + 1], we may choose the same Y j for each z ∈ Z. That is, there is some Y j such that for each z ∈ Z there is y ∈ Y j with y ≤ z. But then Z ∈ U follows from the fact that the meet of each non-empty subset of Y j with at most n elements lies in U .
The above lemma does not hold in all semilattices. Figure 1 shows why. Let U be the upset consisting of the solid dots. Then there is no X ⊆ U such that Y ∈ U for each Y ⊆ 2 X and X ≤ b. There is, however, some such X with X = a, therefore a ∈ [U ] 2 . This fills the empty dot above b, i.e. shows that a ∨ b ∈ [U ] 2 . It immediately follows that b ∈ [U ] 2 . Appending a top element to this semilattice shows that the lemma may even fail in a finite lattice.
Informally speaking, the n-filter generated by X on a general semilattice S is formed by applying the construction of the above lemma ω times.
Recall that in general the n-filters of a meet semilattice S form an algebraic lattice Fi n S. If S is distributive, we can in fact say more about Fi n S.
Fact 2.17. If S is a distributive semilattice, then Fi n S is a distributive lattice. Proof. Let F , G, H be n-filters on S and let G ⊔ H denote the n-filter generated by G ∪ H. Suppose that x ∈ F ∩ (G ⊔ H). Then there is Y ⊆ S such that Y = x and Z ∈ G ∪ H for each Z ⊆ n Y . Since x ∈ F , then in fact Z ∈ F ∩ (G ∪ H) = (F ∩ G) ∪ (F ∩ H), hence Y witnesses that x ∈ (F ∩ G) ⊔ (F ∩ H).
n-filters as intersections of homomorphic preimages
We now show that each n-filter is an intersection of homomorphic preimages of the n-filter P n ⊆ B n .
3.1. Prime n-filters as homomorphic preimages. We first exhibit each prime n-filter as a homomorphic preimage of P n ⊆ B n . In order to better understand how the n-filter P n arises in this context, we recall the notion of the dual product of a family of structures. Here by a structure we simply mean a pair A, F consisting of an algebra A and a set F ⊆ A. Given a family of structures A i , F i with i ∈ I in some common algebraic signature, their (direct) product is defined as
i∈I A i , F i := i∈I A i , i∈I π −1 i [F i ] ,
where π i are the projections. The dual (direct) product of the family A i , F i , studied by Badia and Marcos [1], is less well known. It is defined as the structure
i∈I A i , F i := i∈I A i , i∈I π −1 i [F i ] .
The two constructions are related by De Morgan duality:
A, F ⊗ B, G = A, F × B, G , where A, F := A, A \ F .
The construction A, F → A, A \ F in our case corresponds to switching from an upset (prime n-filter) to a downset (n-prime ideal) and vice versa.
In order to describe prime n-filters and n-prime filters on distributive lattices, we will need to construct homomorphisms into direct products and dual direct products of structures. The following simple observation will be crucial.
Let us recall that a strict homomorphism of structures h :
A, F → B, G is a homomorphism of algebras h : A → B such that F = h −1 [G]. Fact 3.1. Let h i : A, F i → B i , G i
be a family of strict homomorphisms for i ∈ I and let h : A → i∈I B i be the product of these homomorphisms. Then the following are strict homomorphisms:
h : A, i∈I F i → i∈I B i , G i , h : A, i∈I F i → i∈I B i , G i .
If we now apply these two constructions to the structure B 1 := B 1 , {1} , where B 1 is the two-element distributive lattice, we obtain the structures
B n , {1} = i∈[n] B 1 , {1} , B n , P n = i∈[n] B 1 , {1} .
Recall that B n := B n 1 and P n denotes the upset of non-zero elements of B n , with B 0 being the trivial lattice and P 0 = ∅. We shall use the notation
(B 1 ) n := B n , {1} , B n := B n , P n .
Occasionally, we also use the notation B Πn Proof. It is a prime n-filter because it is a union of the n prime filters generated by the atoms of B n . It is not an m-filter for m < n because the meet of each set of m coatoms lies in P 1 but the meet of all coatoms does not.
We say that an upset F on a lattice L is a homomorphic preimage of an upset G on a lattice M if there is a lattice homomorphism h :
L → M such that F = h −1 [G].
An upset F is n-generated if it is the upward closure of a set of cardinality at most n. The following lemma holds equally well for semilattices. (i) for all a 1 , . . . , a n+1 ∈ U there are i = j such that a i ∧ a j ∈ U , (ii) U is a union of at most n filters on L, (iii) U is a homomorphic preimage of an n-generated upset.
Proof. (i) ⇒ (ii): let n be the least number which satisfies (i). If n = 1, then U is a filter, so (ii) holds.
Otherwise, there are b 1 , . . . , b n ∈ U such that b i ∧ b j / ∈ U whenever i = j.
We can now extend the principal filters generated by b 1 , . . . , b n to maximal filters
F 1 , . . . , F n ⊆ U . That is, if x / ∈ F i , then x ∧ f i / ∈ U for some f i ∈ F i . It suffices to show that U = F 1 ∪ · · · ∪ F n . Suppose therefore that x / ∈ F 1 ∪ · · · ∪ F n . Then there are c i ∈ F i such that x ∧ c i / ∈ U .
We may assume without loss of generality that
c i ≤ b i , taking b i ∧ c i instead of c i if necessary. But then c i ∧ c j / ∈ F whenever i = j. The set {x, c 1 , . . . , c n } therefore contradicts the assumption (i). (ii) ⇒ (iii) ⇒ (iv): let U = F 1 ∪ · · · ∪ F k , where F i1 , . . . , a n+1 } there is some b k such that b k ≤ a i and b k ≤ a j for some i = j.
Lemma 3.4 (Homomorphism lemma for P 1 ). Each prime filter on a distributive lattice L is a homomorphic preimage of the prime filter P 1 ⊆ B 1 .
The homomorphism lemma for P 1 immediately extends to P n . Each family of prime filters F i on L for i ∈ I corresponds to a family of strict homomorphisms h i : L, F i → B 1 , {1} , hence the direct product of this family of homomorphisms h : L → i∈I B 1 yields a strict homomorphism h : L, i∈I F i → B n , P n by Fact 3.1.
Lemma 3.5 (Homomorphism lemma for P n ). Let F 1 , . . . , F n be prime filters on a distributive lattice L. Then F 1 ∪ · · · ∪ F n is a homomorphic preimage of the prime n-filter P n ⊆ B n . Theorem 3.6 (Prime n-filters on distributive lattices). The following conditions are equivalent for each upset F of a distributive lattice:
(i) F is a prime n-filter, (ii) F is a prime upset which is a union of at most n filters, (iii) F is a union of at most n prime filters, (iv) F is a homomorphic preimage of the prime n-filter P n ⊆ B n .
Proof. (i) ⇒ (ii): suppose that {a 1 , . . . , a n+1 } ⊆ F . By Lemma 3.3 it suffices to show that a i ∧ a j ∈ F for some i = j. Let b i := j =i a j . Then for each Y ⊆ n {b 1 , . . . , b n+1 } we have a j ≤ Y for some a j . But a j ∈ F and F is an n-filter, thus b 1 ∧ · · · ∧ b n+1 ∈ F . Moreover, b 1 ∧ · · · ∧ b n+1 = j k =j (a j ∧ a k ), therefore a j ∧ a k ∈ F for some j = k because F is prime. (ii) ⇒ (iii): suppose that F is prime and F = F 1 ∪ · · · ∪ F k for k ≤ n,
where F i are filters. We may assume that this union is irredundant, i.e. that for each i there is some
c i ∈ F i such that c i / ∈ F j for j = i. Let a ∨ b ∈ F i . Then (a ∧ c i ) ∨ (b ∧ c i ) = (a ∨ b) ∧ c i ∈ F i , so either a ∧ c i ∈ F or b ∧ c i ∈ F . But a ∧ c i / ∈ F j for j = i, thus a ∧ c i ∈ F i or b ∧ c i ∈ F i . Therefore a ∈ F i or b ∈ F i .
(iii) ⇒ (iv) ⇒ (ii) ⇒ (i): Lemma 3.5 and Facts 2.5 and 2.14.
An analogous fact holds for distributive semilattices.
Theorem 3.7 (Prime n-filters on distributive semilattices). The following conditions are equivalent for each upset F of a distributive meet semilattice:
(i) F is a prime n-filter, (ii) F is a prime upset which is a union of at most n filters, (iii) F is a prime homomorphic preimage of the n-filter P n ⊆ B n .
Proof. The equivalence (ii) ⇔ (iii) and the implication (ii) ⇒ (i) are entirely analogous to the lattice case, taking into account that semilattice filters are precisely the homomorphic preimages of {1} ⊆ B 1 . It remains to show that each prime n-filter F on a semilattice S is a union of at most n filters. To this end, let L be the order dual of the lattice Fi S of filters on S. Since S is distributive, we know that Fi S, hence also L, is a distributive lattice. The map ι : S → L which assigns to each a ∈ S the principal upset generated by a then embeds S into L (as a meet semilattice).
Let F := {G ∈ Fi S | G ⊆ F }.
It is straightforward to show that F is a prime n-filter on L such that ι(a) ∈ F if and only if a ∈ F . Applying the characterization of prime n-filters on distributive lattices, F is a union of n filters on L, therefore F = ι −1 [F ] is a union of n filters on S.
Distributivity is essential in the above theorems. The upset P (M 5 ) of non-zero elements is a prime 2-filter on the five-element diamond M 5 but it is not a union of at most 2 filters, and accordingly it is not a homomorphic preimage of P 2 ⊆ B 2 . On the other hand, the upset generated by the coatoms of the five-element pentagon N 5 is a prime union of 2 filters, but it is not a union of prime filters, and accordingly it is not a homomorphic preimage of P 2 ⊆ B 2 .
Dualizing the above characterization of prime n-filters on distributive lattices yields an analogous characterization of prime n-ideals. But prime n-ideals are precisely the complements of n-prime filters, therefore the theorem can equivalently be phrased in terms of n-prime filters. (i) F is an n-prime filter, (ii) F is a filter and F is an intersection of at most n prime upsets, (iii) F is an intersection of at most n prime filters, (iv) F is a homomorphic preimage of the n-prime filter {1} ⊆ B n .
3.2.
Prime n-filter separation. We now exhibit each n-filter as an intersection of prime n-filters. Theorem 3.9 (Prime n-filter separation). Let F be an n-filter on a distributive lattice which is disjoint from an ideal I. Then F extends to a prime n-filter which is disjoint from I.
Proof. If I = ∅, the claim is true. Otherwise, let θ I be the congruence which collapses all elements of I. That is, a, b ∈ θ I if and only if a ∨ i = b ∨ i for some i ∈ I. If π I : L → L/θ I is the projection map and G ⊆ L/θ I is a proper prime n-filter extending π I [F ], then π −1 [G] is a prime n-filter separating F from I. It thus suffices to prove that each proper n-filter on a distributive lattice with a lower bound extends to a proper prime n-filter.
Consider a lower bounded distributive lattice L with a proper n-filter F on L. Then L, F embeds into an ultraproduct U L i , F i of its finitely generated (hence finite) substructures L i , F i for i ∈ I, where F i is the restriction of F to L i . Suppose that for each F i there is a proper prime n-ideal G i ⊇ F i on L i . Being a prime n-filter can be expressed by a first-order sentence, therefore the ultraproduct U L i , G i satisfies this sentence. Restricting the prime n-filter on this ultraproduct to L (as canonically embedded in U L i ) now yields the desired prime n-filter on L extending F . It therefore suffices to prove that each proper n-filter on a finite distributive lattice extends to a proper prime n-filter.
Each finite distributive lattice L embeds into a finite Boolean algebra B m for some m. The upset generated by a proper n-filter F on L is a proper n-filter G on B m . If G extends to a proper prime n-filter G ′ , then restricting G ′ to L yields a proper prime n-filter F ′ ⊇ F on L. It therefore suffices to prove that each proper n-filter on B m extends to a proper prime n-filter.
Prime filters on B m are principal filters generated by atoms, thus by Theorem 3.6 prime n-filters are precisely the upsets generated by sets of at most n atoms. An upset F on B m thus extends to a prime n-filter if and only if there are atoms a 1 , . . . , a n such that b ∈ F implies a i ≤ b for some a i . Equivalently, F extends to a prime n-filter if and only if there are coatoms c 1 , . . . , c n such that b ∈ F implies b c i for some c i . Contraposition yields that F does not extend to any prime n-filter if and only if for each n-tuple of coatoms c 1 , . . . , c n there is some b ∈ F such that b ≤ c 1 ∧ · · · ∧ c n . If F is an n-filter, this implies that the meet of all coatoms lies in F . An n-filter F on B m which does not extend to any prime n-filter is therefore not a proper n-filter.
Let us also provide a more direct proof of this theorem.
Proof. Let G be a maximal n-filter disjoint from I which extends F . If x, y / ∈ G, then the n-filter generated by F ∪ {x} intersects I and the n-filter generated by F ∪{y} intersects I. That is, there is some i ∈ I such that i ∈ [F, x] n and some j ∈ I such that j ∈ [F, y] n . By the following lemma, i ∨ j ∈ [F, x ∨ y] n , so x ∨ y / ∈ G.
Lemma 3.10. [F, x] n ∩ [F, y] n = [F, x ∨ y] n in every distributive lattice. Proof. Clearly [F, x ∨ y] n ⊆ [F, x] n ∩ [F, y] n . Conversely, let i ∈ [F, x] n ∩ [F, y] n .
There are non-empty finite P, Q ⊆ L such that P ∧ x ≤ i and Q ∧ y ≤ i, where (i) R ⊆ n P and S ⊆ n Q implies R ∈ F and S ∈ F , and (ii) T ⊆ n−1 P and
U ⊆ n−1 Q implies T ∧ x ∈ F and U ∧ y ∈ F . Take z := ( P ∧ x) ∨ ( Q ∧ y). Then z ≤ i. We show that z ∈ [F, x ∨ y] n .
The element z is the meet of a set Z consisting of elements of four types: p ∨ q for p ∈ P and q ∈ Q, p ∨ y for p ∈ P , x ∨ q for q ∈ Q, and x ∨ y. It suffices to prove that the meet of each subset of Z of cardinality at most n lies in F or above x ∨ y. This follows by a straightforward case analysis using the assumptions that R, S, T ∧ x, U ∧ y ∈ F for each R ⊆ n P and each S ⊆ n Q and each T ⊆ n−1 P and U ⊆ n−1 Q. For example, if elements of all four types are represented in a subset of Z of cardinality at most n, we use the bounds p ≤ p ∨ q, p ≤ p ∨ y, x ≤ x ∨ q, and x ≤ x ∨ y (or the bounds q ≤ p ∨ q, y ≤ p ∨ y, q ≤ x ∨ q, and y ≤ x ∨ y) to show that the meet of this subset lies in F or above x ∨ y. As a consequence of prime n-filter separation, we may exhibit each m-prime n-filter F on a finite distributive lattice L as the homomorphic preimages of a m-prime n-filter P n,m ⊆ B n×m , where B n×m , P n,m is the structure defined as
B n×m , P n,m := (B n ) m = i∈[m] j∈[n] B 1 , {1} .
These structures are the m-th direct powers of the n-th dual powers of B 1 . The direct square of the dual square of B 1 is shown in Figure 2 on the left. The structure shown on the right of Figure 2 is, by contrast, the dual square of the direct square of B 1 . This structure provides an example of a 2-filter whose complement is a 2ideal but which is not a 2-prime 2-filter. The reader may verify that it cannot be expressed as an intersection of two prime 2-filters. Proof. Let F be an m-prime n-filter on a finite distributive lattice L. By prime nfilter separation, F is the intersection of a finite family of prime n-filters. Because F is an m-prime n-filter, we may assume that this family consists of m prime n-filters (not necessarily distinct): F = F 1 ∩ · · · ∩ F m . Each of these prime nfilters is in turn a union of at most n prime filters (not necessarily distinct): F i = G i,1 ∪ · · · ∪ G i,n . Finally, each prime filter G i,j on L yields a strict homomorphism L, G i,j → B 1 , {1} . Putting all of this together, Fact 3.1 now yields a strict homomorphism from L, F into B n×m , P n,m .
Conversely, let π i : B n×m → B n for i ∈ [m] be the projection maps for B n×m = (B n ) m . Because P n ⊆ B n is a prime n-filter, so is π −1 i [P n ], hence P n,m is an n-filter by virtue of being a product of n-filters. Moreover, each homomorphic preimage of P n,m is an m-prime n-filter by virtue of being the intersection of the m homomorphic preimages of the prime n-filters π −1 i [P n ]. In the results proved so far, we may replace the two-element distributive lattice B 1 by one of its expansions. By the analysis of Post [8], there are exactly six proper expansions of B 1 up to term equivalence, namely expansions by a constant 0 for the bottom element, by a constant 1 for the top element, by both constants 0 and 1, by co-implication (x − y := x ∧ ¬y), by implication (x → y := ¬x ∨ y), and by negation (¬x). The (quasi)varieties generated by these six expansions are the varieties of lower bounded distributive lattices, upper bounded distributive lattices, bounded distributive lattices, generalized Boolean algebras, dual generalized Boolean algebras, and Boolean algebras.
The results proved so far extend immediately to these six cases, provided that appropriate modifications are made to the definition of n-filters. In the case of upper bounded distributive lattices and dual generalized Boolean algebras, we require that n-filters be non-empty. In the case of lower bounded distributive lattices and generalized Boolean algebras, we require that m-prime filters be proper. Finally, we impose both of these requirements in the case of bounded distributive lattices and Boolean algebras.
Filter classes
Are there other families of upsets of distributive lattices besides n-filters which share the basic properties of filters, i.e. closure under intersections, homomorphic preimages, and directed unions? To investigate this question, we first introduce some terminology and review some relevant results.
Let L be a quasivariety of algebras, i.e. a class of algebras in a given signature closed under subalgebras, products, and ultraproducts. Structures of the form A, F for A ∈ L will be called L-structures. In the following, K will denote a class of L-structures.
A set F ⊆ A ∈ L is called a K-filter if A, F ∈ K.
We say that K forms a filter class if it is closed under substructures, products, and strict homomorphic preimages. Equivalently, K is a filter class if K-filters are closed under homomorphic preimages (not necessarily surjective) and arbitrary intersections. If the class K is moreover closed under strict homomorphic images, we say that K is a logical class. A filter class will be called trivial if it only contains structures A, F where F is the total K-filter, i.e. F = A.
(Recall that a structure A, F is called a strict homomorphic preimage of B, G and conversely B, G is called a strict homomorphic image of A, F if there is a strict surjective homomorphism h : A, F → B, G , i.e. a surjective homomorphism of algebras h :
A → B such that F = h −1 [G].)
A filter class K of L-structures is said to be finitary if K is closed under ultraproducts, or equivalently if the K-filters on each A ∈ L are closed under directed unions. We may also phrase this definition in terms of K-filter generation. The K-filter Fg A K X generated by a set X ⊆ A ∈ L is, of course, the smallest K-filter on A which contains X. The class K is then finitary if and only if the closure operator Filter classes of L-structures admit an equivalent syntactic characterization as classes axiomatized by implications of a certain form. Consider a proper class of variables. An atomic formula has either the form t ≈ u or F(t), where t and u are terms over the given class of variables in the algebraic signature of L. The equality symbol is of course interpreted by the equality relation, while F is interpreted in the structure A, F by the set F ⊆ A. By an implication (more explicitly, a possibly infinitary Horn formula) we shall mean an infinitary formula of the form
Fg A K is finitary for each A ∈ L, i.e. Fg A K X = {Fg A K Y | Y ⊆ Aα i for each i ∈ I =⇒ β,
where β is an atomic formula and α i for i ∈ I is a set of atomic formulas. The implication is said to be finitary if I is finite. In a filter implication, we require that β have the form F(u). In an equality-free implication, we require that all of the formulas α i and β have this form. For example, F is an upset of a distributive latice L if and only if L, F satisfies the implication
F(x) & x ∧ y ≈ x =⇒ F(y).
This is a filter implication, but it is not equality-free.
(The reader concerned about having a proper class of variables may instead consider an infinite set of variables of some cardinality κ and add the following constraint on filter classes and logical classes: if each κ-generated substructure of A, F belongs to K, then so does A, F . With this modification, logical classes are precisely the classes of all models of some logic over the given set of variables in the sense of abstract algebraic logic [4]. The problem of describing the logical classes of L-structures is therefore intimately related to the problem of describing the extensions of the logic of all L-structures.)
We now introduce some notational conventions for talking about implications. When no confusion is likely to arise, we identify the atomic formula F(t) with the term t. Atomic formulas, or the corresponding terms, will be denoted by lowercase Greek letters. A set of atomic formulas of the form F(t), or the corresponding set of terms, will be denoted by Γ, ∆, or Φ, while a set of equations will be denoted by E. We adopt notation common in abstract algebraic logic and write E, Γ ⊢ ϕ instead of E & Γ =⇒ ϕ and E, Γ ⊢ t ≈ u instead of E & Γ =⇒ t ≈ u. For example, the equality-free implication which defines 2-filters becomes simply x ∧ y, y ∧ z, z ∧ x ⊢ x ∧ y ∧ z in this notation. The notation E, Γ ⊢ K ϕ means that the filter implication E, Γ ⊢ ϕ holds in each structure in K.
The following syntactic description of filter classes and logical classes is due to Stronkowski [9, Lemma 2.3 and Theorems 3.7 & 3.8]. The claim for finitary logical classes, as well as its infinitary version in case we restrict to a set of variables, is originally due to Dellunde and Jansana [3].
Theorem 4.4. Let K be a class of L-structures. Then (i) K is a (finitary) filter class if and only if K is axiomatized by some class of (finitary) filter implications. (ii) K is a (finitary) logical class if and only if K is axiomatized by some class of (finitary) equality-free implications.
We shall be interested in cases where the quasivariety L is the class DLat of distributive lattices, the class SLat of meet semilattices, the class uSLat of unital meet semilattices, and the class BA of Boolean algebras. We do not explicitly consider bounded distributive lattices and bounded meet semilattices, but our results easily extend to these classes. (Recall that a unital meet semilattice is an upper bounded meet semilattice expanded by a constant 1 which denotes the top element.)
Distributive lattices equipped with an upset form a filter class DL ∞ of DLatstructures. Distributive lattices equipped with an n-filter then form a filter subclass of DL ∞ denoted DL n for n ∈ ω. Clearly DL i ⊆ DL j ⊆ DL ∞ for i < j, and the structures B n witness that these inclusions are strict. The classes of SLat-structures SL ∞ and SL n are defined in the same way. The classes of uSLat-structures uSL ∞ and uSL n for n ≥ 1, and the classes of BA-structures BA ∞ and BA n for n ≥ 1 are defined similarly, but we restrict to non-empty upsets and non-empty n-filters.
The filter implications valid in DL n can be described in terms of implications valid in DL ∞ , which in turn can be described in terms of implications valid in distributive lattices. We write E ⊢ DLat γ ≤ ϕ if the implication E ⊢ γ ∧ ϕ ≈ γ holds in all distributive lattices. An entirely analogous theorem can be proved relating the filter classes SL n and SL ∞ and implications valid in meet semilattices.
Theorem 4.5. Let E, Γ ⊢ ϕ be a filter implication in the signature of distributive lattices. Then:
(i) E, γ ⊢ DL ∞ ϕ if and only if E ⊢ DLat γ ≤ ϕ. (ii) E, Γ ⊢ DL ∞ ϕ if and only if E, γ ⊢ DL ∞ ϕ for some γ ∈ Γ. (iii) E, Γ ⊢ DLn ϕ if and only if there is some non-empty finite set of terms Φ such that E, Γ ⊢ DL ∞ ∆ for each ∆ ⊆ n Φ and E, Φ ⊢ DL ∞ ϕ.
Proof. The first equivalence is trivial. The third equivalence follows from the nfilter generation lemma (Lemma 2.16) applied to the quotient F DL (X)/θ E of the free distributive lattice F DL (X) generated by the set X of all variables which occur in Γ ∪ {ϕ} by the congruence θ E generated by E. To prove the second equivalence, let Tm be the absolutely free algebra (the algebra of terms) over the set of all variables which occur in Γ ∪ {ϕ} and suppose that E, γ DL∞ ϕ for each γ ∈ Γ. This yields a distributive lattice L γ and a homomorphism h γ : Tm → L γ such that h γ (γ) h(ϕ) for each γ ∈ Γ and h γ (t) = h γ (u) for each equality t ≈ u in E. Let L := γ∈Γ L γ , let h : Tm → L be the product of the maps h γ , and let F be the upset of L generated by {h(γ) | γ ∈ Γ}. Clearly h(γ) h(ϕ) for each γ ∈ Γ, hence h(ϕ) / ∈ F . Also, h(t) = h(u) for each equality t ≈ u in E. The structure L, F thus witnesses that E, Γ DL ∞ ϕ. The right-to-left implication is trivial.
For Boolean algebras and unital meet semilattices, the above equivalences need to be modified slightly. Again, an entirely analogous fact can be proved relating the filter classes uSL n and uSL ∞ and implications valid in unital meet semilattices. Theorem 4.6. Let E, Γ ⊢ ϕ be a filter implication in the signature of Boolean algebras. Then:
(i) E ⊢ BA ∞ ϕ if and only if E ⊢ BA 1 ≤ ϕ. (ii) E, γ ⊢ BA ∞ ϕ if and only if E ⊢ BA γ ≤ ϕ. (iii) E, Γ ⊢ BA ∞ ϕ for Γ non-empty if and only if E, γ ⊢ BA ∞ ϕ for some γ ∈ Γ.
(iv) E, Γ ⊢ BAn ϕ if and only if there is some non-empty finite set of terms Φ such that E, Γ ⊢ BA ∞ ∆ for each ∆ ⊆ n Φ and E, Φ ⊢ BA ∞ ϕ.
In order to understand what happens beyond the finitary case, it remains to introduce one more family of upsets. An upset of a distributive lattice (Boolean algebra, semilattice, unital semilattice) will be called residually finite if it is an intersection of homomorphic preimages of some upsets of finite distributive lattices (Boolean algebras, semilattices, unital semilattices). Observe that an upset of a Boolean algebra A is residually finite if and only if it is residually finite as an upset of the distributive lattice reduct of A: each homomorphism from a Boolean algebra A to a finite distributive lattice B extends to a homomorphism from A to the free Boolean extension of B, which is also finite. Similarly, an upset of a unital semilattice S is residually finite if and only if it is residually finite as an upset of the semilattice reduct of S. Moreover, an upset of a distributive lattice is residually finite if and only if it is an intersection of a family of upsets which are n-filters for some n (not necessarily the same n for the whole family).
The class of all distributive lattices (Boolean algebras, semilattices, unital semilattices) equipped with a residually finite upset is a filter subclass of DL ∞ denoted DL ω (BA ω , SL ω , uSL ω ). Each n-filter on a distributive lattice is a residually finite upset but some residually finite upsets, such as the upset of n≥1 B n , are not n-filters for any n. The class DL ω is a proper subclass of DL ∞ : consider for example the upset P ω of non-zero elements of B ω := (B 1 ) ω . It follows that there is some filter implication which holds in each finite structure in DL ∞ (or equivalently, in each upset which is an n-filter for some n) but which fails in some infinite one. Indeed, one such implication states that if X 1 , X 2 , . . . are sets consisting of at most 2, 3, . . . elements in a distributive lattice such that X 1 = X 2 = · · · = a and the meet of each proper subset of X 1 , X 2 , . . . lies in F , then so does a. This implication holds for each upset which is an n-filter for some n, but it fails in B ω , P ω .
Filter classes of upsets of distributive lattices
We now show that the only finitary filter subclasses of DL ∞ are the classes DL n . The following theorem merely restates the fact that n-filters on distributive lattices are precisely the intersections of homomorphic preimages of P n ⊆ B n . Recall that we are using the notation B n := B n , P n . For n = 0 we take B 0 to be the singleton Boolean lattice and P 0 := ∅.
Theorem 5.1. DL n is generated as a filter class by the structure B n .
Theorem 5.2. DL ω is generated as a filter class by {B n | n ≥ 1}.
An intruiging problem which we leave open is how to axiomatize DL ω . Theorem 5.3. DL ∞ is generated as a finitary filter class by {B n | n ≥ 1}.
Proof. Each structure embeds into an ultraproduct of its finitely generated substructures and each finitely generated structure in DL ∞ is finite. Each finite structure L, F in DL ∞ then embeds into a product of the structures B n , since F is an intersection of prime upsets of L and each prime upset of L is a prime n-filter for some n because L is finite. Proof. We first define an auxiliary construction on distributive lattice terms. Recall that a distributive lattice term in disjunctive normal form is a disjunction of conjunctions of variables. If t is such a term and x is a tuple of variables, we define the term t[x] as follows: if one of the conjunctions in t consists entirely of variables not in x, we leave t[x] undefined, otherwise we remove all variables not in x from t. If t[x] is defined, then the inequality t ≤ t[x] holds in each distributive lattice. If Γ is a set of terms in disjunctive normal form, we define Γ[x] as the set of all t[x] such that t ∈ Γ and t[x] is defined. Now consider an equality-free implication Γ ⊢ u(x) which holds in each B n for n ≥ 1. We may assume without loss of generality that each term in Γ is in disjunctive normal form, i.e. a disjunction of conjunctions of variables. If Γ ⊢ u(x) holds in B n for each n ≥ 1, then in particular it holds if we assign the value 1 to every variable outside of x. It follows that Γ[x] ⊢ u(x) holds in each B n for n ≥ 1. But Γ[x] only contains finitely many variables, therefore up to equivalence of distributive lattice terms Γ[x] is finite and Γ[x] ⊢ u(x) is a finitary equality-free implication. It follows that Γ[x] ⊢ u(x) holds in each structure in DL ∞ . Finally,
Γ ⊢ u(x) follows from Γ[x] ⊢ u(x) because t ⊢ t[x] holds in DL ∞ whenever t[x] is defined.
We can state this is more explicit terms.
Theorem 5.5. Every upset of a distributive lattice is a strict homomorphic image of an intersection of homomorphic preimages of finitely generated upsets.
The free distributive lattice over a set X will be denoted F DL (X) in the following, with F DL (n) := F DL ({x 1 , . . . , x n }). Observe that x 1 ∧· · ·∧x n is the bottom element of F DL (n). Accordingly, P (F DL (n)) := {a ∈ F DL (n) | a > x 1 ∧ · · · ∧ x n }.
Lemma 5.6. The structure B n embeds into F DL (n), P (F DL (n)) .
Proof. F DL (n) is isomorphic to the lattice of non-empty non-total subsets of the poset B n . Each preorder on this poset which extends the partial order of B n determines a subalgebra of F DL (n) and thus a substructure of F DL (n), P (F DL (n)) . The preorder which collapses all the elements strictly below the coatoms into a single point then determines a substructure isomorphic to B n .
Theorem 5.7 (Filter subclasses of DL ∞ ). Let K be a non-trivial filter subclass of DL ∞ . Then either DL ω ⊆ K or K is one of the classes DL n for n ∈ ω.
Proof. Because K is non-trivial, it contains some L, F where F is a proper filter. Then B 0 is a substructure of L, F , so B 0 ∈ K and DL 0 ⊆ K. If K contains some L, F where F is a non-empty proper filter, then B 1 is a substructure of L, F , so B 1 ∈ K and DL 1 ⊆ K. Otherwise K = DL 0 . Now suppose that K DL n for some n ≥ 1. We show that DL n+1 ⊆ K. The theorem then follows at once, taking into account that DL ω ⊆ K if B n ∈ K for each n ≥ 1. Because K DL n , there is some L, F ∈ K such that F is not an n-filter. Since n-filters are defined by an implication involving only n + 1 variables, there is an (n + 1)-generated subalgebra of L, F such that the restriction of F to this subalgebra is not an n-filter. We may therefore assume without loss of generality that L is (n + 1)-generated. Then there is a surjective homomorphism h : F DL (n + 1) → L such that h( j =i x j ) ∈ F but h(x 1 ∧ · · · ∧ x n+1 ) / ∈ F . The structure F DL (n + 1), h −1 [F ] then lies in K. Since each element of F DL (n + 1) is a join of meets of subsets of {x 1 , . . . , x n+1 } and each meet of a proper subset of {x 1 , . . . , x n+1 } lies in h −1 [F ], it follows that each element of F DL (n + 1) other than x 1 ∧ · · · ∧ x n+1 lies in h −1 [F ]. In other words, h −1 [F ] = P (F DL (n + 1)). But then B n+1 is isomorphic to a substructure of F DL (n + 1), h −1 [F ] and therefore B n+1 ∈ K and DL n+1 ⊆ K.
The above theorem holds equally well for lower bounded, upper bounded, and bounded distributive lattices (excluding n = 0 in the last two cases). Corollary 5.10. If F is an n-filter but not an (n + 1)-filter on a distributive lattice L, then L, F generates DL n as a filter class.
Corollary 5.11. If F is an upset but not an n-filter for any n ∈ ω on a distributive lattice L, then L, F generates DL ∞ as a finitary filter class and as a logical class.
The above corollary allows us to determine the finitary filter class generated by any structure of the form L, F where L is a distributive lattice and F is an upset of L. For example, consider the upset F m d of all elements a ∈ B n for n := d + m of height strictly more than m. Equivalently, this is the upset of all a ∈ B n of co-height strictly less than d. Recall that the height (co-height) of an element a in a poset is the size of the longest chain in the principal downset (upset) generated by a, minus one. Equivalently, the height of a ∈ B n is the number of atoms below a, and the co-height of a is the number of co-atoms above a. This convention fits well with our previous notation: P n = F 0 n . Fact 5.12. F m d ⊆ B d+m is a d-filter whose complement is an (m + 1)-ideal. It is not a (d − 1)-filter and its complement is not an m-ideal.
Filter classes of upsets of meet semilattices
We have already seen that not every n-filter on a meet semilattice is an intersection of homomorphic preimages of the n-filter P n . This is of course also witnessed syntactically. The filter implication
x ∧ y ≈ y ∧ z ≈ z ∧ x & F(x) & F(y) & F(z) =⇒ F(x ∧ y ∧ z)
holds in B 2 but not in M 5 , P (M 5 ) , where M 5 is the five-element diamond. Nonetheless, we show that every n-filter is the strict homomorphic image of such an intersection. In other words, SL n and uSL n are generated by B n as logical classes but (unlike in the distributive lattice case) not as filter classes.
The key observation in this respect is that the free unital meet semilattice F uSL (X) over a set of generators X is in fact a distributive lattice. Theorem 6.1. uSL n is generated as a logical class by the structure B n .
Proof. Let F be an n-filter on a unital semilattice S. Then there is a surjective homomorphism h : F uSL (X) → S for some sufficiently large X. Let G := h −1 [F ]. Then h : F uSL (X), G → S, F is a strict surjective homomorphism. Because F uSL (X) is a distributive lattice and G is an n-filter, G is an intersection of homomorphic preimages of P n . Moreover, these homomorphisms preserve the top element of F uSL (X), i.e. they are unital semilattice homomorphisms. Theorem 6.2. uSL ∞ is generated as a logical class by {B n | n ≥ 1}.
Proof. The proof is entirely analogous to the distributive lattice case.
The last two theorems are equally true for SL n and SL ∞ . The classes SL n were in fact already considered by Font and Moraschini [5], who showed that they form an infinite increasing chain. More precisely, they studied the corresponding logics, which they denoted R n . Theorem 6.3 (Logical subclasses of uSL ∞ ). Let K be a non-trivial logical subclass of uSL ∞ . Then either uSL ω ⊆ K or K is one of the classes uSL n for n ≥ 1.
Proof. It suffices to replace F DL (n + 1) in the proof of Theorem 5.7 by F uSL (n + 1) and to observe that F uSL (n + 1) is the unital meet semilattice reduct of B n+1 . The class SL ∞ , on the other hand, admits other finitary logical subclasses in addition to the classes SL n . For example, the implication x ∧ z, y ∧ z ⊢ x ∧ y ∧ z fails in B 2 but it holds in the three-element substructure obtained by removing the top element. This implication is obtained from x, y ⊢ x ∧ y using the substitution x → x ∧ z and y → y ∧ z. An upset F of a semilattice therefore validates this implication if and only if a ∧ b ∈ F holds whenever a ∈ F and b ∈ F and a and b have an upper bound. Observe that if the constant 1 is part of the signature, then the above implication defines ordinary filters: it suffices to substitute 1 for z.
Many other non-equivalent conditions may be obtained by applying such substitutions to n-adjunction, i.e. the implication which defines n-filters:
{ j =i x j | i ∈ [n + 1]} ⊢ x 1 ∧ · · · ∧ x n+1 .
For example, consider the following substitution applied to 5-adjunction:
x i → x i ∧ y for i ∈ {1, 2, 3}, x j → x j ∧ z for j ∈ {4, 5}, x 6 → x 6 .
The corresponding implication states that the meet of each 6-tuple of elements lies in F provided that (i) each non-empty proper submeet lies in F , (ii) three of the six elements have an upper bound, and (iii) so do two of the other three elements. We conjecture that each finitary filter subclass of SL ∞ is axiomatized by such substitution instances of n-adjunction.
Filter classes of upsets of Boolean algebras
The picture for Boolean algebras is substantially more complicated: there are many finitary filter subclasses of BA ∞ in addition to the classes BA n for n ≥ 1. Nonetheless, the classes BA n still play an important role: they are the only ones generated by prime structures, i.e. structures of the form A, F where F is a prime upset. Moreover, as in the case of distributive lattices, each finitary filter subclass of BA ∞ is in fact a logical class.
The proofs of the following theorems carry over from distributive lattices.
Theorem 7.1. BA n is generated as a filter class by the structure B n .
Theorem 7.2. BA ω is generated as a filter class by {B n | n ≥ 1}.
We again leave the problem of axiomatizing BA ω open. We do not even know whether BA ω is a logical class, or more specifically whether it coincides with the logical class generated by the structures B n . It is not the case (as it was for distributive lattices) that BA ∞ is generated as a logical class by {B n | n ≥ 1}. For example, each of the structures B n satisfies the following infinitary equality-free implications:
{(x i ∧ ¬x j ) ∨ y | i, j ∈ ω and i < j} ⊢ y, {(x i ∧ ¬x j ) ∨ y | i, j ∈ ω and i > j} ⊢ y.
These express the fact that the algebras B n are well-partially-ordered and dually well-partially-ordered. Recall that a (dual) well-partial-order is a partial order such that in each sequence x i for i ∈ ω there are i < j with x i ≤ x j (with x i ≥ x j ). But these implications fail in B ω , P ω .
The key to proving that the only finitary filter subclasses of BA ∞ generated by prime structures are the classes BA n is to find suitable splittings of the lattice of all filter subclasses of BA ∞ . In particular, we identify finitary equality-free implications (α n ) such that for each filter subclass K of BA ∞ either (α n ) holds in K or BA n ⊆ K, but not both. We now describe these implications.
Let us pick k ≥ 1 so that 2 k ≥ n. There are 2 k complete conjunctive clauses over k variables x 1 , . . . , x k up to equivalence in Boolean algebras. These are conjunctions of variables and negated variables such that for each of the k variables x i exactly one element of {x i , ¬x i } occurs in the conjunction. Let us call these complete conjunctive clauses π i for i ∈ [2 k ]. Now consider the implication π 1 , . . . , π n ⊢ π n+1 ∨ · · · ∨ π 2 k . (α n ) In case n = 2 k , we interpret this simply as π 1 , . . . , π n ⊢ y. In particular, we may take (α 2 ) to be the rule x, ¬x ⊢ y. We take (α 1 ) to be x ⊢ y.
Lemma 7.4. Let A be a finite Boolean algebra and F be a prime upset of A. If F is an n-filter but not an (n − 1)-filter, then B n embeds into A, F .
Proof. Each finite Boolean algebra is isomorphic to the powerset of some finite set X. Each prime n-filter F on this powerset which is not an (n − 1)-filter is then determined by an n-element set {x 1 , . . . , x n } ⊆ X as follows: U ∈ F if and and only if x i ∈ U for some x i . Each equivalence relation on X determines a subalgebra of the powerset (generated by the equivalence classes). Any equivalence relation where each equivalence class contains exactly one of the elements x i then determines a substructure isomorphic to B n . Lemma 7.5. Let A be a Boolean algebra and F be an upset of A. If (α n ) fails in the structure A, F , then B n embeds into A, F .
Proof. Suppose that (α n+1 ) fails in A, F . This is witnessed by a homomorphism h : F BA (k) → A from the free Boolean algebra over the generators x 1 , . . . , x k . Let the substructure B, G of A, F be the image of h and let H := h −1 [G]. Then h : F BA (k), H → B, G is a strict surjective homomorphism. We have π i ∈ H for each i ∈ [n], while π n+1 ∨ · · · ∨ π 2 k / ∈ H. Here we identify the terms π i with the atoms of the free Boolean algebra F BA (k). This uniquely determines H: each element of F BA (k) either lies below π n+1 ∨ · · · ∨ π 2 k or above some π i for i ∈ [n].
Let I := F BA (k)\H. Then I ⊆ F BA (k) is the principal ideal generated by the element ¬(π n+1 ∧ · · · ∧ π 2 k ). The projection map F BA (k) → F BA (k)/I is a strict homomorphism of structures F BA (k), H → F BA (k)/I, H/I , where we use the notation H/I := {h/I | h ∈ H}. Clearly H/I consists precisely of the non-zero elements of F BA (k)/I. Moreover, F BA (k)/I has exactly n atoms, namely π 1 /I, . . . , π n /I. The structure F BA (k)/I, H/I is thus isomorphic to B n . It follows that H, hence also G, is an n-filter which is not an (n − 1)-filter. But B, G is a finite structure, therefore B n is isomorphic to a substructure of B, G . Lemma 7.6. Let A be a Boolean algebra and F be a prime upset of A. If (α n+1 ) holds in A, F , then F is an n-filter.
Proof. Suppose that F is not an n-filter. We show that (α n+1 ) fails in A, F . There are a 1 , . . . , a n+1 such that j =i a j ∈ F for each i ∈ [n + 1] but i∈[n+1] a i / ∈ F . Let B be the subalgebra of A generated by a 1 , . . . , a n+1 and let G be the restriction of F to B. Then B is finite, therefore G is a prime m-filter for some m > n. (Recall that F is prime.) It follows that there is a strict homomorphism h : B, G → B m . The image of h is some substructure of B m whose upset is not an n-filter. This image is therefore isomorphic to B k for some k > n. In other words, we have a strict surjective homomorphism from B, G to B k for some k > n. But (α n+1 ) fails in B k (because B n+1 is a substructure of B k ), hence also in its strict homomorphic preimage B, G and in the structure A, F . Lemma 7.7. Let K be a filter subclass of BA ∞ . Then either BA n ⊆ K or (α n ) holds in K, but not both.
Proof. The implication (α n ) fails in B n because the upset P n is not an (n− 1)-filter, therefore BA n ⊆ K implies that (α n ) does not hold in K. On the other hand, if (α n ) fails in some structure A, F ∈ K, then B n ∈ K by virtue of being isomorphic to a substructure of A, F . But then BA n ⊆ K. Proof. If for each n the filter subclass K fails to validate (α n ), then B n ∈ BA n ⊆ K for each n ≥ 1, so BA ω ⊆ K. Proof. If K is finitary, then BA ω ⊆ K implies BA ∞ ⊆ K. Theorem 7.10 (Filter subclasses of BA ω ). Let K be a non-trivial filter subclass of BA ω generated by prime structures. Then either BA ω ⊆ K or K is one of the classes BA n for n ≥ 1.
Proof. If BA ω K, then B m / ∈ K for some m ≥ 1, i.e. BA m K. On the other hand, K contains some structure L, F where F is not the total filter. Then B 1 is a substructure of L, F and BA 1 ⊆ K. There is therefore some n ∈ ω such that BA n ⊆ K but BA n+1 K. But then (α n+1 ) holds in K, so K ⊆ BA n . Proof. It suffices to replace filter classes by logical classes and BA ω by BA ∞ in the previous proof.
In the rest of the paper, we consider some further filter subclasses of BA ∞ . One such family of subclasses of BA ∞ may be defined by the implications
x 1 , . . . , x k , ¬(x 1 ∧ · · · ∧ x k ) ⊢ y. (β k )
We take (β 0 ) to be x ⊢ x. Recall that F m d is the upset of all elements a ∈ B n for n := d + m of height strictly greater than k ≥ 0. Proof. The condition x i ∈ F m d states that the co-height of x i is at most d − 1. The co-height of x 1 ∧ · · · ∧ x k is thus at most k(d − 1) = m, or equivalently the height of ¬(x 1 ∧ · · · ∧ x k ) is at most m. But then the co-height of this element is at least (d + m) − m = d and ¬(x 1 ∧ · · · ∧ x k ) / ∈ F m d . The rule (β k ) thus holds in B d+m , F m d . On the other hand, let x i for i ∈ [k + 1] be joins of some pairwise disjoint sets of atoms of cardinality d − 1. Such disjoint sets of atoms exist because (k + 1)(d − 1) = (d − 1) + k(d − 1) < d + k(d − 1) = d + m. Then each x i has height d − 1 and ¬x i ∈ F m d for each x i . By the disjointness condition, x 1 ∨ · · · ∨ x k+1 has height (k + 1)(d − 1), therefore x 1 ∨ · · · ∨ x k+1 ∈ F m d . The rule (β k+1 ) thus fails in B d+m , F m d . Let BA n,k denote the subclass of BA n axiomatized by (β k ). Proof. This follows immediately from Facts 5.12 and 7.12.
Fact 7.14. BA 2 has infinitely many finitary logical subclasses. Theorem 7.15. Each non-trivial filter subclass of BA ∞ other than BA 1 contains the structure B 2 × B 1 .
Proof. Each non-trivial filter subclass K of BA ∞ contains B 1 , so BA 1 ⊆ K. If K BA 1 , then the rule x, y ⊢ x ∧ y fails in some A, F ∈ K. Restricting to an appropriate substructure, we may take A to be 2-generated. It follows that there is a surjective homomorphism h : F BA (2) → A where F BA (2) is the free Boolean algebra over the generators x, y. Taking G := h −1 [F ], the structure F BA (2), G is a homomorphic preimage of A, F where x, y ∈ G but x ∧ y / ∈ G. If ¬x ∈ G or ¬y ∈ G, then B 2 is isomorphic to the substructure of F BA (2), G generated by x or y, therefore BA 2 ⊆ K and B 2 × B 1 ∈ K. We may thus assume Proof. Each finitary filter subclass is generated by its finite structures, since each structure embeds into an ultraproduct of its finitely generated structures. It thus suffices to show that the logical class H −1 S H S SP(K) = H −1 S H S (K) generated by a filter class K has the same finite structures as K. But each finite structure in this logical class is a strict homomorphic preimage of a finite structure A, F which is a strict homomorphic image of some structure B, G in K. Then A, F is in fact a strict homomorphic image of some finitely generated (hence finite) substructure C, H of B, G . By the previous lemma, this makes A, F isomorphic to a substructure of B, G .
It remains an open question to determine how many finitary filter subclasses (or equivalently, finitary logical subclasses) of BA ∞ there are.
Data availability. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Fact 2. 1 .
1An upset F of a semilattice S is an n-filter if and only if j =i
Fact 2. 4 .
4Each upset of a finite semilattice (lattice) is an (m-prime) n-filter for some n (and m). Each element of a finite semilattice is m-prime for some m.
Fact 2 . 5 .
25Let h : S → T be a semilattice homomorphism and F be an n-filter on T. Then h −1 [F ] is an n-filter on S.
Fact 2 . 6 .
26Let h : L → M be a lattice homomorphism and F be a prime n-filter (m-prime filter) on M. Then h −1 [F ] is a prime n-filter (m-prime filter) on M.
Fact 2 . 10 .
210Let S be an ideal subsemilattice of T and let F be an m-prime n-filter on T. Then the restriction of F to S is an m-prime n-filter on S.
Figure 1 .
1Generating n-filters in semilattices a b admissible set. By the definition of F , for each
Fact 3. 2 .
2P n is a prime n-filter on B n . It is not an m-filter for any m < n.
Lemma 3 . 3 .
33The following are equivalent for each upset U of a lattice L:
are filters on L. Consider the embedding ι : L ֒→ Fi 1 (L) of L into the lattice Fi 1 (L) of filters on L ordered by reverse inclusion where ι(a) is the principal filter generated by a. This map is a lattice homomorphism. Then a ∈ F i if and only if ι(a) ⊆ F i , so a ∈ F if and only if ι(a) ⊆ F i for some i ∈ [k]. The upset U is thus the preimage of the upset generated in Fi 1 (L) by the k elements F i . (iii) ⇒ (i): it suffices to show that each upset U generated by some elements b 1 , . . . , b n satisfies (i). But for each set {a
Theorem 3.8 (n-prime filters on distributive lattices). The following conditions are equivalent for each upset F of a distributive lattice:
Corollary 3.11 (n-filters on distributive lattices). The n-filters on a distributive lattice are precisely the intersections of prime n-filters.
Figure 2 .
2The structures (B ⊗2 1 ) Π2 and (B Π2 1 ) ⊗2
Theorem 3 . 12 .
312The m-prime n-filters on a finite distributive lattice are precisely the homomorphic preimages of the m-prime n-filter P n,m ⊆ B n×m .
and Y is finite}. The permutation properties of the class operators in question immediately yield the following theorems, where H −1 S (K), H S (K), S(K), P(K), and P U (K) denote the classes of L-structures which are strict homomorphic preimages, strict homomorphic images, substructures, products, and ultraproducts of structures in K.
Theorem 4 . 1 .
41Let K be a class of L-structures. The filter class generated by K is H −1 S SP(K). The logical class generated by K is H −1 S H S SP(K).
Theorem 4 . 2 .
42The finitary filter class (finitary logical class) generated by K is the filter class (logical class) generated by P U (K).
Corollary 4 . 3 .
43The finitary filter class (finitary logical class) generated by K coincides with the filter class (logical class) generated by K.
Theorem 5 . 4 .
54DL ∞ is generated as a logical class by {B n | n ≥ 1}.
Corollary 5. 8 (
8Finitary filter subclasses of DL ∞ ). The only non-trivial proper finitary filter subclasses of DL ∞ are the classes DL n for n ∈ ω.
Corollary 5. 9 (
9Logical subclasses of DL ∞ ). The only non-trivial proper logical subclasses of DL ∞ are the classes DL n for n ∈ ω.
Proof. F m d is a d-filter by virtue of being the intersection of a family of d-filters. This family consists of all unions of d principal filters generated by atoms of B d+m . (If Y ⊆ X with |X| = n, then |Y | > m if and only if Y intersects each subset of X of cardinality d = n − m.) It is not a (d − 1)-filter because the meet of each set of at most d − 1 coatoms has co-height at most d − 1, but the meet of all coatoms is 0. The other claims follow if we apply the same reasoning to B d+m \ F m d instead of F m d . Fact 5.13. B n+m , F m n generates DL n as a filter class for each m ∈ ω.
Corollary 6 . 4 (
64Finitary logical subclasses of uSL ∞ ). The only non-trivial proper finitary logical subclasses of uSL ∞ are the classes uSL n for n ≥ 1.
Theorem 7 . 3 .
73BA ∞ is generated as a finitary filter class by {B n | n ≥ 1}.
Fact 7 . 8 .
78Each proper filter subclass of BA ω validates some (α n ).
Fact 7 . 9 .
79Each proper finitary filter subclass of BA ∞ validates some (α n ).
Theorem 7 .
711 (Logical subclasses of BA ∞ ). The only non-trivial proper finitary logical subclasses of BA ∞ generated by prime structures are BA n for n ≥ 1.
Fact 7 . 12 .
712Let d ≥ 2, k ≥ 0, and m := k(d − 1). Then B d+m , F m d satisfies (β k ) but not (β k+1 ).
Fact 7 . 13 .
713BA m,i ⊆ BA n,j if and only if either m ≤ n and j ≤ i or n = 1.
is a strict surjective homomorphism, thenr −1 [G] = F , so G = (r • s) −1 [G] = s −1 [r −1 [G]] = s −1 [F ] , i.e. s : A, F → B, G is a strict embedding.Theorem 7.18. Each finitary filter subclass of BA ∞ is a logical class.
The n-filters studied here are unrelated to the n-filters of Lukasiewicz-Moisil algebras[2].
that ¬x, ¬y / ∈ G. Let B, H be the substructure of F BA (2), G generated by {x, x ∧ y}. If ¬x ∨ ¬y / ∈ G, then B, H is isomorphic to B 2 × B 1 . On the other hand, if ¬x ∨ ¬y ∈ G, then B, H is isomorphic to B 3 , F 1 2 , where F 1 2 is the upset (2-filter) generated by the coatoms of B 3 . But(If the coatoms of B 3 are denoted c 1 , c 2 , c 3 , then this embedding sends the two designated coatoms of B 2 × B 1 to c 1 , 1 and c 3 , 1 and the non-designated coatom to c 2 , 0 .) The filter class generated by the structure B m × B n for m > n is axiomatized by the infinite set of rulesWe strongly suspect that this class is not finitely axiomatizable, but we have no proof of this.Theorem 7.16. The filter class generated by B m × B n for m > n consists precisely of m-filters which are either total or are contained in some non-total n-filter. It is axiomatized relative to BA m by the implications (γ n,k ) for k ≥ n.Proof. Let F be an m-filter where (γ n,i ) holds for each i. We show that the nfilter G generated by F does not contain 0, i.e. that F is contained in a nontrivial (prime) n-filter. If G did contain 0, this would be witnessed by a set X =. We could therefore assume without loss of generality that X = {x 1 , . . . , x k , ¬ (x 1 ∧ · · · ∧ x k )}. But the rule (γ n,k ) forbids this. Conversely, if (γ n,k ) fails for some k, then this provides a set witnessing that the n-filter generated by F is trivial. In particular, the upset of B m × B n is an m-filter contained in the n-filter B m × P n (the product of the total filter on B m and the n-filter P n ⊆ B n ), therefore the rules (γ n,k ) hold in B m × B n . Now let F be an m-filter on A contained in some non-trivial n-filter G. It remains to show that A, F lies in the filter class generated by B m × B n . Of course, G extends to a non-trivial prime n-filter H ⊇ G. The m-filter F is the intersection of some prime m-filters F i for i ∈ I. This yields an embedding f : A, F ֒→ i∈I B m . The prime n-filter H corresponds to a strict homomorphism g : A, H → B n , which yields a strict homomorphism g I : A, H → i∈I B n . Taking the product of f and g I yields an embedding of algebras h : A ֒→ i∈I (B m × B n ). It remains to show that this is a strict homomorphism h : A, F → i∈I (B m × B n ): if a ∈ F , then also a ∈ H, so f (a) belongs to the upset of i∈I B m , g(a) belongs to the upset of B n , and h(a) belongs to the upset of i∈I (B m × B n ). On the other hand, if a / ∈ F , then f (a) does not belong to the upset of i∈I B m , so h(a) does not belong to the upset of i∈I (B m × B n ). Therefore h is a strict homomorphism and A, F lies in the filter class generated by B m × B n .The filter class generated by B n+1 × B n lies strictly between the filter classes generated by B n and B n+1 : the upset of B n+1 × B n is an (n + 1)-filter but not an n-filter, and the upset of B n+1 is not contained in a non-trivial n-filter.
On classes of structures axiomatizable by universal d-Horn sentences and universal positive disjunctions. Guillermo Badia, João Marcos, Algebra Universalis. 7941Guillermo Badia and João Marcos. On classes of structures axiomatizable by universal d-Horn sentences and universal positive disjunctions. Algebra Universalis, 41(79), 2018.
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| [] |
[
"TOWARDS RESPONSIBLE QUANTUM TECHNOLOGY SAFEGUARDING, ENGAGING AND ADVANCING QUANTUM R&D",
"TOWARDS RESPONSIBLE QUANTUM TECHNOLOGY SAFEGUARDING, ENGAGING AND ADVANCING QUANTUM R&D"
] | [
"Mauritz Kop ",
"Mateo Aboy ",
"Eline De Jong ",
"Urs Gasser ",
"Timo Minssen ",
"I Glenn Cohen ",
"Mark Brongersma ",
"Teresa Quintel ",
"Luciano Floridi ",
"Ray Laflamme "
] | [] | [] | The expected societal impact of quantum technologies (QT) urges us to proceed and innovate responsibly. This article proposes a conceptual framework for Responsible QT that seeks to integrate considerations about ethical, legal, social, and policy implications (ELSPI) into quantum R&D, while responding to the Responsible Research and Innovation dimensions of anticipation, inclusion, reflection and responsiveness. After examining what makes QT unique, we argue that quantum innovation should be guided by a methodological framework for Responsible QT, aimed at jointly safeguarding against risks by proactively addressing them, engaging stakeholders in the innovation process, and continue advancing QT ('SEA'). We further suggest operationalizing the SEA-framework by establishing quantum-specific guiding principles. The impact of quantum computing on information security is used as a case study to illustrate (1) the need for a framework that guides Responsible QT, and (2) the usefulness of the SEA-framework for QT generally. Additionally, we examine how our proposed SEA-framework for responsible innovation can inform the emergent regulatory landscape affecting QT, and provide an outlook of how regulatory interventions for QT as base-layer technology could be designed, contextualized, and tailored to their exceptional nature in order to reduce the risk of unintended counterproductive effects of policy interventions. | null | [
"https://export.arxiv.org/pdf/2303.16671v1.pdf"
] | 257,804,589 | 2303.16671 | 00b99bdcf70e849de709f75becc52badbed0abe6 |
TOWARDS RESPONSIBLE QUANTUM TECHNOLOGY SAFEGUARDING, ENGAGING AND ADVANCING QUANTUM R&D
Mauritz Kop
Mateo Aboy
Eline De Jong
Urs Gasser
Timo Minssen
I Glenn Cohen
Mark Brongersma
Teresa Quintel
Luciano Floridi
Ray Laflamme
TOWARDS RESPONSIBLE QUANTUM TECHNOLOGY SAFEGUARDING, ENGAGING AND ADVANCING QUANTUM R&D
Page 1 of 22
The expected societal impact of quantum technologies (QT) urges us to proceed and innovate responsibly. This article proposes a conceptual framework for Responsible QT that seeks to integrate considerations about ethical, legal, social, and policy implications (ELSPI) into quantum R&D, while responding to the Responsible Research and Innovation dimensions of anticipation, inclusion, reflection and responsiveness. After examining what makes QT unique, we argue that quantum innovation should be guided by a methodological framework for Responsible QT, aimed at jointly safeguarding against risks by proactively addressing them, engaging stakeholders in the innovation process, and continue advancing QT ('SEA'). We further suggest operationalizing the SEA-framework by establishing quantum-specific guiding principles. The impact of quantum computing on information security is used as a case study to illustrate (1) the need for a framework that guides Responsible QT, and (2) the usefulness of the SEA-framework for QT generally. Additionally, we examine how our proposed SEA-framework for responsible innovation can inform the emergent regulatory landscape affecting QT, and provide an outlook of how regulatory interventions for QT as base-layer technology could be designed, contextualized, and tailored to their exceptional nature in order to reduce the risk of unintended counterproductive effects of policy interventions.
Laying the groundwork for a responsible quantum ecosystem, the research community and other stakeholders are called upon to further develop the recommended guiding principles, and discuss their operationalization into best practices and real-world applications. Our proposed framework should be considered a starting point for these much needed, highly interdisciplinary efforts.
I. INTRODUCTION
More than a hundred years ago scientists discovered that the world at a very small scale behaves very differently from what we are used to in our daily lives. In its inaugural century, quantum science primarily concentrated on understanding the rules and principles that govern physical reality at the scale of atoms. During the first quantum revolution, the theory of quantum mechanics was developed and experimentally validated. The resulting quantum mechanical principles were then used to create first-generation (1G) quantum technologies (QT) such as transistors, lasers, and MRI. More recently, the rapid advances in nanotechnology, optics, high performance computer engineering, and communications have unfolded a myriad of new ways to measure, control, and utilize the quantum properties of light and matter.
We are currently witnessing a second quantum revolution, where quantum mechanical principles and 1G QT are employed to realize a second generation (2G) of quantum technologies. This generation of technologies directly harness quantum mechanical phenomena such as superposition, entanglement, and tunneling (Box 1). 1 The resulting 2G QT highlight the potential for quantum information science to develop into quantum technologies across several domains. 2 Applications of 2G QT include 1) simulating quantum systems to enhance our fundamental understanding of nature and its applications, 3 such as modelling chemical processes in drug development, 2) achieving unprecedented precision in measurement through quantum sensing and metrology, 4 such as Rydberg atom sensors and atomic clocks, 3) solving mathematical and computational problems beyond the reach of classical computing by using quantum computers to formulate and deploy quantum algorithms that leverage quantum superposition and entanglement, 5 and 4) constructing a new generation of secure communication systems. 6 During the early pioneering years of scientific discovery, there was no imminent need for researchers to engage directly with the ethical, legal, social, and policy implications of QT (Quantum-ELSPI). But as we see 2G QT move from pure science to application in real-world we must broaden our lens to considerer the development and use of QT in human and societal contexts. What will it mean for the law and other societal institutions? How should QT be developed and regulated? 7 The introduction of 2G QT in society raises important legal and regulatory questions pertaining to national and economic security, dual use, privacy, product safety and liability, intellectual property, fair competition, and equality. For example, quantum algorithms have the potential to break current cryptography protocols, threatening the information security and data privacy of its users, thereby destabilizing society and undermining trust in its institutions. In this paper we use information security as the illustrative study of the challenges and the application of our proposed framework for Responsible QT.
The current breadth, speed of maturation, and potential impact of 2G QT in human and societal contexts make it an urgent priority to engage with the emerging interdisciplinary research field of Quantum-ELSPI. 8 This approach can help to guide R&D and application of QT into a modus that is ethical and socio-economically sustainable, while also promoting responsible technological advancement and innovation. Steering towards beneficial societal outcomes, we propose a conceptual framework for Responsible QT that is informed by Quantum-ELSPI considerations and responds to key Responsible Research and Innovation (RRI) dimensions.
The article is structured as follows. Part II discusses what makes QT unique and defines quantum mechanical effects such as superposition, entanglement, tunneling, and quantization. Part III conceptualizes the Responsible QT paradigm and provides arguments as to why we need to proactively fill the current responsibility gap. After explaining RRI in terms of Quantum-ELSPI, Part IV then argues that quantum innovation should be guided by a framework for Responsible QT, aimed at jointly safeguarding against risks by proactively addressing them, engaging stakeholders in the innovation process, and continue advancing QT ('SEA'). We further suggest operationalizing the SEA-framework by establishing ten quantum-specific guiding principles. Part V illustrates the importance of such practices by examining the example of information security in the postquantum era, recommending that research on and development of QT should be accompanied by risk-based quantum impact assessments focused on information security risks and implementing 5 quantum-safe information security controls to mitigate such risks. Part VI analyses how our proposed SEA-framework for Responsible QT can inform the emergent regulatory landscape affecting QT, taking as examples the two recent Executive Orders signed May 4 2022 by President Biden, and the Quantum Computing Cybersecurity Preparedness Act that became public law on December 21 2022. Laying the groundwork for a responsible, values-based quantum ecosystem, the conclusion calls upon the collaboration of multidisciplinary teams of diverse quantum stakeholders to discuss and orchestrate normative dimensions of QT futures, and pathways to build towards them.
II. WHAT MAKES QUANTUM TECHNOLOGIES UNIQUE?
For the last 70 years, the transistor has been the fundamental building block in our electronic devices to enable complex computations and information manipulations. It has had a profound impact on the way we work, interact, and think. The rules of how information can be manipulated with classical devices are ingrained in classical physics, which limits the precision of sensors and the problems we can solve efficiently with classical computers. This latter part is encoded in the Strong Church-Turing thesis at the basis of today's classical computer science. 9 This thesis stipulates that all computers are born (roughly) equal because any real-world computation can be translated into an equivalent computation involving a Turing machine, i.e. if a problem is a hard one and thus requiring an exponential amount of computational resources, it will be hard for all computers. 10 For more than 80 years this fundamental tenet was accepted as true, and it is this tenet that QTs challenge. 11 The technology community was surprised when it was demonstrated that quantum mechanics for computing could solve problems for which we do not have efficient classical algorithms such as finding the prime factors of large composite integers. 12 These computationally hard problems for classical computers are the basis of our public-key cryptography infrastructure that secures internet communications, but quantum algorithms have been discovered that could solve these classically intractable problems (e.g., Shor's algorithm for prime factorization). 13 In general, second generation (2G) QT directly harness quantum mechanical phenomena such as quantum superposition, entanglement, and tunneling (discussed in Box 1) to achieve quantum advantage over state-of-the-art technologies in both qualitative and quantitative ways. 14 This includes 9 For an accessible explanation of the Strong Church-Turing thesis for a broader public, see Copeland, B. Jack, "The Church-Turing Thesis", The Stanford Encyclopedia of Philosophy (Summer 2020 Edition), Edward N. Zalta (ed.), https://plato.stanford.edu/archives/sum2020/entries/church-turing/ 10 deploying quantum algorithms for classically intractable problems, simulation of quantum systems beyond the capabilities for our most powerful classical supercomputers, quantum sensing that can sense weak forces due to higher sensitivity and spatial resolution unachievable by classical sensors, and secured quantum communications. In other words, QT take a fundamentally novel approach to computation, sensing and communication, coming with potentially disruptive effects. Imagine the consequences of a first mover being able to crack our ubiquitous RSA encryption and similar public-key cryptosystems in a few seconds by implementing Shor's algorithm in a sufficiently faulttolerant quantum computer. QT are categorically different from the technological improvements that we have seen over the last 70 years because of the nature of the advantages that quantum dominance could entail. Box 1: Superposition, Entanglement, Tunneling, and Quantization in QT Superposition: a concept in quantum mechanics that states that a particle can exist in multiple states at the same time. Superposition is directly harnessed in quantum computing to achieve quantum advantage over classical computation. In a classical computer, data is represented as bits, which can have one of two states 0 or 1. In a quantum computer, quantum information is unfolded by quantum bits (qubits), which can be in a |0> state or in a |1> as well, but also in a superposition of both basis states as a linear combination |ψ>= α|0>+β|1> where α and β are complex numbers corresponding to probability amplitudes. Thus, a quantum computer consisting of n qubits can exist in a superposition of 2 n states enabling unprecedented parallel computation.
Entanglement: a phenomenon in which two or more particles interact in such a way as to become mutually dependent on one another, even when separated by great distances. When two systems are entangled there exists a special connection between them. If two qubits are entangled, it means that, on measurement the results are strongly correlated even if the qubits are physically separated across great distances. For instance, if the first qubit is measured to be in state |0>, the second entangled qubit will also be found to be in state |0>. Entanglement is directly harnessed in quantum computing along with superposition to achieve quantum parallelism resulting in quantum algorithms with exponential speed-up over classical computations. It is also harnessed in quantum communications by taking advantage of unique correlations exhibited by entangled qubits and quantum cryptography, particularly quantum key distribution (QKD).
Tunneling: the ability of quantum systems to go across an energy barrier. A phenomenon in which a particle can pass through a barrier that it does not have enough energy to surmount. Tunneling is directly harnessed in quantum annealing/adiabatic computation to solve quantum simulation and optimization problems. It is also used in applications such as scanning tunneling microscopes, tunneling diodes, and quantum sensors. As the transistors (MOSFETS) used in our classical computers get smaller, they also exhibit quantum mechanical tunnelling from source to gate oxide due to the thickness of the oxide layers and quantum mechanical tunnelling from source to drain when the channel lengths are less than 10 nm. While in classical technologies at the smallscale such as MOSFETS quantum effects are often sources of imperfections to avoid, in 2G QT these quantum mechanical effects are directly harnessed to achieve quantum advantage.
Quantization: a fundamental principle of quantum mechanics that is used to describe the behavior of physical systems at the atomic and subatomic level. The allowed energies of a tightly confined system of particles at quantum scales are restricted to a discrete set. It is an essential part of many technologies including lasers and the physical realization of qubits no matter which qubit approach one takes, such as trapped ions, atoms, photons, quasi particles, or superconducting oscillator circuits. In order to function as a qubit, the discrete energy levels of the superconducting circuit must be carefully controlled and manipulated.
III. CONCEPTUALIZING RESPONSIBLE QUANTUM TECHNOLOGY
We posit that the exceptional nature of QT demands proactive integration of ELSPI considerations throughout the entire QT R&D lifecycle. The potential game changing character of QT, unlocking novel approaches in both research and innovation, comes with the expectation that it will affect our world in a myriad of ways. 15 Pertinent examples of these challenges concern national and economic security, dual use, privacy, product safety and liability, intellectual property, fair competition, and equality.
While we should build upon successes and failures from dealing with -and establishing responsible technology frameworks for-closely related fields such as AI, nanotechnology, biosciences, semiconductors, and nuclear, 16 the exceptionality of 2G QT and their expected societal impact including the nature of the advantages that quantum dominance could cause, demand a tailored approach. 17
It is this expected impact, stemming from the fundamentally novel qualities of QT itself 18 , that urges us to proceed responsibly. 19 A first key step thus is a reflection on that very concept: What does responsible research and innovation amount to in the context of QT? A conceptualization of Responsible QT serves as a touchstone for the nascent Quantum-ELSPI domain.
Founded on the concept of Responsible QT, we call for integrating ELSPI considerations within quantum innovation, taking the societal context into account early on. We propose an overarching framework for responsible quantum innovation (See Fig. 1) and provide suggestions for its operationalization by establishing quantum-specific guiding principles. 20 Our proposition is aimed at researchers, developers, innovators, and regulators, but it may also inspire other stakeholders. The proposed Responsible QT approach should be considered as a starting point for much needed highly interdisciplinary efforts (Box 2).
The concept of Responsible QT is aimed at ensuring that ethical, legal, socio-economic, societal, and philosophical dimensions are identified and discussed while QTs are still shapeable. 21 From a normative perspective, the objective is to capture and promote beneficial opportunities expected from quantum innovation while managing potential downside risk by putting into place technical, organizational, and policy measures appropriate to the risk. At present, many QT applications, such as large-scale fault-tolerant quantum computers 22 or the Quantum Internet are still in the basic research stage, and indeed many societal implications remain unknown. 23 Other applications, such as noisy intermediate-scale quantum (NISQ) computers, and in silico design of new catalysts, materials, and pharmaceuticals by using scalable quantum simulation of molecular energies, are at higher technology readiness levels (TRL). 24 Exactly because of the early stage of the technology and its far-reaching potential, we have a shared opportunity and responsibility to shape its development toward desirable societal outcomes. 25 Responsible QT urges that QT are designed, produced, marketed, and protected in an ELSPI-sensitive manner. The need for Responsible QT becomes strikingly evident when examining prospective trajectories where QT software and hardware structures are developed and commercialized without such considerations. This includes use cases pertaining to national defense and commercial security threats, unwanted disclosure of trade and state secrets, large scale privacy loss, and winner-takes-all dynamics. The next section explores how one could fill these identified responsibility gaps.
IV. A FRAMEWORK FOR RESPONSIBLE QUANTUM INNOVATION
Responsible Research and Innovation (RRI) is an approach that aims to ensure that scientific and technological developments are carried out in a way that is socially desirable, ethically acceptable, and sustainable. 26 RRI is a process that involves a continuous dialogue between researchers, citizens, industry, policy makers, and other stakeholders, in order to actively anticipate and assess the potential social and ethical implications of research, development and innovation.
The concept of RRI emerged as a response to the growing recognition that scientific and technological progress can have unintended negative consequences for society and should be guided by ethical and social considerations. RRI aims to integrate these considerations, norms, and values, into all stages of the R&D and innovation process, from the design of research projects to the dissemination of results.
In accordance with the European Commission, four important dimensions of RRI are anticipation, inclusion, reflection, and responsiveness. 27 Anticipation entails identifying and addressing potential social and ethical issues that may arise. Inclusion implies involving a wide range of stakeholders in the innovation process, including those who may be affected by the outcomes. Reflection concerns ongoing evaluation of the values and assumptions that underpin research and innovation, and considering how they may influence the outcomes. Responsiveness implicates reacting to the feedback and concerns of stakeholders and adapting accordingly. In sum, RRI is a way of ensuring that research, development, and innovation are aligned with the needs and values of society, and that they contribute to the common good.
In response to these key RRI dimensions, we argue that Responsible QT calls for quantum innovation that proactively addresses risks, takes emerging challenges heads-on, and seizes the opportunities that arise with QT. 28 This results in what we call the 'SEA-framework for Responsible Quantum Technology', aimed at safeguarding against risks, engaging stakeholders in the innovation process and advancing QT. 29 Safeguarding, engaging, and advancing QT in unison constitutes the triumvirate of Responsible QT. These objectives can be pursued at three, interrelating levels: (1) the technical level, (2) the ethical level, and (3) the societal, legal and policy level (Fig. 1). 30 The technical level involves setting standards, accountability, and governance mechanisms for QT. The ethical level provides the foundational principles and criteria that should guide both the technical controls as well as the QT societal, legal and policy level, which focuses on the societal impacts of legal frameworks and related policy decisions. Other technologies, including nuclear, biotech, and artificial intelligence (AI) have shown the complexity of developing principles, standards, and regulatory frameworks. 31 Despite its unique characteristics and potential impact, such effort has not been undertaken for QT yet. The methodological framework proposed here seeks to provide a starting point for this highly interdisciplinary endeavor.
Before elaborating on the SEA-framework, we should reflect on three issues in advance. First, as with any ethical framework, categories and principles can conflict. This is inherent to their value ladenness and there is no panacea to avoid tensions. For example, deontological (defining the moral course of action take in terms of right and duties), utilitarian (defining the right in terms of maximizing the good, and in the instance of classic utilitarianism the greatest happiness irrespective of its distribution and understood in terms of pleasure and pain) and consequentialist (justifying actions in the context of results) approaches in ethical decision-making may inform different outcomes. 32 Instead of considering this a weakness of the framework, such tensions reflect the complexity of the real world. Possible conflicts should thus lead to explicit discussions about how to balance different aims and stakes.
Second, the envisioned principles are not necessarily exclusive to QT. By 'Principles for Responsible Quantum Technology' we aim at principles that adequately respond to the SEA-objectives in the context of QT. The resulting set can thus consist of a mix of quantum specific principles and more general principles that are particularly relevant to quantum innovation. 33 Third, we do not intend the framework we propose to be translated into binding law, at least not initially. Its effectiveness will therefore consist in voluntary commitments, at first. For those who seek to practice RRI in the context of QT, it can serve as a useful point of departure. In time, a framework like this could become an effective basis for self-regulation or evolve into instruments of soft and -eventually -hard law, such as a Quantum Governance Act.
As shown in Figure 1, such combined quantum-specific and general principles could be categorized into an overarching framework with distinct functional categories that balance the need to support, protect, and incentivize QT advancements with the need to establish appropriate safeguards, while engaging society. In other words, the technical, ethical, and social/legal/policy levels can be visualized as layers on which the principles that guide responsible quantum innovation can be formulated per SEA category. (1) Safeguarding-principles should address the downside risks arising from QT (Fig. 1). They aim to protect society by taking appropriate measures. 35 For instance, two safeguarding principles could be: i) consider information security as an integral part of QT R&D and ii) proactively anticipate the potential malicious dual use of QT applications. Operationalizing these principles would promote approaches such as implementing risk-based quantum impact assessments -overseen by product and program managers-to minimize security threats, performing risk-reward analysis before granting production and market authorization, or reducing the risks that QT can be used for harmful purposes. Examples of these so-called dual uses include (1) the use of quantum simulation to help develop new drugs, fertilizers or industrial catalysts versus the manufacturing of new biological or chemical weapons using the same foundational technology involving scalable quantum hardware, and (2) using photonics and plasmonics to miniaturize quantum devices to improve energy efficiency -benefitting the planet, versus utilizing miniaturization in quantum sensors to watch people's every move -infringing core human rights. 36 (2) Engaging-principles should enhance societal engagement of QT innovators and other stakeholders (Fig. 1). They should address issues such as the threat of a deepening quantum gap among countries, the need for adequate intellectual property (IP) and fair competition mechanisms for QT, and pursue building diverse quantum communities. 37 For example, IP in this context should be calibrated in a manner that balances incentives, rewards, access, and risks, while addressing associated geostrategic and national security concerns. Moreover, IP policies should work in concert with tailored antitrust regulations to prevent unwanted winner-takes-all market behavior within the emerging quantum ecosystem. 38
(3) Advancing-principles should progress society through QT-based innovation (Fig. 1). Advancing principles should both promote QT R&D -creating a virtuous cycle of progress that fueled Moore's law similar to that of the semiconductor industry-and encourage QT applications for desirable social goals. 39 The 17 UN's Sustainable Development Goals 40 -which are a call for action by both developed and developing countries to adopt a global 2030 agenda for peace and prosperity -might serve as an illustration for objectives to which QT can contribute to areas such as drug discovery, resource optimization, water management, 21-day weather forecasting to improve agriculture, and 35 See for a call to action to address risks arising from QT, Khan, I, Will Quantum Computers Truly Serve Humanity? SCIENTIFIC AMERICAN, 2021, https://www.scientificamerican.com/article/will-quantum-computers-truly-serve-humanity/. 36 40 See https://sdgs.un.org/goals climate modeling. Additionally, QT could help to enhance complementary innovation including quantum-classical synergies such as the variational quantum eigensolver (VQE), and quantum-AI hybrids such as quantum machine learning and specialized quantum co-processors for AI that solve optimization problems that are typically hard for classical systems. 41
Balancing safeguarding, engaging, and advancing (SEA)
In general, engaging principles should help guide and promote a fit-for-purpose QT regulatory framework that balances the safeguarding and advancing principles to promote quantum innovation. Engagement principles should encourage inclusivity and competition. As such, they interact with safeguarding and advancing principles. For instance, engaging principles should help manage the risks of a few dominant QT players achieving monopolistic competitive advantage by restricting access to essential QT infrastructure (safeguarding). That said, the engagement principles should also interact with advancing principles to help promote access to key QT infrastructures by encouraging open access to cloud-based quantum computers, open quantum interoperability standards and protocols, and open-source quantum development tools. 42 Such initiatives lower the education barriers of entry to build a skilled quantum workforce by empowering those with an internet connection to start working with quantum computers at the level of pulses, gates, circuits, and application modules. The ability to program and prototype quantum algorithms using Python, open quantum assembly language (QASM), or utilizing a graphical circuit composer to implement quantum algorithms on a real quantum computer accessible through the cloud, substantially increases engagement and promotes inclusivity. Here, access encourages creativity, experimentation and simply finding out what works and what does not. Finally, engaging principles should encourage awareness of relevant QT issues in society through general education and crosssector dialog with stakeholders across all layers of society, including QT developers, investors, regulators, and the public. 43 Articulating the foundational principles for Responsible QT clustered along the proposed three functional dimensions could be inspired by established best practices of responsible research and innovation (RRI). 44 The proposed framework would need to be operationalized by guiding principles that should be incorporated into the design, architecture, and infrastructure of QT systems, products, and services on a global scale, resulting in Responsible QT by design and default. 45 Crucially, this requires translating the principles into specific R&D design decisions, continuously testing, benchmarking, and validating results, verifying their usefulness, adjusting where deemed 41 This includes quantum/AI hybrids and using quantum resources in AI such as quantum assisted machine learning, hybrid cloud computing, quantum-classical interfaces, and quantum/AI simulation on classical systems. As quantum computing could be a major boost for AI it is necessary to proactively address present day ethical problems pertaining to AI such as bias, representativeness, black boxes, and polluted data, in order to ensure that quantum computing doesn't exacerbate these problems. Here too, our proposed framework can play an important anticipatory role, complementing existing frameworks for AI. That said, these AI issues are primarily related to the data aspects (e.g., the coverage and generality of the training data) as opposed to the computational aspects. appropriate. Concretely, designing quantum computing and sensing hardware architectures, creating quantum software as a service (SaaS) platforms, formulating quantum algorithms, and building a future Quantum Internet must each adhere to the principles of Responsible QT and their underlying norms, standards, and values, aiming for a responsible quantum ecosystem and fostering sustainable innovation.
Connecting both RRI dimensions and ELSPI considerations to responsible quantum R&D, categorized along the lines of the SEA-framework's trifecta of Safeguarding, Engaging, and Advancing QT, this catalogue of principles for Responsible QT could be imagined as follows: 46 As future QT scenarios remain largely unknown, epistemic modesty must be embedded throughout this process. Put differently, as the field is still in its infancy, the framework might have to be adapted to new discoveries. Operationalizing these guiding principles would therefore call for continuous collaborative multi-stakeholder and industry efforts that follow and steward the life cycle of QT systems, products, and services and anticipate novel quantum use cases, 47 for instance involving standard-setting organizations such as ISO and NIST, as well as professional organizations such as the IEEE. Research institutions and research-driven companies would need to dedicate resources in the form of IRB-like bodies (institutional review boards), assessing the ethical implications of a particular QT. 48 Ideally, such the results of such assessments and oversight mechanisms are openly shared to advance a collective learning process and inform evidence-based policy making.
Consider information security as an integral part of QT
Training and continued education programs should help to further develop and implement the principles and drive QT innovation across economic and industrial sectors, such as biopharma, energy, mining, communications, logistics, defense, and space. Guided by the proposed set of principles, these networked communities of practice would identify and address ELSPI challenges in their respective R&D and application contexts, including dual uses of QT, IP, fair competition, product safety and liability, inclusion, complementary innovation, equitable distribution of QT's benefits, innovation externalities, spillovers and trade-offs, capability overhang, and the global quantum race. 49 The importance of such practices can be demonstrated by having a closer look at the example of information security in the post-quantum era.
V. ILLUSTRATING THE CHALLENGE: INFORMATION SECURITY IN THE POST-QUANTUM ERA
Quantum algorithms have the potential to break current cryptography protocols, 50 threatening the information security of existing information technologies (IT) and the privacy of its users. 51 This could destabilize society and undermine trust in its institutions. QT could expose extensive swaths of information currently regarded as private and confidential, ranging from sensitive personal data to financial sector and national security information assets. 52 Concretely, we already have quantum algorithms capable of breaking our widespread public key cryptosystems as soon as the quantum computer hardware is sufficiently mature. Thus, we can foresee potentially disruptive effects to fundamental human rights such as privacy and data protection 53 , including large scale loss of privacy, human identity theft, loss of confidentiality and integrity of digital communication on the Internet, obstruction of commercial transactions, leaking of highly sensitive trade and state secrets, and other unwanted global surveillance disclosures. 54 Given that information security threats are among the most pressing themes in the context of Quantum-ELSPI, one of the safeguarding principles should promote information security to be a central feature of Responsible QT. That said, the challenge is to achieve appropriate safeguarding while continuing to promote advancement of QT such as quantum computing (QC). After all, QC holds great promises for societal benefits because quantum algorithms can solve hard computational problems that are mathematically intractable for classical computers. However, as explained above in section 2, these same problems that are hard for classical computers to solve have been selected as a fundamental building block of our widespread public key cryptosystems precisely because they were believed to be computationally intractable.
The possibility of implementing a "store now, decrypt later" strategy should provide incentives to start replacing our existing cryptosystems for critical information assets with quantum-safe cryptographic systems that are resistant to attacks by quantum algorithms as early as possible. It could take years to replace our cryptosystems. In fact, it has taken nearly two decades to deploy our modern asymmetric key cryptographic infrastructure. Thus, once sufficiently fault-tolerant quantum computer hardware is available, it could be used to reveal information assets previously encrypted with our most common forms of public key cryptography. This calls for researching and investing in quantum-safe cryptography initiatives, as well as advancing and engaging in quantum-safe information security programs. 55 From a safeguarding standpoint we must ensure that the field of post-quantum cryptography is on par, or ahead (advancing) of the realization of a fault-tolerant or error-corrected quantum supercomputer with enough qubits to implement efficient decryption algorithms that leverage quantum parallelism to break our widespread cryptosystems. In short, large scale physical realizations of quantum computers capable of deploying Shor's algorithm to break RSA-2,408 should be paralleled by the development of quantum-resistant cryptographic algorithms. Such algorithms must be safe and secure against cryptanalytic attacks by QC that employ quantum algorithms breaking common asymmetric (public) key cryptographic systems in use today. 56 This is accomplished by exploiting mathematical problems that are computationally intractable, i.e. for which we do not have a known efficient solution using classical or quantum computers. NIST has initiated a broad engaging process to solicit, evaluate, and standardize quantum-resistant public-key cryptographic algorithms. 57 The goal is to create cryptographic systems that are secure against both classical computers and quantum computers. 58 Additionally, these cryptographic systems should be able to interoperate with existing protocols and networks.
From an advancing standpoint, it is noteworthy that QT developments can also help to improve security and data privacy in the sense of safe transfer of information. 59 Some use cases already promise to benefit from quantum cryptography advances such as quantum key distribution (QKD) to achieve secure communications by implementing quantum-based cryptographic protocols to exchange symmetric keys. QKD uses fundamental quantum-mechanical properties to securely exchange keys over an unsecure public channel. Additionally, several protocols have recently emerged to enable private quantum computation. These protocols, such as blind quantum computation (BQC) are designed to secure computation rather than communication. 60 At the operational level, research on and development of QT should be accompanied by risk-based quantum impact assessments focused on information security risks and implementing controls to mitigate such risks. This includes the implementation of state-of-the-art information security management systems (ISMS) such as ISO27001 to protect information assets from a particular QT R&D program. Notably, this requires extending the ISO 27001/27002 controls to include the implementation of quantum-safe information security controls.
VI. CHARTING THE PATHWAY FORWARD: EMERGING REGULATION OF QUANTUM TECHNOLOGY
This section briefly examines how our proposed SEA-framework for Responsible QT can inform the emergent regulatory landscape affecting QT, taking the two recent Executive Orders signed May 4, 2022, by President Biden, and the Quantum Computing Cybersecurity Preparedness Act that became public law on December 21 2022, as examples.
The framework can contribute in three analytically distinct, yet practically intertwined ways. First, it offers an analytical lens to help examine the role, purpose, and process dimensions of emerging regulations, whether general in nature or specifically tailored to QT, to identify and examine such issues in a respective application context and regulatory field. As noted, along recent QT advancements comes the need for careful consideration of legal, regulatory, ethical, and policy issues to ensure that these technologies are safe and ethically sound. 61 This is an area where the SEA-framework can offer not only substantive anchors, but also procedural guide shared among various QT stakeholders to unlock the potential of 'regulation as facilitation'. 62 Regulation of novel technologies should be flexible and dynamic to foster innovation while still providing adequate safeguards to protect against potential harms. Both our dimension of engagement and the concept of 'Regulation as facilitation' emphasize collaboration between regulators, industry, and other stakeholders. They help ensure technology is developed in a way that maximizes their potential benefits while minimizing risks and avoiding unnecessary regulatory burdens.
Second, the SEA-framework can serve as a reference point to evaluate or assess regulatory initiatives regarding the critical question as to what extent such regulatory endeavours strike a balance between safeguarding, engaging, and advancing QT. For instance, it might reveal to what extent proposed future QT regulation in the EU might follow the approach taken in AI by putting strong emphasis on safeguarding QT fueled by the precautionary principle, compared to the US permissionless innovation approach, which is expected to emphasize advancing to retain global tech leadership 63 by actively pursuing a democratic values-based quantum ecosystem in a US-led liberal global order. 64 Here, the frameworks' third contribution becomes visible: It offers at least some initial guideposts of how regulatory interventions for quantum as a base-layer/general-purpose technology could be designed, contextualized, and tailored to their exceptional nature, balancing open innovation and risk control. 65 Specifically, the proposed SEA-framework offers clues on how to balance safeguarding and advancing QT, aiming for Responsible QT by design and default. Similarly, regulating QT can be thought of as a balancing act between under-regulation and over-regulation. 66 The framework can be helpful to raise awareness of important issues in this context. The success of this joint optimization will also depend on the strength of the framework's engaging component which emphasizes the importance of ongoing dialogue between regulators, experts, and industry to ensure regulation remains effective and responsive to emerging technological developments. Without such engagement it is likely that the regulatory focus would seek to primarily optimize one of these dimensions (e.g. safeguarding). The issue with such approach is that optimizing for QT safeguarding would likely result in less safety, since adequate safeguarding is dependent on further QT advances, for instance, in quantum cryptography.
Taken together, the suggested SEA-framework for Responsible QT and its operationalizing principles might inform current and future regulations that share the challenge of optimizing among the different dimensions and issues detailed in this article. Creating horizontal (federal, or international level) norms for QT as a base-layer technology 67 that apply across industrial and economic sectors is a pressing tasks that should be taken on before the technology becomes locked in. 68 But it is also a challenging task, as any regulation must consider the exceptional, counterintuitive traits of applied quantum physical phenomena, its unseen functionality, and their potential for dual use 69 -balancing open innovation, value appropriation, IP, fair trade and competition, and risk control. 70 In addition, the inherent uncertainty underpinning emerging QT 71 applied to real world systems, products, and use cases (e.g., quantum simulation of molecular physics and biochemistry across a wide range of dual-use applications), calls for a risk based approach that incentivizes sustainable innovation in parallel, e.g. via regulatory sandboxes that afford breathing room for experimentation and prototyping. 72 To the extent the proposed SEA-framework for Responsible QT is informed by the technical and physical underpinnings of QT, it might in turn help to ensure that emerging rules and codified laws for QT live up to the same challenge.
Regulatory Approaches to the Emerging Quantum Technology Landscape
The previous section has described in the abstract how the SEA-framework for Responsible QT can inform emerging efforts aimed at regulating QT by offering analytical, evaluative, and design baselines. Recent developments in the US that might mark some of the cornerstones of the future regulatory landscape can serve as a use case to illustrate how the Responsible QT framework might be used in practice when examining emerging QT legal and regulatory norms and approaches.
Recent building blocks of what might be an emerging legal framework include two Executive Orders signed on May 4 2022 by President Biden. The directives are aimed at advancing US quantum information science (QIS) by laying "the groundwork for continued American leadership in an enormously promising field of science and technology […] to foster these advances by furthering the President's commitment to promoting breakthroughs in cutting-edge science and technology," safeguarding "while mitigating the risks that quantum computers pose to America's national and economic security […]", and engaging "it does so by enhancing the National Quantum Initiative Advisory Committee, the Federal Government's principal independent expert advisory body for quantum information science and technology." 73 The safeguarding strategy includes both elements to address the direct proximate risks "plan to address the risks posed by quantum computers to America's cybersecurity", as well as broader risk-based considerations surrounding American IP by urging "Federal agencies to develop comprehensive plans to safeguard American intellectual property, research and development, and other sensitive technology from acquisition by America's adversaries, and to educate industry and academia on the threats they face". Thus, the directives effectively connect IP protection to national and economic security strategy, while emphasizing the importance of advancing responsible and secure R&D in quantum computing. 74 In addition, the Quantum Computing Cybersecurity Preparedness Act of December 2022 requires The Office of Management and Budget by law to give priority to federal agencies' purchases of and transitions to post-quantum cryptographic IT systems, with the goal of safeguarding through advancement of postquantum cryptography. 75 Together with the bipartisan supported America Competes Act and the CHIPS and Science Act 7677 which are applicable to semiconductors including those used in QT, these Acts aim to strengthen supply chains, and counter US adversaries such as China. 78 Collectively, these directives intend to lay the groundwork for continued American competitiveness and leadership in QT, fostering innovation while mitigating risks associated with dual use quantum technology by instigating targeted export controls. See https://www.whitehouse.gov/briefing-room/statements-releases/2022/08/09/fact-sheet-chips-and-science-act-willlower-costs-create-jobs-strengthen-supply-chains-and-counter-china/ and https://dean.house.gov/2022/2/the-house-passesthe-america-competes-act 79 See in this context also Implementation of Certain New Controls on Emerging Technologies Agreed at Wassenaar Arrangement 2018 Plenary, Federal Register 2019, When analyzing and assessing these regulations through the looking glass of the proposed SEAframework for Responsible QT, various components crystallize that map onto three SEA categories. For example, catalyzing investments in domestic advanced semiconductor manufacturing capacity and instigating export controls can be considered an attempt to jointly optimize both safeguarding and advancing US competitiveness in quantum computing. Similarly, the emphasis in investing in post-quantum cryptography illustrates how safeguarding can be achieved by advancing, as opposed to attempting to achieve safeguarding objectives by limiting technological development. Connecting intellectual property strategy to national security policies also illustrates the overarching aim of jointly optimizing the advancing, safeguarding and engagement dimensions. 80 These regulatory objectives to map our suggested catalogue of 10 principles for Responsible QT listed above in section IV, e.g. Principle 5: 'Be as open as possible, and as closed as necessary'. 81 Further, section 1 of the Executive Order on Enhancing the National Quantum Initiative Advisory Committee aims to enable knowledge transfer between industry, academia and government to ensure American leadership in QT, especially quantum information technologies. 82 This relates to the SEA categories of engaging and advancing. The goal is to achieve QT leadership (advancement) through engagement. Section 1 b of the National Security Memorandum on Promoting United States Leadership in Quantum Computing While Mitigating Risks to Vulnerable Cryptographic Systems focuses on the significant risks QC potentially poses to the economic and national security of the United States, which ties to the SEA category of safeguarding. 83 The purpose of the Quantum Computing Cybersecurity Preparedness Act is 'to encourage the migration of Federal Government information technology systems to quantum-resistant cryptography, and for other purposes.' 84 While this mainly qualifies as safeguarding society as described in our case study of information security in the postquantum era, it aims to achieve this safeguarding primarily through advancement in the field of postquantum cryptography. This includes R&D in quantum-based solutions (i.e. protecting against quantum-crypto attacks using quantum technologies), as well as the search for cryptosystems based on problems that are computationally intractable to both classical and quantum computers. Thus, the goal should be to achieve the safeguarding objectives by further advancing QT in the sense that novel algorithms, software, and hardware solutions will have to be developed, increasing economic growth and competitiveness. In addition, the Act offers a chance to build a multidisciplinary, intergenerational quantum workforce as suggested by Principle 6 above, stating to: 'Pursue diverse quantum R&D communities in terms of disciplines and people, engaging people'. Notably, quantum computing brings together the more established American workforces in quantum physics, chip design, and semiconductor manufacturing that has led in global competitiveness from 1950 to the 2000s, with the innovative software workforce that has led the major high-tech developments over the last 20 years.
When moving from the analytical and evaluative dimensions of the SEA-framework to the prospective question of design, additional considerations might come into play. Given the current lack of comprehensive best practices in terms of Responsible QT-oriented policymaking, and considering the need for policymakers to make decisions anyway and under conditions of uncertainty, the framework's normative power to provide specific substantive guidance is limited. However, as discussed in Section IV, the framework alludes to a broad range of options and approaches that policymakers can embrace to steer the development of Responsible QT environments, which offers at least minimum guidance in terms of "asking the right questions" considering the full options available and probing their respective SEA implications. The framework also helps avoid potential detrimental policies-mistakes such as merely optimizing a single dimension (e.g. trying to achieve safeguarding objectives through unbalanced regulatory efforts that hinder further technological development instead of promoting and leveraging QT advancements in order to achieve these underlying safeguarding goals). Further, the framework highlights the important role self-regulation might play across technical, ethical, and societal levels by prescribing pro-active measures such as open-source quantum software and hardware movements, quantum impact assessments, and technological measures to safeguard human rights and freedoms.
Moreover, the SEA-framework indicates that such self-regulatory approaches might work in concert with "hard law", for instance in gestalt of an emerging legal and infrastructural ecosystem consisting of overarching horizontal rules for a particular general purpose technology -analogous to the EU AI Act -flanked by industry-specific legal frameworks. This could cumulate in a binding (International) Quantum Governance Act, or a Global Quantum Treaty 85 , to be enforced by hyperspecialized overseeing bodies such as the FDA or designated notified bodies. Such an extensive set of QT standards would have to interoperate with other areas of the legal system and embedded in existing regulatory structures. 86 This raises some tensions, as we find ourselves in a continuum moving from classical to quantum, where interwoven physical characteristics and legal designations are a matter of degree, and cannot be clearly separated from each other. For example, the (diffuse) technical classification of quantum information in the cloud as data in the classical sense, and defining a quantum/AI hybrid chip as both QT and AI device has consequences for applicable legal regimes. 87 Moreover, these future legal regimes pertaining to quantum information should ideally be applicable to across the range of QT, including quantum computing, simulation, sensing, and communication. 88 The SEA-framework can also inspire ways in which these self-regulatory and "hard laws" could be complemented by operational instruments derived from existing quantum use cases. The framework and its principles point towards a risk-based governance environment, with standardization, certification, production and market authorization, benchmarking, quantum quality management systems (QMS), and life cycle auditing expected to play an important role in fostering sustainable innovation. In this way, equitable access can be ensured, while putting targeted controls and guardrails in place that safely enable scalability of quantum technology, benefitting society at large. 89
VII. CONCLUSION AND FURTHER RESEARCH
This article proposes a framework for Responsible QT that integrates ethical, legal, social, and policy implications into quantum R&D. The aim is to safeguard against risks, engage stakeholders, and continue advancing QT. The proposed SEA approach emphasizes anticipation, inclusion, reflection, and responsiveness as key dimensions of responsible research and innovation. It is aimed at researchers, developers, innovators, investors, regulators and other stakeholders who are involved in the development and commercialization of QT. Our proposed framework should be considered as a starting point for highly interdisciplinary efforts to ensure that ethical considerations are identified and discussed while QT are still shapeable. Overall, we highlight the importance of responsible innovation in the development of quantum technology to ensure its societal impact is positive.
The overarching objective of our interdisciplinary Responsible QT effort is to steer the development and use of QT in a direction not only consistent with a values-based society, but also contributive to addressing some of humankind's most pressing needs and goals. The potential global impacts of QT urge to promote responsible innovation producing responsible technologies, informed by ELSPI considerations.
Following an anticipatory approach that synchronizes precautionary measures with permissionless innovation, a first key step is the conceptualization of what responsible QT should mean and imply. This contribution aims to provide such an initial conceptualization by proposing the contours of a framework to balance safeguarding, engaging, and advancing QT.
We suggested to translate this framework into guiding principles for Responsible QT. Some of these principles will generally go for new and emerging technologies, and some will specifically apply to QT. 90 It is the complete set, however, that should provide guidance to quantum innovation 88 If a general regulatory approach across QT domains is not possible or presents material challenges, a targeted approach should be pursued in the form of quantum domain, or industry specific rules. In that scenario different vertical rules would apply for quantum sensor data than for quantum computer input and output data. This would make sense from a QMS perspective with different applications having their own designated safety and security regimes, comparable to a PC having different CE marking requirements than a drone. In addition, core overarching horizontal rules could apply, analogous to universal human rights of privacy, integrity of the person, and freedom of speech, and inspired by the SEA-framework for responsible quantum innovation's principles. 89 and lead towards Responsible QT. 91 The envisioned principles should thus be understood and applied as a QT-tailored methodological framework that both complements existing frameworks for responsible innovation and translates them to the context of QT. 92 Additionally, these principles should be applicable to technological synergies such as quantum-classical interactions and quantum-AI hybrids. 93 What's more, we envision the principles to inform debates about regulatory interventions in this context, and reduce the risk of unintended counterproductive effects of such policies.
The need for Responsible QT becomes evident when examining prospective trajectories where QT software and hardware structures are developed without adequate consideration of the various aspects of Quantum-ELSPI. We used the impact of quantum computing in information security as a case study to illustrate (1) why we need to consider Quantum-ELSPI and commit to Responsible QT, and (2) how the same ways we handled quantum computing using the proposed framework would be useful for Responsible QT more generally. 94 Looking forward, it will be necessary to complement our proposed framework with principles for responsible quantum innovation. This will require the collaboration of multidisciplinary teams of diverse quantum stakeholders. Although QT's consequences and impact remain largely unknown, we hope this contribution may serve as an open invitation for researchers, innovators, and regulators to discuss and orchestrate normative dimensions of QT futures, and pathways to build towards them.
OF CONTENTS I. INTRODUCTION ............................................................................................................................... 2 II. WHAT MAKES QUANTUM TECHNOLOGIES UNIQUE? .................................................................. 4 III. CONCEPTUALIZING RESPONSIBLE QUANTUM TECHNOLOGY .................................................. 6 IV. A FRAMEWORK FOR RESPONSIBLE QUANTUM INNOVATION ................................................... 7 V. ILLUSTRATING THE CHALLENGE: INFORMATION SECURITY IN THE POST-QUANTUM ERA .. 13 VI. CHARTING THE PATHWAY FORWARD: EMERGING REGULATION OF QUANTUM TECHNOLOGY ................................................................................................................................... 15 VII. CONCLUSION & FURTHER RESEARCH ....................................................................................... 20
22
See e.g., M. Brooks, What's next for quantum computing, MIT TECHNOLOGY REVIEW, (2023), https://www.technologyreview.com/2023/01/06/1066317/whats-next-for-quantum-computing/ 23 See on social implications of QT in connection to EDI-frameworks (Equity, Diversity, and Inclusion), G. Wolbring, Auditing the 'Social' of Quantum Technologies: A Scoping Review, SOCIETIES, 12(2), 41. (2022), https://doi.org/10.3390/soc12020041 24 See e.g., P. O'Malley et al., Scalable Quantum Simulation of Molecular Energies, PHYS. REV. X 6, 031007 (2016), https://journals.aps.org/prx/abstract/10.1103/PhysRevX.6.031007; and Simson L. Garfinkel and Chris J. Hoofnagle, ACM TechBrief: Quantum Computing and Simulation, ASSOCIATION FOR COMPUTING MACHINERY, NEW YORK, NY, USA (2022). 25 E. De Jong, Own the Unknown: An Anticipatory Approach to Prepare Society for the Quantum Age, DIGITAL SOCIETY, QUANTUM-ELSPI TC, 1, SPRINGER NATURE, (2022), https://link.springer.com/article/10.1007/s44206-022-00020-4 Topical Collection: https://link.springer.com/collections/eiebhdhagd 26 For framing and operationalizing RRI into stakeholder's everyday practice, see https://rri-tools.eu/research-community 27 This has been referred to as the AIRR framework. See J. Stilgoe, R. Owen, and P. Macnaghten, Developing a Framework for Responsible Innovation, RESEARCH POLICY 42 (9) (2013): 1568-1580, https://doi.org/10.1016/j.respol.2013.05.008
28 For further reading on the 6 RRI themes (public engagement, open access, gender equality, ethics, science education, and governance) in the context of quantum technologies, see Ten Holter, C., Inglesant, P., & Jirotka, M. Reading the road: challenges and opportunities on the path to responsible innovation in quantum computing, TECHNOLOGY ANALYSIS AND STRATEGIC MANAGEMENT, (2021), https://doi.org/10.1080/09537325.2021.1988070; and Inglesant, P., Ten Holter, C., Jirotka, M., & Williams, R. Asleep at the wheel? Responsible Innovation in quantum computing, TECHNOLOGY ANALYSIS AND STRATEGIC MANAGEMENT, 0(0),
Figure 1 .
1SEA-framework for Responsible Quantum Technology 34 33 The envisioned Principles for Responsible Quantum Technology would likely benefit from general principles that also apply to other technologies, such as AI, nano, biotech, and nuclear, see e.g., M. Kop & M. Brongersma, Integrating Bespoke IP Regimes for Quantum Technology into National Security Policy, August 8, 2021, Stanford Law School, Working Paper, https://law.stanford.edu/publications/integrating-bespoke-ip-regimes-for-quantum-technology-into-national-securitypolicy/; and Gasser & Almeida, supra note 30. 34 This methodological framework builds on the work of Gasser and Almeida on the layered model for technology governance, see Gasser & Almeida, supra note 30.
42 For further reading on quantum technology and standardization, see DeNardis, Laura, Quantum Internet Protocols (August 4, 2022), http://dx.doi.org/10.2139/ssrn.4182865 43 See e.g., Potomac Quantum Innovation Centre, Our Quantum Future: Some Assembly Required, (2022), https://www.quantumworldcongress.com/whitepaper 44 For an overview of RRI literature, see R. Kumar Thapa, T. Iakovleva, L. Foss Responsible research and innovation: a systematic review of the literature and its applications to regional studies, EUROPEAN PLANNING STUDIES, 27:12, 2470-2490, (2019). DOI: 10.1080/09654313.2019.1625871 See in similar vein, Coenen, C., Grinbaum, A., Grunwald, A., Milburn, C., & Vermaas, P. Quantum Technologies and Society: Towards a Different Spin, NANOETHICS, 16, 1-6 (2022), https://doi.org/10.1007/s11569-021-00409-4 45 For ethical design thinking in technology, see B. Friedman, Embodying values in technology: Theory and practice, INFORMATION TECHNOLOGY AND MORAL PHILOSOPHY, 3(6):322-353, CAMBRIDGE PUBLISHER: CAMBRIDGE UNIVERSITY PRESS, (2008).
TABLE
See e.g., Bohr, N. The Quantum Postulate and the Recent Development of Atomic Theory, NATURE 121, 580-590 (1928). https://doi.org/10.1038/121580a0. See also Caltech, What Is Superposition and Why Is It Important? https://scienceexchange.caltech.edu/topics/quantum-science-explained/quantum-superposition, and L. Billings, Explorers of //www.scientificamerican.com/article/explorers-of-quantum-entanglement-win-2022-nobel-prize-in-physics1/ 6 See e.g., M. Aboy, T. Minssen, M. Kop, Mapping the Patent Landscape of QT: Patenting Trends, Innovation and Policy Implications, INTERNATIONAL REVIEW OF INTELLECTUAL PROPERTY AND COMPETITION LAW (IIC), VOLUME 53, PP. 853-882, SPRINGER NATURE, (2022). https://link.springer.com/article/10.1007/s40319-022-01209-3. 7 A pioneering take-up of this challenge is Project Q, an initiative at the University of Sydney, Australia, aimed at investigating the geopolitical and societal implications of quantum innovation in computing, communications and artificial intelligence, see https://projectqsydney.com/ See also C. J. Hoofnagle, S. Garfinkel, Law and Policy for the Quantum Age (BERKELEY, 2021), pp 303-456. 8 For an explanation of key Quantum-ELSPI elements including a selection of relevant Quantum-ELSPI questions, see M. Kop, Quantum ELSPI: Ethical, Legal, Social and Policy Implications of Quantum Technology, DIGITAL SOCIETY (SPRINGER NATURE), July 28, 2021, https://law.stanford.edu/publications/quantum-elspi-ethical-legal-social-and-policy-implicationsof-quantum-technology/.Quantum
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1-13. (2021), https://doi.org/10.1080/09537325.2021.1988557. 29 A 'sea change' refers to a profound or notable transformation. Compare to the 'AREA Framework -Anticipate, Reflect, Engage, Act', which aims to embed RRI practices into emerging technologies, see R. Owen, J. Stilgoe, and P. Macnaghten, A framework for responsible innovation, RESPONSIBLE INNOVATION: MANAGING THE RESPONSIBLE EMERGENCE OF SCIENCE AND INNOVATION IN SOCIETY, 31 (2013): 27-50, https://onlinelibrary.wiley.com/doi/10.1002/9781118551424.ch2 and Stilgoe, J. Why Responsible Innovation?, in Responsible Innovation: Managing the Responsible Emergence of Science and Innovation in Society, edited by R. Owen, J. R. Bessant, and M. Heintz, 306, OXFORD: WILEY (2013). 30 See U. Gasser and V. Almeida, A layered model for AI governance, IEEE INTERNET COMPUTING, vol. 21, no. 6, pp. 58-62, November/December 2017, doi: 10.1109/MIC.2017.4180835. 31 See e.g., C. Emerson, S. James, K. Littler, and F. (Fil) Randazzo, Principles for gene drive research, SCIENCE 358, ISSUE 6367, PP. 1135-1136, 1 DEC (2017) DOI: 10.1126/science.aap9026 32 See e.g., Hursthouse, Rosalind and Glen Pettigrove, Virtue Ethics, THE STANFORD ENCYCLOPEDIA OF PHILOSOPHY (Winter 2022 Edition), Edward N. Zalta & Uri Nodelman (eds.), https://plato.stanford.edu/archives/win2022/entries/ethicsvirtue/
For further reading on quantum technologies becoming instrumental in both security and warfare, see Zhou Q., The subatomic arms race: Mutually assured development, HARVARD INTERNATIONAL REVIEW, (2021), https://hir.harvard.edu/the-subatomic-arms-race-mutually-assured-development/ 37 See Aboy, Minssen & Kop, supra note 6. Compare to M. Kop, Quantum Computing and Intellectual Property Law, 25 BERKELEY TECHNOLOGY LAW JOURNAL 2021, PP 101-115, https://btlj.org/2022/02/quantum-computing-and-intellectualproperty-law/. For an unorthodox yet creative take on empirical IP research in the field of QT, see Zeki Can Seskir & Kelvin W. Willoughby, Global Innovation and Competition in Quantum Technology, Viewed Through the Lens of Patents and Artificial Intelligence, INTERNATIONAL JOURNAL OF INTELLECTUAL PROPERTY MANAGEMENT, 13, 1 (2023), 40-61, https://dx.doi.org/10.1504/IJIPM.2021.10044326 38 See Kop, Aboy & Minssen, Intellectual property in quantum computing and market power: a theoretical discussion and empirical analysis, JOURNAL OF INTELLECTUAL PROPERTY LAW & PRACTICE, VOLUME 17, ISSUE 8, Oxford University Press, August 2022, Pages 613-628, https://doi.org/10.1093/jiplp/jpac060 39 See National Academies of Sciences, Engineering, and Medicine. 2019. Quantum Computing: Progress and Prospects. Edited by Mark Horowitz and Emily Grumbling, Washington, DC: The National Academies Press, pp158-159. https://doi.org/10.17226/25196. A virtuous cycle, or positive feedback loop, requires continuous private and public funding, research, development and engineering efforts, attracting talent, successful commercial QT applications, increasing demand for QT, and economies of scale driving progress, growth and profitability.
, addressing security threats; 2. Proactively anticipate the malicious use of quantum applications, addressing risks of dual use; 3. Seek international collaboration based on shared values, addressing a winner-takes-all dynamic; 4. Consider our planet as the sociotechnical environment in which QT should function, engaging states; 5. Be as open as possible, and as closed as necessary, engaging institutions; 6. Pursue diverse quantum R&D communities in terms of disciplines and people, engaging people; 7. Link quantum R&D explicitly to desirable social goals, advancing society; 8. Actively stimulate sustainable, cross-disciplinary innovation, advancing technology; 9. Create an ecosystem to learn about the possible uses and consequences of QT applications, advancing our understanding of Responsible QT; 10. Facilitate dialogues with stakeholders to better envision possible quantum futures, advancing our collective thinking and education about QT and its impact.
See CHIPS & Science Act of 2022, https://www.commerce.senate.gov/services/files/2699CE4B-51A5-4082-9CED-4B6CD912BBC8 77 See, e.g., McKenzie Prillaman, Billions more for US science: how the landmark spending plan will boost research, NATURE 608, 249 (2022) https://doi.org/10.1038/d41586-022-02086-z79
73 FACT SHEET: President Biden Announces Two Presidential Directives Advancing Quantum Technologies, White House
(May 4, 2021), https://www.whitehouse.gov/briefing-room/statements-releases/2022/05/04/fact-sheet-president-biden-
announces-two-presidential-directives-advancing-quantum-technologies/. See also https://www.quantum.gov/
74 Washington is clearly taking the lead to create the rules of the road for quantum for the rest of the world to follow,
effectively protecting US intellectual property from theft by global competitors such as China and Russia, which are at present
systemic rivals [with incompatible ideologies]. While Brussels is focusing on AI safeguarding, Washington attention is
turning to ensuring technological leadership in the post-quantum era.
75
See https://www.congress.gov/bill/117th-congress/house-bill/7535/actions, and https://fedscoop.com/biden-signs-
quantum-computing-cybersecurity-act-into-law/
76 78
See Kop & Brongersma, supra note 33. 90 For a catalogue of principles relating to quantum computing, see World Economic Forum, Quantum Computing Governance Principles, (WEF Jan. 2022) https://www.weforum.org/reports/quantum-computing-governance-principles. For broader principles pertaining to the entire suite of QT, see Kop, supra note 49.
See e.g., Stephen Witt, The World-Changing Race to Develop the Quantum Computer, The New Yorker, Dec 12, 2022, https://www.newyorker.com/magazine/2022/12/19/the-world-changing-race-to-develop-the-quantum-computer 16 See e.g., International Atomic Energy Agency, Establishing a Code of Ethics for Nuclear Operating Organizations, IAEA NUCLEAR ENERGY SERIES NO. NG-T-1.2, IAEA, VIENNA (2007).17 This is the main reason why we cannot just apply methodological frameworks for AI, biosciences, nuclear fission or nanotechnology to quantum technologythese use cases are categorically different in many dimensions. We should however transplant the parts of those existing frameworks that are relevant to, or of special value for QT use cases, e.g. the parts that address dual use characteristics, or that apply to all general purpose technologies.18 As we explain in box 1 and section 2, QT radically differ from other technologies that rely on, and harness classical physics. The novel nature of QT calls for a fundamental reflection on the direct implications for how we approach ethics, law and policywhich is in its core a philosophical exercise. In this paper, however, we address the novel nature of QT indirectly, by focusing on the expected societal impact that flows from it.19 In parallel, the free world should prioritize environmental, social, and governance (ESG) investing in QT R&D, to avoid losing the competition for technological supremacy from countries with incompatible ideologies.20 Compare to P. Inglesant, M. Jirotka & M. Hartswood, Responsible Innovation in QT applied to Defence and National Security, NQIT, (2018). 21 For further reading on the ethical, legal, socio-economic, societal, and philosophical dimensions of QT, see M. Kop, Ethics in the Quantum Age, PHYSICS WORLD, DEC., 31 (2021), https://physicsworld.com/a/why-we-need-to-consider-the-ethicalimplications-of-quantum-technologies/.
See M. Kop, Establishing a Legal-Ethical Framework for QT, YALE J.L. & TECH. THE RECORD (2021). https://yjolt.org/blog/establishing-legal-ethical-framework-quantum-technology
An example of under-regulation would be to have the market figure out how to self-regulate laissez-faire style, potentially only benefitting a small select group of corporations instead of society at large. 67 Several quantum technologies such as "quantum computers" should be considered "base-layer technologies" akin to a microprocessor (CPU). In the same way that it did not make sense to focus the regulations on the classical microprocessor (the fundamental building block of classical computation) but instead direct the regulation to the upstream devices and applications using these microprocessors, the primary regulatory efforts -with a few exceptions such as of export controlshould not be directed to "quantum processors" or "quantum sensors" but instead target the upstream QT-enabled solutions for specific applications using the legal frameworks available for such domains. For instance, a medical device using a microprocessor is regulated as a medical device due to its intended use satisfying the FDA or EU MDR regulatory definition of medical device while a gaming console using the same microprocessor does not follow these regulations. In the same way, if a medical device were to make use of cloud-based access to a quantum computer it would be regulated as a medical device without raising any new legal issues (i.e., the nature of the processing element is abstracted from the regulation which focuses
ACKNOWLEDGEMENTSTimo Minssen's, Mateo Aboy's, and Glenn Cohen's research for this paper was supported, in part, by a Novo Nordisk Foundation Grant for a scientifically independent International Collaborative Biomedical Innovation & Law Program -Inter CeBIL (grant no. NNF17SA027784). RL thanks Mike and Ophelia Lazaridis for funding.One Sentence Synopsis:The article proposes a conceptual framework for Responsible Quantum Technology by integrating considerations about ethical, legal, social, and policy implications (ELSPI) of quantum technologies and RRI values into quantum R&D.Box 2: Responsible Quantum Technology SummaryWHY -Why the need for responsible quantum technology?• For the last 100 years, scientists have been working on quantum science.• At first, quantum science focused on developing the theory of quantum mechanics to understand the principles and rules that govern physical reality at a fundamental particle level. • Later, these insights were applied to technology development.• The first quantum revolution in engineering brought us 1G QT.• The second quantum revolution in engineering promises 2G QT.• With the introduction of QT into society comes the need for ELSPI-considerations.• The potential impact of QT makes ELSPI-considerations ever more important. A key step is the conceptualization of what "responsible" QT means and implies. • This paper aims to provide such a conceptualization and suggests its operationalization by guiding principles.WHAT -What does responsible quantum technology amount to?• Proposition: the potential societal impact of QT calls for Responsible QT.• We should explore what "responsible" amounts to in the context of QT.• Analytically, Responsible QT is about integrating ELSPI-considerations into R&D processes to ensure responsible quantum innovation. • Normatively, Responsible QT is about minimizing harm and maximizing benefit. • Our proposal: Responsible QT entails an innovation process that proactively addresses risks, takes on challenges and seizes opportunities that come with the development of QT. • We translate this idea into the SEA-framework for Responsible QT, capturing three key aspects of Responsible QT: Safeguarding, Engaging, Advancing. • We propose to approach Responsible QT by the SEA-framework, responding to ResponsibleResearch & Innovation (RRI)-dimensions.HOW -How to pursue responsible quantum technology?• Responsible QT is the aim of responsible quantum innovation.• The SEA-categories should be developed into a set of foundational principles to guide quantum innovation and contribute to Responsible QT. • This could be a mix of quantum-specific principles and generic responsible tech principles that are intrinsically relevant to QT. • The research community is invited to develop these guiding principles and discuss their operationalization. • Ideally, implementing such principles into practice should result in Responsible QT by design and default.
Aboy, Minssen & Kop, supra note 6. See e.g., Aboy, Minssen & Kop, supra note 6.
FACT SHEET: President Biden Announces Two Presidential Directives Advancing QT, White House. FACT SHEET: President Biden Announces Two Presidential Directives Advancing QT, White House (May 4, 2021).
Talking about public good for the second quantum revolution: Analysing quantum technology narratives in the context of national strategies. Id, T See Also Roberson, J Leach, S Raman, 10.1088/2058-9565/abc5abQUANTUM SCIENCE AND TECHNOLOGY. 6225001and Hoofnagle & Garfinkle, supra note 7Id. See also Roberson, T., Leach, J., & Raman, S. Talking about public good for the second quantum revolution: Analysing quantum technology narratives in the context of national strategies, QUANTUM SCIENCE AND TECHNOLOGY, 6(2), 25001, (2021), https://doi.org/10.1088/2058-9565/abc5ab; and Hoofnagle & Garfinkle, supra note 7.
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See NIST Announces First Four Quantum-Resistant Cryptographic Algorithms. NISTSee NIST Announces First Four Quantum-Resistant Cryptographic Algorithms, NIST, July 05, 2022, https://www.nist.gov/news-events/news/2022/07/nist-announces-first-four-quantum-resistant-cryptographic-algorithms
Entanglement-based secure quantum cryptography over 1,120 kilometres. E G See, J Yin, Y Li, S Liao, 10.1038/s41586-020-2401-yNATURE. 582See, e.g., J. Yin, Y. Li, S. Liao et al. Entanglement-based secure quantum cryptography over 1,120 kilometres. NATURE 582, 501-505 (2020). https://doi.org/10.1038/s41586-020-2401-y
For an architecture combining Quantum Random Access Memory (QRAM) and quantum networks, resulting in multiparty private quantum communication, see. Connor T Liu, Liang Hann, Jiang, Quantum Data Center: Theories and Applications. For an architecture combining Quantum Random Access Memory (QRAM) and quantum networks, resulting in multi- party private quantum communication, see.Junyu Liu, Connor T. Hann, Liang Jiang, Quantum Data Center: Theories and Applications, August 1, 2022, https://arxiv.org/abs/2207.14336.
Private quantum computation: an introduction to blind quantum computing and related protocols. E G See, J Fitzsimons, 10.1038/s41534-017-0025-3NPJ QUANTUM INF. 323See, e.g., J. Fitzsimons, Private quantum computation: an introduction to blind quantum computing and related protocols. NPJ QUANTUM INF 3, 23 (2017). https://doi.org/10.1038/s41534-017-0025-3
For further reading on the potential impact of the use of QC in the legal sector, see Jeffery Atik & Valentin Jeutner, Quantum computing and computational law. 10.1080/17579961.2021.1977216LAW, INNOVATION AND TECHNOLOGY. 132For further reading on the potential impact of the use of QC in the legal sector, see Jeffery Atik & Valentin Jeutner, Quantum computing and computational law, LAW, INNOVATION AND TECHNOLOGY, 13:2, 302-324, 2021, DOI: 10.1080/17579961.2021.1977216
Regulation as Facilitation: Negotiating the Genetic Revolution. Julia Black, THE MODERN LAW REVIEW. 615Black, Julia, Regulation as Facilitation: Negotiating the Genetic Revolution, THE MODERN LAW REVIEW 61, NO. 5 (1998): 621-60. http://www.jstor.org/stable/1097126
Multidisciplinary embedded Responsible QT has significant competitive advantages, fostering exponential innovation and trust, while steering toward beneficial societal outcomes. Multidisciplinary embedded Responsible QT has significant competitive advantages, fostering exponential innovation and trust, while steering toward beneficial societal outcomes.
Although the as of yet mostly unregulated quantum sphere offers a once in a lifetime opportunity to harmonize legal-ethical frameworks for QT on the planetary level, we anticipate global divergence in QT regulation between the three tech blocks US, EU, and China, who each have their own values systems, which are culturally sensitive, context-specific, and dynamic. Whoever sets the technical standards will set the rules of the road for quantum for the world to follow. Although the as of yet mostly unregulated quantum sphere offers a once in a lifetime opportunity to harmonize legal-ethical frameworks for QT on the planetary level, we anticipate global divergence in QT regulation between the three tech blocks US, EU, and China, who each have their own values systems, which are culturally sensitive, context-specific, and dynamic. Whoever sets the technical standards will set the rules of the road for quantum for the world to follow.
How to Appropriate Value from General-Purpose Technology by Applying Open Innovation, CALIFORNIA MANAGEMENT REVIEW 2021, 000812562110417, 10.1177/00081256211041787, at 18-20. on the intended use), while access to the same cloud-based quantum computer for education-purposes (e.g., running basic quantum algorithms as part of a university course in quantum computing) or research (e.g., quantum simulation of molecules) will not. Similarly, graphical processing units (GPUs) are the computational building blocks enabling AI. Compare To Yang, & Jialei, Chesbrough, & Henry, Pia Hurmelinna, but also for gaming. Thus, care must be exercised to ensure that regulatory interventions target the correct layer in the stackCompare to Yang, Jialei & Chesbrough, Henry & Hurmelinna, Pia, How to Appropriate Value from General-Purpose Technology by Applying Open Innovation, CALIFORNIA MANAGEMENT REVIEW 2021, 000812562110417, 10.1177/00081256211041787, at 18-20. on the intended use), while access to the same cloud-based quantum computer for education-purposes (e.g., running basic quantum algorithms as part of a university course in quantum computing) or research (e.g., quantum simulation of molecules) will not. Similarly, graphical processing units (GPUs) are the computational building blocks enabling AI, but also for gaming. Thus, care must be exercised to ensure that regulatory interventions target the correct layer in the stack.
M See, ; Kop, Stanford -Vienna, Transatlantic, Law Forum, Stanford Ipr Developments, University, Regulating Transformative Technology in The Quantum Age: Intellectual Property, Standardization & Sustainable Innovation. See M. Kop, Regulating Transformative Technology in The Quantum Age: Intellectual Property, Standardization & Sustainable Innovation (October 7, 2020), STANFORD -VIENNA TRANSATLANTIC TECHNOLOGY LAW FORUM, TRANSATLANTIC ANTITRUST AND IPR DEVELOPMENTS, STANFORD UNIVERSITY, ISSUE NO. 2/2020, https://law.stanford.edu/publications/regulating-transformative-technology-in-the-quantum-age-intellectual-property- standardization-sustainable-innovation/ Compare to Johnson, supra note 47.
The Quantum Governance Stack: Models of Governance for Quantum Information Technologies. E Perrier, DIGITAL SOCIETY, QUANTUM-ELSPI TC. 1SPRINGER NATURESee e.g., E. Perrier, The Quantum Governance Stack: Models of Governance for Quantum Information Technologies, DIGITAL SOCIETY, QUANTUM-ELSPI TC, 1, 22, SPRINGER NATURE, (2022).
. Kop See Also, 49See also Kop, supra note 49.
For further reading on the uncertainty principle in the quantum physics underlying QT, see Feynman et. al., supra note 14. For further reading on the uncertainty principle in the quantum physics underlying QT, see Feynman et. al., supra note 14.
Regulating Uncertain States: A Risk-Based Policy Agenda for Quantum Technologies. See Kop, ; T Martin-Bariteau, F , 10.2139/ssrn.4203758CANADIAN JOURNAL OF LAW AND TECHNOLOGY. 202OTTAWA FACULTY OF LAW WORKING PAPERSee Kop, supra note 68. Compare with Dekker, T. and Martin-Bariteau, F., Regulating Uncertain States: A Risk-Based Policy Agenda for Quantum Technologies (May 1, 2022). (2022) 20:2 CANADIAN JOURNAL OF LAW AND TECHNOLOGY 179, OTTAWA FACULTY OF LAW WORKING PAPER NO. 2022-26, http://dx.doi.org/10.2139/ssrn.4203758
For a podcast discussion about QT's legal and ethical implications on law, economics, and society with a focus on US attempts to incorporate the technology into existing frameworks for intellectual property and international security, see Quantum Computing with Joonas Keski-Rahkonen and Katri Nousiainen. Berkeley Technology Law Journal Podcast: Quantum Computing. For a podcast discussion about QT's legal and ethical implications on law, economics, and society with a focus on US attempts to incorporate the technology into existing frameworks for intellectual property and international security, see Quantum Computing with Joonas Keski-Rahkonen and Katri Nousiainen, August 23, 2022, Berkeley Technology Law Journal Podcast: Quantum Computing, https://btlj.org/2022/08/berkeley-technology-law-journal-podcast-quantum- computing/
. See Also Kop, 46See also Kop et al., supra note 46.
See Executive Order on Enhancing the National Quantum Initiative Advisory Committee. See Executive Order on Enhancing the National Quantum Initiative Advisory Committee, https://www.whitehouse.gov/briefing-room/presidential-actions/2022/05/04/executive-order-on-enhancing-the-national- quantum-initiative-advisory-committee/
See National Security Memorandum on Promoting United States Leadership in Quantum Computing While Mitigating Risks to Vulnerable Cryptographic Systems. See National Security Memorandum on Promoting United States Leadership in Quantum Computing While Mitigating Risks to Vulnerable Cryptographic Systems, https://www.whitehouse.gov/briefing-room/statements-releases/2022/05/04/national-security-memorandum-on-promoting- united-states-leadership-in-quantum-computing-while-mitigating-risks-to-vulnerable-cryptographic-systems/
. See Quantum Computing Cybersecurity Preparedness Act. See Quantum Computing Cybersecurity Preparedness Act, https://www.congress.
More research is required to ascertain whether these rules should have the form of a Presidential executive order, or in the form of a more durable new law from Congress, whether it would work for the tech industry. the Council of Europe's CAI -Committee on Artificial Intelligence is currently drafting a Convention on Artificial Intelligence, Human Rights, Democracy and the Rule of Law. This is a different initiative than the imminent EU AI Act as proposed by the European Commission. and whether it's scope should be broad or narrowIn a similar vein, the Council of Europe's CAI -Committee on Artificial Intelligence is currently drafting a Convention on Artificial Intelligence, Human Rights, Democracy and the Rule of Law, see https://rm.coe.int/cai-2023-01-revised-zero-draft- framework-convention-public/1680aa193f This is a different initiative than the imminent EU AI Act as proposed by the European Commission, see https://digital-strategy.ec.europa.eu/en/policies/european-approach-artificial-intelligence 86 More research is required to ascertain whether these rules should have the form of a Presidential executive order, or in the form of a more durable new law from Congress, whether it would work for the tech industry, and whether it's scope should be broad or narrow.
The authors distinguish between 2 classes of hybrid quantum-classical computing: (1) application agnostic focusing on the quantum computing hardware, and (2) application specific, focusing on quantum computing algorithms. 10.48550/arXiv.2210.15314See note 67 about microprocessors, GPUs, and quantum computing as "base-layer technologies. Efforts to classify hybrid quantum-classical computing paradigms are underway, see Frank Phillipson, Niels Neumann, Robert Wezeman, Classification of Hybrid Quantum-Classical Computing. Such novel technical classifications are relevant for legal classificationsSee note 67 about microprocessors, GPUs, and quantum computing as "base-layer technologies". Efforts to classify hybrid quantum-classical computing paradigms are underway, see Frank Phillipson, Niels Neumann, Robert Wezeman, Classification of Hybrid Quantum-Classical Computing, Oct 27, 2022, https://doi.org/10.48550/arXiv.2210.15314. The authors distinguish between 2 classes of hybrid quantum-classical computing: (1) application agnostic focusing on the quantum computing hardware, and (2) application specific, focusing on quantum computing algorithms. Such novel technical classifications are relevant for legal classifications.
Ideally, one would want to measure, benchmark, validate and certify responsible quantum technologies during their life span, denoting parameters using a data driven approach. Ideally, one would want to measure, benchmark, validate and certify responsible quantum technologies during their life span, denoting parameters using a data driven approach.
The Quantum Governance Stack: Models of Governance for Quantum Information Technologies. E Compare, Perrier, DIGITAL SOCIETY, QUANTUM-ELSPI TC. 1SPRINGER NATURECompare to E. Perrier, The Quantum Governance Stack: Models of Governance for Quantum Information Technologies, DIGITAL SOCIETY, QUANTUM-ELSPI TC, 1, 22, SPRINGER NATURE, (2022).
Hybrid quantum-classical approach to enhanced quantum metrology. SCIENTIFIC REPORTS. 11: 672. PMID. See X Yang, X Chen, J Li, X Peng, R Laflamme, 10.1038/s41598-020-80070-1NATURE. 33436795See X. Yang, X. Chen, J. Li, X. Peng, R. Laflamme, Hybrid quantum-classical approach to enhanced quantum metrology. SCIENTIFIC REPORTS. 11: 672. PMID, NATURE, (2021) 33436795 DOI: 10.1038/s41598-020-80070-1
We are currently moving from the NISQ Era to the Fault Tolerant Era with error corrected quantum computers consisting of 1 million stable logical qubits capable of running novel algorithms expected before 2030, featuring double exponential growth curves. potentially unlocking a new industrial revolutionWe are currently moving from the NISQ Era to the Fault Tolerant Era with error corrected quantum computers consisting of 1 million stable logical qubits capable of running novel algorithms expected before 2030, featuring double exponential growth curves, potentially unlocking a new industrial revolution.
| [] |
[
"Faster Distance-Based Representative Skyline and k-Center Along Pareto Front in the Plane",
"Faster Distance-Based Representative Skyline and k-Center Along Pareto Front in the Plane"
] | [
"Sergio Cabello "
] | [] | [] | We consider the problem of computing the distance-based representative skyline in the plane, a problem introduced by Tao, Ding, Lin and Pei [Proc. 25th IEEE International Conference on Data Engineering (ICDE), 2009] and independently considered by Dupin, Nielsen and Talbi [Optimization and Learning -Third International Conference, OLA 2020] in the context of multi-objective optimization. Given a set P of n points in the plane and a parameter k, the task is to select k points of the skyline defined by P (also known as Pareto front for P ) to minimize the maximum distance from the points of the skyline to the selected points. We show that the problem can be solved in O(n log h) time, where h is the number of points in the skyline of P . We also show that the decision problem can be solved in O(n log k) time and the optimization problem can be solved in O(n log k + n loglog n) time. This improves previous algorithms and is optimal for a large range of values of k. | 10.1007/s10898-023-01280-1 | [
"https://arxiv.org/pdf/2012.15381v1.pdf"
] | 229,923,959 | 2012.15381 | a77ebc799c518ad92acc931a8694167de35ad1b5 |
Faster Distance-Based Representative Skyline and k-Center Along Pareto Front in the Plane
January 1, 2021
Sergio Cabello
Faster Distance-Based Representative Skyline and k-Center Along Pareto Front in the Plane
January 1, 2021skylinePareto frontclusteringalgorithmk-center
We consider the problem of computing the distance-based representative skyline in the plane, a problem introduced by Tao, Ding, Lin and Pei [Proc. 25th IEEE International Conference on Data Engineering (ICDE), 2009] and independently considered by Dupin, Nielsen and Talbi [Optimization and Learning -Third International Conference, OLA 2020] in the context of multi-objective optimization. Given a set P of n points in the plane and a parameter k, the task is to select k points of the skyline defined by P (also known as Pareto front for P ) to minimize the maximum distance from the points of the skyline to the selected points. We show that the problem can be solved in O(n log h) time, where h is the number of points in the skyline of P . We also show that the decision problem can be solved in O(n log k) time and the optimization problem can be solved in O(n log k + n loglog n) time. This improves previous algorithms and is optimal for a large range of values of k.
Introduction
For each point p in the plane, let x(p) and y(p) denote its x-and y-coordinate, respectively. Thus p = (x(p), y(p)). A point p dominates a point q if x(p) ≥ x(q) and y(p) ≥ y(q). Note that a point dominates itself. For a set of points P , its skyline is the subset of points of P that are not dominated by any other point of P . We denote by sky(P ) the set of skyline points of P . Formally sky(P ) = {p ∈ P | ∀q ∈ P \ {p} : x(q) < x(p) or y(q) < y(p)}.
See the left of Figure 1 for an example. The skyline of P is also called the Pareto front of P . A set P of points is a skyline or a Pareto front if P = sky(P ). Note that some authors exchange the direction of the inequalities, depending on the meaning of the coordinates in the application domain.
For each subset of points Q ⊆ sky(P ) we define where d(p, q) denotes the Euclidean distance between p and q. Note that ψ(Q, P ) = ψ(Q, sky(P )) and ψ(sky(P ), sky(P )) = 0. An alternative point of view for ψ(Q, P ) is the Figure 1: Left: Example of point set P with the points of sky(P ) marked as filled dots; the shaded region has to be empty of points of P . Right: if Q is the two-point set marked with squares, then the length of the longest arrow is ψ(Q, P ).
following: it is the smallest value λ such that disks centered at the points of Q with radius λ cover the whole sky(P ). See Figure 1, right. One can consider ψ(Q, P ) as how big error we make when approximating sky(P ) by Q.
In this paper we provide efficient algorithms for the following optimization problem: for a given point set P in the plane and a positive integer k, compute opt(P, k) := min{ψ(Q, P ) | Q ⊆ sky(P ) and |Q| ≤ k}.
A feasible solution is any subset Q ⊆ sky(P ) with |Q| ≤ k. An optimal solution for opt(P, k) is a feasible solution Q * such that ψ(Q * , P ) = opt(P, k). Thus, we use opt(P, k) both as the optimal value and the description of the problem. Note that opt(P, k) = opt(sky(P ), k).
We will also consider the decision problem: given a set P of points in the plane, a positive integer k and a real value λ ≥ 0, decide whether opt(P, k) ≤ λ or opt(P, k) > λ. Motivation. Skylines (or Pareto fronts) can be considered for arbitrary dimensions and play a basic role in several areas. Börzsönyi, Kossmann and Stocker [2] advocated their use in the context of databases, where they have become an ubiquitous tool; see for example [25,26]. They also play a key role in multiobjective optimization and decision analysis; see for example the surveys [8,19,23].
Quite often the skyline contains too many points. The question is then how to select a representative subset of the points in the skyline. The representative may be a final answer or just an intermediary step, as it happens for example in evolutionary algorithms, where diversity plays a key role.
As one can imagine, several different meaningful definitions of best representative subset are reasonable, and different applications advocate for different measures on how good the representatives are. The use of randomization is also relevant in the context of evolutionary algorithms.
The problem we are considering, opt(P, k), was introduced in the context of databases by Tao et al. [27] under the name of distance-based representative of sky(P ). Although we have presented the problem in the plane, it can be naturally extended to arbitrary dimensions. Tao et al. argue the benefit of using such representatives. One of the key properties is that it is not sensible to the density of P , a desirable property when P is not uniformly distributed.
The very same problem, opt(P, k), was considered in the context of optimization by Dupin, Nielsen and Talbi [7], who noted the connection to clustering, multiobjective optimization and decision analysis. They noticed that opt(P, k) is the (discrete) k-center problem for sky(P ). Clustering problems in general, and k-center problem in particular [3,15] are fundamental in several areas; thus it is relevant to find particular cases that can be solved efficiently.
Another very popular and influential measure to select representatives in the context of databases was introduced by Lin et al. [20]. In this case, we want to select k points of sky(P ) so as to maximize the number of points from P dominated by some of the selected points. See our work [5] for a recent algorithmic improvement in the planar case. In the context of evolutionary algorithms one of the strongest measures is the hypervolume of the space dominated by the selected points [1] with respect to a reference point, that can be taken to be the origin. As it can be seen in the survey by Li and Yao [19], an overwhelming amount of different criteria to select representatives from sky(P ) has been considered in the area of multiobjective optimization.
Previous algorithms. We will be using three parameters to analyze the time complexity of algorithms. We will use n for the size of P , k for the bound on the number of points to be selected, and h for the number of points in sky(P ). Unless explicitly mentioned, the discussion is for the plane.
We start reviewing the computation of the skyline. Kung, Luccio and Preparata [18] showed that the skyline of P in the plane can be computed in O(n log n) time, and this is asymptotically optimal. Kirkpatrick and Seidel [17] noted that the lower bound of Ω(n log n) does not hold when h is small, and they provided an algorithm to compute sky(P ) (in the plane) in O(n log h) time. Nielsen [24] provided an algorithm to compute the top k skylines, that is, sky(P ), sky(P \ sky(P )), . . . up
to k iterations. The running time is O(n log H),
where H is the size of the top k skylines together.
Kirkpatrick and Seidel [17] also showed that computing the skyline of P takes Ω(n log h) time in the algebraic decision tree model. In fact, they showed that, for a given h , deciding whether P has exactly h points in the skyline takes Ω(n log h ) time. This implies that finding any k points in the skyline takes Ω(n log k) time. Indeed, if we could find k points in the skyline in o(n log k), then we could run the test for k = h and k = h + 1 and decide in o(n log(k + 1)) = o(n log h ) time whether sky(P ) has exactly h points, contradicting the lower bound. Now we move onto computing opt(P, k). Tao et al. [27] showed that the problem can be solved in O(kh 2 ) assuming that the skyline is available and sorted by x-coordinate. Thus, it takes O(n log h + kh 2 ) time. In the full version of the work [28] they improved the time bound of the second phase to O(kh), which implies a time bound of O(n log h + kh). They showed that the 3d version of the problem is NP-hard.
Dupin, Nielsen and Talbi [7] solve the opt(P, k) problem in O(kh log 2 h) time, again assuming that the skyline is available and sorted. Thus, starting from P , the time bound is O(n log h + kh log 2 h). They also provide a linear-time algorithm for opt(P, 1) and an algorithm with running time O(h log h) for opt(P, 2), again assuming that the skyline is available and sorted by x-coordinate.
Mao et al. [21] provided an exact algorithm for opt(P, k) that takes O(k 2 log 3 h) time, assuming that the skyline is stored for binary searches, and an approximation with error of ε in O(k log 2 h log(T /ε)), where T is the distance between the extreme points of the skyline. Again, if we start from P , we should add the time O(n log h) to both algorithms to compute the skyline. The decision problem is solved in linear time in the work by Yin and Wei and Liu [31], assuming that the skyline is available.
If the points on the skyline are collinear and sorted, we can compute opt(P, k) in linear time using the algorithm of Frederickson [9,10]. See [13] for a recent account that may be easier to understand. The author believes that the same approach can be used to solve opt(P, k) in linear time, once the skyline is available sorted. However, as the details in those algorithms are very complicated, the author is reluctant to make a firm claim.
Our contribution. We show that opt(P, k) can be solved in O(n log h) time. We provide a quite simple algorithm for this. Compared to a possible adaptation of [9,10], we get the added value of simplicity within the same time bound that we need to compute the skyline itself. If the skyline is already available, the running time becomes O(h log h). In all cases we improve all previous works considering algorithms for the problem opt(P, k) explicitly [7,21,27,28].
At first, this may seem optimal because computing the skyline of P takes Ω(n log h) time. However, do we really need to compute the skyline? After all, we only need to select a particular subset of k points from the skyline, and this has a lower bound of Ω(n log k) time, as mentioned above. We show that the decision problem for opt(P, k) can be solved in O(n log k) time. This is asymptotically optimal if we want to find a solution because such a solution has to find k points in sky(P ). In fact, we show that with some additional preprocessing we can solve several decision problems. For example, we can solve O(k 2 ) decision problems in O(n log k) time altogether. We combine the decision problem with parametric search and selection in sorted arrays to show that opt(P, k) can be solved in O(n log k + n loglog n) time. This is asymptotically optimal whenever log k = Ω(loglog n), if we want to find also an optimal solution.
Finally, we provide additional results for the case when k is very small. The driving question here is, for example, how fast can we solve opt(P, 15). We show that a (1 + ε)approximation can be computed in O(kn + n loglog(1/ε)) time. We obtain this by finding a 2-approximation in O(kn) time and then making use of repetitive binary searches using the decision problem.
Besides improving previous algorithms, we believe that the conceptual shift of solving opt(P, k) without computing sky(P ) explicitly is a main contribution of our work. Nevertheless, even when sky(P ) is available, we improve previous works, with the possible exception of [9]. We provide very detailed description of our algorithms to indicate that they are implementable.
We start the paper discussing in detail an optimal algorithm for computing the skyline in O(n log h) time. The algorithm uses the techniques from Chan [4] and Nielsen [24]. The purpose is two-fold. First, we believe that the presentation of this part is pedagogical, slightly simpler than the presentation by Nielsen because we try to solve a simpler problem, and may find interest within the community interested in computing with skylines in the plane. Second, it provides the basic idea for our later approach, and thus it should be discussed to some level of detail in any case. We decided to provide a self-contained presentation of this part.
Organization of the paper. In Section 2 we provide some basic observations. In Section 3 we explain a simple, optimal algorithm to compute the skyline. Section 4 is devoted to the problem of finding the skyline representatives through the computation of the skyline, while Section 5 looks at the problem without computing the skyline. Results for very small k are considered in Section 6. We conclude in Section 7, where we also provide further research questions.
Basic tools
We are usually interested in the order of the points along the skyline sky(P ). For this we store the points of sky(P ) sorted by increasing x-coordinate in an array. We could also use any other data structure that allows us to make binary searches among the elements of sky(P ), such as a balanced binary search tree. Note that sorting sky(P ) by x-or by (decreasing) y-coordinate is equivalent.
Quite often we use an expression like "select the highest point among q 1 , . . . , q t , breaking ties in favor of larger x(·)". This means that we take the point among q 1 , . . . , q t with largest ycoordinate; if more points have the largest y-coordinate y max , then we select among those with y(q i ) = y max the one with largest x-coordinate. This can be done iterating over i = 1, . . . , t, storing the best point q * so far, and updating q * ← q i if and only if y(q * ) < y(q i ) or (y(q * ) = y(q i ) and x(q * ) < x(q i )).
This wording appears to take care of the cases where more points have the same x-or ycoordinate. Let us explain an intuition that may help understanding how ties are broken. Conceptually, think that each point p = (x, y) ∈ P is replaced by the point p ε = (x + yε, y + xε) for an infinitesimally small ε > 0, and let P ε be the resulting point set. A point p ∈ P belongs to sky(P ) if and only if p ε belongs to sky(P ε ). Moreover, no two coordinates are the same in P ε , if ε > 0 is sufficiently small. Thus, whenever we have to break ties, we think what we would do for P ε .
For correctness and making binary searches we will often use the following monotonicity property.
Lemma 1. If p, q, r are points of sky(P ) and x(p) < x(q) < x(r), then d(p, q) < d(p, r).
Proof. From the hypothesis we have y(p) > y(q) > y(r) and therefore
d(p, q) 2 = x(q) − x(p) 2 + y(q) − y(p) 2 < x(r) − x(p) 2 + y(r) − y(p) 2 = d(p, r) 2 .
A first consequence of this lemma is that any disk centered at a point of sky(P ) contains a contiguous subsequence of sky(P ). In particular, if sky(P ) is stored for binary searches along x-coordinates, then we can perform binary searches to find the boundary elements of sky(P ) contained in any given disk centered at a point of sky(P ). (The claim is not true for disks centered at arbitrary points of P .)
For an x-coordinate x 0 and a set of points Q, let succ(Q, x 0 ) be the leftmost point of
succ(Q, x 0 ) = arg min{x(q) | q ∈ Q, x(q) > x 0 }, pred(Q, x 0 ) = arg max{x(q) | q ∈ Q, x(q) < x 0 }.
We will use succ(Q, x 0 ) and pred(Q, x 0 ) when Q is a skyline; see Figure 2. In our use we will also take care that succ(Q, x 0 ) and pred(Q, x 0 ) are well-defined, meaning that there is some point of Q on each side of the vertical line x = x 0 .
When Q is a skyline stored for binary searches along x-coordinates, the points succ(Q, x 0 ) and pred(Q, x 0 ) can be obtained in O(log |Q|) time.
Looking into the case of skylines we can observe that succ(sky(P ), x 0 ) = arg max{y(p) | p ∈ P, x(p) > x 0 }, breaking ties in favor of larger x(·), while using y 0 = max{y(p) | p ∈ P, x(p) ≥ x 0 } we have pred(sky(P ), x 0 ) = arg max{x(p) | p ∈ P, y(p) > y 0 }, breaking ties in favor of larger y(·).
x = x 0 succ(sky(P ), x 0 ) pred(sky(P ), x 0 )
x = x(p) sky(P i ) sky(P j ) succ(sky(P ), x 0 ) pred(sky(P ), x 0 ) sky(P j ) p p i p j Figure 2: Left: succ(sky(P ), x 0 ) and pred(sky(P ), x 0 ) for a set of points P . Right: Computing succ(sky(P ), x 0 ) and pred(sky(P ), x 0 ) when we have sky(P 1 ), . . . , sky(P t ) for P = P 1 ∪ · · · ∪ P t . The points q 1 , . . . , q t are marked with empty squares.
In several of our algorithms we will use a divide and conquer approach. The following observation is one of the simple but key observations. See Figure 2, right.
Lemma 2.
Assume that P is the union of P 1 , . . . , P t . For each real value x 0 , the point succ(sky(P ), x 0 ) is the highest point among succ(sky(P 1 ), x 0 ), . . . , succ(sky(P t ), x 0 ), breaking ties in favor of larger x(·). Moreover, if for each i ∈ {1, . . . , t} the points of sky(P i ) are sorted by x-coordinate for binary searches, then we can compute succ(sky(P ),
x 0 ) in O(t + i log |P i |) time.
Proof. The point succ(sky(P ), x 0 ) is the highest point of P in the halfplane x > x 0 , breaking ties in favor of larger x(·). Since y succ(sky(P ),
x 0 ) = max{y(p) | p ∈ P, x(p) > x 0 } = max i=1,...,t max{y(p i ) | p i ∈ P i , x(p i ) > x 0 } = max i=1,...,t y succ(sky(P i ), x 0 ) },
the claim about the characterization of succ(sky(P ), x 0 ) follows. The algorithmic claim is obtained by making a binary search in each sky(P i ) to obtain succ(sky(P i ), x 0 ) and choosing the highest point; in case of ties we select the one with largest x-coordinate.
We do not provide pseudocode for the simple algorithm of Lemma 2, but the idea will be used inside pseudocode provided later. We also have the following tool, which is slightly more complicated. It tests whether a point belongs to the skyline and computes the predecessor. Pseudocode under the name of TestSkylineAndPredecessor is described in Figure 3.
Lemma 3.
Assume that P is the union of P 1 , . . . , P t and for each i ∈ {1, . . . , t} the points of sky(P i ) are sorted by x-coordinate for binary searches. For any point p ∈ P , the algorithm TestSkylineAndPredecessor solves the following two problems in O(t+ i log |P i |) time:
• decide whether p ∈ sky(P );
• compute pred(sky(P ), x(p)).
Proof. For each i ∈ {1, . . . , t} we use a binary search along sky(P i ) to find the point p i of sky(P i ) with smallest x-coordinate among those with x(p) ≥ x(p i ). Let p 0 be the highest point among p 1 , . . . , p t , breaking ties in favor of those with larger x-coordinate. Note that p 0 is the highest point of sky(P ) in x ≥ x(p); the proof of Lemma 2 can be trivially adapted for this. The point p is in sky(P ) if and only if p = p 0 .
Algorithm TestSkylineAndPredecessor Input: sky(P 1 ), . . . , sky(P t ) stored for binary searches and a point p. Output: A pair (X, q) where X is True if p ∈ sky(P 1 ∪· · ·∪P t ) and False otherwise, and q = pred(sky(P 1 ∪ · · · ∪ P t ), x(p)).
1. for i = 1, . . . , t do 2. p i ← point of sky(P i ) in x ≥ x(p)
with smallest x-coordinate, using binary search 3. p 0 ← highest point among p 1 , . . . , p t , breaking ties in favor of larger x(·) 4. for i = 1, . . . , t do 5.
q i ← point of sky(P i ) in y > y(p 0 ) with smallest y-coordinate, using binary search 6. q 0 ← rightmost point among q 1 , . . . , q t , breaking ties in favor of larger y(·) 7. return (p = p 0 ?, q 0 ) Figure 3: Algorithm described in Lemma 3 for testing whether p ∈ sky(P ) and computing pred(sky(P ), x(p)).
Computing pred(P, x(p)) is similar to computing succ(P, x(p)), if we exchange the roles of x and y-coordinates. See Figure 2, right, for an example. For each i ∈ {1, . . . , t}, we use a binary search along sky(P i ) with respect to the y-coordinate and key y(p 0 ), to find the point q i ∈ sky(P i ) with smallest y-coordinate among those with y(q i ) > y(q 0 ). The point among q 1 , . . . , q t with largest x-coordinate (and breaking ties in favor of the largest y-coordinate) is pred(sky(P ), x(p)). (As seen in the example of Figure 2, right, pred(sky(P ), x(p)) is not necessarily any of the points among pred(sky(P 1 ), x(p)), . . . , pred(sky(P t ), x(p)).)
Algorithm TestSkylineAndPredecessor implements the idea of this proof.
Computing the skyline optimally
In this section we show that the skyline of a set P of n points can be computed in O(n log h), where h = | sky(P )|. The algorithm is simple and reuses the ideas exploited by Chan [4] and Nielsen [24]. We include it because of its simplicity and because we will modify it later on. Compared to Nielsen [24], our algorithm is slightly simpler because we are computing a slightly simpler object. It is well-known and very easy to see that sky(P ) can be computed in O(n log n) time. Indeed, we just need to sort the points lexicographically: a point p precedes a point q if x(p) < x(q) or if x(p) = x(q) and y(p) < y(q). Then we make a pass over the reversed list of sorted points, and maintain the point with largest y(·) scanned so far. The skyline of P is the list of local maxima we had during the procedure. See Figure 4 for the idea. Details are given as Algorithm SlowComputeSkyline in Figure 5. Note that the lexicographic order is important for correctness: if two points p i and p i−1 have the same x-coordinate, then the point p i has larger y-coordinate and thus p i−1 does not make it to the skyline. We will be using that sky(P ) is returned sorted by x-coordinate.
We now describe the algorithm computing sky(P ) in O(n log h) time. The core of the procedure is a subroutine, called ComputeSkylineBounded and described in Figure 6, that takes as input P and an integer value parameter s ≥ 1. The subroutine returns sky(P ) when h ≤ s, and a warning "incomplete", when h > s. The role of s is that of a rough guess for the size of sky(P ). Algorithm SlowComputeSkyline Input: A set P of n points. Output: sky(P ) sorted by x-coordinate.
1. p 1 , . . . , p n ← sort P lexicographically increasing 2. output ← new list with p n 3. p ← p n (* last point added to output *) 4. for i = n − 1, . . . , 1 do 5. if y(p i ) > y(p) then 6. p ← p i 7.
append p i to output 8. return output 9. (* we may reverse output if wished by increasing x-coordinate *) • ComputeSkylineBounded(P, s) returns sky(P ), if | sky(P )| ≤ s,
• ComputeSkylineBounded(P, s) returns " incomplete", if | sky(P )| > s,
• ComputeSkylineBounded(P, s) takes O(n log s) time.
Algorithm ComputeSkylineBounded
Input: A set P of points, a positive integer s. Output: skyline for P , if it contains at most s points, "incomplete" otherwise. 1
. M ← 1 + max p∈P {|x(p)|, |y(p)|} 2. t ← n/s 3. split P into t groups P 1 , . . . , P t , each of at most s points 4. for i = 1, . . . , t do 5.
append the points (−M, M ) and (M, −M ) to P i (* dummy end points *) 6.
S i ← SlowComputeSkyline(P i ) 7.
store S i for binary searches along x-coordinate 8. output ← new empty list 9. p ← (−M, M ) (* dummy starting point *) 10. repeat s + 1 times 11. for i = 1, . . . , t do 12.
p i ← succ(S i , x(p)), using binary search 13. p ← highest point among p 1 , . . . , p t , breaking ties in favor of larger x(·) 14.
if x(p) = M then (* we arrived to p = (M, −M ) *) 15.
return output 16. else 17. append p to output 18. return "incomplete" (* because skyline has more than s points *) Note that Algorithm ComputeSkylineBounded can also be used to decide whether h ≤ s for a given s in O(n log s) time.
To compute the skyline we make repetitive use of ComputeSkylineBounded(P, s) increasing the value of s doubly exponentially: if s is too small, we reset s to the value s 2 , and try again. The intuition is that we want to make an exponential search for the value log h using terms of the form log s. Thus, when we are not successful in computing the whole skyline because s < h, we want to set s to double the value of log s. See the algorithm OptimalComputeSkyline in Figure 7. Algorithm OptimalComputeSkyline Input: A set P of points. Output: skyline for P . 1. output ←"incomplete" 2. s ← 4 3. while output ="incomplete"do 4. output ← ComputeSkylineBounded(P, s) 5.
s ← s 2 6. return output Thus, we finish when r = log 2 (log 2 h) and s 1/2 < h ≤ s. Thus the algorithm uses
log 2 (log 2 h) r=1 O(n log 2 2 r ) = log 2 (log 2 h) r=1 O(n2 r ) = O n2 log 2 (log 2 h) = O(n log h) time.
A bit of thought shows that the exponent 2 in Line 5 of OptimalComputeSkyline can be replaced by any constant larger than 1. Thus, we could replace it by s ← s 3 , for example.
Optimization via computation of the skyline
In this section we show how to solve the problem opt(P, k) in O(n log h) time, independently of the value of k. First we show how to solve the decision problem in linear time, assuming that sky(P ) is available, and then consider the optimization problem.
Decision problem
Assume that sky(P ) is already available and sorted by increasing x-coordinate.
For any point p ∈ sky(P ) and any λ ≥ 0, the next relevant point for p with respect to λ, denoted by nrp(p, λ), is is the point q of sky(P ) that is furthest from p subject to the constraints that d(p, q) ≤ λ and x(q) ≥ x(p). See Figure 8. Note that the next relevant point may be p itself, as it may be the only point of sky(P ) that satisfies the conditions. Because of the monotonicity property of Lemma 1, we can find nrp(p, λ) scanning sky(P ) from p onwards; the running time is proportional to the number of points of sky(P ) between p and nrp(p, λ). (One could also use an exponential search, which is significantly better when k n.) For a given λ ≥ 0 we can decide in O(h) time whether opt(P, k) ≤ λ. The idea is a simple greedy approach that is used for several other "1-dimensional" problems: we iteratively compute next relevant points to compute the centers. Detailed pseudocode is in Figure 9. (* find nrp(S[ a ], λ) *) 6.
while
d(S[ a ], S[i]) ≤ λ and i ≤ h do 7. i ← i + 1 8. c a ← i − 1 (* S[c a ]
is the center for the ath cluster *) 9.
(* find nrp(S[c a ], λ) *) 10.
while d(S[c a ], S[i]) ≤ λ and i ≤ h do 11.
i ← i + 1 12. r a ← i − 1 (* dummy operation for easier analysis *) 13. append S[c a ] to the list output (* S[c a ] is a center that covers S[ a ], . . . , S[r a ]*) 14.
if i > h then (* we finished scanning S because S[i − 1] = S[h] *) 15.
return output 16. return "incomplete" The procedure finishes when we use k centers but the last cluster does not cover S[h] because r k < h, or when we use at most k centers and we are covering S[r a ] = S[h] because i = h + 1. In the former case we have opt(S, k) > λ, while in the latter case we have opt(S, k) ≤ λ and output has the list of centers.
The running time is O(h) because the index i only increases and between consecutive increases we spend O(1) time.
Optimization problem
Let us turn our attention now to the optimization problem of computing opt(P, k) and an optimal solution.
We start computing sky(P ) explicitly and storing it in an array S[1 . . . h] sorted by increasing x-coordinate. This takes O(n log h) time using the optimal algorithm presented in Theorem 5 or the algorithms in [17,24].
Consider the h × h matrix M defined by
[M ] i,j = d(S[i], S[j]) if i < j, −d(S[i], S[j]) if i ≥ j.
Lemma 1 implies that M is a sorted matrix : each row has increasing values and each column has decreasing values. The matrix M is not constructed explicitly, but we work with it implicitly.
Note that opt(P, k) is one of the (non-negative) entries of M because opt(P, k) is an interpoint distance between points in sky(P ). We perform a binary search among the entries of M to find the largest entry λ * in M such that opt(P, k) ≤ λ * . We then have opt(P, k) = λ * .
To perform the binary search, we use an algorithm for the selection problem in sets of real numbers: given a set A and a natural number a with 1 ≤ a ≤ |A|, we have to return the element with rank a when A is sorted non-decreasingly. Frederickson This implies that we can make a binary search among the elements of M without constructing M explicitly. More precisely, we make O(log(h 2 )) = O(log h) iterations of the binary search, where at each iteration we select the ath element λ a in M , for a chosen a, and solve the decision problem in O(h) time using Lemma 6. In total we spend O(h log h) time. It is easy to obtain an optimal solution once we have λ * = opt(P, k) = opt(S, k) by calling DecisionSkyline1(S, λ * ). We summarize.
Theorem 7. Given a set P of n points in the plane and an integer parameter k ≤ h, we can compute in O(n log h) time opt(P, k) and an optimal solution, where h is the number of points in the skyline sky(P ). If sky(P ) is already available, then opt(P, k) and an optimal solution can be found in O(h log h) time. [12] for the selection problem follows very much the same paradigm, where a counter is used instead of a decision problem.
An alternative approach is explained and justified in detail in
Both papers [12,13] include pseudocode. For a practical implementation it is probably better to replace the deterministic linear-time computation of medians by a simple randomized linear-time computation [6, Section 2.4].
Optimization without computation of the skyline
We now describe another algorithm for the decision problem and for the optimization problem where we do not have sky(P ) available, and we do not compute.
Decision problem
Assume that we are given P , a positive integer k and a real value λ ≥ 0. We want to decide whether opt(P, k) ≤ λ.
We use the same idea that was exploited in the algorithm ComputeSkylineBounded. We split the point set into groups and compute the skyline of each group. Then we note that we can use a binary search along the skyline of each group to find the next relevant point on the skyline of each group. First we provide a simple technical lemma and then show how to compute the next relevant point along sky(P ). Details are provided as algorithm NextRelevantPoint in Figure 12.
In our discussion we will be using the infinite curve α(p, λ) obtained by concatenating the infinite upward vertical ray from p + (λ, 0), the circular arc centered at p and radius λ from p + (λ, 0) to p + (0, −λ) clockwise, and the vertical downward infinite ray from p + (0, −λ). See Figure 10. Proof. Note that the set of points from Q to the left of α(p, λ) forms a contiguous subsequence of Q (sorted by x-coordinate). Moreover, for any point q, we can decide in constant time whether it is to the left or the right of α(p, λ). Thus, we can perform a binary search along Q to find where the change occurs. p sky(P j ) Figure 11: Cases in the proof of Lemma 9; the shaded region is empty of points. Left: the case when the segment connecting (x(q), y(q )) to q crosses α(p, λ). Right: the case when the segment connecting q to (x(q), y(q )) crosses α(p, λ).
Proof.
For each i ∈ {1, . . . , t}, let q i and q i be the points along sky(P i ) immediately before and after α(p, λ), respectively. The points q i , q i can be computed with a binary search in O(log |P i |) time; see Lemma 8.
Let q and q be the points of sky(P ) immediately before and after α(p, λ), respectively. The task is to find q = nrp(p, λ). Let γ be the L-shaped curve connecting q to (x(q), y(q )) and then to q . See Figure 11. Let P i be the group that contains q and let P j be the group that contains q . If the horizontal segment connecting (x(q), y(q )) to q crosses α(p, λ), then q is the highest point to the right of α(p, λ) (breaking ties in favor of larger x-coordinates). It follows that q = q j and moreover q is the highest point among q 1 , . . . , q t (breaking ties in favor of larger x-coordinates). If the vertical segment connecting q to (x(q), y(q )) crosses α(p, λ), then there is no point of P to the right of x = x(q) and to the left of α(p, λ). If follows that q = q i and moreover q is the rightmost point among q 1 , . . . , q t (breaking ties in favor of larger y-coordinates) . We have seen that q is the rightmost point among q 1 , . . . , q t (breaking ties in favor of larger y(·)) or q is the highest point among q 1 , . . . , q t , (breaking ties in favor of larger x(·)), or both. As it can be seen in the examples of Figure 11, it may be that only one of the claims is true.
Algorithmically we proceed as follows, using Lemmas 2 and 3 for some of the computations. We set q 0 to be the rightmost point among q 1 , . . . , q t (breaking ties in favor of larger y(·)) and q 0 to be highest point among q 1 , . . . , q t (breaking ties in favor of larger x(·)). Using Lemma 3 we check whether q 0 belongs to sky(P ). If q 0 belongs to sky(P ), then we have q = q 0 and return pred(sky(P ), x(q 0 )), where we use Lemma 3 to compute pred( ). If q 0 does not belong to sky(P ), then we have q = q 0 and return q 0 . Details are provided in the algorithm NextRelevantPoint of Figure 12. Now we can solve the decision problem using at most 2k calls to the function that finds the next relevant point. As we did in previous cases, we add dummy points to avoid corner cases where the binary searches would have to return null pointers. Details are provided as algorithm DecisionSkyline2 in Figure 13 For later use, we distinguish a preprocessing part, which is independent of λ and k, and a decision part that depends on λ and k. Indeed, later on we will be using the preprocessing once and the decision part multiple times.
Algorithm NextRelevantPoint
Input: sky(P 1 ), . . . , sky(P t ) stored for binary searches, a point p ∈ sky(P 1 ∪ · · · ∪ P t ) and a real value λ ≥ 0. Output: nrp(p, λ) for P = P 1 ∪ · · · ∪ P t . 1. for i = 1, . . . , t do 2.
q i ← last point in sky(P i ) to the left of or on α(p, λ), using binary search 3. q i ← succ(sky(P i ), x(q i )) (* q i to the right of α(p, λ) *) 4. q 0 ← rightmost point among q 1 , . . . , q t , breaking ties in favor of larger y(·) 5. q 0 ← highest point among q 1 , . . . , q t , breaking ties in favor of larger x(·) 6. (X, r) ← TestSkylineAndPredecessor((sky(P 1 ), . . . , sky(P t )), q 0 ) 7. if X = True 8. then return r 9. else 10.
return q 0 Figure 12: Algorithm to find the next relevant point using sky(P 1 ), . . . , sky(P t ).
Lemma 10. Given a set P of n points and an integer parameter κ ≤ n, we can preprocess P in O(n log κ) time such that, for any given real value λ ≥ 0 and any positive integer k, we can decide in O(k(n/κ) log κ) time whether opt(S, k) ≤ λ.
Proof. See DecisionSkyline2 in Figure 13. Regarding correctness, the same argument that was used in the proof of Lemma 6 applies. At the end of the ath iteration of the for-loop of lines 17-26 we have the following properties:
• the points c 1 , . . . , c a , 1 , . . . , a , r 1 , . . . r a belong to sky(P );
• the point r a is the rightmost point of sky(P ) with the property that the portion of sky(P ) in x ≤ x(r a ) can be covered with a disks of radius λ centered at points of sky(P );
• the disks centered at c 1 , . . . , c a cover the portion of sky(P ) in x ≤ x(r a ).
This claim holds by induction because NextRelevantPoint(p, λ) computes nrp(p, λ) ∈ sky(P ) whenever p ∈ sky(P ); see Lemma 9.
Setting κ = k in Lemma 10 we obtain one of our main results, where we see that computing the skyline, which takes O(n log h) time, is not needed to solve the decision problem. The result is relevant when log k = o(log h); for example, when k = log h.
Theorem 11. Given a set P of n points in the plane, a positive integer parameter k and a real value λ ≥ 0, we can decide in O(n log k) time whether opt(P, k) ≤ λ.
Proof.
We use the Algorithm of Lemma 10, which is actually DecisionSkyline2 in Figure 13, with κ = k. The preprocessing takes O(n log k) and deciding (the unique) λ takes O(k(n/k) log k) = O(n log k) time.
Algorithm DecisionSkyline2
Input: A set P of points, positive integers k and κ ≤ n and a real value λ ≥ 0. Output: A solution Q ⊆ sky(P ) with |Q| ≤ k and ψ(Q, P ), if opt(S, k) ≤ λ, and "incomplete" if λ < opt(S, k). 1. (* Preprocessing *) 2. p 0 ← highest point of P , breaking ties in favor of larger x(·) 3. q 0 ← rightmost point of P , breaking ties in favor of larger y(·) 4. λ max ← 1 + d(p 0 , q 0 ) (* upper bound for opt(P, k) *) 5. M ← 2λ max + max p∈P {|x(p)|, |y(p)|} 6. t ← n/κ 7. split P into t groups P 1 , . . . , P t , each of at most κ points 8. for i = 1, . . . , t do 9.
append the points (−M, M ) and (M, −M ) to P i (* dummy end points *) 10.
S i ← SlowComputeSkyline(P i ) 11. store S i in an array by increasing x-coordinate 12. (* Decision *) 13. if λ ≥ λ max then 14. return p 0 (* or any other point of sky(P ) *) 15. output ← new empty list 16. 1 ← p 0 (* first non-dummy point of sky(P ) *) 17. for a = 1, . . . , k do 18.
c a ← NextRelevantPoint((S 1 , . . . , S t ), a ) 19.
r a ← NextRelevantPoint((S 1 , . . . , S t ), c a ) 20.
append c a to the list output (* c a is a center that covers the portion of sky(P ) from a to r a *) 21.
(* compute a+1 = succ(sky(P ), x(r a )) *) 22.
for i = 1, . . . , t do 23.
p i ← succ(S i , x(r a )), using binary search 24. a+1 ← highest point among p 1 , . . . , p t , breaking ties in favor of larger x(·) 25.
if x( a+1 ) = M then (* a+1 is the dummy point and we have finished *) 26.
return output 27. return "incomplete" (* because λ < opt(S, k) *) Figure 13: Decision algorithm for testing whether opt(S, k) ≤ λ. The index a is superfluous but helps analyzing the algorithm.
Optimization problem
Now we turn to the optimization problem. We use the paradigm of parametric search; see for example [13,22,29]. Our presentation can be understood without a previous knowledge of the technique.
For convenience we set λ * = opt(P, k).
We simulate running the decision algorithm DecisionSkyline2, analyzed in Lemma 10, for the unknown value λ * . Whenever we run into a computation that depends on the value of λ * , we apply a binary search to find an interval (λ 1 , λ 2 ] with the following properties: (i) (λ 1 , λ 2 ] contains λ * , and (ii) the next step of the algorithm would be the same for any λ ∈ (λ 1 , λ 2 ]. We can then perform the next step of the algorithm for λ * as we would perform it for λ 2 .
For the binary search we will use the following simple algorithm to select the element with a given rank in a collection of sorted arrays. As discussed in the proof, the lemma is not optimal but it suffices for our purpose and its proof is simple enough to implement it.
Lemma 12. Assume that we have t arrays S 1 , . . . , S t , each with numbers sorted increasingly. Set S to be the union of the elements in S 1 , . . . , S t and let n be the number of elements in the union. For any given value λ * , we can find λ = min{x ∈ S | x ≥ λ * } in O(t log 2 n) time plus the time needed to compare O(log n) times some element of S with λ * .
Proof. We use a recursive algorithm. For simplicity we describe the algorithm assuming that the elements of S 1 , . . . , S t are pairwise different. Breaking ties systematically we can assume this; for example replacing each element S i [j] with the triple (S i [j], i, j) and making comparisons lexicographically. We keep an active contiguous portion of each S i ; we store it using indices to indicate which subarray is active. Thus, the recursive calls only takes 2t indices as input.
Let m i be the median of the active subarray of S i ; assign weight w i to m i , where w i is the number of active elements in S i . We can find m i and w i in O(1) time using arithmetic of indices. We select the weighted median M of m 1 , . . . , m t in O(t) time. We compare M against λ * to decide whether λ * ≤ M or M < λ * . If λ * ≤ M , then we clip the active part of each subarray S i to the elements at most M . If M < λ * , then we clip the active part of each subarray S i to the elements at least M . Then we call recursively to search for λ * in the active subarrays.
Note that at some point the active subarray of an array may be empty. This does not affect the approach; we just skip that subarray (or use weight 0 for that). Clipping one single subarray can be done in O(log n) time using a binary search. Each call to the function takes O(t log n) time, plus one comparison between an element of S and λ * , plus the time for recursive calls. Since at each call we reduce the size of the active arrays by at least 1/4, we make O(log n) recursive calls. The result follows.
It should be noted that there is also a very simple randomized solution that probably works better in practice: at each iteration choose one entry uniformly at random among all active subarrays of the arrays, compare the chosen element to λ * , and clip the active subarrays accordingly. One can do faster deterministically [11,12,16], but it does not matter in our application.
We next provide the parametric version of Lemma 9; the algorithm ParamNextRele-vantPoint in Figure 14 is the parametric counterpart of algorithm NextRelevantPoint.
Lemma 13. Assume that P is split into P 1 , . . . P t , and for each i ∈ {1, . . . , t} the points of sky(P i ) are sorted by x-coordinate for binary searches. For any given point p ∈ sky(P ) we can compute the next relevant point nrp(p, λ * ) in O(t log 2 n) time plus the time needed to solve O(log n) decision problems, without knowing λ * .
Proof. Consider the algorithm ParamNextRelevantPoint given in Figure 14 Let Λ = {d(p, q) | q ∈ sky(P ), x(p) ≤ x(q)} and for each i ∈ {1, . . . , t} let Λ i = {d(p, q) | q ∈ sky(P i ), x(p) ≤ x(q)}. Obviously we have Λ ⊆ i Λ i and 0 ∈ Λ. The sets Λ and Λ i are not constructed explicitly, but are manipulated implicitly.
The boundary cases, when λ * = 0 = min Λ and when λ * > max Λ, are treated in lines 5-11. In the remaining case we have 0 < λ * ≤ max Λ.
Consider any index i ∈ {1, . . . , t}. Since p ∈ sky(P ), the point p also belongs to sky(P i ∪ {p}). From Lemma 1 we then obtain that the distances from p increase along sky(P i ), assuming that we are on the same side of the vertical line x = x(p). It follows that In short, the values of Λ i , which are of the form d(p, q), are sorted increasingly when q iterates over S i [1 + a i , . . . , h i ].
∀j, j with a i < j < j ≤ h i : d(p, S i [j]) < d(p, S i [j ]).
Since Λ ⊆ i Λ i , we can perform a binary search in the union of the t sorted lists Λ 1 , . . . , Λ t to find the smallest element λ of i Λ i with λ * ≤ λ . The next relevant point from p for λ * and for λ is the same. Thus, we can return the next relevant point for λ .
To analyze the running time, we note that lines 1 For the case k < n 1/4 , we consider the algorithm ParametricSearchAlgorithm given in Figure 15. In lines 1-12 we make exactly the same preprocessing as in the algorithm DecisionSkyline2 with κ = k 3 log 2 n ≤ n. As stated in Lemma 10, the preprocessing takes O(n log κ) = O(n(log k + loglog n)) time and each subsequent decision problem can be solved in time O(k(n/κ) log κ) = O((n/k 2 log 2 n)(log k + loglog n)) = O(n/k log n).
The rest of the algorithm, lines 13-25, follows the paradigm of the decision part of DecisionSkyline2. We make k iterations, where at each ath iteration we compute points a , c a , r a ∈ sky(P ) such that c a = nrp( a , λ * ) and r a = nrp(c a , λ * ). The point a+1 is then set to be the point after r a along sky(P ). 14. output ← new empty list 15. 1 ← p 0 (* first non-dummy point of sky(P ) *) 16. for a = 1, . . . , k do 17. 19. append c a to the list output (* c a is a center that covers portion of sky(P ) from a to r a *) 20.
c a ← ParamNextRelevantPoint((S 1 , . . . , S t ), a ) 18. r a ← ParamNextRelevantPoint((S 1 , . . . , S t ), c a )
(* compute a+1 = succ(sky(P ), x(r a )) *) 21.
for i = 1, . . . , t do 22. p i ← succ(S i , x(r a )), using binary search 23. a+1 ← highest point among p 1 , . . . , p t , breaking ties in favor of larger x(·) 24.
if x( a+1 ) = M then (* a+1 is the dummy point and we have finished *) 25.
return output Figure 15: Algorithm to find an optimal solution to opt(P, k). The index a is superfluous but helps analyzing the algorithm. Correctness follows the same line of thought as for algorithm DecisionSkyline2 (see Lemma 10), using that ParamNextRelevantPoint correctly finds nrp(p, λ * ) for the unknown value λ * = opt(P, k).
Note that the algorithm finds an optimal solution, but, as written, does not return the value opt(P, k). Assuming, for simplicity, that the returned solution has k points, we then have opt(P, k) = ψ(P, {c 1 , . . . , c k }) = max This quantity can be computed through the iterations of the for-loop. 6 Algorithms for very small k
In this section we show that the problem opt(P, k = 1) can be solved in linear time; note that for this running time we cannot afford to construct the skyline explicitly. We also provide a (1 + ε)-approximation for opt(P, k) that is fast when k is very small. The main tool is the following result.
Lemma 15. Let P be a set of n points in the plane and let p 0 and q 0 be two distinct points of sky(P ) with x(p 0 ) < x(q 0 ). Consider the portion of the skyline S = {p ∈ sky(P ) | x(p 0 ) ≤ x(p) ≤ x(q 0 )}; this portion or the whole skyline is not available. In O(n) time we can compute points r * and r * of S such that r * = arg min p∈S max{d(p, p 0 ), d(p, q 0 )} and r * = arg max p∈S min{d(p, p 0 ), d(p, q 0 )}.
Proof.
We can assume that all the points p of P have x(p 0 ) ≤ x(p) ≤ x(q 0 ) because other points can be ignored. Let β be the bisector of p 0 q 0 ; note that β has positive slope and is not vertical. Let p be the point of sky(P ) to the left of or on β with largest x-coordinate.
Let q be the point of sky(P ) to the right of β with smallest x-coordinate. We show that {r * , r * } ⊆ {p , q } and that p , q can be computed in linear time. The result follows because we can just evaluate max{d(p, p 0 ), d(p, q 0 )} and min{d(p, p 0 ), d(p, q 0 )} for p = p and p = q and select the best. First we show that r * = p or r * = q . Consider any point p ∈ S with x(p) < x(p ); we then have x(p 0 ) ≤ x(p) < x(p ). Because of Lemma 1 and because p, p are to the left of the bisector β we have max{d(p, p 0 ), d(p, q 0 )} = d(p, q 0 ) > d(p , q 0 ) = max{d(p , p 0 ), d(p , q 0 )}, which means that p cannot be the optimal point r * . A symmetric argument shows that any point p ∈ S with x(q ) < x(p) ≤ x(q 0 ) cannot be optimal. We conclude that r * is p or q . A similar argument shows that r * is also p or q . Indeed, for any point p ∈ S with x(p 0 ) ≤ x(p) < x(p ) we have min{d(p, p 0 ), d(p, q 0 )} = d(p, p 0 ) < d(p , p 0 ) = min{d(p , p 0 ), d(p , q 0 )}, and the argument for p ∈ S with x(q ) < x(p) ≤< x(q 0 ) is similar.
It remains to explain how to compute p and q in linear time. Let p 1 be the point to the left of or on β with largest x-coordinate (breaking ties in favor of larger y-coordinate), and let q 1 be the point to the right of β with largest y-coordinate (breaking ties in favor of larger x-coordinate).
The same argument that was used in the proof of Lemma 9 shows that p 1 ∈ sky(P ) or q 1 ∈ sky(P ) (or both). We repeat the argument to make the proof self-contained. See Figure 16 Consider the last point p ∈ sky(P ) to the left of or on β and the first point q ∈ sky(P ) to the right of β. Define γ as the L-shape curve connecting p to (x(p ), y(q )) and to q . If γ crosses β at the vertical segment connecting p to (x(p ), y(q )), then the point p has to be the rightmost point to the left of or on β and thus p = p 1 . If γ crosses β at the horizontal segment connecting (x(p ), y(q )) to q , then q has to be the topmost point to the right of β and thus q = q 1 .
We check whether q 1 belongs to sky(P ) in linear time. Checking this amounts to checking whether y(q 1 ) is the unique maximum among y(q) for all the points q ∈ P with x(q 1 ) ≤ x(q) ≤ x(q 0 ). If q 1 ∈ sky(P ), then it must be q 1 = q and we can compute p using that x(p ) = max{x(p) | x(p) < x(q 1 ), y(p) > y(q 1 )} the case when the segment connecting p to (x(p ), y(q )) crosses β; the square represents a point showing that sometimes q 1 / ∈ sky(P ). Right: the case when the segment connecting (x(p ), y(q )) to q crosses β; the square represents a point showing that sometimes p 1 / ∈ sky(P ).
If q 1 / ∈ sky(P ), we infer that p 1 ∈ sky(P ) and thus p = p 1 . We can compute the point q using that y(q ) = max{y(p) | x(p 1 ) < x(p)}.
The whole procedure to find p and q , as described, can be implemented to take linear time because we only need O(1) scans of the point set P .
The 1-center problem can be solved in linear time using Lemma 15.
Theorem 16. Given a set P of n points in the plane, we can compute in O(n) time opt(P, 1) and an optimal solution.
Proof. Let p 0 be the point with largest y-coordinate (breaking ties in favor of larger xcoordinate) and let q 0 be the point with the largest x-coordinate (breaking ties in favor of larger y-coordinate). The points p 0 and q 0 are the extreme points of sky(P ). Note that for each point p ∈ sky(P ) we have ψ({p}, P ) = max{d(p, p 0 ), d(p, q 0 )} because of Lemma 1.
If p 0 = q 0 , we return p 0 as the solution, which has cost 0. If p 0 = q 0 , we apply Lemma 15 to compute r * = arg min p∈sky(P ) max{d(p, p 0 ), d(p, q 0 )} = arg min p∈sky(P ) ψ({p}, P ) in linear time, and return r * .
We next provide a 2-approximation for opt(P, k) that is relevant when k is very small. As soon as k = Ω(loglog n), Theorem 14 is better, as it can compute an optimal solution. Lemma 17. Given a set P of n points in the plane and a positive integer k, we can compute in O(kn) time a feasible solution Q ⊆ sky(P ) with at most k points such that ψ(Q, P ) ≤ 2 · opt(P, k). In the same time bound we also get ψ(Q, P ).
The theorem implies that for any constant k we can compute a (1 + ε)-approximation in O(n loglog(1/ε)) time.
Discussion
We have improved previous results for computing the distance-based representatives of the skyline in the plane, a problem that is relevant in the context of databases, evolutionary algorithms and optimization. We have shown that such representatives can be computed without constructing the skyline, which is a conceptual shift with respect to previous works and approaches. For example, with a relatively simple algorithm we can solve the decision problem in O(n log k) time, which is asymptotically optimal. We also provided algorithms for the optimization problem that are asymptotically optimal for a large range of values of k: Theorem 7 is optimal when log k = Ω(log h) and Theorem 14 is optimal when k = Ω(log n).
Most of our algorithms are easy enough to be implemented. Specially simple is the new decision algorithm that does not require computing the skyline. In several cases, the decision combined with a trivial binary search may suffice to get reasonable results. A practical implementation would use randomized algorithms for selection, instead of deterministic linear-time median finding.
One may wonder whether using the Euclidean distance is a reasonable choice. That depends on the application. The approach can be modified easily to work with other distances, such as the L ∞ or the L 1 -metric. The main property that we need is that any disk (in the chosen metric) centered at a point of the skyline intersects the skyline in a connected subpiece. From this we can infer monotonicity of the distances (Lemma 1) and we can perform binary searches along the skyline.
We next describe two research directions where progress is awaiting. The first question is whether we can compute opt(P, k) in O(n log k) for all values k. It may be good to start considering the case when k is constant. For example, can we compute opt(P, 15) in O(n) time when h = Θ( √ n)? One can imagine scenarios where we want to solve opt(P, k) for several different values k. Can we exploit correlation between the problems in a non-trivial way? More precisely, what is the computational complexity of the following problem: given a set P of points in the plane and a set K ⊆ {1, . . . , n}, compute opt(P, k) for all k ∈ K.
Q in the open halfplane to the right of the vertical line x = x 0 . Similarly, we denote by pred(Q, x 0 ) the rightmost point of Q in the open halfplane to the left of the vertical line x = x 0 . Thus
Figure 4 :
4Idea in SlowComputeSkyline.
Figure 5 :
5Algorithm to compute sky(P ) in O(n log n) time. The idea of the subroutine ComputeSkylineBounded is as follows. First we compute a value M that is an upper bound on the absolute values of all coordinates. We will use the points (−M, M ) and (M, −M ) as dummy points to avoid having boundary cases. Let P = P ∪ {(−M, M ), (M, −M )}. We split P into t ≈ n/s groups of points, P 1 , . . . , P t , each with roughly s points, and add the dummy points (−M, M ) and (M, −M ) to each group P i . We then compute the skyline of each group P i using an algorithm with time complexity O(n log n), for example SlowComputeSkyline. Each outcome sky(P i ) is stored in an array sorted by increasing x-coordinate, so that we can perform binary searches along sky(P i ). Now we can compute sky(P ) = sky(P ) ∪ {(−M, M ), (M, −M )}. Assuming that we are at a point p of sky(P ), we can compute the next point p = succ(sky(P ), x(p)) along sky(P ) by taking the highest among succ(sky(P 1 ), x(p)), . . . , succ(sky(P t ), x(p)); ties are broken by taking the point with largest x-coordinate. The correctness of this is formalized in Lemma 2. The procedure is started from the dummy point (−M, M ), and iteratively compute the next point s times. If at some point we reach the dummy point (M, −M ), we know that we computed the whole sky(P ) (we do not report (−M, M ) and (M, −M ) as part of the output). If after s + 1 iterations we did not arrive to the dummy point (M, −M ), then we have computed at least s + 1 points of sky(P ) and correctly conclude that | sky(P )| > s.
Lemma 4 .
4Algorithm ComputeSkylineBounded has the following properties:
Figure 6 :
6Algorithm to compute sky(P ) in O(n log s) time, if h ≤ s. Proof. Because of Lemma 2, in lines 11-13 we correctly replace p by succ(sky(P ), x(p)) in each iteration of the repeat loop. If we reach p = (M, −M ) at some point, then we know that we computed the whole sky(P ). If after s + 1 iterations we did not reach p = (M, −M ), then we computed s + 1 points of sky(P ) and the algorithm returns "incomplete". It remains to bound the running time. For each P i we spend O(s log s) time to compute sky(P i ). This means that we spend O((n/s) · s log s) = O(n log s) time to compute all sky(P 1 ), . . . , sky(P t ). In each iteration of the repeat loop we make O(t) = O(n/s) binary searches among O(s) elements, because each S i has O(|P i |) = O(s) elements. Thus, in one iteration we compute the points p 1 , . . . , p t in O(t log s) = O((n/s) log s) time, and selecting p then takes additional O(n/s) time. We conclude that each iteration takes O((n/s) log s) time, and since there are s + 1 iterations, the time bound follows.
Theorem 5 .
5The skyline of a set of n points can be computed in O(n log h) time, where h is the number of points in the skyline.Proof. Consider the algorithm OptimalComputeSkyline inFigure 7. In the calls to ComputeSkylineBounded(P, s) we have s of the form 2 2 r for r = 1, 2, 3 . . . , until h ≤ 2 2 r .
Figure 7 :
7Algorithm to compute sky(P ) in O(n log h) time.
Lemma 6 .Figure 8 :
68Given a skyline S sorted by x-coordinate, an integer k and a value λ, we can decide in O(h) time whether opt(S, k) ≤ λ.Proof. Consider the algorithm DecisionSkyline1 inFigure 9. The input is an array S[1 .. . h] describing the skyline sorted by x-coordinate. We use an index i that scans the skyline S[.. . ]. At the ath iteration of the for-loop, we have an index a and we find the The next relevant point for p with respect to λ is indicated with a squared mark.Algorithm DecisionSkyline1Input: A skyline given as an array S[1 . .. h] sorted by increasing x-coordinate, a positive integer k ≤ h and a real value λ ≥ 0. Output: A solution Q ⊆ S with |Q| ≤ k and ψ(Q, S) ≤ λ, if opt(S, k) ≤ λ, and "incomplete" if λ < opt(S, k). 1. output ← new empty list 2. i ← 1 (* counter to scan S[·] *) 3. for a = 1, . . . , k do 4. a ← i (* S[ a ] is the first point to be covered by the ath center *) 5.
Figure 9 :
9Decision algorithm for testing whether opt(S, k) ≤ λ. The index a is superfluous but helps analyzing the algorithm.largest indices c a , r a such that S[c a ] = nrp(S[ a ], λ) and S[r a ] = nrp(S[c a ], λ). (We use a , c a and r a because of left, center and right for the ath cluster.) With a simple inductive argument we have the following property: after a iterations of the repeat loop, the index i is the maximum index with the property that S[1 . . . (i − 1)] can be covered with a disks of radius λ centered at points of S.
and Johnson [12, Theorem 1] show that the selection problem for the entries of a sorted matrix M of size h × h can be solved in O(h) time looking at O(h) entries of M . This means that, for any given a with 1 ≤ a ≤ h 2 , we can select in O(h) time the ath element of the entries of M .
Frederickson and Zhou [13, Lemma 2.1]. The bottom line is that we can perform a binary search in M using O(h) time and looking at O(h) entries in M , plus the time needed to solve O(log h) decision problems. We provide a high-level description and refer to [13] for the details, including pseudocode. The algorithm is iterative and maintains a family M of submatrices of M . We start with M = {M }. At the start of each iteration we have a family M of submatrices of M that may contain the value λ * , are pairwise disjoint, and have all the same size. In an iteration, we divide each submatrix in M into four submatrices of the same size, and perform two pruning steps, each using a median computation and solving a decision problem. At each iteration the size of the submatrices in M halves in each dimension and the number of submatrices in M at most doubles. After O(log h) iterations, the family M contains O(h) matrices of size 1 × 1, and we can perform a traditional binary search among those values. Actually, the algorithm of Frederickson and Johnson
Figure 10 :
10The curve α(p, λ).
Lemma 8 .
8Given a skyline Q sorted by x-coordinate such that Q has points on both sides of α(p, λ), we can compute in O(log |Q|) time the point of Q to the left of α(p, λ) and maximum x-coordinate and the point of Q to the right of α(p, λ) and minimum x-coordinate.
Lemma 9 .
9Assume that P is split into P 1 , . . . P t , and for each i ∈ {1, . . . , t} the points of sky(P i ) are sorted by x-coordinate for binary searches. Let p be a point of sky(P ). Then we can find the next relevant point nrp(p, λ) in O(t + i log |P i |) time.
Lines 1-11 correspond to the preprocessing, independent of λ and k, while lines 12-27 correspond to the decision problem. The running time for the preprocessing is O(n + i |P i | log |P i |) = O(n + (n/κ)κ log κ) = O(n log κ). In the decision part we perform at most k iterations, where each iteration takes O(t) = O(n/κ) time plus the time for O(1) calls to NextRelevantPoint (Lemma 9) with t = O(n/κ). Thus, each query takes O(k) · O(n/κ) + O((n/κ) log κ) time. The claim about the running times follow.
. The input sky(P i ) is stored in an array S i [1 . . . h i ]. In line 3 we perform a binary search along S i [. . . ] to find the index a i such that S i [1 + a i , . . . , h i ] contains the points in x ≥ x(p).
Figure 14 :
14Algorithm to find the next relevant point for the unknown λ * using sky(P 1 ), . . . , sky(P t ).
-5 perform O(t) binary searches, each with a running time of O(log n), and some other operations using O(t) time. Thus, this part takes O(t log n) time. In lines 8-11 we are making two calls to the decision problem plus O(1) time. In lines 12-13 we use Lemma 12, which takes O(t log 2 n) time plus the time needed to solve O(log n) decision problems.Theorem 14. Given a set P of n points in the plane and an integer parameter k, we can compute opt(P, k) and an optimal solution in O(n log k + n loglog n) time.Proof. If k ≥ n 1/4 , then log k = Θ(log n) and we can use Theorem 7 to obtain an algorithm with running time O(n log h) = O(n log n) = O(n log k).
We are making O(k) calls to the function ParamNextRelevantPoint and, as shown in Lemma 13, each of them takes O(t log 2 n) time plus the time needed to solve O(log n) decision problems. In total we spend in all calls to ParamNextRelevantPoint O(k) · O((n/k 3 log 2 n) log 2 n) + O(log n) · O(n/k log n) = O(n) time. In lines 21-23 we spend additional O(k) · O(t log κ) = O(n) time. The claim about the running time follows.
Figure 16 :
16Cases in the proof of Lemma 15; the shaded region is empty of points. Left:
a=1,...,k {d(c a , a ), d(c a , r a )} .
Proof. We assume k ≥ 2 because the case k = 1 is covered by Theorem 16. Let p 0 be the point with largest y-coordinate (breaking ties in favor of larger x-coordinate) and let q 0 be the point with the largest x-coordinate (breaking ties in favor of larger ycoordinate). The points p 0 and q 0 are the extreme points of sky(P ).Set c 1 = p 0 , c 2 = q 0 , and for i = 3, . . . k, let c i be the point of sky(P ) that is furthest from c 1 , . . . , c i−1 . A classical result by Gonzalez [14] included in textbooks (for example[30,Section 2.2]) shows that C = {c 1 , . . . , c k } is a 2-approximation: we have ψ(C, P ) ≤ 2 · opt(P, k).The points in C can be computed iteratively in O(kn) time as follows. Assume that we computed c 1 , ... , c i and we want to compute c i+1 . The vertical lines though c 1 , ... c i split the region x(p 0 ) ≤ x ≤ x(q 0 ) into slabs. We maintain for each slab σ the points of P that are inside σ. For each slab σ, let c(σ) ∈ {c 1 , ... , c i } be the point that defines its left boundary and let c (σ) ∈ {c 1 , ... , c i } be the point defining its right boundary. Obviously x(c) < x(c ). We can use Lemma 15 to compute the point r * (σ) = arg max p∈sky(P )∩σ min{d(p, c(σ)), d(p, c (σ))}.Because of Lemma 1, for each slab σ and each p ∈ sky(P ) ∩ σ we have max{d(p, c 1 ), d(p, c 2 ), . . . , d(p, c i )} = max{d(p, c(σ)), d(p, c (σ)).If follows that the point c i+1 has to be one of the points r * (σ), where σ iterates over all slabs defined by c 1 , . . . , c i . Thus we take c i+1 to be the point achieving the maximumLet σ * be the slab defining c i+1 ; thus c i+1 = r * (σ * ). Then the slab σ * has to be split into two subslabs with the vertical line x = x(c i+1 ), and the points of P ∩σ * have to be rearranged into the two new subslabs. Since the slabs are interior disjoint, the use of Lemma 15 over all slabs takes linear time. The rest of the work to compute c i+1 and update the split of P into slabs also takes linear time. We conclude that the construction of C takes O(kn) time.Note that with an extra round, we can compute within each slab the point that is furthest from any point of C. This would be the steps to compute c k+1 . Computing the distance from c k+1 to C we obtain ψ(C, P ).From a 2-approximation we can compute a (1 + ε)-approximation using binary search. This is relevant when k = o(loglog n), as otherwise Theorem 14 computes an optimal solution.Theorem 18. Given a set P of n points in the plane, a positive integer k and a real value ε with 0 < ε < 1, we can compute in O(kn + n loglog(1/ε)) time a feasible solution Q with at most k points such that ψ(Q, P ) ≤ (1 + ε) opt(P, k).Proof.We use Lemma 17 to compute a value λ such that λ ≤ 2 opt(P, k) ≤ 2λ. This takes O(kn) time. Then we perform a binary search among the O(1/ε) values λ, λ + ελ 2 , λ + 2 ελ 2 , λ + 3 ελ 2 , λ + 4 ελ 2 , . . . , ≈ 2λ to find the index j such that λ + (j − 1) ελ 2 < opt(P, k) ≤ λ + j ελ 2 . For this we have to solve O(log(1/ε)) decision problems. Once we have the index j, we solve the decision problem for λ + j ελ 2 and return the feasible solution Q that we obtain. This is a (1 + ε) approximation because ψ(Q, P ) ≤ λ + j ελ 2 = λ + (j − 1) ελ 2 + ελ 2 < opt(P, k) + ε opt(P, k).To analyze the running time of this step, we note that we have to solve O(log(1/ε)) decision problems, all for the same number of points k. Using Theorem 10 with κ = k 2 log 2 (1/ε), we spend
] sorted by increasing x-coordinate, and a point p ∈ sky. . . Sky, Algorithm ParamNextRelevantPoint Input: sky(P 1 ). P t ) given as arrays S 1 [1 . . . h 1 ], . . . , S t [1 . . . h t. P 1 ∪ · · · ∪ P tAlgorithm ParamNextRelevantPoint Input: sky(P 1 ), . . . , sky(P t ) given as arrays S 1 [1 . . . h 1 ], . . . , S t [1 . . . h t ] sorted by in- creasing x-coordinate, and a point p ∈ sky(P 1 ∪ · · · ∪ P t ).
Output: nrp(p, λ * ). Output: nrp(p, λ * )
t do 2. q i ← S i. , . , h i ] (* last point of sky(P i ) *for i = 1, . . . , t do 2. q i ← S i [h i ] (* last point of sky(P i ) *)
a i ← number of elements q ∈ sky(P i ) with x(q) < x(p). a i ← number of elements q ∈ sky(P i ) with x(q) < x(p)
h i ] stores {q ∈ sky(P i ) | x(p) ≤ x(q)} *). * S I, a i + 1, . . .* S i [a i + 1, . . . , h i ] stores {q ∈ sky(P i ) | x(p) ≤ x(q)} *).
* binary search in sorted lists i Λ i = i {d(p, q) | q ∈ S i. a i + 1, . . . , h i ]} *(* binary search in sorted lists i Λ i = i {d(p, q) | q ∈ S i [a i + 1, . . . , h i ]} *)
sky(P t )), p, λ ) Algorithm ParametricSearchAlgorithm Input: A set P of points and a positive integer k ≤ n 1/4 . Output: A solution Q ⊆ sky(P ) with |Q| ≤ k. ) , . , returnNextRelevantPoint((sky(P 1. and ψ(Q, P ) = opt(P, kreturnNextRelevantPoint((sky(P 1 ), . . . , sky(P t )), p, λ ) Algorithm ParametricSearchAlgorithm Input: A set P of points and a positive integer k ≤ n 1/4 . Output: A solution Q ⊆ sky(P ) with |Q| ≤ k and ψ(Q, P ) = opt(P, k).
Preprocessing -same as in DecisionSkyline2 *). (* Preprocessing -same as in DecisionSkyline2 *)
p 0 ← highest point of P , breaking ties in favor of larger x(·). p 0 ← highest point of P , breaking ties in favor of larger x(·)
q 0 ← rightmost point of P , breaking ties in favor of larger y(·). q 0 ← rightmost point of P , breaking ties in favor of larger y(·)
P t , each of at most κ points 9. . . , points (−M, M ) and (M, −M ) to P i (* dummy end points *). split P into t groups P 1split P into t groups P 1 , . . . , P t , each of at most κ points 9. for i = 1, . . . , t do 10. append the points (−M, M ) and (M, −M ) to P i (* dummy end points *)
. S I ← Slowcomputeskyline, P iS i ← SlowComputeSkyline(P i )
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| [] |
[
"Radial Duality Part II: Applications and Algorithms",
"Radial Duality Part II: Applications and Algorithms"
] | [
"Benjamin Grimmer "
] | [] | [] | The first part of this work established the foundations of a radial duality between nonnegative optimization problems, inspired by the work of Renegar[38]. Here we utilize our radial duality theory to design and analyze projection-free optimization algorithms that operate by solving a radially dual problem. In particular, we consider radial subgradient, smoothing, and accelerated methods that are capable of solving a range of constrained convex and nonconvex optimization problems and that can scale-up more efficiently than their classic counterparts. These algorithms enjoy the same benefits as their predecessors, avoiding Lipschitz continuity assumptions and costly orthogonal projections, in our newfound, broader context. Our radial duality further allows us to understand the effects and benefits of smoothness and growth conditions on the radial dual and consequently on our radial algorithms.The first part of this work [17] established a theory of radial duality relating nonnegative optimization problems through a projective transformation, extending the ideas of Renegar [38] from their origins in conic programming. We give a minimal overview here of our radial duality theory needed to be- | 10.1007/s10107-023-01974-0 | [
"https://export.arxiv.org/pdf/2104.11185v4.pdf"
] | 233,346,781 | 2104.11185 | fe214a6e829b436c48949958384953a29dce693e |
Radial Duality Part II: Applications and Algorithms
Benjamin Grimmer
Radial Duality Part II: Applications and Algorithms
Received: date / Accepted: dateNoname manuscript No. (will be inserted by the editor)Optimization · Projection-free Methods · Convex · Nonconvex · Nonsmooth · First-Order Methods · Projective Transformations
The first part of this work established the foundations of a radial duality between nonnegative optimization problems, inspired by the work of Renegar[38]. Here we utilize our radial duality theory to design and analyze projection-free optimization algorithms that operate by solving a radially dual problem. In particular, we consider radial subgradient, smoothing, and accelerated methods that are capable of solving a range of constrained convex and nonconvex optimization problems and that can scale-up more efficiently than their classic counterparts. These algorithms enjoy the same benefits as their predecessors, avoiding Lipschitz continuity assumptions and costly orthogonal projections, in our newfound, broader context. Our radial duality further allows us to understand the effects and benefits of smoothness and growth conditions on the radial dual and consequently on our radial algorithms.The first part of this work [17] established a theory of radial duality relating nonnegative optimization problems through a projective transformation, extending the ideas of Renegar [38] from their origins in conic programming. We give a minimal overview here of our radial duality theory needed to be-
gin algorithmically benefiting from it and then a fuller but terse summary in Section 2.3 necessary to derive our radial optimization guarantees.
For a finite dimensional Euclidean space E, our three transformations of interest are the radial point transformation, radial set transformation, and upper radial function transformation, which are denoted by Γ (x, u) = (x, 1)/u,
Γ S = {Γ (x, u) | (x, u) ∈ S}, f Γ (y) = sup{v > 0 | (y, v) ∈ Γ (epi f )}
for any point (x, u) ∈ E × R ++ , set S ⊆ E × R ++ , and function f : E → R ++ , respectively. Here R ++ denotes the extended positive reals R ++ ∪ {0, +∞}. It is immediate that the point and set transformations are dual since
Γ Γ (x, u) = Γ (x, 1) u = (x/u, 1) 1/u = (x, u).
Central to establishing our theory of radial duality is the characterization of exactly when this duality carries over to the function transformation. We say a function f is upper radial if the perspective function f p (y, v) = v · f (y/v) is upper semicontinuous and nondecreasing in v ∈ R ++ . Moreover, it is strictly upper radial if it is strictly increasing in v whenever f p (y, v) ∈ R ++ . The cornerstone theorem of our radial duality [17,Theorem 1] is that
f = f Γ Γ ⇐⇒ f is upper radial.(1)
The duality of the radial function transformation provides a duality between optimization problems (see Section 4 of [17]). For any strictly upper radial function f : E → R ++ , consider the primal problem
p * = max x∈E f (x).(2)
Then the radially dual problem is given by
d * = min y∈E f Γ (y)(3)
and has (argmax f ) × {p * } = Γ (argmin f Γ ) × {d * } . Thus maximizing f is equivalent to minimizing f Γ and solutions can be converted between these problems by applying the radial point transformation Γ or its inverse (which is also Γ by duality). Importantly, the two nonnegative optimization problems (2) and (3) can exhibit very different structural properties. For example, consider maximizing f (x) = 1 − x 2 2+ which takes value zero outside the unit ball and has arbitrarily large gradients and Hessians as x approaches the boundary of this ball. Its radial dual f Γ (y) = 1 + y 2 2 has a full domain with gradients and Hessians bounded in norm by one everywhere. Thus our radial duality theory poses an opportunity to extend the reach of many standard optimization algorithms reliant on such structure. The previous works of Renegar [38] and Grimmer [16] analyzing subgradient methods and Renegar [39] employing accelerated smoothing techniques on a radial reformulation of the objective critically rely on the reformulation being uniformly Lipschitz continuous, which always occurs in the special cases of the radial dual that they consider.
Our Contributions. This work leverages our radial duality theory to present and analyze projection-free radial optimization algorithms in this newfound, wider context than previous works were able to. Finding that a mild condition ensures the radial dual is uniformly Lipschitz continuous, we analyze a radial subgradient method for a broad range of non-Lipschitz primal problems with or without concavity. Observing that constraints radially transform into related gauges, we propose a radial smoothing method that takes advantage of this structure for concave maximization. Further, we find that our radial transformation extends smoothness on a level set of the primal to hold globally in the radial dual, which prompts our analysis of a radial accelerated method. More important than these particular algorithms, this work aims to demonstrate the breadth of applications and algorithms that can be approached using our radial duality theory.
Outline. We begin with a motivating example of the computational benefits and scalability that follow from designing algorithms based on the radial dual (3) in Section 2. Then Section 3 formally establishes algorithmically useful properties of our radial dual, namely Lipschitz continuity, smoothness, and growth conditions. Finally, Section 4 addresses the convergence of our radial algorithms for concave maximization and Section 5 addresses applications and guarantees in nonconcave maximization.
A Motivating Setting of Polyhedral Constraints
We begin by motivating the algorithmic usefulness of our radial duality by considering optimization with polyhedral constraints. Consider any maximization problem with upper semicontinuous objective f : R n → R ∪ {−∞} and m inequality constraints a T i x ≤ b i given by
max x f (x) s.t. Ax ≤ b.(4)
We assume this problem is feasible. Then without loss of generality, we have 0 ∈ int ({x | Ax ≤ b, f (x) > 0}). This can be achieved by computing any point x 0 in the relative interior of {x | Ax ≤ b, f (x) ∈ R} and then (i) translating the problem to place x 0 at the origin, (ii) adding a constant to the objective to ensure f (0) > 0, and (iii) if needed, re-parameterizing the problem 1 to only consider the smallest subspace containing {x | Ax ≤ b, f (x) > 0}. Note that doing this translation suffices to guarantee that any upper semicontinuous, concave f will have f + (x) := max{f (x), 0} be strictly upper radial by [17,Proposition 11]. We will only make the weaker assumption here that f + is strictly upper radial rather than the narrower case of it being concave. Then this problem can be reformulated as the following nonnegative optimization problem of our primal form (2)
max x f + (x) s.t. Ax ≤ b = max x min i f + (x),ι a T i x≤bi (x) whereι a T i x≤bi (x) = +∞ if a T i x ≤ b i 0 if a T i x > b i
is a nonstandard indicator function for each inequality constraint. Note that eachι a T i x≤bi is strictly upper radial since 0 is strictly feasible and so applying [17,Proposition 12] ensures the primal objective min i f + (x),ι a T i x≤bi (x) is strictly upper radial. Then we can compute the radially dual optimization problem (3) using [17,Proposition 13] as min
y max i f Γ + (y), a T i y/b i(5)
since the radial transformation of each nonstandard indicator function is linear
ι Γ a T i x≤bi (y) = sup v > 0 | v ·ι a T i x≤bi (y/v) ≤ 1 = sup v > 0 | a T i (y/v) > b i = (a T i y/b i ) + .
We drop the nonnegative thresholding on a T i y/b i since f Γ + (y) is nonnegative. Importantly, the dual formulation (5) is unconstrained, unlike the primal, since the primal inequality constraints have transformed into simple linear lower bounds on the radially dual objective. This dual further profits from the structure of its objective function as it is often globally Lipschitz continuous (a common property among radial duals that we will show in Proposition 1) and has the simple form of a finite maximum. This radially dual structure gives us an algorithmic angle of attack not available in the primal problem.
Quadratic Programming
To make these benefits concrete, consider solving a generic quadratic program
max x 1 − 1 2 x T Qx − c T x s.t. Ax ≤ b(6)
for some Q ∈ R n×n , c ∈ R n , A ∈ R m×n , and b ∈ R m ++ . Note this satisfies the
needed condition 0 ∈ int ({x | Ax ≤ b, f (x) > 0}) whenever b > 0 as f (0) = 1.
We reformulate this problem as the following nonnegative optimization problem of the form (2)
max x 1 − 1 2 x T Qx − c T x s.t. Ax ≤ b = max x min i (1 − 1 2 x T Qx − c T x) + ,ι a T i x≤bi (x) .
Whenever this primal objective is strictly upper radial, the radial dual of our quadratic program is 2
min y max i c T y + 1 + (c T y + 1) 2 + 2y T Qy 2 + , a T i y/b i(7)
where the first term in our maximum is set to zero if (c T y + 1) 2 + 2y T Qy < 0 as can occur for nonconcave primal objectives. We find that our radial duality holds here whenever 1 2 x T Qx > −1 for all Ax ≤ b. This captures two natural settings: (i) when the primal objective is concave (as Q is positive semidefinite) or (ii) when the primal objective is nonconcave but has a compact feasible region (since we can rescale the objective to be 1 − λx T Qx/2 − λc T x without changing the set of maximizers but ensuring λ 2 x T Qx > −1 everywhere). Section 5.1 shows more generally that any differentiable objective with compact constraints can be rescaled to apply our radial duality theory.
We verify that our primal objective is strictly upper radial (and so our radial duality holds) for this upper semicontinuous objective by checking when f p (y, ·) is strictly increasing on its domain. The partial derivative with respect to v of the perspective function v · min
i (1 − 1 2 (y/v) T Q(y/v) − c T (y/v)) + ,ι a T i x≤bi (y/v) = v 1 − y T Qy 2v 2 − c T y v + if A(y/v) ≤ b 0 otherwise is 1 + y T Qy 2v 2
at every feasible y/v. This is always positive (and hence the perspective function is increasing in v) exactly when every x = y/v with Ax ≤ b has 1 2 x T Qx > −1.
2 Our calculation of the radial dual of the quadratic objective follows by definition as
(1 − 1 2 x T Qx − c T x) Γ + (y) = sup v > 0 | v 1 − y T Qy 2v 2 − c T y v ≤ 1 = sup{v > 0 | v 2 − 1 2 y T Qy − (c T y + 1)v ≤ 0} = c T y + 1 + (c T y + 1) 2 + 2y T Qy 2 + .
Quadratic Programming Numerics
As previously noted, the radially dual formulation (7) is unconstrained and Lipschitz continuous despite the primal possessing neither of these properties. This differs from the structure found from taking a Lagrange dual [11] or gauge dual [14]. As a result, our radial dual is well set up for the application of a subgradient method. We consider the following radial subgradient method with stepsizes α k > 0 defined by Algorithm 1.
Algorithm 1 The Radial Subgradient Method
Require: f : E → R ++ , x 0 ∈ dom f , T ≥ 0 1: (y 0 , v 0 ) = Γ (x 0 , f (x 0 )) Transform into the radial dual 2: for k = 0 . . . T − 1 do 3: y k+1 = y k − α k ζ k , where ζ k ∈ ∂ P f Γ (y k )
Run the subgradient method 4: end for
5: (x T , u T ) = Γ (y T , f Γ (y T ))
Transform back to the primal Further noting that the radially dual problem is a finite maximum of simple smooth Lipschitz functions, we can apply the smoothing ideas of Nesterov [33]. Perhaps the clearest description of these techniques is given by Beck and Teboulle [3]. In particular, for any fixed η > 0, we consider the smooth function given by taking a "soft-max"
g η (y) = η log exp c T y + 1 + (c T y + 1) 2 + 2y T Qy 2η + m i=1 exp a T i y b i η(8)
which approaches our radially dual objective as η → 0. Then we can minimize the radial dual up to accuracy O(η) by minimizing this smoothed objective. Doing so with Nesterov's accelerated method gives the following radial smoothing method defined by Algorithm 2 (a similar radial algorithm was employed by Renegar [39] showing that the transformation of any hyperbolic programming problem also admits a smoothing that can be efficiently minimized).
Algorithm 2 The Radial Smoothing Method
Require: f : E → R ++ , x 0 ∈ dom f , η > 0, Lη > 0, T ≥ 0 1: (y 0 , v 0 ) = Γ (x 0 , f (x 0 )) andỹ 0 = y 0 Transform into the radial dual 2: Let gη(y) denote an η-smoothing of f Γ (y) 3: for k = 0 . . . T − 1 do 4:ỹ k+1 = y k − ∇gη(y k )/Lη
Run the accelerated method 5:
y k+1 =ỹ k+1 + k−1 k+2 (ỹ k+1 −ỹ k ) 6: end for 7: (x T , u T ) = Γ (y T , f Γ (y T ))
Transform back to the primal
The per iteration cost of these radial methods is controlled by the cost of evaluating one subgradient of the radially dual objective (7) or one gradient of our smoothing of the radially dual objective (8). Both of these can be done efficiently in closed form in terms of matrix-vector products with A and Q. Despite this low iteration cost, a feasible primal solution (x k , u k ) = Γ (y k , f Γ (y k )) is known at every iteration. Convergence guarantees for the radial subgradient and smoothing methods for concave maximization are given later in Sections 4.1 and 4.2.
Classic optimization algorithms that preserve feasibility at every iteration tend to have much higher iteration costs. Here we compare with three of the most standard first-order methods that enforce feasibility: projected gradient descent (or rather, projected gradient ascent)
x k+1 = proj {x|Ax≤b} (x + ∇f (x)/L) ,
an accelerated projected gradient method
x k+1 = proj {x|Ax≤b} (x k + ∇f (x k )/L) x k+1 =x k+1 + k−1 k+2 (x k+1 −x k ),
and the Frank-Wolfe method 3 with stepsize sequence β k > 0
x k+1 ∈ argmax x ∇f (x k ) T x | Ax ≤ b x k+1 = x k + β k (x k+1 − x k ).
All three of these methods require solving a subproblem at each iteration. The projected gradient and accelerated gradient methods require repeated projection onto the polyhedron {x | Ax ≤ b}, which is itself an instance of (6) specialized to Q = I. The Frank-Wolfe method requires repeatedly solving a linear program over this polyhedron. Both of these operations are far more expensive than the matrix-vector products required by the radial subgradient and smoothing methods but may allow them to have a greater improvement in objective value per iteration.
To weigh this tradeoff, we consider running these five algorithms on synthetic quadratic programs given by drawing two matrices A ∈ R m×n and P ∈ R n×100 and a vector c ∈ R n with i.i.d. Guassian entries and setting Q = P P T and all b i = 1. Then we run each algorithm for 30 minutes on instances of size (n, m) ∈ {(400, 1600), (800, 3200), (1600, 6400)}. Our numerical experiments are conducted on a four-core Intel i7-6700 CPU using Julia 1.4.1 and Gurobi 9.1.1 to solve any subproblems 4 . For each method, we set x 0 = 0 and use the following choice of stepsizes: the projected and accelerated gradient methods use L = λ max (Q), the Frank-Wolfe method uses an exact
linesearch β k = min ∇f (x k ) T (x k+1 −x k ) P T (x k+1 −x k ) 2 , 1 , the radial subgradient method uses the Polyak stepsize α k = f Γ (y k )−d * ζ k 2
, and the radial smoothing method fixes 3 Quadratic programming was the original motivating setting for Frank-Wolfe [13]. 4 The source code is available at github.com/bgrimmer/Radial-Duality-QP-Example p * of (6), with sizes (n, m) equal to (400, 1600), (800, 3200), (1600, 6400) from left to right, seen by the projected gradient, accelerated gradient, Frank-Wolfe, radial subgradient and radial smoothing methods over 30 minutes.
L η = 0.1 max{ a i /b i 2 }/η and η ∈ {10 −8 , 5×10 −7 , 10 −7 }
for each of our three problem sizes.
The best primal objective value seen by each method is shown in realtime in Figure 1. First, we remark on the total number of iterations completed by each method in the allotted half hour, shown in the following In our largest problem setting (n, m) = (1600, 6400), which has approximately ten million nonzeros, the projected gradient, accelerated gradient, and Frank-Wolfe methods complete 35-152 steps within our time budget whereas our radial methods take hundreds of thousands of steps. For our smallest instance (n, m) = (400, 1600), the accelerated gradient method quickly reaches high accuracy. However, for our moderate-sized instance (n, m) = (800, 3200), the classic methods begin to fall off with the radial smoothing method and accelerated method performing comparably up to accuracy O(η). For our largest instance (n, m) = (1600, 6400), the methods relying on orthogonal projection and linear optimization have their progress substantially slowed due to their high iteration cost. Our radial algorithms appear to provide a more scalable approach. Throughout our experiments, the radial smoothing method outperforms the radial subgradient method by a couple of orders of magnitude. This agrees with our convergence theory showing that the radial subgradient method converges at a O(1/ 2 ) rate while the smoothing technique enables O(1/ ) convergence, presented in Sections 4.1 and 4.2, respectively.
Further comparisons can be made with customized, scalable QP solvers like OSQP [44], which is based on ADMM and provides approximate primal and dual solutions. In Appendix A, we outline how dual solutions can be extracted from the radial smoothing method for the sake of comparison with OSQP. The quality of some primal x and dual v as an approximate KKT solution can be measured in terms of their primal feasibility prim = max{a T i x − b i , 0}, dual feasibility dual = Qx + c + A T v ∞ , and complementary slackness
comp = (Ax − b) · v ∞ .
OSQP always has comp = 0 but guarantees neither primal nor dual feasibility at its iterates. Our radial smoothing method always has a feasible primal solution prim = 0, but guarantees neither dual feasibility nor complementary slackness. As a result, only limited conclusions can be drawn between these two methods. Figure 2 shows the convergence of each OSQP and radial smoothing (with η = 10 −4 ). Although asymptotically OSQP appears to converge faster, throughout the experiment's runtime it is debatable which method's certificates are preferable. Further numerical testing and the design of radial methods/theory focused on KKT attainment are deferred to future works. Note at each iteration, OSQP solves a linear system rather than just relying on matrix multiplications. Numerically, this results in OSQP completing 10, 503 steps whereas our radial smoothing method completed 113, 532.
Broader Computational Advantages from the Radially Dual Problem
We conclude this motivating section with a high-level discussion of the computational advantages we see in optimizing over the radially dual problem.
Maintaining Primal Feasible Iterates Without Costly Projections
Here we generalize the setting of polyhedral constraints considered by (4). After a translation, any convex constraints can be expressed as the intersection of a convex set S ⊆ E with 0 ∈ int S and a subspace T = {x ∈ E | Ax = 0}. Consider any primal problem with strictly upper radial objective f given by
max f (x) s.t. x ∈ S Ax = 0 = max x∈E min{f (x),ι S (x),ι T (x)} whereι S (x) = +∞ if x ∈ S 0 if x ∈ S.
Then the radially dual problem is
min y∈E max{f Γ (y), γ S (y), γ T (y)} = min max{f Γ (y), γ S (y)} s.t. Ay = 0 where γ S (y) = inf{λ ≥ 0 | y ∈ λS} denotes the Minkowski gauge sincê ι Γ S (y) = sup{v > 0 | v ·ι S (y/v) ≤ 1} = sup{v > 0 | y/v ∈ S} = inf{λ > 0 | y ∈ λS} = γ S (y).
The last line above uses that S is convex and contains 0. Having multiple set constraints S 1 . . . S n in the primal max x∈S1∩···∩Sn f (x) simply adds more terms to the dual's maximum of min y∈E max{f Γ (y), γ Si (y)}. This formulation allows algorithms to maintain a feasible primal solution at each iteration without requiring costly subproblems relating to S. Instead, a primal feasible solution can be recovered from any radial dual solution y ∈ E with Ay = 0 as
x = y/ max{f Γ (y), γ S (y)} ∈ S ∩ T since 0 ∈ S ∩ T .
Algorithmically, this replaces the need for orthogonal projections onto the feasible region S ∩ T with the cheaper operations of orthogonally projecting onto the subspace T and evaluating the gauge of S. This computational gain was one of the key contributions identified by [38] and was central to the motivation of [39,16] as well as being a motivation of this work.
Handling Nonconcave Objectives and Nonconvex Constraints
Our calculation of the radial dual for quadratic programming did not fundamentally rely on concavity as it also applies to nonconcave problems with a bounded feasible region. Indeed one of the key insights from the first part of this work was divorcing the idea of radial transformations from relying on notions of convexity or concavity. In Section 5.1, we discuss several nonconcave primal maximization problems where radial duality holds, generalizing the above reasoning to star-convex constraints and covering important areas like nonconvex regularization and optimization with outliers.
Efficiently Evaluating Generic Radial Duals
In many structured settings, we can exactly evaluate the radial transformation with cost comparable to a single evaluation of f . Table 1 gives formulas for the gauge of any norm, halfspace, polynomial, or semidefinite programming constraint. Note that the gauge of an intersection of several of these constraints is simply the maximum of each constraint's gauge formula. Similarly, Table 2 gives formulas for the radial dual of many common function classes.
Outside these common families, f Γ does not have a closed-form formula. However, numerically evaluating f Γ (y) can be done by bisection whenever f is upper radial: Note zero is a trivial lower bound on the value of f Γ (y) and an upper bound can be computed via exponential back-off, finding the first
Set S (assumed star convex with 0 ∈ int S) γ S (y) =ι Γ S (y) Norm Constraints {x | x ≤ b} y /b Halfspace Constraints {x | a T x ≤ b} (a T y/b) + Quadratic Constraints {x | 1 2 x T Qx + p T x ≤ b} p T y+ √ (p T y) 2 +2by T Qy 2c + Polynomial Constraints {x | p(x) ≤ 0} Polynomial Root Finding Semidefinite Constraints {x | Ax − B 0} λmax(B −1 Ay)Function f (assumed upper radial, f (0) > 0) f Γ (y) Norms x ι x ≤1 (y) Linear Functions (a T x + b) + ((1 − a T y)/b) + Quadratic Functions 1 2 x T Qx + p T x + b + 1−p T y+ √ (1−p T y) 2 −2by T Qy 2c + Polynomial Functions p(x) + Polynomial Root Findinginteger i with 2 i · f (y/2 i ) > 1 since all 2 i > f Γ (y) have this property.
Having nondecreasing v · f (y, v) ensures bisection will linearly converge to the unique intermediate value of v where vf (y/v) passes above one. This exponential back-off and subsequent bisection will each require a logarithmic number of function evaluations (both will reach standard machine precision within ≈ 30 steps). Consequently, even when closed-forms are not available, the radial dual is at most only moderately more expensive to calculate than the primal. Even if f is not upper radial, f Γ (y) may still be tractable to compute. For example, any polynomial f has evaluation of f Γ amount to computing a polynomial's largest root. Once f Γ (y) has been computed, its gradient and Hessian can be readily computed from (25) and (26).
Improving Conditioning and Problem Structure
As a final motivating example of the structural advantages of taking the radial dual, consider the following Poisson inverse problem. Given linear measurements with Poisson distribution noise b i ∼ Poisson(a T i x), the maximum likelihood estimator is given by maximizing
L(x) := i b i log(a T i x) − a T i x if all a T i x > 0 −∞ otherwise.
Then given any convex regularizer r(x) and constraint set S ⊆ R n , we formulate a Poisson inverse problem as
max x∈S L(x) − r(x).(9)
This type of problem arises in image processing (see [4] for a survey of applications from astronomy to medical imaging) as well as in network diffusion and time series modeling (see the many references in [20]). Although this problem is concave, the blow-up from the logarithmic terms prevents standard first-order methods from being applied. Provided the regularization r and constraints S are sufficiently simple, customized primal-dual [20] or Bregman methods [1,30] provide a powerful tactic for solving this problem. For generic S and r, our radial duality can be applied. Given any x 0 ∈ int (dom L ∩ S) and u 0 < L(x 0 ) − r(x 0 ), we can reformulate this objective function to be strictly upper radial via a simple translation and truncation. We consider the equivalent problem of
max x∈R n min{(L(x + x 0 ) − r(x + x 0 ) − u 0 ) + ,ι S (x + x 0 )}.
Then we can employ our radial duality machinery using [17,Proposition 11] since our translated and truncated objective is concave with 0 strictly in its domain. The radial dual here is defined everywhere dom f Γ = R n and globally uniformly Lipschitz continuous (see Proposition 1). Moreover if S = R n , r(x) is twice continuously differentiable and L−r has bounded level sets 5 , the radial dual has globally Lipschitz continuous gradient (see Corollary 1). The primal formulation is none of these. Note that different translations of the objective (here corresponding to a different choice of (x 0 , u 0 )) produce different radial duals, which in turn can have very different global Lipschitz and smoothness constants.
Notation and Review
We consider functions f : E → R ++ , where R ++ = R ++ ∪ {0, +∞} denotes the "extended positive reals". Here 0 and +∞ are the limit objects of R ++ , mirroring the roles of −∞ and +∞ in the extended reals. The effective domain, graph, epigraph, and hypograph of such a function are
dom f := {x ∈ E | f (x) ∈ R ++ }, graph f := {(x, u) ∈ E × R ++ | f (x) = u}, epi f := {(x, u) ∈ E × R ++ | f (x) ≤ u}, hypo f := {(x, u) ∈ E × R ++ | f (x) ≥ u}.
We say a function f :
E → R ++ is upper (lower) semicontinuous if hypo f (epi f ) is closed with respect to E × R ++ . Equivalently, a function is upper semicontinuous if for all x ∈ E, f (x) = lim sup x →x f (x ) and lower semicon- tinuous if f (x) = lim inf x →x f (x ). We say a function f : E → R ++ is concave (convex) if hypo f (epi f ) is convex. The set of convex normal vectors of a set S ⊆ E × R at some (x, u) ∈ S is denoted by N C S ((x, u)) := {(ζ, δ) | (ζ, δ) T ((x, u) − (x , u )) ≥ 0 ∀(x , u ) ∈ S}.
5 For example, if either {a i } spans R n or the regularizer r(x) has bounded level sets.
Then the convex subdifferential and convex supdifferential of a function f are
∂ C f (x) := {ζ | (ζ, −1) ∈ N C epi f ((x, f (x)))}, ∂ C f (x) := {ζ | (−ζ, 1) ∈ N C hypo f ((x, f (x)))}.
The proximal normal vectors and differentials of a set S or function f are
N P S ((x, u)) := {(ζ, δ) | (x, u) ∈ proj S ((x, u) + (ζ, δ)) for some > 0}, ∂ P f (x) := {ζ | (ζ, −1) ∈ N P epi f ((x, f (x)))}, ∂ P f (x) := {ζ | (−ζ, 1) ∈ N P hypo f ((x, f (x)))}.
Dual Families of Functions. Most of our theory characterizing the radial transformation relies on the given function being (strictly) upper radial. Recall that [17,Proposition 6] shows an upper semicontinuous function f is upper radial (that is, our radial duality
f Γ Γ = f holds) if and only if all (x, u) ∈ hypo f and (ζ, δ) ∈ N P hypo f ((x, u)) satisfy (ζ, δ) T (x, u) ≥ 0.(10)
Geometrically, this corresponds to the origin lying below all of the hyperplanes induced by proximal normal vectors of the hypograph. Similarly, [17, Proposi-
tion 8] ensures a continuously differentiable function f is strictly upper radial if all x ∈ dom f satisfy (∇f (x), −1) T (x, u) < 0.(11)
For concave functions, being upper radial corresponds to the origin lying in the function's domain. In particular, [17,Proposition 11] ensures an upper semicontinuous concave function f is strictly upper radial if
0 ∈ int {x | f (x) > 0}.(12)
Assuming strict upper radiality holds, the following are radially dual
f is upper semicontinuous ⇐⇒ f Γ is lower semicontinuous,(13)
f is continuous ⇐⇒ f Γ is continuous,
f is concave ⇐⇒ f Γ is convex,
where these follow from [17,Propositions 15,17]. We say a functions mapping into the extended positives is k times differentiable if it is k times differentiable at each x in dom f . For differentiable functions satisfying (11), [17,Proposition 21] shows
f is k times differentiable ⇐⇒ f Γ is k times differentiable,(16)
f is analytic ⇐⇒ f Γ is analytic.
epi f Γ = Γ (hypo f ).(18)
Further, [17,Lemma 3] shows for any continuous strictly upper radial function, the following pair of bijections between graphs and domains hold
graph f Γ = Γ (graph f ),(19)
y
∈ dom f Γ ⇐⇒ y/f Γ (y) ∈ dom f.(20)
Then [17,Propositions 24,25] shows that the radial point transformation relates the maximizers of a strictly upper radial function f to the minimizers of f Γ as well as relates their stationary points
argmin f Γ × {inf f Γ } = Γ (argmax f × {sup f }) ,(21){(y, f Γ (y)) | 0 ∈ ∂ P f Γ (y)} = Γ {(x, f (x)) | 0 ∈ ∂ P f (x)}.(22)
In particular, for any upper semicontinuous, strictly upper radial f , the convex and proximal subgradients of its upper radial transformation are given by
∂ C f Γ (y) = ζ (ζ, δ) T (x, u) | ζ δ ∈ N C hypo f ((x, u)), (ζ, δ) T (x, u) > 0 (23) ∂ P f Γ (y) = ζ (ζ, δ) T (x, u) | ζ δ ∈ N P hypo f ((x, u)), (ζ, δ) T (x, u) > 0(24)
where (x, u) = Γ (y, f Γ (y)) by [17,Propositions 19,20]. Further, if f is continuously differentiable and satisfies (11), [17,Proposition 21] shows the gradient of the upper radial transformation at
y = x/f (x) is ∇f Γ (y) = ∇f (x) (∇f (x), −1) T (x, f (x))
.
If in addition we suppose f is twice continuously differentiable around x, [17,Proposition 22] shows the Hessian of the upper radial transformation is
∇ 2 f Γ (y) = f (x) (∇f (x), −1) T (x, f (x)) · J∇ 2 f (x)J T(26)
where
J = I − ∇f (x)x T (∇f (x),−1) T (x,f (x)) .
Conditioning of the Radially Dual Problem
As we have seen, the radial dual often enjoys structural properties missing from the primal. In the following three subsections, we characterize the Lipschitz continuity, smoothness, and growth conditions of the radially dual problem.
Lipschitz Continuity of the Radially Dual Problem
We say a function f is uniformly M -Lipschitz continuous if for all x, x ∈ E,
|f (x) − f (x )| ≤ M x − x .
For any lower semicontinuous function f : E → R ++ ∪ {∞}, M -Lipschitz continuity is equivalent to all proximal subgradients ζ ∈ ∂ P f (x) having norm bounded by M [9, Theorem 1.7.3]. Lipschitz continuity plays an important role in the analysis of many firstorder methods for nonsmooth optimization. The previous works [38,39,16] critically rely on their radially reformulated objective being uniformly Lipschitz. Here we present a general characterization of when the radial transformation of a function is uniformly Lipschitz. To take advantage of the second characterization of Lipschitz continuity above, we need to ensure f Γ maps into R ++ ∪{∞}. The following simple assumption is equivalent to this (by the definition of the upper radial transformation): for all y ∈ E, lim v→0 v · f (y/v) = 0.
This condition is always the case when f is bounded above as will typically be the case for our primal maximization problem. Under this condition, we find that the Lipschitz continuity of f Γ is controlled by the distance (measured in E) from the origin to each hyperplane defined by a proximal normal vector:
R(f ) = inf{ x | 0 = (ζ, δ) ∈ N P hypo f (x, u), (ζ, δ) T ((x , 0) − (x, u)) = 0}.
The following proposition gives the exact Lipschitz constant in terms of R(f ).
Proposition 1 Consider any upper semicontinuous, strictly upper radial
f where all y ∈ E have lim v→0 v · f (y/v) = 0. Then f Γ is 1/R(f )-Lipschitz continuous.
Proof
The key observation here is that for any (ζ, δ) ∈ N P hypo f ((x, u)),
(ζ, δ) T (x, u) = inf ζ T x | (ζ, δ) T ((x , 0) − (x, u)) = 0 = ζ inf x | (ζ, δ) T ((x , 0) − (x, u)) = 0 ≥ ζ R(f )
where the first equality is trivial and the second uses that the minimum norm point in this hyperplane will be a multiple of ζ. Then the subgradient formula (24) ensures any ζ ∈ ∂ P f Γ (y) must have
ζ = ζ (ζ, δ) T (x, u) ≤ 1/R(f )
for (x, u) = Γ (y, f Γ (y)) and some (ζ, δ) ∈ N P hypo f ((x, u)). Since every radially dual subgradient is uniformly bounded, f Γ is uniformly Lipschitz. Considering a sequence of (ζ, δ) ∈ N P hypo f ((x, u)) approaching attainment of R(f ) makes this argument tight.
The condition (x, u) T (ζ, δ) ≥ R(f ) ζ can be viewed as a natural way to quantify how radial f is by strengthening (10). When f is concave, R(f ) can be simplified.
Lemma 1 For any upper semicontinuous, upper radial, concave f ,
R(f ) = inf{ x | f (x) = 0}.(27)
Proof By the concavity of f , (ζ,
δ) T ((x , 0) − (x, u)) = 0 implies x ∈ cl {x | f (x) = 0}. Hence R(f ) ≥ inf{ x | f (x) = 0}
. Anyx with f (x) = 0 has (x, 0) separated from hypo f by a supporting hyperplane with normal (ζ, δ)
at some (x, u) ∈ hypo f . Hence (ζ, δ) T ((x , 0) − (x, u)) = 0 separatesx from 0. So x ≥ R(f ).
This matches the Lipschitz constants used in the previous works [38,39,16]. This gives a natural way to measure the extent of radiality of a concave function by strengthening (12). From this, we see any concave maximization problem (with a known point in the interior of its domain) can be translated and transformed into a convex minimization problem that is uniformly Lipschitz continuous with constant depending on how interior the known point is to the function's domain.
Smoothness of the Radially Dual Problem
We say a continuously differentiable function f is uniformly L-smooth if its gradient is L-Lipschitz continuous: for all
x, x ∈ dom f ∇f (x) − ∇f (x ) ≤ L x − x .
As an example, consider the radial dual of the continuously differentiable function f (x) = (1 − x T Qx) + , which is upper radial for any matrix Q. This radially transforms into the similarly shaped function
f Γ (y) = sup{v > 0 | v 2 − y T Qy ≤ 1} = (1 + y T Qy) + .
Supposing Q is positive semidefinite and nonzero, our primal is concave and differentiable on its domain but fails to have a Lipschitz gradient since ∇f (x) blows up at the boundary of its domain. However, in this case, the radially dual f Γ is well behaved, being convex and λ max (Q)-smooth.
For generic functions, we cannot hope to find smoothness out of thin air (like we do in the above example or quite generically with Lipschitz continuity in the previous section). This is due to (16) which establishes differentiability is preserved under the radial transformation. In line with this equivalence, we find that when f is L-smooth, f Γ is O(L)-smooth, provided the domain of f is bounded. Let D(f ) = sup{ x | x ∈ dom f } denote the norm of the largest point in the domain of f . Note that since we are primarily taking the radial dual of maximization problems that are bounded above and truncated below to be nonnegative optimization, D(f ) can be viewed as bounding the level set
dom f = {x | f (x) > 0}.
The following proposition shows the operator norm of the radial transformation's Hessian is controlled by the ratio between D(f ) and R(f ) and the norm of the primal Hessian. From this, we conclude for twice differentiable L-smooth functions, the radial dual is also O(L)-smooth.
Proposition 2
Consider any upper radial f with D(f ) < ∞ and R(f ) > 0 and x, y ∈ E satisfying (x, f (x)) = Γ (y, f Γ (y)). If f is twice continuously differentiable around x, then
∇ 2 f Γ (y) ≤ 1 + D(f ) R(f ) 3 ∇ 2 f (x) .
Proof First we verify that (∇f (x), −1) T (x, f (x)) < 0 holds for all x ∈ dom f and so the Hessian formula (26)
applies: if ∇f (x) = 0, (∇f (x), −1) T (x, f (x)) = −f (x) < 0 and if ∇f (x) = 0, (∇f (x), −1) T (x, f (x)) ≤ − ∇f (x) R(f ) < 0.
Then our bound on the Hessian of f Γ follows from the following pair of inequalities. First, the matrix
J = I − ∇f (x)x T (∇f (x),−1) T (x,f (x)) has operator norm bounded by J ≤ 1 + ∇f (x) x |(∇f (x), −1) T (x, f (x))| ≤ 1 + x R(f )
and second, the Hessian formula's coefficient is similarly bounded by
f (x) (∇f (x), −1) T (x, f (x)) = 1 − ∇f (x) T x (∇f (x), −1) T (x, f (x)) ≤ 1 + ∇f (x) x |(∇f (x), −1) T (x, f (x))| ≤ 1 + x R(f ) .
Bounding each term in the Hessian formula (26) gives the claimed result.
Corollary 1 Consider any upper radial, twice continuously differentiable f
with D(f ) < ∞ and R(f ) > 0. If f is L-smooth, f Γ is 1 + D(f ) R(f ) 3 L-smooth.
Proof For a twice continuously differentiable function, having L-Lipschitz gradient is equivalent to having Hessian bounded in operator norm by L. Noting that R(f ) > 0 implies f is strictly upper radial by (11), we have a bijection between the domains of f and f Γ from (20). Hence the Hessian of f Γ is uni-
formly bounded by 1 + D(f ) R(f ) 3 L.
Although this result requires smoothness of the primal objective f to be maximized, it still provides an algorithmically valuable tool due to the symmetry-breaking nature of considering functions on the extended positive reals R ++ . Supposing f is bounded above, this result allows us to extend the smoothness of f on a level set dom f = {x | f (x) > 0} to global smoothness of the dual f Γ on dom f Γ = E.
For example, consider an unconstrained S = R n instance of our previous motivating example of the Poisson likelihood problem (9) which is not defined everywhere (only on {x | a T i x > 0}) with gradients blowing up as x approaches the boundary of this domain. However, provided the measurements {a i } span R n , this objective has bounded level sets. Consequently, for any twice continuously differentiable r(x), our radial duality provides a reformulation that extends the smoothness on the level set {x | L(x) − r(x) > 0} to hold globally.
Growth Conditions in the Radially Dual Problem
For a lower semicontinuous function f : E → R ++ , we say the Lojasiewicz condition holds at a local minimum x * if for some constants r > 0, C > 0 and exponent θ ∈ [0, 1), all nearby x ∈ B(x * , r) have
dist(0, ∂ P f (x)) ≥ C(f (x) − f (x * )) θ .(28)
For an upper semicontinuous function f with local maximum x * , we instead require all nearby x ∈ B(x * , r) have
dist(0, ∂ P f (x)) ≥ C(f (x * ) − f (x)) θ .(29)
These conditions are widespread, holding for generic subanalytic functions [28,29] and nonsmooth subanalytic convex functions [5]. These properties are closely related to the Kurdyka-Lojasiewicz (KL) condition [24] and Hölderian growth/error bounds used by [6,46,41,40], which are known to speed up the convergence of many first-order methods. Under mild conditions, the Lojasiewicz condition is preserved by our radial transformation. Consequently, optimization algorithms based on solving the radially dual problem can enjoy the same improved convergence historically expected in the primal from such conditions. Proposition 3 Consider any upper semicontinuous, strictly upper radial function f with R(f ) > 0, sup f ∈ R ++ and points (y * , f Γ (y * )) = Γ (x * , f (x * )). If f satisfies the Lojasiewicz condition (29) at x * with exponent θ, then f Γ at y * satisfies the Lojasiewicz condition (28) with the same exponent θ.
Proof Let r, C, θ satisfy the Lojasiewicz condition of f at x * and denote the radially dual point as y * = x * /f (x * ). Since f is bounded above, f Γ is 1/R(f )-Lipschitz continuous by Proposition 1. Then every 0 < r < f Γ (y * )R(f ) has y ∈ B(y * , r ) map to x = y/f Γ (y) with
x − x * = y f Γ (y) − y * f Γ (y * ) ≤ y − y * f Γ (y) + y * f Γ (y) − y * f Γ (y * ) = y − y * f Γ (y) + y * |f Γ (y) − f Γ (y * )| f Γ (y)f Γ (y * ) ≤ r f Γ (y) + y * r R(f )f Γ (y)f Γ (y * ) ≤ r f Γ (y * ) − r /R(f ) + y * r R(f )(f Γ (y * ) − r /R(f ))f Γ (y * )
.
Therefore selecting small enough r guarantees that all of the dual points near y * map back to primal points x = y/f Γ (y) in the ball B(x * , r) where the Lojasiewicz condition holds. Further, the Lipschitz continuity of the radial dual allows us to guarantee that all of these primal points have f (x) bounded below by nearly f (x * ) as
f (x) = f Γ Γ (x) ≥ 1/f Γ (y) ≥ 1/(f Γ (y * )−r /R(f )) = (f (x * ) −1 +(R(f )/r ) −1 ) −1 .
Combining this with the assumed upper semicontinuity of f , we have f (x) → f (x * ) as y → y * (despite not assuming continuity of the primal function f ). Then all that remains is to show the Lojasiewicz supgradient norm lower bound from the primal extends to lower bound the norm of the radially dual subgradients. For every y ∈ B(y * , r ), the formula (24) ensures every ζ ∈ ∂ P f Γ (y) has ζ = ζ/(ζ, δ) T (x, u) where (x, u) = Γ (y, f Γ (y)) and (ζ, δ) ∈ N P hypo f ((x, u)). First, suppose δ = 0. Then u = f (x) and −ζ/δ ∈ ∂ P f (x) is a primal supgradient. Consequently, radially dual subgradients have size of at least
ζ = ζ/δ (ζ/δ, 1) T (x, f (x)) ≥ ζ/δ ζ/δ x + f (x) ≥ C(f (x * ) − f (x)) θ C(f (x * ) − f (x)) θ x + f (x) ≥ Cf θ (x)f θ (x * ) C(f (x * ) − f (x)) θ x + f (x) f Γ (y) − f Γ (y * ) θ
where the final inequality uses that f (x) ≥ 1/f Γ (y) and f (x * ) = 1/f Γ (y * ).
Recalling that as y → y * , the related primal point x = y/f Γ (y) → x * and f (x) → f (x * ), the coefficient above must converge to a positive constant
Cf θ (x)f θ (x * ) C(f (x * ) − f (x)) θ x + f (x) → Cf 2θ (x * ) C0 θ x * + f (x * )
.
The boundary case of horizontal normal vectors with δ = 0 follows from the same argument above by passing to a sequence of points ( The case of θ = 0 above is the important special case of sharpness. If this condition holds globally, (28) and (29) correspond to the global error bounds
x i , f (x i )) → (x, f (x)) and proximal normal vectors (ζ i , δ i ) ∈ N P hypo f ((x i , f (x i ))) with (ζ i , δ i ) → (ζ,f (x) ≥ f (x * ) + C x − x *(30)
and
f (x) ≤ (f (x * ) − C x − x * ) +(31)
respectively. This condition has a long history in nonsmooth optimization (see Burke and Ferris [7] as a classic reference establishing the prevalence of sharp minima). The two global sharp error bounds (31) and (30) are dually related.
Proposition 4 For any upper semicontinuous, strictly upper radial f with points (y * , f Γ (y * )) = Γ (x * , f (x * )) satisfying (31) at x * ∈ E with constant C, then f Γ satisfies (30) at y * with constant C/(C x * + f (x * )).
Proof Denote the assumed upper bound on f from sharpness as h(
x) := f (x * ) − C x − x * .
Then h + must be strictly upper radial due to (12) since
h is concave with h(0) = 2h(x * /2) − h(x * ) ≥ 2f (x * /2) − f (x * ) > 0
where the first equality uses that h is linear on the segment [0, x * ], the inequality uses that h(x * /2) ≥ f (x * /2), and the strict inequality uses that f is strictly upper radial. The upper radial transformation h Γ + is lower bounded by our claimed sharpness lower bound for any y ∈ E
h Γ + (y) ≥ 1 f (x * ) + C y − x * /f (x * ) f (x * ) + C x * = f Γ (y * ) + C y − y * f (x * ) + C x * since h p + (y, v) at v = 1 f (x * ) + C y−x * /f (x * ) f (x * )+C x * is at most 1 f (x * ) + C y − y * f (x * ) + C x * f (x * ) − C y − 1 f (x * ) + C y − y * f (x * ) + C x * x * ≤ 1 + C y − y * f (x * ) + C x * f (x * ) − C y − y * + C 2 y − y * x * f (x * ) + C x * = 1 + C y − y * f (x * ) f (x * ) + C x * − 1 + C x * f (x * ) + C x * = 1
where y * = x * /f (x * ) and the inequality uses the reverse triangle inequality.
Using [17,Lemma 4], f ≤ h + implies f Γ ≥ h Γ + , completing our proof.
Radial Algorithms for Concave Maximization
Now we turn our attention to understanding the primal convergence guarantees that follow from algorithms minimizing the radial dual. In this section, we consider concave maximization problems where being strictly upper radial and having R(f ) > 0 hold without loss of generality via a simple translation. We first remark on the natural measure of optimality in the primal that arises from considering the radial dual. Recall the set of fixed points of Γ are exactly the horizontal line at height one
{(y, 1) | x ∈ E} = Γ {(x, 1) | x ∈ E}.
Consequently, a natural way to relate nearly optimal solutions between the primal and radial dual comes from considering when sup f = inf f Γ = 1. In this case, finding a dual point with accuracy f Γ (y k ) − inf f Γ ≤ implies a relative accuracy primal guarantee of
sup f − f (x k ) f (x k ) ≤ . using that 1/f Γ (y k ) ≤ f Γ Γ (x k ) = f (x k ) for x k = y k /f Γ (y k )
on any upper radial f . Following this, we state all of our radial algorithm convergence guarantees in relative terms. Secondly, we remark on the meaning of finding a radially dual solution minimized to zero objective value f Γ (y) = 0. In this case, y certifies that the primal maximization is unbounded as the ray (y, 1)/v ∈ epi f for all v > 0. Note the converse of this is not true: for example, the strictly radial function f (x) = (x + 1) + is unbounded above, but has f Γ (y) > 0 everywhere.
Radial Subgradient Method
We begin by considering the radial subgradient method previously defined in Algorithm 1. This method simply takes the radial dual, applies the classic subgradient method to the resulting minimization problem, and then takes the radial dual again to return a primal solution. Importantly this method is projection-free since any primal constraint set S appears in the radial dual objective through its gauge γ S . This method is very similar to those considered in [38,16] which also apply a subgradient method to a radial reformulation. However, those methods include additional steps periodically rescaling their radial objective. Our algorithm omits such steps while matching the improved convergence guarantees of [16].
The standard subgradient method analysis shows the radial subgradient iterates y k converge in terms of radial dual optimality at a rate controlled by the radially dual Lipschitz constant. Recall that translating a point in the interior of hypo f to the origin ensures R(f ) > 0 (by Lemma 1) and so the radial dual is Lipschitz continuous (by Proposition 1). Consequently, no structure needs to be assumed beyond concavity to analyze the radial subgradient method.
Theorem 1 Consider any upper semicontinuous, concave f with R(f ) > 0 and p * = sup f ∈ R ++ attained on some nonempty set X * ⊆ E. Then the radial subgradient method (Algorithm 1) with stepsizes α k has primal solutions
x k = y k /f Γ (y k ) satisfy min k<T p * − f (x k ) f (x k ) ≤ dist(p * y 0 , X * ) 2 + T −1 k=0 (p * α k /R(f )) 2 2 T −1 k=0 p * α k . Selecting x 0 = 0 and α k = f Γ (y k )/ ζ k 2 for any > 0 ensures T ≥ dist(x 0 , X * ) 2 R(f ) 2 2 =⇒ 1 T T −1 k=0 p * − f (x k ) p * ≤ .
Proof Having R(f ) > 0 ensures f is strictly upper radial by (12). Then f Γ is convex by (15) and has minimum value d * = 1/p * attained on Y * := X * /p * by (21). The classic convex convergence analysis of subgradient methods follows from the fact that: for any y * ∈ Y * ,
y k+1 − y * 2 = y k − y * 2 − 2α k ζ T k (y k − y * ) + α 2 k ζ k 2 ≤ y k − y * 2 − 2α k (f Γ (y k ) − d * ) + α 2 k ζ k 2
and so inductively,
T −1 k=0 α k (f Γ (y k ) − d * ) ≤ y 0 − y * 2 + T −1 k=0 α 2 k ζ k 2 2 .(32)
Noting (x k , u k ) = Γ (y k , f Γ (y k )), the primal iterates have f (x k ) ≥ 1/f Γ (y k ). Then multiplying through by (1/d * ) 2 , which equals (p * ) 2 , yields
T −1 k=0 α k d * p * − f (x k ) f (x k ) = T −1 k=0 α k (d * ) 2 1 f (x k ) − 1 p * ≤ y 0 /d * − y * /d * 2 + T −1 k=0 (α k /d * ) 2 ζ k 2 2 .
Since f Γ is 1/R(f )-Lipschitz (by Proposition 1), every radially dual subgradient is uniformly bounded by ζ k ≤ 1/R(f ). Then selecting y * = proj Y * (y 0 ) gives our claimed primal convergence rate. Observe that setting x 0 = 0 sets
y 0 = x 0 /f (x 0 ) = 0 as well. Then plugging α k = f Γ (y k )/ ζ k 2 into (32) yields dist(x 0 , X * ) 2 2 = dist(y 0 /d * , X * ) 2 2 ≥ T −1 k=0 α k d * f Γ (y k ) − d * d * − 1 2 α k d * ζ k 2 ≥ T −1 k=0 f Γ (y k ) d * ζ k 2 p * − f (x k ) p * − 2 ≥ T −1 k=0 R(f ) 2 p * − f (x k ) p * − 2
where the last line bounds 1/ ζ k 2 below by R(f ) 2 and f Γ (y k )/d * below by one. Rearranging this completes our proof.
Recall for concave f the formula for R(f ) can be simplified to inf{ x | f (x) = 0}, which quantifies how interior the origin is to the set {x | f (x) > 0}. In this light, the constants in this rate agree with those in the guarantees of [16], up to small constants.
The classic convergence rates of the subgradient method improve in the presence of growth conditions like (28) or (30). For example growth with exponent θ = 1/2 corresponds to the case of quadratic growth (generalizing strong convexity) and leads to faster O(1/ ) convergence, see [25] as a simple example. When θ = 0, sharp growth enables the classic subgradient method to converge linearly, as shown by Polyak [36,37] more than 50 years ago. Recalling that these quantities are preserved from primal to radial dual (Propositions 3 and 4), we find the same improvements to hold for our radial subgradient method. The following two theorems establish this speed up when θ = 0 and θ > 0, using the radially dual Polyak stepsize
α k = (f Γ (y k ) − d * )/ ζ k 2 .
Theorem 2 Consider any upper semicontinuous, concave f with R(f ) > 0 and p * = sup f ∈ R ++ attained at x * ∈ E. Fixing α k = (f Γ (y k ) − d * )/ ζ k 2 , if f satisfies the sharp growth condition (31), then the radial subgradient method (Algorithm 1) has x k = y k /f Γ (y k ) satisfy
T ≥ 4 p * + C x * CR(f ) 2 log 2 p * − f (x 0 ) f (x 0 ) =⇒ min k<T p * − f (x k ) f (x k ) ≤ . Proof Plugging the stepsize choice α k = (f Γ (y k ) − d * )/ ζ k 2 into (32) implies T −1 k=0 (f Γ (y k ) − d * ) 2 2 ≤ y 0 − y * 2 2R(f ) 2(33)
where y * = x * /p * and Proposition 1 is used to bound ζ k ≤ 1/R(f ). Then the radially dual sharpness bound from Proposition 4 guarantees y 0 − y * ≤
p * +C x * C (f Γ (y 0 ) − d * ). Hence 1 T T −1 k=0 (f Γ (y k ) − d * ) 2 ≤ (p * + C x * ) 2 (f Γ (y 0 ) − d * ) 2 C 2 R(f ) 2 T . Therefore some k ≤ 4 p * +C x * CR(f ) 2
has halved the dual objective gap, f Γ (y k )− d * ≤ (f Γ (y 0 ) − d * )/2. Repeatedly applying this, we conclude that
T ≥ 4 p * + C x * CR(f ) 2 log 2 f Γ (y 0 ) − d *
implies min k<T f Γ (y k ) − d * ≤ for any > 0. Considering = /p * gives the claimed linear convergence rate.
This generalizes the linear convergence results of [38] for linear programming.
To the best of our knowledge, this is the first first-order method linear convergence guarantee for generic non-Lipschitz, sharp convex optimization.
Theorem 3 Consider any upper semicontinuous, concave f with R(f ) > 0 and p * = sup f ∈ R ++ attained at x * ∈ E. Fixing α k = (f Γ (y k ) − d * )/ ζ k 2 , if f satisfies the Lojasiewicz condition (29) with exponent θ > 0, then the radial subgradient method (Algorithm 1) has x k = y k /f Γ (y k ) satisfy
T ≥ O 1/ 2θ =⇒ min k<T p * − f (x k ) f (x k ) ≤ .
Proof By Proposition 3, the Lojasiewicz condition (28) holds at the dual minimizer y * = x * /p * for some constants r , C with the same exponent θ. Integrating this condition (as done in [6, Theorem 5]) ensures every y ∈ B(y * , r ) has the following local error bound
f Γ (y) − d * ≥ (C (1 − θ) y − y * ) 1/(1−θ) .(34)
The subgradient method must have some y k0 in the ball B(y * , r ) with
k 0 ≤ y 0 − y * (C (1 − θ)r ) 1/(1−θ) R(f ) 2
since (33) ensures the average iterate has objective gap squared at most (C (1 − θ)r ) 2/(1−θ) . Notice that the Polyak stepsize ensures the distance from the iterates y k to y * is nonincreasing as
y k+1 − y * 2 = y k − y * 2 − 2α k ζ T k (y k − y * ) + α 2 k ζ k 2 ≤ y k − y * 2 − 2α k (f Γ (y k ) − d * ) + α 2 k ζ k 2 ≤ y k − y * 2 − (f Γ (y k ) − d * ) 2 ζ k 2 ≤ y k − y * 2 .
Hence all k ≥ k 0 have y k ∈ B(y * , r ) as well. Then our claimed convergence rate follows by bounding the number of iterations required to ensure the objective gap halves f Γ (y k0+k ) − d * ≤ (f Γ (y k0 ) − d * )/2. Applying the local error bound (34) to (33) initialized at y k0 implies
1 T T −1 k=0 (f Γ (y k0+k ) − d * ) 2 ≤ (C (1 − θ)) 2 (f Γ (y k0 ) − d * ) 2(1−θ) R(f ) 2 T .
Thus at some k 1 ≤ k 0 +4 C (1−θ)
R(f ) 2 /(f Γ (y k0 )−d * ) 2θ
, the radially dual objective gap must have halved. Inductively, let k i+1 ≤ k i +4 C (1−θ)
R(f ) 2 /(f Γ (y ki )− d * ) 2θ
denote an iteration with half the dual objective value of k i . Let k j+1 denote the first of these iterations with f Γ (y kj+1 )−d * less than a target accuracy > 0. Then f Γ (y ki ) − d * ≥ 2 j−i for all i ≤ j. Inductively applying the definition of k i implies
k j+1 −k 0 ≤ j i=0 4 C (1 − θ) R(f ) 2 1 (2 j−i ) 2θ ≤ 4 1 − 2 −2θ C (1 − θ) R(f ) 2 1 2θ . Setting = /p * ensures p * −f (x k j+1 ) f (x k j+1 ) ≤ .
The previous pair of convergence theorems relied on using a Polyak stepsize, which requires the often impractical knowledge of d * . This can be remedied by replacing the simple subgradient method in Algorithm 1 with a more sophisticated stepping scheme like [22] or restarting scheme like [46,41,40] which all attain similar convergence guarantees.
Radial Smoothing Method
Now we turn our attention to the radial smoothing method previously defined as Algorithm 2 in the context of smoothing the radial dual of our quadratic program. More generally, we consider maximizing a minimum of several smooth concave functions f j : E → R ∪ {−∞} with bounded level sets over polyhedral constraints Ax ≤ b. Translating any known strictly feasible x 0 in the interior of the domain of f j , we have f j (0) > 0 and b > 0. Then we consider the equivalent nonnegative primal maximization problem
p * = max x min{(f j ) + (x) | j = 1, . . . , m 1 } s.t. a T i x ≤ b i for i = 1, . . . , m 2 (35) which has R((f j ) + ) ≥ R > 0 and D((f j ) + ) ≤ D < ∞ and each b i > 0. Further, since each f j has bounded level sets, f j is L-smooth on the level set {x | f j (x) > 0} for some sup{ ∇ 2 f j (x) | f (x) > 0} ≤ L < ∞.
This objective is strictly upper radial with radial dual
d * = min y∈E max f Γ j (x), (a i /b i ) T y | j ∈ {1, . . . , m 1 }, i ∈ {1, . . . , m 2 } . (36)
Then we consider the smoothing of this objective for any η > 0 given by
g η (y) = η log m1 j=1 exp (f j ) Γ + (y) η + m2 i=1 exp a T i y b i η .(37)
Our radial smoothing method (Algorithm 2) proceeds by minimizing this smoothing with Nesterov's accelerated method to produce a radially dual solution with accuracy O(η). Nearly any other fast iterative method could be employed here instead, which could then avoid needing knowledge of problem constants. Converting this radial dual guarantee back to the primal problem gives the following primal convergence theorem.
Theorem 4 Consider any problem of the form (35). Fixing L η = (1+D/R) 3 L+ max{1/R 2 , ai/bi } η and x 0 = 0, the radial smoothing method (Algorithm 2) has
x k = y k / max{(f j ) Γ + (y k ), (a i /b i ) T y k } feasible with p * − min{f j (x k )} min{f j (x k )} ≤ 2L η (1 + ηp * log(m 1 + m 2 )) 2 D 2 p * (k + 1) 2 + ηp * log(m 1 + m 2 ).
Setting η = /2 log(m 1 + m 2 ) ensures the following O(1/ ) convergence rate
k + 1 ≥ 2(1 + p * /2)D (1 + D/R) 3 L p * + 2 max{1/R 2 , a i /b i 2 } log(m 1 + m 2 ) p * 2 =⇒ p * − min{f j (x k )} min{f j (x k )} ≤ p * .
Proof Observe that all of the m 1 +m 2 functions defining g η are convex (by (15)), max{1/R, a i /b i }-Lipschitz continuous (by Proposition 1) and (1 + D/R) 3 Lsmooth (by Corollary 1). Then [3, Proposition 4.1] ensures g η is convex, is (1 + D/R) 3 L + max{1/R 2 , ai/bi } η -smooth, and closely follows the radially dual objective with every y ∈ E satisfying
0 ≤ g η (y) − max (f j ) Γ + (y), (a i /b i ) T y ≤ η log(m 1 + m 2 ).(38)
Note that for any s > 0, the related primal super-level set is bounded by
sup{ x | f j (x) ≥ s, a T i x ≤ b i } ≤ D.
Recalling epi f Γ = Γ (hypo f ) from (18) bounds every dual sub-level set by
sup{ y | f Γ j (y) ≤ 1/s, (a i /b i ) T y ≤ 1/s} ≤ D/s.
In particular, considering s = p * = 1/d * shows every radial dual minimizer has norm bounded by d * D. Then the upper bound from (38) ensures the d * + η log(m 1 + m 2 ) sub-level set of g η is nonempty and the lower bound from (38) allows us to bound this level set by
sup{ y | g η (y) ≤ d * + η log(m 1 + m 2 )} ≤ (d * + η log(m 1 + m 2 ))D
Therefore the distance from y 0 = 0 to a minimizer of g η is at most (d * + η log(m 1 + m 2 ))D.
Since g η is smooth and has a minimizer, applying the standard accelerated method convergence guarantee [32] guarantees the iterates of our radial smoothing method have
g η (y k ) − inf g η ≤ 2L η (d * + η log(m 1 + m 2 )) 2 D 2 (k + 1) 2 .
Converting this guarantee in terms of our radially dual objective, (38) ensures
max (f j ) Γ + (y k ), (a i /b i ) T y k − d * ≤ 2L η (d * + η log(m 1 + m 2 )) 2 D 2 (k + 1) 2 + η log(m 1 + m 2 ).
Stating this to be in terms of
x k = y k / max{(f j ) Γ + (y k ), (a i /b i ) T y k } yields p * − min{f j (x k )} min{f j (x k )} ≤ 2L η (1 + ηp * log(m 1 + m 2 )) 2 D 2 p * (k + 1) 2 + ηp * log(m 1 + m 2 ).
Renegar [39] uses the same general technique to give accelerated convergence guarantees for solving the broad family of hyperbolic programming problems (which includes semidefinite programming) where the radial dual also admits a natural smoothing. The restarting schemes of [41] and [40] both explicitly consider restarting smoothing methods to attain improved convergence when growth conditions like the Lojasiewicz condition (28) hold. Due to Proposition 3, applying these more sophisticated methods to solve the radially dual problem will give rise to radial algorithms that enjoy the same improved convergence. The analysis of such a method should follow similarly to Theorem 3.
Radial Accelerated Method
Motivated by our example transforming the Poisson likelihood problem (9), algorithms can be designed to take advantage of the radial transformation extending smoothness on a level set to hold globally. Consider maximizing any twice differentiable concave function f : E → R ∪ {−∞} with bounded level sets. Then, without loss of generality, we have 0 ∈ int {x | f (x) > 0} and so f + is strictly upper radial.
Letting L = sup{ ∇ 2 f (x) | f (x) > 0}, Corol- lary 1 ensures f Γ + is (1 + D(f )/R(f )) 3 L-smooth on all of E.
Hence f Γ + can be minimized directly using Nesterov's accelerated method, giving the following radial accelerated method defined by Algorithm 3. This radial algorithm inherits the primal accelerated method's O( Ldist(x 0 , X * ) 2 / ) rate, only requiring L-smoothness on the level set {x | f (x) > 0}.
Algorithm 3 The Radial Accelerated Method
Require: f : E → R ++ , x 0 ∈ dom f , L > 0, T ≥ 0 1: (y 0 , v 0 ) = Γ (x 0 , f (x 0 )) andỹ 0 = y 0
Transform into the radial dual 2: for k = 0 . . . T − 1 do 3:ỹ k+1 = y k − ∇f Γ (y)/(1 + D(f )/R(f )) 3 L Run the accelerated method 4:
y k+1 =ỹ k+1 + k−1 k+2 (ỹ k+1 −ỹ k ) 5: end for 6: (x T , u T ) = Γ (y T , f Γ (y T )) Transform back to the primal Theorem 5 Consider any twice differentiable, concave f with R(f ) > 0, D(f ) < ∞, and p * = sup f ∈ R ++ attained on X * ⊆ E. Fixing x 0 = 0, the radial accelerated method (Algorithm 3) has for any > 0,
k + 1 ≥ (1 + D(f )/R(f )) 3/2 2Ldist(x 0 , X * ) 2 p * =⇒ p * − f (x k ) f (x k ) ≤ .
Proof Recall the f Γ is convex by (15) and is (1 + D(f )/R(f )) 3 L-smooth by Corollary 1. Then Nesterov's classic analysis [32] ensures
f Γ (y k ) − d * ≤ 2(1 + D(f )/R(f )) 3 Ldist(y 0 , Y * ) 2 (k + 1) 2
where Y * = X * /p * . Letting (x k , u k ) = Γ (y k , v k ) yields primal iterates with f (x k ) ≥ 1/f Γ (y k ). Then multiplying through by 1/d * = p * produces
p * − f (x k ) f (x k ) ≤ 2(1 + D(f )/R(f )) 3 Ldist(y 0 /d * , X * ) 2 p * (k + 1) 2 .
Noting that y 0 /d * = x 0 = 0, this gives the claimed convergence guarantee.
A few remarks on this convergence result. The additional coefficient of (1 + D(f )/R(f )) 3/2 is quite pessimistic as many of the examples we have considered have radial dual smoother than the primal, but Corollary 1 fails to capture this potential upside in its O(L) bound. For particular applications, we expect much tighter bounds on the radially dual smoothness are possible. The proposed radial accelerated method unrealistically relies on knowledge of our smoothness constant upper bound (1 + D(f )/R(f )) 3 L. However, this can be remedied by including a linesearch/backtracking as done in [2,34]. Under growth conditions, the convergence of accelerated methods also improves. For example, applying the adaptive accelerated gradient method of [27] to solve the radially dual problem would give a radial method that speeds up in the presence of primal growth conditions by Proposition 3. The analysis of such a method should follow similarly to that of Theorem 3.
Radial Algorithms for Nonconcave Maximization
Our radial duality theory applies beyond concave maximization problems, applying to the broader family of nonconcave but upper radial maximization. Section 5.1 outlines several families of nonconvex settings where upper radiality holds and then Section 5.2 presents a performance guarantee for the radial subgradient method when maximizing a collection of such upper radial nonconcave functions.
Examples of Radial Duality with Nonconvex Objectives or Constraints
We say that a set S ⊆ E is star-convex with respect to the origin if every x ∈ S has the line segment λx ∈ S for all 0 ≤ λ ≤ 1. Geometrically, upper radial functions all have a star-convex-like hypograph with respect to the origin [17,Lemma 1], meaning that all (y, v) ∈ hypo f have λ(y, v) ∈ hypo f for all 0 < λ ≤ 1 6 . Star-convexity has been considered throughout the optimization literature. The structure of optimizing over star-convex constraint sets has been considered as early as [43]. In general, even linear optimization over star-convex bodies is NP-hard [8]. Efficient global optimization of star-convex objectives is possible if star-convexity holds with respect to a global optimizer (see [35,18,26,19,21]).
Star-Convex Constraints. Star convexity w.r.t. the origin is exactly the condition needed to ensure the nonstandard indicatorι S (x) = +∞ if x ∈ S 0 if x ∈ S. is strictly upper radial 7 . Then the radial dual of such a star-convex set's indicator function is given by the gaugê
ι Γ S (y) = sup{v > 0 | v ·ι S (y/v) ≤ 1} = sup{v > 0 | y/v ∈ S} = γ S (y).
Importantly, the gauge γ S (y) is convex if and only if S is convex. As a result, algorithms utilizing the radial dual of star-convex constraints avoid needing difficult nonconvex orthogonal projections, replacing them with evaluating a nonconvex gauge function appearing in the objective.
One important example where star-convex sets arises comes from considering chance constraints [23,31,48]. Given some distribution over potential constraint sets S ξ ⊆ E, a robust problem formulation may ensure that the constraint is satisfied with probability Λ ∈ [0, 1]. Then the chance-constrained feasible region is S = {x | P(x ∈ S ξ ) ≥ Λ}. If each potential constraint set is convex with 0 ∈ S ξ , then S is star-convex w.r.t. the origin.
Optimization over Compact Sets. Now we generalize our previous example from Section 2 where we saw that any nonconcave quadratic program with a compact polyhedral feasible region could be rescaled for our radial duality to apply. Consider maximizing any continuously differentiable function f over a compact set S that is star-convex w.r.t. the origin. Supposing f (0) > 0, this is equivalent to the following maximization problem of the primal form (2) max y∈E min{(1 + λf (x)) + ,ι S (x)} for any λ > 0. We check when this objective is strictly upper radial by considering whether its perspective function is strictly increasing on its domain:
v · min i {(1 + λf (y/v)) + ,ι S (y/v)} = (v + λvf (y/v)) + if y/v ∈ S 0 otherwise.
The partial derivative of this with v at any y/v ∈ S ∩ dom (1 + λf ) + is
1 − λ(∇f (y/v), −1) T (y/v, f (y/v)).
Noting that (∇f (x), −1) T (x, f (x)) is a continuous function on the compact set S ∩ dom (1 + λf ) + , we can select λ > 0 small enough to always have
1 − λ(∇f (y/v), −1) T (y/v, f (y/v)) > 0.
Doing so makes our objective strictly upper radial and so radial duality applies.
Nonconvex Regularization. Many optimization tasks take the additive composite form max y∈E f (x) − r(x) where f is an upper semicontinuous, concave function with f (0) > 0 and r(x) is an added (or rather subtracted since we are maximizing) regularization term. Many sparsity-inducing regularization penalties decompose as a sum over the x's coordinates r(x) = n i=1 σ(x i ) for some simple nonconvex function σ : R → R. For example, q -regularization sets σ(t) = λ|t| q for some 0 < q < 1, bridging the gap between 0 and 1regularization. Many more regularizers are of this form, like SCAD regularization [12], MCP [49], and firm thresholding [15]. See [45] for a wide survey.
These regularizers are all continuous and have r(y/v) nonincreasing in v. These two simple properties suffice to guarantee subtracting r from f will not break its upper radiality since
v(f (y/v) − r(y/v)) + = max{vf (y/v) − vr(y/v), 0}
is a sum of two upper semicontinuous, nondecreasing functions in v. As a result, our radial duality applies to the nonconcave primal (f (x) − r(x)) + .
Optimization with Outliers. Many learning problems take the form of minimizing a stochastic loss function E ξ [f (x, ξ)] using a finite sample approximation with f ( : , ξ) : E → R ∪ {∞}. Given i.i.d. samples ξ 1 , . . . , ξ s , this problem can be formulated as
max x∈E 1 s s i=1 −f (x, ξ i ).
If each −f (·, ξ i ) is concave and a point is known in the interior of each function's domain, a translation can ensure −f (0, ξ i ) > 0 for all i. Hence their sum is concave with a positive value at zero and so 1 s ( s i=1 −f (x, ξ i )) + is upper radial. Hence our radial duality can be applied. In the presence of t outliers in the s samples ξ 1 , . . . , ξ s , this finite sample approximation could be improved to only consider the loss function on the best s − t samples These partial sums are also concave with positive value at zero and hence this whole objective is upper radial by [17, Corollary 2] and so our radial duality applies. The minimax formulation of [47] exactly corresponds to this problem formulation at its equilibrium. By the same corollary, our radial duality also applies to maximizing the (s − t)th largest element of {−f (x, ξ i )} s i=1 . Such an optimization problem captures the classic idea of least median of squares regression [42].
Example Nonconcave Guarantee for the Radial Subgradient Method
In this concluding section, we demonstrate the style of results possible from applying our radial duality to upper radial nonconcave maximization. In particular, we consider the nonconcave, nonsmooth primal problem of maximizing the minimum of a set of twice continuously differentiable, strictly upper radial f j over some convex set S ⊆ E p * = max x min{f j (x) | j = 1, . . . , m} s.t.
x ∈ S = max x∈E min{f j (x),ι S (x)} (39) where each f j has R(f j ) ≥ R > 0 and bounded level sets D(f j ) ≤ D < ∞ and the origin lies in the constraints with B(0, R) ⊆ S. Let L ≥ sup{ ∇ 2 f j (x) | f j (x) > 0, x ∈ S} bound the smoothness of each f j on this compact level set. This primal is strictly upper radial since each function defining the minimum is strictly upper radial. Then the radial dual of this problem is
d * = min y∈E max{f Γ j (y), γ S (y)}.(40)
Note each f Γ j (y) is convex if and only if f j is concave by (15). Hence if our primal (39) is nonconcave, our radial dual (40) will be nonconvex. Regardless, our previously proposed radial subgradient method (Algorithm 1) can still be applied and analyzed.
Recently, convergence theory for subgradient methods without convexity has been developed. Particularly, consider minimizing a nonconvex, nonsmooth function g : E → R that is bounded below. Then [10, Theorem 3.1] ensures that provided g is uniformly M -Lipschitz and ρ-weakly convex (defined as g + ρ 2 · 2 being convex), the subgradient method y k+1 = y k − αζ k for ζ k ∈ ∂ P g(y k ) has some y k that is nearly stationary on the Moreau envelope of g. In particular, [10, (3.9)] this implies that proper selection 8 of α ensures some y k has a nearby y that is nearly stationary T ≥ 16ρM 2 (g(y 0 ) − inf g) 4 =⇒ min k<T { y − y k } ≤ 2ρ with dist(0, ∂ P g(y)) ≤ .
Applying this to the radial dual allows us to ensure a nearly stationary point y near a dual iterate y k exists. Then converting this guarantee back to the primal preserves the above O(1/ 4 ) rate despite not assuming the primal (39) is either Lipschitz or weakly convex (instead assuming it is strictly upper radial).
Theorem 6 Consider any problem of the form (39) with p * ∈ R ++ . Fixing x 0 = 0 and α k = / ζ k 2 , the radial subgradient method (Algorithm 1) with properly chosen constant stepsize α k = α has x k = y k / max{f Γ j (y k ), γ S (y k )} satisfy
T ≥ 16(1 + D/R) 3 L(min{f j (x 0 )} − p * ) R 2 min{f j (x 0 )}p * 4 =⇒ min k<T { x − x k } ≤ 2p * (1 + D/R) 4 L
with dist(0, ∂ P min{f j ,ι S }(x)) ≤ p * 1 − D for some nearby x ∈ E provided 0 < < 1/D.
Proof Observe that each function in the maximum defining the radial dual (36) is 1/R-Lipschitz (by Proposition 1) and each f Γ j is (1 + D/R) 3 L-smooth (by Corollary 1). Then the whole radially dual objective max{f Γ j (y), γ S (y)} is 1/R-Lipschitz and (1 + D/R) 3 L-weakly convex. Hence even though our primal is not assumed to be either Lipschitz or weakly convex, these two properties occur in the radial dual due to each f i having R(f i ) > 0 and smoothness on the level set {x | f j (x) > 0} respectively. Then we can apply (41) implying
that whenever T ≥ 16(1+D/R) 3 L(min{f Γ j (y0)}−d * ) R 2 4
, a nearby y has min k<T { y − y k } ≤ 2(1 + D/R) 3 L and dist(0, ∂ P max{f Γ j , γ S }(y)) ≤ .
First, we show the nearby radial dual solution y maps to a primal solution x = y/ max{f Γ j (y), γ S (y)} that is near the primal iterates. Having dual distance y − y k ≤ 2(1 + D/R) 3 L ensures x − x k is bounded by y max{f Γ j (y), γ S (y)} − y k max{f Γ j (y k ), γ S (y k )} ≤ y − y k max{f Γ j (y), γ S (y)} + y k max{f Γ j (y), γ S (y)} − y k max{f Γ j (y k ), γ S (y k )} = y − y k max{f Γ j (y), γ S (y)} + x k max{f Γ j (y k ), γ S (y k )} max{f Γ j (y), γ S (y)} − 1 ≤ y − y k max{f Γ j (y), γ S (y)} + D y − y k /R max{f Γ j (y), γ S (y)} ≤ p * (1 + D/R) y − y k ≤ 2p * (1 + D/R) 4 L where the first inequality uses the triangle inequality and the second uses the bounded primal level sets and the radially dual 1/R-Lipschitz continuity, and the third uses that d * = 1/p * ∈ R ++ . We complete our proof by relating the stationarity of y to that of x. Let v = max{f Γ j (y), γ S (y)}, u = 1/v and ζ ∈ ∂ P max{f Γ j , γ S }(y) denote a radially dual subgradient with ζ ≤ . Then we can bound (ζ , −1) T (y, v) ≤ ζ y − v ≤ x /u − 1/u ≤ −(1 − D)/p * < 0 using that u ≤ f (x k ) ≤ p * . Note epi max{f Γ j , γ S } = Γ (hypo min{f j ,ι S }) by (18). Then the normal (ζ , −1) ∈ N P epi max{f Γ j ,γ S } (y, v) corresponds to the primal normal (ζ , (ζ , −1) T (y, v)) ∈ N P hypo min{fj ,ι S } (x, u) by [17,Proposition 5]. Hence ζ := ζ /(ζ , −1) T (y, v) ∈ ∂ P min{f j ,ι S }(x) is a primal subgradient with norm at most O( ) as
ζ = ζ (ζ , −1) T (y, v) = ζ |(ζ , −1) T (y, v)| ≤ p * 1 − D .
Fig. 1 :
1The minimum relative accuracy p * −f (x k )
Fig. 2 :
2Convergence of prim , dual , comp on a random QP of size (n, m) = (1600, 6400).
δ) and δ i = 0. The existence of such a sequence is guaranteed by the Horizontal Approximation Theorem [9, Page 67].
,
| S ⊆ {1...s}, |S| = s − t
Table 1 :
1Common families of constraints with closed-form descriptions of their gauge.
Table 2 :
2Common function classes with closed-form descriptions of their radial dual.
Relating Extreme Points, (Sub)Gradients, and Hessians. We recall a few bijections relating functions and their radial transformations. For any strictly upper radial f , [17, Lemma 2] ensures
Instead of using a re-parameterization, one can explicitly include equality constraints in our model. The details of this approach are given in Section 2.2.1, where we see that equality constraints are unaffected by the radial dual.
Note this hypograph is not actually star convex since (0, 0) ∈ hypo f .
This is essentially by definition as v ·ι S (y/v) is nondecreasing in v if and only if S is star-convex w.r.t. the origin. Then it is simple to check this function is upper semicontinuous and is vacuously strictly increasing on its effective domain domι S = ∅, which is empty.
Namely, given the method will be run for T steps, the example analysis of[10] shows selecting α = g(y 0 )−inf g ρM 2 (T +1) suffices to give the claimed rate. Alternative stepsizes could be analyzed by the same proof technique proposed therein resulting in different assumptions on which parameters are known.
Acknowledgements The author thanks Jim Renegar broadly for inspiring this work and concretely for providing feedback an early draft and Rob Freund for constructive thoughts helping focus this work. Additionally, two anonymous referees and the associate editor provided useful feedback much improving this work's presentation and clarity.A LogSumExp Gradients and QP Optimality CertificatesIn our quadratic programming example(8)and our generalized setting(37), we consider smoothings of a finite maximum. Given a smooth convex functions f i : E → R, we considered the smoothing of max{f i } with parameter η > 0 given by fη(x) := η log n i=0 exp(f i (x)/η) . Its gradient is given byComputationally evaluating this requires mild care to avoid precision issues with exponentiating potentially larger numbers. It is numerically stable to instead compute these coefficients via the equivalent formulaNext, we specialize this formula to the setting of quadratic programming for gη in(8). Observe the gradient of the objective component is given byby using the gradient formula(25). The gradients of the transformed constraints are simply ∇a T i y/b i = a i /b i . Then the gradient of the smoothing overall is given byThis gradient can be computed using two matrix multiplications with A: Ay is needed to compute the coefficients λ i , then A T [λ 1 /b 1 . . . λn/bn] is needed for the summation above.This gradient formula indicates a reasonable selection of dual multipliers v i = λ i (1− 1 2 x T Qx) λ 0 b i as we then have gη(y) proportional to Qx + c + A T v.
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| [] |
[
"Exploiting the Partly Scratch-off Lottery Ticket for Quantization-Aware Training",
"Exploiting the Partly Scratch-off Lottery Ticket for Quantization-Aware Training"
] | [
"Journal Of L A T E X Class ",
"Files "
] | [] | [] | Quantization-aware training (QAT) receives extensive popularity as it well retains the performance of quantized networks. In QAT, the contemporary experience is that all quantized weights are updated for an entire training process. In this paper, this experience is challenged based on an interesting phenomenon we observed. Specifically, a large portion of quantized weights reaches the optimal quantization level after a few training epochs, which we refer to as the partly scratchoff lottery ticket. This straightforward-yet-valuable observation naturally inspires us to zero out gradient calculations of these weights in the remaining training period to avoid meaningless updating. To effectively find the ticket, we develop a heuristic method, dubbed lottery ticket scratcher (LTS), which freezes a weight once the distance between the full-precision one and its quantization level is smaller than a controllable threshold. Surprisingly, the proposed LTS typically eliminates 50%-70% weight updating and 25%-35% FLOPs of the backward pass, while still resulting on par with or even better performance than the compared baseline. For example, compared with the baseline, LTS improves 2-bit MobileNetV2 by 5.05%, eliminating 46% weight updating and 23% FLOPs of the backward pass. Code is at https://github.com/zysxmu/LTS. | 10.48550/arxiv.2211.08544 | [
"https://export.arxiv.org/pdf/2211.08544v3.pdf"
] | 253,553,264 | 2211.08544 | cf8a93a8da7eac8ce14443b288a2c061af5916e5 |
Exploiting the Partly Scratch-off Lottery Ticket for Quantization-Aware Training
AUGUST 2021 1
Journal Of L A T E X Class
Files
Exploiting the Partly Scratch-off Lottery Ticket for Quantization-Aware Training
148AUGUST 2021 1Index Terms-Network quantizationQuantization-aware trainingLottery ticket hypothesisWeight frozen
Quantization-aware training (QAT) receives extensive popularity as it well retains the performance of quantized networks. In QAT, the contemporary experience is that all quantized weights are updated for an entire training process. In this paper, this experience is challenged based on an interesting phenomenon we observed. Specifically, a large portion of quantized weights reaches the optimal quantization level after a few training epochs, which we refer to as the partly scratchoff lottery ticket. This straightforward-yet-valuable observation naturally inspires us to zero out gradient calculations of these weights in the remaining training period to avoid meaningless updating. To effectively find the ticket, we develop a heuristic method, dubbed lottery ticket scratcher (LTS), which freezes a weight once the distance between the full-precision one and its quantization level is smaller than a controllable threshold. Surprisingly, the proposed LTS typically eliminates 50%-70% weight updating and 25%-35% FLOPs of the backward pass, while still resulting on par with or even better performance than the compared baseline. For example, compared with the baseline, LTS improves 2-bit MobileNetV2 by 5.05%, eliminating 46% weight updating and 23% FLOPs of the backward pass. Code is at https://github.com/zysxmu/LTS.
In spite of the merits, quantized networks easily suffer from severe performance drops if simply performing the lowbit quantization, since the low-bit data format possesses a very limited representation capacity compared with the fullprecision counterpart. Thus, the quantized network usually needs to be trained for dozens of epochs to compensate for the performance drop, which is known as quantization-aware training (QAT) [11]. Specifically, in the forward pass, weights and activations are quantized to simulate quantized inference. In the back propagation, the weights are updated as usual. All weights of the network are still represented in the fullprecision structure, so that the weights can be updated by small gradients. By continuous updating, the network weights are adjusted to accommodate the quantization effects, which therefore alleviates the performance drop significantly. In QAT, the common experience is that all quantized weights are updated throughout the entire training period. On the contrary, in this paper, we challenge the necessity of continuous weight updating: Whether it is necessary to update all weights in the whole training process of QAT?
To this end, we dive into the training evolution of the quantization level for each weight in QAT. For the first time, we identify an interesting observation that a large portion of network weights quickly reach their optimal quantization levels after a few training epochs. As shown in Fig. 1a, for "layer1.conv1" of the 2-bit ResNet-20 [12] trained on CIFAR-100 [13], over 50% weights have already reached their optimal quantization levels even without any training, and the portion rises up to 80% at epoch 80. For "layer1.conv1" of the 2bit ResNet-20 on CIFAR-10 [13], near 50% over have already reached their optimal quantization levels before training and 80% weights reach their optimal quantization levels at epoch 120 as exhibited in Fig. 1b. Similar observations can be found from Fig. 1c-Fig. 1f across different bit widths and datasets. Inspired by the lottery ticket hypothesis in network pruning [14], in this paper, we term these weights that quickly reach the optimal quantization levels as the partly scratchoff lottery ticket since they appear along with training. We attribute this phenomenon to two characteristics in network quantization. First, a network is usually quantized from a 0000-0000/00$00.00 © 2021 IEEE Mv2-W2A2-ImageNet, s2.u1.conv3 (i) Fig. 1. The ratio of the weights reaching the optimal quantization level w.r.t. training epochs. The red pentagon denotes the best model. "WBAB" indicates the weights and activations are quantized to B-bit. "R-20", "R-18", "R-50", "Mv1", and "Mv2" indicate ResNet-20, ResNet-18, ResNet-50, MobileNetV1, and MobileNetV2, respectively. well-trained full-precision model, which gives good initial values. Second, a discrete quantization level often covers a range of continuous numerical values. Thus, many weights rapidly converge to their best quantization levels although their full-precision version still remains far away from the optimal states.
Such an observation indicates that a portion of weights can be safely frozen and their gradients do not require calculations throughout the whole training process. To find the partly scratch-off lottery ticket, we introduce a simple-yet-effective heuristic method, named lottery ticket scratcher (LTS). Specifically, we first measure the distance between a normalized weight and its corresponding quantization level. Then, the partly scratch-off lottery ticket comprises these weights whose exponential moving average (EMA) distance is lower than a pre-defined threshold. A weight will be permanently frozen once it falls into the partly scratch-off lottery ticket. An illustration of our LTS is presented in Fig. 2. In the forward pass, the full-precision weights and activations are quantized by the quantizer, the results of which are multiplied to derive the outputs. In the backward pass, we freeze these weights within the partly scratch-off lottery ticket by setting their gradients to zero, which actually leads to a sparse weight gradient. This sparse weight gradient indicates a reduction in the training costs, since the related calculations of these gradients can be avoided. Furthermore, such a sparsity in weight gradient incurs an easy-implemented structured computational reduction as stated in Sec. III-D.
We also emphasize that LTS is fundamentally different from sparse training methods in four aspects [15], [16]. First, our LTS focuses on a sparse weight gradient, which involves a structured computational reduction (See Sec. III-D). Second, LTS does not prune any weight while sparse training prunes weights. Third, LTS stops updating frozen weights forever while sparse training usually revives weights, which involves continuous weight gradient calculations [17]. Lastly, LTS is specialized in quantized networks while sparse training is for full-precision models.
By precisely identifying the partly scratch-off lottery ticket, the proposed LTS not only achieves comparable or even better quantizer quantizer ̅ ̅ frozen unfrozen performance than the compared baseline, but also results in a non-negligible reduction in training costs. Generally, LTS achieves comparable performance across various quantization, and even improves the performance when quantizing the network in the same low-bit cases for our LTS effectively solves the weight oscillations problem [18] (Sec. IV-C). At the same time, LTS typically obtains an average of 30%-60% weight gradient sparsity, which eliminates a total of 15%-30% FLOPs of backward propagation. Taking 2-bit ResNet-18 as an example, compared with the normal QAT, our LTS improves performance by 1.41% while eliminating 56% weight updating and 28% FLOPs of the backward pass.
× × × × × × × × × × ̅ × × × × × ( , ) -ℎ row -ℎ col ̅ update STE × × (a) quantizer quantizer ̅ ̅ frozen unfrozen × × × × × × × × × × ̅ × × × × × ( , ) -ℎ row -ℎ col ̅ update STE × × × (b)
II. RELATED WORK
A. Quantization-Aware Training
Network quantization has long been the research focus of the model compression community for its superiority in simultaneously reducing the model size and computational costs. However, the quantized network often suffers from performance degradation. A variety of techniques are explored to overcome this issue. Most existing QAT methods focus on designing quantizers that are more suitable for the network weights or activations [19]- [23]. Other methods instead pay attention to special training strategies for quantized networks [24]- [26], approximating gradients of the quantization functions [9], [27], training regularization [28], [29] or mixprecision quantization [30]- [32]. Although these techniques alleviate performance degradation, they stiffly rely on the training overhead.
B. Lottery Ticket Hypothesis
The lottery ticket hypothesis is first proposed in [14], which indicates that a randomly initialized network contains subnetworks (a.k.a., "winning tickets") that can achieve the test accuracy of the original network when trained independently. Following this conjecture, many empirical studies [33]- [37] have been devoted to finding the lottery ticket. In contrast, the partly scratch-off lottery ticket in this paper is different. We firstly reveal the existence of a sub-network that does not require updating in QAT, while the lottery ticket targets weight pruning weight and full-precision networks.
C. Training Acceleration
Many efforts have been made to accelerate the training DNNs. One common way is to exploit the highly efficient distributed training [38]- [41]. However, the distributed training is towards modern single-and multi-GPU servers, clusters, and even supercomputers, where the acceleration comes at expensive training costs and energy consumption. In contrast, our LTS aims to reduce training costs.
Gradient pruning targets pruning gradients in the computationally-intensive backpropagation [42], [43]. For example, meProp [42] preserves the top-k gradients in the matrix multiplication output. In contrast to gradient pruning which continuously updates weights in the training process, our LTS stops updating these frozen weights forever.
Sparse training imposes sparsity constraints during network training [15]- [17], [44]- [47]. By continuously removing and reviving network weights in the training, related computations in forward and backward propagations can be reduced. Also, these methods are often designed for full-precision networks. Differently, our LTS does not remove or revive any weight and is particularly designed for network quantization.
Another line of work represents gradients in a low-bit data format to reduce gradient computing costs [48], [49].
In compliance with extensive analyses on gradient distribution, most current methods focus on designing a suitable quantizer [50]. For instance, Lee et al. [51] implemented a low-bit representation of gradients in a searching-based style.
[52] reasoned a near-lognormal distribution of gradients and obtained the best clipping value analytically. Our LTS is orthogonal to these methods since we focus on quantizing the network in the forward pass. These methods can be integrated into our LTS to further speed up the network training.
III. METHOD
A. Preliminary
A quantizer is adopted in QAT to quantize the full-precision weights and activations to simulate the quantization inference. Following the settings of [26], [53], we use the uniform quantizer with trainable clipping parameters to implement network quantization. Concretely, given full-precision data x (weights or activations), it is first normalized to a range [0, 1] with two trainable clipping parameters l, u by the following equation:
x n = clip( x − l u − l , 0, 1),(1)
where clip(x n , l, u) = min max(x n , l), u , · , l, u denotes lower bound and upper bound, respectively. Then, the normalized data x n is quantized by the following quantizer:
q = (2 b − 1) × x n ) ,(2)
where · denotes the round function that rounds its input to the nearest integer. The corresponding de-quantized valuex is calculated according to its data type. For weights, the dequantized value is calculated as:
x = 2 × q 2 b − 1 − 0.5 , x ∈ weights.(3)
For activations, the de-quantized value is calculated as:
x = q 2 b − 1 , x ∈ activations.(4)
For activations and weights, we both use layer-wise quantizer. Following [53], for weight, the lower bound l and upper bound u are initialized to negative three times the standard deviation and negative three times the standard deviation of the full-precision weight, respectively. For activations, the lower bound l and upper bound u are initialized to minimum and maximum values of the full-precision activations, respectively.
B. Partly Scratch-off Lottery Ticket
The limited capacity of low-bit representation requires accomplishing network quantization in a quantization-aware training (QAT) manner so as to mitigate the performance gap between the quantized network and its full-precision counterpart. As a common experience, the weights of the quantized network are updated for the entire training process in QAT. On the contrary, in this paper, we challenge the necessity of updating weight throughout the whole training process of QAT, which is barely investigated in the literature.
To this end, we dive into the training evolution of quantization level for each weight and unearth an interesting phenomenon that many network weights converge to their optimal quantization levels in the early training process. Concretely, we calculate the proportion of quantized weights that reach the optimal quantization level along with the network training. Fig. 1 provides the results over various datasets and bit-widths. Taking Fig. 1a as an instance, for "layer1.conv1" of 2-bit ResNet-20 on CIFAR-100, the optimal quantized network is obtained at around epoch 320. However, over 65% of weights reach their optimal quantization levels without any training. And it increases to 80% at epoch 80. A similar observation can be found at other bit-widths. For example, in Fig. 1d, 20% weights already stay in the optimal quantization level before training a 4-bit ResNet-20 on CIFAR-100 and it becomes 40% at epoch 80. Networks on larger datasets such as ImageNet [54] also present a similar observation as that in CIFAR-10/100. For instance, for 2-bit ResNet-18, over 50% weights stay in the optimal quantization level before training and it becomes 60% at epoch 10.
The above phenomenon indicates that the half-trained quantized network contains a quantized subnet consisting of these well-optimized quantized weights. Alike to the lottery ticket in network pruning [14], we term these weights as the partly scratch-off lottery ticket, which means they do not require to be trained from the beginning to end. When diving into a deeper analysis, we attribute the existence of the partly scratch-off lottery ticket to two possible characteristics in network quantization. First, weights of the quantized network are often initialized from a pre-trained full-precision model. It can be expected that many weights start from an optimal or sub-optimal state. As a result, several training epochs lead to the convergence of many weights in their optimal quantization level. Second, compared with the full-precision data format, the quantization levels are discrete. Numerous continuous values in an interval are mapped to the same discrete state. Despite that full-precision weights vary across different training epochs, they remain at the same quantization level. In particular, for the lower bit-width case, more partly scratch-off lottery tickets can be observed since each interval becomes much larger. Fig. 1 demonstrates this potential: the 2-bit ResNet-20 manifests a higher ratio of the optimal quantization level than its 4-bit version in the early training process.
Based on this phenomenon, we realize the unnecessary expense of updating these tickets during most part of the training period since they already fall into their optimal quantization levels. Therefore, if we can well locate these tickets, it can be expected that the performance degradation of the quantized network is negligible. Moreover, as an intuitive byproduct, we can reduce the training costs by creating a weight gradient sparsity due to the gradient calculations of these tickets can be eliminated.
C. Lottery Ticket Scratcher
In this subsection, we introduce lottery ticket scratcher (LTS), a heuristic method, to discover the partly scratchoff lottery. By gradually pulling out weights from network training, the proposed LTS creates a sparse weight gradient, leading to a shrinkage of training costs.
Specifically, our LTS employs a simple-yet-effective heuristic rule to find the partly scratch-off lottery ticket. At first, we train the quantized network for E wm epochs as the warmup stage. Then, the partly scratch-off lottery comprises weights whose EMA distance is lower than a pre-defined threshold t.
Herein, the EMA distance is the exponential moving average distance between the normalized weight w n and its corresponding quantization level. At the i-th training iteration, the EMA distance of the given w n is defined as:
D i wn = mD i−1 wn + (1 − m)D wn ,(5)
where m is the momentum of the EMA function, and D wn is the distance between w n and its corresponding quantization level. Note that D i wn will be reset to the quantization interval ∆ B = 2 2 B once the quantization level of w n is changed. We then compare D i wn with a threshold t to determine whether w n should be frozen. If D i wn is lower than the threshold t, w n will be frozen, meaning that zeroing out the gradient of w n . Otherwise, w n is continuously updated. The threshold t is computed by obtained ∆ B with a rate p:
t = ∆ B · p.(6)
The constant quantization interval ∆ B is determined by the bit-width B. We adjust the rate p to control the ease of frozen weights. The selection of p provides a trade-off between performance and training costs reduction. For example, a small p returns a small threshold t, which means a weight has to be very close to its quantization level if it is frozen. As a result, the identification of the partly scratch-off lottery is more accurate that is able to retain the performance. However, it is more difficult to freeze weights and more weights are prone to be updated. In this case, the weight gradient sparsity is limited and the reduced training costs are also limited. In contrast, a large p indicates a large threshold t. Subsequently, weights are more likely to be misidentified as the partly scratch-off lottery since the condition is loose. This misidentification will lead to performance degradation. However, it would be easier to freeze weights and achieve a higher weight gradient sparsity, so that the training costs can be reduced significantly.
To adjust the value of p, we design three strategies including fixing, linear-growth, and sine-growth. The fixing strategy indicates the rate p is set to a constant c that ranges from 0 to 1. In linear-growth strategy and sine-growth strategy, the rate p gradually increases in a linear manner and a sine manner when the warmup stage finishes, respectively. Supposing the total training epoch is E, the current iteration is i, and an epoch has T iterations. The linear-growth strategy is defined as:
p = i − T · E wm T · E − T · E wm · I(i > T · E wm ),(7)
where I() is the indicator function. The sine-growth is defined as:
p = sin i − T × E wm T · E − T · E wm · π 2 · I(i > T · E wm ). (8)
Both linear-growth strategy and sine-growth strategy increase the rate p from 0 to 1. We employ the linear-growth strategy in our experiments for it achieves a good balance between sparsity and accuracy (Sec. IV-C).
D. Efficient Implementation
In this subsection, we demonstrate that the sparse weight gradients actually lead to a structured computational reduction, which is easy to implement. The illustration of the backward pass is shown in Fig. 2b. As can be seen, the gradient of weight in position (i, j) is computed by multiplying the i-th row of theĀ l and the j-th column of the ∂L ∂Ā l+1 . The computational process is structured since it only involves the multiplication of two vectors. Due to we already know the position of the frozen weight before weight gradient calculations, we can omit the vector multiplications according to the set consisting of positions of frozen weights. We implement it with the CUDA toolkit and the result shows that 50% average weight gradient sparsity gives us roughly 20% time cost reduction in the backward propagation.
IV. EXPERIMENTATION
A. Experimental Settings 1) Implementation: We quantize ResNet-20 [12] for CIFAR-100/10 [13], ResNet-18, ResNet-50 [12], Mo-bileNetV1 [55], and MobileNetV2 [56] for ImageNet [54]. All code are implemented with Pytorch [57]. For all experiments we quantized all layers of networks into 2-, 3-, and 4-bit.
All networks are trained with cross-entropy loss. The gradient of the round function is set to 1 by using the straightthrough estimator (STE) [58]. The SGD optimizer is adopted with a momentum of 0.9. For CIFAR-100/10, the initial learning rate, batch size, weight decay, and total training epochs are set to 0.1, 256, 0.0001, and 400, respectively. For ImageNet, the batch size, weight decay, and total training epochs are set to 256, 0.0001, and 100, respectively. The initial learning rate is set to 0.1 for ResNet-18/50 and 0.01 for MobileNetV1 and MobileNetV2, respectively. The learning rate is decayed by a factor of 0.1 for every 100 epochs on CIFAR-100/10, and every 30 epochs on ImageNet. The pre-trained MobileNetV1 is obtained from pytorchcv and other models are downloaded from the torchvision. For all experiments, we employ the linear-growth strategy for it achieves the best trade-off between sparsity and accuracy. For the hyper-parameters, we use grid search to search for the best configuration for each dataset. We set m as 0.99 for CIFAR-100/10, and as 0.9999 for ImageNet. For E wm , we set it as 80 for CIFAR100/10, and as 30 for ImageNet by using the grid search. Despite they might not be optimal for all networks, we find these values already have provided satisfactory performance.
2) Metrics: We report the top-1 accuracy as the metric. To measure the reduced computational training costs, we report the average weight gradient sparsity, which is the sum of the weight gradient sparsity per iteration divided by the total number of iterations. The weight gradient sparsity per iteration is the ratio of the number of frozen weights to the number of all weights at the current iteration. We also employ the overall reduction in FLOPs of backward propagation to measure training cost reduction. This metric is equal to half of the average weight gradient sparsity since LTS only eliminates the computation of weight gradient.
B. Ablation Study
In this section, we first conduct ablation studies of the proposed three strategies and hyper-parameters of our LTS. All experiments are conducted by quantizing ResNet-20 to 2bit on CIFAR-100. The top-1 accuracy and the reduction in FLOPs of backward propagation are reported. EMA Parameter. The ablation study of EMA parameter m is shown in Tab. II. When m = 0, it is equal to EMA not used. It can be seen that the highest average weight gradient sparsity is achieved when the m = 0, which is 73%. However, compared with m = 0.99, the 7% improvement on average weight gradient sparsity comes at the cost of 1.33% accuracy degradation. As the m increases, the accuracy improvement is slight while the average weight gradient sparsity decreases quickly. Specifically, compared to the results of m = 0.99, results of m = 0.999 only lead to 0.05% accuracy gain but the sparsity decreases by 6%.
Fixing Strategy. The fixing strategy has two hyperparameters including p and E wm . Fig. 3a and Fig. 3b respectively provide the top-1 accuracy of 2-bit ResNet-20 on CIFAR100 w.r.t p and E wm . From Fig. 3a, it can be observed that the reduction in FLOPs of backward propagation increases along with the increase of p, and the best accuracy is obtained when p = 0.05. As presented in Fig. 3b, for fixing strategy, it is hard to find a good trade-off between accuracy and backward propagation FLOPs reduction since the accuracy improvements are accompanied by a low FLOPs reduction.
Thus, compared with the other two strategies, the fixing strategy does not exhibit any advantage. Linear-growth Strategy. As illustrated in Fig. 3c, as the warmup epochs E wm increases, the backward propagation FLOPs reduction consistently decreases. The performance reaches 53.76% and the FLOPs reduction is 33% at E wm = 80. When E wm > 80, the accuracy has not been greatly improved or even decreased.
Sine-growth Strategy. From Fig. 3d, we can find that the accuracy is 53.72% when E wm = 120. After that, the performance improvement is marginal or negative despite the backward propagation FLOPs reduction decreasing rapidly.
In summary, comparing these three strategies, we can observe that the linear-growth strategy and the sine-growth strategy obtain a better performance than the fixing strategy. Moreover, the linear-growth strategy enjoys higher FLOPs reduction than the sine-growth when they have comparable accuracy. Thus, we consider it achieves the best trade-off between performance and the backward propagation FLOPs reduction and thus employ this strategy in all following experiments.
C. Quantitative Comparison 1) CIFAR-100/10: In this subsection, we first evaluate the performance of the proposed LTS CIFAR-100/10 datasets by comparing them against the baseline. The results are presented in Table I. Our proposed LTS achieves significant improvements in terms of average weight gradient sparsity and backward propagation FLOPs reduction across all bit-widths, while maintaining consistently high accuracy compared with the baseline.
On CIFAR-100, our proposed LTS achieves an average weight gradient sparsity of 66%, 68%, and 70% for 2-, 3-, and 4-bit, respectively. This corresponds to a reduction in backward propagation FLOPs of 33%, 34%, and 35%. Notably, the proposed LTS improves the performance by 0.81%, 0.76%, and 0.99% for 2-, 3-, and 4-bit, respectively, compared to the baseline.
Similarly, on CIFAR-10, our proposed LTS yields better or comparable performance compared to the baseline across all bit-widths, while achieving high average weight gradient sparsity and backward propagation FLOPs reduction. Specifically, for 2 and 3-bit, the proposed LTS achieves performance gains of 0.53% and 0.59%, respectively, and only incurs a minor performance drop of 0.01% for 4-bit. For 2-, 3-, and 4-bit, LTS respectively provides 69%, 69%, and 70% weight gradient sparsity, which corresponds to a reduction in backward propagation FLOPs of 34.5%, 34.5%, and 35%.
Overall, experimental results demonstrate the superiority of the proposed LTS over the baseline in terms of both accuracy and efficiency on both CIFAR-100 and CIFAR-10 datasets. The significant improvements in average weight gradient sparsity and backward propagation FLOPs reduction across all bitwidths suggest that the proposed LTS is a promising method for improving the efficiency of quantization-aware training. the baseline, the proposed LTS achieved an average weight gradient sparsity of 49%, 51%, and 47% for 2-, 3-, and 4bits, respectively. This translates to a reduction in backward propagation FLOPs of 24.5%, 25.5%, and 23.5%, respectively. For 3-and 4-bit, LTS only results in a sightly performance drop. Notably, the proposed LTS improves the baseline by 0.28% for 2-bit quantization.
ResNet-50. The results of ResNet-50 are presented in the second part of Tab. III. It can be seen that the proposed LTS in general provides 50% weight gradient sparsity, indicating the effectiveness of our LTS. Moreover, our LTS improves the compared baseline across all bit-widths. Specifically, on 2-, 3-, 4-bit, our LTS improves the performance by 0.74%, 0.43%, and 0.45%, respectively. At the same time, our LTS is able to achieve 52%, 52%, and 53% average of weight gradient sparsity, which gives 26%, 26%, and 26.5% reduction in backward propagation FLOPs, respectively.
MobileNetV1. The results of MobileNetV1 in the third part of Tab. III demonstrate that the proposed LTS method is able to achieve significant weight gradient sparsity (ranging from 36% to 52%) for different bit-widths, indicating that the proposed method is effective in reducing the computational cost of QAT. The proposed LTS achieves an average weight gradient sparsity of 50%, 51%, 52%, 49%, and 36% for 2-, 3-, and 4-bit quantization, respectively, corresponding to a reduction in backward propagation FLOPs of 25%, 25.5%, and 26%. Notably, the proposed method also improves the accuracy by 3.55%, 1.59%, and 0.50% for 2-, 3-, and 4-bit quantization, respectively, compared to the baseline.
MobileNetV2. The fourth part of Tab. III provides the results of MobileNetV2. As can be seen, our LTS improves the performance of MobileNetV2 on 2-, 3-, and 4-bit. Specifically, LTS obtains accuracy gains 5.05%, 0.82%, and 0.60% of on 2-, 3-, 4-bit MobileNetV2, respectively. The improvements are accompanied by a significant weight gradient sparsity, i.e., 46%, 50%, and 52% for 2-, 3-, and 4-bit, respectively. Such sparsity gives 23%, 25%, and 26% reduction in backward propagation FLOPs.
In summary, the proposed LTS either leads to improved performance or comparable performance compared with the baseline even though a high weight sparsity is achieved. Generally, on ImageNet, our LTS is able to obtain 46% -54% average weight gradient sparsity and thus eliminate 23% -27% FLOPs of backward propagation. These findings suggest that the proposed LTS has the potential to be a valuable tool for efficient QAT.
3) Discussion: The performance improvements are most notable in low bit-widths and networks with efficient architecture. This can be attributed to the proposed LTS well solving the weight oscillations problem that is especially obvious for low bit-widths and efficient networks such as MobileNetV1 [18]. The weight oscillations problem refers to the quantized weight periodically oscillating between two quantization levels, which leads to unstable training and inferior performance. Although the original idea is to reduce training costs, our LTS subtly alleviates this problem by freezing weights during the training period and thus improves performance by a large margin.
D. Comparisons With Random Frozen
In this subsection, we compare the proposed LTS with the random frozen mode, i.e., randomly freeze some weights. For a fair comparison, the ratio of randomly freeze weights is the same as the LTS. Tab. IV provides the results of 2-bit ResNet-20 on the CIFAR100 dataset. As can be observed, random freeze weight leads to a 1.50% accuracy drop compared with the baseline, indicating the simple random freeze incurs performance loss. In contrast, our LTS improves the baseline performance by 0.81%, clearly demonstrating the effectiveness of our method.
E. Illustrations of Weight Gradient Sparsity
We present the curve of the weight gradient sparsity of quantized weights w.r.t. training epochs in Fig. 4. It can be seen that the proposed LTS results in a gradual sparsity. The sparsity gradually increases until it reaches an extremely high magnitude. Taking Fig. 4a as an example, for ResNet-20 on CIFAR-100, the weight gradient sparsity increases rapidly after the warmup stage finishes. After epoch 240, the sparsity is very close to 100%, which means most quantized weights are frozen. As shown in Fig. 4b, the results of ResNet-20 on CIFAR-10 are similar to CIFAR-100. As for the large-scale ImageNet dataset, we observe that the sparsity starts to raise from epoch 30 to 33 depending on the bit-width. Then, the sparsity rapidly increases to a very high level. Concretely, Fig. 4c provides the results of ResNet-18. Despite the raise point of sparsity increase being different slightly, our LTS still achieves near 80% at epoch 60 in general. In particular, it can exceed 90% at epoch 65. Fig. 4c-Fig. 4f present the sparsity of ResNet-50, MobileNetV1, and MobileNetV2. Despite the raise points of the sparsity of bit-widths being different, a high sparsity can be achieved around epoch 60 in general. For instance, in Fig. 4e, the sparsity of 2-bit MobileNetV1 starts to increase quickly around epoch 30. While for 4-bit MobileNetV2, its raise point of sparsity increase is around epoch 33. After the raise point, the sparsity raises quickly. For example, ResNet-50 of all bit-widths reaches 80% sparsity around epoch 60. Fig. 4 demonstrates our LTS is able to create a high sparsity for the latter part of the training period.
V. CONCLUSION
In this paper, we challenge the contemporary experience in QAT that all quantized weights require updating throughout the entire training process. We discover a straightforward-yetvaluable observation that a large portion of quantized weights, named as the partly scratch-off lottery ticket, converge to the optimal quantization level after a few training epochs. Thus, weights within the ticket can be frozen, and thus their gradient calculations are zeroed out in the remaining training period to avoid meaningless updating. We develop a heuristic method, dubbed lottery ticket scratcher (LTS), to effectively find the ticket. Specifically, a weight is frozen once the distance between it and its corresponding quantization level is lower than a threshold controlled by the proposed strategy. The introduced LTS is simple but extremely effective in accurately identifying the partly scratch-off lottery ticket and leading to a sparse weight gradient. Extensive experiments demonstrate that the proposed LTS generally achieves 50%-70% weight gradient sparsity and 25%-35% FLOPs reduction of the backward pass, while still exhibiting comparable or even better performance than the compared baseline.
Fig. 2 .
2An illustration of LTS. For a better illustration, the activations are represented in a unfold manner (im2col operation). (a) Frozen and unfrozen weights in the forward pass. (b) Sparsifying weight gradients in the backward pass.
Fig. 3 .
3Influence of the hyper-parameters on the top-1 accuracy of 2-bit ResNet-20 on CIFAR-100. (a) and (b) provide the ablation studies of the fixing strategy. (c) and (d) provide the ablation studies of the linear-growth strategy and sine-growth strategy, respectively. "RD. FLOPs of BP" indicates the reduction in FLOPs of backward propagation.
2 )Fig. 4 .
24ImageNet: We then conduct the experiments on the challenging large-scale ImageNet. The results of ResNet-18, ResNet-50, MobileNetV1 and MobileNetV2 are presented in Tab. III.ResNet-18. Results of the first part of Tab. III show that the proposed LTS achieve the comparable performance of or outperforms the baseline in terms of accuracy and efficiency across all bit-widths (2-, 3-, and 4-bit). The weight gradient sparsity of quantized layers w.r.t. training epochs. "B-bit" indicates the weights and activations are quantized to B-bit. "R-20", "R-18", "R-50", "Mv1", and "Mv2" indicate ResNet-20, ResNet-18, ResNet-50, MobileNetV1, and MobileNetV2, respectively.
TABLE I RESULTS
IOF RESNET-20 ON CIFAR-100/10. "AVG. OF WGS" INDICATES THE AVERAGE WEIGHT GRADIENT SPARSITY. "RD. FLOPS OF BP" INDICATES THE REDUCTION IN FLOPS OF BACKWARD PROPAGATION.Datasets
Networks
W/A
Mode
Accuracy. (%) Avg. of WGS RD. FLOPs of BP
CIFAR-100
ResNet-20
(FP: 65.24%)
2/2
Baseline
52.95
0%
0%
LTS (Ours)
53.76
66%
33%↓
3/3
Baseline
62.91
0%
0%
LTS (Ours)
63.67
68%
34%↓
4/4
Baseline
65.44
0%
0%
LTS (Ours)
66.43
70%
35%↓
CIFAR-10
ResNet-20
(FP: 91.46%)
2/2
Baseline
84.80
0%
0%
LTS (Ours)
85.33
69%
34.5%↓
3/3
Baseline
89.99
0%
0%
LTS (Ours)
90.58
69%
34.5%↓
4/4
Baseline
91.71
0%
0%
LTS (Ours)
91.70
70%
35%↓
TABLE II
ABLATION STUDY OF THE EMA PARAMETER m.
m
Accuracy. (%) Avg. of WGS RD. FLOPs of BP
0
52.43%
73%
36.5%↓
0.9
53.06%
73%
36%↓
0.99
53.76%
66%
33%↓
0.999
53.81%
60%
30%↓
0.9999
53.84%
51%
25.5%↓
TABLE III RESULTS
IIIOF RESNET-18, RESNET-50, MOBILENETV1, AND MOBILENETV2 ON IMAGENET.Datasets
Networks
W/A
Mode
Accuracy. (%) Avg. of WGS RD. FLOPs of BP
ImageNet
ResNet-18
(FP: 69.64%)
2/2
Baseline
55.04
0%
0%
LTS (Ours)
55.32
49%
24.5%↓
3/3
Baseline
66.69
0%
0%
LTS (Ours)
66.28
51%
25.5%↓
4/4
Baseline
68.80
0%
0%
LTS (Ours)
68.37
47%
23.5%↓
ResNet-50
(FP: 76.15%)
2/2
Baseline
65.21
0%
0%
LTS (Ours)
65.95
52%
26%↓
3/3
Baseline
72.43
0%
0%
LTS (Ours)
72.86
52%
26%↓
4/4
Baseline
73.74
0%
0%
LTS (Ours)
74.19
53%
26.5%↓
MobileNetV1
(FP: 73.33%)
2/2
Baseline
45.87
0%
0%
LTS (Ours)
49.42
50%
25%↓
3/3
Baseline
65.01
0%
0%
LTS (Ours)
66.60
51%
25.5%↓
4/4
Baseline
70.00
0%
0%
LTS (Ours)
70.50
52%
26%↓
MobileNetV2
(FP: 71.83%)
2/2
Baseline
40.56
0%
0%
LTS (Ours)
45.61
46%
23%↓
3/3
Baseline
63.32
0%
0%
LTS (Ours)
64.14
50%
25%↓
4/4
Baseline
68.51
0%
0%
LTS (Ours)
69.11
52%
26%↓
TABLE IV COMPARISONS
IVBETWEEN DIFFERENT MODES. RESULTS ARE OBTAINED FROM 2-BIT RESNET-20 ON CIFAR100 DATASET.Mode
Accuracy. (%) Avg. of WGS RD. FLOPs of BP
Baseline
52.95
0%
0%
Random
51.45
66%
33%↓
LTS (Ours)
53.76
66%
33%↓
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M Courbariaux, I Hubara, D Soudry, R El-Yaniv, Y Bengio, arXiv:1602.02830Binarized neural networks: Training deep neural networks with weights and activations constrained to+ 1 or-1. arXiv preprintM. Courbariaux, I. Hubara, D. Soudry, R. El-Yaniv, and Y. Ben- gio, "Binarized neural networks: Training deep neural networks with weights and activations constrained to+ 1 or-1," arXiv preprint arXiv:1602.02830, 2016.
| [
"https://github.com/zysxmu/LTS."
] |
[
"Gyromagnetic factor of rotating disks of electrically charged dust in general relativity",
"Gyromagnetic factor of rotating disks of electrically charged dust in general relativity"
] | [
"Yu-Chun Pynn \nTheoretisch-Physikalisches Institut\nFriedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany\n",
"Rodrigo Panosso Macedo \nTheoretisch-Physikalisches Institut\nFriedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany\n",
"Martin Breithaupt \nTheoretisch-Physikalisches Institut\nFriedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany\n",
"Stefan Palenta \nTheoretisch-Physikalisches Institut\nFriedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany\n",
"Reinhard Meinel \nTheoretisch-Physikalisches Institut\nFriedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany\n"
] | [
"Theoretisch-Physikalisches Institut\nFriedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany",
"Theoretisch-Physikalisches Institut\nFriedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany",
"Theoretisch-Physikalisches Institut\nFriedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany",
"Theoretisch-Physikalisches Institut\nFriedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany",
"Theoretisch-Physikalisches Institut\nFriedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany"
] | [] | We calculated the dimensionless gyromagnetic ratio ("g-factor") of self-gravitating, uniformly rotating disks of dust with a constant specific charge . These disk solutions to the Einstein-Maxwell equations depend on and a "relativity parameter" γ (0 < γ ≤ 1) up to a scaling parameter. Accordingly, the g-factor is a function g = g(γ, ). The Newtonian limit is characterized by γ 1, whereas γ → 1 leads to a black-hole limit. The g-factor, for all , approaches the values g = 1 as γ → 0 and g = 2 as γ → 1. | 10.1103/physrevd.94.104035 | [
"https://arxiv.org/pdf/1609.08604v2.pdf"
] | 119,208,107 | 1609.08604 | 5d8c903468e4dc592dd3aec4d162ac26e8319dbc |
Gyromagnetic factor of rotating disks of electrically charged dust in general relativity
Yu-Chun Pynn
Theoretisch-Physikalisches Institut
Friedrich-Schiller-Universität Jena
Max-Wien-Platz 1D-07743JenaGermany
Rodrigo Panosso Macedo
Theoretisch-Physikalisches Institut
Friedrich-Schiller-Universität Jena
Max-Wien-Platz 1D-07743JenaGermany
Martin Breithaupt
Theoretisch-Physikalisches Institut
Friedrich-Schiller-Universität Jena
Max-Wien-Platz 1D-07743JenaGermany
Stefan Palenta
Theoretisch-Physikalisches Institut
Friedrich-Schiller-Universität Jena
Max-Wien-Platz 1D-07743JenaGermany
Reinhard Meinel
Theoretisch-Physikalisches Institut
Friedrich-Schiller-Universität Jena
Max-Wien-Platz 1D-07743JenaGermany
Gyromagnetic factor of rotating disks of electrically charged dust in general relativity
(Dated: September 28, 2016)PACS numbers: 0440-b, 0440Nr
We calculated the dimensionless gyromagnetic ratio ("g-factor") of self-gravitating, uniformly rotating disks of dust with a constant specific charge . These disk solutions to the Einstein-Maxwell equations depend on and a "relativity parameter" γ (0 < γ ≤ 1) up to a scaling parameter. Accordingly, the g-factor is a function g = g(γ, ). The Newtonian limit is characterized by γ 1, whereas γ → 1 leads to a black-hole limit. The g-factor, for all , approaches the values g = 1 as γ → 0 and g = 2 as γ → 1.
I. INTRODUCTION
To any physical system with a well defined notion for the observables mass M , angular momentum J, electric charge Q and magnetic dipole moment µ B , it is common to introduce the gyromagnetic ratio (g-factor)
g = 2 M Q µ B J .(1)
Such a dimensionless quantity plays an important role in physics. Since this simple measurement is available in both classical and quantum regimes, it allows one to establish connections between several physical theories. In fact, the g-factor was originally introduced in classical electrodynamics [1]. Interestingly, for all classical convective systems (where the ratio of charge and mass density is constant, and where the mass and charge elements have equal velocities, which satisfy v c), one obtains the value g = 1. In quantum mechanics though, a different g-factor is necessary for explaining experimental results from Zeeman spectroscopy. In the non-relativistic Pauli equation, the value g = 2 for the magnetic moment associated with the electron's spin must be imposed ad hoc, while it follows automatically from the Dirac equation, i.e., when relativistic effects are included.
The particular value g = 2 is found in general relativity as well. The most notable example is probably the Kerr-Newman solution, describing a charged and rotating black hole [2]. Later, the authors of [3] generalized this property and showed that any electro-vacuum solution to Einstein-Maxwell's equation obtained by an SU (2, 1) invariance transformation [4][5][6] from a pure vacuum solution also has the value g = 2. The coincidence around the preferred values g = 2 usually motivates one to look for a deeper common root between quantum theory and general relativity (see [7] for a recent review).
More recently, this topic has been further addressed in several physical scenarios. Of particular interest were "intermediate" objects in general relativity, for which the gravitational fields were weaker than for the black-hole solution, but with * [email protected] non-negligible strong field effects. Pfister and King considered the case of a rotating charged mass shell [8]. Apart from generalising previous studies on this matter [9][10][11], they noticed that g ≈ 2 is extremely robust, in the sense that this value is obtained in a big part of the mass shell's parameter space.
However, a different result was obtained in [12]. After constructing numerical equilibrium configurations of rotating neutron stars, the authors always found the value g < 2 within the models considered. In particular, the authors observed values around g ≈ 1 in the Newtonian regime of the solution, while the highest value measured by them was g ≈ 1.9. A discrepancy to the preferred value g = 2 is also found in generalized gravity theories [13][14][15][16][17][18][19], typically due to the absence of a comparable no-hair theorem and the presence of additional fields contributing to the angular momentum of the system.
Electrically charged rotating disks provide us with an interesting scenario to enrich the discussion on this matter. In fact, without taking gravitational effects into account, relativistically rotating disks were dicussed in [20]. Even though Einsteins's equations are not considered in his framework, the author shows that the electromagnetic fields share some similarities with the ones resulting from the Kerr-Newman solution in the limit of vanishing gravitational constant G.
In this work, we consider the complete self-gravitating setup in general relativity and we show that the gyromagnetic ratio of rotating disks of electrically charged dust interpolates smoothly between the classical value g = 1 up until the black-hole value g = 2. Note that this system cannot be obtained directly from the known solution of rotating disks of dust [21,22]. In fact, by performing a Harrison transformation [4][5][6] on the rotating disk solution, one always obtains new (charged) solutions to Einstein-Maxwell's equations with g = 2. Yet, the energy-momentum tensor of those new solutions is, in general, not a physically acceptable source [23].
The construction of our solution follows the strategy from [24][25][26]. Assuming stationarity and axial symmetry, it consists of solving Einstein-Maxwell's equations for a system with an energy-momentum tensor whose contributions come from the dust particles and from the electromagnetic fields. The system is parametrized in terms of a constant specific charge ∈ [−1, 1] and a "relativity parameter" γ ∈ (0, 1]. Based on the algorithm introduced in [27], the authors of [25,26] were able to calculate the solution in terms of a high order post-Newtonian expansion in the parameter γ. In particular, [26] provided strong evidence that, analogous to the uncharged case [28], the limit γ → 1 leads to the extreme Kerr-Newman black hole.
Contrary to the post-Newtonian expansion from [25,26], we here resort to numerical methods in order to obtain a (highly) accurate solution around the black-hole limit γ → 1.
To this end, we make use of a (pseudo-)spectral method, whose algorithm is based on the one described in [29]. This paper has the following structure: section II introduces the physical model. It discusses the field equations and the parameter space of the system. Section III is devoted to the numerical method employed in this work. Section IV then presents our results, while section V summarizes this work and brings some future perspectives. We use the following conventions: boldface letters denote the abstract representation of tensors while latin indices a, b, · · · are used to express their components in a given coordinate basis {∂ a }. Moreover, latin indices in parentheses (a), (b), · · · refer to the components of a tensor in a given tetrad basis e (a) = e (a) a ∂ a . We use units in which G = c = 4π 0 = 1.
II. ROTATING DISK OF CHARGED DUST
A. Geometrical setup
The charged disk is completely described by Einstein's field equations
R ab = 8π T ab − 1 2 g ab T with T = T a a ,(2)
together with Maxwell's equations
∇ b F ab = 4πj a , dF = 0.(3)
With the assumption of stationarity and axial symmetry through the existence of Killing vectors ξ and η, we can globally express the metric in terms of the Weyl-Lewis-Papatetrou coordinates {t, ρ, ζ, φ} as
ds 2 = α 2 dρ 2 + dζ 2 + ρ 2 ν 2 [dφ − ω dt] 2 − ν 2 dt 2 ,(4)
where the unknown functions α, ν and ω depend only on the coordinates {ρ, ζ}. In this adapted coordinated system, the Killing vectors assume the simple form ξ = ∂ t and η = ∂ φ . We remark that the line element (4) has a slightly different representation than the one used in [24][25][26]. Besides, the homogenous Maxwell equation in (3) is trivially satisfied with the introduction of the vector potential A a via
F ab = ∇ a A b − ∇ b A a .
The vector potential can be put in the form
A = A t (ρ, ζ) dt + A φ (ρ, ζ) dφ(5)
due to the axial symmetry. Finally, it will be useful to introduce a tetrad basis {e (a) } as proposed in [21]
e (0) = 1 ν [∂ t + ω ∂ φ ] , e (1) = 1 α ∂ ρ , e (2) = 1 α ∂ ζ , e (3) = ν ρ ∂ φ .(6)
Since e (0) a η a = 0, this tetrad is related to the local inertial frame of zero angular momentum observers.
B. Model of matter
The energy-momentum tensor T ab is composed by a dust and an electromagnetic (EM) contribution, i.e., T ab = T dust ab + T EM ab , with
T dust ab = µu a u b and T EM ab = 1 4π F ac F b c − 1 4 g ab F cd F cd .(7)
In the expressions above, µ is associated to the baryonic mass density of the dust particles, while u a describes their 4−velocity. In the coordinate system {t, ρ, ζ, φ}, we consider the disk at the equatorial plane ζ = 0, with a range ρ ∈ [0, ρ 0 ] and therefore the baryonic mass density assumes the form
µ = σ P α δ(ζ),(8)
with δ(ζ) the Dirac delta and σ P (ρ) the proper surface mass density [29]. The disk's coordinate radius ρ 0 sets the length scale of the system. The 4−velocity is expressed in terms of the Killing vectors ξ and η as
u a = 1 ν √ 1 − V 2 δ a t + Ω δ a φ , with V = ρ ν 2 (Ω − ω).(9)
Here, Ω = dφ dt is the dust particle angular velocity. The quantity V ensures the normalisation u a u a = −1 and it can be physically interpreted as the relative velocity between the dust particle and a zero angular momentum observer 1 . In this work, we are interested in disks with rigid rotation, i.e., with Ω constant.
For the charged particles, we assume a purely convective 4−current density
j a = el u a with el = µ,(10)
i.e., the charge density el is related to the mass density via the constant specific charge ∈ [−1, 1].
The symbols ∇ and ∆ respectively, denote the usual gradient and Laplacian operators in flat space, expressed here in cylindrical coordinates {ρ, ζ, φ}.
Outside the disk range (in the electro-vacuum region), we have σ P = = 0 and the right-hand sides of equations (13)- (16) vanish. Hence, we obtain a coupled system of four elliptic equations for the four variables ν, ω, A t and A φ . Once these fields are known, one can use the remaining equations E (1)(1) and E (1) (2) to obtain α.
In order to uniquely solve the system of elliptic equations (13)-(16), we need to specify boundary conditions that describe the physical scenario we want to model. Concretely, there are four surfaces of interest (see figure 1):
• Region A: the symmetry axis (ρ = 0, ζ = 0) Equations (13)-(16) impose the following regularity conditions
ν ,ρ (0, ζ) = 0, ω ,ρ (0, ζ) = 0, (17) A t,ρ (0, ζ) = 0, A φ (0, ζ) = 0. • Region B: spacelike infinity (r = ρ 2 + ζ 2 → ∞)
We demand the physical condition of asymptotic flatness
lim r→∞ ν(ρ, ζ) = 1, lim r→∞ ω(ρ, ζ) = 0,(18)lim r→∞ A t (ρ, ζ) = 0, lim r→∞ A φ (ρ, ζ) = 0.
• Region C: equatorial plane without matter (ρ > ρ 0 , ζ = 0)
Equatorial symmetry imposes
ν ,ζ (ρ, 0) = 0, ω ,ζ (ρ, 0) = 0,(19)A t,ζ (ρ, 0) = 0, A φ,ζ (ρ, 0) = 0. • Region D: disk of charged dust (ρ ∈ [0, ρ 0 ], ζ = 0)
The surface mass density σ P introduces a discontinuity in the first derivative along ζ. Integrating equations (13)
-(16) along ζ ∈ [−z, +z] with z → 0, we obtain ν ,ζ (ρ, 0 + ) = 2πσ P α 1 + V 2 1 − V 2 , ω ,ζ (ρ, 0 + ) = −8πσ P α ν 2 ρ V 1 − V 2 (20) A φ ,ζ (ρ, 0 + ) = −2πσ P α ρ ν V √ 1 − V 2 , A t,ζ (ρ, 0 + ) + ωA φ ,ζ (ρ, 0 + ) = 2πσ P α ν 1 √ 1 − V 2 .
Since α is decoupled from the other fields, we can eliminate this quantity from the boundary conditions by combing any two of the equations in (20), which yields
ν ,ζ (ρ, 0 + ) = − ρ 4ν 2 1 + V 2 V ω ,ζ (ρ, 0 + ), A φ ,ζ (ρ, 0 + ) = ρ 2 4ν 3 √ 1 − V 2 ω ,ζ (ρ, 0 + ),(21)A t,ζ (ρ, 0 + ) + ωA φ ,ζ (ρ, 0 + ) = − ρ 4ν √ 1 − V 2 V ω ,ζ (ρ, 0 + ).
The boundary conditions (21) are complemented with a relation following from ∇ b T ab = 0. The divergence-free condition of the energy-momentum tensor is easily interpreted if one considers the 4−velocity u a and its associated projection operator h ab = g ab + u a u b . In fact, the contraction u a ∇ b T ab leads to the conservation of the baryonic mass ∇ a (µu a ) = 0, while h ab ∇ c T bc gives
f a = µa a ,(22)
with the acceleration a a = u b ∇ b u a and the Lorentz force
f a = h ab F bc j c . The ρ−component of (22) reads 2 (1+V 2 )ν ,ρ = ν 2 ρ − ρ ν ω, ρ V + 1 − V 2 [A t,ρ + ΩA φ,ρ ] .
(23) As discussed in [24][25][26], one can integrate equation (23) in cases where Ω and are constant to obtain
D := ν 1 − ρ 2 ω 2 ν 4 − (A t + ΩA φ ) = constant. (24)
The value of the constant D is obtained by inspecting the right-hand side of eq. (24) at any value ρ ∈ [0, ρ 0 ] and ζ = 0.
Concretely, at the center of the disk (ρ = 0), we obtain
D = ν c − A c t ,(25)
with ν c = ν(ρ = 0, ζ = 0) and A c t = A t (ρ = 0, ζ = 0). Note that the boundary condition in the differential form (23) provides us with a more generic set-up than the version in eq. (24). In fact, (23) could also be used to model disks with a differential rotation Ω = Ω(ρ), whereas (24) is restricted to the rigid rotation case Ω =constant.
Finally, let us remark that eq. (23) fixes the field ν at the disk up to the integration constant D. In order to solve the equations numerically, it is crucial to assert that the system has a unique solution. Therefore, at the point (ρ = 0, ζ = 0) one would have to fix the value of the integration constant. Equivalently (and more convenient from the physical point of view, see discussion in the next section), one can specify a given value for the quantity ν c .
D. Parameter space and physical quantities
The parameter space of the problem has been identified in the works [24][25][26]. In our system of units, the specific charge assumes values in the range −1 ≤ ≤ 1. Two values of this parameter are of particular relevance. The solution to the (uncharged) disk of dust [21,22,27] is clearly recovered in the case = 0. On the other hand, the case | | = 1 leads to the so-called electrically counterpoised dust configuration (see, e.g., [30]), in which the gravitational attraction is exactly counter-balanced by the electric repulsion.
Apart from the specific charge (without loss of generality, we restrict ourselves to ≥ 0), it is convenient to introduce the relativity parameter
γ = 1 − ν c ,(26)
also used in the study of the uncharged disk [21,22,27,28]. This parameter is related to the redshift Z c of a photon emitted at the centre of the disk and measured at infinity via γ = Z c /(1 + Z c ). As in the uncharged case, one intuitively expects to obtain the Newtonian limit as γ 0, while γ → 1 should lead to a black-hole transition. Indeed, first studies of the post-Newtonian expansion provide a strong indication for this behavior [26].
With such a parametrisation, the angular velocity Ω is not a free quantity that we are allowed to choose. Since Ω depends on the freely specifiable parameters {γ, }, it must be considered as an unknown variable. Therefore, the numerical scheme should be able to account for this extra unknown parameter together with the field variables (see discussion in section III). Apart from Ω, we are interested in the dependence of the following physical quantities upon the parameters {γ, }: the mass M , angular moment J, electric charge Q and magnetic moment µ B . In terms of a spherical-type representation of the coordinates ρ = r sin θ and ζ = r cos θ, these observables are computed out of the far-field behavior of the field variables via
ν ∼ 1 − M r , ω ∼ 2J r 3 , A t ∼ − Q r and A φ ∼ µ B r sin 2 θ.(27)
The gyromagnetic factor g is then directly obtained according to (1). The physical quantities derived from the far-field are connected to the disk quantities by the relation [24][25][26]
M = 2ΩJ + D Q = 2ΩJ + 1 − γ − A c t Q,(28)
with the second line obtained from (25) and (26). Since eqs. (27) and (28) are derived independently from each other, the latter provides us with a solid test for the correctness of our framework.
III. NUMERICAL METHODS
A. Adapted coordinates
In order to use spectral methods to solve the set of equations (13)-(16), we first need to map the original domain
[ρ, ζ] ∈ [0, ∞) × (−∞ × ∞) into a compact region (σ, τ ) ∈ [0, 1] 2 .
The aim is that the regions A, B, C and D are mapped into the boundaries of the numerical domain. This objective is achieved by two coordinate transformations
ρ = ρ 0 1 + ξ 2 1 − η 2 , ζ = ρ 0 ξη and σ = 2 π arctan ξ, τ = η 2 .(29)
The former introduces the elliptic coordinates (ξ, η) ∈ [0, ∞] × [0, 1], while the latter compactifies the ξ−direction.
Note that we exploit the equatorial symmetry and restrict ourselves to the region ζ ≥ 0 (η ≥ 0). Altogether, we obtain the following maps (see figure 1): In appendix A, we explicitly give the corresponding expression for the field equations (13)-(16) and the boundary conditions (17)- (19), (21) and (23) in terms of the spectral coordinates {σ, τ }.
• Region A : η = 1 ⇒ τ = 1 • Region B : ξ → ∞ ⇒ σ = 1
B. Spectral Methods
As already mentioned, we solve the field equations by means of a (pseudo-)spectral method and here we give some details on the techniques used. Let us recall that, apart from the functions ν(σ, τ ), ω(σ, τ ), A t (σ, τ ) and A φ (σ, τ ), we also must include the parameter Ω as an unknown in our scheme. As usual in any spectral algorithm, we first fix a resolution N σ and N τ and consider a vector X composed of all the variables of the system
X = ν ij ω ij A ij t A ij φ | Ω T for i = 0 · · · N σ , j = 0 · · · N τ .(30)
In the above expression, we use the notation 3 f ij = f (σ i , τ j ) to denote the function values at the Chebyschev-Lobatto grid points given by
σ i = 1 2 1 + cos π i N σ , τ j = 1 2 1 + cos π j N τ .(31)
For each function f stored in X, we can compute its corresponding Chebyshev coefficients c mn by inverting the relation 3 With f denoting either ν, ω, At or A φ Finally, we compute spectral approximations of first and second derivatives in the σ− and τ −directions at all grid points (31) which we perform by applying specific differentiation matrices to the vector X, see [31,32]. With all the discrete quantities available, we evaluate the field equations (A1)-(A3) and boundary conditions (A4)-(A7) at the grid points (31). This set of equations+boundary conditions forms a system for determining the field variables ν, ω, A t and A φ . We still need one extra condition to fix the parameter Ω uniquely, which is achieved by explicitly imposing the value of ν c = 1 − γ at the center of the disk [see eq. (A8)]. Altogether, we obtain a non-linear system of algebraic equations F ( X) of order n total = 4(N σ + 1)(N τ + 1)+1. This system is solved with a Newton-Raphson scheme. Note that within the Newton-Raphson scheme, one must solve a linear system involving the Jacobian matrixĴ = ∂ F /∂ X. As detailed in [29], this linear system is solved with the iterative BiCGStab method, with a pre-conditioner based on a finite difference representation of the algebraic system of equations.
f ij = Nσ m=0 Nτ n=0 c mn T m (2σ i − 1) T n (2τ j − 1).(32)
In order to cover all the parameter space {γ, }, we start with parameters γ ∼ 0 and = 0 and provide the solver with a initial guess X 0 constructed out of the lowest post-Newtonian approximation
ν 2 = 1 + 2U , Ω 2 = γ 1 − γ 2 ω = A t = A φ = 0.
The potential U corresponds to the exact solution for the gravitational potential of the uncharged disk of dust in the Newtonian theory of gravity
U = − 4 3π Ω 2 ρ 2 arccot ξ + 3 4 ξ − ξ 2 + 1 3 arccot ξ (1 − 3η 2 ) .
Once a solution is available, we use it as an initial-guess for a modified set of parameters {γ, }. By slowly increasing γ and we are able to cover the region (γ, ) ∈ (0, 1) × [0, 1) in the parameter space.
We end this section by mentioning that near the ultrarelativistic limit γ = 1, the functions develop strong gradients around the boundary σ = 1. In order to avoid a massive increase in the resolution N σ (which in turn signficantly slows down the speed of the solver), we implement the analytical mesh-refinement
σ = 1 − sinh[κ(1 −σ)] sinh(κ) with κ ∼ | ln(1 − γ)| (33)
introduced in [29] and successfully applied in many different contexts [33,34]. (34). Starting from γ ≈ 0.9, the analytical mesh-refinement is applied to rectify the gradient problem of the field equations around the boundaries. As shown in the inset, high numerical accuracy is assured for values of γ → 1, i.e, in the black-hole limit.
IV. RESULTS
A. Numerical Accuracy
We begin the results section with a technical discussion on the performance of the numerical solution. Note that eq. (28) provides us with a neat accuracy test to check our results. Indeed, the equation relates far field observables (out of which the gyromagnetic factor is constructed) with quantities defined on the disk. Furthermore, it includes the angular velocity Ω, which is an unknown variable within the numerical code on its own. Thus, we introduce an error measurement for the numerical solution via the relative deviation
Error = 1 − 2ΩJ M + 1 − γ − A c t Q M .(34)
The error dependence on the parameter γ is shown in fig. 2 for some representative values of the specific charge ( = 0.01, 0.5, 0.99). The numerical solutions were obtained with a resolution N σ = N τ = 50. We observe that the error is of order 10 −7 in a large range of the parameter space. From γ ≈ 0.9 onwards, the error increases significantly due to strong gradients in the fields around the boundary σ = 1. As discussed in section III B, we apply the analytical meshrefinement (33) to subdue this problem. As shown in the inset of the same figure, this technique is essential to keep the accuracy at 10 −7 without a massive increase of the numerical resolution.
The saturation of the numerical resolution at this order of magnitude is limited by the machine precision and it is compatible with the measured observables. Note that the angular momentum is given by J = lim r→∞ r 3 ω/2. When expressed in terms of the coordinates {σ, τ }, the limit can be explicitly performed and it involves third derivatives ω ,σσσ . The final accuracy is, hence, restricted to the numerical errors on the performance of third derivatives with spectral methods.
B. Gyromagnetic factor
With the numerical solution under control, we proceed and study the dependence of the gyromagnetic factor on the "relativity parameter" γ. Here again, we concentrate ourselves on representative values = 0.01, 0.5, 0.99 for the specific charge. Fig. 3 confirms that the g-factor of rotating disks of electrically charged dust interpolates smoothly between the classical value g = 1 and the black-hole limit g = 2. Moreover, we obtain a very mild dependence on . As shown in the figure, the slightly charged case = 0.01 and the near electrically counterpoised case = 0.99 do not deviate drastically from each other.
It is interesting to note that the g-factor has a (nonvanishing) finite limit in both cases → 0 and → 1. The former corresponds to the uncharged rotating disk with Q = µ B = 0, while the latter leads to the electrically counterpoised case with J = µ B = 0. That the gyromagnetic ratio has a finite limiting case, in spite of vanishing observables, can be best appreciated with the help of the post-Newtonian expressions. In [26], it is shown that the charge, the angular momentum and the magnetic moment scale as Q ∼ , J ∼ √ 1 − 2 and µ B ∼ √ 1 − 2 respectively. Therefore, the ratio µ B /(QJ) is finite in both limits. Figure 3 brings a further inset, where we zoom in on the ultra-relativistic limit γ ≈ 1. Note that the g-factor monotonically approaches the black-hole limit and the value g = 2 is achieved with slope zero, i.e., lim γ→1 g ,γ = 0
First hints for this behavior can be alluded from a highorder post-Newtonian expansion [26]. Yet, the ultrarelativistic limit is rather delicate. Hence, an ultimate conclusion regarding this issue requires the use of more powerful techniques. Indeed, we compared our numerical results with the one obtained via a post-Newtonian expansion up to the 9th order 4 . Here we show results for the case = 0.5.
The left panel of fig. 4 shows that, as expected, the post-Newtonian method reproduces the behavior at low γ, in particular the evidence of g = 1 in the Newtonian limit γ 1. However, the post-Newtonian expansion is not so accurate in the prediction of the g-factor at larger γ (see lower inset). A more detailed comparison is depicted in the right panel of fig. 4, where we display the difference between the numerical g-factor g numeric and the post-Newtonian g PN .
Note that in [26], the extrapolation for larger γ values is addressed with two techniques. One either directly calculates the series expansion in the parameter γ or one makes use of a Padé extrapolation with the obtained coefficients. Both methods are displayed in fig. 4. Confirming our previous explanation, the numerical results deliver a more accurate description as we increase γ. In the post-Newtonian best performance (post-Newtonian expansion together with Padé extrapolation), the numerical solution is more accurate from γ ≈ 0.5 onwards and the error in the black-hole limit is of the order 10 −3 .
V. DISCUSSION
In this work we calculated the gyromagnetic ratio of rotating disks of electrically charged dust. The disk is parametrized by a specific charge and a parameter γ controlling the strength of relativistic effects. The system is modelled with 4 In appendix B we present concretely the expansion of the g−factor up to the 4th order.
an energy-momentum tensor composed of dust and an electromagnetic contributions and the resulting Einstein-Maxwell equations are solved numerically with spectral methods.
This system provides us with a nice scenario to study and discuss the g−factor of electrically charged rotating objects in both Newtonian and relativistic regimes. Indeed, our highly accurate numerical results showed that the g−factor approaches the classical value g = 1 in the Newtonian limit γ 1 while the black-hole value g = 2 is obtained in the ultrarelativistic limit γ → 1. In particular, these two values are connected smoothly and monotonically through the parameter γ. The dependence on , on the other hand, is rather mild.
While in this work we focused on a rigidly rotating disk, we would like to stress that our approach imposes no restriction to this feature and one could use the same setup (field equations and boundary conditions) to obtain numerical solutions with differential rotation Ω = Ω(ρ). In a broader perspective, it would be interesting to address the question under which general conditions the relation 1 ≤ g ≤ 2 holds. Studies in these directions are planned for future work. In this first appendix we display the equations numerically implemented in terms of the spectral coordinates {σ, τ } [see eq. (29)]. For the numerical solution, it is convenient to intro-duce the re-scaled fields ω = ρ 0 ν c ω, andà t = ρ 0 ν c A t .
In the electro-vacuum region (σ, τ ) ∈ (0, 1) 2 , the field equations (13)-(16) read
Eq ν : − 1 2 ρ 0 2 (1 − τ )νF Dust ν + F EM ν = 0, with (A1) F Dust ν = (1 − 3τ )ν ,τ + 2∆ ν − 2 ν (∇ ν) 2 − (1 − τ )(ν c ) 2 cos 2 π 2 σ ν 3 (∇ ω) 2 , F EM ν = cos 2 π 2 σ ν 4 + (ν c ) 2 (1 − τ )ω 2 (∇ A ϕ ) 2 + (ν c ) 2 (1 − τ ) 2ω∇ A ϕ ∇ Ã t + ∇ Ã t 2 Eqω : − ν c ρ 0 2 (1 − τ ) 8 cos 2 π 2 σ ν 2 F Dust ω + F EM ω = 0, with F Dust ω = 1 π sin(πσ)ω ,σ + (1 − 5τ )ω ,τ − 8 ν ∇ ν∇ ω + 2∆ ω, F EM ω = ν c ∇ A ϕ ∇ Ã t +ω (∇ A ϕ ) 2 ,
Eqà t :
1 2 (1 − 3τ )νν c ωA ϕ,τ +Ã t,τ − 2ν cω ∇ ν∇ A ϕ + ν c ν∇ ω∇ A ϕ −2ν c ∇ ν∇ Ã t + ν c νω∆ A ϕ + ν c ν∆ Ã t = 0. (A2)
Eq A φ : 1 2π cos 2 π 2 σ ν 4 sin(πσ)A ϕ,σ − π(1 − τ )A ϕ,τ −2 cos 2 π 2 σ ν 3 ∇ ν∇ A ϕ + (ν c ) 2 (1 − τ )ω∇ ω∇ A ϕ +(ν c ) 2 (1 − τ )∇ ω∇ Ã t − cos 2 π 2 σ ν 4 ∆ A ϕ = 0.
The action of the operators ∆ and ∇ onto two generic function a(σ, τ ) and b(σ, τ ) is, respectively, ∇ a∇ b := 1 π 2 cos 2 π 2 σ a ,σ b ,σ + τ (1 − τ )a ,τ b ,τ , ∆ a := 1 π 2 cos 2 π 2 σ a ,σσ + τ (1 − τ )a ,τ τ .
Moreover, the equivalent to the boundary conditions (17)- (19), (21) and (23) • Region C : τ = 0, σ ∈ [0, 1) ν ,σ =ω ,σ = A φ,σ =Ã φ,σ = 0 (σ = 0)
Eq ν | τ =0 , Eqω| τ =0 , Eq A φ τ =0 , Eqà t τ =0 (else)(A6)
• Region D : σ = 0, τ ∈ (0, 1]:
4νṼ ν ,σ = −ν c 1 + V 2 ω ,σ ,
4ν 3 A φ,σ = ρ 0 ν c (1 − τ ) 1 − V 2ω ,σ ,(A7)
4νṼ Ã t,σ +ωA φ,σ = − ρ 0 1 − V 2ω ,σ ,
ν(1 + V 2 )ν ,τ = − ν 2Ṽ + 2ν c (1 − τ )ω ,τ Ṽ +ν ρ 0 A φ,τ ν cω +Ṽ ν 2 + ν c A t,τ 1 − V 2 .
Note that we introduced
V = V √ 1 − τ = Ωρ 0 − ν cω ν 2
in the expressions above.
In order to complete the system, we must also fix ν c at the center of the disk. This value is related to the relativity parameter γ via (26). Thus, at (σ = 0, τ = 1), we impose an extra condition ν(0, 1) = ν c = 1 − γ.
(A8)
FIG. 1 . 0 •
10The rotating disk of charged dust shown in the Weyl coordinates (left) and the compactified coordinates (right). The thick line denotes the infinitely thin disk with radius ρ0. The areas illustrated in the figure imply each particular part of the boundary conditions: A: ζ-axis, B: Infinity, C: Equatorial plane outside of the disk, D: On the disk surface.• Region C : η = 0 ⇒ τ = Region D : ξ = 0 ⇒ σ = 0.
FIG. 2 .
2Accuracy test of the physical quantities using
FIG. 3 .
3Gyromagnetic factor g from the Newtonian to the ultrarelativistic limit with different specific charge . The lower right window shows the result near the ultrarelativistic limit, with all curves tending monotonically to g = 2.
FIG. 4 .
4Left: Comparison of the numerical and post-Newtonian result. The upper left window depicts the detail of the g-factor near the Newtonian limit, whereas the lower right shows the result near the ultrarelativistic limit. Right: The comparison of the accuracy between the numerical and the post-Newtonian/Padé approximation.
Eqà t τ =1 , A φ = 0. (A4) • Region B : σ = 1, τ ∈ [0, 1] ν = 1, ω = A t = A φ = 0.(A5)
In fact, in terms of the tetrad basis (6), u results from the boost u = 1 √ 1 − V 2 e (0) + V e(3) .
This condition also follows from a convenient combination of the equations E (1)(1) , E (1)(2) and E (a) (a) .
We encountered a misprint in eq. (A.23) from[26]. One of the terms proportional to ψ 4 should read 9/22400 instead of 9/86400.
ACKNOWLEDGMENTSThe authors would like to thank Marcus Ansorg, Michael Kalisch, Andreas Kleinwächter and Gernot Neugebauer for valuable discussions, and Christopher J Pynn for proofreading the manuscript. This work was supported by the Deutsche Forschungsgemeinschaft (DFG-GRK 1523/2). R.P.M. was supported by CNPq under the programme "Ciência sem Fronteiras". Y.-C.P. is additionally supported by the Ministry of Education, Taiwan under the program "Government Scholarship to Study Abroad (GSSA)".Appendix B: Post-Newtonian expansion for the g-factorWith the help of studies in the post-Newtonian approximation from[24][25][26], we expand the g−factor in the form 5Here, we present the coefficients up to 4th order in γ: .As already mentioned, we clearly recover the classical result g = 1 as γ = 0. Moreover, we also obtain a finite limit in both uncharged ( = 0) and electrically counterpoised case ( = 1).
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| [] |
[
"AN ARTIFICIALLY-DAMPED FOURIER METHOD FOR DISPERSIVE EVOLUTION EQUATIONS",
"AN ARTIFICIALLY-DAMPED FOURIER METHOD FOR DISPERSIVE EVOLUTION EQUATIONS"
] | [
"Anne Liu ",
"Thomas Trogdon "
] | [] | [] | Computing solutions to partial differential equations using the fast Fourier transform can lead to unwanted oscillatory behavior. Due to the periodic nature of the discrete Fourier transform, waves that leave the computational domain on one side reappear on the other and for dispersive equations these are typically high-velocity, high-frequency waves. However, the fast Fourier transform is a very efficient numerical tool and it is important to find a way to damp these oscillations so that this transform can still be used. In this paper, we accurately model solutions to four nonlinear partial differential equations on an infinite domain by considering a finite interval and implementing two damping methods outside of that interval: one that solves the heat equation and one that simulates rapid exponential decay. Heat equation-based damping is best suited for small-amplitude, high-frequency oscillations while exponential decay is used to damp traveling waves and high-amplitude oscillations. We demonstrate significant improvements in the runtime of well-studied numerical methods when adding in the damping method. | 10.1016/j.apnum.2023.05.023 | [
"https://export.arxiv.org/pdf/2301.05789v1.pdf"
] | 255,942,091 | 2301.05789 | 9f76220fd3dd9b7d0669f178b2d1d80317dd04d1 |
AN ARTIFICIALLY-DAMPED FOURIER METHOD FOR DISPERSIVE EVOLUTION EQUATIONS
Anne Liu
Thomas Trogdon
AN ARTIFICIALLY-DAMPED FOURIER METHOD FOR DISPERSIVE EVOLUTION EQUATIONS
Computing solutions to partial differential equations using the fast Fourier transform can lead to unwanted oscillatory behavior. Due to the periodic nature of the discrete Fourier transform, waves that leave the computational domain on one side reappear on the other and for dispersive equations these are typically high-velocity, high-frequency waves. However, the fast Fourier transform is a very efficient numerical tool and it is important to find a way to damp these oscillations so that this transform can still be used. In this paper, we accurately model solutions to four nonlinear partial differential equations on an infinite domain by considering a finite interval and implementing two damping methods outside of that interval: one that solves the heat equation and one that simulates rapid exponential decay. Heat equation-based damping is best suited for small-amplitude, high-frequency oscillations while exponential decay is used to damp traveling waves and high-amplitude oscillations. We demonstrate significant improvements in the runtime of well-studied numerical methods when adding in the damping method.
Introduction
Distortions to numerical solutions of partial differential equations (PDEs) on the line caused by employing the periodic Fourier method are often mitigated by increasing the size of the computational domain. While this does mitigate the errors caused by the periodic copies generated by the discrete Fourier transform, it also increases the runtime. In this paper, we outline a method of artificial damping that allows for a smaller computational domain and significantly shorter runtimes. We demonstrate the effectiveness and generality of the method by showing how it applies to the Korteweg-de Vries (KdV) equation, the nonlinear Schrödinger (NLS) equation, a Riemann problem for the KdV equation, and a Riemann problem for the Kawahara equation 1 . The Jupyter notebooks used for computations in this paper, as well as a template for general implementation of this damping method, are provided in a GitHub repository [21].
We also present a heuristic error analysis of the method as applied to the linearized KdV equation. Artificially damping of solutions of PDEs has been discussed in other contexts and referred to by many other names. In [5], Carmigniani and Violeau model solutions to water wave problems in an open ocean by adding terms that introduce linear dissipative forces according to a sponge layer function. Unlike our method, which is intended for general dispersive evolution equations, this method is intended for full Navier-Stokes solvers. In [4], Cao, Beck, and Schultz discuss the modeling of free surface waves on an infinite domain by adding a damping term to the free surface boundary condition. They refer to the region of damping as an absorbing beach. In this work numerical results for a damping term added to Bernoulli's equation are given, but the discussion of the method's generality and accuracy for nonlinear PDEs is limited.
For our method, we use the fast Fourier transform [28], described in Section 2, and the standard fourth-order Runge-Kutta method [20] to evolve the Fourier coefficients forward in time. Fourier series are inherently periodic, so when the Fourier transform of an aperiodic function on R is computed by restricting it to an interval 2 [−L, L], the inverse transform on the whole real line becomes periodic with a period of 2L. This is illustrated in Figure 2. This forced periodicity distorts solutions and becomes increasingly problematic at larger times. Typically, a large enough interval is used to minimize interference from the periodic copies of the solution outside the interval. However, depending on the problem, a prohibitively large interval may be required. For this reason, we seek to artificially damp behavior outside the interval of interest, [ We have two primary methods of artificial damping. The first is to solve the heat equation using Strang-splitting 3 outside [−R, R]. This technique gradually damps oscillations and is best suited for small-amplitude, high-frequency oscillations. Our second method is to add exponential decay to solutions outside [−R, R]. This technique is more aggressive and is best suited for solitons, traveling waves, and high-amplitude oscillations. Both methods are discussed in more detail in Section 3.
In Section 4, we solve the KdV equation [18], which has a number of physical applications, including shallow-water waves [33], internal solitary waves in the ocean [19], and ion acoustic waves [17]. It is the first known PDE that is solvable with the inverse scattering transform [11] and has also been solved using numerical inverse scattering without any boundary approximation [30] and we use this method to compute a reference, or benchmark, solution. We consider a solution with the initial condition, q(x, 0) = 1.3e −x 2 , that produces a dispersive tail and a single soliton. We numerically compute solutions without the damping technique on intervals of varying sizes to demonstrate the pitfalls of the periodic Fourier method. We then artificially damp the solution outside the interval of interest and are able to compute an accurate solution on a smaller computational domain, thus significantly decreasing the runtime. This section also includes a discussion of how the accuracy and runtime of the method depends on the damping parameters, as well as some error bound analysis on the damping method for the linearized KdV equation (10).
In Section 5, we solve the NLS equation [23], which can be used to model Bose-Einstein condensation and plasma physics [1], deep water waves [7], and nonlinear optics [10]. There exist many numerical methods for solving the NLS equation, including numerical Fourier methods [32], a relaxation scheme [2], and numerical inverse scattering [30] which is again used to compute a reference solution. We compute solutions with the initial condition, q(x, 0) = (1+x)e ix−0.7x 2 , which produces solitons traveling in both directions and high-amplitude oscillations. Remark 1.1. We consider the integrable KdV and NLS equations because of the existence of the numerical inverse scattering transform that enables one to compute high-accuracy reference solutions that do not suffer from boundary approximation issues. This is analogous to numerically evaluating the Fourier transform solution for a linear PDE on the line. Thus, one can estimate the true error of a numerical method applied to these equations to a high degree of accuracy. This approach first appeared in [3].
In Section 6, we compute solutions to a Riemann problem for the KdV equation, motivated by a problem discussed by, for example, Grava and Klein in [12]. Solutions to this problem are dispersive shock waves, which have been observed in plasmas [15], optical fibers [25], and viscously deformable media [22]. We use an aperiodic step-like function for the initial condition, q(x, 0) = 1 1+e 10x , which has a nearly periodic derivative. Inspired by [9,27], we modify the PDE to instead solve for the derivative of the desired solution.
Similarly, in Section 7, we solve the Kawahara equation with an aperiodic step-like initial condition, q(x, 0) = 1 1+e 10x − 1, also producing a dispersive shock wave. Therefore, we utilize the 3 Lower-order or higher-order splitting methods could be considered but we use Strang-splitting due to its wide use.
approximately periodic derivative of the initial condition in the same way as with the Riemann problem for the KdV equation. The problem we choose to solve is motivated by work done by Sprenger and Hoefer in [27]. Physical applications of the Kawahara equation include magnetoacoustic waves in plasmas [16] and shallow water waves with surface tension [14]. By solving these four nonlinear dispersive PDEs, we demonstrate the effectiveness and generality of the damping technique, which we believe can be applied to a wide class of dispersive PDEs with the appropriate modifications.
The Fourier method
We begin with an elementary overview of the classical method on which we expand. We consider a kth-order partial differential equation for q(x, t) of the form
(1) q t + Lq + N (q, q x ) = 0, where L = α 1 ∂ x + α 2 ∂ 2 x + · · · + α k ∂ k
x is a linear combination of the spatial derivatives and N is a nonlinear function 4 of q and q x . We assume that if f (x) = e ikx , then Lf (x) = ω(k)f (x), where ω : R → iR is purely imaginary. We use the Fourier method [28] with approximations 5
q(x, t) ≈ F −1 J (c(t))(x) := J j=−J c j (t)e ijπ L x , c(t) = (c −J (t), c −J+1 (t), · · · , c J (t)), and c j (t) ≈ 1 2L L −L q(x, t)e − ijπ L x dx.
Here, F J is the discrete Fourier transform for functions on [−L, L]. The approximate coefficients of the initial condition, c j (0), found using the FFT, are mathematically equivalent to using the trapezoidal rule [26] and discretizing the computational domain, [−L, L], using m = 2J + 1 evenlyspaced grid points as follows:
(2) c j (0) = 1 2J + 1 m i=1 q j (x i , 0)e − ijπ L x i , where (3) x i = −L + 2L i − 1 m . M = α 1 D J + · · · + α n D k J , D J = iπ L diag(−J, −J + 1, · · · , J).
Here we use the notation that a function g : R n → R extends to a function g : R m × · · · × R m n times → R m , 4 In effect, we require this function to be a polynomial. This term could depend on higher derivatives of q but this may prevent one from using explicit time stepping, complicating matters significantly. 5 In principle, one should use a power of 2 for the number of Fourier modes to improve the efficiency of the FFT.
We will use 2J + 1 modes to simplify the presentation.
by applying g entrywise
g(x (1) , x (2) , . . . , x (n) ) = g(x (1) 1 , x (2) 1 , . . . , x (n) 1 ) . . . g(x (1) m , x (2) m , . . . , x (n) m ) .
This convention will be used throughout what follows. We then rewrite the system as
e −M t d dt e M t c(t) = F (c(t)) ,
and define a(t) = e M t c(t) so that
(4) a (t) = e M t F e −M t a(t) .
Note that our assumptions on L imply that M is diagonal with purely imaginary diagonal entries.
To compute solutions, we obtain the Fourier coefficients, c(0), of the initial condition q(x, 0) and advance them forward in time according to (4). We use the standard RK4 method [20], as outlined in Algorithm 1. The algorithm takes in a vector a (n) that approximates a(t n ) and computes the vector a (n+1) to approximate a(t n+1 ) = a(t n + t), where t is the size of one time step. Once we have obtained the approximate Fourier coefficients of the solution at the desired time, we use the inverse Fourier transform to compute the approximation of q(x, t) on the grid (x i ) m i=1 .
Algorithm 1 Fourth-order Runge-Kutta 1: function rk4(a (n) , t n , t) 2:
f 1 = e M t F (e −M tn a (n) ) 3: f 2 = e M (tn+ t 2 ) F (e −M (tn+ t 2 ) (a (n) + t 2 f 1 )) 4: f 3 = e M (tn+ t 2 ) F (e −M (tn+ t 2 ) (a (n) + t 2 f 2 )) 5: f 4 = e M (tn+ t) F (e −M (tn+ t) (a (n) + tf 3 )) 6: a (n+1) = a (n) + t 6 (f 1 + 2f 2 + 2f 3 + f 4 ) 7:
return a (n+1) 8: end function
Artificial damping
Due to the periodic nature of the Fourier method, the approximate solution will have a period of 2L. As time evolves, non-zero behavior from neighboring solutions will enter the computational domain, [−L, L], and distort the solution. Errors worsen over time as more collisions occur at the edges of the interval -a result of the dispersive nature of the PDEs. This phenomenon is discussed in more detail in Section 4. In order to mitigate this problem, we introduce damping at the edges of the interval of computation 6 : [−L, −P − ] and [P + , L]. We modify the PDE in (1) to introduce damping:
(5) q t + Lq + N (q, q x ) = k 1 (σ(x)q x ) x − k 2 (1 − γ(x))q.
The terms added to the right-hand side are effectively zero outside of the damping regions:
k 1 (σ(x)q x ) x − k 2 (1 − γ(x))q ≈ 0, |x ∓ P ± | 1.
We solve (5) using splitting methods, as described in [24]. This requires two additional components.
(1) We solve the heat equation,
q t = k 1 (σ(x)q x ) x ,
with a diffusion coefficient, k 1 σ(x), that is chosen to be non-zero where damping is desired and zero elsewhere. Using the Fourier transform, we obtain an approximate ODE system for the Fourier coefficients of q(x, t),
c (t) = k 1 D J F J σ(x)F −1 J (D J c(t))
. Here, x = (x 1 , · · · , x m ) is the vector of m evenly-spaced grid points on [−L, L] as in (3). We discretize the system using the trapezoidal method [20]. This yields the following system of equations for c (n) ≈ c(t n ),
(6) Bc (n+1) = Ac (n) ,
where B and A are matrices defined by
Bx = x − k 1 t 2 D J F J ΣF −1 N (D J x)
and
Ax = x + k 1 t 2 D J F J ΣF −1 N (D J x) ,
with Σ = diag(σ(x)). We solve (6) using the conjugate gradient algorithm [13], performing this diffusion every f 1 time steps. We review the conjugate gradient algorithm in Algorithm 2, where a = Ac (n) and is a parameter that determines the accuracy of the approximation. We do not use a preconditioner or an initial guess in Algorithm 2 and the runtime of solutions could be improved through these additions.
q t = k 2 (1 − γ(x))q.
We simulate rapid exponential decay as k 2 → ∞ by judiciously multiplying the solution values by γ(x) every f 2 time steps, where again γ(x) is chosen to enforce damping in the desired regions. This procedure is outlined in Algorithm 3, where we define q (0) ∈ R m to be a vector of q(x, 0) evaluated on x = (x 1 , · · · , x m ). Note that when calling rk4 in the algorithm, we use rk4(c (n) , 0, t), as opposed to rk4(a (n) , t, t). The PDEs we are solving are autonomous, so evolving the solution from t to t + t gives the same result as evolving it from 0 to t. The latter is simpler computationally.
Algorithm 3 Artificial damping
Input: A vector of the function values of the initial condition, q (0) .
Output: A vector of the function values at time T , q (N ) .
1: c (0) = F J (q (0) ) 2: N = T t 3: for (n = 0; n < N; n++) do 4: if n f 1 is an integer then σ(x) damping 5: a (n+ 1 2 ) = rk4(c (n) , 0, t 2 ) 6:c = e −M t 2 a (n+ 1 2 )
Strang-splitting 7:c = cg(B, A(c), ) 8:
a (n+1) = rk4(c, 0, t 2 ) 9: c (n+1) = e −M t 2 a (n+1)
10:
else No damping 11:
a (n+1) = rk4(c (n) , 0, t) 12: c (n+1) = e −M t a (n+1) 13: end if 14: if n f 2 is an integer then γ(x) damping 15: c (n+1) = F J γ(x) · F −1 J (c (n+1) ) 16:
end if 17: end for 18:
q (N ) = F J (c (n) ) 19: return q (N )
Korteweg-de Vries equation
We now apply our damping technique to the Korteweg-de Vries (KdV) equation [18], (7) q t + 6qq x + q xxx = 0, with the initial condition q(x, 0) = 1.3e −x 2 . This produces leftward traveling oscillations (i.e., a dispersive tail) and a rightward traveling soliton, as shown in Figure 1.
The Fourier method for the KdV equation.
We seek to apply the Fourier method described in Section 2. The KdV equation is a third-order nonlinear PDE, so the general functions in (1) become L(q) = q xxx and N (q, q x ) = 6qq x . Therefore, M = D 3 J and the system of ordinary differential equations in (4) becomes
a (t) = e D 3 J t F (e −D 3 J t a(t))
where
a(t) = e D 3 J t c(t), F (c) = −6F J F −1 J (c(t)) · F −1 J (D J c(t))
, and · denotes entrywise multiplication of vectors. As was briefly discussed in Section 2, the Fourier method forces the solution to be periodic on the whole real line. In Figure 2 takes 13 seconds and produces a maximum error of 0.99. We can clearly see that the Fourier method has introduced oscillations and a soliton that do not appear in the true solution. In order to achieve a more accurate solution, we increase the interval of computation to [−10000, 10000], as shown in Figure 4. This solution still falls short of a reasonable accuracy goal of of 4 · 10 −6 . Computing it 0.0001 (305 sec) · L = 10000, m = 2 16 4 · 10 −6 (869 sec) · L = 20000, m = 2 17 1 · 10 −6 (2812 sec) · L = 40000, m = 2 18 1 · 10 −6 (4260 sec) · with a better error can become prohibitively time-consuming to do on most computers. For this reason, we introduce artificial damping.
4.3.
Artificially-damped solutions of the KdV equation. As in Section 3, we solve a modified version of the KdV equation (7),
q t + 6qq x + q xxx = k 1 (σ(x)q x ) x − k 2 (1 − γ(x))q.
We damp the small-amplitude dispersive tail on the left side of the interval by solving the heat equation with Strang-splitting. We choose k 1 = 1 and
(8) σ(x) = σ(x; 1 , 2 ) = 1 − 1 2 (tanh(x − 1 ) + 1) + 1 2 (tanh(−x − 2 ) + 1)) ,
where 1 = −L + L 2 − 10 and 2 = L − 5. We perform this diffusion at every time step (f 1 = 1). The soliton on the right side of the interval requires a more aggressive damping technique. We simulate rapid exponential decay as k → ∞ by multiplying the solution values by (9) γ
(x) = 1 − σ(−x)
every 1000 time steps (f 2 = 1000). We compute an artificially-damped solution on [−600, 600] with 2 12 grid points and a time step of 0.01. This solution produces errors that are better than the undamped solution in Figure 4, its error is 4 · 10 −8 ; however, it only takes 157 seconds (≈ 2.6 minutes) to compute. To summarize, we have achieved an error that is 100 times smaller in 18% of the runtime. Further comparisons of solutions computed with and without damping are in Table 1. We see that the errors for the damped solutions saturate on the order of 10 −8 as the size of the interval increases. This is because we are using a fourth-order time-stepping method (RK4) and a time step of 0.01.
4.4.
A note on damping parameters. The choices of k 1 , k 2 , f 1 , f 2 , σ(x), and γ(x) will determine the severity of damping and its region. We always consider k 2 → ∞ to simulate rapid exponential decay, but the other parameters can be chosen judiciously. For this problem, we found that setting k 1 = 1, f 1 = 1, and f 2 = 1000, as well as choosing σ(x) to be non-zero on the left and γ(x) to be zero on the right, produced an accurate solution on a smaller interval. However, it is important to note that different choices are valid and may be advantageous depending on the problem. When using our method, experimentation with these parameters is encouraged to determine values that best suit the specific problem. In Figure 5, we demonstrate how the error is affected by the damping parameters. The first plot shows the effect of altering k 1 and f 1 , while holding f 2 constant and implementing γ(x) damping every 1000 time steps. In the second plot, we do not use σ(x) damping and instead use an even γ(x), as shown in Figure 8. We alter f 2 to demonstrate the importance of choosing an optimal value. Several observations are in order.
f 2 = 1000
(1) There exists an important trade-off between accuracy and runtime of σ(x) and γ(x) damping, where use of only γ(x) damping is faster, but including σ(x) damping improves errors. (2) There appears to be a balance between frequency and amplitude of σ(x) and γ(x) damping that maintains a shorter runtime while achieving high accuracy.
q t + q xxx = 0, in order to more concretely, but still heuristically, demonstrate the soundness of our numerical damping method. We introduce damping by considering a modified equation, where again σ(x) is a non-zero diffusion coefficient that determines the region of damping, as shown in (6).
q t +q xxx = (σ(x)q x ) x ,
We consider an initial condition that is a wave packet with fundamental frequency k centered at the origin. This wave packet travels with approximate velocity −3k 2 and at time T = P k 2 , it arrives at 9 x = −P and damping begins. It remains in the damping region for a time τ = ω k 2 and experiences damping of magnitude e −σ 0 k 2 τ = e −σ 0 ω =: ε 0 . We know that the true solution to the linearized KdV equation can be expressed as (11) q(x, t) = 1 2π ∞ −∞ e ikx+ik 3 tq 0 (k)dk whereq 0 (k) is the Fourier transform initial condition:
q 0 (k) = ∞ −∞ e −ikx q(x, 0)dx.
While many calculations below are general, the final conclusions will be valid for the choicê q 0 (k) = e −k 2 .
We are led to conjecture that the damped solution can be approximated as
(12)q(x, t) ≈ 1 2π k: |3k 2 t|<P e ikx+ik 3 tq 0 (k)dk + ε 0 2π k: |3k 2 t|>P e ikx+ik 3 tq 0 (k)dk.
Consider the periodic problem,
Q t + Q xxx = 0, Q(x + 2L, t) = Q(x, t), Q(x, 0) ≈ q 0 (x), x ∈ [−L, L].
We construct a solution by discretizing the integral in (11): 9 Here, we change the notation from P− to P for simplicity and because we are only performing damping on the left, so distinction between P− and P+ is no longer necessary.
(13) Q(x, t) = 1 2L ∞ j=−∞ e ikx+ik 3 tq 0 (k j ), k j = π L j.
q(x, t) (11)q(x, t) (12) Figure 7. In this section, we compare the undamped (13) and damped (14) periodic approximations of the true solution. The damped approximation is a periodic approximation of the damped solution on the whole line (12).
Q(x, t) (14) Q(x, t) (13) (15) (28)(23)
From this, we might expect that we can approximate a damped solution of
Q t +Q xxx = σ(x)Q xxQ (x + 2L, t) =Q(x, t), Q(x, 0) ≈ q 0 (x), x ∈ [−L, L]. using (14)Q(x, t) ≈ 1 2L ∞ j=−∞ |3k 2 j t|<P e ik j x+ik 3 j tq 0 (k j ) + ε 0 2L ∞ j=−∞ |3k 2 j t|>P e ik j x+ik 3 j tq 0 (k j ).
This provides a heuristic for us to analyze what the damping method is doing. However, we must first confirm that (14) is indeed a good approximation of what our numerical method produces. We compute solutions to the linearized KdV equation at T = 150 using (14) and our numerical method and measure their maximum difference, as shown in (2). We chooseq 0 (k) = e −k 2 and define σ(x) as in (8). We use a time step of 0.01 for the numerical solutions. From the table, we argue that (14) is a reasonable heuristic for what our numerical damping method is producing. We begin analysis of how well (14) approximates the true solution (11) by first considering the damped approximation on the whole line (12) as an approximation of the true solution (11):
|(11) − (12)| = 1 2π ∞ −∞ e ikx+ik 3 tq 0 (k)dk − 1 2π k: |3k 2 t|<P e ikx+ik 3 tq 0 (k)dk − ε 0 2π k: |3k 2 t|>P e ikx+ik 3 tq 0 (k)dk ≤ (1 − ε 0 ) 2π k: |3k 2 t|>P e ikx+ik 3 tq 0 (k) dk ≤ (1 − ε 0 ) 2π k: |3k 2 t|>P |q 0 (k)| dk.
Since ε 0 is small, (1 − ε 0 ) will not be. Therefore, we need the integral in
(15) (1 − ε 0 ) 2π k: |3k 2 t|>P |q 0 (k)| dk
to be small to have a reasonable approximation. Throughout the rest of this section, we assume that |q 0 (k)| ≤ e −k 2 , and use the rough estimate
k: |3k 2 t|>P e −k 2 dk ≤ ∞ P 3t e −k dk.
If we desire that this contribution is bounded by ε 1 , we arrive at the following condition:
(16) P 3t ≥ ln 1 ε 1
Next, we consider the difference between the damped periodic approximation (14) and the damped whole-line approximation (12). We begin by considering whether
(17) 1 2L ∞ j=−∞ |3k 2 j t|<P e ik j x+ik 3 j tq 0 (k j )
is a good approximation of 1 2π
k: |3k 2 t|<P e ikx+ik 3 tq 0 (k)dk (i.e. whether the first term in (14) is a good approximation of the first term in (12)). Since the second terms are both multiplied by ε 0 , which is small, we ignore the effect of these terms in the comparison. We are seeking a bound for
(18) 1 2π P 3t − P 3t e ikx+ik 3 tq 0 (k)dk − 1 2L M j=−M e ik j x+ik 3 j tq 0 (k j ) .
We first note that the grid of k j values may not line up perfectly such that k −M = − P 3t and k M = P 3t . Generally, k −M will be the largest negative k j value that is greater than − P 3t and k M will be the largest k j that is less than P 3t . Then, the integral that the Riemann sum is actually approximating is
(19) 1 2π k M k −M e ikx+ik 3 tq 0 (k)dk.
This differs from the integral we want to approximate by at most
(20) 2π L max |k|> P 3t − π L |q 0 (k)|,
since k −M and k M cannot differ from the integral bounds by more than the size of the grid spacing, π L . This is one component of the error in (18) and, because of the condition on P and t (16), we assume this contribution is small and ignore its effect on the overall error bound. The second component of the error is more significant and comes from how well the Riemann sum (17) approximates (19).
Theorem 4.1 (The exponentially convergent trapezoidal rule for periodic integrals [29]). An integral,
(21) I = T 0 v(θ)dθ,
can be approximated by a trapezoidal rule approximation,
I N = 2π N N k=1 v(θ k ),
with an error bound,
(22) |I − I N | ≤ 2T A e 2πAN T − 1 , for any N ≥ 1 if v(θ)
is T-periodic, analytic, and satisfies |v(θ)| ≤ A in the strip −a < Im θ < a for some a > 0.
We use (4.1) on the integral in (19) by assuming thatq 0 (k) → 0 as |k| → ∞ sufficiently quickly such that it is approximately periodic on [k −M , k M ]. Defining θ = k+k M and assuming k −M ≈ −k M puts (19) in the desired form of (21) with
v(θ) = 1 2π e i(θ−k M )x+i(θ−k M ) 3 tq 0 (θ − k M )
and T = 2k M . Since we have assumed |q 0 (k)| ≤ e −k 2 , this allows us to compute a specific upper bound for |v(θ)| = |v(E + iη)| for |η| < a:
|v(θ)| ≤ 1 2π e aR+a 3 t+a 2 e (3ηt−1)(E−k M ) 2
This leads us to impose that a = 1 3t such that 3ηt − 1 < 0 and the exponential is bounded. From this, we conclude that
e (3ηt−1)(E−k M ) 2 ≤ 1
and arrive at an upper bound on |v(θ)|:
A = 1 2π e aR+a 3 t+a 2 ,
where a = 1 3t . Now we apply (22), with T = 2k M and N = 2k M L π , to arrive at the following error bound:
(23) |(19) − (17)| 2k M π e aR+a 3 t+a 2 −2aL .
Assuming P is sufficiently large according to (16) and ε 0 is small, we ignore the other smaller contributions to the error from (20) and the terms multiplied by ε 0 in (11) and (14). We consider (23) to be an upper bound on |(11) − (14)|. If an error of is desired, then L and t must satisfy
(24) L ≥ 3 2 t ln 1 + 3 2 t ln 2 π P 3t + 1 2 R + 2 9t ,
where we have used the fact that k M ≈ P 3t . Our goal is to make a theoretical comparison between damped and undamped solutions, so we consider
(25) |(11) − (13)| = 1 2π ∞ −∞ e ikx+ik 3 tq 0 (k)dk − 1 2L ∞ j=−∞ e ik j x+ik 3 j tq 0 (k j ) ,
an upper bound on the error for the undamped periodic approximation of the solution.
Theorem 4.2 (The exponentially convergent trapezoidal rule for whole-line integrals [29]). An integral,
I = ∞ −∞ ω(k)dk,
can be approximated by a trapezoidal rule approximation,
I [n] h = h M j=−M ω(k j ),
with an error bound,
(26) I − I [n] h ≤ 2A e 2πa h − 1 ,
if ω(k) is analytic in the strip −a < Im k < a for some a > 0, ω(k) → 0 uniformly as |k| → ∞ in the strip, and for some A,
∞ −∞ |ω(E + iη)|dE ≤ A for all η ∈ (−a, a).
We use (4.2) by defining ω(k) = 1 2π e ikx+ik 3 tq 0 (k) and h = k = π L to derive an upper bound for (25). We begin by seeking an upper bound on the integral of ω(E + iη):
1 2π ∞ −∞ e i(E+iη)x e i(E+iη) 3 tq 0 (E + iη) dE ≤ 1 2π ∞ −∞ e η|x| e (η 3 +3E 2 η)tq 0 (E + iη) dE ≤ 1 2π e aR+a 3 t ∞ −∞ e 3ηtE 2q 0 (E + iη) dE.
With |q 0 (k)| ≤ e −k 2 , we have that
A = C 2π e aR+a 3 t+a 2 where (27) C = ∞ −∞ e (3ηt−1)E 2 dE = π 1 − 3ηt for η < 1 3t .
We impose that |η| < a = 1−δ 3t for 0 < δ < 1 to ensure that C is finite. Now, we obtain an error bound of the form (26), (28) |(11) − (13)| C π e aR+a 3 t+a 2 −2aL which we use to derive a similar relationship between L and t for a desired error of ,
(29) L ≥ 3 2(1 − δ) t ln 1 + 3 2(1 − δ) t ln 1 π C + 1 2(1 − δ) R + 2 9t(1 − δ)
.
Note that this differs from the damped error bound (24) in that all of the coefficients are larger by a factor of 1 1−δ and that in the second term we previously had ln 2 π P 3t
whereas now we have ln 1 π C . The value of C can be decreased by increasing δ, however this also increases the coefficients of every term in the bound. On the other hand, P is a damping parameter that, although it must be sufficiently large according to (16), can be chosen somewhat judiciously. Through our analysis of the damped and undamped errors, the benefit of artificial damping is made clearer. The introduction of the additional parameter P allows us to impose a less restrictive relationship on the size of the computational domain, L, as it relates to the final time, t. This provides a partial explanation for our observations when solving the nonlinear Korteweg-de Vries equation in (7), which is that artificiallydamped solutions achieved a given error with a smaller computational domain than undamped solutions. However, we believe that this theoretical analysis is pessimistic in its projection of the advantages of our damping method. The numerical data suggests far less restrictive conditions on L are required by the artificial damping method than the small improvement predicted by the error bound.
One possible indication of this analysis, something that might hint at a ceiling on the improvements of the proposed method, is that the lower bounded for L in both cases are, to leading order, linear in t. So, it is likely that introducing damping does not change this order, keeping it linear, but introducing damping reduces the leading constant.
Nonlinear Schrödinger equation
In this section, we compute solutions to the nonlinear Schrödinger (NLS) equation [23], (30) iq t + q xx + 2|q| 2 q = 0, with an initial condition, q(x, 0) = (1+x)e ix−0.7x 2 . The real part of the solution has small-amplitude oscillations traveling in both directions.
The Fourier method for the NLS equation.
For the nonlinear Schrödinger equation, the general functions in (1) become Lq = −iq xx and N (q, q x ) = −2i|q| 2 q. Therefore, M = −iD 2 J and the system of ordinary differential equations in (4) becomes
a (t) = e −itD 2 J F (e itD 2 J a(t)), for a(t) = e −itD 2 J c(t) where F (c) = 2iF J (F −1 J (c) · |F −1 J (c)| 2
) and, again, · denotes the entrywise multiplication of two vectors.
5.2.
Artificially-damped solutions of the NLS equation. We compute solutions to (30) at t = 150 that are accurate (errors on the order of 10 −8 ) on the interval [−99.85, 100.05], using a time step of 0.01 for all computations. We effectively solve the modified equation, (31) iq t + q xx + 2|q| 2 q = −k 2 (1 − γ(x))q, by multiplying the solution values by an even γ(x), defined as
(32) γ(x) = 1 − (σ(x) + σ(−x)),
where σ(x) is the same as in (8). We chose an even γ(x) function, shown in (8), due to the more symmetric nature of the real part of the solution. As in the previous section, we perform this multiplication every 1000 time steps. We do not include the k 1 (σ(x)q x ) x term in the modified equation because of the increase in runtime that using Strang-splitting to solve the heat equation causes. For the KdV equation, we found that using this form of damping significantly improved the accuracy of solutions on smaller intervals, but for the NLS equation, we were able to achieve the desired accuracy without this term. The real part of the solution shown in Figure 9 is computed on [−1200, 1200] with 2 13 grid points. This takes 50 seconds and has a maximum error 10 of 10 −8 . Without damping, an interval of [−2500, 2500] and 2 14 grid points is required to achieve the same order of accuracy. This solution takes 91 seconds to compute, increasing the runtime by 82% in comparison to the damped solution. Table 3 contains further results comparing the errors and runtimes of damped and undamped solutions computed on varying intervals. Again, we see that the error saturates on the order of 10 −8 due to our use of fourth-order Runge-Kutta and a time step of 0.01.
Parameters
Undamped 13 1 · 10 −5 (46 sec) 1 · 10 −8 (50 sec) L = 2500, m = 2 14 1 · 10 −8 (91 sec) 5 · 10 −8 (95 sec) Now having shown that our damping technique can improve the runtime of computing accurate solutions to the Korteweg-de Vries equation and the nonlinear Schrödinger equation, we turn our attention to a slightly more complicated problem.
The Riemann problem for the KdV equation
We consider the Riemann problem for the KdV equation (7), (33) q t + qq x + 2 q xxx = 0, with = 10 −1.5 and the initial condition, q(x, 0) = 1 1+e 10x . A similar problem is discussed in [12]. Since the Fourier method requires periodicity and q(x, 0) is not periodic, as described in [27] and [9], we use the fact that the derivative of q(x, 0) is nearly periodic and obtain the Fourier coefficients c(t) of u(x, t) = q x (x, t), which satisfies the following nonlinear PDE, 6.1. The Fourier method for the Riemann problem. Applying the Fourier method in Figure 2 to the problem, we have that the general functions in (1) are L(q) = 2 q xxx and N (q, q x ) = qq x . Therefore, M = 2 D 3 J and the system of ordinary differential equations in (4) becomes
(34) u t + (∂ −1 x u)u x + u 2 + 2 u xxx = 0.a (t) = e 2 D 3 J t F (e − 2 D 3 J t a(t))
with a(t) = e 2 D 3 J t c(t) and
F (c) = −F J F −1 J (D J · c) · H(c) + F −1 J (c) 2 ,
where H(c) computes the antiderivative of u(x, t).
6.2.
Defining H(c). We begin by expressing q(x, t) as the integral of the approximate Fourier series of u(x, t).
q(x, t) ≈
q = c 0 m (x + L) + F −1 J (S) 6: q = q − q −L + C − 7:
return q 8: end function 6.3. Solutions to the Riemann problem. Unlike in previous problems, we do not have a groundtruth solution to compare our computations to because the numerical inverse scattering transform for this problem currently unable to evaluate solutions [6] at a sufficiently large time t. But we do know what solutions should look like based on [12], for example. We begin by computing a solution 11 to (33) at t = 25 on the interval [−40, 40] with 2 12 grid points. A higher density of grid points than in previous sections is required due to the solution forming a shock wave. Without any damping, this computation takes 68 seconds and produces the plot in Figure 10. Wrap-around oscillations have caused the entire shock wave to shift upwards unnaturally. We can improve the solution quality by increasing the computational domain to [−60, 60], shown in Figure 11, or introducing artificial damping, shown in Figure 12. Following Section 3, we damp the shock wave at the edges of the domain by effectively solving u t + (∂ −1
x u)u x + u 2 + 2 u xxx = −k 2 (1 − γ(x))u and defining γ(x) as in (32). Again, we choose to set k 1 = 0 and do not use Strang-splitting to solve the heat equation because we found it possible to get a qualitatively accurate solution at a faster time without it. Both methods solve the problem, but increasing the computational domain nearly doubles the runtime to 120 seconds, while using damping maintains a runtime of 68 seconds.
Our technique relies on integrating the derivative of the solution at each time step. In Figure 13, we plot the derivative of the solution at t = 25. The derivative looks like the solution scaled up by 10 -a result of the chosen initial condition, q(x, 0) = 1 1+e 10x . We also observe that despite the high frequency and amplitude of the derivative, our integrating function, H(c), produces a qualitatively accurate shock wave solution.
The Kawahara equation
Finally, we use our damping methods to solve the Kawahara equation [16],
(35) q t + qq x + q xxx + q xxxxx = 0,
with the initial condition q(x, 0) = 1 1+e 10x − 1. Again, we have an aperiodic initial condition so we rewrite (35) to solve for u(x, t) = q x (x, t), 7.1. The Fourier method for the Kawahara equation. In the notation of Section 2, we define Lq = q xxx + q xxxxx and N (q, q x ) = qq x such that the Kawahara equation is in the form of (1). From this, we have M = D 3 J + D 5 J and the system of ordinary differential equations in (4) becomes an approximate finite-dimensional ODE system, [27] with the initial condition, q(x, 0) = 0 x < 0, −1 x ≥ 0, and the same method described in Section 6. Motivated by their work, we compute a reference solution at t = 24 on [−10 5 , 10 5 ]. We use a time step of 0.0005 and m = 2 22 grid points; we approximate their discontinuous initial condition with q(x, 0) = 1 1+e 10x − 1. This computation took over 30 hours 12 . We now compute solutions with and without damping on [−1000, 1000] with 2 15 grid points. We introduce damping by solving a modified version of (36), (37) u t + (∂ −1 x u)u x + u 2 + u xxx + u xxxxx = −k 2 (1 − γ(x))u, where γ(x) is still defined to be an even function as in (32). We multiply the solution values by γ(x) every 100 time steps (f 2 = 100). More aggressive damping is required for this problem due to the fifth-order term that causes higher velocity dispersion. We compare the damped and undamped solutions in Figure 14. The plots on the right show a smaller subset of the interval to highlight the effect of damping on unwanted oscillations. The damped solution took 444 seconds to compute and has a maximum error of 0.001 from the reference solution. The undamped solution had a shorter runtime of 492 seconds, but produced a maximum error of 0.03.
(36) u t + (∂ −1 x u)u x + u 2 + u xxx + u xxxxx = 0.a (t) = e (D 3 J +D 5 J )t F (e −(D 3 J +D 5 J )t a(t)),
Summary of results
We have developed a general method for modeling nonlinear dispersive PDEs on the line with a finite computational domain. Our technique utilizes the Fourier method which is a very efficient numerical tool; however, it forces periodicity onto the solution. In order to accurately model the dispersion of solutions, we modify the PDE to include damping terms, k 1 (σ(x)q x ) x and k 2 (1 − γ(x))q). The former solves the heat equation in the desired damping region, specified by the choice of σ(x), to diffuse the solution near the edges of the interval. The latter causes rapid exponential decay in the region determined by γ(x). We found that σ(x) damping works particularly well for dispersive tails, while γ(x) damping is better suited to damping solitons. Different choices of damping parameters will vary the accuracy and computation of approximate solutions.
Appendix A. Further applications
In this section, we include plots of solutions to different problems computed using our damping method. We begin by solving the KdV equation (7) with an initial condition that produces a twosoliton solution, shown in (15). In (16), we solve the Riemann problem for the KdV equation (33) with an initial condition motivated by an example in [12]. Qualitative comparison of the solution to that of Grava and Klein confirms its accuracy. We then solve the Riemann problem for the KdV equation with a soliton on top of a shelf for the initial condition in (17). In (18), we plot the real part of the solution to the Eckhaus equation [8]. The Eckhaus equation models wave propagation in dispersive media and falls within the nonlinear Schrödinger class of equations. (a) Initial condition, q(x, 0) = − 1 cosh 2 (x) (as in [12]).
−R, R]. Damping the solution on [−L, −P − ] and [P + , L] creates a buffer between periodic solutions and stops neighboring solutions from interfering with the desired solution on [−R, R].
2
A note on intervals: [−L, L] is the interval on which the solution is computed, [−R, R] is the interval on which we will compare the solution to the true solution, and [−L, −P−] and [P+, L] are the damping regions at the edges of the computational domain. The sizes of the intervals are related as follows: −L < −P− < −R and R < P+ < L.
Algorithm 2
2Conjugate gradient algorithm 1: function cg(B, a, ) 2: c (n+1) = 0; r = a; p = r; second component of damping comes from solving
Figure 1 .
1Solution to the KdV equation with initial condition q(x, 0) = 1.3e −x 2 at t = 150.
, we plot the inverse transform of the Fourier coefficients of a solution to the KdV equation computed on [−100, 100]. Outside of the computational domain, the solution has become periodic. At the bounds of the
Figure 2 .
2Solution to the KdV equation computed on [−100, 100] with initial condition q(x, 0) = 1.3e −x 2 at time t = 5. interval, the solution collides with the neighboring periodic solutions. The dispersive tail of the solution on [100, 300] flows into the interval from the right and introduces oscillations that are not present in the true solution. At a later time, the effective soliton in the solution on [−300, −100] will enter the interval from the left and cause large errors.4.2.Undamped solutions of the KdV equation. Without artificial damping, the primary way to mitigate this problem is to increase the size of the interval of computation. This improves the accuracy of the solution by creating a buffer between the periodic instances of the solution. However, this means that observing the long-term behavior of a solution requires a sizable interval and thus a large runtime. In this section, we compute an accurate solution to (7) with q(x, 0) = 1.3e −x 2 at t = 150.7 We compare it to the true solution 8 on [−99.85, 100.05], seeking a maximum error on the order of 10 −8 . To demonstrate the effect of the periodic Fourier method on the accuracy of the solution, we use an interval of computation,[−200, 200], as shown inFigure 3. This computation
Figure 3 .
3Undamped solution to the KdV equation computed on [−200, 200] with 2 10 grid points.
Figure 4 .
4Undamped solution to the KdV equation computed on [−2500, 2500] with 2 14 grid points.took 869 seconds (≈ 14.5 minutes). Computing a solution on a larger interval, at a later time, or Heat eq. damping coefficient, σ(x).Exponential decay damping factor,
Figure 5 .
5Errors for solutions to (7) at t = 150, computed on [−200, 200] with m = 2 10 grid points and a time step of 0.01. Error is defined as the absolute difference between the true solution and the artificially-damped solution at each grid point.
Figure 6 .
6General diffusion coefficent σ(x).
Table 2 .
2Solutions are evaluated on an interval [−L, L] with m grid points. Error is calculated as the maximum absolute difference between the numerical method and (14) on the computational domain, excluding the damping region.
Figure 8 .
8Even damping factor, γ(x).
) L = 600, m = 2 12 0.002 (26 sec) 3 · 10 −5 (27 sec) L = 1200, m = 2
Figure 9 .
9Damped real part of the solution to the NLS equation computed on [−1200, 1200].
ds + C − ,We normalize the antiderivative of u(x, t) at −L by defining the integration constant as C − = q(−L, 0). Pulling out the n = 0 term from the sum and integrating it separately yields q(x, t) ≈ C − + c 0 (t)(x + L)
=
c 0 (t)(x + L) + C − − N n=−N,n =0
L
inπ c n (t)e −inπ + N n=−N,n =0
L
inπ c n (t)e inπ L x = c 0 (t)(x + L) n (t)e −inπ n = 0. We can then define the function H(c) through Algorithm 4, which takes in a vector of the Fourier coefficients of u(x, t) and computes a vector of values of q(x, t) along the interval [−L, L] with m evenly spaced grid points. In this algorithm we use the notation that for a diagonal matrix D, D † is defined by (D † ) ij = D −1 ij if i = j, D ii = 0 and (D † ) ij = 0 otherwise.
Figure 10 .
10Undamped solution of the Riemann problem problem computed on [−40, 40].
Figure 11 .
11Undamped solution of the Riemann problem computed on [−60, 60].
Figure 12 .
12Damped solution of the Riemann problem computed on [−40, 40].
Figure 13 .Figure 14 .
1314Derivative of the damped solution of the Riemann problem computed on [−40, 40]. Solutions to the Kawahara equation at t = 24 computed on [−1000, 1000].
F
(c) = −F J F −1 J (D J · c) · H(c) + F −1 J (c) 2 . H(c) isstill defined through Algorithm 4. 7.2. Solutions to the Kawahara equation. Sprenger and Hoefer solve the Kawahara equation (35) in
Initial condition, q(x, 0) = 6e −x 2 .(b) Solution at t = 5.
Figure 15 .
15Solution to the KdV equation(7). Parameters: L = 200, m = 2 11 , t = 0.001, σ(x) = (8), γ(x) = (9), k1 = 1, f1 = 1, f2 = 1000. Runtime: 28 seconds.
(b) Solution at t = 5.
Figure 16 .
16Solution to the Riemann problem for the KdV equation (33). Parameters: L = 40, m = 2 12 , t = 0.001, σ(x) = (8), γ(x) = (32), k1 = 0, f2 = 1000. Runtime: 15 seconds.
Solution at t = 5.
Figure 17 .
17Solution to the Riemann problem for the KdV equation (33). Parameters: L = 40, m = 2 12 , t = 0.001, σ(x) = (8), γ(x) = (32), k1 = 0, f2 = 1000. Runtime: 18 seconds.(a) Initial condition, q(x, 0) = e −x 2 . (b) Solution at t = 10.
Figure 18 .
18Solution to the Eckhaus equation [8]: iqt + qxx + 2|q| 2 x q + |q| 4 q = 0. Parameters: L = 200, m = 2 10 , t = 0.01, σ(x) = (8), γ(x) = (32), k1 = 0, f2 = 1000. Runtime: 5 seconds.
Table 1 .
1Solutions evaluated on an interval [−L, L] with m grid points.
Different choices of parameters result in different oscillatory frequencies of the errors across the computational domain. It may be advantageous to choose parameters that display less frequent oscillations. (4) In the left plot, errors spike at the left side of the computational domain. This could be due to the high-frequency dispersive tail that is difficult to approximate numerically, or the rightward traveling soliton that has wrapped around to the left side of the domain.Note that the blue curve corresponding to the most aggressive σ(x) damping on the plot
minimizes this spike in the error.
4.5. On the linearized Korteweg-de Vries equation. We consider the linearized Korteweg-de
Vries equation,
Table 3 .
3Solutions evaluated on an interval [−L, L] with m grid points.
For all computations in this paper, P− = P+, but we leave this distinction for generality.
All solutions in this section are computed with a time step of 0.01.8 The solution used as the ground truth in this section comes from[31].
The true solution in this section comes from[30].
All solutions in this section are computed with a time step of 0.01.
This computation was done on an Intel® Core™ i7-6700 Processor with 32 GB RAM, a 3.40GHz CPU, and the CentOS operating system.
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| [
"https://github.com/anneliu7/"
] |
[
"Q-balls in K-field theory",
"Q-balls in K-field theory"
] | [
"Aníbal Faúndez \nInstituto de Física\nPontificia Universidad Católica de Valparaíso\nAv. Brasil 2950ValparaísoChile\n",
"Radouane Gannouji \nInstituto de Física\nPontificia Universidad Católica de Valparaíso\nAv. Brasil 2950ValparaísoChile\n"
] | [
"Instituto de Física\nPontificia Universidad Católica de Valparaíso\nAv. Brasil 2950ValparaísoChile",
"Instituto de Física\nPontificia Universidad Católica de Valparaíso\nAv. Brasil 2950ValparaísoChile"
] | [] | We study the existence and stability of Q-balls in noncanonical scalar field theories, K(|Φ| 2 , X) where Φ is the complex scalar field and X is the kinetic term. We extend the Vakhitov-Kolokolov stability criterion to K-field theories. We derive the condition for the perturbations to have a wellposed Cauchy problem. We find that K,X > 0 and K,X +XK,XX > 0 are necessary but not sufficient conditions. The perturbations define a strongly hyperbolic system if (K,X − 2φ ′2 K,XX )(K,X + 2ω 2 φ 2 K,XX ) > 0. For all modifications studied, we found that perturbations propagate at a speed different from light. Generically, the noncanonical scalar field can lower the charge and energy of the Q-ball and therefore improves its stability. | 10.1103/physrevd.107.104058 | [
"https://export.arxiv.org/pdf/2301.05890v2.pdf"
] | 255,942,427 | 2301.05890 | 47ef6714d889bf5a96aae13fba37bea8fbb74c9b |
Q-balls in K-field theory
30 May 2023
Aníbal Faúndez
Instituto de Física
Pontificia Universidad Católica de Valparaíso
Av. Brasil 2950ValparaísoChile
Radouane Gannouji
Instituto de Física
Pontificia Universidad Católica de Valparaíso
Av. Brasil 2950ValparaísoChile
Q-balls in K-field theory
30 May 2023(Dated: May 31, 2023)
We study the existence and stability of Q-balls in noncanonical scalar field theories, K(|Φ| 2 , X) where Φ is the complex scalar field and X is the kinetic term. We extend the Vakhitov-Kolokolov stability criterion to K-field theories. We derive the condition for the perturbations to have a wellposed Cauchy problem. We find that K,X > 0 and K,X +XK,XX > 0 are necessary but not sufficient conditions. The perturbations define a strongly hyperbolic system if (K,X − 2φ ′2 K,XX )(K,X + 2ω 2 φ 2 K,XX ) > 0. For all modifications studied, we found that perturbations propagate at a speed different from light. Generically, the noncanonical scalar field can lower the charge and energy of the Q-ball and therefore improves its stability.
I. INTRODUCTION
Q-balls are pseudolike particles that could be defined as lumps of a singularity-free scalar field with finite energy. They have been originally discovered in [1] and independently rediscovered in [2]. Contrary to solitons, they do not have a topological charge but a Noether charge based originally on the U (1) global symmetry, and therefore they belong to the class of nontopological solitons. The scalar field is captured in some region of space because of nonlinear self-interaction, therefore forming a pseudolike particle carrying charge and energy.
Q-balls can be produced via many mechanisms, which makes them very interesting in particular in cosmology. Indeed, they could be produced from inflationary models, such as natural inflation [3,4], where if a complex scalar field with a global symmetry is spontaneously broken, we end up with the inflaton as the goldstone boson and a naturally flat potential due to the shift symmetry. Also in supersymmetric extensions of the standard model (see e.g. [5]), Q-balls emerge naturally where the global charge could be assumed by the baryon or the lepton number. For example, the Affleck-Dine mechanism [6,7] uses the supersymmetric flat directions to generate baryogenesis. In this context, some of these flat directions (scalar field) can be parametrized as a complex field, which is in general a condensate of squarks, sleptons and Higgs field. This condensate can be unstable and form Q-balls [8].
Of course, the most interesting property of Q-balls is their stability, because they could then be considered dark matter candidates [9,10]. For that reason, it will be our main focus in this paper along with some interesting properties related to their existence. The analysis of the classical stability was studied in [11,12] where they found that considering a Q-ball of frequency ω and charge Q, stability is similar to the condition dQ/dω < 0. It was shown in [13] that the stability of gauged Q-balls is not related to this condition. It would be interesting to see * [email protected] † [email protected] the extension of this criteria to global charge Q-balls but in modified gravity theories. We will study three types of stability conditions that appear in the literature [15], namely, classical stability as we have previously mentioned, absolute stability, and stability against fission [12].
In most of the papers, a canonical scalar field is assumed, which appears naturally at low energies of various theories. But studying Q-balls in the early universe might modify this simple picture. Indeed, e.g. higher dimensions naturally produce scalar fields with nonlinear kinetic terms such as D3-brane [16] or in the context of braneworld gravity [17]. Also in string theory, a rolling tachyon has a Dirac-Born-Infeld (DBI) type of action [18]. It is therefore natural to look to noncanonical scalar fields. Q-balls in the DBI type of kinetic term was studied in [19] along with its stability using catastrophe theory [20]. In this context, we will study Q-balls in the context of a complex K-field also known as K-inflation [21] or K-essence [22].
The plan of the paper is as follows. We introduce the model before discussing the stability conditions encountered in the literature. In the next section, we analyze the range of existence of the Q-balls and define the energy conditions for these solutions. Finally, we will study numerically the properties of the Q-balls before studying the equation of perturbation. We analyze the strong hyperbolicity of these equations along with the stability of the Q-ball before conclusions.
II. Q-BALLS
Let us consider the density Lagrangian
L = K(|Φ| 2 , X) (2.1)
where K is a generic function of a complex scalar field Φ and the kinetic term X = −∂ µ Φ∂ µ Φ * . The equation of motion is
∇ µ (K ,X ∂ µ Φ) + ΦK ,|Φ| 2 = 0 (2.2)
where we have used the notation K ,A ≡ ∂K/∂A.
The model admits a global U(1) symmetry with which the associated Noether current is
j µ = iK ,X Φ * ∂ µ Φ − Φ∂ µ Φ * (2.3)
This current is conserved ∂ µ j µ on-shell. The corresponding conserved scalar charge (or total particle number) is
Q = d 3 xj 0 = i d 3 xK ,X (ΦΦ * −ΦΦ * ) (2.4)
To obtain the energy, we define the canonical conjugate momenta to the variables Φ and Φ * ,
π Φ = ∂L ∂Φ = K ,XΦ * (2.5) π Φ * = ∂L ∂Φ * = K ,XΦ (2.6)
so the Hamiltonian density is
H = π ΦΦ + π Φ * Φ * − L = 2|Φ| 2 K ,X − K (2.7)
The energy of the system is then
E = d 3 x 2|Φ| 2 K ,X − K (2.8)
We are looking for solutions that minimize the energy for a given charge Q. For that, we define the functional
E ω = E + ω Q − i d 3 xK ,X (ΦΦ * −ΦΦ * ) (2.9)
where ω is a Lagrange multiplier which enforces the given charge Q. We have
E ω = ωQ + d 3 x K X 2|Φ| 2 − iω(ΦΦ * −ΦΦ * ) − K = ωQ + d 3 x K X |Φ − iωΦ| 2 + K X (|Φ| 2 − ω 2 |Φ| 2 ) − K (2.10)
In the case of a canonical scalar field, K = X − V (|Φ| 2 ), we have
E ω = ωQ + d 3 x |Φ − iωΦ| 2 − ω 2 |Φ| 2 + | ∇Φ| 2 + V (|Φ| 2 ) (2.11) where we used that X = −∂ µ Φ∂ µ Φ * = |Φ| 2 − | ∇Φ| 2 .
We can therefore conclude that for a given charge Q, the energy is minimized whenΦ − iωΦ = 0 which means for Φ(t, x) = φ( x)e iωt [12]. This simple argument for the canonical scalar field cannot be easily generalized to the K-field. But we observe that in the general case, if
Φ(t, x) = φ( x)e iωt , E ω = ωQ − d 3 x K (2.12)
which implies that the extrema of the energy (for fixed charge) coincide with the extrema of the action. Therefore solutions of the following type Φ(t, x) = φ( x)e iωt extremize the energy. Even if we do not know of the existence of other solutions that could also extremize the energy functional, we will assume in the future for this paper this time-dependent phase of the solution. For a given model, the only parameter that characterizes the energy E and the charge Q is the parameter ω. Therefore we can consider that energy and charge are functions of ω, thus differentiating the energy, and we get
dE dω = d 3 x 2ωφ 2 K ,X + 4ω 3 φ 4 K ,XX (2.13)
Performing the same differentiation of the charge Q, we found dE dω = ω dQ dω (2.14)
which extends to K-field results from [11]. When dQ dω = 0 also dE dω = 0 which corresponds to the existence of extremum of the charge and the energy at the same time. They will correspond to the cusps in the diagram E(Q).
When dQ dω = 0, we obtain dE dQ = ω (2.15)
which corresponds to the generic relation found for a U (1) Q-ball.
III. STABILITY
Usually, three different stability criteria are discussed in the literature. The first condition considers that a given Q-ball should not decay into smaller Q-balls, sometimes referred to as stability against fission [12]. In that case, the stability translates into
E(Q 1 + Q 2 ) < E(Q 1 ) + E(Q 2 ) (3.1)
and if taking derivatives with respect to both charges (Q 1 , Q 2 ), we obtain the equivalent condition d 2 E dQ 2 < 0; by using Eq.(2.15) it reduces to dQ dω < 0. Notice the similarity with the more generic Vakhitov-Kolokolov stability criterion [23] (or spectral stability). Of course, because of Eq.(2.14), we could equivalently consider dE dω < 0. The second stability criterion considers decay into free particles of mass M = V ′′ (0) 2 . To avoid the decay of a Q-ball into Q free particles with the rest masses M , we need to consider E(Q) < M Q.
Finally, the last stability considers the time evolution of small perturbations, the so-called classical stability that we will analyze later.
Notice that from the catastrophe theory, a simple criteria of stability has been proved [24]. Indeed, considering the diagram E(Q), the lowest branch corresponds to the stable soliton while the upper branch is unstable. This condition will be found to be equivalent to the linear stability.
IV. EXISTENCE
In this section, we briefly summarize the conditions of the existence of Q-balls. These conditions are obtained by constraining the shape of the potential.
Considering a flat spherically symmetric spacetime, and Φ = φ(r)e iωt ; Eq.(2.2) becomes
K ,X φ ′′ (r) + 2 r φ ′ (r) + ω 2 φ(r) + φ ′ (r)X ′ (r)K ,XX + φ ′ (r) 2 K ,φX + 1 2 K ,φ = 0 (4.1) with X = ω 2 φ(r) 2 − φ ′ (r) 2 .
Let us first consider the canonical case, namely K = X − V (φ). The equation of motion reduces to
φ ′′ (r) + 2 r φ ′ (r) + ω 2 φ(r) − V ′ (φ) 2 = 0 (4.2)
which can be written as
φ ′′ (r) + 2 r φ ′ (r) − V ′ eff (φ) = 0 (4.3) with V eff (φ) = (V (φ)− ω 2 φ 2 )/2.
We see that the ω 2 term acts as a tachyonic contribution to the mass of the field, which will produce solitonic solutions otherwise absent for ω = 0. Considering only solutions with finite energy, the energy functional (2.
8) E = d 3 x(φ ′ (r) 2 + ω 2 φ 2 + V (φ)) implies that (φ, φ ′ ) → 0 for r → ∞ and V (0) = 0 (we assumed V (φ) > 0).
It is easier to use the analogy with a particle in Newtonian mechanics, namely replacing φ → x and r → t which givesẍ
+ 2 tẋ + W ′ eff (x) = 0, where W eff (x) = −V eff (x)
. Looking for a trajectory φ(r) or equivalently x(t), we need to impose x(∞) = 0 to obtain a finite energy solution. Therefore, the problem reduces to classifying the different trajectories of the equivalent particle giving finite energy. It is easy to show [2] that we need to im-
pose W ′′ eff (0) < 0 and W eff (φ) > 0 around φ(r = 0). These conditions translate into V ′′ (0) > 2ω 2 as well as min V (φ) φ 2
≤ ω 2 . Thus, nonrenormalizable potentials have to be considered and the simplest could be V (φ) = m 2 φ 2 − bφ 4 + λφ 6 . The previous constraints reduce to
0 < m 2 − b 2 4λ < ω 2 ≤ m 2 (4.4)
The positivity of m 2 − b 2 /4λ is imposed by demanding that V (0) is a global minimum. In this paper, we will normalize [25] the parameters such as λ = 1 and b = 2 which implies m > 1. Therefore we will consider m 2 = 1.1 which implies 0.32 < ω ≤ 1.05. The Q-ball will exist only in this range of frequencies. It is important to mention that this range will change for K-fields. For example, in a model where K = X + αX 2 − V (φ), we have around r = 0, and using the condition φ ′ (r = 0) = 0,
φ ′′ (r) + W ′ ef f (φ) ≃ 0 with W ′ ef f = ω 2 φ − m 2 − 2bφ 2 + 3λφ 4 1 + 2αω 2 φ 2 φ (4.5)
Therefore the condition W ef f > 0 for some range of the scalar field, implies a different value for the minimum of ω. For our parameters, we found that with good accuracy, ω min ≃ (1 + α/30)/ √ 10 while ω max remains unchanged.
Another important condition for the existence of the Q-ball is the nature of the differential equation. We have an equation
K ,X − 2φ ′2 K ,XX φ ′′ + F (φ, φ ′ ) = 0 (4.6)
To avoid singular points, we need to impose K ,X − 2φ ′2 K ,XX = 0. Therefore, for any model, smoothly connected to the canonical case,
K ,X − 2φ ′2 K ,XX = 1, we should impose K ,X − 2φ ′2 K ,XX > 0. Considering the model K = X + αX 2 − V (φ 2 ), we have 1 + 2αω 2 φ 2 − 6αφ ′2 > 0.
Around the origin, we have φ ′ = 0, which implies the condition 1 + 2αω 2 φ 2 0 > 0 and therefore large negative values of α will not be allowed.
V. ENERGY CONDITIONS
For these type of models, the fluid interpretation is not suitable because the kinetic term does not have a definite sign. But, it is mostly positive in the interior of the Q-ball and becomes negative near the surface of the Q-ball. Therefore, deep inside the Q-ball, we can use the hydrodynamical interpretation of the scalar field, by defining the energy-momentum tensor
T µν = Kg µν + K ,X (∂ µ Φ∂ ν Φ * + ∂ µ Φ * ∂ ν Φ) (5.1)
from which we define the energy density ρ = 2|Φ| 2 K ,X − K = 2ω 2 φ(r) 2 K ,X − K, the radial pressure P r = 2φ ′ (r) 2 K ,X + K and finally the tangential pressure P t = K. These quantities can be converted into the pressure P = (P r +2P t )/3 and the shear force S = P r −P t . Notice that the energy defined from E = d 3 xT 00 corresponds to Eq.(2.8).
The hydrodynamical approach helps to obtain easily the energy conditions such as the strong energy condition (SEC)
K ,X ≥ 0 , K + (ω 2 φ 2 + φ ′2 )K ,X ≥ 0 (5.2)
the dominant energy condition (DEC)
K ,X ≥ 0 , (ω 2 φ 2 − φ ′2 )K ,X − K ≥ 0 (5.3)
the weak energy condition (WEC)
K ,X ≥ 0 , 2ω 2 φ 2 K ,X − K ≥ 0 (5.4)
and the null energy condition (NEC)
K ,X ≥ 0 (5.5)
We notice that K ,X ≥ 0 is common to all energy conditions.
VI. NUMERICAL ANALYSIS
As we have mentioned, Q-balls are finite energy objects and therefore with a finite space extension, which imposes the asymptotic condition φ(∞) = 0. Therefore we have used a shooting method for each value of the frequency ω with mixed boundary conditions φ ′ (0) = 0 and φ(∞) = 0. In practice, we have integrated the system from r = 10 −30 to some value, r max , and demanded that the solution remains unchanged if we increase r max . In Fig. 1, we have considered the standard model K(X) = X − V (|Φ| 2 ) with the potential defined in Sec. IV. For lower frequencies, or the thin wall limit, the scalar field is constant and at some radius (often considered as the Q-ball radius) the scalar field drops rapidly to zero, while for larger values of ω, also known as the thick wall limit, the scalar field is more shallow. The latter will be unstable. In the same graphics, we have represented the energy and the charge. The energy and charge seem to diverge for the frequencies ω min and ω max . Also E(ω) and Q(ω) reach their minimum for the same frequency, defining therefore a cusp in the energy vs charge graphics. We show also the stability conditions of the Q-balls. The stability criteria against decay is stronger than the fission stability condition. In the (Q, E) plot, it is easy to determine the stable Q-ball. Indeed, for every given charge Q, two Q-balls exist, and the one with the smallest energy corresponds to the solution stable under fission. We will see later, that it corresponds also to the stable solution under linear perturbations. Q-balls have also excited states that correspond to solutions with nodes but with the same limit at infinity, namely φ(∞) = 0. In Fig. 2, we show the first and second excited modes for a given frequency ω. To fulfill the boundary conditions, for excited states, the initial conditions must be extremely fine-tuned. The excited states have as expected larger energy but also charge. We found that the frequency corresponding to dE/dω = 0 becomes larger with the number of nodes. For example, for the fundamental mode, we have a minimal energy for ω = 0.972, while ω = 1.015 for the first excited mode and ω = 1.025 for the second excited mode.
All these solutions are easily generalized to K-field the-ories. We will consider the simplest model where the action is modified by a single parameter, K = X + αX 2 − V (|Φ|) where α is the new parameter of the model 1 . Generically, we found that the structure of the solutions will not change. Q-balls exist for a certain range of frequency which depends on the parameter α. We see from energy for large positive values of the parameter α, because the radius decreases. Notice that the critical value, (E ′ (ω) = 0), of the energy and charge is also lowered for larger values of α. Therefore, for a given frequency, the modified model with α > 0 produces Q-balls with lower charge and energy. The modification by the K-field allows one to build Q-balls with small charge and energy or on the contrary with larger energy and charge. Finally, we found that for all values of the parameter α, in the limit of ω → ω max , or the thick-wall limit, we have the scaling solution E = ωQ γ with γ = 1 ± 10 −4 . This expression generalizes results found in [15]. In Fig. 4, we show the energy versus the frequency for different values of α but with the information on the violation of the energy conditions. We see that NEC is never violated. This condition corresponds to 1 + 2α(ω 2 φ 2 − φ ′2 ) > 0. It could be violated for very negative values of α, but the construction of Q-balls for α < −0.5 becomes very challenging and often impossible. In general, the larger and positive α, the lower the probability to violate an energy condition, except for the SEC which is violated for any α.
VII. PERTURBATIONS
To study the mechanical stability, we decompose our field as Φ(t, r, θ, ϕ) = φ(r)e iωt + ℓ,m δΦ lm (t, r)e iωt Y m ℓ (θ, ϕ)
where φ(r) is the background scalar field studied in the previous sections, δΦ ℓm is the scalar field perturbation, e iωt in the second term is included for convenience and Y m ℓ are spherical harmonics. Because of the symmetries of the Q-balls, the perturbations will be independent of the azimuthal number m and therefore the spherical harmonics reduce to Legendre polynomials. We will fix m = 0. Notice that the different modes, ℓ, do not couple and therefore we will omit this index. At second order in perturbations, and after integrating over the angle variables, the action reduces to
S = dtdr r 2 K ,XΨ 2 1 − r 2 (K ,X − 2φ ′2 K ,XX )Ψ ′2 1 + r 2 (K ,X + 2ω 2 φ 2 K ,XX )Ψ 2 2 − r 2 K ,X Ψ ′2 2 − 2ωr 2 φφ ′ K ,XX Ψ 1 Ψ ′ 2 + Ψ ′ 1Ψ2 + A Ψ 1 Ψ 2 − Ψ 1Ψ2 − M 2 1 Ψ 2 1 − M 2 2 Ψ 2 2 (7.1)
where we have decomposed the perturbation into its real and imaginary parts, δΦ = Ψ 1 + iΨ 2 , and
A = −2ωr 2 d d(φ 2 ) φ 2 K ,X − ω d dr r 2 φφ ′ K ,XX M 2 1 = λK ,X − r 2 2 K ,φφ − d dr r 2 φ ′ K ,Xφ M 2 2 = λK ,X − r 2 K ,φ 2 + ω 2 K ,X λ = ℓ(ℓ + 1) (7.2)
From this action, we obtain the two coupled equations for linear perturbations
− K ,XΨ1 + (K ,X − 2φ ′2 K ,XX )Ψ ′′ 1 + 2ωφφ ′ K ,XXΨ ′ 2 + F 1 (r, Ψ 1 , Ψ 2 , Ψ ′ 1 ,Ψ 2 ) = 0 (7.3) − (K ,X + 2ω 2 φ 2 K ,XX )Ψ 2 + K ,X Ψ ′′ 2 + 2ωφφ ′ K ,XXΨ ′ 1 + F 2 (r, Ψ 1 , Ψ 2 , Ψ ′ 2 ,Ψ 1 ) = 0 (7.4)
with F 1 and F 2 some functions defined by the perturbations and their first derivative. These equations form a set of two coupled differential equations representing the evolution of the perturbations in an effective metric. Indeed, if we consider, e.g. Eq. (7.3), and in the absence of coupling between Ψ 1 and Ψ 2 , i.e., ω = 0, the equation would reduce to the generic form h µν ∇ µν Ψ 1 + · · · = 0, with h 00 = −K ,X and h 11 = K ,X − 2φ ′2 K ,XX , from which we obtain the stability conditions, i.e., a Lorentzian effective metric h 00 < 0 and h 11 > 0. These conditions are equivalent to the Hamiltonian of field perturbations to be positive definite; indeed, as seen from Eq.(7.1), the Lagrangian (of Ψ 1 in the case of ω = 0) reduces to L = r 2 (−h 00Ψ2 − h 11 Ψ ′2 ) and therefore to a Hamiltonian H = r 2 (−h 00Ψ2 + h 11 Ψ ′2 ). The Hamiltonian is bounded from below [26,27] if we satisfy the conditions for an effective Lorentzian metric
K ,X > 0 K ,X − 2φ ′2 K ,XX > 0 ⇔ K ,X + 2XK ,XX > 0 (ω = 0)
But as nicely stated in [28], one should be careful, because Hamiltonian unboundedness is not always equivalent to stability. A Hamiltonian can be unbounded only because of our set of variables chosen. Therefore, stability should be imposed only from the existence of a future causal cone defined by the effective metric. In conclusion, to study stability, we need to ensure that the problem is well-posed. For that, we will derive the conditions of weak and strong hyperbolicity. Broadly speaking, the weak hyperbolicity condition forbids any solution to grow exponentially in time while the strong hyperbolicity condition imposes a stronger bound than the exponential growth and therefore is equivalent to local well-posedness of the Cauchy problem. In the case of a strong hyperbolic system, F 1 and F 2 will not be relevant while they could change the behavior of the system if weakly hyperbolic. We define the vector u = (Ψ 1 , Ψ 2 ) T and the system (7.3) and (7.4) becomes
u ,tt = Au ′′ + Bu ′ ,t + · · · (7.5)
where · · · indicates the lowest derivative terms, and
A 11 = K ,X − 2φ ′2 K ,XX K ,X (7.6) A 22 = K ,X K ,X + 2ω 2 φ 2 K ,XX (7.7) B 12 = 2ωφφ ′ K ,XX K ,X (7.8) B 21 = 2ωφφ ′ K ,XX K ,X + 2ω 2 φ 2 K ,XX (7.9)
while other elements of the matrices are zero. We consider wave solutions u(t, r) = e ikrû (t, k) and obtain u ,tt = −k 2 Aû + ikBû ,t + · · · (7.10)
This system can be reduced to first order by defining the variablev
=û ,t /(i|k|) v u ,t = i|k|P v u (7.11) withP = 0 k |k| B 12 A 11 0 k |k| B 21 0 0 A 22 1 0 0 0 0 1 0 0 (7.12)
The well-posedness of this system is reduced to the analysis of the matrixP (see e.g. [29]). If, for all k, the eigenvalues ofP are real, the system is weakly hyperbolic. The eigenvalues are ±1, ± K ,X − 2φ ′2 K ,XX K ,X + 2ω 2 φ 2 K ,XX (7.13) Therefore, we conclude that, if K,X −2φ ′2 K,XX K,X +2ω 2 φ 2 K,XX ≥ 0, the system is weakly hyperbolic. Additionally, when
K ,X − 2φ ′2 K ,XX K ,X + 2ω 2 φ 2 K ,XX
> 0 (7.14)
the system is strongly hyperbolic because the eigenvectors form a complete set. In that case, the two perturbations propagate at the speed
c 1 = 1 , c 2 = K ,X − 2φ ′2 K ,XX K ,X + 2ω 2 φ 2 K ,XX (7.15)
As we have shown in Sec. IV, we consider the condition K ,X − 2φ ′2 K ,XX > 0 which implies K ,X + 2ω 2 φ 2 K ,XX > 0. Summing these two conditions, we find a weaker condition, viz. K ,X > 0 and K ,X + XK ,XX > 0. Notice that for a real scalar field (ω = 0), the condition K ,X + 2ω 2 φ 2 K ,XX > 0 reduces to K ,X > 0 along with the condition K ,X −2φ ′2 K ,XX > 0 (K ,X +2XK ,XX > 0), and they correspond to the stability conditions obtained in [27].
Notice that the conditions of well-posedness of the system are independent of the energy conditions derived previously (5.2), (5.3), (5.4) and 5.5). In Fig. 5, and for the model K = X + αX 2 − V (φ), we have found that for a certain range of the parameters (ω, α), the Cauchy problem is not well-posed which never corresponds to α > 0. Also we found that for any α < 0, the perturbations are superluminal in some region of space. Even if the classical theory is well-posed, the superluminal propagation of the perturbations could be an obstacle to a quantum version of the theory. For example, requiring UV completion for K-essence (real scalar field analog to the case studied in this paper) imposes subluminal propagation [30]. A similar situation should be expected in our case [14]. Even if not equivalent, we found numerically, for all parameters (ω, α) of Fig. 5, that a system which violates the weak energy condition does not have a well-posed Cauchy problem. The converse is not true.
Restricting our analysis to the cases where the Cauchy problem is well-posed, we can study the mechanical stability of our solutions. For that, we assume the following form for the perturbation:
δΦ(t, r) = η(r) r n e iρt + χ * (r) r n e −iρ * t (7.16)
The system (7.3) and (7.4) reduces to two ordinary coupled differential equations for η(r) and χ(r). We have included a factor r n for numerical stability. In general, n = ℓ provides faster numerical results. In the canonical case where K ,X = 1, the stability analyses shows that any instability corresponds to ρ = −ρ * [13] which implies the condition dQ dω < 0. We could not extend this analysis to K-field theories and therefore we will study the perturbations by numerical means. For that, our system can be written as four first order differential equations for the variable Ψ ≡ (η, χ, η ′ , χ ′ ) T , Ψ ′ = BΨ where the matrix B is given in Appendix . Considering the conditions at r = 0 on the scalar field, φ ′ = 0, it is easy to show that perturbations behave as η(r ≃ 0) = c 0 r ℓ+n (7.17) χ(r ≃ 0) = c 1 r ℓ+n (7.18) which implies
Ψ(0) = c 0 r ℓ+n−1 r 0 ℓ + n 0 + c 1 r ℓ+n−1 0 r 0 ℓ + n (7.19)
Therefore, we can perform two numerical integrations from r = 0 with initial conditions η = r ℓ+n , χ = 0 and η = 0, χ = r ℓ+n respectively. The general solution will be a linear combination of these two solutions with coefficients (c 0 , c 1 ). Similarly, we perform an integration from infinity to r = 0. We have also a system with two free parameters (c 3 , c 4 ). We can integrate it from a large radius with initial conditions or
χ = e −r − K ,φ 2 (0,0) K ,X (0,0) −(ρ−ω) 2 r 1−n , η = 0 (7.21)
Having the solution integrated from both boundaries with four free parameters (c 1 , c 2 , c 3 , c 4 ), we can match them at a given radius, using the four continuity conditions of (η, χ, η ′ , χ ′ ). Notice that, because our system is linear, we can always fix one of the parameters, e.g. c 1 = 1. Therefore, we end with a system of four conditions and three parameters, the fourth parameter will determine the value of ρ. In conclusion, only a certain number of discrete values of ρ can solve our problem. In Fig. (6), we show |φ + δΦ| 2 , for ω = (0.5, 1) and α = 0. For each case, we have found the parameter ρ and using Eq. pendence of the solution. In the case, where ω = 0.5, the radius of the Q-ball is oscillating, and ρ is real. The energy of this solution is constant in time, while for ω = 1, the energy grows exponentially as well as the radius of the Q-ball. The solution is unstable and ρ is purely imaginary. Therefore, the strategy is simple, for each Q-ball, we search in the complex plane for values of the ρ solution to our previous problem.
For the excited states, all frequencies ω were unstable. But for various frequencies, the unstable modes were not always purely imaginary but also with a nonzero real part.
For the fundamental solution, Fig. 6 shows two cases where α = 0 and ω = (0.5, 1). The first frequency corresponds to a stable solution for which we see an oscillation of the radius of the Q-ball while the energy remains perfectly constant in time. The second case, corresponds to an unstable solution for which the radius increases and the energy grows exponentially.
Generically, we found that the stability region corresponds to dQ/dω < 0 for all ω, generalizing results that were known in the canonical case. In the unstable region, the timescale of the instability is of the order 1/Im(ρ). We found that Im(ρ) and therefore the timescale of the instability depends on the mode ℓ. For example, for α = 0, Im(ρ) is of the order 10 −1 for ℓ = 0 and of the order 10 −4 for ℓ = 1. Therefore, we will focus mainly on the spherical mode of perturbations ℓ = 0.
In Fig. 7, we show the unstable modes for three values of α. For each α, the instability starts when dQ/dω = 0. We notice also that even if for a given frequency, such as ω = 1.03, the Q-ball is unstable for all values of the parameter α, the instability is slower to develop (lower value of Im(ρ)), for larger positive values of α, which is consistent with the previous section where we found that the energy is lowered.
In Fig. 8, we summarize the various stability condi- tions. The quantum stability condition, namely the stability against fission is, as expected, stronger than the classical stability condition. We have also represented regions where the energy conditions are violated. The NEC is never violated in the region of analysis of the model while the WEC is violated only in the region where the Cauchy problem is not well-posed. The violation of the SEC and the DEC are totally independent of the stability conditions.
VIII. CONCLUSION
In this work we studied Q-balls in noncanonical scalar field theory. We derived the general equations of existence and stability for these theories. We found that the stability against fission and the linear mechanical stability are equivalent and reduce to Q ′ (ω) < 0 (see Table I). On the other hand, the condition for decay into free particles is stronger. TABLE I. Summary of the three stability conditions studied in this paper and extended to K-field theories.
We found that perturbations have a well-posed Cauchy problem if K,X −2φ ′2 K,XX K,X +2ω 2 φ 2 K,XX > 0. When the perturbations are strongly hyperbolic, we found that perturbations are superluminal or subluminal. In the particular case, K = X + αX 2 − V (|Φ| 2 ), perturbations are subluminal and luminal for α > 0 while they are superluminal and luminal for α < 0. We found that a Q-ball with α > 0 lowers its energy for larger values of α. Even in the unstable region, the timescale of this instability becomes larger and therefore more stable. The frequency at which Q-balls become unstable increases with α. It would be interesting to find models for which all Q-balls are stable irrespectively of their frequency.
Finally, we have studied the different energy conditions such as the SEC, DEC, WEC, NEC. We found that NEC is never violated and none of these conditions can be related to mechanical stability. (1)
31 = A K ,X − φ ′2 K ,XX r 2 K ,X (K ,X − 2φ ′2 K ,XX ) − ω r 4 φφ ′3 K 2 ,XX ′ 2r 4 K ,X (K ,X − 2φ ′2 K ,XX ) − ωφ ′2 (φ ′2 − φφ ′′ )K 2 ,XX
2K ,X (K ,X − 2φ ′2 K ,XX ) (A.8)
B
(1) 32 = ω φφ ′ K ,XX ′ K ,X − 2φ ′2 K ,XX + 2ωφφ ′ K ,X K ,XX rK ,X (K ,X − 2φ ′2 K ,XX ) + 2ωrφ ′2 K ,XX K ,X + ω 2 φ 2 K ,XX + φ 2 K ,Xφ 2 rK ,X (K ,X − 2φ ′2 K ,XX ) (A.9)
B
(1)
33 = − 2ωφφ ′3 K 2 ,XX
K ,X (K ,X − 2φ ′2 K ,XX ) (A.10)
B
(1) 34 = 2ωφφ ′ K ,XX K ,X − φ ′2 K ,XX K ,X (K ,X − 2φ ′2 K ,XX ) (A.11)
B
(2) 31 = 2ω 2 φ 2 φ ′2 K 2 ,XX − K 2 ,X − XK ,X K ,XX K ,X (K ,X − 2φ ′2 K ,XX ) (A.12)
B
(2) 32 = K ,XX
ω 2 φ 2 K ,X − φ ′2 K ,X − 2φ ′2 K ,XX (A.13)
and (A, M 2 1 , M 2 2 ) are defined by Eq. (7.2). These equations are given in the case of n = 0.
FIG. 1 .
1Left: The field φ(r) is shown as a function of the radial coordinate for different values of ω. For each value of ω, φ(0) is adjusted such that φ(∞) = 0. Center: The energy E and the charge Q are shown as a function of the frequency ω with the critical frequency (change of colors) defined by the condition dQ/dω = 0. Right: The energy is shown as a function of the charge. For all graphics, in green we have stable configurations according to the fission stability criteria, while in red we have unstable solutions. In the first figure, the solution for the critical frequency is shown in blue and in the third graphics, we have added the decay stability criteria that is shown by a red solid line and red dashed line for the unstable solutions while the fission unstable configurations are represented only by red solid line.
FIG. 2 .
2The field φ(r) is shown as a function of the radial coordinate for the fundamental mode (green curves), the first (purple curves) and the second (blue curves) radial excited mode for ω = 0.7. We also show the evolution of the energy as a function of the frequency. The dashed region corresponds to the unstable solutions according to the fission stability criteria.
FIG. 3 .FIG. 4 .
34The energy is shown as a function of charge for different values of the parameter α which runs from α = −0.5 in red to α = 0.5 in purple with an incrementation of 0Energy versus frequency for K-field model with α running from −0.5 (in red) to +0.5 (in purple) with a step of 0.1. For each panel, we have represented in dotted lines the regime where some energy condition is violated. From top left to bottom right, we show the violation of the SEC, DEC, WEC, NEC.
FIG. 5 .
5In gray, the region of parameter space (ω, α) where the Cauchy problem is not well-posed and in cyan the region of superluminal propagation.
FIG. 6 .
6Spacetime diagram of |Φ| 2 . The upper diagram shows the stability of the background solution with ω = 0.5 and the lower case shows an unstable solution for ω = 1. For both solutions, we have considered α = 0.
FIG. 7 .
7(7.16), we obtain the time and space de-Existence of Im(ρ) as a function of ω for α = (−0.5, 0, +0.5). The existence of such a mode implies an instability of the background solution. The dotted line corresponds to unstable modes but in a region where the Cauchy problem is not well-posed and therefore should be excluded from the analysis.
FIG. 8 .
8Space of parameters (ω, α) within the region where the Cauchy problem is well-posed. We have represented regions of quantum stability (against fission) and classical stability as well as regions where the energy conditions such as the SEC and DEC are violated. We have kept the cyan and white colors for, respectively, superluminal and subluminal propagation.
numerically to be the same condition for K = X +αX 2 −V (|Φ| 2 )
The matrix of the system Ψ ′ = BΨ can be decomposedas B = B (0) + ρB (1) + ρ 2 B (2) log K ,X (K ,X − 2φ ′2 K ,XX ) (A.6) K ,X K ,X − 2φ ′2 K ,XX (A.7) B
We assume our model corresponds to the low energy effective field theory where a small-X expansion is possible and therefore terms X n with n ≥ 3 are negligible. This is the complex analogous of[14] where α −1/4 is a cut off scale.
ACKNOWLEDGEMENTSThe work of A.F. is supported by ANID/CONICYT No. 21171262 while R.G. is supported by ANID FONDE-CYT Regular No. 1220965.Appendix: Perturbation equations
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| [] |
[
"Finding Biclique Partitions of Co-Chordal Graphs",
"Finding Biclique Partitions of Co-Chordal Graphs"
] | [
"Bochuan Lyu \nDepartment of Computational Applied Mathematics and Operations Research\nRice University\nUnited States of America\n",
"Illya V Hicks \nDepartment of Computational Applied Mathematics and Operations Research\nRice University\nUnited States of America\n"
] | [
"Department of Computational Applied Mathematics and Operations Research\nRice University\nUnited States of America",
"Department of Computational Applied Mathematics and Operations Research\nRice University\nUnited States of America"
] | [] | The biclique partition number (bp) of a graph G is referred to as the least number of complete bipartite (biclique) subgraphs that are required to cover the edges of the graph exactly once. In this paper, we show that the biclique partition number (bp) of a co-chordal (complementary graph of chordal) graph G = (V, E) is less than the number of maximal cliques (mc) of its complementary graph: a chordal graph G c = (V, E c ). We first provide a general framework of the "divide and conquer" heuristic of finding minimum biclique partitions of co-chordal graphs based on clique trees. Furthermore, a heuristic of complexity O[|V |(|V | + |E c |)] is proposed by applying lexicographic breadth-first search to find structures called moplexes. Either heuristic gives us a biclique partition of G with size mc(G c ) − 1. In addition, we prove that both of our heuristics can solve the minimum biclique partition problem on G exactly if its complement G c is chordal and clique vertex irreducible. We also show that mc(G c ) − 2 ≤ bp(G) ≤ mc(G c ) − 1 if G is a split graph. | 10.1016/j.dam.2023.05.001 | [
"https://export.arxiv.org/pdf/2203.02837v2.pdf"
] | 256,900,626 | 2203.02837 | 9c18105c066a17b0d0f6a9c2b4a5a0e755d78da5 |
Finding Biclique Partitions of Co-Chordal Graphs
16 Feb 2023
Bochuan Lyu
Department of Computational Applied Mathematics and Operations Research
Rice University
United States of America
Illya V Hicks
Department of Computational Applied Mathematics and Operations Research
Rice University
United States of America
Finding Biclique Partitions of Co-Chordal Graphs
16 Feb 2023biclique partitionsco-chordal graphsclique vertex irreduciblesplit graphs
The biclique partition number (bp) of a graph G is referred to as the least number of complete bipartite (biclique) subgraphs that are required to cover the edges of the graph exactly once. In this paper, we show that the biclique partition number (bp) of a co-chordal (complementary graph of chordal) graph G = (V, E) is less than the number of maximal cliques (mc) of its complementary graph: a chordal graph G c = (V, E c ). We first provide a general framework of the "divide and conquer" heuristic of finding minimum biclique partitions of co-chordal graphs based on clique trees. Furthermore, a heuristic of complexity O[|V |(|V | + |E c |)] is proposed by applying lexicographic breadth-first search to find structures called moplexes. Either heuristic gives us a biclique partition of G with size mc(G c ) − 1. In addition, we prove that both of our heuristics can solve the minimum biclique partition problem on G exactly if its complement G c is chordal and clique vertex irreducible. We also show that mc(G c ) − 2 ≤ bp(G) ≤ mc(G c ) − 1 if G is a split graph.
Introduction
The biclique partition number (bp) of a graph G is referred to as the least number of complete bipartite (biclique) subgraphs that are required to cover the edges of the graph exactly once (however, the vertices can belong to two or more bicliques). The set of such biclique subgraphs is called a biclique partition of G. Graham and Pollak first introduced this concept in network addressing [11] and graph storage [12]. Their famous Graham-Pollak Theorem proves a result about the biclique partition numbers on complete graphs and it draws much attention from algebraic graph theory [7,17,19,25,26]. However, no purely combinatorial proof is known to the result [18]. Rawshdeh and Al-Ezeh [21] extended Graham-Pollak Theorem to find biclique partition numbers on line graphs and their complements of complete graphs and bicliques. The biclique partition also has a strong connection with biclique cover number (bc), where the edges of a graph are covered by bicliques but not necessarily disjointed.
Pinto [20] showed that bp(G) ≤ 1 2 (3 bc(G) − 1). Moreover, finding a biclique partition with the minimum size is NP-complete even on the graphs without 4-cycles [15]. In this work, we focus on studying the biclique partition number on co-chordal (complement of chordal) and its important subclass, split graphs 1 .
There are also many related research studies around biclique partitions. Motivated by a technique for clustering data on binary matrices, Bein et al. [3] considered a biclique vertex partition problem on a bipartite graph where each vertex is covered exactly once in a collection of biclique subgraphs. De Sousa Filho et al. [9] also studied the biclique vertex partition problem and its variant bicluster editing problem, where they developed a polyhedral study on biclique vertex partitions on a complete bipartite graph. Groshaus et al. [13] gave a polynomial-time algorithm to determine whether a graph is a biclique graph (an intersection graph of the bicliques) of a subclass of split graphs. Shigeta and Amano [24] provided an explicit construction of an ordered biclique partition, a variant of biclique partition, of K n of size n 1/2+o (1) , which improved the 1 Almost all chordal graphs are split graphs. It means that as n goes to infinity, the fraction of n-vertex split graphs in n-vertex chordal graphs goes to 1 [4]. Since the complement of a split graph is also split, a split graph is also co-chordal.
O(n 2/3 ) bound shown by Amano [2].
Another related graph characteristic is bp k (G) where the edges can be covered by at least one and at most k biclique subgraphs [1] and Alon showed that the minimum possible number for bp k (K n ) is Θ(kn 1/k ), where K n is a complete graph with n vertices. Recent work by Rohatgi et al. [22] showed that if each edge is exactly covered by k bicliques, the number of bicliques required to cover
K n is (1 + o(1))n.
In this paper, we study a biclique (edge) partition problem on co-chordal graphs and split graphs using clique trees, moplexes, and lexicographic breadthfirst search (LexBFS) defined in Section 2. We introduce a new definition, partitioning biclique, which can naturally naturally partition a graph into two induced subgraphs with no shared edges in Section 3. In Section 4, we provide a heuristic to find a biclique partition on a co-chordal graph given a clique tree of the complement of the co-chordal graph. We also prove the correctness of the heuristic and show the size of the biclique partition is exactly equal to the number of maximal cliques in the complementary graph of the co-chordal graph minus one. We also provide a corollary that states that the biclique partition number of a co-chordal graph is less than the number of maximal cliques of its complement. In Section 5, we provide an efficient heuristic to obtain biclique partitions of co-chordal graphs by finding moplexes [5], defined later, using LexBFS. We also show that two heuristics provide biclique partitions of the same size. In Section 6, we prove that both heuristics can find a minimum biclique partition of G if its complement G c is chordal and clique vertex irreducible. We also derive a lower bound of the biclique partition number of split graphs and show that our heuristics can obtain a biclique partition on any split graph with a size no more than the biclique partition number plus one. In Section 7, we summarize the contribution of our work and point out some future directions.
Preliminaries
A simple graph is a pair G = (V, E) where V is a finite set of vertices and E ⊆ {uv : u, v ∈ V, u = v}. We use V (G) and E(G) to represent the vertex set and edge set of the graph G. Two vertices are adjacent in G if there is an edge between them. A subgraph
G ′ = (V ′ , E ′ ) of G is a graph where V ′ ⊆ V and E ′ ⊆ {uv ∈ E : u, v ∈ V ′ }. Given A ⊆ V , the subgraph of G induced by A is denoted as G(A) = (A, E A ), where E A = {uv ∈ E : u, v ∈ A}.
A clique is a subset of vertices of an graph G such that every two distinct vertices are adjacent in G. The neighborhood of a vertex v of a graph G is the set of all vertices, other than v, that are adjacent with v and is denoted
as N G (v). The closed neighborhood of v is denoted as N G [v] = N G (v) ∪ {v}.
For simplicity, we also define the neighborhood of a vertex set A of a graph
G as N G (A) = {u ∈ N G (v) : ∀v ∈ A} \ A and closed neighborhood of A as N G [A] = N G (A) ∪ A. A maximal clique of G is a clique of G such that it is not
a proper subset of any clique of G. Note that a vertex set with only one vertex K 1 is also a clique. We denote the number of maximal cliques of G as mc(G) and we use K G or K to denote the set of all maximal cliques of G. A maximum clique of G is a clique of G with the maximum number of vertices and we denote that number as the clique number of G, ω(G). An independent set of a graph G is a set of vertices that are pairwise nonadjacent with each other in G. Similarly, a maximal independent set is an independent set that is not a proper subset of any independent set and a maximum independent set is an independent set with the maximum number of vertices. Note that an independent set can be empty or only have one vertex.
A graph C n = (V, E) is a cycle if the vertices and edges:
V = {v 1 , v 2 , . . . , v n } and E = {v 1 v 2 , v 2 v 3 , . . . , v n−1 v n , v n v 1 }.
A graph is a tree if it is connected and does not have any subgraph that is a cycle.
Given two vertex sets U and V , we denote U × V to be the edge set {uv :
u ∈ U, v ∈ V }. A bipartite graph G = (L ∪ R, E)
is a graph where L and R are disjointed vertex sets with the edge set E ⊆ L × R. A biclique graph is a complete bipartite graph G = (L ∪ R, E) where E = L × R and we denote it as {L, R} for short. A biclique partition of a graph G is a collection of biclique subgraphs of G such that every edge of G is in exactly one biclique of the collection. The minimum biclique partition problem on G is to find a biclique partition with the minimum number of bicliques in the collection and we denote that value to be bp(G).
We denote that n = {1, 2, . . . , n} where n is a positive integer. A vertex is simplicial if its neighborhood is a clique. An ordering v 1 , v 2 , . . . , v n of V is a perfect elimination ordering if for all i ∈ n , v i is simplicial on the induced subgraph G({v j : j ∈ {i, i + 1, . . . , n}}). An ordering function σ : n → V is defined to describe the ordering of vertices V . A clique tree T K for a chordal graph G is a tree where each vertex represents a maximal clique of G and satisfies the clique-intersection property: given any two distinct maximal cliques K 1 and K 2 in the tree, every clique on the path between K 1 and K 2 in T K contains
K 1 ∩ K 2 .
We also define the middle set of edge e ∈ T K , mid(e), to be the intersection of the vertices of cliques on its two ends.
A graph G is clique vertex irreducible if every maximal clique in G has a vertex which does not lie in any other maximal clique of G [16].
A split graph is a graph whose vertices can be partitioned into a clique and an independent set. A graph is chordal if there is no induced cycle subgraph of length greater than 3. Note that a split graph is also chordal. A graph is chordal if and only if it has a clique tree [6]. Also, a graph is chordal if and only if it has a perfect elimination ordering [10]. A perfect elimination ordering of a chordal graph can be obtained by lexicographic breadth-first search (LexBFS) described in Algorithm 1 [8,23], where lexicographical order is defined in Definition 1.
Note that the LexBFS algorithm in Algorithm 1 can be implemented in lineartime: O(|V | + |E|) with partition refinement [14].
Definition 1. [Lexicographical Order] Let X be a set of all vectors of real num-
bers with a finite length. Then, we can define a lexicographical order on X where 1. ∅ ∈ X and for any x ∈ X, ∅ x.
2. (x 1 , x 2 ) (y 1 , y 2 )
if and only if one of the following holds:
(a) x 1 < y 1 (b) x 1 = y 1 and x 2 y 2 where x 1 , y 1 ∈ R and x 2 , y 2 ∈ X.
Algorithm 1 Generic Lexicographic Breadth-First Search [8,23]. Select an unnumbered vertex u with lexicographically the largest label. 6:
σ(i) ← u. 7: for each unnumbered vertex w in N G (u) do 8: label(w) ← (label(w), i). ⊲ (·, ·)
here is a concatenation operation. In our work, we study biclique partitions on co-chordal graphs, so most of the G c 's in the following sections are chordal graphs. We use K c and T K c to denote the set of all maximal cliques and a clique tree of G c .
A module of a graph G = (V, E) is a vertex set A ⊆ V such that all the vertices in A share the same neighborhood in V \ A. A separator of G = (V, E) is a set of vertices, say S, such that G(V \ S) is disconnected. A separator S is minimal if no proper set of S is a separator. A moplex X of G is both a
clique and a module such that N G (X) is a minimal separator (see Figure 1). Berry and Bordat [5] discovered that LexBFS can be applied to find a moplex of a general graph G in linear time, which motivates us to propose the biclique partition algorithm in Section 5.
Proposition 1 (Theorem 5.1 [5]). Given a graph G and σ generated by Algorithm 1, σ(1) belongs to a moplex.
X 1 X 2 Figure 1: X 1 is a module but not a moplex. X 2 is a moplex.
Partitioning Biclique
We will introduce a new definition: partitioning biclique, which can naturally partition the edges of a graph into two edge disjoint subgraphs. We first start with a definition of partitioning biclique. It is worth to noticing that {L, R} is not necessarily equal to the induced subgraph G(L ∪ R). It means that L and R might not be independent sets of G.
We then show that an arbitrary biclique subgraph can divide the edges of the original graphs into three parts: the biclique and two induced subgraphs.
graph of G. Then, E = E({L, R}) ∪ E(G(V \ L)) ∪ E(G(V \ R)). Proof. Since {L, R}, G(V \L), G(V \R) are all subgraphs of G, then E({L, R})∪ E(G(V \ L)) ∪ E(G(V \ R)) ⊆ E. Let C = V \ (L ∪ R) and uv be an arbitrary edge in G. Thus, V \ L = R ∪ C and V \ R = L ∪ C. If u, v ∈ (L ∪ C), then uv ∈ E(G(V \ R)). If neither of u and v is in (L ∪ C), then u, v ∈ R and uv ∈ E(G(V \ L)
). If one of u and v is in L ∪ C and the other one is not, then we can assume that u ∈ (L ∪ C) and
v ∈ R without loss of generality. Then,
either uv ∈ E(G(V \ L)) (u ∈ C) or uv ∈ {L, R} (u ∈ L). Hence, E({L, R}) ∪ E(G(V \ L)) ∪ E(G(V \ R)) = E.
Next, we show that a biclique subgraph is a partitioning biclique of a graph Proof
. Denote C = V \ (L ∪ R). In the backward direction, suppose that C is an independent set in G. By Lemma 1, we know that E = E({L, R}) ∪ E(G(V \ L)) ∪ E(G(V \ R)). Then, we need to show that E({L, R}), E(G(V \ L)), E(G(V \ R)) are disjointed. Since every edge in {L, R} is between vertices in L and R, then E({L, R}) ∩ E(G(V \ L)) = ∅ = E({L, R}) ∩ E(G(V \ R)). Since
C is an independent set of G and
[(V \ L) × (V \ L)] ∩ [(V \ R) × (V \ R)] = C × C, we know that E(G(V \ L)) ∩ E(G(V \ R)) = ∅.
In the forward direction, the proof is by contrapositive. Suppose that C isn't an independent set in G. Then, there exists an edge uv such that u, v ∈ C,
where uv is in both subgraphs G(V \ L) and G(V \ R). Hence, C must be an independent set.
Note that an independent set can be empty. Since the partitioning biclique can partition the original graph into a biclique subgraph and two induced subgraphs, it can be used to design a heuristic to find a biclique partition of a graph. In the next two sections, we will focus on a class of graph, co-chordal,
where partitioning bicliques are easy to find since the complementary graph is chordal.
A Heuristic Based on Clique Trees
In this section, we want to design a heuristic to find a biclique partition of a co-chordal graph G. Since G c is chordal, one of the good ways to represent G c is its clique tree where each vertex represents a maximal clique of G and satisfies the clique-intersection property. We demonstrate a heuristic with an input of a clique tree T K c of G c and an output of a biclique partition of G in Algorithm 2. We also show that the size of that biclique partition is equal to mc(G c ) − 1, which provides us an upper bound of the biclique partition number of co-chordal graphs. Select an arbitrary edge e to cut T K c into two components T K c 1 and T K c 2 .
8:
L = K∈V (T K c 1 ) K \ mid(e); R = K∈V (T K c 2 ) K \ mid(e). 9: return {{L, R}} ∪ FindPartition(T K c 1 ) ∪ FindPartition(T K c 2 ) 10: end function
We prove that the output of Algorithm 2 is a biclique partition of a cochordal graph G by showing that at each recursion a nonempty partitioning biclique {L, R} is found and two subtrees T K c 1 and T K c 2 are also clique trees of two induced subgraphs G c (V \ L), and G c (V \ R) of G c respectively. Note that the edge e can be selected arbitrarily in Algorithm 2.
Proposition 3. Given a chordal graph G c = (V, E c ) and one of its clique trees Proof. Since every edge in a tree is a cut, e can partition T K c into two sets of vertices, K c 1 and K c 2 , in T K c . Let the two ends of edge e in T K c to be K ′ and K ′′ . Since mid(e) = K ′ ∩ K ′′ and both K ′ and K ′′ are clique subgraphs of G c , then mid(e) is an independent set of G.
T K c = (K c , E) where V (T K c ) > 1,
Given an arbitrary u ∈ L = K∈K c 1 K \ mid(e) and v ∈ R = K∈K c 2 K \ mid(e), we assume that uv ∈ E(G c ). Otherwise, {L, R} is a biclique of G and our result has been proved.
Since uv ∈ E(G c ), we know that there exists some K ∈ K c such that {u, v} ⊆ K. Without loss of generality, we assume that u ∈ K 1 and {u, v} ⊆ K 2 where K 1 ∈ K c 1 and K 2 ∈ K c 2 . Since K ′ and K ′′ both on the path between K 1 and K 2 , K 1 ∩ K 2 ⊆ mid(e). Therefore, u ∈ mid(e), which is a contradiction. Hence, uv is not an edge of G c and {L, R} is a biclique of G.
In Claim 2, we prove part (2) of Proposition 3. We first remark that any subgraph of a chordal graph is chordal and then use it to prove part (2) in Claim 2. Thus, there exist clique trees for both G c (V \ L) and G c (V \ R). Without loss of generality, we only need to prove T K c 1 is a clique tree of G c (V \ R). Since T K c 1 is a subtree of T K c , we only need to show that K c 1 is the set of all maximal cliques of G c (V \ R).
First, we want to prove that K∈K c 1 K = V \ R. We proved in Claim 1, Hence, K 1 is a maximal clique of G c (V \ R) for any K 1 ∈ K c 1 . By the definition of clique tree, given an arbitrary
{L, R} = { K∈K c 1 K \ mid(e), K∈K c 2 K \ mid(e)}K 2 ∈ K c 2 K 2 ∩ (V \ R) = K 2 ∩ K∈K c 1 K = K∈K c 1 (K 2 ∩ K) ⊆ mid(e).
Since mid(e) ⊂ K ′ for some K ′ ∈ K c 1 , then K 2 ∩(V \R) cannot be a maximal clique of G c . Therefore, K c 1 is the set of all maximal cliques of G c (V \ R) and T K c 1 is its clique tree.
Next, we show that the edge set of biclique {L, R} is not empty. Proof. Let the two ends of edge e in T K c to be K ′ and K ′′ . Since both K ′ and K ′′ are maximal cliques of G c and mid(e) = K ′ ∩ K ′′ , then both K ′ \ mid(e) and K ′′ \ mid(e) are not empty. We can complete the proof since K ′ ∈ K c 1 and K ′′ ∈ K c 2 .
Next, we show that the output of Algorithm 2 is a biclique partition of a co-chordal graph G.
Theorem 1. Given a co-chordal graph G and clique tree T K c of its complement
G c , the output of FindPartition(T K c ) is a biclique partition of G.
Proof. We will use induction to prove Theorem 1. In the base step, if T K c only has one vertex, then G c is a complete graph and G is an empty graph. Thus, bp(G) = 0.
In the induction step, supposed that FindPartition(T K c ) is a biclique partition of G if |V (T K c )| < k. Then, we consider the scenario that T K c has k vertices. By Proposition 3, an arbitrary edge e cuts T K c into two components T K c 1 and T K c 2 , where we can construct a partitioning biclique {L, R} as in Algorithm 2 that partitions the edges of G into the edge sets of {L, R}, G(V \L), and G(V \R). Moreover, T K c 1 and T K c 2 are clique trees of G c (V \R), and G c (V \L) respectively. Then, FindPartition(T K c 1 ) returns a biclique partition of G(V \ R) and FindPartition(T K c 2 ) returns a biclique partition of G(V \ L). Therefore,
{{L, R}} ∪ FindPartition(T K c 1 ) ∪ FindPartition(T K c 2 ) is a biclique partition of G.
Finally, we prove that the size of the output of Algorithm 2 is equal to one less than the number of maximal cliques of G c .
Theorem 2. Given a co-chordal graph G (with at least one vertex) and clique
tree T K c of its complement G c , the output of FindPartition(T K c ) is a biclique partition of G with size mc(G c ) − 1.
Proof. By Theorem 1, we know that the output of FindPartition(T K c ) is a biclique partition of G. We then want to prove that the size of the output of
FindPartition(T K c ) is mc(G c ) − 1 by induction.
In the base step, |V (T K c )| = 1, then mc(G c ) = 1 since G c is a complete graph. The output is an empty set, which has size of 0.
In the induction step, assume that the size of the output of
FindPartition(T K c ) is |V (T K c )| − 1 if |V (T K c )| ∈ {1, 2, . . . , k − 1}. Then, if |V (T K c )| = k, the out- put is {{L, R}} ∪ FindPartition(T K c 1 ) ∪ FindPartition(T K c 2 ) where L, R, T K c 1 and T K c 2 are defined in Algorithm 2. Since |V (T K c 1 )| + |V (T K c 2 )| = k and |V (T K c 1 )|, |V (T K c 2 )| > 0, we know that |V (T K c 1 )| < k and |V (T K c 2 )| < k.
Also both L and R are not empty by Claim 3, the size of the output of FindPartition(T K c )
is 1 + |V (T K c 1 )| − 1 + |V (T K c 2 )| − 1 = k − 1.
Theorem 2 leads us to a direct result that the biclique partition number of a co-chordal graph G is less than the number of maximal cliques of G c .
A Heuristic Based on Finding Moplexes
Although Algorithm 2 can give a biclique partition of a co-chordal graph G, it requires a clique tree of its complementary chordal graph G c and computes unions of some maximal cliques, which could include some redundant computations. Instead of using clique tree, we can use a moplex in G c to find a biclique partition. Figure 3 motivates the idea via a simple example. In Algorithm 2, we can select an arbitrary edge in the clique tree to find a biclique. Thus, we can always select an edge that is incident to a leaf node of the tree. For example, if we select the edge with middle set {x 3 }, the biclique built in the current call of
FindPartition(T ) has L = {x 4 , x 5 , x 6 , x 7 } and R = {x 1 , x 2 }, where {x 1 , x 2 }
is a moplex of G c . We can also see that selecting the other two edges can also lead to bicliques with a moplex as L or R. It motivates us to design another heuristic to find biclique partitions on co-chordal graphs by finding moplexes, which can be found efficiently by LexBFS. In this section, we show that Algorithm 3 can return a biclique partition of a co-chordal graph G with a same size Let σ be the perfect elimination ordering of G ′ obtained by LexBFS (Algorithm 1).
as Algorithm 2 in O[|V |(|V | + |E c |)] time. {x 1 , x 2 , x 3 } {x 3 , x 4 , x 5 } {x 3 , x 5 , x 6 } {x 4 , x 7 } {x 4 } {x 3 } {x 3 , x 5 }
7:
L ← V (G ′ ) \ N G ′ [σ(1)]. 8: R ← {u ∈ N G ′ [σ(1)] : N G ′ (u) ∩ L = ∅}.
⊲ R is a moplex containing σ(1).
9:
If L is not empty, bp ← bp ∪{{L, R}}.
10:
Remove vertices in R from G ′ .
11: end while 12: return bp Note that Xu et al. [27] also apply lexicographical depth-first search (LexDFS) to find moplexes, which can be an alternative method to find moplexes than LexBFS in Algorithm 3.
Before we can prove that Algorithm 3 returns a biclique partition of a cochordal graph, we need to show that {L, R} is a partitioning biclique of G ′c with vertices V ′ . In addition, one of the two induced subgraphs, G ′c (V ′ \ L), has an empty edge set.
E[G ′c (V ′ \ R)].
Proof. Since σ is obtained by LexBFS (Algorithm 1), σ is a perfect elimination ordering. Then, the closed neighborhood
N G ′ [σ(1)] is a clique of G ′ . Thus, V ′ \ (L ∪ R)
is an independent set of G ′c . By Proposition 2, {L, R} is a partitioning biclique of G ′c . Thus, the edges of G ′c is partitioned into
E({L, R}), E[G ′c (V ′ \ L)], and E[G ′c (V ′ \ R)]. Since V ′ \ L = N G ′ [σ(1)] is a clique of G ′ , E[G ′c (V ′ \ L)] = ∅. Hence, the edges of G ′c is partitioned into E({L, R}) and E[G ′c (V ′ \ R)].
Next, we show that the output of Algorithm 3 is a biclique partition of a co-chordal graph G with a size of mc(G c ) − 1.
N G ′ (u) ∩ L = ∅} and G ′ R = G ′ (V ′ \ R).
Then, R is a moplex of G ′ and mc(G ′ ) = mc(G ′ R ) + 1.
Proof. The conclusion that R is a moplex of G can be drawn from Proposition 1 (Theorem 5.1 [5]). The reader is referred to [5] for the detailed proof. We want to note that R is both a module and a clique.
Since σ is obtained by LexBFS in Algorithm 1, σ is a perfect elimination
ordering. Then, (V ′ \ L) is a maximal clique of G ′ . Since σ(1) ∈ V (G ′ R ), G ′ (V ′ \ L)
is not a subgraph of G ′ R . Given an arbitrary maximal clique K of G ′ that is not (V ′ \L), there exists v ′ in clique K such that v ′ is not σ(1) or its neighbor. Thus, u ∈ R for every u ∈ K.
Hence, K is also a maximal clique subgraph of G ′ R . Thus, mc(G ′ R ) ≥ mc(G ′ )−1.
Since R is a moplex,
S = N G ′ [σ(1)] \ R is a minimal separator of G ′ . Note that (V ′ \ L) = N G ′ [σ(1)] is a maximal clique of G ′ . Since G ′ is chordal, there must exists a maximal clique K of G ′ that is not (V ′ \ L) such that S ⊆ K (See
Theorem 4.1 [6]). Note that K is a maximal clique of G ′ R . Therefore, S is not a maximal clique of G ′ R and every maximal clique in G ′ R is also a maximal clique of G ′ . Hence, mc(G ′ ) = mc(G ′ R ) + 1. Proof. By Proposition 4, the output of Algorithm 3 is a biclique partition of G.
By Proposition 5, the number of bicliques in the biclique partition is mc(G c )−1.
We want to note that L = ∅ if mc(G c ) = 1, so the number is mc(G c ) − 1 instead of mc(G c ).
We also show that Algorithm 3 is efficient:
O[|V |(|V | + |E c |)] time.
Lower Bounds of Biclique Partition Numbers on Some Subclasses of Co-Chordal Graphs
In this section, we will first prove that both Algorithms 2 and 3 find a minimum biclique partition of G if its complement G c is both chordal and clique vertex irreducible. We start with the well-known Graham-Pollak Theorem, which provides a lower bound on the biclique partition number for complete graphs.
Theorem 6 (Graham-Pollak Theorem [11,12]). The edge set of the complete graph, K n , cannot be partitioned into fewer than n − 1 biclique subgraphs.
We first show that the biclique partition number of a graph is no less than the biclique partition number of its induced subgraph.
Lemma 2. For every induced subgraph G ′ of a graph G, bp(G ′ ) ≤ bp(G). Proof. Suppose that {{L i , R i }} bp(G) i=1
is a minimum biclique partition of graph
G. Then, it is trivial to show that {{L i ∩V ′ , R i ∩V ′ }} bp(G) i=1
is a biclique partition of graph G ′ . Thus, bp(G ′ ) ≤ bp(G).
Then, we can show that the biclique partition number of an arbitrary graph is no less than its clique number minus one.
Proposition 6. Given a graph G = (V, E), bp(G) ≥ ω(G) − 1.
Proof. Let K be a maximum clique of G with size ω(G) and G K be the corresponding induced subgraph. By Graham-Pollak Theorem, we know that bp(G K ) ≥ ω(G) − 1. By Lemma 2, bp(G) ≥ bp(G K ) ≥ ω(G) − 1.
Next, we prove that Algorithms 2 and 3 are exact when the input graph G has a chordal and clique vertex irreducible complement.
Lemma 3. If a graph G is clique vertex irreducible, then mc(G) = ω(G c ).
Proof. Since G is clique vertex irreducible, there exists a vertex v i in maximal clique K i such that K i is the only maximal clique in G contains v i . Then,
{v i } mc(G) i=1
is an independent set in G so it is a clique in G c . Thus, mc(G) ≤ ω(G c ). Furthermore, assume that ω(G c ) > mc(G) and let K c be a maximum clique of G c . Then, there exist two distinct vertices u, v in K c such that u
and v are in the same maximal clique of G. It implies uv ∈ E(G), which is a contradiction.
Theorem 7. Given a graph G where its complement G c is chordal and clique vertex irreducible, bp(G) = mc(G c ) − 1.
Proof. By Corollary 1, bp(G) ≤ mc(G c ) − 1 since G is a co-chordal graph.
Since G c is clique vertex irreducible, then mc(G c ) = ω(G). By Proposition 6, bp(G) ≥ mc(G c ) − 1.
Then, we derive a lower bound of the bp number on any split graph and
prove that mc(G c ) − 2 ≤ bp(G) ≤ mc(G c ) − 1 if G is a split graph.
Lemma 4. Given a split graph G = (V, E), V can be partitioned into two sets V 1 , V 2 such that V 1 is a clique and V 2 is an independent set. Then, |V 2 | ≤ mc(G) ≤ |V 2 | + 1.
Proof. Since V 2 is an independent set, any maximal clique of G can include at most one vertex in V 2 . Also, each vertex in V 2 is in at least one maximal clique.
Then, |V 2 | ≤ mc(G).
We then want to prove that given an arbitrary vertex v ∈ V 2 , there is at most one maximal clique of G containing v. Assume that there exists two distinct maximal cliques of G, K 1 and K 2 , containing v. Then, it is safe to conclude that both K 1 \ {v} ⊆ V 1 and K 2 \ {v} ⊆ V 1 . Since V 1 is a clique in G, then we can construct a larger clique subgraph of G by including all the vertices in K 1 and K 2 , which is a contradiction.
There is at most one maximal clique of G not including any vertex in V 2 , which is V 1 . Therefore, mc(G) ≤ |V 2 | + 1.
Note that if V 1 is a maximal clique, then mc(G) = |V 2 | + 1. Then, we use Lemma 4 to prove mc(G c ) − 1 ≤ ω(G) and eventually bp(G) ≥ mc(G c ) − 2 for an arbitrary split graph G.
Theorem 8. Given a split graph G, then bp(G) ≥ mc(G c ) − 2.
Proof. We first claim that the clique number of G, ω(G), is no less than the number of maximal cliques, mc(G c ) − 1, i.e. ω(G) ≥ mc(G c ) − 1. Then, by Proposition 6, bp(G) ≥ ω(G) − 1 ≥ mc(G c ) − 2.
To prove the claim, we first start with the definition of split graphs. Since G = (V, E) is a split graph, then V can be partitioned into two sets V 1 , V 2 such that V 1 is a clique and V 2 is an independent set of G. Then, we know that
|V 1 | ≤ ω(G).
Furthermore, G c is also a split graph, V 1 is an independent set of G c , and V 2 is a clique in G c . By Lemma 4, we know that mc(G c ) ≤ |V 1 | + 1. Therefore, mc(G c ) − 1 ≤ |V 1 | ≤ ω(G).
In the proof of Theorem 8, we show that ω(G) ≥ mc(G c ) − 1 if G is a split graph. In Figure 4, we show that it is possible that ω(G) = mc(G c ) − 1. In this case, we only have mc(G c )−2 ≤ bp(G) ≤ mc(G c )−1. However, we can have the exact value of bp(G), bp(G) = mc(G c ) − 1, if ω(G) = mc(G c ), or, equivalently, the vertices of G can be partitioned into vertex sets V 1 and V 2 such that V 1 is a maximal (maximum) clique and V 2 is an independent set but not maximal.
V 1 V 2 Figure 4: A split graph G with vertex sets V 1 (a maximum clique) and V 2 (a maximum independent set), where ω(G) = 4 but mc(G c ) = 5.
Remark 2.
If the vertices of a split graph G can be partitioned into vertex sets V 1 and V 2 such that V 1 is a maximal (maximum) clique and V 2 is an independent set but not maximal, then bp(G) = mc(G c ) − 1.
Final remarks
If a graph G = (V, E) is a co-chordal graph, the biclique partition number of G is less than the number of maximal cliques of its complement G c . Additionally, we provided two heuristics, one based on clique trees and one based on finding moplexes, to find an explicit construction of a biclique partition with a size of mc(G c )−1. We also showed that the computational time of the moplex heuristic
is O[|V |(|V | + |E c |)].
If a graph G where its complement G c is both chordal and clique vertex irreducible, then bp(G) = mc(G c ) − 1. If a graph G is a split graph, another subclass of co-chordal, then we have mc(G c ) − 2 ≤ bp(G) ≤ mc(G c ) − 1.
In Section 4, we showed that given a co-chordal graph G and a clique tree of
1 :
1Input: A graph G = (V, E) and an arbitrary selected vertex v in V .
2 :
2Output: An ordering function σ : n → V of the vertices V .
3: label(v) ← (n) where n = |V |, and label(u) ← () for all u ∈ V \ {v}. 4: for i ∈ {n, n − 1, . . . , 1} do 5:
Definition 2
2(partitioning biclique). Given a graph G = (V, E), a biclique subgraph of G, {L, R}, is a partitioning biclique subgraph of G if the edge sets of {L, R}, G(V \ L), and G(V \ R) partition the edges of G, i.e. each edge of G is exactly in one of {L, R}, G(V \ L), and G(V \ R).
See Figure 2
2for a visualization of this idea: the edges of graph G are partitioned into the edges of {L, R}, G(V \ L), and G(V \ R).
Lemma 1 .
1Given a graph G = (V, E), let {L, R} be an arbitrary biclique sub-
GFigure 2 :
2if and only if the vertices in G that are not in the biclique form an Given a graph G = (V, E), {L, R} is a partitioning biclique of G, where each edge in E is exactly in one of {L, R}, G(V \ L), and G(V \ R).Proposition 2. Given a graph G = (V, E) and a biclique subgraph of G: {L, R}, then {L, R} is a partitioning biclique subgraph of G if and only if V \ (L ∪ R) is an independent set in G.
Algorithm 2
2Find a biclique partition of a co-chordal graph G given a clique tree of G c . 1: Input: A clique tree T K c of a chordal graph G c . 2: Output: A biclique partition of the complementary graph G of G c . 3: function FindPartition(T K c )
any edge e of T K c can partition K c into K c 1 and K c 2 (trees T K c 1 and T K c 2 respectively) such that (1) The edges of {L, R} = { K∈K c 1 K \ mid(e), K∈K c 2 K \ mid(e)}, G(V \ L), and G(V \R) partition the edges in G where G is the complementary graph of G c , i.e. {L, R} is a paritioned biclique subgraph of G.
(2) T K c 2 and T K c 1
1are clique trees of chordal graphs G c (V \ L), and G c (V \ R) respectively.
( 3 )
3Both L and R are not empty. Proof. (1) is proved by Claim 1 and definition of partitioning biclique, (2) is proved by Claim 2, and (3) is proved by Claim 3. We next prove that {L, R} is a partitioning biclique of G. Note that L and R do not have to be independent sets. Claim 1. {L, R} is a partitioning biclique of the complementary graph G = (V, E) of G c .
Remark 1 .
1Given a chordal graph G c = (V, E c ), any induced subgraph of G c is chordal.Claim 2. T K c 2 and T K c 1 are clique trees of chordal graphs G c (V \ L), and G c (V \ R) respectively.Proof. By Remark 1, we know that both G c (V \ L), and G c (V \ R) are chordal.
is a biclique. Thus, L and R are disjoint vertex sets. Since V = K∈K c K and L ∪ R = K∈K c K \ mid(e), then mid(e) = V \ (L ∪ R). Thus, K∈K c 1 K = L ∪ mid(e) = V \ R.
Claim 3 .
3Both L and R are not empty.
Corollary 1 .
1If G is a co-chordal graph, then bp(G) ≤ mc(G c ) − 1. The total time complexity of Algorithm 2 is O[|V | 3 ], which includes both of taking the complement of a co-chordal graph G and computing a clique tree of G c . Theorem 3. Algorithm 2 runs in O[|V | 3 ]. Proof. Taking the complement of a graph requires O(|V | 2 ) time. A clique tree of G c can be computed in O(|V | 3 ) using an algorithm described in Figure 4.2 [6]. Since the number of maximal cliques in the chordal graph G c is at most |V |, the number of edges in T K c is O(|V |). Each e in T K c is selected once so that the resulting subtrees are trees with one vertex. Thus, the total number of FindPartition(T K c ) called is O(|V |). Since the size of a maximal clique in G c is at most |V |, it takes O(|V | 2 ) to compute L and R in FindPartition(T K c ).
Figure 3 :
3A clique tree T of some chordal graph G c with maximal cliques on the vertices and middle sets on the edges.Algorithm 3 A moplex heuristic to find a biclique partition on co-chordal graph G. 1: Input: Co-chordal graph G = (V, E).2: Output: A biclique partition bp of G. 3: Initialize bp ← {}. 4: G ′ ← G c so it is a chordal graph.
5: while |V (G ′ )| ≥ 1 do 6:
Proposition 4 .
4Given a chordal graph G ′ = (V ′ , E ′ ) with at least two maximal cliques and σ obtained by LexBFS, let L = V ′ \ N G ′ [σ(1)] and R = {u ∈ N G ′ [σ(1)] : N G ′ (u) ∩ L = ∅}. Then, the edges of G ′c , the complement of G ′ , can be partitioned into edges in a biclique E({L, R}) and edges in a chordal graph
Proposition 5 .
5Given a chordal graph G ′ = (V ′ , E ′ ) with at least two maximal cliques and σ obtained by LexBFS, let L = V ′ \ N G ′ [σ(1)], R = {u ∈ N G ′ [σ(1)] :
Theorem 4 .
4Algorithm 3 returns a biclique partition of G with a size of mc(G c )− 1 given G is a co-chordal graph.
Theorem 5 .
5Algorithm 3 runs in O[|V |(|V | + |E c |)] time. Proof. Taking the complement of a graph requires O(|V | 2 ) time. The outer while loop has at most O(|V |) iterations. Finding a perfect elimination ordering σ of a chordal graph G c can be done in O(|V | + |E c |) time by LexBFS with partition refinement. The construction of L takes O(|V |) time. Computing R in Line 8 takes O(|E c |), where each edge in G ′ can be traversed in constant times in the worst case.
G
c , T K c , Algorithm 2 can return a biclique partition with a size of mc(G c ) − 1 no matter which edge of T K c is selected in each recursion. An open question is whether bp(G) = mc(G c ) − 1 if G is a co-chordal graph or split graph. If it is the case, it is an extension of Graham-Pollak theorem to a more general class of graphs.
AcknowledgementsThe authors would like to thank the reviewers and Bo Jones for their helpful and insightful comments.
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| [] |
[
"Quantum Correlations in the Minimal Scenario Dedicated to the memory of Boris Tsirelson",
"Quantum Correlations in the Minimal Scenario Dedicated to the memory of Boris Tsirelson"
] | [
"Thinh P Le ",
"Chiara Meroni ",
"Bernd Sturmfels ",
"Reinhard F Werner ",
"Timo Ziegler "
] | [] | [] | In the minimal scenario of quantum correlations, two parties can choose from two observables with two possible outcomes each. Probabilities are specified by four marginals and four correlations. The resulting four-dimensional convex body of correlations, denoted Q, is fundamental for quantum information theory. It is here studied through the lens of convex algebraic geometry. We review and systematize what is known and add many details, visualizations, and complete proofs. A new result is that Q is isomorphic to its polar dual. The boundary of Q consists of three-dimensional faces isomorphic to elliptopes and sextic algebraic manifolds of exposed extreme points. These share all basic properties with the usual maximally CHSH-violating correlations. These patches are separated by cubic surfaces of non-exposed extreme points. We provide a trigonometric parametrization of all extreme points, along with their exposing Tsirelson inequalities and quantum models. All non-classical extreme points (exposed or not) are self-testing, i.e., realized by an essentially unique quantum model.Two principles, which are specific to the minimal scenario, allow a quick and complete overview: The first is the pushout transformation, the application of the sine function to each coordinate. This transforms the classical polytope exactly into the correlation body Q, also identifying the boundary structures. The second principle, self-duality, reveals the polar dual, i.e., the set of all Tsirelson inequalities satisfied by all quantum correlations. The convex body Q includes the classical correlations, a cross polytope, and is contained in the no-signaling body, a 4-cube. These polytopes are dual to each other, and the linear transformation realizing this duality also identifies Q with its dual.We also discuss the sets of correlations achieved with fixed Hilbert space dimension, fixed state or fixed observables. | 10.22331/q-2023-03-16-947 | [
"https://arxiv.org/pdf/2111.06270v2.pdf"
] | 243,986,086 | 2111.06270 | d2d1a5285714d9fb770782d9dd639b21a57749df |
Quantum Correlations in the Minimal Scenario Dedicated to the memory of Boris Tsirelson
Thinh P Le
Chiara Meroni
Bernd Sturmfels
Reinhard F Werner
Timo Ziegler
Quantum Correlations in the Minimal Scenario Dedicated to the memory of Boris Tsirelson
In the minimal scenario of quantum correlations, two parties can choose from two observables with two possible outcomes each. Probabilities are specified by four marginals and four correlations. The resulting four-dimensional convex body of correlations, denoted Q, is fundamental for quantum information theory. It is here studied through the lens of convex algebraic geometry. We review and systematize what is known and add many details, visualizations, and complete proofs. A new result is that Q is isomorphic to its polar dual. The boundary of Q consists of three-dimensional faces isomorphic to elliptopes and sextic algebraic manifolds of exposed extreme points. These share all basic properties with the usual maximally CHSH-violating correlations. These patches are separated by cubic surfaces of non-exposed extreme points. We provide a trigonometric parametrization of all extreme points, along with their exposing Tsirelson inequalities and quantum models. All non-classical extreme points (exposed or not) are self-testing, i.e., realized by an essentially unique quantum model.Two principles, which are specific to the minimal scenario, allow a quick and complete overview: The first is the pushout transformation, the application of the sine function to each coordinate. This transforms the classical polytope exactly into the correlation body Q, also identifying the boundary structures. The second principle, self-duality, reveals the polar dual, i.e., the set of all Tsirelson inequalities satisfied by all quantum correlations. The convex body Q includes the classical correlations, a cross polytope, and is contained in the no-signaling body, a 4-cube. These polytopes are dual to each other, and the linear transformation realizing this duality also identifies Q with its dual.We also discuss the sets of correlations achieved with fixed Hilbert space dimension, fixed state or fixed observables.
Introduction
The quantum correlation set is a fundamental object in quantum information theory. The key point is that some correlations predicted by quantum theory cannot be modeled within classical probability, more precisely under the constraints of "local realism". This far-reaching insight can be gained from just a single example, the singlet state of two quantum bits with a particular measurement setup. The correlations predicted by quantum theory were demonstrated experimentally [2] to high precision, and recently also while carefully closing some loopholes that persisted for decades [30,57]. The non-classical nature is certified by the violation of the CHSH Bell inequality [4,13]. The singlet state also has the property of "self-testing", which means that its realization is essentially unique. Just from the correlation measurement, one can infer not only the quantum state, but also the action of the measurement devices.
In this situation it is natural to ask: Where exactly lies the boundary between the classical and the quantum? What other correlations or quantum states exhibit the same features? In a correlation setting, N parties share some quantum state, so that each party can choose from M different measurements, each of which can have K different outcomes. So what exactly is the set of correlation data that can arise from either classical probability or quantum? This is a hard problem even for fairly small N, M, K, as seen on the open problems website [48,Problems 1,26,27,32,33,34]. The singlet state has the minimal set of parameters N M K = 222. Here, characterizations have been known since Tsirelson's seminal work [74], particularly in the "zero-marginals" case 222|0, defined by the property that each outcome by itself, without considering the results of other parties, is equidistributed. This is the scenario indicated by the adjective "minimal" in our title. We provide an overview of the literature below in Sect. 1.3. Known results are scattered. The connections between different characterizations are rarely given, the overall structure of the boundary is not analyzed, and no attempts at a full geometric understanding or visualization are made. Moreover, the self-duality of the body seems to have escaped notice altogether. All this will be provided in the present article, along with self-contained proofs of all assertions.
Our paper arose from a project of T.P.L. and R.F.W. aimed at a better understanding of self-testing. T.Z. joined as a Masters student. When we realized that even the 222-case was not clear in reasonable generality, we focused on that and could win the help of C.M. and B.S. from the mathematical side. The 222|0 case was to serve as the "well-understood example". Only that it was not understood at the desired level of detail. Furthermore, most of the available techniques for 222|0 do not apply to the full-marginal 222 case, from Tsirelson's correlation matrices to the cosine parametrization and the pushout principle to the semidefinite hierarchy. Doing justice to these techniques would have been a distraction in the full 222 context, so we decided to separate it, and organize the material into this 222|0 review with a geometric flavor. As work progressed, we realized that there was more in the works of Boris Tsirelson than we had recollections of. Tsirelson was writing at a time when the relevant community was very small (and included one of us, R.F.W., who should have remembered more). Back then long proofs of exotic material were hard to publish, which may be why he often chose to state his results without proof. But he was clearly a pioneer, coming more than a decade before the surge of interest with the rise of Quantum Information. Boris Tsirelson died last year, so we felt it was fitting to dedicate this paper to him, and include the proofs he left out. We like to think that he would have enjoyed our presentation.
This article is very much a two-communities paper. We ask experts from the quantum side to bear with us when we cover standard material, just as we ask patience from geometers when we explain basic notions. Aiming for completeness on a well-researched subject means that it is largely a review. But we hope that even those experienced with quantum correlations will find new connections, just as we have.
Our presentation is organized as follows: We first set the scene with a brief introduction to quantum correlations. This is followed by a mathematical discussion which states main results in a concise form. We then also give a brief summary of previous work. Sect. 2 offers a more extended description of the correlation body Q, from basic visualization and overall properties to a detailed classification of boundary points. In Sect. 3 we focus on the dual body and its connection to Q. These geometric and algebraic features are related back to quantum issues in Sect. 4. We also discuss constrained sets in which some of the quantum data are fixed, namely Hilbert space dimension (Sect. 4.5), state (Sect. 4.6), or the observables (Sect. 4.7). In these descriptive sections we give no proofs. Proofs are collected in Sect. 5. Every statement of a proposition or theorem begins with a clickable pointer such as (→Sect. 5) to the subsection containing the proof. An exception to this rule are statements that are clear from the context, and merely summarize a narrative just given. The proof section is organized in logical order, and should be readable from beginning to end without forward references. Naturally, this order differs from the narrative in Sect. , and also from the theorems in Sect. 1.2.
Source i j A i B j x A = ±1 x B = ±1 c ij = x A x B
Background from Physics: Quantum Correlations
In a correlation experiment, several parties carry out measurements on a shared quantum system. We consider N = 2 causally disconnected parties, conventionally called Alice and Bob. Each of them chooses from M = 2 possible measurements, labeled i, j = 1, 2, with K = 2 possible outcomes, labeled ±1 (see Fig. 1). Thus there are four experiments, labeled by the pairs (i, j) of choices for Alice and Bob. The correlation c ij ∈ [−1, 1] is the probability of equal outcomes minus that of different outcomes. Equivalently, c ij is the expectation of the product of the outcomes, when these are labeled as ±1. The c ij are not sufficient to reconstruct the full statistics. That would give an 8-dimensional convex body, whose coordinates are the marginals for single outcomes, plus one correlation for every pair (i, j). This count incorporates the no-signalling condition, namely that the marginals do not depend on the setting chosen at the other site. Restricting to the 4-tuples c = (c 11 , c 12 , c 21 , c 22 ) corresponds to a projection of the 8-dimensional body. We can realize this projection geometrically by taking the equal weight mixture of the given model with one in which all outcomes are flipped to their negatives. This operation changes the sign of the marginals, but not of the correlations c ij . Therefore, we can alternatively think of the 4-dimensional body as that section of the 8-dimensional body, in which the marginals are set equal to zero. This explains why we call our scenario in the 4-dimensional c-space the zero marginals case.
We are interested in the set Q of correlations c = (c 11 , c 12 , c 21 , c 22 ) that are consistent with quantum theory. Quantum systems are described in some separable Hilbert space H over C. The source is given by a positive Hermitian operator ρ acting on H. It satisfies tr(ρ) = 1 and is called the density operator. The measurements are characterized by Hermitian operators A 1 , A 2 , B 1 , B 2 on H that satisfy the hypotheses
[A i , B j ] = 0 and − 1 ≤ A i , B j ≤ 1 for 1 ≤ i, j ≤ 2.(1)
If H = C m then A i and B j are Hermitian m×m matrices and 1 is the identity matrix. Condition (1) says that A i commutes with B j and that the eigenvalues of all these matrices are in [−1, 1]. The commutation condition represents the hypothesis that the two parties are causally disconnected, i.e., all measurements by Alice can be executed jointly with those of Bob. In contrast, the commutators [A 1 , A 2 ] and [B 1 , B 2 ] are usually nonzero, i.e., the two measurement choices of each party individually are not commensurate. The correlations are computed from the operators above by taking traces:
c ij = tr(ρA i B j ) for 1 ≤ i, j ≤ 2.(2)
The correlation body Q consists of all points c in the cube [−1, 1] 4 that admit such a representation. There is an analogous set C in classical probability theory, where the A i and B j are ±1-valued random variables, with joint probability distribution ρ. Writing angle brackets for expectations, the formula is
c ij = A i B j for 1 ≤ i, j ≤ 2.(3)
The classical set C consists of all points c in the cube [−1, 1] 4 that admit such a representation. We note that (3) is the special case of (2) when all A i and B j commute. All matrices can then be taken to be diagonal, and the diagonal entries of ρ form a probability distribution. (Analogous statements hold in infinite dimensional Hilbert spaces, where the A i , B j may have continuous spectra). Hence C ⊆ Q.
The whole point of our correlation body is that the reverse inclusion is false. A prominent example is c = 1 √ 2 (1, 1, 1, −1) ∈ Q \ C.
It is easy to build a quantum representation, and this has been realized experimentally to very high precision. So the realizability of this c is a well-confirmed experimental fact. On the other hand, c cannot be classical, because the Clauser-Horne-Shimony-Holt version of John Bell's inequality holds for all c ∈ C:
CHSH(c) :
The point c in (4) is not classical because the left hand side of (5) equals √ 2, and this exceeds 1. This is a remarkable result, the basis of an experimentum crucis ruling out a whole mode of describing Nature.
The experiments put quantum theory to a sharp test: The value √ 2 is an upper bound for all quantum correlations. If a value significantly larger than √ 2 had been found, then this would refute the quantum way of describing Nature, in just the same way as classical theories are excluded by a violation of Bell's CHSH inequality. This inequality and all linear inequalities bounding Q are called Tsirelson inequalities.
Tsirelson's bound has led to speculations about super-quantum correlations in families of theories ("generalized probabilistic theories"), and to the desire to view the quantum case in a larger context. The only constraint then would be that Alice choosing a measurement device makes no detectable difference for the probabilities of outcomes seen by Bob alone, i.e., without comparing outcomes with Alice. This is the no signalling set of correlations, denoted by N . It satisfies C ⊂ Q ⊂ N . Since in this paper we ignore marginals seen by only one partner, the remaining constraint is that c lies in the cube, so N = [−1, 1] 4 .
When N, M, K are larger than 2, 2, 2, computing optimal bounds for Q is a hard problem. One source of difficulty is that there are multiple definitions of Q, depending on how the separation of parties is required to be encoded in a tensor product structure, or just commutativity between Alice and Bob. This is known as Tsirelson's problem [21,36], and was recently resolved in [33]. However, it does not arise in the 222 case. Another source of difficulty is that no bound on the Hilbert space dimension can be assumed. Restricting the dimension to be finite, in general, gives a set that is not closed, so some limiting correlations require infinite dimension [63]. Again this subtlety does not occur in the minimal case. The minimal case considered in this paper is the only one for which a sharp characterization has been achieved. This is the principal reason for undertaking the detailed geometric study that is to follow.
A crucial property for quantum key distribution is that a maximal violation of the CHSH inequality can be achieved in an essentially unique way. Thus, by just verifying such correlations, without any knowledge about the construction of the devices, one can reconstruct ρ, A i , B j up to trivial enlargements. This property is called self-testing. It implies that any further system will be uncorrelated, so an eavesdropping third party could never learn anything about the data collected by Alice and Bob. We extend this property to all non-classical extreme points of Q in Thm. 19, and explain the cryptographic background in Sect. 4.2.
A View from Mathematics: Convex Algebraic Geometry
The theory of polytopes is a mature subject in mathematics [83]. Based on linear algebra, it leads to rich geometric objects classified by discrete combinatorial structures and a beautiful duality theory. In its guise as linear programming, i.e, optimization over a polytope with linear objective, it has found many applications. The natural next step in complexity is to go from linear algebra to nonlinear algebra [46]. If we replace convex sets and objectives described by linear inequalities with those described by polynomials, the basic geometric appeal and duality theory are still there. But we now enter into the world of algebraic geometry. Much richer ways of combining sets, partial descriptions and geometric constraints have to be considered. Examples play an important role in exploring these possibilities. Our study of quantum correlations serves as a showcase for nonlinear phenomena that occur in convex algebraic geometry [7,64].
The correlation body Q is compact and convex. Compactness is not obvious but follows from Thm. 1. For convexity, let c,c ∈ Q have realizations (2) by matrices of size m andm. Any convex combination λc + (1 − λ)c is realized by block matrices of size m +m, namely λρ ⊕ (1 − λ)ρ, A i ⊕Ã i and B j ⊕B j . By contrast, if we were to fix m then compactness is easy to see but convexity generally fails. For instance, fixing m = 1, the image under (2) is the set of 2 × 2 matrices of rank ≤ 1 with each entry in [−1, +1].
The body Q lies between two polytopes. First, Q is contained in the 4-cube N = [−1, 1] 4 , which has 16 vertices, 32 edges, 24 ridges and 8 facets. Second, Q contains the classical set C, which is also a polytope. Namely, C is the demicube, which is the convex hull of the eight even vertices of N . The 4-dimensional demicube C is combinatorially dual to the cube N . It coincides with the cross polytope [83, Example 0.4], so it has 8 vertices, 24 edges, 32 ridges and 16 facets. This census of the faces of C is of direct relevance for our description of the boundary of the convex body Q, to be given in Prop. 7.
The correlation body Q is semialgebraic: It can be described by a Boolean combination of polynomial inequalities. Here a phenomenon arises that is unfamiliar from polytope theory. It is not sufficient to use a conjunction of polynomial inequalities. In other words, Q is not a basic semialgebraic set. Moreover, while both polynomials g and h in the description below are needed, only h is determined by Q, as the unique algebraic description of a part of the boundary, while there is some freedom of choice for g.
A main source of convex semialgebraic sets is the cone of positive semidefinite matrices. Quantum theory is entirely based on this cone. Its states, observables, and channels are all defined in terms of it. The intersection of the semidefinite matrix cone with an affine-linear space is called a spectrahedron. The set Q arises from a spectrahedron by projection, and it is thus in the class of spectrahedral shadows [59].
Each of the themes described in the previous paragraphs can be used to characterize the set Q. This leads to six descriptions that look different at first glance. We summarize these in the following theorem.
Theorem 1 (→Sect. 5.1) The following six items all describe the same subset Q in R 4 :
(a) The set of quantum correlations c, as defined in Sect. 1.1, i.e., the c ij from (2) satisfying (1).
(b) The convex hull of the hypersurface (cos α, cos β, cos γ, cos δ) ∈ R 4 | α + β + γ + δ ≡ 0 mod 2π .
(c) The image of the demicube C under the homeomorphism sin :
N → N , c → sin π 2 c ij 1≤i,j≤2 . (d) The semialgebraic set c ∈ N | g(c) ≥ 0 or h(c) ≥ 0 , where g(c) = 2 − (c 2 11 + c 2 12 + c 2 21 + c 2 22 ) + 2c 11 c 12 c 21 c 22 (6) h(c) = 4(1 − c 2 11 )(1 − c 2 12 )(1 − c 2 21 )(1 − c 2 22 ) − g(c) 2 .(7)
(e) The spectrahedral shadow consisting of all points (c 11 , c 12 , c 21 , c 22 ) ∈ R 4 such that the matrix
C = 1 u c 11 c 12 u 1 c 21 c 22 c 11 c 21 1 v c 12 c 22 v 1 (8)
is positive semidefinite for some choice of u, v ∈ R.
(f ) The scalar products of pairs of unit vectors a i , b j in some Euclidean space: c ij = a i ·b j for i, j = 1, 2.
One way to describe a convex body is by the maxima of all linear functionals. This is the support function
φ(f ) = sup f · c | c ∈ Q .(9)
Here f ∈ R 4 and "·" denotes the scalar product in R 4 . The following theorem gives an explicit formula.
Theorem 2 (→Sect. 5.4) Consider the following expressions in four variables f = (f 11 , f 12 , f 21 , f 22 ):
k(f ) = (f 11 f 22 −f 12 f 21 )(f 11 f 12 −f 21 f 22 )(f 11 f 21 −f 12 f 22 )(10)p(f ) = f 11 f 12 f 21 f 22 (11) φ C (f ) = max{|f 11 +f 12 +f 21 +f 22 |, |f 11 +f 12 −f 21 −f 22 |, |f 11 −f 12 +f 21 −f 22 |, |f 11 +f 12 −f 21 −f 22 |} (12) m(f ) = min i,j |f ij | i,j |f ij | −1 .(13)
The support function of the correlation body Q equals
φ(f ) = k(f ) p(f ) if p(f ) < 0 and m(f ) > 2, φ C (f ) otherwise.(14)
The case distinction in (14) is between the "classical" case and the "quantum" case. Indeed, as the notation suggests, the piecewise linear expression φ C in (12) is the support function of the cross polytope C. Hence φ C represents inequalities for classical correlations. On the other hand, in first case of (14), the maximizers are non-classical correlations, which share all essential properties with CHSH: for fixed f the maximizer is unique (see Prop. 17) and given by a unique quantum model (see Sect. 4).
Let K be a convex body that contains the origin in its interior. Then its dual or polar is a convex body K • that represents the linear inequalities satisfied by K. In symbols, we have
K • = {f | f ·c ≤ 1, c ∈ K}.
If K is a polytope then so is K • , and the face numbers of K • are the reversal of the face numbers of K.
Consider the sequence of inclusions C ⊂ Q ⊂ N we discussed above. Then we have N • ⊂ Q • ⊂ C • . Here C and N • are cross polytopes, while N and C • are 4-cubes. However, even stronger statements are true: C is affinely isomorphic to N • , and N is affinely isomorphic to C • . Moreover, these isomorphisms extend to the middle term in the inclusion C ⊂ Q ⊂ N , i.e., the correlation body Q is self-dual:
Theorem 3 (→Sect. 5.2) There is an orthogonal transformation H on R 4 such that C • = 1 2 HN , N • = 1 2 HC and Q • = 1 2 HQ.
The proofs of the three theorems are presented in Sect. 5. While proving that they agree, we write
Q (a) , Q (b) , Q (c) , Q (d) , Q (e)
, Q (f ) for the six sets in Thm. 1. All objects and assertions are explained in detail in Sect. 2. along with lots of additional information. For instance, Prop. 7 describes the stratification of the boundary of Q into various patches. Readers might start with Fig. 2, Fig. 3
Short Review of Previous Work
The correlation body Q first came into focus in Tsirelson's work [73]. That paper gives no proofs, but some of them were supplied in [74]. This includes the characterization Q (a) = Q (f ) in the more general 2M2|0 case ([73, Thm. 1]=[74, Thm. 2.1]). Thereby the study of Q, whose definition also allows infinite dimensional Hilbert spaces, is reduced to a finite dimensional problem. A semialgebraic description for the 222|0 case is given in [74,Thm. 2.2], along with an expression for the support function [74,Thm. 2.2]. This is our Thm. 2. Tsirelson calls these results 'elementary' consequences of (f), and does not provide a proof. He also thought about issues not covered in our review, like the full 222 case, multipartite scenarios (N > 2) [74,Sect. 5], and violations of CHSH inequalities by position and momentum ( [74], see also [38]).
The spectrahedral shadow Q (e) first appeared in Landau's work [40] as a relaxation of the correlation body. That the relaxation is tight follows from Tsirelson's theorem [73]. Landau also gave a nearly semialgebraic characterization of Q (see (21) below), which only misses the semialgebraic standard form by containing a square root. He almost achieved the description (c).
The pushout (c) was found by Masanes [42], who stated that it identifies C and Q. He also considered the cosine-parametrized manifold of correlations in (b). The pushout was used implicitly in [47,76], in the form of a characterization of Q by linear inequalities applied to the inverse pushout. However, it was not pointed out that the linear inequalities just characterize C. Of course, spectrahedral shadows are used as outer approximants to Q in the semidefinite hierarchies [17,47]. This is an important technique for higher NMK, even though one gets the convex body exactly only in the minimal case.
The uniqueness of quantum models, now known as self-testing [44], was also studied by Tsirelson [74]. Independently, [66] obtained the self-testing property for the CHSH inequality. A covering of the set of extreme points ∂ e Q, which results in the cosine parametrization (b) and its analog in the N22|0-case, were found in [78], see also [43]. The exact identification of the set of extreme points was found much later in [72,76]. The minimal case is an important example in many applications such as quantum nonlocality [9,27], self-testing [68], and quantum cryptography [60,70].
Description of the Correlation Body
The convex body Q has dimension four. Our aim is to describe it in every detail. Naturally, the geometric description will strain our 3-dimensional intuition. As always, the solution is to build the geometric intuitions (German "Anschauung", visualization) on analytic notions, such as sections, projections, affine submanifolds, extreme points, and faces, which have clear definitions, but also on low-dimensional instances on which intuitions can be grounded. In the case at hand, the dimension gap is not too large, and we will see that some three-dimensional sections faithfully display important features of the fourdimensional body. We will also point out where this becomes too misleading. One general cautionary remark is that extreme points of a section usually fail to be extremal in the higher-dimensional body.
This section is divided into subsections, which are organized by geometric features, from overall properties to the classification of boundary points and their explicit description, to the dual inequalities, and finally the quantum realizations. This is different from the logical ordering in a proof. A complete set of proofs will be given only later in Sect. 5, which is accordingly organized in logical progression.
Gallery
We visualize the 4-dimensional body Q by showing 3-dimensional sections. Some of these will be at the same time projections. For example, the zero marginal case arises from the full marginal case either by ignoring the marginals (a projection) or by taking the subset with zero marginals (a section by a linear subspace). Such sections/projections often arise by averaging over a symmetry group [75].
We begin in Fig. 2 with correlations that are symmetric under exchanging Alice and Bob, i.e., c 12 = c 21 . The corresponding projection is represented by (c 11 , c 12 , c 21 , c 22 ) → (c 11 , c 22 , (c 12 + c 21 )/2). The corresponding sections of the polytopes N and C are a cube and an octahedron, respectively. The extreme points of the section of C are not all extremal in four dimensions: The point (1, 1, 0), where two parabolas meet in Fig. 2, has c 12 = c 21 = 0. It is not extremal in C. But it is the midpoint between the extreme points (1, 1, −1, 1) and (1, −1, 1, 1) of C, which lie outside the section shown. The nonlinear boundary in Fig. 2 (right) is a quartic surface, obtained by setting c 12 = c 21 in the sextic h(c) and cancelling a factor (c 11 − c 22 ) 2 . These geometric features have 4-dimensional counterparts, to be described later.
In Fig. 3 we show the sections parallel to the facets of the cube N . These are obtained by fixing the value of one coordinate, say c 11 , at a number t in the interval [−1, 1]. This family of pictures gives a full description of Q. The corresponding projections are non-informative: they are equal to the full 3-cube. The special sections c 11 = ±1 are facets of Q. This 3-dimensional shape is known as the elliptope.
Other cutting directions, which can be expected to have an interesting symmetry are sections orthogonal to the main diagonals of N . Like the vertices, they come in two kinds, either connecting two classical correlations or connecting two PR-boxes. In Fig. 4 we show, on the left, a cut very close to the hyperplane c 11 + c 12 + c 21 + c 22 = 0, which is orthogonal to the diagonal connecting the classical point (1, 1, 1, 1) and its antipode. On the right in Fig. 4, we see the cut given by c 11 + c 12 + c 21 − c 22 = 0. The polytopes: the cube N has black edges, and the octahedron C has blue edges, with two kinds of facets distinguished: CHSHfacets (yellow) defined by saturation of a CHSH-Bell inequality and "N -facets" extending to facets of N (gray). Right: The correlation body Q. Its boundary consists of a strictly convex surface of exposed extreme points, together with N -faces extending those of C, whose boundaries are outlined in red. The arrow points to the CHSH point c = (1, 1, 1, −1)/ √ 2. The 16 facets of the demicube C come in two classes. These are distinguished by how they sit inside the cube N . A facet in the first class extends to a facet of N . It is the intersection of C with a hyperplane like c 11 = 1. There are eight such facets, which we call N -facets of C. The more interesting kind is called a CHSH-facet, because it saturates a Clauser-Horne-Shimony-Holt inequality (5). There are eight such inequalities: Any odd number of minus signs can multiply the four correlations.
Enclosing and enclosed polytopes
All of this can be seen also in the 3-dimensional cut along the hyperplane c 12 = c 21 shown on the left in Fig. 2. The two polytopes are the intersection of C and N with this hyperplane. The CHSH-facets and the N -facets are marked in different colors. Other slices of C (blue frame) and N (black frame) are shown in Figures 3 and 4. The cut through the origin, orthogonal to the long diagonal connecting classical vertices (approximately as in the left panel in Fig. 4) has the intersections with N and C equal to the same octahedron. Since C ⊂ Q ⊂ N , this cut also makes our convex body Q look like a polytope.
Of course, in every cut we see the inclusions C ⊂ Q ⊂ N . More precisely, the N -facets of C extend to facets of Q, which we also call N -facets. This is typically a strict extension. The CHSH-facets are no longer faces of Q. Instead, they become the basis of a bulging part of Q, above which we find a single vertex of N . In fact, this is a feature of any body between C and N . We record this fact in the following proposition, which helps with keeping track of the parts of Q.
Proposition 4 (→Sect. 5.3.3) Every non-classical correlation c ∈ N \C violates exactly one of the eight CHSH-inequalities.
A feature which will play a major role later, and is characteristic of the minimal case, is the duality between the polytopes C and N . We saw this in Thm. 3, and it will be considered in detail in Sect. 3.
Pushout
The connection between the boundary structures of Q and C can be raised from a qualitative observation to a precise mathematical statement. There is a natural homeomorphism between the convex bodies. To this end, we define a transformation sin : N → N of the cube, which we call the pushout operation:
sin c ij = sin π 2 c ij .(15)
This is the coordinatewise application of a suitably scaled sine function. Since the sine maps the interval [−π/2, π/2] bijectively and continuously onto [−1, 1], we see that sin is bijective, continuous and has a continuous inverse. This is relevant because of the following astonishingly simple characterization of Q.
Proposition 5 (→Sect. 5.1.5) sin C = Q.
As a connection between convex bodies this is quite strange: The sine is neither convex nor concave on [−π/2, π/2], so the sin transformation applied to a convex set normally does not give a convex set. Moreover, the sin function is transcendental but both sets have an algebraic description. So why does this work? What is the general principle? The pushout property is inherited by the sections in It also connects C-sections and Q-sections in Fig. 3 when the fixed c 11 -coordinate is appropriately transformed. The closest we can come to a general principle is related to the fact that the pushout of the tetrahedron is the elliptope (see also [41]). This fact also underlines the cosine parametrization in Sect. 2.6. The threefolds of extreme points are parametrized by angles satisfying a linear constraint. A similar connection arises between two families of curves inside the 3-cube, on one hand Lissajous knots, i.e., cosine-parametrized closed rational curves, and, on the other, billiard knots, i.e., closed piecewise linear curves bouncing from the boundaries by specular reflection; see [35], [39, Fig. 3]. Another feature can be understood from the pushout characterization: At the edges of a polytope extending all the way to the boundary we get a rounded surface with continuous tangents. This is explained in Fig. 5. (15). The rectangle on the left gets mapped to the ellipse on the right. To verify this compare the scaled grid lines. The yellow semicircle shows that convexity is generally not preserved under the pushout map.
Symmetry
The symmetry group of the regular 4-cube N = [−1, 1] 4 is the hyperoctahedral group B 4 . This group has order 384 = 2 4 × 4!. All of these symmetries are given by rotations and reflections in R 4 . Each symmetry either preserves the parity of the vertices of N , or it swaps the eight even vertices and the eight odd vertices. The symmetry group G of the demicube C is an index two subgroup of B 4 . It consists of symmetries of N that preserve the parity. It follows from Prop. 5, and the fact that the pushout map commutes with coordinate permutations and sign changes, that Q has the same symmetry group G as C.
Proposition 6
The common symmetry group of Q and C has order 192. It is the semidirect product G = (Z/2Z) 3 S 4 , where S 4 is the symmetric group on four elements. The first factor swaps labels.
It is noteworthy that this group is larger than one would expect from the definition of Q. Indeed, some obvious symmetries arise from changing the conventions for describing the correlations: Which party is called Alice, which is Bob? Which outcome is +1 or −1, and which settings get the labels 1 or 2? Changing any of these conventions defines a transformation that clearly leaves each of the correlation sets invariant. The resulting group is visualized in Fig. 6, and acts on the tuples (c 11 , c 12 , c 21 , c 22 ) by sign changes (first row) and permutations (second row) giving only 64 transformations. Not all sign changes can be obtained, only even ones, and the permutations cannot break pairs of diagonally opposing pairs in the square form in which we arranged the c ij . As the additional "non-trivial" transformation needed to generate the group given in Prop. 6 A group that played a key role in analyzing the classical polytope in the N22|0 case [78] will also play a special role below for establishing self-duality. It is familiar to quantum physicists as a discrete version of phase spaces, describing a quantum or classical system in terms of "position" and "momentum" operators [24,77] and to mathematicians as the standard case of symplectic self-duality [52]. The position variables are given by a locally compact abelian group X, and the momenta by elements of the dual group X. The Hilbert space of the system is h = L 2 (X), on which X acts by transformation of the arguments, and X by multiplication. This determines a projective representation of X × X on h by the Weyl operators. The case of mechanics has X = R f ∼ = X, where f is called the number of degrees of freedom. For a single qubit, X is the two element group, and the Weyl operators are the Pauli matrices. For our problem we obtain the Klein four-group: a product of two copies of the two element group. The elements of this group are the indices of c ij , with the convention that 1 is the identity and 2·2 = 1. Thus the correlation body is a subset of h. The 16 Weyl operators are the products of the first four elements of the each of the two lines in Fig. 6, the top row giving the momentum translations and the bottom row the position translations. A standard automorphism of this structure is the Fourier transform that takes functions on X to functions on X. This is not a symmetry of Q, but, as we will see, takes Q to its dual convex body.
Boundary, faces, and extreme points
A face F of Q is a subset such that a convex decomposition of a point in F with non-zero weights must have all components in F . We use the term facet for faces of codimension 1. An extreme point of Q is a point c in Q such that {c} is a face. A face is called exposed if it is the zero set of an affine-linear function which is non-negative on Q. Since the pushout is a homeomorphism, it identifies the boundary of Q with the boundary of C. The facets of the demicube C thus provide a partition of the boundary of the correlation body Q. However, the image of a facet may now become curved, making interior points of the facet extremal in Q. This happens exactly for the CHSH-facets of C, as the following proposition shows. In the following we always list the interiors of faces because the boundaries already belong to another face. Here by "interior" we mean the relative interior, i.e., the interior in the affine span of the face.
Proposition 7 (→Sect. 5.3.4) The convex body Q is the disjoint union of the following semialgebraic sets:
(Qcx) 8 classical exposed points c ∈ ∂ e C.
(Qce) 24 exposed edges, i.e., the interiors of line segments that connect pairs of classical exposed points.
(Qnx) 32 surfaces of non-exposed extreme points.
Each is the pushout of a triangle in ∂C, which is the intersection of an N -facet and a CHSH-facet.
(Qqx) 8 threefolds of exposed extreme points. Each threefold is the curved pushout of the interior of a CHSH-facet of C, which is a tetrahedron.
(Qei) 8 elliptopes' interiors, i.e., the interiors of the facets that arise from N -facets of C, as in Fig. 2.
(Qin) The interior of Q.
The stratification of Q shown in Prop. 7 mirrors the stratification of the demicube C into relatively open faces. Indeed, the numbers seen in Prop. 7 count the faces of various dimension of C. Namely, C has 8 vertices, 24 edges, 32 two-dimensional faces etc. In short, the f-vector of C equals (8, 24, 32, 8+8, 1).
The closures of sets of type (Q) are said to be of type [Q]. Types [Qqx] and [Qei] suffice to cover the boundary. In the literature in convex algebraic geometry [12,51], boundary strata of codimension one are now called patches. Thus, our body Q has 16 patches, eight of type [Qqx] and eight of type [Qei].
The difference between the types (Qqx) and (Qei) will be important for what follows. It also relates to the two types of maximal non-trivial faces of Q. The only two types are the singletons in (Qqx), and the N -facets [Qei]. These maximal faces are also exposed. Exposedness properties will be discussed later in their natural context: duality. An intuitive geometric understanding of the non-exposedness of type (Qnx) can be gained from considering the pushout mechanism: This converts the junction between a CHSH-facet and an N -facet of C to a junction with matching tangents. The prototype for this is shown in Fig. 5. The red ellipse is tangent to the boundary of the square at the four points of intersection. For (Qnx)-points the same happens in higher dimension.
Since the (Qqx) extreme points will be discussed in more detail, let us briefly describe the [Qei] facets. These elliptopes are visualized in Fig. 7, an enlarged version of the first panel in Fig. 3. Analytically, the elliptope, say the subset of correlations c = (−1, x, y, z) ∈ Q, is described by the single inequality 1 − x 2 − y 2 − z 2 + 2xyz ≥ 0 in the cube; cf. [46, eqn (1.1)]. Since the pushout map restricts to N -facets, it is the pushout of the C-like tetrahedron that forms its skeleton. Indeed, the edges of C are marked by two of the c ij being ±1, and the other two equal up to a sign. This condition is invariant under pushout, so the edges of C are invariant as sets, but not pointwise fixed, and become exactly the [Qce]-edges of Q. How do all these pieces fit together? Once again this is readily answered by looking at the pushout identification of Q with C. Consider the bicoloring of the facets of the cross polytope C. The two colors distinguish CHSH and N types. Only tetrahedra of different types intersect in a surface, and each surface In order to visualize the boundary structure we can use a stereographic projection from the origin. Each boundary point is first mapped to the line through the origin it generates, and then to the intersection of that line with a fixed reference hyperplane, to be identified with R 3 . This defines a rational map from the boundary of our convex body onto R 3 . It is 2-to-1, because it identifies points ±c, but this hardly matters because this is a symmetry of Q and C. If the reference hyperplane supports the body, then this will produce a local image of the boundary. Stereographic projection takes line segments to line segments, so the boundary of a polytope like C induces a subdivision of R 3 into polyhedra. This is shown in the top row of Fig. 8, where the reference plane has been chosen as {c 11 = 1}. Points with c 11 = 0 are mapped to infinity, so facets containing such points are represented in two parts, connected via some infinite points, but forming a single polyhedron in projective geometry. This partitions R 3 into 8 projective tetrahedra, all of which have the same four vertices. This reminds us that the "line segment between two given points" is intrinsically ambiguous in projective geometry. The term "projective polyhedron" is therefore often used for the whole tessellation, and not just for "connected region in projective space bounded by finitely many hyperplanes" as in this paragraph.
The resulting pictures are inside-out versions of Schlegel diagrams [83,Chapter 5], which represent 4-dimensional polytopes by stereographic projection from a point just outside a reference facet. When the projection point is close enough to that facet, the whole boundary is mapped to a polyhedral subdivision of the facet, thus avoiding infinite points and the identification of antipodes. The construction easily generalizes to other convex bodies. In this generalized setting convexity still guarantees that the ray from the projection point will cut the boundary in at most two points. But it may happen that one The common edges are shown as blue lines. The surfaces separate N -facets from CHSHfacets. Their coloring for C corresponds to Fig. 2. For Q the surfaces are the boundary sets of type (Qnx). Left panels: all boundaries. Center panels: The N -face at the center, together with another N -face. The latter is displayed in two pieces, connected via the plane at infinity. Right panels: a CHSH-face, resp. a (Qqx) curved tetrahedron.
cannot arrange for one of these to be in the plane of the reference facet. Our body Q demonstrates this point. The only facets are N facets, and these are bounded by non-exposed extreme points, whose unique tangent plane is the plane of the reference facet. Therefore, no matter how close the projection point is chosen to the reference facet, the cone generated by the facet does not contain the whole body Q. So the Schlegel map does not map ∂Q into the facet, and has non-trivial double images. Therefore we opted for the projection from the center, which is shown in the second row of Fig. 8.
Since the boundaries ∂Q and ∂C are identified via pushout, their stereographic projections are topologically equivalent partitions. Stereographic projection suggests another identification of ∂C and ∂Q, namely by identifying points on the same ray through the origin. Such points are simply the same in Fig. 8, and that the stereographic projections of the boundaries are distinct shows the non-linearity of the pushout map. Indeed there are points c in a CHSH face, so that the multiple λc ∈ ∂Q lies in an N -face. Such c are readily found already in Fig. 2.
Curved tetrahedra
Prop. 7 allows us to turn an affine parametrization of a CHSH-face into a trigonometric parametrization of a curved tetrahedron (Qqx). For later purposes, we find cosines a bit more convenient than sines. where
α + β + γ + δ = 0 mod 2π and(16)∆ = sin α · sin β · sin γ · sin δ < 0.
The angle parameters can be taken as the triples (α, β, γ), with δ = −(α + β + γ). In this 3-space, the sign of ∆ marks a partition into two kinds of subsets: On the one hand, we have the curved tetrahedra with ∆ < 0 considered in Prop. 8. A prototype which contains the CHSH-point (π/4, π/4, π/4) is given by the conditions α, β, γ, α + β + γ ∈ (0, π).
On the other hand, consider the angles with ∆ > 0. Adding multiples of π to any of α, β, γ (and hence implicitly to δ) corresponds to an even sign change on the c ij , and hence a symmetry of Q. Therefore, it suffices to consider the cube (0, π) 3 . In this cube only sin δ can be negative, so ∆ > 0 means α + β + γ ∈ (π, 2π). This is an octahedron, as shown in Fig. 9. By taking cosines, these points end up in the interior of Q. Figure 9: Left: The cube (0, π) 3 containing the octahedron given by α + β + γ ∈ (π, 2π), on which ∆ > 0. The (red and black) tetrahedra built on its faces have ∆ < 0, and are hence mapped to the curved tetrahedra (Qqx) according to Prop. 8. Right: An affine transformation of the left panel making the symmetry of permuting angles and the regularity of the tetrahedra and octahedra more evident.
Finally, note that the angle parameters (α, β, γ) with ∆ = 0 correspond to further boundary elements from Prop. 7, as follows. The symmetries also help to reduce each of the classes in the following boundary version of the parametrization to a single case, which is readily checked.
Semialgebraic description
One possibility to decide quickly, for a given c ∈ R 4 , whether c ∈ Q, is to apply the inverse pushout map, and to check whether the result lies in C. This involves a transcendental function, and requires the checking of 16 linear inequalities. Here we consider an alternative, which only requires checking the positivity of two polynomials in c. In other words, we will describe our body Q as a semialgebraic set.
There is a standard method to obtain relevant polynomials from the parametrization given in Prop. 8. Namely, one represents each angle variable η by the point (cos η, sin η) on the unit circle, i.e., one introduces new variables s ij with c 2 ij + s 2 ij = 1, and expresses the constraint on the sum of angles by trigonometrically expanding. Then one eliminates the s ij -variables. Computer algebra systems, such as Mathematica or Macaulay2 [28] handle such tasks routinely and, in this instance, in no time. The result is the identity h(c) = 0 with h the sextic polynomial in (19) below. But also the condition ∆ < 0 has to be transcribed, for which we use the following polynomial g, which on (Qqx) satisfies g(c) = ∆. We wrote two formulas for h, because (18) shows that h is invariant under the full symmetry group in Sect. 2.4, and (19) clarifies that the degree of h is six and not eight, as suggested by (18). Then we have:
g(c) = 2 − (c 2 11 + c 2 12 + c 2 21 + c 2 22 ) + 2c 11 c 12 c 21 c 22 .(17)h(c) = 4(1 − c 2 11 )(1 − c 2 12 )(1 − c 2 21 )(1 − c 2 22 ) − g(c) 2 (18) = 4(c 11 c 22 − c 12 c 21 )(c 11 c 21 − c 12 c 22 )(c 11 c 12 − c 21 c 22 ) −(19)Proposition 10 (→Sect. 5.1.3) A point c in the cube N lies in Q if and only if h(c) ≥ 0 or g(c) ≥ 0.
While the polynomial h is an intrinsic feature of Q, there is some freedom in the choice of g. Indeed, knowing a small piece of the boundary suffices to determine h. This is expressed by saying that h, together with the linear polynomials 1±c ij describe the algebraic boundary of Q. At each boundary point of Q one of these polynomials vanishes. In algebraic geometry language [46, Chapter 2], the threefold {h(c) = 0} is the Zariski closure of the curved tetrahedra in (Qqx), or pieces thereof. This threefold has unbounded pieces outside the cube N , but taking its convex hull after the intersection with N gives exactly Q. This is visualized in Fig. 10 (left) which shows the surface {h(c) = 0} in the 3-space {c 12 − c 21 = 0}. The unbounded pieces arise because the algebraic elimination process works just as well in the complex domain. So the circle c 2 + s 2 = 1, as a complex variety, also contains real points with imaginary s, corresponding to angles α = ir or α = π + ir with r ∈ R, and hence cos α = ± cosh r in Prop. 8.
What is the role of the second polynomial g? This polynomial is needed to make Prop. 10 true. Fig. 10 (left) shows the zero set of h, so h is negative on the outside of the yellow surface. This extends well into the cube, where c ∈ Q, and even c ∈ C, e.g., near the origin. Hence, "c ∈ N and h(c) ≥ 0" would produce many false negatives. The disjunction with g(c) ≥ 0 captures the convex hull of the threefold inside the cube N . The surface defined by g in the hyperplane {c 12 − c 21 = 0} is shown in Fig. 10 (right). Fig. 11 is a two-dimensional representation that shows the geometry of the relevant intersections. Fig. 10 intersect. The convex hull of the orange curve represents Q. The disjunction h(c) ≥ 0 or g(c) ≥ 0 describes Q. Note that Q is not a basic semialgebraic set, i.e., not a conjunction of polynomial inequalities.
We remark that more compact forms than the characterization by two polynomials can be given, if we allow the use of absolute values or roots. Such conditions can be converted to polynomial expressions. However, they typically generate a case distinction, hence some overhead in the logical part of a semialgebraic description. For example, by taking a root in (18) and combining with Prop. 10, we can see that c ∈ Q is equivalent to c ∈ N together with the single inequality
g(c) ≥ −2 (1 − c 2 11 )(1 − c 2 12 )(1 − c 2 21 )(1 − c 2 22 ).(20)
The following reformulation, due to Landau [40], is not invariant under the full symmetry group of Q:
(1 − c 2 11 )(1 − c 2 12 ) + (1 − c 2 21 )(1 − c 2 22 ) ≥ |c 11 c 12 − c 21 c 22 |.(21)
Note that these inequalities must be combined with the hypothesis c ∈ N . This excludes unbounded solutions with a product of two negative factors under the square root.
Spectrahedral shadow
For any quantum correlations, given by a density operator ρ and observables A 1 , A 2 , B 1 , B 2 , and complex numbers ξ 1 , . . . , ξ 4 , we consider the operator
X = ξ 1 A 1 + ξ 2 A 2 + ξ 3 B 1 + ξ 4 B 2 . Since X * X is positive semidefinite, we conclude that tr(ρX * X) = 4 ν,µ=1 ξ ν C νµ ξ µ ≥ 0, where we introduced the 4 × 4 matrix C = (C νµ ) = d 1 u c 11 c 12 u d 2 c 21 c 22 c 11 c 21 d 3 v c 12 c 22 v d 4 .(22)
The entries other than c ij are
u = tr(ρA 1 A 2 ), v = tr(ρB 1 B 2 ), and d 1 = tr(ρA 2 1 ) ≥ 0, etc.(23)
The positivity stated above is C ≥ 0, our notation for C being positive semidefinite. The existence of u, v and d i with d 2 i ≤ 1 making C ≥ 0 is thus a necessary condition for c ∈ Q. This is the bottom level of the semidefinite hierarchy [47] for quantum correlations. In the case at hand, the necessary condition is also sufficient. We can further assume that u and v are real and that the diagonal entries are all 1.
Proposition 11 (→Sect. 5.1.1) A point c lies in the convex body Q if and only if there exist numbers u, v ∈ R such that C ≥ 0 in (22) with d 1 = d 2 = d 3 = d 4 = 1.
Hence, Q is characterized by a semidefinite matrix completion problem. This is essentially the completion problem for the 4-cycle, as discussed in [46,Example 12.16]. Our boundary polynomial h(c) is obtained from the degree eight polynomial given there by setting the diagonal entries to be 1. The matrix inequality C ≥ 0 defines a spectrahedron in R 6 . Deleting the matrix entries u and v specifies a projection R 6 → R 4 . The correlation body Q is the image of the spectrahedron {C ≥ 0} under this projection. Thus, Prop. 11 furnishes an explicit realization of Q as a spectrahedral shadow [59].
The fiber over any interior point under our projection {C ≥ 0} → Q is a spectrahedron in the (u, v)plane. Fig. 12 shows the union of these 2-dimensional fibers over a line that cuts through Q. The fiber over any boundary point is a single point. In other words, the matrix completion problem has a unique solution C for c ∈ ∂Q. We record the ranks of these matrices C for the various families in Prop. 7. (2) The resulting unique matrix C has rank 1 if c is of type (Qcx), it has rank 2 for types (Qce), (Qnx) and (Qqx), and it has rank 3 for type (Qei).
(3) The point c is in the interior (Qin) if and only if there is some completion with rank C = 4.
We close with one remark regarding item (3). It is not true that all the completions C of a given interior point c ∈ (Qin) have rank 4. For instance, the midpoint c = 0 clearly allows C = 1, but also the extension with u = v = 1, which is the direct sum of two rank 1 operators, and hence has rank 2.
Volume
The volume of Q is a fundamental geometric invariant. It could be understood as the probability of quantum correlations in an ensemble, for which c ∈ N is distributed according to Lebesgue measure. This ensemble of generalized probabilistic theories has no operational meaning, so the volume has no direct physical relevance. However, the probabilistic interpretation suggests a way to compute it stochastically: The semialgebraic description gives us a fast way to decide whether c ∈ Q, for points c in a random ensemble with each c ij independent and equidistributed in [−1, 1]. A run of 10 6 samples led to
V (Q) V (N ) ≈ 0.925898.(24)
On account of √ N -fluctuations, this can be expected to be accurate to within three digits. On the other hand, we can compute the volume exactly, by integrating the pushout over C. Since the pushout acts coordinatewise, the Jacobi matrix is diagonal and the functional determinant is readily determined. The resulting trigonometric integrals can be solved, giving the overall result
V (Q) V (N ) = 3π 2 2 · 1 16 ≈ 0.9252754126.(25)
For the surface area, we get the volume of the N -faces as the well-known elliptope volume π 2 /2 [34]. For the volume of the curved tetrahedra we did not find a closed expression. Testing the numerical value for being a simple fraction times a low power of π by continued fraction expansion also did not seem to give a simple expression. The (Qnx) boundaries are directly expressed by the surface area of the elliptope. This area is known to be 5π, by a direct calculation found on math.stackexchange.com.
Description of the Dual Body
The polar K • = {f | f ·c ≤ 1, c ∈ K} of a convex body K provides the description of K by the affine inequalities it satisfies. Since K •• = K, by [58, Thm. IV.1.5], this is a symmetric relation. We could ask all the questions we studied so far about the three bodies C ⊂ Q ⊂ N also about their polars. For the polars the inclusion is reversed, so N • ⊂ Q • ⊂ C • . The big surprise, which is a rather special feature of the minimal case, is that this dual chain of inclusions is essentially the same as the original chain. We will spell this out in detail. For now it just means the good news that much of the work is already done.
The duality transform
We first noticed the duality in the polytopes C ⊂ N from the observation that their face counts by dimension (the f-vectors) are reversals of each other. Strengthening this to an isomorphism C ∼ = N • goes as follows. We start from the inequalities describing N . These come from the 8 vertices of N • , which are
± (1, 0, 0, 0), ±(0, 1, 0, 0), ±(0, 0, 1, 0), ±(0, 0, 0, 1).(26)
These have to be identified with the vertices of C, i.e., the 8 even vertices of N itself. They are
± (1, 1, 1, 1), ±(1, −1, 1, −1), ±(1, 1, −1, −1), ±(1, −1, −1, 1).(27)
The following transformation H maps the points in (26) to those in (27).
So we get C = 2HN • with H = 1 2 1 1 1 1 1 −1 1 −1 1 1 −1 −1 1 −1 −1 1 .(28)
Here we included the factor 1/2 so that H 2 = 1. Since H * = H, this matrix is then also unitary, i.e., a Hadamard matrix [69] (note different conventions for the normalization of such matrices though). This makes it easy to write down the consequences of C = 2HN • from dualization or multiplication with H:
N • = 1 2 HC and C • = 1 2 HN .(29)
To visualize this duality, consider the left panel of Fig. 13. Since the section with the hyperplane {c 12 = c 21 } is also a projection, and duality swaps projection and intersection operations, this 3-dimensional picture faithfully represents the 4-dimensional polarity relations. The outer blue frame represents C • , the dual of the inner frame in Fig. 2 (Left). It is a cube whose vertices correspond to the facets of C. They can thus correspond either to a CHSH-facet (marked in yellow) or to a N -facet (marked in black). Overall, the figure looks like a rotation of its counterpart in Fig. 2. In 4 dimensions the required H is exactly an orthogonal reflection. In the 3-dimensional section it is still a submatrix of a rotation, and looks like a rotation because human 3D perception is highly capable of ignoring uneven affine stretching.
The right panel in Fig. 13 shows the pertinent section of the dual body Q • . We can draw it using the parametrized points f from Prop. 14. This mimics the right panel of Fig. 2, by the same transformation as the one used for the polytopes. Indeed, this is true exactly, also for the full 4-dimensional body: This is Thm. 3, but with the matrix H spelled out. The linear transformation H is far from unique. Indeed, if S 1 , S 2 are matrices representing any of the symmetries from Sect. 2.4, then so are their transposes. Hence H = S 1 HS 2 , incidentally always again a Hadamard matrix, also maps the inclusion chain to its dual. Note that S → S = HSH −1 is an automorphism of the symmetry group which changes the semidirect product decomposition, so multiplication by an even number of signs can become a permutation and conversely. In fact, H is just the Fourier transform in the discrete phase space representation mentioned in Sect. 2.4, and in this picture it is the swap between position shifts and momentum shifts.
Proposition 13 (→Sect. 5.2) Q • = 1 2 HQ.
In the terminology of axiomatic quantum mechanics, this form of self-duality of a convex set is called "weak self-duality" [3], as opposed to stronger forms with a canonical isomorphism between the set and its dual, that is characteristic for Jordan algebra state spaces and in particular the quantum state space. Can one strengthen Prop. 13 in that direction? Indeed, we already have a canonical mapping taking each point c of a (Qqx) patch to the unique maximizer f , via Prop. 8 and Prop. 14. By multiplying H with a symmetry of Q we can achieve that it takes the patch of c to the patch of f , but can we do it for all patches simultaneously? The answer is no: A map with that property would have to commute with all symmetries. Since our representation of the group G on R 4 is irreducible, we would conclude that H is a multiple of the identity. Another context of interest are normed spaces: Since −Q = Q, our convex body Q is the unit ball of a norm in R 4 and the dual normed space has unit ball Q • . So we have an example of a normed space that is isomorphic to its dual, a subject studied more generally, e.g., in [81].
After the completion of this work, we became aware of the T. Fritz's results which extend the duality of C and N to many parties [20]. There it was remarked that the self-duality of Q was also known to C. Palazuelos and I. Villanueva in a private communication.
Parametrized extreme points of Q •
The boundary patches (Qqx) of exposed extreme points have a unique maximizing functional f ∈ Q • . The parametrization by angles can be taken over directly.
Proposition 14 (→Sect. 5.3.2) Let α, β, γ, δ and c satisfy the conditions of Prop. 8. Define
f = (f 11 , f 12 , f 21 , f 22 ) = 1 K 1 sin α , 1 sin β , 1 sin γ , 1 sin δ ,(30)
where K = cot α + cot β + cot γ + cot δ. Then f · c ≤ 1 for all c ∈ Q with equality if and only if c = c.
Moreover, the vector f is uniquely determined by this property.
Applying the duality transform to the point f in (30) gives again a point 2Hf ∈ Q of type (Qqx), which in turn can be parametrized by angles. The resulting map Φ from one tetrahedron of angles to another expresses the duality of boundary points.
For a concrete computation let T be the tetrahedron defined by α, β, γ, −δ = α + β + γ ∈ (0, π). Write c(θ) and f (θ) for the images in Prop. 8 and Prop. 14. We obtain a self-map Φ :
T → T with the property 2H f (θ) = c(Φ(θ)).(31)
By definition, this map will commute with the permutations of vertices of T , which extend to symmetries of Q. By self-duality it is also its own inverse. This map is visualized in Fig. 14.
By symmetry, these properties uniquely fix the map also for other (Qqx)-patches. When doing this concretely, one has to observe that whereas the association of f with the unique maximizer c is a property of Q, the concrete parametrization of the tetrahedra involves a convention, and depends on the choice of self-duality operator H. Therefore in solving (31) for Φ(θ), on any of the tetrahedra in Fig. 9, one has to carefully pick the branches of the arccos function. Figure 14: The map Φ from the tetrahedron T to itself: The triangular meshes on the left are mapped to the surfaces on the right. The figure on the right has been truncated so that the image surfaces can be seen. Since Φ commutes with vertex permutations, analogous surfaces can be drawn between any tetrahedron face and the opposing vertex. Since Φ 2 is the identity, the diagrams can also be read from right to left.
The map Φ is continuous and even analytic on the open tetrahedron. But it does not have a continuous extension to the closure of the tetrahedron. Indeed, a glance at (30) shows that when one of the angles in θ goes to zero, and the others to values not in {0, π}, the image Φ(θ) approaches the opposite vertex.
Hence the whole open part of the bottom face on the left goes to a single point. This is evident from Fig. 14 in the form that most of the triangles in the evenly spaced triangulation close to the base triangle end up close to the top vertex. Similarly, when θ approaches a point on the edge, the limit of Φ(θ) depends on the direction from which the edge is approached.
Dualized descriptions
We can now apply the duality transform to each of the previous subsections to get analogues for Q • of all the statements made about Q. There is no simple analogue of the pushout. The cosine parametrization of the curved tetrahedra was already given an analogue in Prop. 14 (not via duality transform).
Consider the polynomials (17) and (19) of the semialgebraic description. Since f = (f 11 , f 12 , f 21 , f 22 ) ∈ Q • is equivalent to 2Hf ∈ Q, we need to consider these polynomials after substituting c → 2Hf . Note that such a substitution takes polynomials which are invariant under the symmetry group to polynomials with the same property. Moreover, by linearity of the substitution, homogeneous polynomials of some degree go to homogeneous polynomials of the same degree. This constrains the number of polynomials we need to consider. The invariant polynomials of degree two are proportional to |c| 2 = c 2 11 + c 2 12 + c 2 21 + c 2 22 , and since H is orthogonal, this goes to itself under substitution. Among the quartics we need to consider the product p(c) = c 11 c 12 c 21 c 22 that already appeared in Thm. 2. Under substitution it becomes
q(f ) = p(2Hf ) = (f 11 +f 12 +f 21 +f 22 )(f 11 −f 12 +f 21 −f 22 )(f 11 +f 12 −f 21 −f 22 )(f 11 −f 12 −f 21 +f 22 ). (32)
This is the quartic part in (19). The sextic part of h is the polynomial k from Thm. 2. This is self-dual in the sense that k(2Hf ) = 64k(f ). With these building blocks, the polynomials describing Q and Q • are
h(c) = 4k(c) − q(c), h • (f ) := 1 256 h(2Hf ) = k(f ) − p(f ), g(c) = 2 − |c| 2 + 2p(c), g • (f ) := 1 2 g(2Hf ) = 1 − 2|f | 2 + q(f ).(33)
We conclude that f ∈ Q • if and only if 2Hf
∈ N and ( g • (f ) ≥ 0 or h • (f ) ≥ 0 ).
Describing Q • as a spectrahedral shadow is done in general by the dual semidefinite hierarchy [17]. Indeed, the dual of a spectrahedral shadow is again of this form [7,Remark 5.43]. This holds because the operations of linear section and projection change roles under dualization, and the semidefinite cone is anyhow self-dual. In terms of matrix completion problems, where the primal matrix has unspecified entries (a subspace constraint), the dual matrix has zeros (resulting from the dual projection). In the case at hand, the unspecified entries u, v in C become zeros in the dual matrix, which we call F . The condition that the diagonal entries of C are 1 dualizes to constraint on the trace of F . This gives:
Proposition 15 (→Sect. 5.2) A point f = (f 11 , f 12 , f 21 , f 22 ) lies in the dual body Q • if and only if there exist positive real numbers p 1 , p 2 , p 3 , p 4 with 4 i=1 p i = 2 such that F = p 1 0 −f 11 −f 12 0 p 2 −f 21 −f 22 −f 11 −f 12 p 3 0 −f 21 −f 22 0 p 4 ≥ 0.(34)
Moreover, there exists a completion satisfying the above constraints, and additionally p 1 +p 2 = p 3 +p 4 = 1. This condition is automatically satisfied for all boundary points.
Indeed, one checks that tr CF = 2−2f ·c holds for C from (22), so tr CF ≥ 0 is equivalent to f ·c ≤ 1.
The normal cycle
The definition of the polar suggests to look at the incidence relation between points c ∈ Q and f ∈ Q • for which the inequality c · f ≤ 1 is tight. The resulting set is called the normal cycle [12,51]. Given any convex body K in R d , which contains the origin in its interior, the normal cycle is defined as
N(K) = (c, f ) ∈ ∂K × ∂K • | c · f = 1 .(35)
This integrates K and K • into a single structure, but it is useful far beyond this role. Importantly for our project, it gives a unified description of curved manifolds and polyhedral kinks, both of which are features of the correlation body Q. It appears that this structure is not so well known in the quantum community. Therefore we begin with a brief description for an arbitrary K before specializing to Q. Fixing c to range over a subset A ⊂ ∂K leaves f to range over a closed face of K • . We denote by
A ⊥ = {f ∈ K • | ∀c ∈ A : c·f = 1 } the set incident to A.
We use the orthogonality symbol, and call A ⊥ the face orthogonal to A, although the relation is not given by the vanishing of a scalar product, unless we work in projective geometry, with homogenous coordinates and cones instead of convex sets (see [51]). The relation behaves in many ways like an orthogonality relation:
Since A ⊂ B implies B ⊥ ⊂ A ⊥ , and A ⊂ B ⊥ ⇔ B ⊂ A ⊥ , the map A → (A ⊥ ) ⊥ is a closure operation, taking A to the smallest face of the form B ⊥ containing it. This implies A ⊥⊥⊥ = A ⊥ . An exposed face is by definition a set of the form {x} ⊥ .
Let us look at the two extreme cases of convex sets: First, we have strictly convex sets with smooth boundary whose only faces are exposed singletons {c}. Here, N(K) is the graph of a homeomorphic identification of ∂K with ∂K • . At each point we can introduce coordinates so that ∂K is the graph of a convex function. To second order it is given near c by a quadratic form, containing the curvature information. Dually, ∂K • is given near f by the Legendre transform, hence the inverse quadratic form.
Physicists are familiar with this reciprocity from the kinetic energy forms of Lagrange and Hamilton mechanics. In the limit, where curvature goes to zero in some directions, the dual curvature form diverges, corresponding to kinks of the dual manifold. Carrying this further, one gets to a situation where all curvature is concentrated on lower dimensional manifolds. This happens for polytopes.
For a polytope K, the lattice of closed faces is neatly ordered by dimension, and A → A ⊥ is an order reversing bijection of face lattices. Although the normal cycle is far from being a convex set, given that
c · f = 1 is nonlinear, for every k-dimensional face F , the set F × F ⊥ ⊂ N(K) is a polytope of dimension k + (d − k − 1) = d − 1.
Consider the 4-cube N and the demicube C = N • . Their common normal cycle N(N ) = N(C) consists of 80 strata, one for each of the 8 + 24 + 32 + 16 pairs (F, F ⊥ ), where F ⊂ ∂C and F ⊥ ⊂ ∂N are faces. For instance, if dim(F ) = 2 then dim(F ⊥ ) = 1 and F × F ⊥ is a triangular prism.
Let us clarify the word strata. Consider an arbitrary semialgebraic set S. As a consequence of the Cylindrical Algebraic Decomposition (see, e.g., [8, Theorem 2.3.6]), the set S can be decomposed as a finite disjoint union of strata
S = ∪ N i=1 C i with dim C i = d i . Each stratum C i is semialgebraically homeomorphic to (0, 1) di . Moreover, the closure of C i in S is given by the union of C i with some other C j such that d j < d i .
Note that the decomposition of S into these strata is far from unique.
The power of the normal cycle lies in the uniform treatment covering the whole range of convex bodies from polytopes to smooth. Indeed, for any compact convex d-dimensional set K containing the origin in its interior, N(K) is always a (d − 1)-dimensional (Lipschitz Legendrian) submanifold of R 2d [22]. Moreover, the map K → N(K) is continuous in the Hausdorff metric for sets. This makes the normal cycle a remarkably stable structure under approximations either by smooth manifolds or by polytopes.
The normal cycle plays the role of the normal bundle for more general geometric objects. It was defined by Federer [18] for sets of positive reach. These include convex bodies but also much crazier sets. It is an important tool from geometric measure theory, used for defining curvature measures [80,82]. This is related to the classic result of Steiner, who noted that the volume of the smooth approximation to a body, which one gets as the union of all balls of small radius r centered on the body, is a polynomial in r, whose coefficients relate to curvature of different dimensionality. On the other hand, stable polyhedral approximations are needed in visualization, computer graphics and computational anatomy (see [14,56,65]). Recently, the normal cycle has also emerged as a key player in convex algebraic geometry [12,51].
If a convex body K ⊂ R d is semialgebraic, then its normal cycle N(K) is semialgebraic as well, by [51,Theorem 1.7]. Moreover, N(K) admits a finite semialgebraic stratification whose top-dimensional strata are (d − 1)-dimensional. To recognize the nonlinear nature of the normal cycle, we also introduce the algebraic normal cycle N(K). This is, by definition, the Zariski closure of the normal cycle. It is a (d − 1)dimensional subvariety of the complex space C 2d , modeled by conormal varieties as in [51,Section 2]. The radical ideal of the algebraic normal cycle is the intersection of the prime ideals associated to the various strata. If K is a polytope then each such prime ideal is generated by the linear polynomials in c that vanish on F , the linear polynomials in f that vanish on F ⊥ , plus one bilinear polynomial c · f − 1.
Our goal is to describe the normal cycle N(Q) of the correlation body Q. We describe a stratification of N(Q) that mirrors the stratification of Q in Prop. 7. The symbol (Qnn,mm) will represent the set of points (c, f ) in N(Q) such that c ∈(Qnn) and f ∈(Qmm), for the families from Prop. 7.
Proposition 16
The normal cycle N(Q) is divided into 24 = 8 + 8 + 8 strata of dimension 3 whose types are (Qqx,qx), (Qei,cx), (Qcx,ei). These threefolds are separated by 88 = 32 + 32 + 24 surfaces (Qnx,cx), (Qcx,nx), (Qce,ce). The eight strata of type (Qqx,qx) belong to the same irreducible component of the algebraic normal cycle N(Q), so the radical ideal of N(Q) is the intersection of 17 = 1+8+8 prime ideals.
All these features are made more explicit in Table 1: Classification of boundary points c ∈ ∂Q, in the notation of Prop. 7, and what they correspond to in the normal cycle N(Q). The first three columns give the full-dimensional strata of N(Q). The last two columns are two of the three strata of dimension 2. The fact that the stratum (Qcx,nx) does not appear here reflects the non-exposed nature of (Qnx).
The extreme points (Qqx) and their duals are both zero dimensional, hence maximally violate the typical counts for dual faces of polytopes. This is made up for by having a manifold of such points, so the local dimension of N(Q) is still d − 1 = 3. It is also clear, from the pushout transformation, that a tradeoff between manifold dimension and the dimensions of individual face pairs must be possible.
We now briefly discuss the strata (Qce) and (Qnx) in the fourth and fifth column of Table 1. These are surfaces in the threefold N(Q). Points c on the surfaces (Qnx) are supported only by the linear functional that supports the entire elliptope, which is {c} ⊥⊥ . Hence there are 32 surfaces (Qnx,cx) in N(Q). Dually, we also get 32 surfaces of type (Qcx,nx). These do not appear in Table 1 because the faces of type (Qnx) are not exposed and thus cannot be realized as {f } ⊥ . This feature is highlighted by the jump in the dimension of {c} ⊥⊥ for c ∈(Qnx). All 64 surfaces arise as the intersection of the closure of (Qqx,qx) with the closure of a stratum (Qei,cx) or (Qcx,ei). Each intersection produces four such surfaces.
The exposed edges (Qce) of Q are supported by one-dimensional families of normal directions. These are the exposed edges (Qce) on Q • . We therefore have 24 squares of type (Qce,ce) in N(Q). Each square separates two 3-dimensional strata of type (Qei,cx) and (Qcx,ei). In addition to the 64 curved triangles of type (Qnx,cx) or (Qcx,nx), this accounts for all 2-dimensional cells in our stratification of N(Q).
The 3-dimensional strata of N(Q) give the irreducible components of the algebraic normal cycle N(Q). We begin with the most nonlinear stratum, denoted (Qqx,qx). This stratum is characterized by Prop. 8 and Prop. 14. It is parametrized by angles α, β, γ, δ that add up to 0 modulo 2π, and it consists of pairs (c, f ) where c satisfies (16) ,
h • = f 11 f 12 f 21 f 22 − (f 11 f 22 −f 12 f 21 )(f 11 f 21 −f 12 f 22 )(f 11 f 12 −f 21 f 22 ).
The remaining 14 generators of our prime ideal are the following polynomials:
c 2 11 f 2 11 − c 2 22 f 2 22 − f 2 11 + f 2 22 , c21f11f12f21 + c22f11f12f22 + c11f11f21f22 + c12f12f21f22, c 2 11 f11f12 − c 2 21 f21f22 − c12c21f12f22 + c11c22f12f22 − f11f12 + f21f22, c 2 11 f11f21 − c 2 12 f12f22 − c12c21f21f22 + c11c22f21f22 − f11f21 + f12f22, c 2 12 f11f12 − c 2 21 f21f22 − c11c21f11f22 + c12c22f11f22 − f11f12 + f21f22, c 2 12 f12f21 − c 2 11 f11f22 − c11c21f21f22 + c12c22f21f22 − f12f21 + f11f22, c 2 21 f11f21 − c 2 12 f12f22 − c11c12f11f22 + c21c22f11f22 − f11f21 + f12f22, c 2 21 f12f21 − c 2 11 f11f22 − c11c12f12f22 + c21c22f12f22 − f12f21 + f11f22, (c11c 2 12 + c11c 2 21 + c11c 2 22 − 2c12c21c22)f11 + c 3 12 f12 + c 3 21 f21 + c 3 22 f22 − 1, c11c12f12f21 − c21c22f12f21 + c12c21f21f22 − c11c22f21f22 + c 2 12 f12f22 − c 2 22 f12f22, c12c21f11f21 − c11c22f11f21 + c11c21f11f22 − c12c22f11f22 + c 2 21 f21f22 − c 2 22 f21f22, c12c21f11f12 − c11c22f11f12 + c11c12f11f22 − c21c22f11f22 + c 2 12 f12f22 − c 2 22 f12f22, (c 3 12 −c 2 11 c12−c12c 2 21 −c12c 2 22 +2c11c21c22)f12 − (c 3 22 −c 2 11 c22−c 2 12 c22−c 2 21 c22+2c11c12c21)f22, (c 3 21 −c 2 11 c21−c 2 12 c21−c21c 2 22 +2c11c12c22)f21 − (c 3 22 −c 2 11 c22−c 2 12 c22−c 2 21 c22+2c11c12c21)f22.
This list was generated by computer algebra as follows. We start from a list of polynomials that cuts out N(Q) as a subset of C 8 . That list consists of , h, h • and the twelve 2 × 2 minors of the two matrices c 11 c 12 c 21 c 22 ∂h • /∂f 11 ∂h • /∂f 11 ∂h • /∂f 21 ∂h • /∂f 21 and f 11 f 12 f 21 f 22 ∂h/∂c 11 ∂h/∂c 11 ∂h/∂c 21 ∂h/∂c 21 .
The respective rows are linearly dependent for any pair (c, f ) of supporting linear functions. The resulting ideal has the desired prime ideal as its radical, by Hilbert's Nullstellensatz. We computed that radical. We now describe the other components of the variety N(Q). The eight strata (Qei,cx) consist of points (c, f ) where one of the entries of f is ±1 and the others are 0. This linear functional f exposes one of the elliptopes in ∂Q. For example, consider f = (1, 0, 0, 0). The prime ideal of this component is
f 11 − 1, f 12 , f 21 , f 22 , c 11 − 1 .
The corresponding stratum in the semialgebraic set N(Q) satisfies the additional cubic inequality
c 2 12 + c 2 21 + c 2 22 − 2c 12 c 21 c 22 ≤ 1.(36)
The boundary of the elliptope, seen inside N(Q), separates (Qei,cx) from the nonlinear stratum (Qqx,qx). By duality, there are also eight strata (Qcx,ei) in the normal cycle N(Q). Now, the elliptopes appear in the f -coordinates and c is one of the 8 classical extreme points (Qcx). These are obtained by exchanging the roles of f and c, using the linear transformation H. The ideal of one of the components (Qcx,ei) is
c 11 − 1, c 12 − 1, c 21 − 1, c 22 − 1, f 11 + f 12 + f 21 + f 22 − 1 .
The semialgebraic description of this stratum is obtained by setting c = 2Hf in the cubic inequality (36).
Support function and gauge function
Two real valued functions are commonly used to describe a convex set K. The first is the gauge function γ K of the set [58]. It measures how far we have to go along a ray in some direction until we leave the set. The best known example is a norm as determined from its unit ball. The gauge function γ K is the reciprocal of another well-known function in convex geometry: the radial function of the body.
The other standard measure is the maximum of a given linear functional over the given set. This is called the support function φ K . In short, for any convex body K with 0 in its interior, we consider:
γ K (x) = inf {λ ≥ 0 | x ∈ λK},(37)φ K (f ) = sup {f · x | x ∈ K}.(38)
Both functions are homogeneous of degree 1. From (37) we recover K = {x | γ K (x) ≤ 1}, and dually
K • = {f | φ K (f ) ≤ 1}.
Hence, the support function of K is the gauge functional of K • . By self-duality, the two functions are closely related for K = Q. In the sequel we study φ = φ Q , and we drop the subscript. Since the set Q has two markedly different kinds of boundary points, the support function requires a binary distinction concerning a functional f : Will max c∈Q f · c be attained at a classical point or at an exposed extreme point? The answer is the same for all multiples of f , including −f , so we are really asking about the projective geometry of the boundary points of Q • , which is visualized for Q in Fig. 8. The separating surfaces are the images of the [Qnx] surfaces. Note that (after the identification of f with −f ) there are four such surfaces. We can therefore not expect a simple algebraic relation to mark the distinction: Each of the elliptope boundaries has a Zariski completion with multiple additional irrelevant components overlapping the division Fig. 8, so some careful sifting of inequalities is needed.
Once we know the nature of the maximizer, however, the computation of φ is straightforward. In the first case, it is the maximum of an affine functional over the 8 vertices of C. Namely,
φ(f ) = φ C (f ) with φ C (f ) = max{f · c | c ∈ ∂ e C} = max{|f 11 +f 12 +f 21 +f 22 |, |f 11 +f 12 −f 21 −f 22 |, |f 11 −f 12 +f 21 −f 22 |, |f 11 +f 12 −f 21 −f 22 |} = 2Hf ∞ ,(39)
In the second line, we grouped the maxima over a pair of antipodal classical extreme points by writing an absolute value. In the third line, · ∞ denotes the maximum norm on R 4 . This case applies exactly when the ray R f intersects the boundary of Q • in an N -facet.
The other possibility is that the ray intersects the boundary in a curved tetrahedron. In this case we think of φ as the gauge function of Q • . We need to determine the intersection point of the ray {λf } with ∂Q • . This point is in the zero set of h • . The equation h • (λf ) = 0 in (33) is readily solved for λ, using the splitting of h • into homogeneous parts. We get
λ 6 k(f ) − λ 4 p(f ) = 0, so in this case λ = φ(f ) is φ(f ) = k(f ) p(f ) .(40)
This function is homogeneous of degree 1, as required. The above heuristics is made precise in the following proposition, which extends Thm. 2:
Proposition 17 (→Sect. 5.4) For any f ∈ R 4 \{0}, the following conditions are equivalent:
(1) The ray {λf } intersects the boundary of Q • in a point of type (Qqx).
(2) Some point c ∈ Q maximizing f · c is of type (Qqx).
(3) We have p(f ) < 0 and
m(f ) = min i,j=1,2 |f ij | i,j=1,2 |f ij | −1 > 2.(41)
(4) We have p(f ) < 0 and
m(f ) = 1 f11 + 1 f12 + 1 f21 + 1 f22 1 f11 + 1 f12 − 1 f21 − 1 f22 1 f11 − 1 f12 + 1 f21 − 1 f22 1 f11 − 1 f12 − 1 f21 + 1 f22 < 0.
In this case the support function of Q is φ(f ) = φ(f ) from (40), otherwise φ(f ) = φ C (f ) from (39).
In the domain described by Prop. 17, the maximizer of φ(f ) is a unique exposed point c * ∈ Q. An explicit formula f → c * would correspond to solving for c given f in the system of 17 polynomials defining the stratum (Qqx,qx) in N(Q). In terms of the angle parametrization of boundary pieces, it is essentially the self-dual counterpart of the map Φ sketched in Fig. 14.
Quantum Connections
We now return to Sect. 1.1, and we discuss what our geometric findings on Q mean for quantum theory. We begin by exhibiting explicit quantum models for the extremal points c of type (Qqx). In Sect. 4.2 we examine what geometric features of some point c ∈ Q make it suitable for quantum key distribution. Actually, the extremal correlations have much stronger uniqueness properties, known as self-testing. We state these in Sect. 4.3. This is followed by a short reflection on the historical role of the elliptope case.
Quantum models
We introduced the convex body Q as the set of quantum correlations. A necessary condition implied by this was used at the beginning of Sect. 2.8. For showing sufficiency of any condition, it is necessary to construct explicit quantum models. In this section we write out a parametrized family of quantum models, that realizes all points described in Prop. 8. That such a simple family already exhausts the extreme points of Q is, of course, a special feature of the minimal 222 case. In a sense which we will describe subsequently, and in terms familiar to the quantum community, this family is actually the simplest one containing non-classical correlations. This will also facilitate computing c for each model.
Using the notation introduced in Sect. 1.1, we set m = 4 and H = C 4 . Algebraically, a quantum state is a positive semidefinite 4 × 4 matrix ρ with trace 1. For the density operator, we fix it as the rank one matrix with entries ρ αβ = Ψ α Ψ β , defined by the unit vector
Ψ = 0 , 1 √ 2 , − 1 √ 2 , 0 T .(43)
The measurements of the two parties are represented by the following real symmetric 4 × 4 matrices:
A 1 = cos(α) 0 sin(α) 0 0 cos(α) 0 sin(α) sin(α) 0 − cos(α) 0 0 sin(α) 0 − cos(α) , A 2 = cos(γ) 0 − sin(γ) 0 0 cos(γ) 0 − sin(γ) − sin(γ) 0 − cos(γ) 0 0 − sin(γ) 0 − cos(γ) , B 1 = −1 0 0 0 0 1 0 0 0 0 −1 0 0 0 0 1 , B 2 = − cos(α + β) − sin(α + β) 0 0 − sin(α + β) cos(α + β) 0 0 0 0 − cos(α + β) − sin(α + β) 0 0 − sin(α + β) cos(α + β) .
These matrices satisfy the hypotheses stated in (1). We now compute the four correlations c ij in (2).
Lemma 18
The point c with parameters (α, β, γ, δ) in Prop. 8, even without the inequality constraint on ∆, satisfies c ij = tr(ρA i B j ).
This is a formal identity, for all 1 ≤ i, j ≤ 2, and can be checked using computer algebra. However, this takes time and some careful typing. Instead we offer a proof that can be verified without aids. This will be facilitated by first describing how this family is the simplest possible.
When either Alice or Bob holds a classical system, for instance if the observables A 1 and A 2 commute, no entangled states are possible. This even characterizes classical systems [54]. Therefore only classical correlations can be constructed. Hence Alice's and Bob's subsystems need at least 2-dimensional Hilbert spaces. Since [A i , B j ] = 0, these subsystems must be combined in a tensor product, so H = C 2 ⊗C 2 = C 4 . Actually, we can take the A i and B j , as well as the density operator ρ, to be real rather than complex, so the Hilbert space is actually R 4 . Of course, for almost all of quantum physics, e.g., the Schrödinger equation, this would be an untenable restriction. As shown only recently, there are even correlation experiments of just the type considered here (only not minimal) that prove that "real quantum mechanics" is insufficient [55]. With the further simplification that A 2 i = B 2 j = 1, these matrices must be given on each side by reflections in planar geometry, a one parameter family. Explicitly, the reflections are M (τ ) := cos(τ ) sin(τ ) sin(τ ) − cos(τ ) = cos(τ ) · σ 3 + sin(τ ) · σ 1 for τ ∈ [0, 2π].
The matrices above are then A 1 = M (α)⊗1, A 2 = M (−γ)⊗1, B 1 = 1⊗M (π), and B 2 = 1⊗M (α+β+π). The state ρ must be entangled. We choose a maximally entangled one, which is by definition pure on the whole system, ρ = |Ψ Ψ|, and maximally mixed on the subsystems, i.e. for any matrix A we have
Ψ|(A ⊗ 1)Ψ = Ψ|(1 ⊗ A)Ψ = tr(A)/2.(45)
This would be too special for the full marginals case [15]. But here we can get away with it, and even fix the vector Ψ in (43) for the whole family. This is the unique vector (up to scaling) which is antisymmetric with respect to the exchange of the two tensor factors. For any matrix A, the vector (A ⊗ A)Ψ is again antisymmetric. Hence (A ⊗ A)Ψ = ε(A)Ψ, for some homogeneous quadratic function ε, which is also multiplicative, i.e., equals the determinant. Inserting for A a one-parameter subgroup of SU(2), i.e., A = exp(itM ) for traceless M , we find that ε(A) = 1 is constant. Hence, in first order in t, we have (M ⊗ 1 + 1 ⊗ M )Ψ = 0. Applying this to the matrices from (44) and combining with (45) we thus find
tr ρ M (u) ⊗ M (v) = Ψ|M (u) ⊗ M (v)Ψ = Ψ|(M (u)M (v)) ⊗ 1Ψ = tr M (u)M (v) /2 = − cos(u − v).(46)
The last identity is seen by observing that tr(AB T ) is the standard scalar product in matrix space, and using the addition theorem for the cosine, or, geometrically that the product of reflection across lines at an angle u − v is the rotation by that angle. Inserting the choice of angles after (44) gives the result.
Geometric aspects of quantum key distribution
Quantum key distribution (QKD) is an important task in quantum information technology [5]. It furnishes the main practical reason for studying the body Q. Here we discuss geometric features that are fundamental for that task. The goal of QKD is for two distant parties, Alice and Bob, to utilize quantum correlations for generating a key which is guaranteed to be secret from any eavesdropper, here called Eve. Eve is only assumed to be constrained by the laws of quantum mechanics, but otherwise enjoys every possible freedom. In particular, she is allowed to manipulate the correlated systems on which the scheme is based, the quantum channels by which they are transmitted, and even the measurement devices. She also gets a copy of the communications exchanged between Alice and Bob. However, she can only read these but not change them. One must also assume that once the data collection starts, Eve cannot reach into Alice's and Bob's lab and access their measurement settings or outcomes. Indeed, if Eve could do that, she would not even need to bother with the whole quantum setup, or she could play a trivial woman-in-the middle attack, and secrecy would be obviously impossible. So the rules of the game force her to gain at least some information from the quantum systems. According to the laws of quantum mechanics, this introduces a disturbance detectable by Alice and Bob. When they do detect such deviations from the expected statistics the key distribution has failed. Eve can always achieve that, but this is counted as a failure for her, because she will also not learn any secrets. We want to show here that the minimal setup in this paper is already sufficient to support QKD. Moreover, the main security argument is based directly on the geometry of Q. In undisturbed operation the setup leads to some correlations c ∈ Q. Alice and Bob will use a random sample of their particles to verify this via the public classical channel, and will abort the process if they find significant deviations from c. We claim that QKD is possible whenever c is a non-classical extreme point (cf. [19])
Suppose Alice and Bob test their correlations and find them to be such a point c ∈ ∂ e Q. What could Eve know about their measurement results? Let ε be a random variable that summarizes her findings. The conditional correlation c ε is the 2×2 matrix that pertains to those cases where Eve found ε. We have c ε ∈ Q since Eve is constrained by quantum mechanics. Combining the data with the probabilities p ε for observing ε, we get c = ε p ε c ε . But since c is extremal, all c ε that appear with nonzero probability must be equal to c. That is the same as saying that ε is statistically independent of Alice's and Bob's information. So while Eve knows ε, she learns nothing about c.
Note, however, that the argument applies equally to classical extreme points. Only, in that case the extreme points are completely deterministic. In the case of deterministic agreement probabilities indeed factorize, say c ij = a i b j . With c ij ∈ 0, 1 and a i , b j ∈ [0, 1], this factorization would be 1 · 1 = 1 or 0 · 0 = 0. Of course, this is utterly useless for drawing a secret key. So what Alice and Bob use to generate the key are the non-trivial correlations that may be present in non-classical extreme points. Any non-classical correlation is fine for that purpose, since this may be further distilled into perfect agreement.
Note also that in case of non-classical extremal points, any pair (out of four) of measurement settings can be chosen to be the "key (generating) basis" due to the very definition of cryptographic security [19]. Weakening this to only the existence of one pair of measurement settings that is uncorrelated with Eve still supports a perfect QKD protocol. This necessitates moving to the full 8-dimensional body where a non-extremal point P has been shown to exhibit this phenomenon [37]. Intuitively, this means while the behaviour P admits a nontrivial convex decomposition P = p ε P ε , Eve does not gain additional information in the key basis: P ε (a, b|key) = P (a, b|key) for all a, b, ε; she only has an advantage in other pairs of settings where the outcomes are not used for making a pair of secret key.
The full analysis of QKD takes not only error correction into account, but also the overhead of statistically verifying that the given source is really described by c. This necessarily involves errors, and the experimental implementation will have additional errors of its own. The analysis, done carefully also for c sufficiently close to ∂ e Q, results not in a blanket statement that Eve will know "nothing", but in a quantitative bound on how much she might know in the worst case. So, in addition to error correction (getting the keys to be really the same) one needs "privacy amplification", a process that had already been studied in purely classical settings [6] prior to the advent of QKD. The traditional information theoretic view focuses on rates in the asymptotic regime, i.e., for a large number of exchanged raw key bits. This systematically neglects the overhead of reliably estimating c. This can be considerable in real, and therefore finite, runs. A usable QKD security proof always has to include the finite key analysis, and all imperfections. This is far beyond the current paper, and we refer to [61,71]. To connect with the literature, we emphasize that here we have described "device independent" QKD, for which the experimental entrance ticket is a "loophole free" Bell test, which has been achieved only recently [26,30,62]. The quality of these experiments is still not in the range where the data collection could be done in the lifetime of a lab. Nevertheless, recent advances on the theoretical side [61,70,71] have brought this into feasible range. On the other hand, systems not realizing the ideal of device independence are already commercially available (see [79,Section 3.2]).
Coming back to the geometric features of Q relevant for QKD, we can first see some of the tradeoffs that enter the choice of c ∈ Q. Choosing c of type (Qnx) might seem advantageous, because then the outcomes of Alice and Bob agree for one of the settings, making the error correcting step superfluous. However, this comes at the price of moving closer to C, so that relatively low noise may make privacy amplification harder. The traditional working point therefore has been the c from (5), maximally violating a CHSH inequality. As an exposed point this seems to simplify the tomography, because only one combination of correlations needs to be estimated. But this actually leaves unnecessary leeway in the tangent directions, which are only fixed to √ ε, when the correlation is fixed to order ε. It is therefore better to use all the correlation data, rather than focusing on just one linear combination.
We further have to correct a simplification in the above argument for non-classical extreme points. It really applies only to the full statistics of Alice's and Bob's measurements including marginals, i.e., the 8-dimensional body of which Q is a projection. The key must be generated from the local measurement data, not just the combined outcomes entering a correlation. So it will be important to get extremality of c ∈ Q also when it is extended to 8 dimensions with zero marginals. This will be done in the following section, in particular in items (4),(5) of Thm. 19. This means enlarging H by an additional tensor factor where the measurement acts trivially. If we use H ⊗ H ν with A i ⊗ 1, B j ⊗ 1, and a state ρ whose partial trace over the second factor is ρ, then the correlation do not change. Writing the tensor product as a direct sum with respect to a basis of H ν makes this a direct sum of possibly correlated copies of the given model.
Uniqueness of quantum models
The best we can hope for is uniqueness of the quantum model for c up to these three operations. This is Condition (2) in Thm. 19 below. Condition (1) is the cryptographic security discussed in Sect. 4.2, i.e., anything an eavesdropper might know is statistically independent of c. Remarkably, the two conditions are equivalent. Moreover, cryptographic security is extended in (4) to all measurements made with observables in the algebra A generated by A 1 , A 2 , B 1 , B 2 . The central condition is (3), the uniqueness of a certain kind of model, defined by removing the redundancy of the operations of expansion and adding multiplicity. These cyclic models are defined by the property that ρ = |Ψ Ψ| is pure and Ψ is cyclic for the operators A i , B j , i.e., we get a dense subspace of H by acting with these operators repeatedly on Ψ.
Uniqueness results like Thm. 19 appear in the literature under the keyword self-testing [44,68]. This indicates that the correlations can be used cryptographically without first verifying that the devices act as they should, or that the prepared state is as planned: Security is based directly on the observed correlations. Current definitions of self-testing [27,44,68,76] implicitly use some of the equivalences below. Unlike these definitions and statements therein, our formulation below does not make purifying assumptions on the unknown models, i.e. "going to the church of the larger Hilbert space", nor require additional ancillas into which the ideal system is swapped.
Theorem 19 (→Sect. 5.5) Fix a point c ∈ Q. Let (ρ, A i , B j ) be any quantum model in a Hilbert space H for the correlations c, and write A for the norm closed operator algebra generated by A i , B j . When applicable, denote by (ρ, i ,B j ) the specific model for c given in Sect. 4.1. The following are equivalent:
(1) c is nonclassical and extremal in Q.
(2) There is a unique cyclic model for c. Explicitly, when ρ = |Ψ Ψ| is pure and cyclic for A (i.e., A|Ψ is dense in H), then it is unitarily equivalent to the model specified for c in Sect. 4.1.
(3) The quantum model is unique up to unitary transformation, expansion and adding multiplicity. That is, by a unitary transformation any model can be brought into canonical form
H = (Ĥ ⊗ H ν ) ⊕ H 0 , ρ =ρ ⊗ ρ ν ⊕ 0, A i =Â i ⊗ 1 ν ⊕ A 0 i , and ditto for B j .
(4) c has a unique extension to a quantum behavior p (necessarily extreme) in the 8-dimensional body including both correlations and marginals.
(5) c is cryptographically secure, i.e. its unique extension p is cryptographically secure (cf. [19]) (6) c is algebraically secure, i.e. tr(ρEX) = tr(ρE) tr(ρX) for X ∈ A and any E commuting with A.
We stress that these equivalences are claimed only for the minimal scenario assumed in this paper. For larger parameters NMK, the condition (1) may well be weaker than the others. This was noted by Tsirelson who treated the 2M2|0 case and showed [74, Thm. 3.3] that for "odd rank" there are two inequivalent cyclic representations (but connected by transposition, a non-unitary operation). Recently, [37] gave another example thanks to the existence of inequivalent MUBs (this time even up to transposition). Parallel to Tsirelson's work, uniqueness was shown in [66,67]. At the time, the authors were not aware of cryptographic applications, and R.F.W. (the second author of [66]) thought of this result as a fortuitous algebraic property of the CHSH expression, and unlikely to generalize. After an earlier attempt in [19], he only realized with the current work that the methods of [66] provide uniqueness for largest scope possible within the 222 scenario.
In cryptography, one generally has only an approximation to c. Luckily, the uniqueness result holds robustly. As shown first in [67], one can find a nearby set of operators and a state realizing c precisely.
The role of the elliptope
Historically, the elliptope has played an important role for quantum correlations. John Bell's first inequality [4] concerned three measurement settings on each side, a 232 setting. The underlying quantum model was the singlet state ρ above, and the settings on the two sides pairwise equal, so that the outcomes where pairwise perfectly anticorrelated, hence opposite with probability one. Even without accepting Bell's conclusion that they are therefore "predetermined", or going into a discussion what that means, this simplifies the analysis to a discussion of just three possibilities. Although Bell does not include the picture, the quantum body is exactly the elliptope shown in Fig. 7, and the classical body is the embedded tetrahedron. Bell's original scenario and the elliptope continue to serve as a simplified example for explaining quantum correlations, from Mermin's classic article [45] to today [32].
However, perfect anticorrelation, while easily verified in the model, is much harder to get in the quantum experiments. It was the decisive advance due to Clauser, Horne, Shimony and Holt [13] to eliminate this experimentally doubtful assumption, and to identify the 222 case as the minimal setting.
Our analysis of the convex body Q shows that the elliptope still occurs in the 222 case, and that it suffices to assume (or experimentally verify) full correlation between only two of the observables. The geometric picture of Bell's analysis (he does not draw it, though) then holds. This is of interest in cryptography, since drawing key from the perfectly correlated settings eliminates the error correction step, albeit at the expense of the distance from the classical polytope, and hence harder noise requirements.
We close with the remark that the elliptope is iconic in many branches of the mathematical spectrum, notably in convex optimization, as the feasible region of a semidefinite program, and in Gaussian statistics, as the set of 3 × 3 correlation matrices. See the pointers surrounding [46, Figure 1.1] and [64, Figure 1].
Fixed Hilbert space dimension
The models above use a 4-dimensional Hilbert space. This is an option, that happens to be sufficient for getting the extreme points of Q, but no assumption like this is made in the definition of Q. As will be seen in Sect. 4.2, this is vital for QKD. However, one can ask how such an assumption would change Q. For non-minimal settings, the question what inference about dimension can be drawn from an observed correlation has been studied extensively (see, e.g., [10,11,16]). So what can be said in the minimal setting? Here we just collect some basic observations, that can be skipped without loss for other sections.
Let us denote by Q m the (warning: nonconvex!) set of quantum correlations that are obtainable by models as in Sect. 1.1 under the additional assumption dim H = m. We also write C m ⊂ Q m for the corresponding set of classical correlations realized in a sample space of m points. Since we do not require A 2 i = B 2 j = 1, or tensor product separation, the set Q m is increasing with m, and so is C m . Since building a model for a convex combination of correlations requires the direct sum of models, and this increases m, these sets are not in general convex. However, we know that extreme points of the full bodies can be realized in fixed dimensions, so ∂ e C ⊂ C 1 and ∂ e Q ⊂ Q 4 . Thus Carathéodory's Theorem allows us to put a bound on the required m: Since in an n-dimensional convex body every point is the convex combination of at most n + 1 extreme points, and n = 4, we conclude that C ⊂ C 5 and Q ⊂ Q 20 .
The Carathéodory bound is typically tight only for polytopes, where most points require n + 1 of the vertices for a convex representations. When there is a continuum of extreme points, that Carathéodory number is usually smaller. One extreme example for this are Euclidean balls of any dimension, in which every point is the convex combination of two antipodal boundary points. Another familiar example is the quantum state space with dim H = m, which has dimension m 2 − 1, although every density operator has a spectral resolution into m pure states. Our body Q will be an intermediate case in this respect.
Proposition 20 (→Sect. 5.6) Let Q m be the set of quantum correlations obtainable from models with m-dimensional Hilbert space. Then Q m ⊂ C = Q for m ≤ 3, and Q m = Q for m ≥ 4.
The analogous set C m for the cross polytope C is also interesting and worth further study. Obviously, C 1 is the subset of rank 1 matrices c ∈ R 2×2 , a condition which is invariant under the natural symmetry group of correlation bodies but not under the extended symmetry group discussed in Sect. 2.4.
In both cases an interesting variant of the problem is to look at the subset of models with A 2 i = B 2 j = 1 and/or a pure state ρ. Also it may be of interest to fix the observables, and consider the set of correlations resulting from varying ρ.
Fixed State
For many applications it is crucial that no assumption about the quantum state is made: the linear bounds given by f ∈ Q • hold for any quantum state. In some applications, however, the state is well- known and the question is what correlations might be realized on that basis. The question really only makes sense, when the Alice-Bob split is also known, so we ask that A i ∈ A, B j ∈ B, where A and B are commuting operator algebras, typically even a tensor product split A = B(H A ) ⊗ 1 and B = 1 ⊗ B(H B ). We thus consider
Q(ρ) := {c ij ∈ R 4 : c ij = trρA i B j , −1 ≤ A i , B j ≤ 1, A i ∈ A, B j ∈ B} .
This is, in general, not a convex set. For example, when ρ is a product state, c ij = a i b j factorizes, and Q(ρ) = Q 1 = C 1 in the sense of the previous section (see also Fig. 15). For entangled states close to that, the non-convexity will persist, even though there are some proper quantum correlations. A closed form or complete algebraic description of Q(ρ) can hardly be expected, so we just make a few remarks.
First, if we fix a a linear functional f , finding the maximal quantum value is a bilinear optimization problem. We can fix all observables except one, e.g. A i , and isolate the linear functional by which it enters, i.e. find X i such that j f ij trρ(A ⊗ B j ) = trX i A or equivalently X i = j f ij tr B (ρ1 ⊗ B j ). Maximizing over A i , is now the problem of finding the A with −1 ≤ A ≤ 1 for which tr(AX i ) is maximized. Clearly, that is A i = sign(X i ) in the functional calculus of the hermitian operator X i . The maximum is then the trace norm of X i , which would seem to eliminate the A i . However, it is better to just fix the optimal A i , and optimize again with respect to the B j in the next step, in a see-saw style. This works nicely in practice, and improves the target functional in every step. However, this iteration has no convergence guarantee.
The second remark is that we can give a "closed solution" in the case of a 2 ⊗ 2-dimensional Hilbert space, the minimal dimension where nonclassical correlations could manifest by Prop. 20. This is well- known [31] for the case of the CHSH inequality, but it actually holds for the whole correlation body:
Proposition 21 (→Sect. 5.7) Let ρ be a density operator on H = C 2 ⊗ C 2 , and let λ 1 ≥ λ 2 ≥ 0 be the two largest singular values of the 3 × 3-matrix with entries R k = trρ σ k ⊗ σ for Pauli matrices σ 1 , σ 2 , σ 3 . Then co Q(ρ) = co Q st (λ 1 , λ 2 ) ∪ C , where Q st (λ 1 , λ 2 ) is the set of correlations of the form c ij = λ 1 cos(α i ) cos(β j ) + λ 2 sin(α i ) sin(β j ). (47) Moreover, we have Q(ρ) ⊂ C if and only if λ 2 1 + λ 2 2 ≤ 1.
The map of possibilities is shown in Fig. 16. The singular values for a vector with Schmidt spectrum s 1 ≥ s 2 ≥ 0, e.g., ψ = s 1 |++ + s 2 |−− are λ 1 = s 2 1 + s 2 2 = 1 and λ 2 = 2s 1 s 2 , which parametrizes the line segment PM. The fact that all these states (other than P) give correlations outside C, and hence violate the CHSH inequality, is known as Gisin's Theorem [25].
Fixed Observables
Dually we can fix the operators A i , B j , so that the resulting compact convex semialgebraic set Q(A i , B j ) is just an affine image of the quantum state space, a spectrahedron. Still an explicit description may be difficult. As in the case of a fixed state, the boundary in any single direction, i.e., the maximization of any affine functional f reduces to an eigenvalue problem, namely the largest eigenvalue of the operator
T (f ) = ij f ij A i B j .
Giving an algebraic solution might be harder because this corresponds to computing an algebraic formula for the largest root of a polynomial of high degree, depending on the dimension of the underlying Hilbert space.
Q st (tλ 1 , tλ 2 ) = tQ st (λ 1 , λ 2 ), which for t ≤ 1/ √ 2 ≈ .7 is included in C.
The general case −1 ≤ A i , B j ≤ 1 can, in principle, be dealt with by a minimal dilation U : H → K so that Q(A i , B j ) is the affine image of a specific isometric copy of the state space of H in K. This is harder to work out, so we give a characterization only for the special case A 2 i = B 2 j = 1. There is a subtlety here when A i , B j are "genuinely infinite dimensional". Then the Prop. 22 holds, if we interpret the convex hull as the closed convex hull and Q(A i , B j ) as the closure of the set of correlations attainable with arbitrary density operators. But for points in the continuous spectrum (not eigenvalues), certain boundary points will be unattainable by density operators in the given space. This is a straightforward consequence of the uniqueness theorem Thm. 19. When some self-testing correlation inequality f is attained, the maximum must be an eigenvector of the corresponding T f . Such points can only be attained by singular states, i.e. positive normalized linear functionals not induced by density operators, because the continuous spectrum can be approximated by eigenvectors.
Q(A i , B j ) = co{Q obs (u, v) := S(u)O(2)S(v) | u ∈ Σ A , v ∈ Σ B }, where S(u) = 1 0 u √ 1 − u 2 ,(48)
Since O(2) is the basic model for the fixed-observable correlation sets, we consider it in more detail. (2) Both circles consist of extreme points, and any line connecting a pair of points from different circles is an exposed edge of Q obs (0, 0). The union of these edges is the entire boundary.
(3) The necessary and sufficient semialgebraic condition for c ∈ Q obs (0, 0) is that
(c) := 1 − (c 2 11 + c 2 12 + c 2 21 + c 2 22 ) + (det c) 2 ≥ 0 and − 1 ≤ det c ≤ 1.(49)
Since the observables used by Alice and Bob for Q obs (0, 0) are the same, the Alice-Bob swap symmetry leaves this set invariant, and so its intersection with the plane {c 12 = c 21 } is at the same time the projection (see the double cone in Fig. 19). On the other hand, for general values of u and v this is not true, as shown in Fig. 18.
The above Proposition suggests that quantum points with | det c| = 1 are self-testing.
Proofs
In this section we prove all theorems and propositions seen so far. Since many results in our paper have appeared previously in the literature, we could give many proofs by citation. However, we aim to make our text self-contained. Where the pedestrian argument, tailored to the case at hand, can be understood as an example of a more general theory, we provide this background as well. As a help for monitoring the logical flow we included some summaries of what has been shown up to some point. These are formatted like the definitions and propositions throughout the paper, and numbered consecutively with these.
There are, of course, many ways to organize the proofs. We now briefly describe our overall strategy and the structure of Sect. 5. We begin in Sect. 5.1 with the equivalences of Thm. 1, centered around the matrix completion problem. Self-duality (Thm. 3) follows in Sect. 5.2. On this path we already need some information about the boundary (e.g., in Sect. 5.1.4), which is extended to the detailed classification of boundary points in Sect. 5.3. The study of boundary points is perhaps a bit more detailed than necessary, since it uses a local criterion for excluding certain points from the extreme boundary of the convex hull of a variety. This technique might be helpful more generally. Sect. 5.4 contains the computation of the support function (Prop. 17), which proved to be more subtle than expected, even with a full understanding of the boundary. Finally the quantum properties, self-testing and all that, are established in Sect. 5.5.
Proof of Theorem 1
For the sake of this proof let us denote by Q (x) ⊂ R 4 the set characterized by item (x) in Thm. 1. We have to show that these are all equal. The backbone of the proof is the chain
Q (a) ⊂ Q (e) = Q (d) = Q (b) ⊂ Q (a)(50)
and separate arguments for Q (e) = Q (f ) and Q (b) = Q (c) . The main work will be getting the solution set Q (e) of the matrix completion problem in great detail. The boundary information coming out of that, in particular for rank C = 2, will then be used to get the further equivalences and finally go back to Q (a) .
Making matrix completion real
The inclusion Q (a) ⊂ Q (e) is an important step, because the definition of Q (a) admits infinite Hilbert space dimension, while Q (e) only allows finite dimension. This reduction step works for more parties, settings, and outcomes, as well, which is the whole point of the semidefinite hierarchies [17,47]. But outside the minimal scenario the inclusion is strict. We saw the inclusion Q (a) ⊂ Q (e) at the beginning of Sect. 2.8. From the quantum model we naturally get a positive definite matrix with some unknown complex entries, and diagonal not equal to 1. These assumptions are part of the description of Q (e) , however. In order to prove Q (a) ⊂ Q (e) , we thus have to make sure that the additional assumptions do not make Q (e) smaller. This is the content of Prop. 11.
Proof of Prop. 11:
Consider a matrix C ≥ 0 of the form (22) with diagonal entries d i ≤ 1. We also fix the matrix C = eC, the entrywise real part of C. Then since the complex conjugate of a positive semidefinite matrix is a again semidefinite, C ≥ 0. Add to this the matrix with diagonal entries 1 − d i to get C . Then C ≥ C ≥ 0 is a matrix with the same off-diagonal c ij , but in the standard form with u, v real and d i = 1.
Solving the real completion problem
In this and the following two sections, we proceed to actually solve the matrix completion problem defining Q (e) , i.e. decide which c permit real values u, v that make the following matrix positive semidefinite:
C = 1 u c 11 c 12 u 1 c 21 c 22 c 11 c 21 1 v c 12 c 22 v 1 (51)
The solution uses Sylvester's criterion. We use the standard notation C I for the principal submatrix selecting the rows and the columns specified by the indices in I, and m I = det C I for the corresponding principal minor. Sylvester's necessary and sufficient criterion for C ≥ 0 is that m I ≥ 0 for all 2 n − 1 index sets I. For C > 0 it suffices to have m 1 , m 12 , m 123 , m 1234 > 0. The positivity of principal 2 × 2-minors gives c 2 ij ≤ 1 or equivalently Q (e) ⊆ N as a necessary condition. Consider next the principal 3 × 3-minors
C 123 = 1 u c 11 u 1 c 21 c 11 c 21 1 ≥ 0 and C 124 = 1 u c 12 u 1 c 22 c 12 c 22 1 ≥ 0.(52)
These that do not involve v, and they give a condition for u. We show that (52) is all we need to consider:
Lemma 25 Suppose that, for some u ∈ R, the matrices in (52) are positive semidefinite. Then one can find v such that in (51) we have C ≥ 0, i.e., c ∈ Q (e) .
Proof We consider first the case of strict positivity, i.e., C 123 > 0 for some fixed u, which necessarily satisfies |u| < 1. By Silvesters criterion, we only need to find v such that m 1234 = det C > 0. Note that det C is a quadratic polynomial in v with a negative leading coefficient. There if the desired v exists, the v maximizing this polynomial will do just as well. This is v = (c 11 c 12 + c
Here we only used m 124 ≥ 0, but with strict inequality would also get C > 0. Now suppose that we only have m 123 ≥ 0. The we apply the argument to C which arises from C by multiplying every off-diagonal element by 0 < λ < 1. Since C = C/λ + (1 − 1/λ)1, this operation makes C 123 > 0 and C 124 > 0, and hence provides a choice v λ making C ≥ 0. Since |v λ | ≤ 1, this has a convergent subsequence as λ → 1, and the resulting C converge to a completed C ≥ 0.
Alternatively, we could invoke a general result on semidefinite matrix completion [29,Theorem 7]: Consider the graph G whose edges represent the given entries of a partial matrix whose completion we seek. The obvious necessary conditions for completability are that for those subsets of vertices, where all matrix entries are specified (cliques of G), the corresponding submatrices are positive semidefinite. Then, if the graph is chordal (meaning any cycle of length ≥ 4 allows a shortcut), this condition is also sufficient. Moreover, this holds for both strict positive definiteness and semidefiniteness. In the case at hand, the graph given for Q (e) is a 4-cycle shown in Fig. 20. Assuming the existence of u with C 123 , C 124 ≥ 0 adds a diagonal, making the graph chordal. Hence no further condition needs to be considered. Lem. 25 eliminates v and leaves us with the nonnegativity of the following three principal minors:
m 12 = 1 − u 2 (A) m 123 = 1 − c 2 11 − c 2 21 + 2c 11 c 21 u − u 2 (B) m 124 = 1 − c 2 12 − c 2 22 + 2c 12 c 22 u − u 2 (C)(54)
We have c ∈ Q (e) if and only if these are simultaneously satisfied for the same u. By Helly's Theorem in R 1 , the three positivity intervals have a common point if and only if they intersect pairwise. Hence we can consider pairwise intersections. The maximum of (B) is
max u m 123 = (1 − c 2 11 )(1 − c 2 21 ) = m 13 m 23 ≥ 0,(55)
attained at u = c 11 c 21 ∈ [−1, 1], so its positivity interval intersects that of (A), and similarly for the pair (A, C). Hence only the pair (B, C) needs to be considered. We conclude:
Summary 26
If c ∈ N , then c ∈ Q (e) if and only if (B) and (C) are both non-negative for some u ∈ R.
Joint positivity of two parabolas
We next examine the criterion in Sum. 26 independently of the specific context. We do the quantifier elimination carefully, because the structure of the solution explains the disjunction in Thm. 1 (d). Thus at the end of this section we will achieve the equality Q (e) = Q (d) .
The configurations of two parabolas in the following lemma are shown in Fig. 21. Both are given by quadratic polynomials f (x) = b − (x − a) 2 with the same negative quadratic term. The parameters are chosen so that (x, y) = (a, b) is the location of the maximum. This function is positive in the interval
[a − √ b, a + √ b].
The question is when for two such parabolas the positivity intervals overlap. It is clear that the problem is invariant under shifts (adding a constant to both a 1 and a 2 ), and (a 1 − a 2 ) 2 just sets a scale for the bs. Hence we could choose a 1 = 0 and a 2 = 1. Fig. 21 is drawn with this choice.
Lemma 27
Given two quadratic polynomials f i (x) = b i −(x−a i ) 2 , i = 1, 2, the following are equivalent:
(1) There exists a point u ∈ R such that f 1 (u) ≥ 0 and f 2 (u) ≥ 0.
(2) b 1 ≥ 0 ∧ b 2 ≥ 0 ∧ b 1 + b 2 − (a 1 − a 2 ) 2 ≥ 0 ∨ 4b 1 b 2 − (b 1 + b 2 − (a 1 − a 2 ) 2 ) 2 ≥ 0 .
If one demands strict inequality then the following are also equivalent:
(1') There exists a point u ∈ R such that f 1 (u) > 0 and f 2 (u) > 0. Proof This is based on Fig. 21. Clearly, both positivity ranges must be non-empty. So, unless b 1 , b 2 ≥ 0, there is nothing to prove. We can also trivially take care of the cases with a 1 = a 2 , because then the positivity intervals are contained in each other. The maximum of each parabola is clearly in the positivity interval. So if b 1 ≥ (a 1 − a 2 ) 2 the maximum of the second parabola is in the positivity range of the first, and so there is a non-empty intersection. Symmetrically, this holds for b 2 ≥ (a 1 − a 2 ) 2 . This gives two closed rectangles in the (b 1 , b 2 )-plane with non-empty intersection.
(2') b 1 > 0 ∧ b 2 > 0 ∧ b 1 + b 2 − (a 1 − a 2 ) 2 ≥ 0 ∨ 4b 1 b 2 − (b 1 + b 2 − (a 1 − a 2 ) 2 ) 2 > 0 .
Next consider the intersection of the two parabolas. Their unique intersection point is
x s , f i (x s ) = a 1 + a 2 2 − b 1 − b 2 2(a 1 − a 2 ) , 4b 1 b 2 − b 1 + b 2 − (a 1 − a 2 ) 2 2 4(a 1 − a 2 ) 2 .(56)
Now suppose f i (x s ) ≥ 0. Then x s is in the intersection of the positivity ranges. The corresponding region, defined by the positivity of the numerator of f i (x s ) in (56) is the closed parabola in Fig. 21. Hence we have non-zero intersection for the union of the two rectangles and the parabola in Fig. 21. This can be simplified as the union of just two regions, namely the parabola and the region {b 1 + b 2 ≥ (a 1 − a 2 ) 2 }, which contains the two rectangles plus a triangle, which is contained in the parabola.
To complete the proof of the first part, we need to show that, for any point outside this region, the positivity ranges have empty intersection. This complement is defined by the conditions 0 ≤ b i < (a 1 − a 2 ) 2 and f i (x s ) < 0. Now the first condition implies |b 1 − b 2 | < (a 1 − a 2 ) 2 . Hence the second term for x s in (56) is bounded by |a 1 − a 2 |/2, so x s lies between a 1 and a 2 . Therefore one of the maxima lies to the right of x s and the other one lies on its left, and therefore the same holds for the positivity ranges of the parabolas. Since x s < 0 the ranges do not intersect. This completes the proof of the first part.
The primed statements characterize the interior of the parameter set, i.e., the region just described without the boundary points, for which b 1 = 0, or b 2 = 0 or f i (x s ) = 0.
We now apply Lem. 27 to the quadratic polynomials (B,C) in (54). The parameters are
a 1 = c 11 c 21 , a 2 = c 12 c 22 , b 1 = (1 − c 2 11 )(1 − c 2 21 ), b 2 = (1 − c 2 12 )(1 − c 2 22 ).(57)
With this, the two polynomials in Lem. 27 (2) are g in (17) and h in (19). The combination of Sum. 26 and Lem. 27 now establishes the equivalence of (d) and (e) in Thm. 1. We record this as follows:
Summary
Properties of the boundary and extreme points
Sum. 28 characterizes membership in the spectrahedral shadow Q (e) . We now come to its boundary, beginning with an extended version of the first statement in Prop. 12.
Lemma 29
Given any point c ∈ Q (e) , the following three conditions are equivalent:
(1) c ∈ ∂Q (e) .
(2) The matrix completion problem for c has a unique solution (u, v).
(3) Either ( c ∈ ∂N and g(c) ≥ 0 ) or ( g(c) < 0 and rank(C) ≤ 2 for every completion C of c).
Proof Suppose that c is in the interior of Q (e) . Then the fiber over c of the projection from the 6dimensional spectrahedron {C ≥ 0} onto the 4-dimensional body Q (e) is two-dimensional. Hence (2) is false. Also, both conditions in (3) are false. The first condition fails because int(Q (e) ) ⊆ int(N ) and the second condition fails because c has a preimage C that is positive definite and hence has rank 4. It remains to shown that (1) implies both (2) and (3). Let c ∈ ∂Q (e) . We have g(c) ≥ 0 or h(c) ≥ 0, by Lem. 27 (2') applied to (57). We distinguish two cases, namely c ∈ ∂N and c / ∈ ∂N . If c ∈ ∂N then c ij = ±1 for some i, j. Up to symmetry we may assume c 11 = −1. From (18) we see that h(c) = −g(c) 2 and hence g(c) ≥ 0, so (3) holds. For (2) we note that m 123 (u) = −(u + c 21 ) 2 and m 134 (v) = −(v + c 12 ) 2 . Nonnegativity of principal minors requires u = −c 21 and v = −c 12 , so the matrix completion is unique. Now assume c / ∈ ∂N . Then we have h(c) = 0, and this implies g(c) < 0, so that the parabolas from Lem. 27 have a unique point of intersection at level 0. This point is given either by
u = a 1 − √ b 1 = a 2 + √ b 2 , or u = a 2 − √ b 2 = a 1 + √ b 1 .
Analogously there is also a unique v. These unique choices u, v make all third order principal minors m 123 , m 124 , m 234 , m 134 vanish. This means rank(C) ≤ 2. Hence (2) and (3) hold in this case as well.
The next lemma shows how the trigonometric functions arise in the parametrization of the boundaries.
Lemma 30 (1) If c is extremal in Q (e) then it must have a unique matrix completion with rank C ≤ 2.
(2) It must then be of the form c ij = a i · b j for unit vectors a 1 , a 2 , b 1 , b 2 in the Euclidean space R 2 ,
(3) This means that it can be written as c ij = cos(α i − β j ) for some α 1 , α 2 , β 1 , β 2 ∈ R.
Proof (1) Of course, extreme points are always part of the boundary, because any line segment through an extreme point necessarily contains some points outside the body. Hence the previous Lemma applies. So by part (3) of the previous Lemma they either have rank C ≤ 2 anyway, or else belong to a face of the cube N . Up to symmetry this means c 22 = 1. Then C 124 ≥ 0 forces u = c 12 , so the extendability is just the elliptope condition det C 123 = 1 − c 2 12 c 2 23 c 2 31 − 2c 12 c 23 c 31 ≥ 0. Since we are not just assuming c to be on the boundary, but even extremal, this inequality must be tight, and so we also get rank C ≤ 2.
(2) By the spectral theorem, every positive semidefinite d×d matrix can be written as a Gram matrix, i.e, C αβ = w α · w β for vectors w 1 , . . . , w d in some Euclidean space R r . Here r = rank C is the number of non-zero eigenvalues. In our case the diagonal matrix entries are 1, so these vectors are unit vectors. Moreover, the c ij are themselves matrix elements of C, so we just need to rename the vectors according to whether the dimension belongs to Alice or to Bob, i.e., a 1 = w 1 , a 2 = w 2 , b 1 = w 3 , and b 2 = w 4 .
(3) Unit vectors in R 2 lie on the unit circle, parameterized by angles. Scalar products between such vectors are the cosines of the enclosed angle. Setting w i = (cos α i , sin α i ) for some α i ∈ R, we thus have
C = 1 cos(α1 − α2) cos(α1 − α3) cos(α1 − α4) cos(α1 − α2) 1 cos(α2 − α3) cos(α2 − α4) cos(α1 − α3) cos(α2 − α3) 1 cos(α3 − α4) cos(α1 − α4) cos(α2 − α4) cos(α3 − α4) 1 .(58)
Renaming according to the Alice/Bob distinction, i.e. α 3 = β 1 and α 4 = β 2 , we obtain the claim.
Part (2) of Lem. 30 says that ∂ e Q (e) ⊂ Q (f ) , and part (3) that ∂ e Q (e) ⊂ Q (b) . Since Q (f ) is convex by a direct sum construction, and Q (b) is anyhow defined to be convex, this also means that Q (e) ⊂ Q (f ) and Q (e) ⊂ Q (b) . The first of these inclusions can be inverted trivially, because we can just set u = a 1 · a 2 and v = b 1 · b 2 to get a matrix completion from the unit vectors a i , b j . To revert the second inclusion, we have to combine Sum. 28 This leaves only one part of Thm. 1, namely the equality with Q (c) . This will be shown in Sect. 5.1.5. Also, we have established one part of Prop. 8, the cosine parametrization, in a slightly different parametrization. The parametrization used in Prop. 8 and Prop. 9 follows from the form (58) by setting
α = α 1 − α 3 , β = α 4 − α 1 , γ = α 3 − α 2 , δ = α 2 − α 4 .(59)
Note that the signs were chosen (not changing the respective cosine) such that α + β + γ + δ = 0. This constraint reflects the fact that only differences α i − α j enter. What we have not shown yet, however, is the criterion ∆ < 0 for a tuple of angles to give an extreme point. This will be done in Sect. 5.3.2.
The pushout characterization
The continuous map t → sin(πt/2) takes the interval [−1, 1] to itself, and it has a continuous inverse. By applying this map coordinate-wise, we conclude that the pushout map in (15) is a homeomorphism from the cube N = [−1, 1] 4 to itself. Here we shall establish Prop. 5, which states that Q = sin(C). Note that in the literature of matrix completion, this is often presented as Q = cos(πMET(K 2,2 )) which is obvious if one realizes that the metric polytope MET(K 2,2 ) of the complete bipartite graph K 2,2 is isomorphic to the classical polytope C. See [41] and references therein.
Our strategy is to show this for the boundaries, i.e., sin(∂C) = ∂Q. Suppose that we know this, then notice that sin maps connected components of N \ ∂C to connected components of N \ ∂Q. Since 0 is in both the classical polytope and the quantum set, and sin(0) = 0 we get that C is precisely mapped to Q. Hence equality of the boundaries is sufficient.
We examine the boundary of the demicube C facet by facet. This task is greatly reduced by symmetry, given that sin commutes with all symmetry operations as explained in Sect. 2.4. Only one CHSH face and one N -face need to be considered. Consider first the CHSH facet {(x, y, z, w) ∈ [−1, 1] 4 : x+y+z−w = 2}. The images of the points on this facet under the pushout map sin are sin(x, y, z, w) = sin π 2
x , sin π 2 y , sin π 2 z , sin π 2 w = (cos α, cos β, cos γ, cos δ),
where α = π(1−x)/2, β = π(1−y)/2, γ = π(1−z)/2, δ = π(w−1)/2. This gives α + β + γ + δ = π − (x + y + z − w)π/2 = 0. Moreover, x, y, z, w ∈ [−1, 1] implies α, β, γ ∈ [0, π] and δ ∈ [−π, 0]. These inequalities imply ∆ = sin α sin β sin γ sin δ ≤ 0. Hence we get exactly those parametrized patches (Qqx) from Prop. 8, for which α, β, γ > 0 > δ. The other seven patches (Qqx) are obtained by symmetry.
Next consider an N -face of C, say that defined by w = 1. The pushout map preserves this equation. We can now go through the same considerations as above, but in one dimension lower. The geometric statement is that the pushout of the tetrahedron equals the elliptope (see Fig. 7). This also follows from the fact that sin identifies their boundaries. We know this from the above discussion of the CHSH facets.
Together with Sum. 31 this completes the proof of Thm. 1.
Self-duality
Self-duality is a special feature of the pair Q, Q • . It is not the kind of property that tends to hold for any sufficiently simple case. Indeed, in one dimension smaller, the elliptope E 3 (see Fig. 7) is not self-dual: It has extreme points with a pointed normal cone, which translate to flat faces in (E 3 ) • . But E 3 has no such faces, and hence is not self-dual. For an illustration of this phenomenon see [7, Figure 5.2].
With the boundary information obtained so far one could try to derive self-duality extreme point by extreme point. We take a more global approach based on the duality of the semidefinite matrices C and F , and their representations as Gram matrices. We begin with the characterization of the F matrices.
Proof of Prop. 15:
The primal/dual characterization of spectrahedral shadows described briefly before the statement of Prop. 15 is a standard result, so we omit its proof and are left with the additional property p 1 +p 2 = p 3 +p 4 . We can take p 1 + p 2 and p 3 + p 4 both non-zero, since otherwise we have the trivial case f = 0, F = 1 2 1. Now observe that F = ΛF Λ, where Λ is the diagonal matrix with diagonal (λ, λ, λ −1 , λ −1 ) is also positive, and has the same off-diagonal block f ij . The diagonal is changed to (p 1 , p 2 , p 3 , p 4 ) = (λ 2 p 1 , λ 2 p 2 , λ −2 p 3 , λ −2 p 4 ). In order to satisfy the sum constraint we need λ 2 = (p 3 + p 4 )/(p 1 + p 2 ). Then (F + F )/2 is the desired extension with p 1 + p 2 = p 3 + p 4 . This is different from F , if λ 2 = 1.
Finally, using a similar argument, we show that if the point f has a completion F for which the condition p 1 + p 2 = p 3 + p 4 = 1 does not hold, then f is not extreme. Using the diagonal matrix with (λ A , λ A , λ B , λ B ), and assuming, without loss that x = p 1 +p 2 −1 > 0, the normalization condition becomes
λ 2 A (1 + x) + λ 2 B (1 − x) = 2.
Then choosing both terms equal to 1 maximizes λ A λ B = (1 − x 2 ) −1/2 > 1. We conclude that f ij = λ A λ B f ij is again in Q • , so f is not a boundary point.
We now consider matrices C and F as Gram matrices of suitable vectors in a real Euclidean space.
That is, we choose a 1 , a 2 , b 1 , b 2 from the characterization (f) in Thm. 1, and similarly four vectors x 1 , x 2 , y 1 , y 2 whose scalar products give F , and −f ij = x i · y j . The conditions for these vectors for C to be of the form (51) and F to be of the form (34) with p 1 + p 2 = 1 are a 1 = a 2 = 1 and
x 1 ⊥ x 2 , x 1 2 + x 2 2 = 1,(61)
and analogous conditions for b j and y j . The sets of vectors can be mapped to each other by
a 1 = x 1 + x 2 , a 2 = x 1 − x 2 and x 1 = (a 1 + a 2 )/2, x 1 = (a 1 − a 2 )/2.(62)
With the analogous relations for the b j and y j we get a bijective correspondence between the allowed vectors. Expressing the relation by the 2 × 2-Hadamard matrix H 2 = 2 −1/2 1 1 1 −1 and inserting the relations into c ij = a i · b j this correspondence relates the 4-vector c = (c 11 , c 12 , c 21 , c 22 ) to the corresponding f ∈ R 4 by H = H 2 ⊗ H 2 , i.e., (28). This proves the duality theorems Prop. 13 and Thm. 3.
Further boundary properties
The pushout characterization makes the disjoint union structure of Prop. 7 obvious, but we need to verify various geometric properties of strata mentioned in that proposition.
Classification of boundary points by rank
We now show the remainder of Prop. 11, the rank statements in Prop. 12, and the parametrization of boundary points in Prop. 9. Each of these propositions has items referring to different parts of the boundary. The proofs will be not be organized by these, but by the classification of Prop. 7. The ordering of these items was modified to give a better flow of the arguments. C will be a matrix completion for c.
(Qin) ⇔ rank C may be 4. Any interior point c can be written as a convex combination in which the origin c = 0 has positive weight. There exists a matrix completion C in which 1 has a positive weight, so C has full rank. Conversely, if C has rank 4, a small change of c translates to a small change of C, leaving the other matrix entries unchanged. As full rank positive matrices are open, this will not lead out of Q. Hence c is an interior point.
(Qei) ⇔ c ∈ ∂Q and rank C = 3. From now on all c will be in the boundary, and so by Lem. 29 c has a unique completion C. Whenever one of the |c ij | = 1, the i th and the (j + 2) th rows/columns are equal up a sign, so the rank is reduced by 1. Deleting one of these row/column pairs leaves a 3 × 3 correlation matrix, whose semidefiniteness describes the elliptope. This has full rank precisely in the interior.
(Qnx),(Qce) ⇔ c ∈ ∂Q exactly one, resp. two |c ij | = 1. From now on, we have rank C ≤ 2, and so by Lem. 30(3), the matrix C lies in the cosine-parametrized family shown in (58). If at least one |c ij | equals 1 then the point is in the boundary case of an elliptope. It is clear from the embedding of the elliptope into the cube that the non-classical points are exactly the ones that are not in another cube face. The edges lie in exactly 2 cube faces. This directly translates to the corresponding angles in the parametrization being multiples of π, as claimed in Prop. 8.
(Qcx) ⇔ c ∈ ∂Q rank C = 1 Of course, this continues, so if three |c ij | = 1, it must actually be four, and hence we have a classical extreme point. It is easy to see that this implies rank 1: Up to symmetry this is the case c ij = 1 for all i, j = 1, 2, which clearly has the rank 1 matrix C ij ≡ 1 as a completion. For the converse, note that rank C = 1 and C ≥ 0 imply C ij = x i x j for some vector x, and c ij = x i x j+2 . If we now insist on the special form of C from Prop. 11, i.e., unit diagonal, we must have an extreme point just from the rank condition. However, we get even many interior rank 1 points, and c ∈ ∂Q needs to be imposed as well.
(Qqx) ⇔ c ∈ ∂Q ∩ (int N ) ⇔ ∆ < 0 in Prop. 8
From the semialgebraic description of Q in Prop. 10 this type consists of the points c ∈ N with h(c) = 0 and g(c) < 0. But for the cosine parametrized correlations with δ = −(α + β + γ) we get the identity g(cos α, cos β, cos γ, cos δ) = 2∆.
This completes the proofs of Prop. 11, Prop. 12, and Prop. 9.
Exposing functionals: first order analysis
Proof of Prop. 14:
Since c is a boundary point, we have c · f = max{c · f |f ∈ Q • } = 1 for some f ∈ Q • . Indeed, if the maximum over the compact set Q • were M < 1, we would have c/M ∈ Q •• = Q, and c would be in the interior. We fix some such f . Thus c · f has a global maximum at c = c, and therefore a local maximum on the surface c ij = cos θ ij with ij θ ij = 0 where the θ ij are the given angles. We analyze this extremum problem by introducing a Lagrange multiplier λ. The critical equations are
0 = d ij f ij cos(θ ij ) + λθ ij = ij −f ij sin(θ ij ) + λ dθ ij .(64)
Because ∆ < 0, all sin(θ ij ) = 0, and f ij = λ/ sin(θ ij ). The multiplier is determined from c · f = 1 to be λ −1 = ij cot θ ij ≡ K. Hence a maximizing f is unique, and given by the formula.
This proof utilizes our knowledge from other sources that such c are extremal. When computing the convex hull of a parametrized surface (as in [12]) this is the tricky part. We can compute the tangent hyperplane at every point of such a surface, but is the local extremum a global maximum? It is then natural to look first at the local criterion and to rule out saddle points. That is, by looking at the parametrization in second order Taylor approximation, we can determine whether the surface is locally on one side of the tangent plane. If not, the point can be omitted from the list of potential extreme points. We carried this out for the case at hand, and found that all points with ∆ > 0 are saddles.
When a variety is not given by an parametrization, but by implicit equations, the first order analysis is just the definition of the dual variety. From the self-duality of the convex body we thus conclude that {h = 0} is a self-dual variety. For the study of Q this is not only modified by intersecting with the cube, but also by eliminating the branch of the variety in the interior of Q. Under dualization these operations are connected, i.e., the duals of the normalized tangents of interior points end up outside the dual body.
Unique CHSH violation
It is clear that each curved tetrahedron, being the pushout of a CHSH-face, violates exactly that CHSH inequality. That this is true for all non-classical correlations is an elementary fact that we have not used otherwise. Its self-dual version has likewise nothing to do with Q, but only with the enclosing polytopes C and N . It states that for any non-trivial Bell inequality, i.e., for any affine inequality valid for all classical correlations (f ∈ C • ), which is not true by virtue of positivity constraints alone (f / ∈ N • ) there is a unique non-local box (c ∈ ∂ e N ) making this evident (c · f > 1).
Proof of Prop. 4:
The 8 CHSH inequalities are ±c 11 ± c 12 ± c 21 ± c 22 ≤ 2, where the product of the signs is −1. Given two distinct inequalities of this sort, the signs cannot be equal in all four places (then the inequalities would be the same), and also not in three places (because this would imply equality on the fourth place). Adding two violated inequalities (· · · > 2) thus gives ±2c ij ± 2c kl > 4, which contradicts the inequality |c ij | ≤ 1 following from non-signalling. Hence at most one such inequality is violated for c ∈ N . On the other hand, for c / ∈ C at least one is violated.
Face orthogonality
We now consider the normal cycle and the orthogonality relations of faces. Some claims about exposedness are already made in Prop. 7. These and the entries in Table 1 will now be treated in detail. As in Sect. 5.3, statements are grouped by boundary types. For points {c} of each type we must identify the functionals f ∈ Q • attaining the maximal value 1 at c, i.e., the face {c} ⊥ . Note that the orthogonal face operation
S ⊥ = {f ∈ Q • | ∀c ∈ S ⊂ Q, c · f = 1}(65)
also depends on the whole convex body in which this is taken (here: Q). We will write S ⊥ C and S ⊥ N when we take orthogonal complements with regards to C and N respectively. Obviously, the operation is monotone in the sense that for S ⊂ C we have S ⊥ N ⊂ S ⊥ ⊂ S ⊥ C . Since these characterizations are taken over by self-duality, let us look at 2Hf for f ∈ {c} ⊥ . The first component (2Hf ) 11 is just c · f = 1; the others (with maybe a sign added) are the scalar products of f with all the other classical points, which need to be ≤ 1 for f ∈ Q • . Hence 2Hf is indeed in the standard cube, with (2Hf ) 1 = 1. The condition 2Hf ∈ Q singles out an elliptope satisfying the third order inequality, which in terms of f is the elementary symmetric polynomial appearing in (42), albeit with the opposite inequality. Note that {c} ⊥ is thus the closed elliptope [Qei], which also contains boundary points of type (Qcx),(Qce),(Qnx). Computing their complements, as we will do presently, will give faces including c.
(Qei) ⊥ = (Qcx):
Starting from an interior elliptope point c, self-duality gives us essentially the dual of the previous paragraph. More explicitly, set c = (1, x, y, z) with 1 − x 2 − y 2 − z 2 + 2xyz ≥ 0, and x 2 , y 2 , z 2 < 1, to avoid edges and classical extreme points. For an interior point, the equality c · f = 1 clearly extends from c to the face generated by this. Hence f must be (1, 0, 0, 0), and 2Hf = (1, 1, 1, 1) is the classical extreme point considered in the previous paragraph.
(Qnx) ⊥ = (Qcx): More care is needed for elliptope boundary points, because, in principle, this could allow more freedom for f . We will use the local extremality in the sense of Sect. 5.3.2, with angles θ 11 = 0, and sin(θ ij ) = 0 for the other (i, j) = (1, 1). Then the equation f ij sin(θ ij ) = λ implies λ = 0 · f 11 = 0; since the other sines are = 0, then once again the only possibility is f = (1, 0, 0, 0). This shows that the (Qnx)-points are indeed non-exposed extreme points as claimed in Prop. 7.
(Qce) ⊥ = [Qce] = (Qce)∪(Qcx):
Now consider a point c on an edge, but not an endpoint. This forces exactly two angles, say θ 11 = θ 12 , to be zero (cf. Prop. 9) and by the same reasoning used in the previous paragraph, λ = 0, and f 21 = f 22 = 0. However, f 11 and f 12 remain unconstrained and merely have to add up to 1. Remarkable here is that we get a drop of expected dimension from C, where the complement of and edge is a 2-dimensional face:
{(1, 1, 0, 0)} ⊥ C = (1 + x)/2, (1 − x)/2, y/2, −y/2 |x| + |y| ≤ 1 .(66)
Geometrically, 3-dimensional faces of C containing this edge meet at an angle, whereas the corresponding elliptopes are tangent. This difference is explicitly seen comparing the rows of Fig. 8.
(Qqx) ⊥ = (Qqx):
Here points go to points, as discussed in detail in Sect. 5.3.2. Table 1 also gives the manifold dimensions of each boundary type. We only have one continuous family of type (Qqx), of dimension 3. All other types occur only in discrete instances, i.e., with dimension zero.
Support function
As described in Sect. 3.5, we seek the criteria for a ray to hit the boundary of Q • either in an exposed extreme point or in an N -face. The relevant geometry is shown in Fig. 8. A point in these diagrams represents a ray, and the separating surfaces precisely mark the distinction we need to study here. The problem becomes almost trivial, however, if we already know by which face the ray leaves the surrounding cube: Then we just have to check whether the boundary point is inside or outside the elliptope, for which we have a convenient third order criterion. The following proof is based on the case distinction by cube faces. But since these are all connected by symmetries, it boils down to just considering one case.
Proof of Prop. 17:
(1)⇔(2): We use the normal cycle: The set of pairs (c, f ) ∈ N(Q) such that both c and f are uniquely determined when the other is fixed is just the stratum (Qqx,qx). Hence, if f is of type (Qqx), so is c.
Reduction by symmetry:
Applying a symmetry to f , i.e., a permutation of the components or an even number of sign changes, clearly does not change the validity of (1), (2), (3), or (4), while (5) respects the symmetry by requiring the necessary transformation to be made first. Hence it suffices prove for f in a standard form achievable by symmetry transformation. Since all extreme points of C are connected by symmetry we can assume, as required in (5) that a maximizer for c · f in ∂ e C is c = (1, 1, 1, 1). Now suppose that two or more coordinates f ij < 0. Then, by applying an even sign change to c we could increase c·f . So our assumption actually rules out more than one negative sign. By the same argument, if there is a negative sign, this must be on an f ij which minimizes |f ij |. By a permutation we may assume that this element is f 22 .
We can quickly handle the case that all f ij ≥ 0. In that case, max{c · f |c ∈ C} = ij f ij = ij |f ij | = max{c · f |c ∈ N }. So the maximum over Q is attained at a classical point, p(f ) ≥ 0 and the product in (42) cannot be < 0, so all conditions evaluate to false. So we may assume that there is just one negative sign and sort the remaining f ij . That is, from now on we take
f 11 ≥ f 12 ≥ f 21 ≥ |f 22 | > 0 > f 22 .(67)
not unitarily equivalent. More generally, any classical correlation c ∈ C allows a model with a 1 = +1, and another with a 1 = −1. So no classical correlation can have property (2). Any c ∈ Q has some cyclic model: By definition of Q, it has a model. The state induced on the algebra A generated by all A i , B j in that model then has a GNS-representation, which is cyclic. Moreover, it has a finite dimensional cyclic representation. Indeed, by Carathéodory's Theorem we can represent c as a finite convex combination of extreme points. Since we have provided finite dimensional models for each of these, we can combine them into a single cyclic model by the direct sum construction
H = k H k with A i = k A (k) i , B j = k B (k) j , and Ψ = k √ p k Ψ (k) ,(69)
where p k is the convex weight of the k th contributing extreme point. This could fail to be cyclic, for example, if two of the extreme points used were actually (needlessly) the same. In that case one can take the cyclic subspace generated from Ψ instead. In any case, from Carathéodory's bound we can get by with 5 terms, so the overall dimension can be chosen to be ≤ 20.
The direct sum construction can be reversed: If P k are orthogonal projections in the Hilbert space of some cyclic model commuting with all A i , B j , then setting H k = P k H,
p k = Ψ|P k Ψ , Ψ (k) = √ p k P k Ψ, and A (k) i , B (k) j
the restrictions of A i , B j to H k we have just written the given model in the form (69).
Proof of Thm. 19:
(1)⇒(2): Let c be a non-classical extreme point, and let Ψ be the cyclic vector of some quantum model. By Prop. 12, there is a unique matrix completion C with −1 < u, v < 1 and rank 2. The real part of the Gram matrix of the four vectors χ 1 , . . . , χ 4 = A 1 Ψ, A 2 Ψ, B 1 Ψ, B 2 Ψ is such a completion, so it must equal C. Thus e A 1 Ψ|A 2 Ψ = u, but it is not immediately obvious that this scalar product has to be real. Since rank C = 2 we have two linearly independent real vectors ξ in the kernel of C. Any such vector satisfies ij ξ i χ i |χ j ξ j + ij ξ i χ i |χ j ξ j = 0. Since both terms are positive, they must vanish separately, and j ξ j χ j = 0. Thus the linear relations between the χ i are given by real coefficients. Since the 1, 2 and 3, 4 submatrices of C are both nonsingular, either (χ 1 , χ 2 ) or (χ 3 , χ 4 ) can serve as a basis of this subspace, and the two are related by a non-singular real 2 × 2-matrix γ, so
B j Ψ = i γ ji A i Ψ(70)
In particular, c ij = A i Ψ|B j Ψ = A i Ψ|A k Ψ γ jk . Since the matrices c and γ are real, so is the scalar A 1 Ψ|A 2 Ψ = u. Thus, there was no need after all to take the real part in the first paragraph of this proof. It follows from (70) that non-commutative polynomials in A 1 , A 2 acting on Ψ already span the whole space. By cyclicity of Ψ this is true for polynomials involving also the B j . Now we can successively get rid of the factors B j in any polynomial acting on Ψ: In any monomial consider the rightmost factor B j , so we have an expression of the form
M B j M A Ψ = M M A B j Ψ = i γ ji M M A A i Ψ,(71)
where M is a monomial involving factors A i , B k and M A is a monomial containing only factors A i . In the evaluation we used that B j commutes with all A i , hence with M A , and (70). By downwards induction on the number of B j -factors we get that the vectors M A Ψ span the space. Symmetrically the same is true for polynomials in B. Next suppose that f (A 1 , A 2 , 1)Ψ = 0 for some non-commutative polynomial.
Then even the operator equation f (A 1 , A 2 , 1) = 0 holds, because we can multiply with any polynomial in B, commute through and use the cyclicity just established. The first application is to f = 1 − A 2 1 , which is positive, and has zero expectation Ψ|f Ψ because C 11 = 1. This implies f Ψ = 0, and hence f = 0. We conclude that
A 2 1 = A 2 2 = B 2 1 = B 2 2 = 1. Similarly, Ψ = B 2 j Ψ = B j i γ ji A i Ψ = i γ ji A i B j Ψ = i γ ji A i 2 Ψ = (γ 2 j1 + γ 2 j2 )Ψ + γ j1 γ j2 (A 1 A 2 + A 2 A 1 )Ψ.(72)
If γ j1 γ j2 = 0 then (A 1 A 2 + A 2 A 1 )Ψ is a multiple of Ψ, and so (A 1 A 2 + A 2 A 1 ) is a multiple of the identity, and from C 12 this multiple is 2u. This conclusion does not depend on j, so it is valid whenever γ j1 γ j2 = 0 holds either for j = 1 or for j = 2. Now if γ j1 γ j2 = 0, one of the coefficients is zero, and by (70) B j Ψ is proportional to some A i Ψ. By taking the norm, the factor (the non-zero γ ji ) is ±1 and c ij = ±1. That is, the correlation is on an N -facet. But this cannot hold for the other B j at the same time except for a classical c. Hence in either case
(A 1 A 2 + A 2 A 1 ) = 2u1.(73)
Using this identity and A 2 i = 1, every word in A 1 and A 2 simplifies to a linear combination of 1, A 1 , A 2 , and A 1 A 2 . Hence the *-algebra generated by A 1 and A 2 is at most four dimensional. It is noncommutative, because A 1 A 2 = A 2 A 1 would imply 4u 2 1 = (A 1 A 2 +A 2 A 1 ) 2 = 41, hence |u| = 1. Therefore that algebra is isomorphic to C 2×2 , the smallest noncommutative finite dimensional C*-algebra.
Our next step is to evaluate the state given by the vector Ψ on our algebra:
Ψ|M Ψ = 1 2 trM(74)
when M is a polynomial in the A i . Indeed
0 = Ψ|M B j Ψ − B j Ψ|M Ψ = i γ ji Ψ|(M A i − A i M )Ψ .(75)
Since γ is non-singular, we conclude Ψ| (74). Since A 2 j = 1 and A * j = A j , we must either have A j = ±1, which would entail classical correlation, or else trA j = 0. The trace of (73) gives trA 1 A 2 = u and so we have evaluated Ψ|M Ψ on all polynomials in A i . By the reduction process used above we can also compute the expectations involving polynomials in both A i and B j . Hence by the GNS construction the cyclic model is unique. Moreover, it is equivalent to the model in Sect. 4.1 via a unique unitary isomorphism that maps the linearly independent (since |u| < 1) vectors Ψ, A 1 Ψ, A 2 Ψ, A 1 A 2 Ψ to the corresponding vectors of the standard model (then also cyclic). This finishes the proof. We remark that the family of standard models was defined for points other than those in (1) may very well be cyclic but not unique for these points.
(2)⇒(1): Assume c has a unique cyclic model. Any property that holds in some cyclic model must be true for this unique one. For example, the unique model must be finite dimensional. Similarly, dilation theory tells us that A 2 1 = 1. To this end, we first decompose the algebra into irreducible summands. In each summand the algebras generated by A 1 , A 2 and B 1 , B 2 must themselves be irreducible, so full matrix algebras. Hence the algebras are combined in a tensor product, and are represented on a Hilbert space tensor product H A ⊗ H B . We can write A 1 = V * A 1 V ⊗ 1 B , where V : H A → H A is isometric, and A 2 1 = 1. This construction preserves the commutativity of A i and B j , and mapping the cyclic vector by V ⊗ 1 B to the larger space we get a model with A 2 1 = 1. Restricting to the cyclic subspace preserves this property. Hence the unique cyclic model must also satisfy it, and similarly for the other operators. Now suppose there is a factor λ > 1 such that λc ∈ Q. Then using a cyclic model for λc and scaling down the A i by a factor 1/λ we get a cyclic model with A 2 1 = 1. Hence c does not have a unique model. It follows that c must be on the boundary of Q.
Consider now the boundary classification of Prop. 7. The classical types (Qcx) and (Qce) anyhow fail to have unique models, as mentioned in the beginning of this section. So in order to show that c is of type (Qnx) or (Qqx), a non-classical extreme point, we only need to exclude points in an elliptope interior (type (Qei)). Let us assume without loss that c 11 = 1. Consider the one parameter family of correlations (1, λc 12 , c 21 , λc 22 ) with λ increasing from 1. This will intersect the elliptope boundary for some λ > 1. Starting from a model at that point, and scaling B 2 → B 2 /λ we can obtain a model with B 2 < 1. As argued before, this contradicts the uniqueness of the cyclic model.
(3)⇒ (2): Consider a cyclic model of the type described in (2). In ρ = |Ψ Ψ| the state vector must be of the form Ψ = Ψ ⊗ Ψ ν ⊕ 0. The cyclic subspace contains only vectors of the form (XΨ) ⊗ Ψ ν ⊕ 0, so H 0 = {0} and H ν = CΨ ν ∼ = C, and this tensor factor can be omitted.
(2)⇒(3): Given a model H for c, let us denote by A the norm closed algebra generated by the A i , B j , by A its commutant, i.e., the algebra of operators commuting with every element of A. By K ρ , we denote the vector space of φ ∈ H such that |φ φ| ≤ λρ for some factor λ. The closure of K ρ is called the support of ρ, but in infinite dimension may be larger. Then we define H 0 as the subspace orthogonal to all vectors of the form XY φ with X ∈ A, Y ∈ A , and φ ∈ K ρ . Since this H 0 is an invariant subspace for all X ∈ A, every X ∈ A splits into a direct sum of a component on H 0 , about which we can say nothing at all, and a rest, which we will need to characterize. So we may assume H 0 = {0} in the sequel.
For any unit vector of the form Ψ 1 = Y φ with Y ∈ A and φ ∈ K ρ consider the cyclic representation subrepresentation of A on AΨ 1 . Then for positive X ∈ A we get
Ψ 1 |XΨ 1 = φ|Y * √ X 2 Y φ = φ| √ XY * Y √ Xφ ≤ Y 2 φ| √ X √ Xφ = Y 2 tr(|φ φ|X) ≤ λ Y 2 trρX.
Thus as a functional on A the state defined by Ψ 1 is dominated by a multiple of ρ, hence is a convex component of ρ. Since we have already established from (2) that c is extremal, Ψ 1 defines again a cyclic model for c, and is hence unitarily isomorphic to the standard model for c. In particular, all algebraic identities of that model hold on AY φ, and since such vectors span dense subspace, they hold on all of H.
There is now a polynomial G in the generators A i , B j that in the standard model is equal to the one-dimensional projection onto the state vector. WhenX ∈ B(Ĥ) and X ∈ A are given by the same polynomial in the generators, we therefore have GXG = Ψ |XΨ G. Moreover, all vectors Y φ from the previous paragraph are in GH. We can therefore define H ν = GH, and get a unitary operator U :Ĥ⊗H ν → H defined by U (XΨ⊗Φ ν ) = XΦ ν . Since ρ has support in GH we have U * ρU = |Ψ Ψ |⊗ρ ν for some state ρ ν ∈ B(H ν ), as claimed in the standard form (3).
(3)⇒(6): From the explicit form of the model it is clear that any operator E commuting with A is of the form E = 1 ⊗ E 1 ⊕ E 0 . Hence tr(ρEX) = tr(ρ ν E 1 ) Ψ |XΨ = tr(ρE) tr(ρX).
(6)⇒(5): This is trivial, because the factorization is no longer demanded for all X ∈ A, but only for those X actually measured in the experiment (including the marginals!).
(5)⇒(4): Since the marginals are now included, their expectations, which define p must also be those of the unique model, i.e., zero.
(4)⇒(1): First we eliminate the classical extreme points as they do not have unique extensions p, as explained in the beginning of this Section. If c is not extremal, i.e., on an edge, in an elliptope interior or the interior of Q, then c = λc + (1 − λ)c for some λ > 0 and c classical. So we get distinct extensions by fixing the extension of c but changing that of c .
Dependence on Hilbert space dimension
We now return to Sect. 4.5, and we prove the remaining statement on the Hilbert space dimension.
Proof of Prop. 20:
(1) Let c ∈ Q m be a non-classical correlation in a finite dimensional Hilbert space. Then the algebra A generated by A i , B j can be decomposed into irreducible components, represented on orthogonal subspaces H α , of which we consider one. By definition, any operator in the algebra generated by A 1 and A 2 will also commute with A, and hence be a multiple of 1 on H α . Hence on H α the algebra generated by the A i has trivial center, and is thus a full matrix algebra, say the n × n matrices. Similarly, we get a matrix dimension n for the algebra generated by the B j . Since the two sets of operators commute, and even generate commutants of each other, dim H α = nn . Hence, by assumption, nn ≤ m ≤ 3, so either n = 1 or n = 1. Taking the first case without loss, A i = a i 1 and the contribution from this summand is c ij = tr α (ρA i B j ) = a i tr α (ρB j ), which is classical. Since this holds for all (up to three) summands, c ∈ C.
(2) By monotonicity, we only need to show that Q ⊂ Q 4 . For any given model c, and λ ∈ [0, 1], the correlation λc can be realised in the same dimension, by taking A i = λA i , B j = B j , and the same state. So if c is a correlation, for which the ray Rc leaves Q at a point of type (Qqx), then c ∈ Q 4 . By the same argument, it remains to show that all N -facets are in Q 4 ; without loss consider the facet c 11 = 1. Since the boundary of the elliptope is in Q 4 , we can apply the scalings A 1 = A 1 , B 1 = B 1 , A 2 = √ λ A 2 , and B 2 = √ λ B 2 to conclude that the interior is likewise in Q 4 .
Fixed state
Proof of Prop. 21:
The extreme points of co Q(ρ) must have observables with A 2 i = B 2 j = 1, i.e. A i = a i · σ = k a i,k σ k with a i is a unit vector. Moreover, in two dimensions, A i must therefore a multiple of the identity, so a i has one zero component. If any of the A i , B j are ±1, the resulting correlation is classical, and all extremal classical correlations can be obtained in this way, for any state ρ. For the non-classical extreme points, our parametrization gives
c ij = trρA i ⊗ B j = a i · R b j ,(76)
where the R is the matrix of Pauli expectations given in the Proposition. The map R → U RV for orthogonal matrices U, V does not change the correlation c, and therefore the set of such correlations. What matters is the singular values of R: λ 1 ≥ λ 2 ≥ λ 3 ≥ 0. The same logic applies if we constrain the vectors on each side to a two-dimensional subspace, with the only difference that no third singular value λ 3 appears. Moreover, in this case we can just set a i = cos(α i ), sin(α i ) . Hence, with such a constraint we have c ∈ Q st (λ 1 , λ 2 ), for λ 1 , λ 2 the two singular values of the R matrix restricted (as a quadratic form) to the chosen subspaces. It remains to optimize the two-dimensional subspaces containing the a i , b j . The family of sets Q st (λ 1 , λ 2 ) is increasing in each variable λ 1 and λ 2 . Therefore the optimum over the two dimensional subspaces becomes maximal when the singular values for the subspaces become largest, which is the case, when we just pick the two largest singular values of the 3 × 3-matrix R.
Finally, we have to check when Q(ρ) becomes classical. This happens iff all CHSH inequalities are satisfied, also in Q st (λ 1 , λ 2 ). By symmetry and from (5) we have
CHSH(c) = 1 2 a 1 · R( b 1 + b 2 ) + a 2 · R( b 1 − b 2 ) ,(77)
which is clearly maximized with respect to the unit vectors a i as 1 2 R( b 1 + b 2 ) + R( b 1 − b 2 ) . Since the b 1 ± b 2 are orthogonal vectors, we can write b 1 ± b 2 = λ ± e ± , where e + and e − are orthogonal unit vectors and λ 2 + + λ 2 − = b 1 + b 2 2 + b 1 − b 2 2 = 4. Thus by Schwarz inequality max CHSH(c) = 1 2 λ + R e + + λ − R e − ≤ R e + 2 + R e −
The choice of orthogonal unit vectors e ± maximizing this expression is the eigenvectors of R * R for its to largest eigenvectors, which are λ 2 1 and λ 2 2 .
Fixed observables
Proof of Prop. 22: By Jordan's Lemma [43] or C*-algebra of two projections [53], any choice of operators A 1 , A 2 and B 1 , B 2 is a representation this algebra and so it decomposes into irreducibles. We first show that Q(A i , B j ) = co{Q obs (u, v)| u ∈ Σ A , v ∈ Σ B } for some correlation sets Q obs (u, v) to be defined later. Indeed, let Π i denote the central projection onto the i th irreducible summand, then restricted to this summand the operator A 1 A 2 + A 2 A 1 = 2u1 for some u ∈ Σ A because it commutes with all generators. This can be used to drastically simplify polynomials in A 1 , A 2 , so that they generate the algebra of 2 × 2-matrices, except at the endpoints u = ±1, where they form a commutative subalgebra. That is, up to isomorphism we can take A 1 = σ 1 , and A 2 = cos α σ 1 + sin α σ 3 , so that u = cos α. The same holds for Bob with central parameter v, and the operators are combined as A i ⊗ B j . Whenever λ i = tr(ρΠ i ) = 0, we can define the state ρ i = λ −1 i Π i ρΠ i , and the corresponding correlation c i ∈ Q obs (u, v), the affine image of the state space of C 2 ⊗ C 2 under ρ → tr(ρA i (u) ⊗ B j (v)). Thus, c ∈ Q(A i , B j ) implies that it is a convex hull of points in Q obs (u, v). The converse is also true.
The sets Q obs (u, v) are affinely isomorphic to each other except when u, v = ±1. To see this, observe that A i (u) = i S(u) ii A i (0), likewise for B j (v), and consequently c(u, v) = S(u)c(0, 0)S(v) where both c(u, v) ∈ R 4 and c(0, 0) ∈ R 4 are formed with the same density operator ρ, namely c ij (u, v) = trρA i (u) ⊗ B j (v). This gives Q obs (u, v) = S(u)Q obs (0, 0)S(v) , which is sufficient for the purpose of this Proposition. Note, in addition, that Q obs (u, v) are all non-isometrically isomorphic to Q obs (0, 0) since S(u) are invertible whenever −1 < u < 1.
It remains to show that ∂ e Q obs (0, 0) = O(2). Since all A i (0) ⊗ B j (0) commute with U = σ 2 ⊗ σ 2 , we can assume without loss that ρ commutes with U . Moreover, extreme points must come from pure states ρ = |ψ ψ|, which by the argument in Sect. 5.1.1 we can choose as given by a real vector ψ. Together these imply ψ + = (cos t, sin t, sin t, − cos t)/ √ 2 or ψ − = (cos t, sin t, − sin t, cos t)/ √ 2, which gives two families of extreme points − cos 2t sin 2t sin 2t cos 2t and cos 2t − sin 2t sin 2t cos 2t ,
belonging to two orthogonal subspaces of R 4 . These are exactly all the 2 × 2 orthogonal matrices: the reflections and rotations.
Proof of Prop. 23:
The general element of Q obs (0, 0) can be written as c = λ cos α − sin α sin α cos α + µ cos β sin β sin β − cos β ,
where λ, µ ≥ 0, and λ+µ < 1. The two extremal circles are, by definition, the points with {λ, µ} = {0, 1}. This covers convex combinations form different circles, when λ + µ = 1. If some contributions to a convex combination come from the same circle, this just reduces the corresponding coefficient. From this description it is clear that all points with λ + µ < 1 belong to the interior of the body. The respective planes are given by c 11 − c 22 = c 12 + c 21 = 0, and c 11 + c 22 = c 12 − c 21 = 0, respectively. They are clearly Euclidean orthogonal. The body is symmetric under rotations in any of these planes, so all connecting edges are symmetry equivalent, and one can also check directly that they are all exposed. Note that the image of a cylinder that three-dimensional geometric intuition maybe suggests as the convex hull of two circles is entirely misleading here: The interiors of the circles are not faces but in the interior of the body.
To get the semialgebraic description, note that, with s = ij c 2 ij ,
So in order to get a polynomial condition for the c ij we need to express the necessary and sufficient condition λ + µ ≤ 1 in terms of λ 2 and µ 2 . By successive squaring we get first 2λµ ≤ (1 − λ 2 − µ 2 ), and then 0 ≤ (1 − 2λ 2 − 2µ 2 + (λ 2 − µ 2 ) 2 ). Using s = 2(λ 2 + µ 2 ) and det c = (λ 2 − µ 2 ) gives the inequality for given in the proposition. On the boundary λ + µ = 1 these inequalities are equalities. So this polynomial is clearly the algebraic description of the boundary. However, the second squaring step is not reversible, so we get some unwanted elements in the Zariski closure. As for Q, where we had to augment the boundary polynomial h by an additional (non-unique) polynomial inequality g ≥ 0, there are different ways of achieving this. Fig. 22 is the map for this purpose. We see that we can either supplement the polynomial inequality λ 2 + µ 2 ≤ 1, meaning s ≤ 2, or by the determinant inequality given in the proposition.
Proof of Prop. 24:
We have shown in Prop. 24 that on Q obs (0, 0) the determinant inequality holds, and by Prop. 22 and the observation that det S(u) ≤ 1, the same holds for all components Q(A i , B j ) with different singleton spectra Σ A , Σ B . This would finish the argument if det were a convex function. However, it clearly is not. We therefore take a more global approach using the detailed knowledge compiled in the paper. First of all, it suffices to show the determinant inequality on the boundary ∂Q, because the determinant is homogenous and every point c ∈ Q can be written as c = λc with λ ≤ 1 and c ∈ ∂Q. This shows that also the cases of equality can only lie on the boundary. For the curved tetrahedra we can use the cosine parametrization Prop. 8. The determinant is easily evaluated and simplifies to det c = − sin(α + β) sin(α + γ) ∈ [−1, 1]. Figure 22: The condition λ + µ ≤ 1 is expressed in terms of λ 2 and µ 2 as the gray shaded region near the origin. The bounding parabola extends further, so the inequality which it expresses needs to be supplemented by further conditions to exclude the red shaded areas.
We choose the intersection with the green strip, which amounts to a condition on det c. But other choices, e.g., the triangle bounded by the dashed blue line would also work. The latter would express that Q obs (0, 0) is contained in a ball of radius √ 2.
On an N -facet, say {c 11 = 1}, the determinant is an affine function of c 22 . Hence its maximum will be attained on the bounding elliptope, where the previous argument applies. For the case of equality it suffices to find points in the parametrized surface with det c = −1, since the sign can be reversed by a symmetry transformation. From (82) we find, up to an overall sign for all angles and modulo 2π that α + β = π/2 = α + γ. So we get β = γ, making c Alice-Bob symmetric. Moreover, the fourth angle is δ = −(α + β + γ) = −α − 2(π/2 − α) = α − π, so that c 22 = −c 11 . This singles out the circle of reflection matrices, shown in red in Fig. 19. 6 Outlook 6.1 What more of Q?
There are some geometric aspects of Q we did not discuss, although they might be natural and interesting.
• Robustness of self-testing
In the practice of quantum key distribution, the correlation c is determined by statistical evaluation.
The assumption that c is an extremal point is never exactly verifiable. Therefore, one needs explicit bounds of the sort: When c is known up to accuracy ε (confidence intervals), then the eavesdropper cannot know more than δ about the bits used for key generation. Various definitions of "accuracy" and "know more" can be given. In any case, the concrete bound δ as a function of ε will depend also on the local geometry of Q, particularly the curvature. Giving more details here would have turned this into a paper on key distribution. Generally speaking, self-testing is a robust phenomenon: Near-extremal correlations allow the conclusion that the observables involved can be deformed (norm-)slightly so that they turn into the minimal two-qubit example [67].
• Integral curvatures As mentioned in Sect. 3.4, the Steiner volume polynomial contains information about curvature integrals of the boundary. In Sect. 2.9 we merely determined the volume, but no further coefficients.
• Weakly self-dual geometry The Hadamard matrix defines an indefinite pseudo-Euclidean metric on R 4 , for which Q is self-dual. Some natural questions come with this structure, but it is unclear whether it sheds any light on Q.
• Constrained Hilbert space dimension The basics were discussed in Sect. 4.5, but a full characterization is still lacking.
How does this generalize?
Many of the techniques described above were originally drafted to address more general situations. Let us briefly indicate their natural levels of generality.
• The full 222 case, and the C*-algebra generated by two projections In the minimal 222 case, without the zero marginals condition, the quantum body Q is a convex body in R 8 . The semidefinite matrix completion point of view is not directly effective for breaking this down to a finite dimensional problem. As has been noted also by Masanes [43] (actually more generally for the N22 case) this can be achieved by the representation theory of the universal C*-algebra generated by two projections [53]. This provides a description of the extreme points parametrized by the product of spheres S 1 ×S 1 ×S 3 , analogous to Prop. 8. A Macaulay2 computation reveals that this variety has degree 40 and its prime ideal is generated by 28 polynomials whose degrees are 5, 6, 7 and 8. This may be a starting point for the analysis of the 222 case.
• Higher universal correlation bodies It is clear that the analog of Q can be defined for larger N , M , or K, or any specification of parties settings and outcomes. However, the construction is notoriously non-constructive; see [1].
• Semidefinite hierarchies We used this extensively, and got a complete characterization of the minimal Q out of it, just using Level 1. In fact, this characterizes the minimal case: If the upper bound provided by this method is tight, we must be in the 222|0 case. For larger N , M , or K, this gap can be expected to become rapidly larger. Hierarchies are still the key tool for getting upper bounds, but tightness is too much to ask. It should also be noted that in spite of the proven convergence of the hierarchy it is computationally unfeasible to really push this to high levels. It is unclear (to us) for which other cases a tight bound at some higher but finite hierarchy level holds.
• Parametrized extreme points Although this is perhaps best understood via the two-projections theory, another parametrization of the extreme points (not explicitly using this theory) in the N22 case was given in [78].
• Duality
The duality in Thm. 3 clearly depends on minimality: The duality of N and C requires at least the dimensions of these sets to coincide, which fails for M K > 4.
• Correlation matrices and Clifford algebras All the above extensions drop the 0-marginal condition or increase N . We stress that Tsirelson's technique of correlation matrices is an extension in another direction, i.e., to 2M2|0, with general M .
• Algebra Algebraic methods are expected to apply once a reduction to finite dimension has been achieved by other means. We found them directly useful in the full 222 case (work in progress), but there are clearly also applications to N22 and 2M2 cases. However, the complexity of algebraic characterizations can be expected to increase very rapidly. Even worse are the case distinctions and inequalities of a semi-algebraic description. We can understand this from the classical case, the characterization of C. Here the algebra is linear, but the inequality part, i.e., the determination of all Bell inequalities, is a family of convex hull problems, which is known to grow badly [50]. So no general results can be expected, but as in the classical case [48, Problem 1] one can look for trails into the wilderness, e.g., infinite families of models defined by special properties or symmetries.
• Algebraic statistics Consider the ±1-valued random variables A 1 , A 2 , B 1 , B 2 satisfying (1). Their statistical model is the graphical model whose graph G is the 4-cycle with edges
A 1 − B 1 , B 1 − A 2 , A 2 − B 2 , B 2 − A 1 .
This is [23,Example 4] up to relabelling. The sufficient statistics of this model are obtained by applying the linear map A(G) in [23,Example 4]. The image of this map is our classical polytope C. In particular this map is a bijection between the model and C and inverting it is called maximum likelihood estimation (MLE). We can recover also N from this construction, and more in general such polytopes can be defined for any toric model. For an undirected graphical model G we have that C = N if and only if the graph is decomposable; the four-cycle model is the smallest nondecomposable model so we can deduce also from this argument that the inclusion is strict. It would be particularly interesting to examine the general 222 case through the lens of algebraic statistics.
from Physics: Quantum Correlations . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 A View from Mathematics: Convex Algebraic Geometry . . . . . . . . . . . . . . . . . . . 6 1.3 Short Review of Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Description of the Correlation Body 9 2.1 Gallery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Enclosing and enclosed polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Pushout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.5 Boundary, faces, and extreme points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.6 Curved tetrahedra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.7 Semialgebraic description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.8 Spectrahedral shadow . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 20 2.9 Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3 Description of the Dual Body 22 3.1 The duality transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Parametrized extreme points of Q • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3 Dualized descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.4 The normal cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.5 Support function and gauge function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4 Quantum Connections 30 4.1 Quantum models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 Geometric aspects of quantum key distribution . . . . . . . . . . . . . . . . . . . . . . . . 32 4.3 Uniqueness of quantum models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.4 The role of the elliptope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.5 Fixed Hilbert space dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.6 Fixed State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.7 Fixed Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5 Proofs 42 5.1 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.1.1 Making matrix completion real . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.1.2 Solving the real completion problem . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.1.3 Joint positivity of two parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.1.4 Properties of the boundary and extreme points . . . . . . . . . . . . . . . . . . . . 46 5.1.5 The pushout characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.2 Self-duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.3 Further boundary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.3.1 Classification of boundary points by rank . . . . . . . . . . . . . . . . . . . . . . . 49 5.3.2 Exposing functionals: first order analysis . . . . . . . . . . . . . . . . . . . . . . . 50 5.3.3 Unique CHSH violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.3.4 Face orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.4 Support function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.5 Quantum representations of extreme points . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.6 Dependence on Hilbert space dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.7 Fixed state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.8 Fixed observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6 Outlook 60 6.1 What more of Q? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.2 How does this generalize? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Figure 1 :
1A correlation experiment: Alice chooses setting i and Bob chooses setting j. The outcomes of A i and B j are ±1. One measures the correlation c ij of these outcomes.
1 2
1(c 11 + c 12 + c 21 − c 22 ) ≤ 1.
Figure 2 :
2The Alice-Bob symmetric subsets of correlations (c 12 = c 21 ), both a projection and a section of their 4-dimensional counterparts. Left:
Figure 3 :
3Parallel sections c 11 = t through the body Q, for t = −1, −0.8, 0, 0.8, 1.
Figure 4 :
4Sections through Q perpendicular to a long diagonal of the cube N . There are two symmetry classes of diagonals: those connecting two classical correlations (left), and those connecting two PR-boxes (right). The cut on the right goes through the origin. On the left we would get the full octahedron. Our cut is thus slightly off-center.
cube N consists of the tuples c = (c 11 , c 12 , c 21 , c 22 ) with −1 ≤ c ij ≤ 1. It has 2 4 = 16 vertices, namely the points in {−1, 1} 4 . The classical distributions are the mixtures of uncorrelated c, i.e., c ij = a i b j for some a i , b j ∈ [−1, 1]. A classical correlation is extremal when a i , b j = ±1. Hence, the extreme points of C are also extreme points of N , but which? This is decided by a sign: For a classical extreme point we have c 11 c 12 c 21 c 22 = a +1. So the classical extreme points of C are just the 8 even vertices of N . The odd vertices are the so-called Popescu-Rohrlich (PR-)box correlations. They have neither classical nor quantum realizations: In (5) they satisfy CHSH(c) = 2, which clearly exceeds the quantum maximum.
Fig. 2 and by the N -facets because the hyperplanes {c 12 = c 21 } and {c 11 = −1} are invariant under sin.
Figure 5 :
5The 2-dimensional version of the pushout map in
, one can take the swap (c 11 , c 12 , c 21 , c 22 ) → (c 12 , c 11 , c 21 , c 22 ).
Figure 6 :
6Symmetries that preserve the definition of quantum correlations. First row: Sign patterns. Second row: The eight permutations arising from a swap of partners or relabeling. The symmetry group G of Q, as described in Prop. 6, is larger: it allows all 24 permutations.
Figure 7 :
7Every 3-dimensional face of Q is an elliptope. Points of type (Qcx) are vertices of the blue tetrahedron, (Qce) give blue edges, (Qnx) is the orange surface. Extreme points of the elliptope are exposed in 3 dimensions, but not in 4. Type (Qei) comprise the interior.
[Qnx] is the intersection of a [Qqx] and a [Qei]-set. The triangular surfaces (Qnx) are thus faithfully portrayed in Fig. 7. The intersection of two [Qqx]-surfaces is lower dimensional: If they are not disjoint opposites, they intersect in a straight edge [Qce].
Figure 8 :
8Stereographic projection of the boundaries of C (top row) and Q (bottom row).
Proposition 8
8(→Sect. 5.1.4) The threefolds (Qqx) of exposed extreme points on Q are parametrized by c = (c 11 , c 12 , c 21 , c 22 ) = (cos α, cos β, cos γ, cos δ)
Proposition 9
9(→Sect. 5.3) In the cosine parametrization of Prop. 8, taking ∆ = 0 parametrizes further boundary strata of lower dimension. Specifically, we get points of type (Qcx) if and only if α, β, γ, δ are all multiples of π, (Qce) if and only if exactly two of these angles are multiples of π, and (Qnx) if and only if exactly one of these angles is a multiple of π.
−
(c 11 +c 12 −c 21 −c 22 )(c 11 −c 12 +c 21 −c 22 )(c 11 −c 12 −c 21 +c 22 )(c 11 +c 12 +c 21 +c 22 ).
Figure 10 :
10Inside the hyperplane {c 12 = c 21 }, we consider the two quartic surfaces {g = 0} (right) and {(c 11 − c 22 ) −2 h = 0} (left). The two surfaces intersect transversally in the red curves that are seen in the boundary of the cube.
Figure 11 :
11A slice that illustrates how the two surfaces in
matrix completion problem has a unique solution (u, v) if and only if c ∈ ∂Q.
Figure 12 :
12Nonuniqueness of matrix completion for a family of correlations c = tc CHSH +(1− t)c center with c center = (−1, 0, 0, 0) along the line segment from the center of an elliptope facet (top) to the CHSH-point (bottom). The horizontal cuts for fixed t (black meshes) represent the pairs (u, v) that make (22) positive semidefinite. These slices are 2-dimensional convex bodies, shrinks to a point at both boundaries. The figure is a quartic spectrahedron [49].
Figure 13 :
13Polar duals for the Alice-Bob symmetric sections shown in Fig. 2. The marked vertices of C • (blue cube frame) correspond to the facets of C in Fig. 2. The figure of Q • (right panel) is an affine transformation of Q in Fig. 2. This expresses the self-duality.
and f satisfies(30). Its Zariski closure is an irreducible threefold in C 8 . Itsprime ideal is generated by 17 polynomials. The first three of these 17 generators are familiar: = c 11 f 11 + c 12 f 12 + c 21 f 21 + c 22 f 22 − 1, h = 4(c 11 c 22 − c 12 c 21 )(c 11 c 21 − c 12 c 22 )(c 11 c 12 − c 21 c 22 ) −(c 11 +c 12 −c 21 −c 22 )(c 11 −c 12 +c 21 −c 22 )(c 11 −c 12 −c 21 +c 22 )(c 11 +c 12 +c 21 +c 22 )
( 5 )
5Perform a symmetry transformation (even number of sign changes) so that the maximum (39) of f · c over C is attained at the extreme point c = (1, 1, 1, 1), i.e., φ C (f ) = f 11 +f 12 +f 21 +f 22 . Then f 11 f 12 f 21 + f 11 f 12 f 22 + f 11 f 21 f 22 + f 12 f 21 f 22 < 0.
can we deduce about the uniqueness of quantum realization of points in Q? Prima facie there is no reason for any uniqueness. In Sect. 4 we made special choices to realize the extreme points somehow. Why m = 4 in Lem. 18? Why not m = 3? What about m = ∞? That there is limited choice if we try to find realizations in minimal dimension says very little about the non-uniqueness if we allow more spacious Hilbert spaces. We begin by noting that there are some trivial ways in which uniqueness fails: • Unitary transformation This refers to a change of basis in the Hilbert space H. It makes the quantum model looks different but does not change the correlations c ij . They are invariants of the action of the unitary group U (m) on quintuples of matrices (ρ, A 1 , A 2 , B 1 , B 2 ) as in Sect. 1.1, with H = C m . Two unitarily equivalent models, i.e., two quintuples in the same U (m)-orbit, are considered to be "the same". • Expansion This means enlarging the Hilbert space H of the model by an additional summand H 0 where the states act trivially. If we use the space H ⊕ H 0 , the state ρ ⊕ 0 and the observables A i ⊕ A 0 i and B j ⊕ B 0 j , with A 0 i and B 0 j arbitrary, provided (1) still holds, then the correlations do not change. • Adding multiplicity
Figure 15 :
15The set C 1 ≡ Q(ρ A ⊗ ρ B ) = N ∩ {c : det(c) = 0} in the Alice-Bob symmetric plane. The black bounding cube is N , and the edges of C are given as blue lines. The section of C 1 with this plane is the (purple) double cone c 11 c 22 = c 2 12 emanating from the origin. The projection contains additional points bounded between the double cone and the (top and bottom) surfaces 2|c 12 | − 1 = c 11 c 22 .
Figure 16 :
16The map of singular value pairs (λ 1 , λ 2 ) in Prop. 21. In the shaded region C, Q st (λ 1 , λ 2 ) ⊂ C. The labelled points are: M: the maximally entangled case, P: a pure product state, all pure two-qubit states, parametrized for this purpose by their Schmidt spectra, lie on the segment PM. The diagonal gives scaled versions of Q. At S the set just touches C from within.
Figure 17 :
17The projection of the set Q st (1, 0.5) of Prop. 21 to the AB-symmetric plane. This is a 3-dimensional connected nonconvex semialgebraic set, the midpoint of a continuous family connecting Q st (1, 0) = C 1 fromFig. 15with Q st (1, 1) = Q fromFig. 2. The twoparameter family is homogeneous:
Proposition 22 (
22→Sect. 5.8) Let A i , B j be hermitian operators on a finite dimensional Hilbert space with [A i , B j ] = 0 and A 2 i = B 2 j = 1. Denote the spectrum of (A 1 A 2 + A 2 A 1 )/2 by Σ A , and define Σ B analogously. Then
and O(2) is the set of 2 × 2 orthogonal matrices, i.e. the circles of rotations and reflections about some axes.
Figure 18 :
18The fixed observable set Q obs (0.7, −0.9) of Prop. 22 in the Alice-Bob symmetric plane. The section with that plane is the orange body, while the projection is larger and is generated by the two circles shown. The red circle corresponds to maximal (positive) determinant, the blue one to minimal determinant.Proposition 23 (→Sect. 5.8) (1) The set Q obs (0, 0) = co O(2) is the convex hull of two circles. The circles both have the origin as their center, and lie in orthogonal planes of R 4 .
Proposition 24 (
24→Sect. 5.8) For c ∈ Q, | det c| ≤ 1 with equality if and only if c is an orthogonal matrix. In that case, in the unique cyclic quantum model for such correlations the identities A 1 A 2 = −A 2 A 1 , and B 1 B 2 = −B 2 B 1 hold, i.e., up to unitary isomorphism all measurements operators A i , B j are Pauli matrices.
Figure 19 :
19The determinant function in an Alice-Bob symmetric plane. Green: The two surfaces det c = −1 (one-sheeted hyperboloid, cut open for visibility) and det c = +1 (one sheeted hyperboloid touching N only in two points). Orange: the boundary of Q. Double cone inside Q is both section and projection of Q obs (0, 0). The intersection of all three sets is the red line, together with the two cone tips.
21 c 22 − (c 11 c 22 + c 12 c 21 )u)/(1 − u 2 ), and at that point we can compute the determinant and get max v∈R m 1234 = (1 − u 2 ) −1 m 123 m 124 ≥ 0.
Figure 20 :
20The black graph is the 4-cycle associated to Q (e) with vertices 1, 3, 2, 4 starting from the upper-left corner. It is not chordal. On the other hand, adding the orange edge, which corresponds to assuming the existence of u satisfying (52), makes it chordal.
Figure 21 :
21The configurations of two parabolas f i (x) = b i −(x−a i ) 2 with a 1 = 0 and a 2 = 1. Top Left: Parameter plane for (b 1 , b 2 ). Shaded: region, where intersection of positivity ranges is nonempty. Points A,B,C: parameters for the other panels. Black Parabola: line at which the intersection of the parabolas lies on the x-axis.
28 Q (e) = Q (d) : For c ∈ R 4 , we have c ∈ Q (e) if and only if c ∈ N and g(c) ≥ 0 or h(c) ≥ 0 . Furthermore, we have c ∈ int(Q (e) ) if and only if c ∈ int(N ) and g(c) ≥ 0 or h(c) > 0 .
, the inclusion Q (a) ⊂ Q (e) from Sect. 5.1.1, and the concrete quantum models from Sect. 4.1, which realize all cosine parametrized c, and thus show Q (b) ⊂ Q (a) . Altogether this gives Summary 31 Q (e) = Q (f ) , Q (e) = Q (d) , and Q (a) ⊂ Q (e) ⊂ Q (b) ⊂ Q (a) , i.e., all these sets are equal.
(
Qcx) ⊥ = [Qei] = (Qei)∪(Qnx)∪(Qce)∪(Qcx): Consider a classical extreme point in (Qcx); without loss of generality c = (1, 1, 1, 1). Then {c} ⊥ ⊂ {c} ⊥ C , i.e., {c} ⊥ is a face contained in a facet of the cube C • . This was our definition of an N -face.
(A i M − M A i )Ψ = 0. By induction on the degree of an arbitrary other polynomial M , we have Ψ|(M M − M M )Ψ = 0. This property uniquely identifies the tracial state
4λ 2 =
2(c 11 + c 22 ) 2 + (c 12 − c 21 ) 2 = s + 2 det c 4µ 2 = (c 11 − c 22 ) 2 + (c 12 + c 21 ) 2 = s − 2 det c.
, and Fig. 4.
Table 1 .
1The labels refer to the strata of ∂Q (cf. Prop. 7).c is a point of type
(Qqx)
(Qcx) (Qei) (Qce) (Qnx)
{c} ⊥ is a set of type {f }, f ∈(Qqx) [Qei] (Qcx) [Qce] (Qcx)
{c} ⊥⊥ is a set of type
{c}
{c}
[Qei]
[Qce]
[Qei]
dim{c} ⊥⊥
AcknowledgementsWe would like to thank M. Navascués, N. Gisin, A. Acín, J. Kaniewski, L. Masanes, and S. Pironio for helpful comments regarding the literature. T.P.L gratefully acknowledges support from the Alexander von Humboldt Foundation and the Austrian Academy of Sciences (project number M 2812-N).(1) means that c = 2Hf is a multiple of a (Qqx) point. We know that the component with the largest absolute value is the first, c 11 = ij f ij . Thus the point where the ray Rc intersects the boundary of N is (1, c 12 /c 11 , c 21 /c 11 , c 22 /c 11 ) = (1, x, y, z). This is in an N -facet of Q if and only if 1−x 2 −y 2 −z 2 +2xyz ≥ 0. Otherwise, this intersection is already outside of Q, and hence the ray intersects ∂Q at a (Qqx) point. So the necessary and sufficient condition for (1) is that the cubic is < 0. Multiplying by c3 11, which we know to be positive, and rearranging the resulting homogeneous polynomial in the f ij , gives condition (5). It is interesting to see what a purely algebraic approach as in[7,Section 5.3] would say about the situation. First of all, the support function asks us to compute a maximum of a linear functional over a variety. Evaluating just the first order conditions, we get an expression for the maximum for each patch of ∂Q. For the curved (Qqx) patches this will be just φ. So we get little progress over the simple story told in Sect. 3.5. In many well-known problems of duality, we can simply take the maximum of the different branches of the algebraic function. For example when computing the Legendre transform (LF )(p) := sup x {p·x−F (x)} of a non-convex function like F (x) = x 4 −x 2 one gets a multi-valued function defined by eliminating x from the cubic p = dF (x). This algebraic version of the Legendre transform is easily corrected by taking the supremum from the convex definition just given as a maximum over the branches. One might therefore guess (and we tried that out in an early stage of this work) that our support function is simply the maximum of the two branches φ and φ C . Suffice it to say that this fails, and one easily finds points where φ(f ) > φ C (f ) = φ(f ). To get this to work, one must ensure that the associated maximizers are contained in the quantum set.Quantum representations of extreme pointsWe now prove Thm. 19. A central role will be played by the uniqueness of a cyclic model. Recall that every state of a C*-algebra has a unique cyclic model known as the GNS (Gelfand-Naimark-Segal) representation. What makes uniqueness of a cyclic model non-trivial for correlations c ∈ Q is that the algebra itself is not known from expectation values: We can usually not infer the multiplication rules for the A i , B j from just these expectations, nor the expectation of, say, A 1 B 2 2 . If the cyclic model is unique up to unitary isomorphism, as condition (2) of Thm. 19 asserts, then all algebraic relations are fixed.According to Thm. 19 uniqueness fails for the classical extreme points, so these provide a key example. The correlations are c ij = a i b j with a i , b j = ±1. These numbers constitute a model with one dimensional Hilbert space, which is obviously cyclic. The models related by a i = −a i and b j = −b j clearly give the same c, but the unitary operator connecting the model Hilbert spaces cannot take a 1 to a 1 , so these are
We use the same classification of boundary points for Q and Q • via duality transform. So References⇔(5): We use the same classification of boundary points for Q and Q • via duality transform. So References
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G M Ziegler, Lectures on Polytopes. BerlinSpringerAuthors' addressesG. M. Ziegler. Lectures on Polytopes. Springer, Berlin, 1995. Authors' addresses:
. P Thinh, Le, Vienna, Austria [email protected] Chiara Meroni. MPI-MiS Leipzig [email protected] P. Le, IQOQI Vienna, Austria [email protected] Chiara Meroni, MPI-MiS Leipzig [email protected]
. Bernd Sturmfels, Mpi-Mis Leipzig, Berkeley, mpg.deBernd Sturmfels, MPI-MiS Leipzig and UC Berkeley [email protected]
. Reinhard F Werner, Leibniz Universität Hannover [email protected] Timo Ziegler ; Leibniz Universität Hannover [email protected] F. Werner, Leibniz Universität Hannover [email protected] Timo Ziegler, Leibniz Universität Hannover [email protected]
| [] |
[
"Twisted bilayer graphene at charge neutrality: competing orders of SU(4) Dirac fermions",
"Twisted bilayer graphene at charge neutrality: competing orders of SU(4) Dirac fermions"
] | [
"Nikolaos Parthenios \nMax Planck Institute for Solid State Research\nD-70569StuttgartGermany\n",
"Laura Classen \nMax Planck Institute for Solid State Research\nD-70569StuttgartGermany\n\nDepartment of Physics\nTechnical University of Munich\nD-85748GarchingGermany\n"
] | [
"Max Planck Institute for Solid State Research\nD-70569StuttgartGermany",
"Max Planck Institute for Solid State Research\nD-70569StuttgartGermany",
"Department of Physics\nTechnical University of Munich\nD-85748GarchingGermany"
] | [] | We study possible patterns for spontaneous symmetry breaking in a Dirac fermion model, which is applicable to twisted bilayer graphene at charge neutrality. We show how a chiral SU(4) symmetry emerges and construct the corresponding low-energy model that includes a Fierz-complete set of symmetry-allowed four-fermion interactions. We employ an unbiased renormalization group treatment to identify the critical points that describe transitions into different ordered phases. The resulting phase diagram depends on the number of fermion flavours and we show that the coupling between ordering channels prevents many of the possible mean-field orders from being accessible at relevant, small flavour numbers. We argue that, as a consequence, twisted bilayer graphene is governed by a quantum Hall state or an SU(4) manifold of insulating spin-valley orders with emergent Lorentz symmetry that contains inter-valley coherent, spin Hall, and valley Hall states. We study how SU(4)-breaking perturbations affect the accessibility and can additionally stabilize symmetry-broken (semi-)metallic states. arXiv:2305.06949v1 [cond-mat.str-el] | null | [
"https://export.arxiv.org/pdf/2305.06949v1.pdf"
] | 258,615,580 | 2305.06949 | ce3618c1973e46811e34eab3c363ed09fc287902 |
Twisted bilayer graphene at charge neutrality: competing orders of SU(4) Dirac fermions
Nikolaos Parthenios
Max Planck Institute for Solid State Research
D-70569StuttgartGermany
Laura Classen
Max Planck Institute for Solid State Research
D-70569StuttgartGermany
Department of Physics
Technical University of Munich
D-85748GarchingGermany
Twisted bilayer graphene at charge neutrality: competing orders of SU(4) Dirac fermions
We study possible patterns for spontaneous symmetry breaking in a Dirac fermion model, which is applicable to twisted bilayer graphene at charge neutrality. We show how a chiral SU(4) symmetry emerges and construct the corresponding low-energy model that includes a Fierz-complete set of symmetry-allowed four-fermion interactions. We employ an unbiased renormalization group treatment to identify the critical points that describe transitions into different ordered phases. The resulting phase diagram depends on the number of fermion flavours and we show that the coupling between ordering channels prevents many of the possible mean-field orders from being accessible at relevant, small flavour numbers. We argue that, as a consequence, twisted bilayer graphene is governed by a quantum Hall state or an SU(4) manifold of insulating spin-valley orders with emergent Lorentz symmetry that contains inter-valley coherent, spin Hall, and valley Hall states. We study how SU(4)-breaking perturbations affect the accessibility and can additionally stabilize symmetry-broken (semi-)metallic states. arXiv:2305.06949v1 [cond-mat.str-el]
I. INTRODUCTION
Spontaneous symmetry breaking of Dirac fermions plays an important role in many systems, ranging from the chiral phase transition in quantum chromodynamics to quantum critical points in semimetals 1-4 . Interactions must be sufficiently strong for such phase transitions to occur because the density of states of massless Dirac fermions vanishes at charge neutrality. Furthermore, if the phase transition takes place at zero temperature and is continuous, fermionic quantum critical behavior is expected which does not possess any classical analogue 3 .
As a well-known example, spontaneous symmetry breaking in graphene was intensely studied. However, interactions in graphene are estimated to be slightly too small to induce a phase transition 5 .
The recent discovery of strongly correlated moiré materials 6,7 provides a new opportunity for the investigation of quantum phase transitions in two-dimensional Dirac systems, where the relative interaction can be tuned via a twist angle [8][9][10] . This includes, for example, symmetrically twisted few-layer graphene systems 11 , or twisted Γ-valley transition metal dichalcogenides 12,13 . In particular, the most prominent moiré material -twisted bilayer graphene (TBG) -also hosts Dirac fermions at charge neutrality. Although the Dirac velocity vanishes in TBG at a so-called magic angle, strong interactions are expected in its vicinity where Dirac fermions remain present 14,15 .
Experiments on TBG report both insulating [16][17][18][19][20][21][22][23] and semimetallic 6,7,[24][25][26][27][28][29][30][31][32][33] behavior near charge neutrality with the differences potentially coming from twist-angle disorder, strain, or substrate alignment. In particular, ultralow strain samples show the emergence of a gap 34 . Theoretically, several ground states were proposed for TBG near charge neutrality. [35][36][37][38][39][40][41] It was argued that different ordered states lie close in energy because the interaction of the continuum model for TBG possesses a U(4) symmetry, or even U(4)×U(4) in the chiral limit, which relates the ordered states 36,[42][43][44] . Symmetry-lowering effects of the full model then determine the ground state on smaller energy scales, while external perturbations (such as disorder, stain, or substrates) can change the selection [45][46][47][48][49] . However, internal processes, such as the mutual feedback between correlations of different ordering channels, can be equally decisive in this situation. The near-degeneracy of ordered states that was demonstrated in Hartree-Fock calculations [36][37][38][47][48][49][50]50 highlights the need for unbiased treatments of competing orders. Recently, Monte Carlo simulations 39,40 and renormalization group calculations 41 addressed this competition, but within different models for the narrow bands in TBG, which affects the delicate interplay of orders.
In this work, we present a complementary analysis that investigates competing orders as instabilities of itinerant Dirac electrons. Focusing on the Dirac spectrum and employing an unbiased renormalization group approach allows us to draw universal conclusions and assess relevant orders. Our rationale comes from the observed signatures of a Dirac dispersion in TBG, and the relative reduction of interaction effects due to the vanishing density of states of Dirac electrons.
As a point of departure, we use an effective low-energy model valid around charge neutrality that respects the symmetries and topology of the narrow bands of TBG in the vicinity of the magic angle 51 . We argue that a chiral SU(4) symmetry emerges in this low-energy Dirac fermion model, which we generalise to an arbitrary number of flavours (Dirac points) N f . We study possible phase transitions based on all symmetry-allowed local interactions via the perturbative renormalisation group (RG). To do so, we analyse the fixed point structure of the RG equations for a Fierz-complete basis of the fermionic interactions. Our analysis is valid beyond the concrete application to TBG for other two-dimensional Dirac systems and thereby complements, e.g., previ-ous fermionic RG investigations of phase transitions in graphene [52][53][54][55][56][57][58] . We classify possible quantum phase transitions based on flavour and Lorentz symmetry of their order parameter manifolds, and if they dynamically generate a mass that gaps out the Dirac spectrum or if they distort the spectrum but maintain (semi-)metallicity.
We find that, although there are many accessible transitions on the single-channel mean-field level, which is equivalent to large flavour numbers, the corresponding fixed points can disappear for small flavour numbers due to the coupling between different ordering channels. As a result, only a few transitions remain accessible for small flavour numbers, which is relevant to TBG. They include transitions with emergent Lorentz symmetry to a quantum anomalous Hall state and a gapped state with an SU(4) order parameter manifold that contains spin Hall, valley (spin) Hall and timereversal-symmetric intervalley-coherent states. We also determine how relaxing SU(4) symmetry affects the possible phase transitions. We show that the instabilities descending from the order parameter manifold of insulating states remain accessible separately. In addition, we argue that perturbing SU(4) stabilizes instabilities towards spin-(valley-)-polarized, spin nematic, and Kramers intervalley-coherent states.
II. SU(4) DIRAC ELECTRONS
A. Dirac fermion model of twisted bilayer graphene Our starting point is a low-energy effective model for Dirac fermions around charge neutrality in TBG 51 with the action at zero temperature
S 0 = d 3 q (2π) 3 ψ † (−iω + ρ x τ z q x + ρ y q y )ψ ,(1)
where ψ is a 16-component spinor and σ γ , ρ γ , τ γ , µ γ denote unity (γ = 0) and Pauli (γ = x, y, z) matrices in spin, sublattice, valley, and mini-valley space. The sublattice and valley degrees of freedom form the Clifford algebra of the Dirac spinors (see below for a Lorentz-invariant formulation), while the kinetic part of the action S 0 is diagonal in spin and mini-valley space. The mini-valley degree of freedom originates from the Dirac points in the same valley of the two different graphene layers, i.e. the two mini-valleys per valley possess the same chirality.
We generalize the mini-valley degrees of freedom to an arbitrary number of flavours for the Dirac fermions ψ → ψ α with α = 1, 2, . . . N f . In the case of TBG N T BG f = 2, i.e., N f counts the number of eight-component spinors 85 . The Dirac fermion model (1) implements the emergent C 2 symmetry of the continuum model of TBG via R C2 = ρ x τ x µ x , → −q. and time-reversal via Θ = σ y τ x µ x K, q → −q. The threefold rotation C 3 is promoted to a full U(1) rotational symmetry R rot = exp[−iϕρ z τ z ] in the low-energy limit around charge neutrality. Furthermore, since the hybridization between layers can be treated as a perturbation for states around the Dirac cones 15 , the spin-valley SU(2)×SU(2)×U(1) symmetry of TBG is enhanced to an emergent SU(4) in the low-energy limit, similar to the chiral limit 44,59 . It is generated by the set of matrices
T = {σ c , τ z σ γ , ρ y τ x σ γ , ρ y τ y σ γ }(2)
with c = {x, y, z} and γ = {0, x, y, z}. The generalization of the flavour index leads to a unitary flavour symmetry acting via a transformation U f ∈ U(N f ). Together with the chiral SU(4) symmetry, the theory is then invariant under SU(4N f ).
On the level of interactions, we include all symmetryallowed local four-fermion couplings according to the symmetries above. They will be generated by fluctuations, even if zero in the microscopic Hamiltonian so that it is important to include them for an unbiased analysis.
In particular, this means that we must include couplings that break the U(4)×U(4) symmetry of the interactions in TBG in the chiral limit because it is partially broken by the dispersion 36,[42][43][44] . We discuss effects that break chiral SU(4) symmetry further in Sec. III F. We also retain the emergent flavour symmetry in the interactions, i.e. we do not consider symmetry breaking within mini valleys (which would translate to layer polarization or translational symmetry breaking on the scale of the moiré lattice.) We then find that there are six distinct interaction channels allowed by these symmetries so that the interacting Lagrangian is given by
L int = g 1 N f (ψ α † ψ α ) 2 + v 1 N f 15 i=1 (ψ α † T i ψ α ) 2 + g 4 N f (ψ α † ρ z τ z ψ α ) 2 + v 4 N f 15 i=1 (ψ α † ρ z τ z T i ψ α ) 2 + g 2 N f (ψ α † νψ α ) 2 + v 2 N f 15 i=1 (ψ α † ω i ψ α ) 2(3)
where T i ∈ T , ν = (ρ x τ z , ρ y ), and ω i = (ρ x τ z T i , ρ y T i ).
In the following we study the full effective action
Γ = S 0 + d 3 ⃗ xL int(4)
The 64 matrices describing the couplings in Eq.
(3) form a complete basis set for 8×8 matrices. Furthermore, we only included flavour-diagonal terms in L int because Fierz identities allow us to appropriately rewrite any flavor-symmetry-allowed terms as a linear combination of the ones considered in Eq.
(3) (see App. A). Thus, the six couplings form a Fierz-complete basis for interacting SU(4N f )-symmetric Dirac fermions with N f > 1.
The inclusion of all terms allowed by the Fierz identities is important for the correct identification of critical points 4,60 . For N f = 1, there are additional Fierz identities which reduce the number of couplings to three (see App. A).
B. Ordered states
Taken on their own, each of the couplings can induce an instability towards a state with spontaneously broken symmetry if strong enough. The ordered states are characterized by the condensation of the corresponding bilinears φ M = ⟨ψ α † M ψ α ⟩ ≠ 0 with M ∈ {T i , ρ z τ z , ρ z τ z T i , ν, ω i } (see also Tab. I). Note that n = ⟨ψ α † ψ α ⟩ is fixed by the density and does not break any symmetry. The condensation of φ ρzτz corresponds to the spontaneous formation of a quantum anomalous Hall state (QAH) and generates a gap in the spectrum. A finite φ ρzτzTi also gaps out the Dirac fermions. Its SU(4) order parameter manifold contains quantum spin Hall (QSH) ∼ ρ z τ z σ c , valley Hall (VH), and valley spin Hall (VSH) ∼ ρ z σ γ states, as well as inter-valley coherent order 35 ∼ ρ x τ x,y σ γ symmetric under spinless timereversal (T-IVC). In contrast, the SU(4) order parameter manifold described by Φ Ti splits the degeneracy of the Dirac spectrum, resulting in a metallic state (except the splitting becomes on the order of the bandwidth). It contains spin-polarized ∼ σ c , valley-polarized ∼ τ z , and spin-valley-polarized ∼ τ z σ c orders, as well as the Kramers inter-valley coherent order 36 (K-IVC) ∼ ρ y τ x,y σ γ . Finally, condensation of φ ν and φ ωi spontaneously breaks rotation symmetry. This shifts the position of the Dirac points away from the corners of the Brillouin zone and preserves the semi-metallic state. A finite order parameter φ ν couples to the fermions in the same form like a vector potential ∼ (ρ x τ z , ρ y ) which leads to the integer quantum Hall (IQH) effect 61 . The SU(4) order parameter manifold φ ωi contains a spinful variant of this ∼ (ρ x τ z σ c , ρ y σ c ) (S-IQH), in addition to nematic ∼ (ρ x , ρ y τ z ) (NEM) and spin-nematic ∼ (ρ x σ c , ρ y τ z σ c ) (S-NEM) orders, as well as nematic and spin-nematic intervalley coherent states (N-IVC and S-N-IVC). The latter possess matrix order parameters built of two vectors under rotation (ρ z τ y σ γ , τ x σ γ ), (ρ z τ x σ γ , τ y σ γ ) which are related by valley U(1) v 62,63 .
C. Lorentz-invariant SU(4)-symmetric model
As Dirac fermions are relevant in a variety of systems besides TBG, we reformulate the action in an explicitly Lorentz-invariant form. To this end, we rewrite L 0 as
L 0 =ψγ µ q µ ψ (5) whereψ = ψ † γ 0 , γ 1 = γ 0 ρ x τ z , γ 2 = γ 0 ρ y .
With the requirement that the γ matrices satisfy the Euclidean Clifford algebra {γ µ , γ ν } = 2δ µν , there are four choices of γ 0 ∈ {ρ z τ z , ρ z , ρ x τ x , ρ x τ y }. For γ 0 = ρ z τ z , the γ-matrices form a reducible representation of the two-dimensional Clifford algebra, for the other choices, they form a fourdimensional representation. To impose Lorentz invariance on the interacting Lagrangian (3), we additionally require g 1 = −g 2 =∶ g and v 1 = −v 2 =∶ v. This yields
L L int = g N f (ψ α γ µ ψ α ) 2 + v N f 15 i=1 (ψ α γ µ T i ψ α ) 2 + g 4 N f (ψ α ψ α ) 2 + v 4 N f 15 i=1 (ψ α T i ψ α ) 2(6)
for the 2D representation, or
L L int = g N f (ψ α γ µ ψ α ) 2 + v N f 15 i=1 (ψ α γ µ T i ψ α ) 2 + g 4 N f (ψ α γ 35 ψ α ) 2 + v 4 N f 15 i=1 (ψ α γ 35 T i ψ α ) 2(7)
for the 4D representations, where γ 35 = iγ 3 γ 5 is formed by the two additional matrices γ 3 , γ 5 which anti-commute with γ µ , i.e. γ 35 commutes with γ µ . In analogy to the discussion of the ordered states above, the chiral condensates ⟨ψ α ψ α ⟩ and ⟨ψ α T i ψ α ⟩ (⟨ψ α γ 35 ψ α ⟩ and ⟨ψ α γ 35 T i ψ α ⟩) spontaneously generate a mass in the fermion spectrum, while the vector condensates ⟨ψ α γ µ ψ α ⟩ and ⟨ψ α γ µ T i ψ α ⟩ spontaneously break Lorentz symmetry.
III. RENORMALIZATION GROUP ANALYSIS
A. RG flow and beta functions
To study the possible ordering tendencies, we employ a perturbative renormalization group scheme. Within the RG, we successively integrate out modes above a cut-off scale k and express the scale evolution of the couplings λ ∈ {g 1 , g 2 , g 4 , v 1 , v 2 , v 4 } via the differential equations β λ = k∂ k λ. This defines an effective action Γ k at scale k, and for k → 0, we recover the full effective action that includes all quantum corrections. We show in App. B in terms of a functional RG formulation that the one-loop β functions are independent of the cut-off scheme. For example, a Wilsonian scheme can be used that integrates over modes within a momentum shell. We obtain the flow equations
β g1 = g 1 − 4 N f [g 2 1 (4N f − 1) − 2g 1 g 2 − g 1 g 4 − 15g 1 (v 1 + 2v 2 + v 4 ) − 4g 2 g 4 − 60v 2 v 4 ](8)β v1 = v 1 + 4 N f [g 1 v 1 + 2g 2 (v 1 + 2v 4 ) + g 4 v 1 + 4g 4 v 2 − 4N f v 2 1 − 9v 2 1 + 30v 1 v 2 − v 1 v 4 + 16v 2 2 + 24v 2 v 4 − 8v 2 4 ](9)β g4 = g 4 + 4 N f (4g 1 g 2 − 3g 1 g 4 − g 2 2 + 6g 2 g 4 + 12g 2 4 N f − 3g 2 4 + 90g 4 v 2 − 45g 4 v 1 − 45g 4 v 4 + 60v 1 v 2 − 30v 2 2 )(10)β v4 = v 4 + 4 N f [4g 1 v 2 − 3g 1 v 4 + g 2 (4v 1 − 4v 2 + 6v 4 ) − 3g 4 v 4 + 12N f v 2 4 + 24v 1 v 2 − 13v 1 v 4 − 12v 2 2 + 26v 2 v 4 + 3v 2 4 ] (11) β g2 = g 2 − 4 N f [g 1 (g 2 − 2g 4 ) − 4g 2 2 N f + g 2 g 4 + 15v 1 − 15v 4 ) + 30v 4 (v 2 − v 1 )](12)β v2 = v 2 − 4 N f [g 1 v 2 − 2g 1 v 4 + 2g 2 v 4 + g 4 (v 2 − 2v 1 ) − 4N f v 2 2 − 8v 2 1 + 3v 1 (5v 2 − 4v 4 ) − 32v 2 2 + 13v 2 v 4 − 8v 2 4 ](13)
where we rescaled the couplings k d−2 l f λ → λ with spacetime dimension d = 2+1 and loop integral l f (see App. B). For Wilson's momentum-shell cut-off,
l f = 2π d 2 Γ[d 2]
is the area of the unit sphere in d = 2 dimensions. For the Lorentz-invariant model, we again impose g = g 1 = −g 2 and v = v 1 = −v 2 . The Lorentz symmetry is maintained along the flow, therefore β g = β g1 = −β g2 as well as β v = β v1 = −β v2 .
B. Fixed points and stability
We are interested in fixed points of the RG equations because they are connected to the possible quantum phase transitions that can be induced by strong couplings. Thus, we look for solutions λ * = (g * 1 , . . . , v * 4 ) where all the beta functions vanish
β λi (λ * ) = 0 .(14)
To identify which solutions correspond to critical points, we consider the linearized flow of the beta functions around a fixed point and evaluate the stability matrix
R ij = − ∂β λi ∂λ j λ=λ *(15)
which describes how the scale evolution of the couplings is attracted to or repelled from the fixed point. The eigenvalues of the stability matrix determine the critical exponents of the corresponding second-order phase transition. They are universal quantities which do not depend on the microscopic details of the model. We are interested in stable fixed points that can be accessed by tuning only one parameter because they are associated to critical points. This means the spectrum of the stability matrix must have all negative eigenvalues except one, which defines a relevant repulsive direction in coupling space. This largest critical exponent determines the correlation-length exponent. The second largest exponent describes the corrections to scaling and decides over the stability of the fixed point. Fixed points with more than one relevant direction are considered multicritical or unstable. The trivial Gaussian fixed point (λ = 0), which defines a non-interacting theory in the IR, has a fully negative spectrum, reflecting the need of strong couplings to induce a phase transition. Any nontrivial solution describes an interacting fixed point with finite values of the couplings. If the bare couplings lie beyond the threshold set by the interacting fixed points, their flow diverges along the relevant direction towards the infrared indicating the formation of an ordered state with spontaneously broken symmetry (see Fig. 1).
C. Susceptibility analysis
The characterization of the possible ordered phases based on the stable fixed points is non-trivial for finite flavour numbers. Since the fixed points' coordinates generally have non-zero values, the identification of the ordering tendencies is complicated by multiple divergent couplings. In order to gain further insight within this fermionic description, we calculate the flow of susceptibilities. This is done by adding an infinitesimal test vertex to the effective action that explicitly breaks the symmetry along a specific ordering channel
Γ k → Γ k + h i ψ α † M i ψ α ,(16)
where
M i ∈ {T i , ρ z τ z , ρ z τ z T i , ν, ω i }.
These terms represent the coupling of fermions to an external field, which is set to zero at the end. We define the corresponding susceptibilities as
χ i = ∂ 2 Γ k ∂h 2 i hi=0 .(17)
In the vicinity of the fixed points, the fields and susceptibilities scale according to
h i ∝ k βi (18) χ i ∝ k γi(19)
with γ i = 2β i + 1 2 (see App. C). If γ i < 0, the corresponding susceptibility diverges. In d = 2 + 1 dimensions, this means that β < − 1 2 . We associate the fixed point with the ordering channel whose susceptibility shows the strongest singularity.
We extract the exponents from the flow equations for the infinitesimal fields
∂ t h g1 = 2 N f (g 1 + 2g 2 + g 4 − 8g 1 N f + 15v 1 + 30v 2 + 15v 4 )h g1 (20) ∂ t h v1 = − 2 N f (v 1 + 8N f v 1 + 2v 2 + v 4 − g 1 − 2g 2 − g 4 ) h v1(21)∂ t h g4 = − 6 N f (g 1 − 2g 2 + g 4 − 8g 4 N f + 15v 1− 30v 2 + 15v 4 )h g4 (22) ∂ t h v4 = 6 N f (2g 2 − g 4 + v 1 − 2v 2 + v 4 + 8N f v 4 − g 1 ) h v4 (23) ∂ t h g2 = 2 N f (g 4 + 8g 2 N f − 15v 1 + 15v 4 − g 1 ) h g2 (24) ∂ t h v2 = 2 N f (g 4 + v 1 + 8N f v 2 − v 4 − g 1 ) h v2(25)
at the fixed point solution. We can see that in the linear response regime the introduction of a symmetry breaking term along a certain channel renormalizes only the respective field with no feedback to the others, i.e the flow equations are decoupled so that we can evaluate them separately. Additionally, we note that these equations are Lorentz invariant if we again impose g = g 1 = −g 2 and v = v 1 = −v 2 .
D. RG flow of the Lorentz invariant system
We first analyze the RG flow of the Lorentz invariant system, i.e. we consider the case where g = g 1 = −g 2 and v = v 1 = −v 2 . Our results for the system without Lorentz invariance and relaxed SU(4) symmetry are presented in the next sections. We perform the fixed point analysis described above for several values of the fermionic flavor number N f . In the large-N f limit, the RG analysis becomes equivalent to a single-channel, mean-field treatment. But for finite N f , the feedback between ordering channels is important and can qualitatively change possible ordering tendencies as we show below. We start with the large-N f limit, where the beta functions decouple and allow for an unambiguous characterization of the possible ordered phases. We identify four stable fixed points that in this limit are characterized by only one coupling of the scalar (S) or vector (V) channels being non-zero
FP S ∶ g * 4 = − 1 48 + O(1 N f )(26)FP S−SU(4) ∶ v * 4 = − 1 48 + O(1 N f )(27)FP V ∶ g * = 1 16 + O(1 N f )(28)FP V−SU(4) ∶ v * = 1 16 + O(1 N f ) .(29)
Upon lowering N f , the coupling between the different ordering channels starts to affect the fixed points and their properties in several ways. With the exception of FP S , other couplings become non-zero at the fixed points, necessitating the susceptibility analysis for an unambiguous identification of the corresponding instability. The scalar fixed point FP S is located at
[g * 4 , v * 4 , g * , v * ] = − 1 12 N f N f − 1 , 0, 0, 0(30)
for general N f , and the vector fixed point FP V at
[g * 4 , v * 4 , g * , v * ] = [g * 4,V (N f ), 0, g * V (N f ), 0](31)with g * 4,V (N f ), g * V (N f ), 0) ≥ 0.
We give explicit analytical expressions in App. D. For FP S−SU (4) and FP V−SU(4) , the fixed-point values of all couplings become non-zero for general N f , see Fig. 2. We find λ * i,V−SU(4) ≥ 0 for all λ ∈ {g, v, g 4 , v 4 }, while g * S−SU (4) , v * 4S−SU(4) ≥ 0, and v * S−SU(4) , g * 4S−SU(4) ≤ 0. As we described above, the location of the fixed points determines the regime of strong coupling where the flow becomes unstable signalling an instability towards spontaneous symmetry breaking. The flow to strong coupling governed by FP S and FP S−SU (4) is shown in Fig. 1. Importantly, we observe that not all of the fixed points are accessible for any N f . Specifically, we find critical values of the flavor number N c f at which they either disappear or at which they become multi-critical. These changes occur via a "collision" with other (multi-critical) fixed points. In the first case, the fixed point ceases to exist in the space of real-valued couplings at the critical N f , and instead a pair of complex conjugate solutions moves into the complex plane. We find that the fixed-point solution FP S−SU(4) exhibits this behavior and disappears at N c S−SU(4) = 1.89. In the second case, the second largest eigenvalue of the stability matrix θ 2 changes its sign at the critical value of the flavor number so that the fixed point becomes unstable. The largest eigenvalue of the stability matrix θ 1 = 1 independent of the critical point and N f as expected for critical four-fermion models 60 . The second largest critical exponent approaches θ 2 → −1 for large N f , but generally varies as function of N f . We show the exponent θ 2 along with θ 3 for all four critical points in Fig. 3. The quantum critical points labeled as FP S and FP V−SU(4) remain accessible, i.e. θ 2 < 0, for all values 1 < N f < ∞, as does FP S−SU (4) in the range where it exists N f > 1.89. In contrast, FP V becomes multi-critical at N c V = 3.0 (see App. D for analytical expression).
Furthermore, we underline the importance of including the analysis of the susceptibilities (Sec. III C) in the proper characterization of a fixed-point solution. We find that no divergent channel exists in the case of FP V−SU (4) for values of N f < 15.5 where all β i > −1 2 (see Fig. 4) even though it is a critical point with one relevant direction.
In summary, we find divergent susceptibilities at small values of N f ≈ 2 for the instabilities governed by critical points FP S , and FP S−SU (4) . These both correspond to states with a gap in the symmetry-broken regime. In the case of TBG, they describe a QAH state, or a state with SU(4) order parameter manifold connecting QSH, VH, VSH, and T-IVC phases.
E. RG flow of Dirac fermion model of TBG Since a physical system like TBG is a priori not Lorentz-invariant, we are interested in extending the above discussion to the full Lagrangian spanned by all six couplings in Eq.(3) and compare to what is captured in the Lorentz-invariant case. We thus look to find the fixed point solutions that describe quantum phase transitions for the group of beta functions in Eqns. (8)(9)(10)(11)(12)(13). We find in total seven fixed points that are stable for some range of N f , but generally display a varied behavior as a function of the fermionic flavor number N f . In the large-N f limit, the RG equations again decouple and we can identify six of the seven stable fixed points corresponding to the separate ordering channels
FP S ∶ g 4 = − 1 48 + O(1 N f )(32)FP S−SU(4) ∶ v 4 = − 1 48 + O(1 N f )(33)
FP n ∶ g 1 = 1 16
+ O(1 N f )(34)FP SM ∶ g 2 = − 1 16 + O(1 N f )(35)FP M−SU(4) ∶ v 1 = 1 16 + O(1 N f )(36)FP SM−SU(4) ∶ v 2 = − 1 16 + O(1 N f ) ,(37)
where all other couplings are of order O(1 N f ), respectively. We recover the fixed points FP S and FP S−SU(4) of the scalar channels from the Lorentz-invariant case. The two remaining Lorentz-invariant fixed points
FP V ∶ g 1 = −g 2 = 1 16 + O(1 N f )(38)FP V−SU(4) ∶ v 1 = −v 2 = 1 16 + O(1 N f )(39)
are unstable for N f → ∞. Instead, we find the four fixed points FP n , FP SM , FP M−SU (4) , and FP SM−SU (4) to be stable at large N f . FP n describes the density channel, which does not correspond to any symmetry breaking. A divergence of g 1 signals the singular response of the chemical potential to a density change at the Dirac point where the inverse compressibility diverges. The semi-metallic (SM) instabilities of FP SM and FP SM−SU(4) correspond to the breaking of rotation symmetry with or without an SU(4) order parameter manifold, and FP M−SU(4) describes polarizing instabilities which yield a metallic (M) state for small order parameters (see Sec. II B).
In a similar manner to the Lorentz invariant case, the existence and the stability of the fixed points can change when N f becomes small, and a single-channel mean-field description breaks down see App. D for the evolution of fixed-point couplings and critical exponents with N f ). Exceptions are the fixed point related to a QAH instability FP S given by Eq. (30), and the fixed point related to the density channel FP n given by g * 1 = N f 4(4N f − 1), which both possess just one non-zero coupling. A general overview of the of the fixed points' behavior as a function of N f is provided in Fig. 5.
We find that FP S and FP n remain accessible for any 1 < N f < ∞. As in the Lorentz-invariant case, FP S−SU (4) exists and is stable for 1.89 < N f < ∞. This means that Lorentz symmetry emerges in the vicinity of the associated phase transition towards the insulating SU(4) order. In contrast, the Lorentz-invariant fixed point FP V is always multi-critical if Lorentz invariance is not enforced. Interestingly, another fixed point with emergent Lorentz symmetry becomes critical when lowering N f , which is not part of the stable large-N f fixed points. The Lorentz invariant solution FP V−SU(4) collides with the semi-metallic SU(4) solution FP SM−SU (4) at N c f = 13.6 leaving FP SM−SU(4) unstable and FP V−SU(4) stable at smaller N f . At slightly smaller N f ≈ 12.89, FP SM−SU (4) and FP M−SU(4) collide and vanish into the complex plane. Finally, FP SM becomes multi-critical at N f ≈ 2.55. We also calculate the exponents characterizing the divergence of susceptibilities for the seven fixed points (see App.C). We find that all the divergent channels can be connected to the large-N f limit, with the exception of FP V−SU (4) , which displays no divergent susceptibility in the regime where it is critical.
Therefore, a picture similar to the Lorentz-invariant case arises in the Dirac fermion model of TBG at small N f ≈ 2. We obtain instabilities related to FP S and FP S−SU (4) , which are related to symmetry-broken phases with a gap due to QAH states or SU(4)-related VH, VSH, QSH and T-IVS states. Interestingly, there is neither an instability towards polarizing orders (SP, VP, K-IVC) associated with FP M−SU (4) or FP V−SU(4) , nor towards nematic orders associated with FP SM or FP SM−SU(4) in the strict SU(4)-symmetric model. If however, this symmetry is reduced, several of these solutions become accessible also in the small-N f limit because the critical flavour number of the separate instabilities is lowered. This is further elaborated on in the next section.
F. Relaxing SU (4) As described in Sec. II A, the Dirac fermion model of TBG possesses an emergent SU(4) symmetry, leading to a fourfold degeneracy of the Dirac fermions at charge neutrality. Since three out of the six quantum critical points are related to orders containing an SU(4) symmetric manifold, we are interested in studying whether these fixed-point solutions are robust under a possible breaking of this symmetry. The breaking can occur on the level of the interactions due to high-energy corrections above the UV cutoff of the Dirac fermion model, which lower the symmetry from SU(4) to SU(2)×SU(2)×U(1), i.e., independent spin rotations in both valleys.
We thus split the initial manifold into five new channels that are invariant by the lowered symmetry
T (1) = {σ c } T (2) = {τ z σ c } T (3) = τ z T (4)
= {ρ y τ x , ρ y τ y } T (5) = {ρ y τ x σ c , ρ y τ y σ c } and study the Lagrangian that accounts for the new interaction terms. The above channel separation means that five new couplings are to be introduced for each term ∝ v i that contained an SU(4) manifold, yielding in total eighteen interaction terms in the lower symmetry Lagrangian
L ′ int = g 1 N f (ψ α † ψ α ) 2 + i,j v (i) 1 N f (ψ α † T (i) j ψ α ) 2 + g 4 N f (ψ α † ρ z τ z ψ α ) 2 + i,j v (i) 4 N f (ψ α † ρ z τ z T (i) j ψ α ) 2 + g 2 N f (ψ α † νψ α ) 2 + i,j v (i) 2 N f (ψ α † ω (i) j ψ α ) 2(40)
where ω
(i) j = (ρ x τ z T (i) j , ρ y T (i) j )
. We derive the flow equations for the new set of couplings (see App. E) and investigate how the fixed-point structure changes.
To study the robustness of the SU(4) symmetry, we evaluate the eigenvalues of the stability matrix at the coupling coordinates of the SU(4) symmetric quantum critical points. We find that the fixed points related to FP S , FP n and FP SM retain the same behavior as in the previous cases since they are not related to any SU(4) manifold. Furthermore, we consider the stability matrix for N f = 2, 14, 20 for the SU(4) fixed points FP M−SU(4) , FP SM−SU(4) and FP S−SU (4) , which all become multi-critical. The number of relevant directions is increased to four for FP S−SU(4) and FP SM−SU(4) and to five for FP M−SU (4) . They are primarily along couplings of the channel they descend from, i.e along v (4) . As such, we observe that the emergent SU(4) symmetry of the Dirac fermion model is not robust against perturbations. Mechanisms which induce a breaking of the SU(4) symmetry select a specific ground state out of the SU(4) order parameter manifolds.
To determine possible selections, we follow the N fevolution of the eighteen single-channel solutions that occur as quantum critical points at large N f in this case. The result is summarized in Tab. I. In total, out of the eighteen large-N f stable fixed points we find that seven remain critical down to small N f ≈ 2. Besides FP S and FP n , fixed-point solutions characterizing the transition to QSH, VH, VSH, and spinless T-IVC orders are stable. These originate from the FP S−SU(4) fixed point solution in the SU(4) symmetric case. Lorentz invariance is again emergent at these solutions. Furthermore, a fixed point that originates from inaccessible FP M−SU(4) is now stable for N f = 2 in the SU(2)×SU(2)×U(1) case. It is related to the transition to a spin-valley polarized state. Interestingly, the critical N c f for several other fixed points that describe transitions to orders previously related by SU(4) symmetry is also considerably lowered. The effect is the strongest for spin-polarization, K-IVC, and N-IVC order.
Their critical N f as part of the SU(4) order parameter manifold N c f ≈ 13 is lowered to N c f ≈ 3. 66, N c f ≈ 3.2, and N c f ≈ 5.2, respectively. In this sense, these ordering tendencies are stabilized by perturbations that break SU(4).
IV. CONCLUSION
In the present work we studied competing orders in the Dirac fermion model of TBG at charge neutrality in a universal, unbiased way. As such, our considerations generally apply to Dirac fermions with approximate SU(4) symmetry in 2 + 1 space-time dimensions.
We determined a Fierz-complete set of all symmetry-preserving, local 4-Fermi interactions, which we could classify in eighteen, six, or four different channels in the case of SU(2)×SU(2)×U(1), chiral SU(4), or Lorentz invariant SU(4) symmetry, respectively. We calculated the perturbative RG flow of the couplings and investigated their fixed-point structure with the aim of identifying quantum critical solutions that are associated with instabilities towards different ordered states. We diagnosed the instabilities by a divergence in the corresponding susceptibilities.
The model contains a control-parameter in the form of the fermion flavor number. This allowed us to investigate the interplay of multiple interaction channels, which becomes important in the small-N f regime relevant to TBG. We connected the results to the large-N f limit, where the RG equations decouple and reduce to a single-channel mean-field approach.
We found a rich landscape of critical fixed points, which display a varied behavior as a function of N f . Importantly, we showed that the inter-channel feedback makes many of the single-channel mean-field solutions unstable at small N f , either because the solutions disappear or because they become multi-critical. We note that, while the critical values of N f where this happens cannot be determined quantitatively on the oneloop level, the qualitative findings which solutions become unstable are usually robust.
We showed that the instabilities that gap out the Dirac spectrum, and thus can gain much condensation energy, are particularly stable at any N f and symmetry setups. They correspond to quantum anomalous Hall, quantum spin Hall, valley (spin) Hall, and time-reversal symmetric inter-valley coherent states. These solutions possess emergent Lorentz symmetry and thus can already be described via a Lorentz-invariant version of the Dirac fermion model of TBG. The fixed point associated with QAH order remains stable through the entire range of 1 < N f < ∞, as well as upon breaking SU(4) symmetry. Order parameters of QSH, VH, VSH, and T-IVC form a degenerate manifold in the SU(4)-symmetric case since they are related by symmetry. The fixed point of the SU(4) manifold FP S−SU (4) disappears at N f ≈ 1.89 and we found it to be unstable when SU(4) is broken. How-ever, the separate solutions associated with QSH, VSH, VH, and T-IVC that descend from this channel in the SU(2)×SU(2)×U(1) case remain stable in the entire range 1 < N f < ∞. The only exception is the fixed point that describes a phase transition to a spinful T-IVC state, which disappears at N f ≈ 2. 75.
In addition, we found that relaxing SU(4) symmetry stabilizes several orders which generically do not gap out the Dirac spectrum. In particular, the critical flavour number of spin-(valley-)polarized, Kramers intervalley coherent, and spin nematic ordering tendencies is strongly reduced. Among them, a spin-valley-polarized state is stable at N f = 2.
Overall, our results provide an unbiased assessment of possible instabilities of multi-flavour Dirac fermions motivated by TBG at charge neutrality. The instabilities we found that gap out the Dirac spectrum potentially realize the sought-after chiral phase transition in a 2D Dirac material 1-4,52,53 . The corresponding quantum critical behavior falls into (generalized) Gross-Neveu universality classes [64][65][66][67][68][69][70][71][72][73][74][75][76][77][78][79][80][81][82] . Interestingly, we also found several critical points whose behavior around a quantum phase transition was not yet studied. With respect to selecting the ground state in TBG, we argued that it is crucial to account for the competition between ordering tendencies because several symmetry-broken states lie close in energy. In this context and in light of recent experiments 34,83 , it is interesting to note that the leading instability is not necessarily the one expected based on symmetries in the mean-field picture [36][37][38][47][48][49][50] due to inter-channel renormalizations of the interaction. It will be enlightening to extend our unbiased treatment via allowing the breaking of additional symmetries either spontaneously or externally in future studies.
In principle, the generalized flavour symmetry permits terms of the form ∑
N 2 f −1 i=0 [Ψ † (M ⊗ κ i )Ψ] 2
in the interacting Lagrangian, where Ψ = (ψ 1 , . . . , ψ N f ) T is a collective 8N f -component spinor, M is an 8 × 8 matrix acting in spin-valley-sublattice space, and κ i are generators of U(N f ), which replace the Pauli matrices µ γ of mini-valley space for general N f . To make the flavour index explicit, we can use the completeness relation
∑ i κ αβ i κ γδ i = N f δ αδ δ βγ and rewrite ∑ N 2 f −1 i=0 [Ψ † M ⊗κ i Ψ] 2 = (ψ α † M ψ β )(ψ β † M ψ α ) with 8-component spinors ψ α of flavour α.
Furthermore, we can connect the terms nondiagonal in flavour (ψ α † M ψ β )(ψ β † M ψ α ) to the flavourdiagonal ones via Fierz identities using that the 64 matrices M form a basis for 8×8 matrices. The general form of a Fierz identity is
ψ †a M X ψ b ψ †c M X ψ d = Y F XY ψ †a M Y ψ d ψ †c M Y ψ b .
(A1) The above equation effectively means that we can write any fermionic bilinear as a linear combination of others provided we know the coefficients F XY . To see how diagonal and non-diagonal flavour terms are related, we need to set a = b and c = d.
The Fierz identities can be condensed in the form of a matrix whose entries are exactly the coefficients F XY . By labelling the 4-Fermi terms based on the six possible matrix channels M , we can define
X s t = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 s t T i s t ρ z τ zs t ρ z τ z T i s t ν s t ω s t ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (A2) where M s ≡ (ψ α † M ψ α ) 2 and M t ≡ (ψ α † M ψ β )(ψ β † M ψ α ), and Eq.
(A1) can be cast in matrix form
F X t = X s (A3) with F = − 1 8 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .(A4)
In the special case where N f = 1, the Fierz matrix relates flavour-diagonal terms and the number of independent couplings can be reduced to three, which is the degeneracy of the eigenvalue 1 of the Fierz matrix F .
For the Lorentz-invariant case, where the number of independent couplings is reduced to four, the Fierz identity is given by
F L X L t = X L s ,(A5)X L = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ γ µs t γ µ T i s t 1 s t T i s t ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦(A6)
and
F L = − 1 8 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 1 3 3 −15 1 45 −3 1 1 1 1 15 −1 15 −1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (A7)
Appendix B: Derivation of RG equations
Our starting point is the Wetterich equation 84
∂ t Γ k = − 1 2 Tr (Γ (2) k + R k ) −1 ∂ t R k (B1)
where ∂ t = k d dk and Γ
(2) k is the matrix of functional derivatives of the effective action with respect to the fields, defined as
Γ (2) k = ⃗ δ δΨ T (−p) Γ k ⃗ δ δΨ(q)(B2)
with Ψ = (ψ, ψ † T ) T . The essence of Eq.(B1) is that it describes the evolution of an effective action as a function of a scale variable k ∈ [0, Λ] with Λ being a UV-cutoff. This evolution is encoded in the regulator function R t , which defines the way fluctuations in the interval [k, Λ] are integrated out. At k = 0, all fluctuations are integrated out, which yields the full quantum effective action and the complete solution to the problem.
Based on the discussion of Sec. II A we make an ansatz for the scale dependent action by introducing a scale dependence on all interactions and the regulator function. We neglect the wavefunction renormalization coefficient as diagrams that contribute to the anomalous dimension in a purely fermionic 1-loop approximation vanish in the regime of point-like interactions. To extract the beta functions for the scale-dependent couplings, we first redefine the scale derivative to only act on the regulator and rewrite Eq. (B1) as
∂ t Γ k = − 1 2∂ t Tr ln (Γ (2) t + R k ) −1 .(B3)
We split the inverse full propagator into a fieldindependent part Γ
(2) 0 and ∆Γ k which incorporates the effects of fluctuations. We expand the logarithm around this point
∂ t Γ k = −∂ t ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 1 2 Tr ⎛ ⎝ ∆Γ k Γ (2) 0 ⎞ ⎠ − 1 4 Tr ⎛ ⎝ ∆Γ k Γ (2) 0 ⎞ ⎠ 2 + . . . ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ . (B4)
We then insert our ansatz and compare right and left hand side of Eq. (B4) to identify the beta functions for the running couplings. This is analogous to a 1-loop, Wilsonian RG scheme for a specific choice of the regulator function. However our analysis will turn out to be independent of the choice of the regulator. Following the procedure detailed above the right hand side of Eq (B4) evaluates to
∂ t Γ k = −∂ t p i,j 2λ i λ j [(ψ † M i G k M j ψ)(ψ † M j G k M i ψ) + 2(ψ † M i ψ)(ψ † M j G k M i G k M j ψ) − Tr(G k M i M j )(ψ † M i ψ) 2 − (ψ † M i G k M j ψ) 2 ] ,(B5)
where p labels the internal momentum integration variable, M i,j are any of the 64 matrices considered in the interacting Lagrangian and G k is the scale dependent fermionic propagator defined as
G k = (Γ (2)0 + R k ) −1 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 iω+qxρxτz+qyρy q 2 (1+r(k)) iω+qxρxτz+ρ T y qy q 2 (1+r(k)) 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (B6)
where r(k) is a dimensionless function encoding the regulator scheme R k = (iω + q x ρ x τ z + q y ρ y )r(q 2 k 2 ). Evaluating the matrix products in Eq. (B5) yields the beta functions for the six couplings.
The quantity l f defined as
l f = − 1 3 k 2−d∂ t d 3 p 2π 3 1 (1 + r(p 2 k 2 )) 2 p 2 (B7)
is the threshold function. By rewriting our couplings asλ i = λ i l f , the threshold function is absorbed in the rescaling and thus the beta functions become regulatorindependent.
Appendix C: Susceptibilities
As mentioned in the main text, to gain insight into the ordered phases related to the quantum critical points, we analyze the divergence of the test-vertex susceptibilities. We start from an effective action
Γ k → Γ k + rh i ψ α † M i ψ α + χ i h 2 i .(C1)
The quantities r and χ will flow according to:
∂ t r i = ∂ ∂h i ⃗ δ δψ µ † ∂ t Γ k ⃗ δ δψ ν hi=0,ψ µ † =0,ψ ν =0 (C2) ∂ t χ i = ∂ 2 (∂ t Γ k ) ∂h 2 i hi=0,ψ µ † =0,ψ ν =0 ,(C3)
where the flow of Γ k is given by Eq. (B1). We perform an expansion of the effective action, while noting that the field-independent part now gets a contribution from the inclusion of the linear vertex term. Keeping only terms that contribute to the flow of the above quantities we get
∂ t r i = −l f C ij λ j r i (C4) ∂ t χ i = l f 4 r 2 i ,(C5)
where C ij is a matrix containing all constant prefactors for each term that appears in the beta functions and l f is given by Eq. (6). We rescale the couplingsλ = k d−2 λl f , with d being the space-time dimension of the model. At the fixed point solutions,λ i =λ * i , we can solve the differential equations to get the explicit dependence as a function of k:
r i = r 0 k Λ Cijλ * j (C6)
We define β i ≡ C ijλi . Then, the dependence of the susceptibility can be written as
χ i = χ 0 k 2β+d−2 Λ 2β . (C7)
We thus relate the two exponents via γ i = 2β i +d−2. Setting d = 3, we get the condition for a divergent susceptibility used in the text β i < − 1 2 . The treatment described above is condensed in Fig. 4 in the main text and Figs. 6 and 7 in this appendix for the Lorentz invariant model (Eq. 6) and full SU(4) model (Eq. 3) respectively. We reiterate the importance of this analysis to identify which of the stable fixed points can describe a quantum phase transition. Specifically, as can be seen in Fig. 7 none of the susceptibilities of FP V−SU(4) diverges in the regime of N f where it is stable.
FP V ∶ [g * 4 , v * 4 , g * , v * ] = N f (−7 + 7N f − 4N 2 f + (1 + 4N f ) 4 + N f (14 + N f )) 4(−1 + 4N f )(5 + 4N f + 8N 2 f ) , 0, 3N 2 f 4 + 2N f + 16N 2 f + 2 4 + N f (14 + N f )
, 0 (D3) Fig. 8 shows the disappearance of physical solutions into the complex plane related to FP S−SU (4) , which constitutes one of the two possibilities that renders solutions inaccessible. Fig. 9 shows the behavior of the other two stable fixed points of the Lorentz invariant model and Fig. 10 shows the evolution of the couplings for the fixed points in the full SU(4) symmetric model. These complement Fig. 2 of the main text. Importantly, in Fig. 10 (4)), semi-metallic SU(4), vector SU(4) (V-SU(4)), density (n), and scalar SU(4) (S-SU(4)) condensates.
Appendix E: Flow equations for the case of SU(4) symmetry relaxation
We provide the beta functions for the eighteen couplings in the case of lower symmetry here
∂ t g 1 = g 1 − 4 N f g 2 1 (4N f − 1) − g 1 (2g 2 + g 4 + 3v
(1)
1 + v (2) 1 + 3v (3) 1 + 2v (4) 1 + 6v (5) 1 + 6v (1) 2 + 2v (2) 2 + 6v (3) 2 + 4v (4) 2 + 12v(5)
2 + 3v
(1)
4 + v (2) 4 + 3v (3) 4 + 2v(4)4 + 6v(5) 4 )
− 4(g 2 g 4 + 3v
(1)
2 v
(1)
4 + v(2)
2 v
(2)
4 + 3v (3) 2 v (3) 4 + 2v(4)
2 v
4 + 6v (5) 2 v (5) 4 ) (E1) ∂ t v(4)
(1)
1 = v (1) 1 + 4 N f g 1 v (1) 1 + 2g 2 (v (1) 1 + 2v
(1)
4 ) + g 4 v (1) 1 + 4g 4 v (1) 2 − 4(N f + 3)(v (1) 1 ) 2 + v (1) 1 v (2) 1 − v (1) 1 v (3) 1 + 2v (1) 1 v (4) 1 − 2v (1) 1 v (5) 1 + 6v (1) 1 v (1) 2 + 2v (1) 1 v (2) 2 − 2v (1) 1 v (3) 2 + 4v (1) 1 v (4) 2 − 4v (1) 1 v (5) 2 − v (1) 1 v (1) 4 + v (1) 1 v (2) 4 − v (1) 1 v (3) 4 + 2v (1) 1 v (4) 4 − 2v (1) 1 v (5) 4 − 2(v (3) 1 ) 2 + 8v (3) 1 v (3) 2 − 4(v (5) 1 ) 2 + 16v (5) 1 v (5) 2 − 4(v (1) 2 ) 2 + 4v (2) 2 v (3) 4 − 4(v (3) 2 ) 2 + 4v (3) 2 v (2) 4 + 8v (4) 2 v (5) 4 − 8(v (5) 2 ) 2 + 8v (5) 2 v (4) 4 − 2(v (1) 4 ) 2 − 2(v (3) 4 ) 2 − 4(v (5) 4 ) 2 (E2) ∂ t v (2) 1 = v (2) 1 + 4 N f g 1 v (2) 1 + g 4 v (2) 1 + 3v (1) 1 v (2) 1 − (4N f − 1)(v (2) 1 ) 2 + 3v (2) 1 v (3) 1 − 2v (2) 1 v (4) 1 − 2(v (4) 1 ) 2 − 6v (2) 1 v (5) 1 − 6(v (5) 1 ) 2 + 6v (2) 1 v (1) 2 + 4g 4 v (2) 2 + 2v (2) 1 v (2) 2 + 6v (2) 1 v (3) 2 − 4v (2) 1 v (4) 2 + 8v (4) 1 v (4) 2 − 4(v (4) 2 ) 2 − 12v (2) 1 v (5) 2 + 24v (5) 1 v (5) 2 − 12(v(5)
2 ) 2 + 3v
(2)
1 v (1) 4 + 12v (3) 2 v (1) 4 + v (2) 1 v (2) 4 + 2g 2 (v (2) 1 + 2v (2) 4 ) + 3v (2) 1 v (3) 4 + 12v (1) 2 v (3) 4 − 2v (2) 1 v (4) 4 − 2(v (4) 4 ) 2 − 6v (2) 1 v (5) 4 − 6(v (5) 4 ) 2 (E3) ∂v (3) 1 = v 3 1 + 4 N f g 1 v (3) 1 + g 4 v (3) 1 − 5v (1) 1 v (3) 1 + v (2) 1 v (3) 1 − (4N f + 1)(v (3) 1 ) 2 − 2v (3) 1 v (4) 1 + 2v (3) 1 v (5) 1 − 4v (4) 1 v (5) 1 + 6v (3) 1 v (1) 2 + 2v (3) 1 v (2) 2 + 4g 4 v (3) 2 + 8v (1) 1 v (3) 2 − 2v (3) 1 v (3) 2 − 8v (1) 2 v (3) 2 − 4v (3) 1 v (4) 2 + 8v (5) 1 v (4) 2 + 4v (3) 1 v (5) 2 + 8v (4) 1 v (5) 2 − 8v (4) 2 v (5) 2 − v (3) 1 v (1) 4 + 4v (2) 2 v (1) 4 + v (3) 1 v (2) 4 + 4v (1) 2 v (2) 4 − v (3) 1 v (3) 4 − 4v (1) 4 v (3) 4 + 2g 2 (v (3) 1 + 2v (3) 4 ) − 2v (3) 1 v (4) 4 + 2v (3) 1 v (5) 4 + 16v (5) 2 v (5) 4 − 4v (4) 4 v (5) 4 (E4) ∂ t v (4) 1 = v (4) 1 + 4 N f g 1 v (4) 1 + g 4 v (4) 1 + 3v (1) 1 v (4) 1 − 3v (2) 1 v (4) 1 − 3v (3) 1 v (4) 1 − 4N f (v (4) 1 ) 2 − 6v (3) 1 v (5) 1 + 6v (4) 1 v (1) 2 + 2v (4) 1 v (2) 2 − 6v (4) 1 v (3) 2 + 12v (5) 1 v (3) 2 + 4g 4 v (4) 2 + 4v (2) 1 v (4) 2 − 4v (2) 2 v (4) 2 + 12v (3) 1 v (5) 2 − 12v (3) 2 v (5) 2 + 3v(4)
1 v
(1)
4 + 12v (5) 2 v (1) 4 − v (4) 1 v (2) 4 − 3v (4) 1 v (3) 4 − 2v (2) 4 v (4) 4 + 2g 2 (v (4) 1 + 2v (4) 4 ) + 12v (1) 2 v (5) 4 − 6v (3) 4 v (5) 4 (E5) ∂ t v (5) 1 = v (5) 1 + 4 N f g 1 v (5) 1 + 2g 2 v (5) 1 + g 4 v (5) 1 − 5v (1) 1 v (5) 1 − 3v (2) 1 v (5) 1 − 4N f (v (5) 1 ) 2 + 6v (5) 1 + 4v (4) 1 v (3) 2 + 2v (5) 1 v (3) 2 − 4v (3) 2 v (4) 2 + v (3) 1 (−2v (4) 1 + v (5) 1 + 4v (4) 2 ) + 4g 4 v (5) 2 + 8v (1) 1 v (5) 2 + 4v (2) 1 v (5) 2 − 8v (1) 2 v (5) 2 − 4v (2) 2 v (5) 2 − v (5) 1 v (1) 4 + 4v (4) 2 v (1) 4 − v (5) 1 v (2) 4 + v (5) 1 v (3) 4 + 8v (5) 2 v (3) 4 + 4v (1) 2 v (4) 4 − 2v (3) 4 v (4) 4 + 4g 2 v (5) 4 + 8v (3) 2 v (5) 4 − 4v (1) 4 v (5) 4 − 2v (2) 4 v (5) 4 (E6) ∂ t g 4 = g 4 + 4 N f 4g 1 g 2 − 2g 2 2 − 3g 1 g 4 + 6g 2 g 4 − 3g 2 4 + 12g 2 4 N f − 9g 4 v (1) 1 − 3g 4 v (2) 1 − 9g 4 v (3) 1 − 6g 4 v (4) 1 − 18g 4 v (5) 1 + 18g 4 v (1) 2 + 12v (1) 1 v (1) 2 − 6(v (1) 2 ) 2 + 6g 4 v (2) 2 + 4v (2) 1 v (2) 2 − 2(v (2) 2 ) 2 + 18g 4 v (3) 2 + 12v (3) 1 v (3) 2 − 6(v (3) 2 ) 2 + 12g 4 v (4) 2 + 8v (4) 1 v (4) 2 − 4(v (4) 2 ) 2 + 36g 4 v (5) 2 + 24v (5) 1 v (5) 2 − 12(v (5) 2 ) 2 − 9g 4 v (1) 4 − 3g 4 v (2) 4 − 9g 4 v (3) 4 − 6g 4 v (4) 4 − 18g 4 v (5) 4 (E7) ∂ t v
(1)
4 = v (1) 4 + 4 N f 4g 1 v (1) 2 + 4v(3)
1 v
(2) 2 + 4v
(2)
1 v (3) 2 − 4v (2) 2 v (3) 2 + 8v(5)
1 v (4) 2 + 8v
(4) 1 v (5) 2 − 8v (4) 2 v (5) 2 − 3g 1 v (1) 4 − 3g 4 v (1) 4 − v (1) 1 v (1) 4 − 3v(2)
1 v
(1)
4 + 3v(3)
1 v
(1)
4 − 6v(4)
1 v
(1)
4 + 6v(5)
1 v
(1)
+ 2v
(1)
2 v
(1)
+ 6v
(2)
2 v
(1)
4 − 6v(3)
2 v
(1) 4 + 12v (4) 2 v
(1)
4 − 12v(5)
2 v
(1)
4 + 3(v (1) 4 ) 2 + 12N f (v(1)
4 ) 2 + g 2 (4v
(1)
1 − 4v (1) 2 + 6v
(1)
) − 3v
(1)
4 v (2) 4 − 4v (3) 1 v (3) 4 + 12N f (v(5)
4 ) 2 + g 2 4v (5) 1 − 4v (5) 2 + 6v ∂ t g 2 = g 2 − 4 N f g 1 (g 2 − 2g 4 ) − 4g 2 2 N f + g 2 (g 4 + 3v
(1)
1 + v (2) 1 + 3v (3) 1 + 2v (4) 1 + 6v (5) 1 − 3v (1) 4 − v (2) 4 − 3v (3) 4 − 2v (4) 4 − 6v (5) 4 ) + 2(−3v (1) 1 v (1) 4 + 3v (1) 2 v (1) 4 − v (2) 1 v (2) 4 + v (2) 2 v (2) 4 − 3v (3) 1 v (3) 4 + 3v (3) 2 v (3) 4 − 2v (4) 1 v (4) 4 + 2v (4) 2 v (4) 4 − 6v (5) 1 v (5) 4 + 6v (5) 2 v (5) 4 ) (E13) ∂ t v (1) 2 = v (1) 2 + 4 N f 2g 4 v (1) 1 + 2(v (1) 1 ) 2 + 2(v (3) 1 ) 2 + 4(v (5) 1 ) 2 − g 1 v (1) 2 − g 4 v (1) 2 − 3v (1) 1 v (1) 2 − v (2) 1 v (1) 2 + v (3) 1 v (1) 2 − 2v (4) 1 v (1) 2 + 2v (5) 1 v (1) 2 + 4(v (1) 2 ) 2 (N f + 2) − 4v (3) 1 v (3) 2 + 8(v (3) 2 ) 2 − 8v (5) 1 v (5) 2 + 16(v (5) 2 ) 2 + 2g 1 v (1) 4 − 2g 2 v (1) 4 − v (1) 2 v (1) 4 + 2(v (1) 4 ) 2 + 2v (3) 1 v (2) 4 + v (1) 2 v (2) 4 − 2v (3) 2 v (2) 4 + 2v (2) 1 v (3) 4 − v (1) 2 v (3) 4 − 2v (2) 2 v (3) 4 + 2(v (3) 4 ) 2 + 4v (5) 1 v (4) 4 + 2v (1) 2 v (4) 4 − 4v (5) 2 v (4) 4 + 4v (4) 1 v (5) 4 − 2v (1) 2 v (5) 4 − 4v (4) 2 v (5) 4 + 4(v (5) 4 ) 2 (E14) ∂ t v (2) 2 = v (2) 2 + 4 N f 2g 4 v (2) 1 + 2(v (4) 1 ) 2 + 6(v (5) 1 ) 2 − g 1 v (2) 2 − g 4 v (2) 2 − 3v (1) 1 v (2) 2 − v (2) 1 v (2) 2 − 3v (3) 1 v (2) 2 + 2v(4)
1 v
(2) 2 + 6v
1 v
(2)
2 + 4N f (v (2) 2 ) 2 − 4v(4)
1 v
2 + 8(v(4)
2 ) 2 − 12v
1 v
2 + 24(v
2 ) 2 + 6v
(3)
1 v
(1)
+ 3v
(2)
2 v
(1)
4 − 6v (3) 2 v (1) 4 + 2g 1 v (2) 4 − 2g 2 v (2) 4 + v (2) 2 v (2) 4 + 6v (1) 1 v (3) 4 − 6v (1) 2 v (3) 4 + 3v (2) 2 v (3) 4 − 2v (2) 2 v (4) 4 + 2(v (4) 4 ) 2 − 6v (2) 2 v (5) 4 + 6(v (5) 4 ) 2 (E15) ∂ t v (3) 2 = v (3) 2 − 4 N f −4v (4) 1 v (5) 1 + 4v (3) 1 v (1) 2 + g 1 v (3) 2 + v (2) 1 v (3) 2 − v (3) 1 v (3) 2 − 2v (4) 1 v (3) 2 + 2v (5) 1 v (3) 2 − 16v (1) 2 v (3) 2 − 4N f (v(3)
2 ) 2 + g 4 (−2v
(3)
1 + v (3)
2 ) + 4v (5) 1 v (4) 2 + 4v (4) 1 v (5) 2 − 16v (4) 2 v (5) 2 − 2v
(2)
1 v
(1)
+ 2v
(2)
2 v
(1)
4 + v (3) 2 v (1) 4 + v (1) 1 (−4v (3) 1 + 3v (3) 2 − 2v
(2) 4 ) + 2v
(1)
2 v
(2)
4 − v (3) 2 v (2) 4 − 2g 1 v (3) 4 + 2g 2 v (3) 4 + v (3) 2 v (3) 4 − 4v (1) 4 v (3) 4 + 2v (3) 2 v (4) 4 − 8v (5) 1 v (5) 4 − 2v (3) 2 v (5) 4 + 8v (5) 2 v (5) 4 − 4v (4) 4 v (5) 4 (E16) ∂ t v (4) 2 = v (4) 2 − 4 N f −6v (3) 1 v (5) 1 + 2v(4)
1 v
(2) 2 + 6v
(5) 1 v (3) 2 + g 1 v (4) 2 + 3v (1) 1 v (4) 2 − 3v (3) 1 v (4) 2 − 8v (2) 2 v (4) 2 − 4N f (v(4)
2 ) 2 + g 4 (−2v
(4) 1 + v(4)
2 ) + v
(2) 1 (−2v
(4) 1 + v(4)
2 ) + 6v
(3)
1 v (5) 2 − 24v (3) 2 v (5) 2 − 6v (5) 1 v (1) 4 − 3v(4)
2 v
(1)
4 + 6v (5) 2 v (1) 4 + v(4)
2 v
(2)
4 + 3v (4) 2 v (3) 4 − 2g 1 v (4) 4 + 2g 2 v (4) 4 − 2v (2) 4 v (4) 4 − 6v (1) 1 v (5) 4 + 6v (1) 2 v (5) 4 − 6v (3) 4 v (5) 4 (E17) ∂ t v (5) 2 = v (5) 2 − 4 N f −4v (1) 1 v (5) 1 − 2v (2) 1 v (5) 1 + 4v (5) 1 v (1) 2 + 2v (5) 1 v (2) 2 + 2v (4) 1 v (3) 2 − 8v (3) 2 v (4) 2 + g 1 v (5) 2 + 3v (1) 1 v (5) 2 + v (2) 1 v (5) 2 − 16v (1) 2 v (5) 2 − 8v (2) 2 v (5) 2 − 4N f (v(5)
2 ) 2 + g 4 (−2v
(5) 1 + v(5)
2 ) − 2v (4) 1 v
(1)
4 + 2v (4) 2 v (1) 4 + v (5) 2 v (1) 4 + v (5) 2 v (2) 4 − 4v (5) 1 v (3) 4 + 3v (5) 2 v (3) 4 − 2v (1) 1 v (4) 4 + 2v (1) 2 v (4) 4 − 2v (3) 4 v (4) 4 + v (3) 1 (−2v (4) 1 + 2v (4) 2 + v (5) 2 − 4v (5) 4 ) − 2g 1 v (5) 4 + 2g 2 v (5) 4 + 4v (3) 2 v (5) 4 − 4v (1) 4 v (5) 4 − 2v (2) 4 v (5) 4 (E18)
FIG. 1 :
1Flow of couplings according to the beta functions βg 4 , βv 4 and βv 1 along a plane in coupling space illustrating the behavior around the critical fixed points FP S , FP S−SU(4) and FP M−SU(4) (red) and the trivial non-interacting Gaussian fixed point (black). If the bare values of the couplings are larger in magnitude than the values set by the critical fixed points, they flow to strong coupling in the infrared signaling a potential instability towards symmetry breaking.
FIG. 2 :
2The couplings of the scalar (FP S−SU(4) ) and vector (FP V−SU(4) ) SU(4) fixed points for several values of the fermion flavor number N f . For general N f , all fixed-point couplings are non-zero demonstrating that different ordering channels are coupled. For large values of N f only one coupling is non-zero and a single-channel, mean-field description is possible.FIG. 3: Second largest critical exponent θ2 of the scalar (FP S , FP S−SU(4) ) and vector (FP V , FP V−SU(4) ) fixed points of the Lorentz-invariant Dirac fermion model. The largest critical exponent θ1 = 1 (and is omitted for clarity). The sign of θ2 dictates the stability of the fixed point solution. It can be clearly seen that FP V becomes unstable at N c V = 3, where θ2 changes sign.
FIG. 4 :
4The exponents β that determine the divergence of susceptibilities Eq.(18) for the vector SU(4) (FP V−SU(4) ) and scalar (FP S ) fixed points in the Lorentz invariant case. If β < − 1 2 , the corresponding suceptibility diverges signaling a phase transition as marked by the orange dashed line. None of the susceptibilities of FP V−SU(4) diverges for small flavour numbers, even though it is a critical point.FIG. 5: Schematic overview which summarizes the varied behavior of the different fixed point solutions as function of flavor number N f , where N f = 2 in the Dirac fermion model of TBG. Density (FPn) and semi-metallic (FP SM ) fixed points become multi-critical for small N f , the Lorentz-invariant vector fixed point (FP V ) is multi-critical for any N f , and the SU(4) vector fixed point (FP V−SU(4) ) at large N f (red dashed line). Metallic (FP M−SU(4) ), semi-metallic (FP SM−SU(4) ), and insulating (FP S−SU(4) ) SU(4) fixed points lie in the complex plane below a critical flavor number (blue dashed line). Only the scalar fixed point is stable for any N f . It is associated with a QAH phase transition.
FIG. 6 :
6The exponents β determining the susceptibilities of the scalar SU(4) FP S−SU(4) and vector FP V fixed points in the Lorentz invariant case as function of flavor number N f . If β < −1 2 the susceptibility of the corresponding order diverges. The exponents for the two other critical fixed points FP S and FP V−SU(4) is shown in the main text. FIG. 7: The exponents β that determine the behavior of the susceptibilities for all critical fixed point of the SU(4) symmetric Dirac fermion model of TBG as function of flavor number N f . Values of β < − 1 2 can lead to second order phase transitions. All of the fixed point solutions exhibit a divergence in a certain channel with the exception of FP V−SU(4) .
FIG. 8 :
8, we can observe the recovery of the Lorentz invariant solution FP S−SU(4) in the non-Lorentz invariant case. Furthermore, FP V−SU(4) , while accessible for all values of N f in the Lorentz invariant case, becomes multicritcal in the case of the full SU(4) model (Eq. 3) at N c f ≈ 13.89 as mentioned in the main text. As such, it does not describe a phase transition in the interval of N f where it is stable. An example of a complex conjugate collision of the scalar SU(4) fixed point FP S−SU(4) with another multicritical fixed point leading to the creation of a pair of real valued fixed point solutions. The points correspond to the coupling values for different N f . FIG. 9: The couplings of the scalar (FP S ) and vector (FP V ) SU(4) fixed points for several values of the fermion flavor number N f . As mentioned in the main text , for general N f , several fixed-point coupling values are non-zero so that different ordering channels are coupled. For large values of N f only one coupling is non-zero and a single-channel, mean-field description is possible. Out of the four stable, Lorentz invariant fixed point solutions, FP V does not emerge as a quantum critical point in the non Lorentz-symmetric model.
FIG. 10 :
10Coupling values as a function of the flavor number N f of the potentially critical fixed points in the SU(4) symmetric Dirac fermion model for TBG. The red, dashed line indicates the critical value N c f = 13.6 for which FP V−SU(4) becomes multicritical. Fixed points FPi are associated with semi-metallic (SM), metallic SU(4) (M-SU
TABLE I :
ISummary of possible orders introduced in Sec. II B and their tensor structure specified by Pauli matrices in sublattice ρ, valley τ and spin σ space (c = x, y, z). We also list the corresponding single-channel coupling in the limit of large flavour numbers N f and the critical values N c f in the SU(2) × SU(2) × U(1) case. The dash indicates stability for the whole interval of N f studied. Double lines group orders related by SU(4) symmetry Eq. (2).Order
Tensor structure Large-N f channel N c
f
density
1
g1
1.55
IQH
ρxτz
ρy
g2
2.46
QAH
ρzτz
g4
-
SP
σc
v
(1)
1
3.66
SVP
τzσc
v
(2)
1
1.48
VP
τz
v
(3)
1
4.49
K-IVC
(ρyτx, ρyτy)
v
(4)
1
3.23
S-K-IVC (ρyτxσc, ρyτyσc)
v
(5)
1
9.14
S-IQH
ρxτzσc
ρyσc
v
(1)
2
5.21
S-NEM
ρxσc
ρyτzσc
v
(2)
2
2.30
NEM
ρx
ρyτz
v
(3)
2
6.83
N-IVC
ρzτy
τx
,
ρzτx
τy
v
(4)
2
5.23
S-N-IVC
ρzτyσc
τxσc
,
ρzτxσc
τyσc
v
(5)
2
19.9
QSH
ρzτzσc
v
(1)
4
-
VSH
ρzσc
v
(2)
4
-
VH
ρz
v
(3)
4
-
T-IVC
ρxτx, ρxτy
v
(4)
4
-
S-T-IVC
ρxτxσc, ρxτyσc
v
(5)
4
2.75
T. Wehling, A. Black-Schaffer, and A. Balatsky, Advances in Physics 63, 1 (2014). 2 O. Vafek and A. Vishwanath, Annual Review of Condensed Matter Physics 5, 83 (2014). 3 R. Boyack, H. Yerzhakov, and J. Maciejko, The European Physical Journal Special Topics 230, 979 (2021).
C. Wetterich, Phys. Lett. B 301, 90 (1993). 85 Note that this deviates from other common definitions which count two-or four-component spinors.
AcknowledgmentsWe are thankful for valuable discussions with Jens Braun, Holger Gies, Lukas Janssen, Walter Metnzer, Shouryya Ray, Fabian Rennecke, Michael Scherer, and Mathias Scheurer.Appendix A: Fierz identitiesAppendix D: Fixed point analysis Generally, due to the large number of couplings considered, the fixed point solutions do not allow for an analytic form as a function of N f . However in the case where one or more couplings are zero, the expressions simplify considerably permitting analytic expressions for the quantum critical points. Specifically this holds for FP S , FP n and FP V , with the respective expressions
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Imaging inter-valley coherent order in magic-angle twisted trilayer graphene. H Kim, Y Choi, Étienne Lantagne-Hurtubise, C Lewandowski, A Thomson, L Kong, H Zhou, E Baum, Y Zhang, L Holleis, K Watanabe, T Taniguchi, A F Young, J Alicea, S Nadj-Perge, arXiv:2304.10586cond-mat.str-elH. Kim, Y. Choi,Étienne Lantagne-Hurtubise, C. Lewandowski, A. Thomson, L. Kong, H. Zhou, E. Baum, Y. Zhang, L. Holleis, K. Watanabe, T. Taniguchi, A. F. Young, J. Alicea, and S. Nadj- Perge, "Imaging inter-valley coherent order in magic-angle twisted trilayer graphene," (2023), arXiv:2304.10586 [cond-mat.str-el] .
| [] |
[
"Characterizing qubit channels in the context of quantum teleportation",
"Characterizing qubit channels in the context of quantum teleportation",
"Characterizing qubit channels in the context of quantum teleportation",
"Characterizing qubit channels in the context of quantum teleportation"
] | [
"Arkaprabha Ghosal \nCentre for Astroparticle Physics and Space Science (CAPSS)\nBose Institute\nBlock EN, Sector V, Salt Lake700 091KolkataIndia\n",
"Debarshi Das \nS. N\nBose National Centre for Basic Sciences\nBlock JD, Sector III, Salt Lake700 106KolkataIndia\n",
"Subhashish Banerjee \nIndian Institute of Technology Jodhpur\n342 037JodhpurIndia\n",
"Arkaprabha Ghosal \nCentre for Astroparticle Physics and Space Science (CAPSS)\nBose Institute\nBlock EN, Sector V, Salt Lake700 091KolkataIndia\n",
"Debarshi Das \nS. N\nBose National Centre for Basic Sciences\nBlock JD, Sector III, Salt Lake700 106KolkataIndia\n",
"Subhashish Banerjee \nIndian Institute of Technology Jodhpur\n342 037JodhpurIndia\n"
] | [
"Centre for Astroparticle Physics and Space Science (CAPSS)\nBose Institute\nBlock EN, Sector V, Salt Lake700 091KolkataIndia",
"S. N\nBose National Centre for Basic Sciences\nBlock JD, Sector III, Salt Lake700 106KolkataIndia",
"Indian Institute of Technology Jodhpur\n342 037JodhpurIndia",
"Centre for Astroparticle Physics and Space Science (CAPSS)\nBose Institute\nBlock EN, Sector V, Salt Lake700 091KolkataIndia",
"S. N\nBose National Centre for Basic Sciences\nBlock JD, Sector III, Salt Lake700 106KolkataIndia",
"Indian Institute of Technology Jodhpur\n342 037JodhpurIndia"
] | [] | We consider a scenario where a party, say, Alice prepares a pure two-qubit (either maximally entangled or non-maximally entangled) state and sends one half of this state to another distant party, say, Bob through a qubit (either unital or non-unital) channel. Finally, the shared state is used as a teleportation channel. In this scenario, we focus on characterizing the set of qubit channels with respect to the final state's efficacy as a resource of quantum teleportation (QT) in terms of maximal average fidelity and fidelity deviation (fluctuation in fidelity values over the input states). Importantly, we point out the existence of a subset of qubit channels for which the final state becomes useful for universal QT (having maximal average fidelity strictly greater than the classical bound and having zero fidelity deviation) when the initially prepared state is either useful for universal QT (i.e., for a maximally entangled state) or not useful for universal QT (i.e., for a subset of non-maximally entangled pure states). Interestingly, in the latter case, we show that non-unital channels (dissipative interactions) are more effective than unital channels (non-dissipative interactions) in producing useful states for universal QT from non-maximally entangled pure states. | 10.1103/physreva.103.052422 | [
"https://export.arxiv.org/pdf/2102.02054v2.pdf"
] | 231,786,544 | 2102.02054 | cc71bc4d3dc92a23c880d45cd774dd23be795bd3 |
Characterizing qubit channels in the context of quantum teleportation
18 May 2021
Arkaprabha Ghosal
Centre for Astroparticle Physics and Space Science (CAPSS)
Bose Institute
Block EN, Sector V, Salt Lake700 091KolkataIndia
Debarshi Das
S. N
Bose National Centre for Basic Sciences
Block JD, Sector III, Salt Lake700 106KolkataIndia
Subhashish Banerjee
Indian Institute of Technology Jodhpur
342 037JodhpurIndia
Characterizing qubit channels in the context of quantum teleportation
18 May 2021
We consider a scenario where a party, say, Alice prepares a pure two-qubit (either maximally entangled or non-maximally entangled) state and sends one half of this state to another distant party, say, Bob through a qubit (either unital or non-unital) channel. Finally, the shared state is used as a teleportation channel. In this scenario, we focus on characterizing the set of qubit channels with respect to the final state's efficacy as a resource of quantum teleportation (QT) in terms of maximal average fidelity and fidelity deviation (fluctuation in fidelity values over the input states). Importantly, we point out the existence of a subset of qubit channels for which the final state becomes useful for universal QT (having maximal average fidelity strictly greater than the classical bound and having zero fidelity deviation) when the initially prepared state is either useful for universal QT (i.e., for a maximally entangled state) or not useful for universal QT (i.e., for a subset of non-maximally entangled pure states). Interestingly, in the latter case, we show that non-unital channels (dissipative interactions) are more effective than unital channels (non-dissipative interactions) in producing useful states for universal QT from non-maximally entangled pure states.
I. INTRODUCTION
Developments in the understanding of quantum correlations along with the superposition principle, have bench-marked advances in the field of quantum information. Quantum entanglement is one of the most prominent of quantum correlations which is empowered due to the superposition principle. Entanglement plays a pivotal role in the success of a number of quantum information protocols, in particular, Quantum Teleportation (QT).
QT [1] can be realized as a strategy between two spatially separated parties where a sender (say, Alice) transfers an unknown quantum state to the receiver (say, Bob) using Local Operations and Classical Communications (LOCC) and shared entanglement without any physical transmission of quantum systems. QT has played a crucial role in the advancement of quantum communication. Motivated from QT, innumerable quantum information theoretic and communication tasks ranging from quantum repeaters [2], quantum gate teleportation [3] to measurement-based computing [4] have been proposed. The idea of QT has been extended to multipartite systems [5] and continuous variable systems [6]. Experimental demonstrations of QT have been reported [7][8][9] which also include QT over large distances [10] or QT from ground to satellite [11].
The standard figure of merit for QT is the average fidelity [12][13][14][15]. It represents the average closeness or overlap between the input state at the sender's end and the output state at the receiver's end, where the average is taken over all possible input states. In case of * [email protected] † [email protected] ‡ [email protected] perfect QT, the output state is exactly equal to the input state for all possible input states. Maximally entangled states are the necessary resource for perfect QT [1]. However, in reality one cannot expect maximally entangled states due to environmental interactions leading to imperfect QT, where the average fidelity is strictly less than one [12]. In such cases, all input states may not be teleported equally well and dispersion or fluctuation in fidelity over the input states may arise [16,17]. Although the average fidelity being the standard quantifier for QT, it does not contain any information about the fluctuation in fidelity or fidelity deviation [16][17][18][19][20][21]. Hence, average fidelity associated with fidelity deviation can completely characterize QT. The maximal average fidelity is the maximal value of average fidelity achievable over all possible local unitary operations within the standard teleportation protocol [12]. On the other hand, fidelity deviation is something that one would like to minimize while keeping the average fidelity to the maximal possible value [17,18]. Any two-qubit entangled state, for which the maximal fidelity is strictly greater than the classical bound, is known as an useful state for QT [12,13]. A useful state for QT is called useful for Universal Quantum Teleportation (UQT) if and only if the state shows vanishing fidelity deviation [17]. In other words, a UQT implies that all input states can be teleported equally well with the same fidelity value equals to the maximal average fidelity. Hence, the concept of fidelity deviation can be used as a filter to select the optimal states for QT [18]. Quantifying the success of QT based only an average fidelity has some limitations in practical cases, for example, in the context of quantum circuit [19,22,23] consisting of QT as an intermediate step. In such realistic scenarios, one may need to teleport a finite number of input states without considering all possible input states. In such cases, the knowledge of fidelity deviation plays a crucial role to estimate the individual fidelities for different input states. In particular, if the entangled channel is useful for UQT, then any input state is teleported with fidelity equal to the average fidelity.
The necessary resource for QT is entanglement which must be shared between the sender and the receiver. Sharing of entanglement can be done by a simple process; Alice prepares an entangled pair in her lab and sends one half to Bob via a quantum channel. A perfect QT requires a maximally entangled state which can only be established via noiseless quantum channels. In practical scenarios, the channels are noisy and, hence, these studies need to be considered by taking into account of the effects of environmental interactions. This can be done using the tools of open quantum systems [24,25].
A quantum channel [26][27][28] is a Completely Positive and Trace Preserving (CPTP) map Φ with the operator sum representation Λ(ρ) = i K i ρK † i . Here K i are the Kraus operators [26,27] obeying the completeness condition, i K † i K i = I. Any quantum channel is characterized by the following properties: (a) linearity, (b) Hermiticity preserving, (c) positivity preserving, and (d) trace preserving. The evolution modeled by a quantum channel could be, in general, unital or non-unital.
In the present study, we consider the scenario where Alice prepares either a maximally entangled two-qubit state or a non-maximally entangled two-qubit pure state and sends one half of it via a unital or non-unital qubit channel. The final state shared between Alice and Bob is used as the teleportation channel. Since the maximally entangled two-qubit states are useful for UQT [17], in this case we find out the set of unital as well as non-unital channels for which the final state still remains useful for UQT. On the other hand, the non-maximally entangled two-qubit pure states being not useful for UQT [17], we work out on finding the subset of unital and non-unital channels which convert the initial non-maximally entangled two-qubit pure states into useful states for UQT by acting on one half of the states. Our results indicate that both unital and non-unital qubit channels can decrease the fidelity deviation (even can eliminate it completely).
Environmental interactions and the effects of quantum channels inevitably degrade the efficacy of a quantum resource. Hence, from foundational point of view as well as from information theoretic perspective it is important to find out the set of quantum channels that preserve the effectiveness of any quantum resource. QT being one of the primitive quantum information processing protocols, analyzing the set of quantum channels preserving the resources for QT is of paramount significance. Our present study is motivated to address this practical issue. Most importantly, our results effectively filter out the set of qubit channels which can be used in practical scenario for realizing UQT.
The paper is organized as follows. Sec. II is dedicated to the preliminary ideas and definitions useful for our paper. In particular, we briefly discuss the concept of concurrence for two-qubit states, the Hilbert-Schmidt representation and canonical representation of an arbitrary two-qubit state, the concept of maximal average fidelity and fidelity deviation for a two-qubit state, and the qubit channels. In Sec. III, we summarize the results obtained in this paper. Next, in Sec. IV, we present the results when the initial state is a Bell state followed by the results in Sec. V with non-maximally entangled twoqubit pure states as the initial states. An analysis of physically motivated noise models is made in Sec. VI. Finally, we conclude with a brief discussion in Sec. VII. Some of the technical details are relegated to two Appendices.
II. PRELIMINARIES
In this section, we will discuss the basic definitions and preliminaries of quantum entanglement, Hilbert-Schmidt representation and canonical representation of a two-qubit state, maximal average fidelity and fidelity deviation in QT with a two-qubit state and qubit channels.
A. Concurrence of a two-qubit state
Entanglement is a fundamental aspect of quantum correlation present in compound quantum systems. There exist a number of well-known measures of quantum entanglement. In the present paper, we restrict ourselves to the concurrence measure [29]. For a two qubit state ρ, the concurrence C(ρ) is defined as [29],
C(ρ) = max{0, λ 1 − λ 2 − λ 3 − λ 4 }.(1)
Here λ 1 ≥ λ 2 ≥ λ 3 ≥ λ 4 are square roots of the eigenvalues of ρρ, whereρ = (σ y ⊗ σ y )ρ * (σ y ⊗ σ y ) is the Pauli rotated state with σ y being the Pauli bit-phase flip matrix and ρ * being the complex conjugation of ρ in the computational basis.
B. Hilbert-Schmidt representation and the canonical form of a two-qubit state
The Hilbert-Schmidt representation of a two-qubit density matrix ρ is given by [12,30,31],
ρ = 1 4 I 4 + R · σ ⊗ I 2 + I 2 ⊗ S · σ + 3 i,j=1 T ij σ i ⊗ σ j .
(2) The terms R, S represent local vectors in R 3 in each respective marginal and R(S) · σ = T ρ . Then there always exits a product unitary operation U 1 ⊗ U 2 that transforms ρ → ρ C such that [13,30,31]
ρ C = (U 1 ⊗ U 2 )ρ(U 1 ⊗ U 2 ) † = 1 4 I 4 + r · σ ⊗ I 2 + I 2 ⊗ s · σ + 3 k=1 λ k |t kk | σ k ⊗ σ k ,(3)
with λ k ∈ {−1, +1} for k = 1, 2, 3; r, s represent local vectors in R 3 in each respective marginal and r(s
) · σ = 3 i=1 r i (s i )σ i . Now, one can further choose the product unitary U 1 ⊗ U 2 such that (i) if det(T ρ ) ≤ 0, then λ k = −1 for |t kk | = 0, k = 1, 2, 3; (ii) if det(T ρ ) > 0, then λ i , λ j = −1, λ k = +1 for |t kk | = 0 for any choice of i = j = k ∈ {1, 2, 3} with |t ii | ≥ |t jj | ≥ |t kk |.
This transformed ρ C is known as the canonical form of ρ [17,32].
C. Maximal average fidelity and fidelity deviation in QT with a two-qubit state
Perfect QT is achieved if and only if the shared state is maximally entangled. In this case, the output state of QT is exactly equal to the input state. However, in practice, the available states are mixed entangled and hence, QT will not be perfect. The standard figure of merit for QT is expressed through the concept of average fidelity [12][13][14][15], which signifies the closeness between the input and the output states.
The average teleportation fidelity for a two-qubit state ρ is defined as [12] f ρ = f ψ,ρ dψ,
where f ψ,ρ = ψ|χ|ψ is the fidelity between the inputoutput pair (|ψ ψ|, χ). The above integration is taken over a uniform distribution of all possible pure qubit input states |ψ (normalized Haar measure, dψ = 1). In other words, this integration is over the parameters appearing in |ψ . In Bloch representation, an arbitrary pure qubit input state is given by |ψ ψ| = 1 2 (I 2 +â· σ), where the unit vectorâ represents the Bloch vector of the input state and it is given by,â = (sin θ cos φ, sin θ sin φ, cos θ). With such parametrization of an arbitrary input state, we have dψ = sin θdθdφ. Note that the average fidelity f ρ is defined when the standard protocol for QT proposed by Bennett et al. [1] is adopted. As mentioned earlier, f ρ = 1 is possible if and only if ρ is maximally entangled.
Fidelity deviation δ ρ is a secondary quantifier of QT which measures fluctuations in fidelity over the input states. It is defined as the standard deviation of fidelity values over all possible input states given by [16][17][18][19],
δ ρ = f 2 ρ − f ρ 2 ,(5)
where f 2 ρ = f 2 ψ,ρ dψ, and 0 ≤ δ ρ ≤ 1 2 . For a given two-qubit state ρ, the maximal average fidelity (or, maximal fidelity) F ρ is defined as the maximal value of average fidelity obtained over all strategies under the standard protocol and local unitary operations [12,32],
F ρ = max LU f ρ ,(6)
where the maximization is done over all possible local unitary (LU) strategies. The protocol which maximizes the average fidelity is known to be the 'optimal protocol'. Now, one can show that [32] F ρ = f ρC .
The above Eq. (7) indicates that an optimal protocol consists of two steps: (i) transforming ρ → ρ C using an appropriate local unitary operation, and then (ii) using ρ C for QT following the standard protocol proposed by Bennett et al. [1].
Since the primary motivation of QT is to maximize the average fidelity, fidelity deviation should be analyzed for optimal protocol. The fidelity deviation corresponding to the optimal protocol can be defined as [17,18]
∆ ρ = δ ρC .(8)
A two-qubit state ρ is useful for QT iff F ρ > 2 3 [12,13], where 2 3 is the maximum average fidelity obtained in classical protocols. On the other hand, a two-qubit state ρ is universal for QT iff ∆ ρ = 0 [17]. If a two-qubit state satisfies the above universality condition, then all input states will be teleported with the same fidelity.
It has been shown earlier [31,32] that useful states for QT form a subset of the states with the property det(T ρ ) < 0. The analytical expressions of maximal fidelity and fidelity deviation for two-qubit states with det(T ρ ) < 0 can be written as [17,32]
F ρ = 1 2 1 + 1 3 3 i=1 |t ii | , ∆ ρ = 1 3 √ 10 3 i<j=1 (|t ii | − |t jj |) 2 .(9)
From the above equations, it follows that a two-qubit state ρ is useful for Universal Quantum Teleportation (UQT) (i.e., useful and universal for QT) if and only if |t 11 | = |t 22 | = |t 33 | > 1 3 [17]. Hence, in order to theoretically determine usefulness and universality of a two-qubit state ρ in the context of QT, we only need to find out the eigenvalues of the correlation matrix T ρ , we don't need to find out the optimal protocol or the canonical form ρ C .
D. Qubit channels
If any qubit state χ is passed through a channel Λ, the output state χ Λ can be written as [26][27][28],
Λ(χ) = χ Λ = rΛ−1 i=0 K Λ i χK Λ † i .(10)
This type of representation is known as the operator sum representation or the Kraus representation [26,27] where {K Λ i } are known as the Kraus operators. The Kraus operators always satisfy the completeness property given by,
rΛ−1 i=0 K Λ † i K Λ i = I 2 ,(11)
where the quantity r Λ represents the number of Kraus operators.
In general, there is no unique representation of the Kraus operators corresponding to a particular qubit channel [26,27,33]. For a qubit channel Λ, one can find another Kraus
representations {K Λ i } related with {K Λ i } by the relation [34],K Λ i = j W ij K Λ j ,(12)
where W ij is any unitary transformation such that
W † ij W ij = W ij W † ij = I 2 .
Hence, there exist infinite number of possible Kraus representations for any given qubit channel Λ.
Any qubit channel Λ can be categorized in two classes-(i) unital class of channels Λ u and (ii) non-unital class of channel Λ nu . Unital channels always preserve the identity operator, i.e., Λ u (I 2 ) = I 2 . Whereas, for any nonunital channel Λ nu one has Λ(I 2 ) = I 2 . Consequently, for any unital map Λ u , the equality i K Λu i K Λ † u i = I 2 always holds. However, for any non-unital map Λ nu , the above equality does not hold, i.e., i K Λnu
i K Λ † nu i = I 2 .
It can be shown that any convex combination of any two unital qubit channels also represents a unital qubit channel [35]. Hence, they form a convex set with four possible extreme points which are the four Pauli channels. In other words, any unital qubit channel can be expressed as a convex combination of four Pauli channels. The Kraus operators associated with the extreme points
are {U σ 0 V, U σ 1 V, U σ 2 V, U σ 3 V } [35]
, where U and V are unitaries. The action of a unital qubit channel Λ u on the state χ can be expressed as [33,35]
Λ u (χ) = χ Λu = U 3 i=0 p i σ i V χV † σ i U † , 0 ≤ p i ≤ 1 ∀ i, .(13)with 3 i=0 p i = 1.
When a qubit channel is realized with only one Kraus operator, then that channel must be unital [35]. In this case, the only Kraus operator will be unitary and such a channel is called a unitary channel.
In general, the number of Kraus operators r Λ has no specific upper bound. However, the lower bound on the number of operators, i.e., r min Λ becomes important while representing any channel. The minimum number of Kraus operators for any given channel can be understood from the concept of Choi states. Let us consider a bipartite scenario where Alice prepares the Bell state |Φ 1 and sends one half to Bob via any Λ. Here |Φ 1 is one of the states in Bell basis given by
|Φ 1 = 1 √ 2 (|00 + |11 ) , |Φ 2 = 1 √ 2 (|01 + |10 ) , |Φ 3 = 1 √ 2 (|01 − |10 ) , |Φ 4 = 1 √ 2 (|00 − |11 ) .(14)
The final state ρ Λ,Φ1 shared between Alice and Bob after the channel interaction is known as the Choi state or dual state of Λ given by [28,36,37],
ρ Λ,Φ1 = (I ⊗ Λ)|Φ 1 Φ 1 | = Φ Λ |Φ 1 Φ 1 |.(15)
A qubit channel Λ is completely positive, if and only if its Choi state ρ Λ,Φ1 is non-negative [28]. The tracepreserving condition of Λ implies that the marginal of Alice for ρ Λ,Φ1 is always maximally mixed, i.e., Tr 2 (ρ Λ,Φ1 ) = I 2 .
Eq.(15) represents the Choi-Jamiolkowski isomorphism [28,36] between a channel Λ and its dual state ρ Λ,Φ1 . Hence, it is obvious that the inherent geometry of the state ρ Λ,Φ1 ∈ L(C 2 ⊗ C 2 ) will be similar with the geometry of I ⊗ Λ ∈ L(C 2 ⊗ C 2 ).
Let us describe the Hilbert-Schmidt decomposition of a Choi state. Up to unitary rotations, the state ρ Λ,Φ1 can be written in the following canonical form [33,38,39],
ρ Λ,Φ1 = 1 4 I 4 + I 2 ⊗ s · σ + k λ k |t kk | σ k ⊗ σ k ,(16)
where s ≡ (s 1 , s 2 , s 3 ) is the local vector at Bob's side and λ k |t kk | are the eigenvalues of the correlation matrix T ρΛ,Φ 1 such that λ k ∈ {−1, +1}. The rank of the Choi state ρ Λ,Φ1 is given by the the minimal number r min Λ of Kraus operators associated with the channel Λ. For qubit channels, we have 1 ≤ r min Λ ≤ 4. For a unital channel Λ u , up to local unitary rotations, the Choi state has the following Bell-diagonal form [35,40],
ρ Λu,Φ1 = 1 4 I 4 + k λ k |t kk | σ k ⊗ σ k ,(17)
which implies that Tr 1 (ρ Λu,Φ1 ) = I 2 for any unital channel Λ u . These Choi states form a convex set with maximally entangled states being the extreme points. On the other hand, for a non-unital channel Λ nu , up to unitary rotations, the Choi state ρ Λnu,Φ1 has the form given by Eq.(16) with s = (0, 0, 0), i.e., Tr 1 (ρ Λnu,Φ1 ) = I 2 .
Next, we will summarize the results obtained in this paper followed by detailed proofs and analysis of the results.
III. SUMMARY OF THE RESULTS
In the present study we consider two scenarios.
Scenario 1: Alice prepares a maximally entangled twoqubit state and sends one half to Bob through a qubit channel.
Here the initial state is useful for UQT [17] and our aim is to find out the set of quantum channels for which the final state remains useful for UQT. The channel interaction can either be dissipative (non-unital) or nondissipative (unital). In case of unital channels, we show that the final state is useful for UQT if and only if the channel is unitary for a particular single parameter channel associated with a rank-four Choi state. On the other hand, for non-unital channels, we find out that the final state remains useful for UQT if and only if the channel belongs to a strict subset associated with rank-three and rank-four Choi states. In this case, we also derive the most general form of orthogonal Kraus operators associated with the non-unital channels that preserve the usefulness and universality.
Next, we consider another scenario:
Scenario 2:
Alice prepares a pure non-maximally entangled two-qubit state and sends one half of it to Bob through a qubit channel.
In this case, the initial state is not useful for UQT (this state is useful for QT, but has non-vanishing fluctuation in fidelity) [17]. We want to find out whether there exists any quantum channel for which the final state becomes useful for UQT. When the channel is unital, we show that the final state is useful for UQT for a strict subset of unital channels if and only if the concurrence of the initial state is strictly greater than 1 2 . On the other hand, we demonstrate that the final state is useful for UQT for a strict subset of non-unital channels when the concurrence of the initial state is strictly greater than a critical value. The critical value in this case is less than 1 2 , im-plying an advantage of non-unital channels over unital interactions. Hence, these results indicate that the local interaction of unital as well as non-unital channels can eliminate the fluctuation in fidelity values. Moreover, these results also point out that non-unital interactions are more effective than unital interactions in producing desirable states for UQT. Finally, we supplement our studies with some quantum channels motivated from physical noise models.
IV. BELL STATE AS THE INITIAL STATE
Here we consider the scenario 1, where Alice prepares a two-qubit Bell state |Φ 1 = 1 √ 2 (|00 + |11 ) and sends half of this state to Bob through a qubit channel Λ. Hence, in this case, the initially prepared state is useful for UQT [17]. Here our goal is to find out the class of qubit channels for which the final state will also be useful for UQT. That is, we want to characterize the qubit channels that preserve usefulness and universality in the context of QT.
We will start by analyzing the rank of Choi states associated with the qubit channels Λ that preserve usefulness and universality. Proposition 1. If Alice sends one half of a Bell state |Φ 1 through any qubit channel Λ 1 associated with rank-one Choi state, then the final shared state will always be useful for UQT.
Proof. Any quantum channel Λ 1 associated with rankone Choi state can be implemented with only one Kraus operator and that Kraus operator must be a unitary operator [35]. Hence, sending one half of a Bell state |Φ 1 through any quantum channel Λ 1 associated with rankone Choi state is equivalent to applying local unitary operation on the Bell state. The final shared state, therefore, will be a maximally entangled state, which is useful for UQT [17].
Proposition 2.
If Alice sends one half of a Bell state |Φ 1 through any qubit channel Λ 2 associated with rank-two Choi state, then the final shared state will never be useful for UQT.
Proof. This proof mainly follows from results presented in [33]. Let Alice prepares a Bell state |Φ 1 and sends one half of that state through any qubit channel Λ 2 . Then the finally shared two-qubit state ρ f is nothing but the Choi state ρ Λ 2 ,Φ1 of the channel, i.e.,
ρ f = ρ Λ 2 ,Φ1 = 1 4 1 +s 3 +t 33s1 − is 2 0t 11 −t 22 s 1 + is 2 1 −s 3 −t 33t11 +t 22 0 0t 11 +t 22 1 +s 3 −t 33s1 − is 2 t 11 −t 22 0s 1 + is 2 1 −s 3 +t 33 ,(18)
where the above matrix is written in the computational basis {|00 , |01 , |10 , |11 }. Henceforth, all 4 × 4 matrices will be written in this basis.
Since the channel Λ 2 is associated with rank-two Choi state, rank of ρ f will be two. Hence, linear combinations of 3 × 3 minors of ρ f must be zero, which implies the following three conditions [33],
s 3 t 33 +t 11t22 = 0,s 2 t 22 +t 11t33 = 0,s 1 t 11 +t 22t33 = 0.(19)
These conditions, together with the fact that diagonal elements of a positive semi-definite matrix are always greater than the elements in the same column, lead to the conclusion that alls k but one have to be equal to zero if ρ f is rank-two [33]. Without loss of generality, one can chooses 1 =s 2 = 0 and parameterizet 11 = cos α, t 22 = cos β. Hence, we havet 33 = − cos α cos β. The state ρ f will be useful for UQT if and only if |t 11 | = |t 22 | = |t 33 | > 1 3 , wheret 11 = cos α,t 22 = cos β,t 33 = − cos α cos β. This will be satisfied if and only if | cos α| = | cos β| = 1. But this implies that the state ρ f is rank-one. Hence proved.
Next, we provide an example which supports Proposition 2. Let us consider the dephasing channel Λ dephasing with the following Kraus operators: K
Λ dephasing 0 = √ p I, K Λ dephasing 1 = √ 1 − p σ 3 with 0 < p < 1. Note that this channel is unital as 1 i=0 K Λ dephasing i K Λ dephasing i † = I 2 .
The final shared state between Alice and Bob in this case is given by,
ρ f dephasing = 1 i=0 (I ⊗ K Λ dephasing i )|Φ 1 Φ 1 |(I ⊗ K Λ † dephasing i ) = p|Φ 1 Φ 1 | + (1 − p)|Φ 4 Φ 4 |.(20)
The maximal fidelity and fidelity deviation for ρ f dephasing is given by,
F ρ f dephasing = 2p + 1 3 > 2 3 when 1 2 < p < 1, 2 3 when p = 1 2 , 1 − 2p 3 > 2 3 when 0 < p < 1 2 , ∆ ρ f dephasing = 2(1 − p) 3 √ 5 = 0 when 1 2 < p < 1, 1 3 √ 5 = 0 when p = 1 2 , 2p 3 √ 5 = 0 when 0 < p < 1 2 .
Hence, the final shared state in this case is useful, but not universal for QT. Proof. We will prove this proposition by presenting two examples. Let us consider the qubit channel Λ 3 associated with a rank three Choi state with the following Kraus operators,
K Λ 3 0 = √ 1 − p 0 0 0 , K Λ 3 1 = 0 √ 1 − p 0 0 , K Λ 3 2 = √ p 0 0 √ p ,(21)
with 0 < p < 1. The above matrices are written in the basis {|0 , |1 }. Henceforth, all 2 × 2 matrices will be expressed in this basis. Here one should note that
i (K Λ 3 i )(K Λ 3 i ) † = I,
which implies that the above channel is non-unital. In this case, the final shared state between Alice and Bob is given by,
ρ f Λ 3 = 2 i=0 (I ⊗ K Λ 3 i )|Φ 1 Φ 1 |(I ⊗ K Λ 3 † i ) = p|Φ 1 Φ 1 | + (1 − p) I 2 2 ⊗ |0 0|.(22)
One can easily check that the maximal average fidelity
F ρ f Λ 3
and fidelity deviation ∆ ρ f Λ 3
of the above state are given by,
F ρ f Λ 3 = 1 + p 2 > 2 3 when 1 3 < p < 1, ∆ ρ f Λ 3 = 0 ∀ p ∈ (0, 1).
Hence, in this case, the final shared state is useful for UQT for a particular range of the channel parameter. This particular example shows that in the whole set of qubit channels associated with rank-three Choi states, for a subset of channels the final state will be useful for UQT. Next, we will now show that this subset is strict by giving another example where the final state is useful but not universal. Let us consider the qubit (unital) channelΛ 3 associated with a rank three Choi state with the following Kraus operators,
KΛ 3 0 = √ p 0 I, KΛ 3 1 = √ p 1 σ 1 , KΛ 3 2 = √ p 2 σ 2 (23) where 2 i=0 p i = 1 and 1 > p 0 ≥ p 1 ≥ p 2 > 0. The final
shared state between Alice and Bob in this case is given by,
ρ f Λ 3 = 2 i=0 p i |Φ i+1 Φ i+1 |.(24)
When 0 < p 0 ≤ 1 2 , the above final state is not entangled and, hence, is not useful for UQT [17]. When 1 2 < p 0 < 1, the maximal average fidelity and fidelity deviation for this state are given by,
F ρ f Λ 3 = 2p 0 + 1 3 > 2 3
,
∆ ρ f Λ 3 = 0.
Hence, in this case, the final state is never useful for UQT. Proof. Here also, we will prove the proposition by presenting two examples. Consider the qubit (unital) channel Λ 4 associated with a rank four Choi state having Kraus operators given by,
K Λ 4 0 = √ p I, K Λ 4 i = 1 − p 3 σ i , i = 1, 2, 3,(25)
with 0 < p < 1. In this case, the final shared state is a rank-four Werner state given by,
ρ f Λ 4 = p|Φ 1 Φ 1 | + 1 − p 3 4 i=2 |Φ i Φ i |(26)
When 1 2 < p < 1, the maximal fidelity and the fidelity deviation of ρ f Λ 4 is given by,
F ρ f Λ 4 = 2p + 1 3 > 2 3
,
∆ ρ f Λ 4 = 0.
Hence, the final shared state is useful and universal for QT for a specific range of p.
Next, consider another qubit channelΛ 4 associated with a rank-four Choi state having Kraus operators,
KΛ 4 0 = √ p 0 I, KΛ 4 1 = √ p 1 σ 1 , KΛ 4 2 = √ p 2 σ 2 , KΛ 4 3 = √ p 3 σ 3(27)
with 3 i=0 p i = 1 and 1 > p 0 > p 1 > p 2 > p 3 > 0. The final shared state is given by,
ρ f Λ 4 = 3 i=0 p i |Φ i+1 Φ i+1 |.(28)
When 0 < p 0 ≤ 1 2 , the above final state is not useful for UQT [17]. On the other hand, for 1 2 < p 0 < 1, the maximal fidelity and fidelity deviation can be written as
F ρ f Λ 4 = 2p 0 + 1 3 > 2 3
,
∆ ρ f Λ 4 = 0.
Hence, the final state is not useful for UQT in this case. The above two examples complete the proof.
Next, we will characterize the set of unital as well as non-unital channels for which the final shared state will be useful for UQT.
A. Alice sends one half of a Bell state via a unital channel
Here we consider the scenario where Alice prepares the two-qubit Bell state |Φ 1 and sends half of this state to Bob through a unital channel Λ u . In this case, we present the following proposition. Proof. If one half of the Bell state Φ 1 is sent via a unital channel Λ u , then, up to local unitary rotations, the shared state after the channel interaction is given by [33,35],
ρ f u = (I ⊗ Λ u )|Φ 1 Φ 1 | = i (I ⊗ K Λu i )|Φ 1 Φ 1 |(I ⊗ K Λu i ) † = 3 i=0 p i (I ⊗ σ i )|Φ 1 Φ 1 |(I ⊗ σ i ) = 3 i=0 p i |Φ i+1 Φ i+1 |,(29)with 0 ≤ p i ≤ 1 ∀ i, 3 i=0 p i = 1.
Hence, the final state is Bell-diagonal up to local unitary rotations. Note that the above state ρ f u is nothing but the Choi state associated with Λ u . Now, rank-four Werner states (with a particular range of the state parameter) and rank-one maximally entangled pure states are the only useful and universal states within Bell-diagonal class of states [17]. The final state (40) will be rank-four Werner state if and only if Λ u satisfies p i = p j = p k = 1−p l 3 for any choice of i = j = k = l ∈ {0, 1, 2, 3}. Here, the rank-four Werner state will be useful and universal for QT when 1 2 < p l < 1. That is, in this case, the unital channel is a one-parameter channel with rank-four Choi state having p i = p j = p k =
B. Alice sends one half of a Bell state via non-unital channel
Here we focus on non-unital qubit channels. When the initially prepared state is the Bell state |Φ 1 , we can state the following proposition, Proposition 6. If Alice sends one half of a Bell pair |Φ 1 through any non-unital channel Λ nu , then the final shared state will be useful and universal for QT if and only if Λ nu belongs to a strict subset of non-unital qubit channels associated with rank-three and rank-four Choi states.
Proof. Let Alice sends one half of a Bell state |Φ 1 through a non-unital channel Λ nu to Bob. The final shared state ρ f nu between Alice and Bob will be nothing but the Choi state ρ Λnu,Φ1 associated with the channel Λ nu . The generic structure of ρ f nu can be written as
ρ f nu = (I ⊗ Λ nu )|Φ 1 Φ 1 | = 1 4 I 4 + I ⊗ s · σ + 3 i=1 t ii σ i ⊗ σ i ) ,(30)
Note here that the marginal at Bob's end is given by
Tr 1 (ρ f nu ) = 1 2 I + 3 i=1 s i σ i ,
where {s 1 , s 2 , s 3 } are local vector components with |s| = s 2 1 + s 2 2 + s 2 3 = 0. Now, ρ f nu cannot be rank-one as channels associated with rank-one Choi states are unital [35]. Next, if ρ f nu is a rank-two state, then Proposition 2 tells that it will not be useful and universal for QT. Hence, ρ f nu will be useful for UQT if it is a rank-three or rank-four state. That is, the final state will be useful and universal for QT if Λ nu is associated with rank-three or rank-four Choi states. The set of channels Λ nu , for which the final state ρ f nu is useful for UQT, is always a strict subset of non-unital qubit channels associated with rank-three and rank-four Choi states. The reason for the strictness is very simple. When Alice sends one half of a Bell state to Bob through an arbitrary non-unital channel associated with rank-four/rank-three Choi state, the final state given by Eq.(30) may not satisfy the condition |t 11 | = |t 22 | = |t 33 |. Among them, the set of useful and universal states must satisfy |t 11 | = |t 22 | = |t 33 | = t > 1 3 . Next, let us evaluate the orthogonal Kraus operators of the most general non-unital channels for which the final state will be useful for UQT.
For non-unital channels with rank four Choi states: At first, we consider Λ nu associated with rank-four Choi states. In this case, the final state given by Eq.(30) will be useful for UQT if and only if |t 11 | = |t 22 | = |t 33 | = t > 1 3 . Without any loss of generality, let us assume the canonical representation of ρ f nu for which t 11 = t 22 = t 33 = −t and 1 3 < t ≤ 1. Since, the Choi states of Λ nu are rank-four states, the final states will also be rank four states. This will be satisfied when ρ f nu satisfies the following inequality (see Appendix A for details),
|s| < 1 − t.(31)
From the above condition and from the condition of nonunital channel: |s| > 0, one can say that t cannot be equal to one. Henceforth, we will consider 1 3 < t < 1. Therefore, in case of non-unital channels with rank-four Choi states, when the final state is universal, it cannot have maximal average fidelity equal to unity. Now, from the spectral decomposition of the final state For non-unital channels with rank three Choi states: Next, we consider Λ nu associated with rank-three Choi states. In this case also, the final state given by Eq.(30) will be useful for UQT if and only if |t 11 | = |t 22 | = |t 33 | = t > 1 3 . Without any loss of generality, we consider the canonical representation of ρ f nu for which t 11 = t 22 = t 33 = −t. The final state ρ f nu will be rank-three state if the following condition is satisfied (for details, see Appendix A),
K Λ 4 nu 0 = x 0 is 1 + s 2 s 3 − 2t − |s| 2 + 4t 2 i(|s| 2 + s 3 (s 3 + 2 |s| 2 + 4t 2 )) |s| 2 − s 2 3 + 4s 3 t −i is 1 − s 2 −s 3 + 2t + |s| 2 + 4t 2 , K Λ 4 nu 1 = x 1 −i(|s| + s 3 ) s 1 + is 2 −i −i i(s 3 − |s|) s 1 − is 2 , K Λ 4 nu 2 = x 2 is 1 + s 2 s 3 − 2t + |s| 2 + 4t 2 i(|s| 2 + s 3 (s 3 − 2 |s| 2 + 4t 2 )) |s| 2 − s 2 3 + 4s 3 t −i −is 1 + s 2 s 3 − 2t + |s| 2 + 4t 2 , K Λ 4 nu 3 = x 3 is 1 + s 2 |s| + s 3 −i −i i(|s| + s 3 ) s 1 − is 2 ,(32)where 1 3 < t < 1; 0 < |s| = s 2 1 + s 2 2 + s 2 3 < 1 − t and x0 = (2t − s3 + |s| 2 + 4t 2 ) 2 √ 2 (1 + t + |s| 2 + 4t 2 ) |s| 2 + 2t(2t + |s| 2 + 4t 2 ) , x1 = 1 2 √ 2 (|s| 2 − s 2 3 )(1 + |s| − t) |s| , x2 = (s3 − 2t + |s| 2 + 4t 2 ) 2 √ 2 (1 + t − |s| 2 + 4t 2 ) |s| 2 + 2t(2t − |s| 2 + 4t 2 ) , x3 = 1 2 √ 2 (|s| 2 − s 2 3 )(1 − |s| − t) |s| .|s| = s 2 1 + s 2 2 + s 2 3 = 1 − t,(33)
Since, |s| > 0, we have t < 1, i.e., the final state cannot have maximal average fidelity equal to one. From the above condition, one can parametrize {s i } as follows,
s 1 = (1 − t) sin θ cos φ, s 2 = (1 − t) sin θ sin φ, s 3 = (1 − t) cos θ,
where θ ∈ [0, π] and φ ∈ [0, 2π]. With these, one can construct a the following complete set of three orthogonal
Kraus operators {K Λ 3 nu i } (see Appendix A), K Λ 3 nu 0 = y 0 i(1 − t) sin θ e −iφ (1 − t) cos θ − 2t − (1 − t) 2 + 4t 2 i (1 − t) 2 (1 + cos 2 θ) + 2(1 − t) cos θ (1 − t) 2 + 4t 2 (1 − t) 2 sin 2 θ + 4t(1 − t) cos θ −i i(1 − t) sin θe iφ −(1 − t) cos θ + 2t + (1 − t) 2 + 4t 2 , K Λ 3 nu 1 = y 1 −i(1 + cos θ) sin θe iφ −i −i −i(1 − cos θ) sin θe −iφ , K Λ 3 nu 2 = y 2 i(1 − t) sin θe −iφ (1 − t) cos θ − 2t + (1 − t) 2 + 4t 2 i (1 − t) 2 (1 + cos 2 θ) − 2(1 − t) cos θ (1 − t) 2 + 4t 2 (1 − t) 2 sin 2 θ + 4t(1 − t) cos θ −i −i(1 − t) sin θe iφ (1 − t) cos θ − 2t + (1 − t) 2 + 4t 2 ,(34)
where 1 3 < t < 1 and
y 0 = 2t − (1 − t) cos θ + (1 − t) 2 + 4t 2 2 √ 2 (1 + t + (1 − t) 2 + 4t 2 ) (1 − t) 2 + 2t(2t + (1 − t) 2 + 4t 2 ) , y 1 = sin θ √ 1 − t 2 , y 2 = (1 − t) cos θ − 2t + (1 − t) 2 + 4t 2 2 √ 2 (1 + t − (1 − t) 2 + 4t 2 ) (1 − t) 2 + 2t(2t − (1 − t) 2 + 4t 2 ) .
Here also, the above matrices given by Eq. The above class of channels represents the most general non-unital channels with rank-three Choi states for which the final state will be useful for UQT.
Note that the set of Kraus operators associated with any non-unital channel, for which the final state will be useful for UQT, are always unitarily connected with the set of orthogonal Kraus operators given by Eq.(32) or Eq.(34). Now, we will present specific examples of non-unital channels associated with rank three and rank four Choi states. Let us consider the non-unital quantum channel associated with rank-four Choi state having the following four orthogonal Kraus operators,
K Λ 4 nu 0 = 6 + √ 17 (17 − √ 17) 0 1 1 − √ 17 4 0 , K Λ 4 nu 1 = √ 3 2 √ 2 0 0 0 , K Λ 4 nu 2 = 6 − √ 17 (17 + √ 17) 0 1 1 + √ 17 4 0 , K Λ 4 nu 3 = 0 0 0 1 2 √ 2 .(35)
In this case, the maximal fidelity and fidelity deviation of the final state ρ f Λ 4 nu are given by,
F ρ f Λ 4 nu = 3 4 > 2 3 , ∆ ρ f Λ 4 nu = 0,
which imply that the final state is useful for UQT.
Next we will show an example non-unital qubit channel with rank-four Choi state which does not preserve the universality criterion. For example, consider the generalized amplitude-damping channel Λ GADC [41,42] with the following Kraus operators,
K GADC 0 = √ 1 − N 1 0 0 √ 1 − γ , K GADC 1 = √ 1 − N 0 √ γ 0 0 , K GADC 2 = √ N √ 1 − γ 0 0 1 , K GADC 3 = √ N 0 0 √ γ 0 ,(36)
where γ, N ∈ [0, 1]. This channel is non-unital when γ(2N − 1) = 0, i.e., for γ = 0 and N = 1 2 . Note that the above Kraus operators are not orthogonal, but it can be checked that the Choi state associated with this Channel is rank-four. In this case, the maximal fidelity and fidelity deviation of the final state ρ f Λ GADC are given by,
F ρ f Λ GADC = 1 2 + 2 √ 1 − γ + (1 − γ) 6 , > 2 3 when γ < 2( √ 2 − 1), ∆ ρ f Λ GADC = √ 1 − γ 3 √ 5 1 − 1 − γ ,
where ∆ ρGADC = 0 holds if and only if γ = 0 or γ = 1.
But the final state becomes useless for QT when γ = 1.
On the other hand, for γ = 0, the channel does not remain to be non-unital. Hence, the final state is not useful for UQT in case of non-unital GADC. The above two examples illustrate that the set of nonunital qubit channels with rank-four Choi states, for which the final state remains to be useful for UQT, forms a strict subset of all non-unital channels associated with rank four Choi states.
Next, let us consider the non-unital channel with rankthree Choi state having the Kraus operators given by Eq. (21). As shown earlier, the final state in this case is useful for UQT for a particular range of the channel parameter.
Next, let us present another example of non-unital channel with rank-three Choi state having the following three orthogonal Kraus operators,
, K Λ 3 nu 1 = −3 i 2 √ 10 −3 i 2 √ 10 −3 i 2 √ 10 −3 i 2 √ 10 , K Λ 3 nu 2 = 3 i 20 5 − 7 5 17 i 20 65 − 107 5 17 −i 20 65 − 107 5 17 −3 i 20 5 − 7 5 17 .(37)
In this case, the maximal fidelity and fidelity deviation of the final state ρ f Λ 3 nu are given by,
F ρ f Λ 3 nu = 11 20 < 2 3 , ∆ ρ f Λ 3 nu = 0.
Hence, this final state is universal, but not useful for QT, i.e., this state is not useful for UQT.
V. TWO-QUBIT PURE NON-MAXIMALLY ENTANGLED STATE AS THE INITIAL STATE
Here we consider the scenario where Alice prepares a non-maximally entangled two-qubit pure state given by, |Ψ a = √ a|00 + √ 1 − a|11 with 1 2 < a < 1 and sends half of this state to Bob through a qubit channel Λ. The concurrence of the initial state |Ψ a is given by, C(|Ψ a ) = 2 a(1 − a) with 0 < C(|Ψ a ) < 1. In this case, the initially prepared state is useful and but not universal for QT [17]. Here, we want to analyse in details the characteristics of the final states in terms of maximal fidelity and fidelity deviation. We start with the following result.
Proposition 7. When Alice sends one half of a nonmaximally entangled two-qubit pure state to Bob via any qubit channel, then maximal fidelity of the final state will be less than or equal to that of the initial state.
Proof. Suppose Alice prepares a pure entangled state |Ψ a = √ a|00 + √ 1 − a|11 , such that 1 2 < a < 1. The maximal fidelity of this initial state is given by [12,17],
F |Ψa = 2 + C(|Ψ a ) 3 .(38)
When Alice sends one half of the state |Ψ a to Bob via any qubit channel, then the final state shared between Alice and Bob is denoted by ρ f and its concurrence is denoted by C(ρ f ). Now, we have the following relation, C(ρ f ) ≤ C(|Ψ a ) as concurrence cannot be increased under local operations and classical communication [29]. It is well known that the maximal average fidelity F ρ f is always upper bounded by [43],
F ρ f ≤ 2 + C(ρ f ) 3 .(39)
Hence, using relations (38), (39), we have
F ρ f ≤ 2 + C(|Ψ a ) 3 = F |Ψa .
When the channel is an unitary channel (i.e., a particular unital channel), the above upper bound is saturated.
Next, we focus on unital qubit channels.
A. Alice sends one half of a non-maximally entangled two-qubit pure state through a unital channel At first, we present the following proposition which addresses the issue of usefulness and universality of the final state. Proposition 8. If Alice sends one half of any non-maximally entangled two-qubit pure state |Ψ a (with concurrence C(|Ψ a ) ∈ (0, 1)) to Bob through any unital channel, then the final shared state will be useful for UQT for a strict subset of the unital channels if and only if 1 2 < C(|Ψ a ) < 1.
Proof. If one half of the state |Ψ a is sent via an unital channel Λ u , then, up to local unitary rotations, the shared state after the channel interaction is given by [33,35],
ρ f u = (I ⊗ Λ u )|Ψ a Ψ a | = i (I ⊗ K Λu i )|Ψ a Ψ a |(I ⊗ K Λu i ) † = 3 i=0 p i (I ⊗ σ i )|Ψ a Ψ a |(I ⊗ σ i ),(40)with 0 ≤ p i ≤ 1 ∀ i, 3 i=0 p i = 1.
Up to local unitary transformations, without any loss of generality, we can assume that p 0 ≥ p j ∀ j ∈ {1, 2, 3}. Now, it can be checked that the state ρ f u belongs to the X-class of states [44,45]. The expression of concurrence for any X-class two-qubit state is known [45]. Using this, the concurrence of the state ρ f u can be written as,
C(ρ f u ) = max[0, (|p 1 − p 2 | − p 0 − p 3 )C(|Ψ a ), (p 0 − p 1 − p 2 − p 3 )C(|Ψ a )].(41)
Now, we have (|p 1 − p 2 | − p 0 − p 3 ) ≤ 0 as long as 0 ≤ p i ≤ 1 ∀ i ∈ {0, 1, 2, 3} and p 0 ≥ p j ∀ j ∈ {1, 2, 3}. Hence, C(ρ f u ) > 0 if and only if p 0 > p 1 + p 2 + p 3 . When p 0 > p 1 + p 2 + p 3 does not hold, the final state will not be entangled and, therefore, will not be useful for QT. Henceforth, we will consider p 0 > p 1 + p 2 + p 3 .
It can be verified that the correlation matrix of ρ f u is diagonal with the following eigenvalues,
t 11 = (p 0 + p 1 − p 2 − p 3 ) C(|Ψ a ) = |t 11 |, t 22 = −(p 0 − p 1 + p 2 − p 3 ) C(|Ψ a ) = −|t 22 |, t 33 = (p 0 − p 1 − p 2 + p 3 ) = |t 33 |.(42)
Next, we use the relation, p 3 = 1 − p 0 − p 1 − p 2 . Now, the final state ρ f u will be useful for UQT if and only if |t 11 | = |t 22 | = |t 33 | = t > 1 3 . Using Eq.(42), it can be checked that the condition: |t 11 | = |t 22 | = |t 33 | = t is satisfied if and only if
p 1 = p 2 = 1 + (1 − 2p 0 ) C(|Ψ a ) 4 + 2C(|Ψ a ) .(43)
With these, we have
|t 11 | = |t 22 | = |t 33 | = t = (4p 0 − 1)C(|Ψ a ) 2 + C(|Ψ a )(44)
Now, t > 1 3 will be satisfied if and only if 1 4 < C(|Ψ a ) < 1 and
1 + 2C(|Ψ a ) 6C(|Ψ a ) < p 0 ≤ 1(45)
With the above conditions, one can check that the following conditions hold: 0 ≤ p i ≤ 1 ∀ i ∈ {0, 1, 2, 3} and p 0 > p 1 + p 2 + p 3 if and only if
1 2 < C(|Ψ a ) < 1 and 1 + 2C(|Ψ a ) 6C(|Ψ a ) <p 0 ≤ 1 2 − C(|Ψ a )
.
The above conditions imply that, for 1 2 < C(|Ψ a ) < 1, the condition |t 11 | = |t 22 | = |t 33 | = t > 1 3 for the final state will not be satisfied for arbitrary choices of p 0 , p 1 , p 2 , p 3 . Only when p 0 , p 1 , p 2 , p 3 satisfy some specific conditions, the final state will be useful for UQT. This completes the proof.
The above proposition implies that for an arbitrary initial state |Ψ a , one may not find any unital channel for which the final state is useful for UQT. But when the concurrence of the initial state is strictly greater than 1 2 , then one can always find a strict subset of the unital channels, for which the final state will be useful for UQT. Hence, when one half of a non-maximally entangled two-qubit pure state with 1 2 < C(|Ψ a ) < 1 is subjected to a particular unital channel, then there will be one disadvantage and one advantage. The disadvantage is that the maximal average fidelity of the final state will be less than or equal to that of the initial state. On the other hand, the advantage is that the final state will be useful for UQT though the initial state is not useful for UQT. Therefore, interaction of unital channel can reduce the fluctuation in fidelity which may have important information theoretic implications.
Next, we present two examples in support of the above proposition. Let us consider the unital channel Λ 4 u associated with rank-four Choi state having the following Kraus operators,
K Λ 4 u 0 = 2 3 I, K Λ 4 u 1 = 3 − p 6(2 + p) σ 1 , K Λ 4 u 2 = 3 − p 6(2 + p) σ 2 , K Λ 4 u 3 = 2p − 1 3(2 + p) σ 3 ,(47)
where 1 2 ≤ p < 1. Note that the above Kraus operators do not represent any CPTP map for p < 1 2 as the associated Choi state becomes negative in this range. Now, one half of the non-maximally entangled two-qubit pure state |Ψ a = √ a|00 + √ 1 − a|11 with C(|Ψ a ) = 2 a(1 − a) = p is sent through the above channel. The maximal average fidelity and fidelity deviation of the final shared state are given by,
F ρ f Λ 4 u = 3 + 4 C(|Ψ a ) 6 + 3 C(|Ψ a ) > 2 3 when 1 2 < C(|Ψ a ) < 1, ∆ ρ f Λ 4 u = 0 ∀ C(|Ψ a ) ∈ (0, 1).
Hence, the final state in this case is useful for UQT when the concurrence of the initial state is strictly greater than 1 2 .
Next, consider the unital channel Λ 2 u associated with rank-two Choi state having the following Kraus operators,
K Λ 2 u 0 = √ p I, K Λ 2 u 1 = 1 − p σ 3 , 0 < p < 1. (48)
One half of the state |Ψ a = √ a|00 + √ 1 − a|11 with C(|Ψ a ) = 2 a(1 − a) ∈ (0, 1) is sent through the above unital channel. In this case, the maximal average fidelity and fidelity deviation of the final state are given by,
F ρ f Λ 2 u = 2 + C(|Ψ a )(1 − 2p) 3 > 2 3 when 0 < p < 1 2 , 2 3 when p = 1 2 , 2 + C(|Ψ a )(2p − 1) 3 > 2 3 when 1 2 < p < 1, ∆ ρ f Λ 2 u = 1 − C(|Ψ a )(1 − 2p) 3 √ 5 = 0 when 0 < p < 1 2 , 1 3 √ 5 = 0 when p = 1 2 , 1 − C(|Ψ a )(2p − 1) 3 √ 5 = 0 when 1 2 < p < 1,
Hence, the final state is not useful for UQT for any value of a ∈ ( 1 2 , 1).
B. Alice sends one half of a non-maximally entangled two-qubit pure state through a non-unital channel
Here, we consider that Alice prepares a nonmaximally entangled two-qubit pure state |Ψ a = √ a|00 + √ 1 − a|11 with 1 2 < a < 1 and sends one half of it to Bob through a non-unital qubit channel.
At first, let us take the non-unital channel Λ nu having the Kraus operators given by Eq. (21). It can be checked that the maximal average fidelity and fidelity deviation of the final shared state are given by,
F ρ f Λnu = 1 6 3 + (1 − p) 1 − [C(|Ψ a )] 2 + p + 2p C(|Ψ a ) > 2 3 when 0 < C(|Ψ a ) < 1 and 2 + C(|Ψ a ) − 2 1 − [C(|Ψ a )] 2 4 + 5C(|Ψ a ) < p < 1, ∆ ρ f Λnu = 0 ∀ C(|Ψ a ) ∈ (0, 1) and ∀ p ∈ (0, 1).
Hence, the final state is not useful for UQT for all values of the concurrence of the initial state.
Next, let us take another example of non-unital channel Λ nu having the following Kraus operators,
K Λnu 0 = 1 √ 2 1 − p 2 − p 2 √ 1 − p 1 √ 1 + p 1 0 0 0 1 , K Λnu 1 = 1 √ 2 1 − p 2 + p 2 √ 1 − p 1 √ 1 + p 1 1 0 0 0 , K Λnu 2 = 1 + p 1 + p 2 + p 1 p 2 − p 2 √ 1 + p 1 √ 5 + 3p 1 5 + 3p 1 + √ 1 − p 1 √ 5 + 3p 1 0 1 √ 1 − p 1 + √ 5 + 3p 1 2 √ 1 + p 1 0 , K Λnu 3 = 1 + p 1 + p 2 + p 1 p 2 + p 2 √ 1 + p 1 √ 5 + 3p 1 5 + 3p 1 − √ 1 − p 1 √ 5 + 3p 1 0 1 √ 1 − p 1 − √ 5 + 3p 1 2 √ 1 + p 1 0 ,(49)
where 0 < p 1 < 1 and 0 < p 2 < 1 + p 1
1 + p 1 + 1 − p 2 1
. In this range, the above Kraus operators form a CPTP map corresponding to non-unital channel. Suppose, one half of the non-maximally entangled two-qubit pure state |Ψ a = √ a|00 + √ 1 − a|11 with C(|Ψ a ) = 2 a(1 − a) = p 1 is sent through the above channel. The maximal average fidelity and fidelity deviation of the final shared state are given by,
F ρ f Λnu = 1 + p 2 C(|Ψ a ) 2 > 2 3 when √ 17 − 1 6 < C(|Ψ a ) < 1 and 1 3C(|Ψ a ) < p 2 < 1 + C(|Ψ a ) 1 + C(|Ψ a ) + 1 − [C(|Ψ a )] 2
,
∆ ρ f Λnu = 0 ∀ p 1 ∈ (0, 1) and ∀ p 2 ∈ 0, 1 + C(|Ψ a ) 1 + C(|Ψ a ) + 1 − [C(|Ψ a )] 2 .
Hence, the final state in this case is useful for UQT when the concurrence of the initial state is strictly greater than √ 17 − 1 6 ≈ 0.52.
Next, we consider another non-unital channel Λ * nu having the following Kraus operators,
K Λ * nu 0 = 1 − γ(p 1 ) 0 0 1 , K Λ * nu 1 = 0 0 γ(p 1 ) 0 ,(50)
where γ(
p 1 ) = 1 + 1 − p 2 1 − 3p 2 1 − 2 + 2 1 − p 2 1 2 + 2 1 − p 2 1
and 0 < p 1 < 1. Note that the above Kraus operators are obtained from the generalized amplitude-damping channel [41] given by Eq.(36) by putting N = 1. Now, one half of the non-maximally entangled two-qubit pure state |Ψ a with concurrence C(|Ψ a ) = 2 a(1 − a) = p 1 is sent through the above channel. The maximal average fidelity and fidelity deviation of the final shared state in this case are given by,
F ρ f Λ * nu = 3 − 1 − [C(|Ψ a )] 2 + 3 [C(|Ψ a )] 2 − 2 + 2 1 − [C(|Ψ a )] 2 4 > 2 3 when 2 3 4 + √ 3 1 − 1 6 4 + √ 3 < C(|Ψ a ) < 1 ∆ ρ f Λ * nu = 0 ∀ C(|Ψ a ) ∈ (0, 1).
Hence, the final shared state is useful for UQT when the concurrence of the initial state is strictly greater than
2 3 4 + √ 3 1 − 1 6 4 + √ 3 ≈ 0.41.
We have analyzed with a number of analytical and numerical examples of non-unital channels. However, we have not found any non-unital qubit channel for which the final state is useful for UQT when the concurrence of the initial state is less than or equal to 0.41. Hence, we can conjecture the following Conjecture 1. If Alice sends one half of any pure entangled state |Ψ a to Bob through any non-unital channel, then the final shared state will be useful for UQT for a strict subset of the non-unital channels if and only if C Critical < C(|Ψ a ) < 1, where 0 < C Critical ≤ 0.41.
We could not determine the precise value of C Critical ; further investigations are needed.
This conjecture demonstrates that when one-half of the state |Ψ a = √ a|00 + √ 1 − a|11 with concurrence 0.41 < C(|Ψ a ) = 2 a(1 − a) ≤ 1 2 is sent through a qubit channel, then the final state will be useful for UQT if and only if the channel is non-unital. This result is in some sense counter-intuitive as it shows the advantage of non-unital channels or dissipative interactions over unital or non-dissipative interactions in the context of UQT.
The examples presented here demonstrate that, similar to the case of unital channels, non-unital channels can decrease the fidelity deviation while acting upon onehalf of a non-maximally entangled two-qubit pure state.
VI. ANALYSIS WITH SOME PHYSICAL NOISE MODELS
Here we supplement our above developed results to quantum channels motivated by physical noise models. Quantum channels modeling these effects can be Markovian or non-Markovian, both unital as well as nonunital. We consider that the Bell state |Φ 1 is prepared and half of this entangled state is sent to a distant location through any of these channels. In this context, we want to find out the channels for which the final shared state will remain useful for UQT.
Among the examples of unital channels, we take the Random Telegraph Noise (RTN), the modified Ornstein-Uhlenbeck (OUN), Power law noises (PLN), depolarising, phase damping (PD) and non-Markovian dephasing (NMD). As representatives of non-unital channels, we study the amplitude damping and the Unruh channels. The Kraus representations of these channels along with the channel parameters, their impact on QT are presented in the tables of Appendix B.
Random Telegraph Noise (RTN) [46] is a local non-Markovian dephasing noise [47]. The effect of RTN on the dynamics of quantum systems, specifically quantum correlations and control of open system dynamics has been studied in [48][49][50][51]. The autocorrelation function for the RTN, represented by the stochastic variable Γ(t), is given by
Γ(t)Γ(s) = a 2 e −|t−s|γ ,(51)
where a has the significance of the strength of the system-environment coupling and γ is proportional to the fluctuation rate of the RTN. The RTN channel poses a well defined Markovian limit [47]. The modified Ornstein-Uhlenbeck noise (OUN) is a well-known stationary Gaussian random process [52] which is in general a non-Markovian process. This could model, for example, the spin of an electron interacting with a magnetic field subjected to random fluctuations [53]. Power law noise (PLN), also called 1/f α noise is a non-Markovian stationary noise process, where α is some real number. A functional relationship exists between the spectral density and the frequency of the noise [54]. Both the OUN and PLN have well-defined Markovian limits [55].
The non-Markovian depolarising channel is a generalization of the depolarizing channel to the case of colored noise [56]. Phase Damping channel models the phenomena where decoherence occurs without dissipation (loss of energy) [57]. Non-Markovian dephasing channel is an extension of the dephasing channel to non-Markovian class, identified with the breakdown in complete-positivity of the map [58].
A canonical non-unital channel which models both decoherence as well as dissipation is the amplitude damping channel (ADC). This can be modelled by a standard Lindblad type of master equation [25,42] describing the evolution in the Markovian regime. When the decoherence rate is time-dependent, with a damped oscillatory behavior, then the amplitude damping noise is non-Markovian [55]. With an appropriate change in the parameters, the Markovian limit of the channel can be easily derived. As another example of a non-unital channel, we take up the Unruh channel [59]. To an observer undergoing acceleration, the Minkowski vacuum appears as a warm gas emitting black-body radiation at temperature, related to the acceleration, and called the Unruh temperature. This is known as the Unruh effect.
From the analysis (see Appendix B) it can be seen that, among the channels considered, only for the unital depolarizing (Markovian or non-Markovian) noise channel, the final state is useful for UQT. The final state in this case is nothing but a Werner class of state. In contrast, all the non-unital channels considered here create ranktwo final states. As discussed earlier, these rank-two final states cannot be useful for UQT.
VII. CONCLUSION
Apart from numerous applications, the idea of QT has refined various fundamental concepts of quantum information theory. Hence, from practical as well as from a foundational perspective, it is relevant to address the is-sue of suitably realizing quantum channels that can be used as the resource of QT. Motivated by this, in the present study we have considered a scenario where an observer, say, Alice prepares a two-qubit pure entangled state and sends one half of it to a distant observer, say, Bob through a quantum channel. The shared entangled state thus prepared is then used as a resource for QT. In this scenario, we have characterized the set of qubit channels in terms of the final shared state's performance in QT.
In order to characterize the efficacy of the shared state in the context of QT, we have used two quantifiersmaximal average fidelity [12][13][14][15] and fidelity deviation [17,18]. These two quantifiers together help to introduce the notion of "useful states for Universal Quantum Teleportation (UQT)". A two-qubit state is called useful for UQT if and only if the maximal average fidelity is strictly greater than 2 3 (i.e., the classical bound) and all input states are teleported with the same fidelity [17]. Let us now explain the significance of UQT from practical point of view. In real experiments, quantum teleportation can be realized as a single shot quantum gate operation with an input and an output [22]. In such realistic situations, it is not expected that teleportations of all possible input states will be performed. Rather, teleportations of some particular input states are executed. In such cases, if the fidelity deviation of the two-qubit entangled channel is non-zero, then those particular input states may be teleported with fidelity much less than the desired maximal average fidelity. However, two-qubit entangled channels with zero fidelity deviation can overcome such drawback-all input states will be teleported equally well. Most importantly, when the maximal average fidelity of a two-qubit entangled channel lies near the classical-quantum boundary (i.e., when the maximal average fidelity is greater than, but close to 2 3 ) and fidelity deviation is non-zero, then some input states may be teleported with fidelity less than 2 3 . That is why the states with zero fidelity deviation should be preferred over states with nonzero fidelity deviation, especially near the quantum-classical boundary. Hence, with twoqubit entangled states having zero fidelity deviation, the gate operation will be universal or fluctuation free. Otherwise, one has to implement different gate operations for different choices of input states, which is problematic.
Using the above notions, we have shown that when half of a Bell state (which is useful for UQT) is sent through a unital or non-unital channel, then the final state is useful for UQT for a strict subset of channels. We further completely characterize these channels for which the final states are useful for UQT. Hence, these results indicate that a subset of unital as well as non-unital qubit channels can preserve the usefulness and universality of a maximally entangled state in the context of QT while acting upon one half of the state.
If one-half of a non-maximally entangled two-qubit pure state (which is not useful for UQT) is sent to Bob through a unital or non-unital channel, then we have demonstrated that the final state may become useful for UQT. It thus signifies that the action of a qubit channel on one half of a non-maximally entangled two-qubit pure state can turn it into useful for UQT. In case of unital channels, we have completely characterized such channels. On the other hand, we have shown the above in case of non-unital channels by presenting some specific examples. However, a complete characterization of nonunital channels for which the final state is useful for UQT when the initial state is a non-maximally entangled twoqubit pure state merits further investigation.
The set of channels which converts a pure nonmaximally entangled two-qubit state into a state useful for UQT becomes more crucial when the input state is weakly entangled. The reason is that the maximal average fidelity of a weakly entangled pure two-qubit state lies near the quantum-classical boundary (i.e., the maximal average fidelity is slightly greater than 2 3 ). On the other hand, weakly entangled pure two-qubit states possess a large amount of fidelity deviation. Hence, some input states in these cases are teleportated with fidelity less than 2 3 , though the maximal average fidelity is greater than 2 3 . Thus, in these cases, it is much more desirable to choose quantum channels which completely eliminate the fidelity deviation and also keep the state useful for QT.
The present study opens up several other directions for future research. Firstly, one should consider a different scenario where a two-qubit pure entangled state is initially prepared by an observer, say, Charlie and then one qubit is sent to Alice, another one is sent to Bob (where Alice, Bob, Charlie-all are spatially separated from each other) through two different quantum channels. In this scenario, it is worth characterizing the set of qubit channels for which the final state will be useful for UQT. Apart from pure input states, the initially prepared state can be a mixed entangled state. It has been shown [32] that some specific non-unital channel interactions can increase the maximal average fidelity of some mixed two-qubit states. Therefore, it would be interesting to characterize quantum channels which not only increase maximal average fidelity but also eliminate fidelity deviation. Next, extending the present study to higher dimensional systems is another area for future research. Finally, we believe that our results will help in the experimental implementation of QT in realistic contexts.
ACKNOWLEDGEMENT
AG and DD acknowledge fruitful discussions with Somshubhro Bandyopadhyay. AG acknowledges Bose Institute, Kolkata for financial support. AG acknowledges his visit at Indian Institute of Technology Jodhpur, where a part of this work was done. DD acknowledges Science and Engineering Research Board (SERB), Government of India for financial support always hold. Now substituting x = 2t and y = |s|, we obtain the three conditions (A5, A6, A7). Hence, the ordering of the eigenvalues are given by, q 0 > q 1 > q 2 > q 3 . Therefore, the positivity of ρ f nu implies that
q 3 ≥ 0 =⇒ |s| ≤ 1 − t.
Now, if the final state is rank four, then we have q 3 > 0 implying |s| < 1 − t whereas for rank-three final state we have q 3 = 0 implying |s| = 1 − t.
Estimation of the orthogonal Kraus operators: Next, we will evaluate the orthogonal Kraus operators associated with the non-unital Channels preserving the useful and universal condition while acting on one half of a Bell state |Φ 1 following the approach mentioned in [13,33].
After the action of an arbitrary non-unital channel, any eigenvector of the final state ρ f nu can be written as
|χ i = 1 m,n=0
a (i) mn |m |n , with 1 m,n=0 |a (i) mn | 2 = 1.
Now, one can define a 2 × 2 complex matrix A i given by,
A i = √ 2 a
It can be easily checked that
|χ i = (A i ⊗ I)|Φ 1 = (I ⊗ A T i )|Φ 1 ,(A9)
where A T i is the transposition of A i . Hence, from Eq.(A9), we can write
ρ f nu = 3 i=0 q i |χ i χ i | = 3 i=0 (I ⊗ √ q i A T i )|Φ 1 Φ 1 |(I ⊗ √ q i A T i ) † ,(A10)
where q i are real positive numbers denoting the eigenvalues of ρ f nu with i q i = 1; {|χ i } are the orthonormal eigenvectors of ρ f nu . Here { √ q i A T i } denotes the Kraus operators of the channel Λ nu . Note that the Kraus operators { √ q i A T i } correspond to completely positive trace preserving maps [13,33]. Next, one can easily varify the following condition,
Tr √ q i A T i † √ q j A T j = q i δ ij ,(A11)
where δ ij is the Kronecker delta function. In the above, we have used the fact that the eigenvectors {|χ i } are orthonormal. Hence, the Kraus operators { √ q i A T i } are orthogonal Kraus operators. Now, using the expressions of eigenvalues and eigenvectors given by (A3) and (A4), and using the condition |s| < 1 − t for non-unital channels with rank-four Choi states and the condition |s| = 1 − t for non-unital channels with rank-three Choi states, one can evaluate the most general form of orthogonal Kraus operators { √ q i A T i } of nonunital channels that preserve the usefulness and universality condition.
Proposition 5 .
5If Alice sends one half of a Bell state |Φ 1 via any unital channel Λ u , then the final shared state will be useful for UQT if and only if the channel is either unitary or a single parameter unital channel associated with rank-four Choi state (for a particular range of the channel parameter).
choice of i = j = k = l ∈ {0, 1, 2, 3} and 1 2 < p l < 1. On the other hand, the final state (40) will be a rank-one maximally entangled pure state if and only if Λ u satisfies p i = 1 for any choice of i ∈ {0, 1, 2, 3}, which is nothing but a unitary channel.
ρ f nu with the condition (31), one can construct the set of four orthogonal Kraus operators {K Λ 4 nu i }. The explicit expressions of these Kraus operators are given by (see Appendix A for details),
=)
The above matrices given by Eq.(32) representing the Kraus operators are expressed in the basis {|0 , |1 }. Any non-unital channel with rank-four Choi state, for which the final state will be useful for UQT, belongs to the set of Channels with the above four orthogonal Kraus operators. It can be easily checked that the above Kraus operators satisfy the completeness prop-I. The Choi states associated with set of the above Kraus operators are nonnegative. Hence, these Kraus operators represent † = I holds as long as |s| > 0.
)
(34) representing the Kraus operators are expressed in the basis{|0 , |1 }. These Kraus operators {K † = I holds. The Choi states associated with the above set of Kraus operators are non-negative. Hence, the set of Kraus operators given by Eq.(34) represent CPTP maps associated with non-unital channels.
Proposition 3.If Alice sends one half of a Bell state |Φ 1 through any qubit channel Λ 3 associated with rank-three Choi state, then the final shared state will be useful for UQT when Λ 3 belongs to a strict subset of all qubit channels associated with rank-three Choi states.
Proposition 4.If Alice sends one half of a Bell state |Φ 1 through any qubit channel Λ 4 associated with rank-four Choi state, then the final shared state will be useful for UQT when Λ 4 belongs to a strict subset of all qubit channels associated with rank-four Choi states.
i=1 R i (S i )σ i with σ i (i = 1, 2, 3) being the Pauli matrix; T ij = Tr(ρσ i ⊗ σ j )are the elements of the 3 × 3 correlation matrix T ρ , where i, j = 1, 2, 3. Let t 11 , t 22 , t 33 are the eigenvalues of
Appendix A: Non-unital channels preserving the useful and universal condition while acting on one half of a Bell state Alice prepares a Bell state |Φ 1 and sends one half to Bob via any non-unital channel Λ nu resulting the final state ρ f nu . Here ρ f nu is nothing but the Choi state of the channel Λ nu . Now, ρ f nu cannot be rank-one as channels associated with rank-one Choi states are unital[35]. Moreover, Proposition 2 tells that ρ f nu cannot be a rank-two state if it is useful for UQT. Henceforth, we will consider that ρ f nu is either rank-three or rank-four state. If ρ f nu is useful for UQT, then the canonical density matrix of the state ρ f nu is given by,where |s| = s 2 1 + s 2 2 + s 2 3 > 0 (as the Channel is non-unital) and we have taken t 11 = t 22 = t 33 = −t with 1 3 < t ≤ 1. The spectral decomposition of ρ f nu is given by,where {q i } is the set of eigenvalues of ρ f nu given by,And the set of eigenvectors {|χ i } of ρ f nu are given by,where {n 0 , n 1 , n 2 , n 3 } are the normalization factors of the eigenstates. The set of the normalized eigenstates {|χ i } forms an orthonormal basis. Let us now define δ 1 = q 0 − q 1 , δ 2 = q 1 − q 2 and δ 3 = q 2 − q 3 . Since the parameters |s| and t are always non-negative numbers, one can verify that the conditionsalways hold. The proof is straightforward. For any two positive real numbers x and y with 0 < x < 1 and 0 < y < 1, one can always verify that the set of inequalitiesx + x 2 + y 2 > y, y + x 2 + y 2 > x, and x + y > x 2 + y 2Appendix B: Details of physical noise models and their impact on QTThe details of some physically motivated noise models, as discussed in Sec. VI, are summarized in the below table.Type of noise Nature Kraus operators Channel parameters Next, we will present the maximal fidelity and fidelity deviation of the final states when one half of the state |Φ 1 is subjected to the aforementioned channels. Note that if a state ρ has det(T ρ ) ≥ 0 (with T ρ being the correlation matrix of ρ), then that state is not useful for QT[31,32]. That is why, in the following, we will present the the expressions for maximal fidelity and fidelity deviation of the final states only for those ranges of channel parameters where the final states satisfy det(T ρ ) < 0. In the below tabel, 'M' stands for Markovian channels and 'NM' stands for non-Markovian channels.Maximal fidelityif and only if = 0 when Noif and only if when Nowhere wherewhere where p(t) = exp Gt(gt + 2) 2(gt + 1) 2 p(t) = exp Gt(gt + 2) 2(gt + 1) 2 OUN (NM) 2p(t) + 1 3 > 2 3 when 1 2 < p(t) < 1, 2(1 − p(t)) 3 √ 5 = 0 when 1 2 < p(t) < 1,
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| [] |
[] | [
"Sang-Ho Kim [email protected]†e-mail:[email protected] \nAsia Pacific Center for Theoretical Physics (APCTP)\n790-784PohangGyeongbukRepublic of Korea\n",
"Hyun-Chul Kim \nDepartment of Physics\nInha University\n402-751IncheonRepublic of Korea\n\nSchool of Physics\nInstitute for Advanced Study (KIAS)\n130-722SeoulKorea, Republic of Korea\n",
"Atsushi Hosaka \nResearch Center for Nuclear Physics (RCNP)\nOsaka University\n567-0047IbarakiOsakaJapan\n\nJ-PARC Branch\nKEK Theory Center\nInstitute of Particle and Nuclear Studies\nKEK\n319-1106Tokai, IbarakiJapan\n"
] | [
"Asia Pacific Center for Theoretical Physics (APCTP)\n790-784PohangGyeongbukRepublic of Korea",
"Department of Physics\nInha University\n402-751IncheonRepublic of Korea",
"School of Physics\nInstitute for Advanced Study (KIAS)\n130-722SeoulKorea, Republic of Korea",
"Research Center for Nuclear Physics (RCNP)\nOsaka University\n567-0047IbarakiOsakaJapan",
"J-PARC Branch\nKEK Theory Center\nInstitute of Particle and Nuclear Studies\nKEK\n319-1106Tokai, IbarakiJapan"
] | [] | In this study, we investigate the J/ψ production induced by pion beams off the nucleon, particularly the heavy pentaquarks P c (4380) and P c (4450) in intermediate states, based on a hybridized Regge model. The process involving ρ and π meson exchange in the t channel is considered as background, and the heavy pentaquark exchange is included in the s channel. The coupling constants such as the ρN N and πN N vertices are taken from the N N potentials, whereas those for the J/ψρπ and J/ψππ vertices are determined by using experimental data based on the branching ratios. In order to estimate the P c (4380) and P c (4450) coupling constants, we use the experimental upper limit on the total cross section as a guide for the πN → J/ψN reaction. The background total cross section is the order of 10 −4 − 10 −3 nb. In the vicinity of the heavy pentaquark masses, the total cross section reaches about 1 nb. * | 10.1016/j.physletb.2016.10.061 | [
"https://arxiv.org/pdf/1605.02919v2.pdf"
] | 118,369,611 | 1605.02919 | b82312610f19f355afa20aaf45b7ef6b85c0f9b8 |
20 Oct 2016
Sang-Ho Kim [email protected]†e-mail:[email protected]
Asia Pacific Center for Theoretical Physics (APCTP)
790-784PohangGyeongbukRepublic of Korea
Hyun-Chul Kim
Department of Physics
Inha University
402-751IncheonRepublic of Korea
School of Physics
Institute for Advanced Study (KIAS)
130-722SeoulKorea, Republic of Korea
Atsushi Hosaka
Research Center for Nuclear Physics (RCNP)
Osaka University
567-0047IbarakiOsakaJapan
J-PARC Branch
KEK Theory Center
Institute of Particle and Nuclear Studies
KEK
319-1106Tokai, IbarakiJapan
20 Oct 2016(Dated: October 21, 2016)Heavy pentaquark states P c (4380) and P c (4450) in the J/ψ production induced by pion beams off the nucleon
In this study, we investigate the J/ψ production induced by pion beams off the nucleon, particularly the heavy pentaquarks P c (4380) and P c (4450) in intermediate states, based on a hybridized Regge model. The process involving ρ and π meson exchange in the t channel is considered as background, and the heavy pentaquark exchange is included in the s channel. The coupling constants such as the ρN N and πN N vertices are taken from the N N potentials, whereas those for the J/ψρπ and J/ψππ vertices are determined by using experimental data based on the branching ratios. In order to estimate the P c (4380) and P c (4450) coupling constants, we use the experimental upper limit on the total cross section as a guide for the πN → J/ψN reaction. The background total cross section is the order of 10 −4 − 10 −3 nb. In the vicinity of the heavy pentaquark masses, the total cross section reaches about 1 nb. *
I. INTRODUCTION
Finding new exotic hadrons is one of the most important issues for hadron and particle physics. Recently, the LHCb Collaboration announced the observation of two heavy pentaquarks in Λ b → J/ψK − p decays [1], where the quark content is uudcc. The significance of these pentaquarks is more than 9σ. The masses and widths were reported as: M Pc = (4380 ± 8 ± 29) MeV and Γ Pc = (205 ± 18 ± 86) MeV for the lower state, whereas M Pc = (4449.8 ±1.7 ±2.5) MeV and Γ Pc = (39 ±5 ±19) MeV for the higher state. Thus, it is very important to confirm these pentaquark states in other possible reactions. For example, the energy of the pion beam at the Japan Proton Accelerator Research Complex (J-PARC) facility is sufficient to observe them during the πN → J/ψN process. The photon beam at the Thomas Jefferson National Accelerator Facility (Jefferson Lab) can also be used to measure the P c states in J/ψ photoproduction [2][3][4].
In fact, a recent theoretical study of the π − p → J/ψn reaction with neutral charm pentaquarks P 0 c [5] employed the effective Lagrangian method. Lu et al. [5] assumed that the branching ratios of P c → J/ψN and P c → πN are about 10 % and 1 %, respectively. The background total cross section is in the order of 10 − 100 nb and the total cross section near the pentaquark masses increases to around 1 µb. This indicates that the magnitude of the total cross section for J/ψ production in the vicinity of the pentaquark masses is almost comparable to that of the πN → φN reaction.
In the present study, we consider the contribution of pentaquark resonances to J/ψ production by using previous experimental information on the upper limit of the πN → J/ψN reaction [6,7] and by employing a hybridized Regge model that incorporates the heavy pentaquark states. To estimate the contribution of the heavy pentaquarks to the πN → J/ψN reaction, it is crucial to know the background contribution. Recently, the open charm production π − p → D * − Λ + c was analyzed based on a comparison with the associated strangeness production π − p → K * 0 Λ [8-10] (see Fig. 1(a)), and thus the parameters for the π − p → D * − Λ + c reaction can be plausibly estimated. A similar approach was also applied to various reactions such aspp →Ȳ c Y c andpp →M c M c , where Y c and M c denote Λ + c , Σ + c and D, D * [11]. We adopt the same strategy to study the hidden charm process π − p → J/Ψn together with the strangeness process π − p → φn (see Fig. 1(b)). However, the hidden charm (strangeness) reactions are distinguished from the open charm (strangeness) reactions. In the case of the open charm processes, the exchanged meson in the t channel should be different from that in the open strangeness process, as shown in Fig. 1(a). In addition, ρ meson and pion exchanges play similar roles in both the hidden charm and strangeness reactions because of the Okubo-Zweig-Iizuka (OZI) suppression, which is illustrated in Fig. 1(b). This allows us to obtain the coupling constants more explicitly by using the corresponding experimental data rather than relying on model calculations.
p Λ + c (Λ) π − D * − (K * 0 ) d u u u d d c(s) c(s) u d π − d u u u d p d u d n c(s) c(s) J/ψ(φ) (a) (b) D * (K * ) ρ
This Letter is organized as follows. In Section II, we explain the general formalism of a hybridized Regge model and we show how to determine the relevant parameters for the π and ρ Reggeons and the P c pentaquark states. In Section III, we first examine the assumption made in previous studies regarding the branching ratios of P c → J/ψN and P c → πN, where we compute the total cross section for the πN → J/ψN. We also present numerical results for the total cross-section and differential cross sections. In Section IV, we give our conclusions and a summary of the present study.
II. GENERAL FORMALISM
In this study, we employ a hybridized Regge model for the charm production π − p → J/ψn in order to consider the contribution of the charm pentaquark states P 0 c together with the strangeness production π − p → φn. Figure 2(a) shows a generic t-channel tree diagram for π − p → V (φ, J/ψ)n, and Fig. 2(b) depicts P 0 c exchange in the s channel only for the π − p → J/ψn process. The initial momenta of the pion and the proton are denoted by k 1 and p 1 , respectively, and the final momenta of the vector meson and the neutron are denoted by k 2 and p 2 , respectively.
p p(p 1 ) π − n n(p 2 ) π − (k 1 ) J/ψ V (k 2 ) ρ + , π + P 0 c (a) (b) FIG. 2. Tree-level Feynman diagrams for (a) π − p → V (φ, J/ψ)n in the t channel and (b) π − p → J/ψn in the s channel.
The effective Lagrangians for the exchanges of the ρ and π mesons are expressed as
L V ρπ = g V ρπ M V ε µναβ ∂ µ V ν ∂ α ρ β · π,(1)L V ππ = −ig V ππ (π − ∂ µ π + − ∂ µ π − π + )V µ ,(2)
where V = (φ, J/ψ) and π, ρ, φ, and J/ψ denote the fields corresponding to the π(140, 0 − ), ρ(770, 1 − ), φ(1020, 1 − ), and J/ψ(3097, 1 − ) mesons, respectively. The coupling constants for ρ exchange are determined by using the decay width of the corresponding vector mesons [12] Γ(V → ρπ) = 1 4π
1 M 2 V | k| 3 g 2 V ρπ ,(3)where | k| = [M 2 V − (M ρ + M π ) 2 ][M 2 V − (M ρ − M π ) 2 ]/(2M V ).
Similarly, the coupling constants g V ππ are derived from experimental data on the decay width [12] Γ(V → π + π − ) = 1 6π
1 M 2 V | q| 3 g 2 V ππ ,(4)
where | q| = M 2 V − 4M 2 π /2. All of the relevant numerical values for the couplings of φ and J/ψ mesons [12] are summarized in Table I. In fact, the Particle Data Group provides the branching ratio Γ(φ → (ρπ + π + π − π 0 ))/Γ(φ) = 15.32 % [12]. However, we ignore this part of the 3π decay in the data, where we assume that it is smaller than the ρπ channel. The ρ meson has a rather large width, but we find that its effect is very small, as stated previously [13]. Hence, we neglect the effect of the ρ meson decay width when calculating the coupling constant g V ρπ . The effective Lagrangians for the ρNN and πNN vertices are as follows
V Γ V [MeV] Γ V →ρπ /Γ V [%] Γ V →ρπ [keV] g V ρπ Γ V →π + π − /Γ V [%] Γ V →π + π − [eV] g VL ρN N = −g ρN N N γ µ τ N − κ ρN N 2M NN σ µν τ N∂ ν · ρ µ , L πN N = −ig πN NN γ 5 τ · πN,(5)
where N denotes the nucleon and the coupling constants are taken from the NN potentials, i.e., g ρN N = 3.36, κ ρN N = 6.1 and g πN N = 13.3 (e.g., see [14]). The invariant amplitudes are derived in the form of
M =ū n M µ ǫ * µ u p ,(6)
where
M µ ρ = I ρ g V ρπ g ρN N t − M 2 ρ ǫ µναβ γ ν − iκ ρN N 2M N σ νλ (k 2 − k 1 ) λ k 2α k 1β , M µ π = −2I π ig V ππ g πN N t − M 2 π γ 5 k µ 1 .(7)
u p and u n denote the Dirac spinors of the incoming and outcoming nucleons, respectively, and ǫ µ is the polarization vector of the final vector meson. The isospin factors are given by I ρ = I π = √ 2. We now replace the Feynman propagators in Eq. (7) with the Regge propagators R(t) as [15] 1
t − M 2 ρ → R ρ (t) = s s ρ αρ(t)−1 1 sin[πα ρ (t)] πα ′ ρ Γ[α ρ (t)] , 1 t − M 2 π → R π (t) = s s π απ (t) 1 sin[πα π (t)] πα ′ π Γ[1 + α π (t)] ,(8)
4
where the Regge trajectories are given by α ρ (t) = 0.55 + 0.8t and α π (t) = 0.7(t − M 2 π ) [15]. The energy scale parameters are selected as s ρ = s π = 1 GeV 2 for simplicity. For the signature factor, we select a constant phase in both the Regge propagators because the ρ and π mesons are degenerate in pion photoproduction within a Regge model [15]. Though a phase factor, exp(−iπα ρ(π) (t)), can also be included, we find that the results change only slightly.
The Regge amplitude T R can be expressed in terms of the individual invariant amplitudes combined with the Regge propagators
T R = M ρ · (t − M 2 ρ ) · R ρ (t) · C ρ (t) + M π · (t − M 2 π ) · R π (t) · C π (t),(9)
where C ρ (t) and C π (t) are called scale factors [10,11], which are employed to fit the experimental data based on the π − p → φn reaction at high energies. Now, we consider the resonance contribution from the pentaquark states P 0 c (4380) and P 0 c (4350). The exact quantum numbers of these two pentaquark states are not known. However, a previous study [1] suggested that the spins and parities for P c (4380) and P c (4450) should be (3/2 − , 5/2 + ), respectively, for the best fit in partial-wave analysis. In addition, the combinations of (3/2 + , 5/2 − ) and (5/2 + , 3/2 − ) provide acceptable solutions for the pentaquark states. We also consider the case of (5/2 − , 3/2 + ) in our calculation.
The effective Lagrangians for the P c Nπ vertex shown in Fig. 2(b) are given as [16]
L 3/2 ± PcN π = g PcN π M πN Γ (∓) τ · ∂ µ πP µ c + H.c., L 5/2 ± PcN π = i g PcN π M 2 πN Γ (±) τ · ∂ µ ∂ ν πP µν c + H.c.,(10)
where we ignore the off-shell part of the Rarita-Schwinger fields because the resonances are almost on mass shell. The following notations are used
Γ (±) = γ 5 1 , Γ (±) µ = γ µ γ 5 γ µ .(11)
After we obtain the branching ratio of P c → Nπ, we can easily determine the coupling constants g PcN π from the decay widths [16]
Γ(P 3/2 ± c → Nπ) = g 2 PcN π 4π p 3 N M 2 π M Pc (E N ± M N ), Γ(P 5/2 ± c → Nπ) = 2 5 g 2 PcN π 4π p 5 N M 4 π M Pc (E N ∓ M N ),(12)
where
E N = (M 2 Pc +M 2 N −M 2 π )/(2M Pc ) and p N = E 2 N − M 2 N .
Unfortunately, the branching ratios are not known at present, so we have to rely on previous experimental information on the upper limit on the total cross section for πN → J/ψN [6,7].
The effective Lagrangians for the P c NJ/ψ vertex can be expressed as [17]
L 3/2 ± PcN ψ = iP cµ g 1 2M N Γ (±) ν N ∓ ig 2 (2M N ) 2 Γ (±) ∂ ν N ± ig 3 (2M N ) 2 Γ (±) N∂ ν ψ µν + H.c., L 5/2 ± PcN ψ = P cµα g 1 (2M N ) 2 Γ (∓) ν N ± ig 2 (2M N ) 3 Γ (∓) ∂ ν N ∓ ig 3 (2M N ) 3 Γ (∓) N∂ ν ∂ α ψ µν + H.c..(13)
5
The heavy pentaquark states play a dominant role near the threshold region, so we only consider the first term in Eq. (13). Thus, the decay widths for the heavy pentaquarks are written as [16]
Γ(P 3/2 ± c → NJ/ψ) = g 2 PcN J/ψ 12π p N M Pc (E N ∓ M N ) × [2E N (E N ± M N ) + (M Pc ± M N ) 2 + 2M 2 J/ψ ], Γ(P 5/2 ± c → NJ/ψ) = g 2 PcN J/ψ 60π p 3 N M Pc (E N ± M N ) × [4E N (E N ∓ M N ) + (M Pc ∓ M N ) 2 + 4M 2 J/ψ ],(14)
which we can use to determine the coupling constants for the P c s. The kinematic variables E N and p N in Eq. (14) are defined as
E N = (M 2 Pc + M 2 N − M 2 J/ψ )/(2M Pc ) and p N = E 2 N − M 2 N .
Finally, we have the following expressions for the s channel
M µ Pc(3/2 + ) =I Pc ig PcN J/ψ 2M N g PcN π M π 1 s − M 2 Pc γ 5 γ ν (k α 2 g µν − k ν 2 g µα )∆ αβ (P c , k 1 + p 1 )k β 1 , M µ Pc(3/2 − ) =I Pc ig PcN J/ψ 2M N g PcN π M π 1 s − M 2 Pc γ ν (k α 2 g µν − k ν 2 g µα )∆ αβ (P c , k 1 + p 1 )k β 1 γ 5 , M µ Pc(5/2 + ) = − I Pc ig PcN J/ψ (2M N ) 2 g PcN π M 2 π 1 s − M 2 Pc γ ν k α 2 2 (k α 1 2 g µν − k ν 2 g α 1 µ ) × ∆ α 1 α 2 ;β 1 β 2 (P c , k 1 + p 1 )k β 1 1 k β 2 1 γ 5 , M µ Pc(5/2 − ) = − I Pc ig PcN J/ψ (2M N ) 2 g PcN π M 2 π 1 s − M 2 Pc γ 5 γ ν k α 2 2 (k α 1 2 g µν − k ν 2 g α 1 µ ) × ∆ α 1 α 2 ;β 1 β 2 (P c , k 1 + p 1 )k β 1 1 k β 2 1 ,(15)
where the different spins and parities of the P c states are assumed. The isospin factors are given by I Pc = √ 2. Given the decay widths of the P c states, the propagators of the pentaquark states should be modified to M Pc → (M Pc − iΓ Pc /2). Previous studies [17,18] provided the explicit expressions for ∆ αβ and ∆ α 1 ,α 2 ;β 1 β 2 in Eq. (15). The relevant hadrons are spatially extended, so we consider the phenomenological form factors in the s channel
F Pc (s) = Λ 4 Λ 4 + (s − M 2 Pc ) 2 2 ,(16)
where the cutoff masses are selected as Λ = 1.0 GeV. The cutoff masses actually play no crucial roles in this calculation because the pentaquark states lie almost near the threshold region.
III. RESULTS
First, we study the background contribution (ρ and π Reggeon exchanges) to the total cross sections for both the π − p → φn and πN → J/ψN reactions. Figure 3 shows the effects of ρ and π Reggeon exchanges on both the reactions. Note that the total cross π × 10 3 ρ BG (ρ + π) π ρ BG (ρ + π) π − p → ϕn sections are drawn as a function of s/s th so we can easily compare the total cross section of the π − p → φn reaction with that of J/ψ production. s th denotes the threshold value of s, i.e., s s th = (M φ + M n ) 2 = 3.84 GeV 2 and s c th = (M J/ψ + M n ) 2 = 16.3 GeV 2 for the π − p → φn and π − p → J/ψn reactions, respectively. The ρ Reggeon exchange dominates the π Reggeon exchange, which is expected due to the relatively smaller values of g V ππ compared with those of g V ρπ , where V generically represents φ and J/ψ mesons (see Table I). The Regge approach is known to describe the experimental data well at higher energies, so we fit the scale factor C ρ(π) (t) defined in Eq. (9) such that it explains the total cross section for π − p → φn [19] in the higher energy region. We use the form of C ρ(π) (t) = 0.5/(1 − t/Λ 2 ) 2 with the cutoff mass Λ = 1 GeV fixed in Eq. (16) to avoid additional ambiguity. We use the same form of the scale factor to obtain the total cross section for the π − p → J/ψn reaction. The results of the background contribution lie below the experimental upper limit [6,7] on the total cross section of the π − p → J/ψn reaction.
π − p → J/ψn
As shown in Fig. 3, we find that the magnitude of the background contribution to the total cross section for π − p → J/ψn is about 10 6 times smaller than that for π − p → φn. This is due mostly to the greatly suppressed value of g J/ψρπ /M J/ψ in Eq. (1) compared with g φρπ /M φ in the case of dominant ρ-meson Reggeon exchange: (g φρπ /M φ ) 2 ≃ 2.5·10 5 ×(g J/ψρπ /M J/ψ ) 2 . It is interesting to compare the current results with those of previous theoretical studies. For example, Kodaira and Sasaki [20] estimated the total cross section for the π − p → J/ψn reaction by using generalized Veneziano models many years ago. The results were obtained as σ(π − p → J/ψn) = (1.1, 0.44) pb at p lab = (50, 100) GeV/c, respectively, and we obtained (0.17, 0.071) pb at the corresponding momenta. Thus, the orders of magnitude appear to be similar to each other. However, Wu and Lee [21] predicted about 1.5 nb near the threshold region (W ∼ 4.2 GeV) within a coupled-channel model. Lu et al. [5] computed the total cross section for the π − p → J/ψn reaction by considering the heavy pentaquark states, where they determined the background contribution to the total cross section in the vicinity of the P c resonances as in the order of 10 − 100 nb, which is about 10 4 − 10 5 times larger than those obtained in the present study.
In the case of the πN → φN reaction, the contributions of the nucleon resonances have roles at low energies, as studied by Xie et al. [22], who focused on the role of N * (1535). Similarly, the heavy pentaquark resonances will make a specific contribution to the π − p → J/ψn reaction near the threshold. When considering the heavy pentaquark resonances, we [23]. The spin and parity are given as J P = (3/2 − , 5/2 + ) for (P 0 c (4380), P 0 c (4450)), respectively. The notations for the experimental data are the same as those given in Fig. 3. encounter a problem with determining the coupling constants due to the unknown branching ratios of the P c states relative to any other channels. Wu et al. [23] predicted new resonances with the cc component based on a unitarized coupled-channel formalism before observations of the P c states were reported. The branching ratios of the predicted states for the J/ψN and πN decay channels were proposed as about 40 % and 6 %, respectively. Assuming that they are the heavy pentaquark states announced recently, we find that the magnitude of the total cross section reaches about 10 4 nb, as depicted in Fig. 4, which is even closer to the magnitude of the total cross section for the π − p → φn reaction. Moreover, it exceeds the experimental upper limit on the π − p → J/ψn reaction [6,7] by approximately 10 4 times. Note that we have set the quantum numbers of the P c states as J P = (3/2 − , 5/2 + ), which are mostly favorable. We consider different combinations of the spin and parity for the P c states, but we cannot avoid this large result for the total cross section.
We may follow another suggestion proposed by Wang et al. [2], where B(P + c → J/Ψp) = 5 % was assumed for the study of J/ψ photoproduction. As a result, Wang et al. [2] were able to describe the old experimental data even when the heavy pentaquark states were considered. Following the same line of reasoning, we consider the experimental upper limit for the πN → J/ψN reaction, which is given as around 1 nb [6]. This implies that the branching ratio of P c → πN should be very small: B(P 0 c → πN) ∼ 10 −5 . Thus, we can obtain the corresponding coupling constants using Eqs. (12) and (14). The results are listed in Table II. Coupling State 3/2 + 3/2 − 5/2 + 5/2 − g PcN π P c (4380) 2.74 · 10 −4 4.23 · 10 −4 4.47 · 10 −5 2.89 · 10 210 MeV), so its peak overlaps with that of P c (4450), where the width is Γ Pc(4450) ≈ 40 MeV. 9 Thus, it may be very difficult to directly distinguish P c (4380) from P c (4450) based on the π − p → J/ψn reaction.
We note that a larger value of the branching ratio could be considered for the P c → J/Ψ decay, where we suppress that of P 0 c → πN to examine the dependence of the total cross section on them. For example, we can choose B(P 0 c → J/ΨN) = 0.5, which is 10 times larger than the value used in the present study. By contrast, we may take B(P 0 c → πN) = 10 −6 , which is 10 times smaller than the value in the present study. However, we obtain almost the same numerical results.
In addition to the hidden-charm production process investigated in this study, it is also very interesting to investigate the production of the heavy pentaquarks in the opencharm channel. Previously, we performed studies based on the π − p → D * − Λ + c [10] and π − p → D − Λ + c [24] reactions without considering the pentaquark states. We compare the corresponding results for the total cross sections with those obtained in the present study in Fig. 6. As expected, due to OZI suppression, the total cross section for J/Ψn production is approximately 10 2 − 10 4 smaller than those for D * − Λ + c and D − Λ + c production, excluding the resonance region. This indicates that if the pentaquark states are strongly coupled to the D * − Λ + c and/or D − Λ + c channels, it might be easier to find evidence for the existence of the heavy pentaquarks in open-charm processes. (color online) Total cross section for π − p → J/ψn. The notation is the same as that employed in Fig. 5. The spin and parity are given as J P = (3/2 − , 5/2 + ) for (P 0 c (4380), P 0 c (4450)), respectively. The experimental data on the upper limit come from Jenkins et al. [6] (blue circle) and Chiang et al. [7] (blue square). The black triangle represents the upper limit on D * − Λ + c production [25]. The dotted curve depicts the results of the total cross section for π − p → D − Λ + c , and the dot-dashed curve illustrates that for π − p → D * − Λ + c . The contribution of the heavy pentaquark is not included in the results for these two reactions.
Assuming that ρ-meson exchange is dominant, we find that the other isospin channels for the πN → J/ψN reactions are related to each other by the isospin factors given as follows σ(π − p → J/ψn) = σ(π + n → J/ψp) = 2 σ(π 0 n → J/ψn) = 2 σ(π 0 p → J/ψp).
However, considering that J/ψ cannot decay into two neutral pions and charged mesons are not allowed to be exchanged, then the mechanism of the π 0 N → J/ψN reactions should differ from that of the π ± N → J/ψN processes. Note that the π 0 beam is not suitable for use in the experimental production of any hadrons because of its neutral nature and short lifetime. However, the π + n → J/ψp reaction provides an opportunity to study the existence of the charged heavy pentaquark P + c , and the present study considers the neutral P 0 c . In Fig. 7, we show the results of differential cross sections for the π − p → J/ψn reaction at four different CM energies W near the threshold. As expected from the results obtained for the total cross section, the contribution of the P c states has a dominant influence on the differential cross section in the vicinity of the energies corresponding to the P c resonances, so the differential cross section is almost independent of the scattering angle at W = 4.38 GeV. As W increases, the magnitude of the differential cross section drops drastically. At W = 4.75 GeV, which is above the energies corresponding to the P c resonances, the differential cross section exhibits a forward peak, where the dependence on θ is rather weak as θ increases.
IV. CONCLUSION AND SUMMARY
In the present study, we aimed to investigate the production mechanism for the π − p → J/ψn reaction based on the hybridized Regge model by including the contributions of the two heavy pentaquark P c (4380) and P c (4450) states. First, we considered the effects of ρ and π Reggeon exchanges, which yielded the background contribution to the total cross section for the π − p → J/ψn reaction. The scale factor for the π − p → φn was determined by fitting it to experimental data on the total cross section. In order to avoid ambiguity, we employed the same form for the scale factor to describe the π − p → J/ψn reaction by using the same numerical values for the parameters used. Thus, we found that the background contribution to the total cross section for the π − p → J/ψn process is approximately 10 6 times smaller than that for the π − p → φn reaction. This is mainly explained by the highly suppressed value of the coupling constant g J/ψρπ compared with that of g φρπ in the case of dominant ρ-meson Reggeon exchange.
No information is available regarding the branching ratios of the decays for heavy pentaquark resonances, so we carefully studied the assumptions and suggestions proposed in previous theoretical studies. First, we examined the branching ratios proposed by Wu et al. [23], where we assumed that the predicted states in Wu et al. [23] correspond to the heavy pentaquark states observed by the LHCb Collaboration. Wu et al. [23] proposed branching ratios for the pentaquark states of the J/ψN and πN decay channels of about 40 % and 6 %, respectively. Considering these values for the branching ratios, the results for the total cross section for π − p → J/ψn near the resonance region are 10 4 nb, which is approximately of the same order as that for π − p → φn. However, if we use the branching ratio proposed by Wang et al. [2], the results obtained for the total cross section decrease dramatically, where the maximum values are in the order of 1 nb. This is consistent with the experimental upper limit [6,7] on the total cross section for the π − p → J/ψn reaction. The results are not sensitive to the selection of different spins and parities for the heavy pentaquark states.
In the present study, we examined the two neutral pentaquark states P 0 c , but the other isosymmetric reaction π + n → J/ψp is as appropriate for the study of charged pentaquark states P + c as for the intermediate states. The total cross section of the π + n → J/ψp process is simply the same as that of π − p → J/ψn. However, for J/ψ photoproduction, we could have a large isospin asymmetry because of the different photocouplings between P + c and its neutral partner. We also obtained the results for differential cross sections at four different energies in the CM frame in the vicinity of the energies corresponding to the heavy pentaquark resonance states.
FIG. 1 .
1Quark diagrams for (a) open and (b) hidden charm (strangeness) production. (a) π − p → D * − Λ + c (K * 0 Λ) and (b) π − p → J/ψn(φn).
FIG. 3 .
3(color online) Background (BG) contributions to the total cross sections for π − p → (φ, J/ψ)n with ρ and π Reggeon exchanges. The experimental data on π − p → φn are taken from Dahl et al.[19] (red circles, squares, and triangles) and the data on the upper limit of the π − p → J/ψn reaction are taken from Jenkins et al.[6] (blue circle) and Chiang et al.[7] (blue square).
FIG. 4 .
4(color online) Total cross section for π − p → J/ψn under the assumption that B(P 0 c → J/ΨN ) = 40% and B(P 0 c → πN ) = 6% extracted from Wu et al.
− 5
5P c (4450) 1.17 · 10 −4 1.79 · 10 −4 1.86 · 10 −5 1.21 · 10 −5 g PcN J/ψ P c . Coupling constants for the heavy pentaquark states with each J P assignment. The branching ratios for the pentaquarks are assumed to be B(P 0 c → πN ) = 10 −5 and B(P 0 c → J/ΨN ) = 0.05.
Figure 5 FIG. 5 .
55shows our results for the total cross section as a function of the center of mass (CM) energy W , where different combinations of the spin and parity are considered for the heavy pentaquark states. As illustrated inFig. 5, the results are not sensitive to the selection of the spin and parity for the heavy pentaquark states. The peaks of the P c states reach the experimental upper limit, i.e., about 1 nb. P c (4380) has a broad width (Γ Pc(4380) Total cross sections for π − p → J/ψn. The spin-parity selections of (a-d) are J P = (3/2 − , 5/2 + ), (3/2 + , 5/2 − ), (5/2 + , 3/2 − ), (5/2 − , 3/2 + ) for (P 0 c (4380), P 0 c (4450)), respectively. The experimental data come from Jenkins et al.[6] (blue circle) and Chiang et al.[7] (blue square).
FIG. 6. (color online) Total cross section for π − p → J/ψn. The notation is the same as that employed in Fig. 5. The spin and parity are given as J P = (3/2 − , 5/2 + ) for (P 0 c (4380), P 0 c (4450)), respectively. The experimental data on the upper limit come from Jenkins et al. [6] (blue circle) and Chiang et al. [7] (blue square). The black triangle represents the upper limit on D * − Λ +
FIG. 7 .
7Differential cross sections for π − p → J/ψn at four different CM energies. The spin-parity is selected as J P = (3/2 − , 5/2 + ) for (P 0 c (4380), P 0 c (4450)).
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| [] |
[
"Pricing Optimal Outcomes in Coupled and Non-Convex Markets: Theory and Applications to Electricity Markets",
"Pricing Optimal Outcomes in Coupled and Non-Convex Markets: Theory and Applications to Electricity Markets"
] | [
"Şeref Mete ",
"Martin Ahunbay ",
"Johannes Bichler ",
"Knörr "
] | [] | [] | According to the fundamental theorems of welfare economics, any competitive equilibrium is Pareto efficient. Unfortunately, competitive equilibrium prices only exist under strong assumptions such as perfectly divisible goods and convex preferences. In many real-world markets, participants have non-convex preferences and the allocation problem needs to consider complex constraints. Electricity markets are a prime example, but similar problems appear in many real-world markets, which has led to a growing literature in market design. Power markets use heuristic pricing rules based on the dual of a relaxed allocation problem today. With increasing levels of renewables, these rules have come under scrutiny as they lead to high out-of-market side-payments to some participants and to inadequate congestion signals. We show that existing pricing heuristics optimize specific design goals that can be conflicting. The trade-offs can be substantial, and we establish that the design of pricing rules is fundamentally a multi-objective optimization problem addressing different incentives. In addition to traditional multi-objective optimization techniques using weighing of individual objectives, we introduce a novel parameter-free pricing rule that minimizes incentives for market participants to deviate locally. Our theoretical and experimental findings show how the new pricing rule capitalizes on the upsides of existing pricing rules under scrutiny today. It leads to prices that incur low make-whole payments while providing adequate congestion signals and low lost opportunity costs. Our suggested pricing rule does not require weighing of objectives, it is computationally scalable, and balances trade-offs in a principled manner, addressing an important policy issue in electricity markets. | 10.48550/arxiv.2209.07386 | [
"https://export.arxiv.org/pdf/2209.07386v2.pdf"
] | 252,283,947 | 2209.07386 | cd4cb919f4a2904b61519543d2380350bd981c97 |
Pricing Optimal Outcomes in Coupled and Non-Convex Markets: Theory and Applications to Electricity Markets
May 24, 2023
Şeref Mete
Martin Ahunbay
Johannes Bichler
Knörr
Pricing Optimal Outcomes in Coupled and Non-Convex Markets: Theory and Applications to Electricity Markets
May 24, 2023
According to the fundamental theorems of welfare economics, any competitive equilibrium is Pareto efficient. Unfortunately, competitive equilibrium prices only exist under strong assumptions such as perfectly divisible goods and convex preferences. In many real-world markets, participants have non-convex preferences and the allocation problem needs to consider complex constraints. Electricity markets are a prime example, but similar problems appear in many real-world markets, which has led to a growing literature in market design. Power markets use heuristic pricing rules based on the dual of a relaxed allocation problem today. With increasing levels of renewables, these rules have come under scrutiny as they lead to high out-of-market side-payments to some participants and to inadequate congestion signals. We show that existing pricing heuristics optimize specific design goals that can be conflicting. The trade-offs can be substantial, and we establish that the design of pricing rules is fundamentally a multi-objective optimization problem addressing different incentives. In addition to traditional multi-objective optimization techniques using weighing of individual objectives, we introduce a novel parameter-free pricing rule that minimizes incentives for market participants to deviate locally. Our theoretical and experimental findings show how the new pricing rule capitalizes on the upsides of existing pricing rules under scrutiny today. It leads to prices that incur low make-whole payments while providing adequate congestion signals and low lost opportunity costs. Our suggested pricing rule does not require weighing of objectives, it is computationally scalable, and balances trade-offs in a principled manner, addressing an important policy issue in electricity markets.
Introduction
Low transaction costs on electronic markets have led to an increased use of market mechanisms to allocate scarce resources. The promise of markets rests on fundamental theoretical results such as the welfare theorems (Arrow and Debreu 1954). These theorems state that in markets with convex preferences, a Walrasian equilibrium will maximize social welfare and that every welfaremaximizing allocation can be supported by Walrasian equilibrium prices. At those prices no market participant would want to deviate from what they are assigned, and the outcome is envy-free and budget-balanced. The welfare theorems provide the theoretical rationale for using markets in a wide variety of applications ranging from the sale of frequency spectrum to treasury bills.
However, more recent literature on competitive equilibrium theory for markets with multiple indivisible goods shows that Walrasian equilibrium prices only exist under restrictive assumptions on the valuation functions (Bikhchandani and Mamer 1997, Gul and Stacchetti 1999, Bikhchandani and Ostroy 2002, Baldwin and Klemperer 2019. Most real-world markets have non-convex value and cost functions leading to non-convex welfare maximization problems, where Walrasian equilibrium prices do not exist in general. Electricity markets are a prime example of such non-convex markets, where ramp-up costs for generators or block bids of buyers lead to significant non-convexities. Similar characteristics can be found in industrial procurement, spectrum auctions, and transportation markets (Cramton et al. 2006). As a result, pricing in non-convex markets has become a central topic in market design (Milgrom 2017).
While the fundamental problems that arise in pricing goods in non-convex markets are independent of the domain, electricity market design is particularly challenging due to the fact that supply and demand need to be balanced at all times and due to specific bid languages. These bidding languages account for operational restrictions and complex preference functions of market participants and thereby imply non-convexities (Herrero et al. 2015, Kuang et al. 2019). More importantly, electricity markets consist of coupled markets where trade happens on interlinked nodes in the electricity grid. As a result, congestion on network links needs to be reflected in the prices as well. These specifics lead to one of the most challenging market design problems, one that is receiving renewed attention due to the energy transition that is currently ongoing world-wide (Tian et al. 2022). Currently, market operators in U.S. electricity markets determine the welfare-maximizing dispatch by solving a non-convex allocation problem, including network constraints, and obtain nodal prices through convexification of the original problem (e.g., Convex Hull or IP pricing, cf. Section 4). The outcome does not constitute a Walrasian equilibrium and out-of-market sidepayments are required to compensate for incentives to deviate from the dispatch. In fact, current pricing rules have come under scrutiny since these side-payments to some of the market participants have become very high. The U.S. FERC recently argued that "the use of side-payments can undermine the market's ability to send actionable price signals." 1 The resulting price signals are considered intransparent, and they can set flawed incentives as they do not reflect marginal costs anymore.
Multiple Pricing Objectives
We consider the following auction mechanism: a market operator collects bids and asks from market participants and solves for the optimal, i.e. welfare-maximizing, allocation of different items (the allocation problem). Subsequently, the market operator determines prices of items (the pricing problem) and makes allocation and prices public. In a Walrasian equilibrium it is possible to determine linear and anonymous prices that align all incentives such that no participant wants to deviate from their outcome (aka. envy-freeness) and the market is budget balanced. Unfortunately, such equilibria generally only exist in convex markets (Starr 1969, Bikhchandani andMamer 1997). Therefore, in markets with non-convexities, one must resort to other pricing rules that violate either envy-freeness or budget-balance. We refer to incentives to deviate from the optimal outcome as global lost opportunity costs, GLOCs. They describe the difference between each participant's profits under the welfare-maximizing allocation and the individual profit maxima each participant could obtain given the prices. Most prior work on pricing in non-convex electricity markets has focused on minimizing GLOCs and thereby approximating a Walrasian equilibrium. This pricing rule is referred to as Convex Hull (CH) pricing (Gribik et al. 2007, Hogan andRing 2003). The market operator then implements the optimal outcome and either compensates participants for some or all of their GLOCs (violating budget-balance) or accepts that participants have incentives to deviate (violating envy-freeness).
This paper is based on the observation that in non-convex electricity markets a market operator needs to consider multiple incentives to deviate from the optimal allocation. In particular, while GLOCs capture all incentives to deviate from the optimal allocation, market participants are typically not compensated for their entire GLOCs. Instead, additional compensations in the form of individual side-payments are only paid out to those market participants that incur a loss under the given prices, addressing individual rationality. 2 These side-payments are called make-whole payments, MWPs. As MWPs have become very high, they have been a concern to market operators and regulators alike. Developing pricing rules that reduce MWPs has been the focus of much recent research (Bichler et al. 2022, O'Neill et al. 2019. We refer to this pricing rule as minimum MWP (Min-MWP) pricing. MWPs are a subset of GLOCs, but neither does Convex Hull pricing always lead to small MWPs nor does minimum MWP pricing lead to small GLOCs, as we will show.
A specific aspect of electricity markets is that they are based on a network with possibly thousands of spatially distributed nodes connected via transmission lines, operated by transmission system operators. To put it differently, we have a set of coupled markets where supply and demand are equalized by participants who only provide transmission. Thus, we also need to price transmission lines in such coupled markets appropriately (Lété et al. 2022). For example, prices should only differ across a pair of nodes if there is congestion in the network and no further power may be transmitted from the node with the higher price to that with a lower price. Violations of this condition, i.e. price differences across uncongested branches, may result in a product revenue shortfall for transmission operators which would require compensation. Unfortunately, neither Convex Hull pricing nor minimum MWP pricing satisfy this requirement (Schiro et al. 2016, Bichler et al. 2022. We therefore introduce a third class of incentives to deviate from the optimal allocation. Local lost opportunity costs, LLOCs, measure incentives to deviate from the optimal allocation under fixed commitment. 3 In other words, LLOCs assume that the commitment decisions have been made but that generators can deviate from their assigned volumes in an attempt to improve their payoff. Minimizing LLOCs also provides for the desired congestion signals in the network, in the sense that prices reflect the marginal value of additional transmission capacity (Yang et al. 2019). A pricing rule that yields zero LLOCs is the well-known integer programming (IP) pricing (O'Neill et al. 2005). Being widely used in U.S. electricity markets, this pricing rules involves solving for the optimal allocation, fixing binary commitment variables to their optimal values, and solving the dual of the resulting linear program to obtain prices.
Contributions
It is well known that in a convex market, one can derive Walrasian equilibrium prices from the dual variables of the welfare maximization problem and these prices have zero GLOCs, LLOCs and MWPs. First, we generalize the welfare theorems and show that they hold in coupled markets as long as the preferences of buyers and sellers are convex. We prove the theorem for any convex cost or concave valuation functions, which do not need to be differentiable. This delineates environments where we can expect to find a Walrasian equilibrium from those (non-convex) markets where this is not the case.
Second, we focus on non-convexities in electricity spot markets and reconsider the pricing problem based on the specifics of these coupled and non-convex markets. Minimizing envy (in the form of GLOCs) has been the guiding principle of Convex Hull pricing, which is widely considered as central approach to electricity market pricing. However, pure minimization of GLOCs as in Convex Hull pricing is computationally intractable and can lead to unreasonably high LLOCs and MWPs. This causes high side-payments and budget-balance is often violated. More importantly, wrong congestion signals are a major concern as they can trigger demand response in cases where this is not necessary or vice versa. Congestion signals refer to the ability of prices to reflect congestion on the transmission lines between the nodes. More accurately, prices between nodes should only differ when congestion occurs in the network and the price difference reflects the marginal value of transmission capacity.
More generally, low GLOCs, LLOCs, and MWPs are conflicting design goals and optimizing only one of these classes comes at the expense of another. If MWPs are minimized, GLOCs can be unreasonably high and congestion signals become distorted leading to high LLOCs. Focusing only on LLOCs can be equally harmful and lead to very high MWPs for some participants, a phenomenon that has caught the interest of policy makers. In a second contribution, we establish these trade-offs for market operators and formalize the pricing problem as a multi-objective optimization problem. This breaks with prior literature that mainly focuses on GLOCs in electricity market pricing. By using our convex optimization framework, we also make explicit the link between previously proposed pricing rules and the classes of incentives they optimize.
In our third contribution we introduce the join pricing rule, which balances trade-offs between MWPs and LLOCs to minimize local incentives of participants to deviate from the efficient outcome as well as incentives to exit the market. As discussed earlier, the first objective addresses adequate congestion signals, while the second objective avoids high side-payments by the market operator. We prove that the join pricing rule always achieves lower MWPs than IP pricing, and lower LLOCs than any minimum MWP solution if zero MWPs are attainable. In addition, we show that prices computed via the join result in a participant-wise Pareto-optimal outcome, such that different prices cannot jointly reduce MWPs and LLOCs of this participant. As a practical advantage, the join does not require to specify weights and is thus a parameter-free pricing rule. The join is different to other techniques that have been discussed in the extensive literature on multi-objective optimization, and contributes to this literature as well (Jahn et al. 2009).
In addition to the formal characterization, we show in extensive numerical experiments that prices computed via this join require significantly less MWPs than traditional IP pricing and retain good congestion signals with very low LLOCs at the same time. Besides, the approach can be computed efficiently and it requires no fine-tuning of objective weights, as pointed out earlier.
In electricity markets, where global incentives to deviate (GLOCs) can be enforced by penalizing deviations, the join strikes a desirable balance, and the experiments show that the remaining incentives to deviate are low. Our findings have implications for the ongoing electricity pricing discussions in the U.S. and in the EU, and we make a novel proposal that promises significant advantages regarding MWPs and the quality of congestion signals.
Organization of the Paper
The remainder of this paper is structured as follows. Section 2 summarizes relevant literature on competitive equilibrium theory and on electricity market design. In Section 3, we introduce a generic market and revise central findings for convex and non-convex settings. In Section 4, we outline how current pricing rules each optimize different objectives, and how these objectives can be in substantial conflict with each other. To that end, we propose a multi-optimization perspective in Section 5 and introduce principled ways to balance the trade-offs. Subsequently, we tailor these pricing rules to an exemplary electricity market in Section 6. We present explicit formulations and numerical findings that illustrate the advantages of our pricing rules. Section 7 provides a summary and conclusions.
Related Literature
The literature on competitive equilibrium has a long history. In this section, we summarize the main theoretical findings before we discuss the related literature on electricity market design. These streams of literature are often considered separately, while we aim to connect the contributions of economics and engineering.
Competitive Equilibrium Theory
Early in the study of markets, general equilibrium theory was used to understand how markets could be explained through the demand, supply, and prices of multiple commodities. The Arrow-Debreu model shows that under convex preferences, perfect competition, and demand independence there must be a set of competitive equilibrium prices (Arrow and Debreu 1954). Market participants are price-takers, and they sell or buy goods in order to maximize their total utility. General equilibrium theory assumes divisible goods and convex preferences and the well-known welfare theorems do not carry over to markets with indivisible goods and complex (non-convex) preferences and constraints.
More recently, competitive equilibria with indivisible objects were studied and the idea of a quasilinear utility function was widely adopted (Baldwin and Klemperer 2019, Bikhchandani and Mamer 1997, Bikhchandani and Ostroy 2002. In these competitive equilibrium models, buyers and sellers with a quasilinear utility maximize their respective payoffs at the prices, resulting in an outcome that is stable (i.e. no participant wants to deviate from their resulting trade). A large part of the literature focuses on Walrasian equilibria, i.e. efficient market outcomes with linear (itemlevel) and anonymous prices, where all participants maximize payoff. If such prices exist, then the outcome is allocatively efficient, i.e. it maximizes welfare, as can be shown via linear programming duality. In general, Walrasian equilibria for markets with indivisible goods only exist for restricted valuations such as strong substitutes (Baldwin and Klemperer 2019, Bikhchandani and Mamer 1997, Kelso and Crawford 1982. These conditions lead to a concave aggregate value function, the allocation problem can be solved in polynomial time, and linear and anonymous (Walrasian) competitive equilibrium prices clear the market (Bichler and Waldherr 2019). Importantly, under these conditions the welfare theorems hold for markets with indivisible objects (Blumrosen andNisan 2007, Bichler et al. 2020).
Unfortunately, these conditions are very restrictive and in most markets goods can be substitutes and complements such that no Walrasian equilibria exist. Besides, most allocation problems that have been analyzed in operations research and the management sciences (e.g., various types of scheduling or packing problems) lead to NP-hard combinatorial optimization problems, which can neither be solved in polynomial time nor do they exhibit simple Walrasian equilibrium prices. This has led to significant research on non-convex combinatorial markets, which allow bidders to specify package bids, i.e. a price is defined for a subset of the items Goeree 2017, Milgrom 2017). The specified bid price is only valid for the entire package and the package is indivisible such that bidders can express complex (quasilinear) preferences for general valuations including complements and substitutes.
The generality of these markets comes at a price. First, the winner determination problem becomes an NP-hard optimization problem. Second, competitive equilibrium prices need to be non-linear and personalized to allow for full efficiency (Bikhchandani and Ostroy 2002). Bichler and Waldherr (2017) show that the core of the game can even be empty such that no competitive equilibrium prices exist. Milgrom and Watt (2022) recently introduced near-efficient mechanisms for non-convex markets that are computationally scalable, nearly incentive-compatible, and produce two price vectors (one for sellers and one for buyers). However, in practice, non-linear and personalized prices would convey little information other than that a bidder lost or won. Besides, if prices should serve as a baseline for futures trading, this is hardly possible with non-linear prices that differ among participants. In other words, anonymity and linearity are important requirements for many markets, electricity markets being the prime example.
Pricing on Electricity Markets
Electricity markets typically comprise a sequence of markets. The characteristic network structure of these markets has a major influence on market design decisions (Ehrenmann and Neuhoff 2009). While forward (Dent et al. 2011, Peura andBunn 2021) and reserve markets (Papavasiliou et al. 2011) are highly important, we focus on the central wholesale spot markets that are based on auctions. Here, market participants submit supply and demand bids according to a certain bid language which translates into a central allocation problem. As market participants often exhibit start-up costs, minimum generation requirements, or other technical constraints, bid languages typically imply some form of non-convexities (Herrero et al. 2020). With the advent of variable energy sources, demand response becomes increasingly important (Papavasiliou and Oren 2014). To adequately reflect flexibility on the demand side, market operators need new bid formats that likely lead to additional non-convexities and price-sensitive demand (Bichler et al. 2022). Further, the multi-period nature of the clearing problem adds complexity (Cho and Papavasiliou 2022). Market operators around the world use mixed-integer programming (MIP) to address these non-convexities and to determine the efficient allocation or dispatch (Hobbs et al. 2001). Even stochastic versions of such models have been proposed Oren 2013, Zavala et al. 2017). However, computing (electricity) prices in the presence of non-convexities remains a fundamental problem.
If Walrasian equilibrium prices exist, no market participant will have an incentive to deviate from the optimal allocation. In other words, no participant would bear GLOCs. A natural extension to non-convex markets -where Walrasian equilibrium prices do not exist in general -is to minimize these GLOCs but maintain linearity and anonymity of prices. This is referred to as Convex Hull (CH) pricing, originally explored by Gribik et al. (2007) based on Hogan and Ring (2003). Convex Hull pricing replaces the non-convex feasible region of the combinatorial allocation problem by its convex hull, and obtains prices from the dual of the resulting convex problem. We refer to Schiro et al. (2016) for a comprehensive and critical overview.
However, obtaining exact Convex Hull prices is computationally expensive. A common approach involves solving the (convex but non-smooth) Lagrangian dual of the original non-convex problem, which -under mild assumptions -is equivalent to optimizing the convex envelope of the original cost functions over the convex hull of the feasible region (Falk 1969, Hua andBaldick 2017). Several algorithms have been introduced, including subgradient methods (Ito et al. 2013), extreme-point subdifferential methods (Wang et al. 2013b), and many more (Wang et al. 2013a, Goffin and Vial 2002, Yu et al. 2020, Knueven et al. 2022, Stevens and Papavasiliou 2022. Convex Hull pricing has also been shown to be robust when exact solutions to the (NP-hard) welfare maximization problem cannot be computed (Eldridge et al. 2020). Overall, there has been significant progress in this field, but the methods are not yet used in practice and computational complexity remains a major concern. In practice, market operators resort to different heuristics in order to price electricity on realworld markets. Most practical implementations are based on integer programming (IP) pricing, where the non-convex allocation problem is first solved to optimality, and then solved again with integer variables being fixed to their optimal values. The IP prices are derived from the dual solution of the latter convex program (O'Neill et al. 2005). IP pricing has become popular as it follows the notion of marginal cost pricing in non-convex markets and furthermore provides accurate congestion signals when applied on an electricity network. However, IP prices do not constitute competitive equilibrium prices and can come with high lost opportunity costs. In an attempt to address this, some U.S. markets have started to apply Extended Locational Marginal pricing (ELMP), where prices are based on the dual variables of the linear programming relaxation of the non-convex allocation problem (MISO 2019). The motivation to use ELMP is based on the finding that under certain assumptions on the bidding formats, ELMP corresponds to the primal approach to compute Convex Hull prices (Hua and Baldick 2017). However, this property generally does not hold, and Convex Hull prices as well as ELMP can imply high MWPs. As a result, new pricing rules have been proposed in an attempt to reduce MWPs (O'Neill et al. 2019, Bichler et al. 2022, either via price differentiation or by minimizing them directly. Recently, Yang et al. (2019) proposed to consider congestion signals in the design of pricing rules, which we do in our proposal. A comprehensive discussion of pricing rules as they have been proposed in the academic literature is beyond this paper. We point the interested reader to Liberopoulos and Andrianesis (2016), who provide an excellent overview of the relevant literature.
Preliminaries
In this section, we introduce a model for a non-smooth and coupled market which allows for Walrasian equilibria. Much of the prior literature on pricing in non-convex markets relies on linear programming (LP) and LP duality Mamer 1997, Liberopoulos andAndrianesis 2016). Our paper draws in large parts on the more general field of convex optimization allowing for general convex and not just linear cost or value function. Therefore, we first provide a brief introduction on the relevant notions from convex and non-smooth optimization (Bagirov et al. 2014).
Basics of Convex Optimization
We work with functions that take values in the extended real line, R = R ∪ {−∞, ∞}. Addition and multiplication are commutative binary operations on the extended real line, defined as usual for any real number, and x + ∞ = ∞, x − ∞ = −∞ for any x ∈ R.
Definition 1 (Convexity, closedness and properness). A function f : R n → R is called 1. convex if for any x, y ∈ R n , for any λ ∈ [0, 1], f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y), 2. closed (or lower semi-continuous) if for any x ∈ R n , lim inf y→x f (y) ≥ f (x), and 3. proper if f (x) ̸ = −∞ for any x ∈ R n , and f ̸ = ∞ identically.
In turn, a set S ⊆ R n is called convex if its characteristic function χ S is convex, where χ S (x) = 1 if x ∈ S and 0 otherwise.
A function f might not be closed or convex, but it is always possible to consider its convex closure. The convex closure of a function f , conv(f ), is the pointwise maximum closed and convex function that underestimates it, conv(f )(x) = max{g(x)|g ≤ f is closed and convex}.
A closed, proper and convex function f admits information on the change in the value of the function in response to small changes at each point in its domain. This quantity is given in the form of an affine minorant, and is the analogue of a gradient for differentiable functions.
Definition 2 (Affine minorants and subgradients). A function
f : R n → R is said to have an affine minorant if ∃p ∈ R n , c ∈ R, ∀x, f (x) ≥ p T x + c. If c = f (x * ) − p T x * for some x * ∈ R n , i.e. if f (x) ≥ f (x * ) + p T (x − x * ) ∀ x ∈ R n ,
then p is said to be a subgradient of f at x * . The set of all subgradients of f at a given point x is called the subdifferential of f at x, and is denoted ∂f (x).
For a function f that has an affine minorant, a transformation of f encodes information on the set of its affine minorants. This transformation is obtained by evaluation of a maximization problem.
Definition 3. For a function f : R n → R, the Legendre-Fenchel transformation (or the convex conjugate) of f is the function
f * (p) = sup x∈R n p T x − f (x).
In particular, for a proper function f , f * (p) < ∞ if and only if there exists c such that p T x + c is an affine minorant of f . In this case, f * is closed, convex and proper. Furthermore, the biconjugate f * * then equals conv(f ), the convex closure of f (Rockafellar 2015). Convex conjugation provides a connection between operations on functions f, g : R n → R and their conjugates f * , g * . We remark that for a function f taking values in a primal space, its conjugate f * has its arguments in the corresponding dual space. For instance, in our economic setting, as we shall see in (4), if f (x) corresponds to the costs a generator incurs for supplying x MWh then f * (p) is the maximum profit attainable by this generator given a price vector p -i.e. the generator's indirect utility.
The following propositions are standard in convex analysis, and we need them to prove the welfare theorems for coupled markets and in the analysis of the pricing rules we consider in this paper. The first proposition lists the rules for the algebraic manipulation of functions and their conjugates, which we employ to study the link between pricing rules and their associated convex market models. )). Let f, g : R n → R, α, β ∈ R such that α > 0, and v ∈ R n , then the following hold: 1. (Translation by a vector) Addition of a linear function to f corresponds to a constant shift in the argument of its conjugate, (f − v T (·)) * = f * ((·) + v).
2. (Addition of a number) Addition of a constant to f corresponds to subtracting the same constant from its conjugate, (f + β) * = f * − β.
ω(σ) = min Ax+By=σ f (x) + g(y)
for some m × n matrices A, B. Then for any p ∈ R m the conjugate of ω is given by
ω * (p) = f * (A T p) + g * (B T p).
(Partial closure)
The convex closure of the minimum of two functions f, g has as its conjugate the pointwise maximum of the conjugates f * , g * , (conv min{f, g}) * = max{f * , g * }.
Specifically, in the next section we will see that optimal market clearing subject to supplydemand equivalence is a maximization problem, which can be interpreted as the value of a welfare function defined by a convolution. Thus Proposition 3.1.4 implies that the conjugate of this welfare function additively separates over participants' indirect utility functions. Pricing problems in practice turn out to be approximations of this conjugate, and the rest of Proposition 3.1 allows us study their corresponding convex models.
The next proposition shows that if f is closed and convex, then its convex conjugate encodes information on its subgradients: Proposition 3.2 (Fenchel-Young (in)equality). Suppose that f is closed, convex and proper. Then for any p, x ∈ R n ,
f * (p) + f (x) ≥ p T x.
Moreover, the inequality holds with equality if and only if p ∈ ∂f (x).
Thus for any
x, ∂f (x) = arg max p p T x − f * (p).
In particular, if f (x) is given as the solution to a linear (minimization) program, then the subdifferential ∂f (x) is a polyhedron whose form may be noted down explicitly.
We conclude this section by noting that the results above may be extended immediately to concave functions. A function f is called concave if −f is convex, and upper semi-continuous if −f is closed. The concave closure conc(f ) is then simply −conv(−f ), and the definitions of affine majorants and supragradients follow analogously to the discussion above.
Dual Pricing in Convex Markets
Our goal in this paper is to gain an understanding of previously proposed pricing rules through the lens of convex optimization to reveal the corresponding design objectives they optimize. Towards this end, in this section, we first introduce our market model and then formalize the notion of a dual pricing problem via analysis of a convex market, where there exists a canonical pricing rule for optimal outcomes -Walrasian equilibria.
Definition 4. A coupled market consists of a set of goods M , a set of transmission network parameters F and a set of market participants L = B ∪ S ∪ R, partitioned into the set of buyers B, set of sellers S and the set of transmission operators R. Each market participant ℓ ∈ L has preferences over bundles in R M ∪F , i.e. each buyer b has a valuation function v b : R M ∪F → R, each seller s has a cost function c s : R M ∪F → R, and each transmission operator r has a cost function
d r : R M ∪F → R.
The set of goods M may also include identical products at different points in time, e.g., electricity at the same node but for different hours. Transmission operators perform trades between different nodes in the network and are modeled as a separate class of market participants. We encode any feasibility constraint for market participants in the domain of cost functions, i.e. buyers have valuation −∞ and sellers / transmission operators have cost +∞ for an infeasible bundle. For now we assume concave valuation and convex cost functions (e.g., sellers incur no fixed costs when selling a positive amount). We typically write x b ≥ 0 for a bundle purchased by buyer b, y s ≥ 0 for a bundle supplied by seller s and f r ∈ R for an exchange enacted by transmission operator r (the sign of f r indicates the direction of the exchange). To make the notation more concise, we write z ℓ for the allocation of an arbitrary market participant ℓ ∈ L. Finally, we assume that buyers and sellers only have values for the consumption or supply of goods in M while interactions over the transmission network are performed exclusively by transmission operators, i.e. only bundles in R M × {0} F are feasible for buyers and sellers.
As a result of encoding feasibility in costs, allocations to market participants are assumed to be constrained only by the set of supply-demand equivalence constraints
s∈S y s − b∈B x b + r∈R B r f r = 0,(1)
where B r is some matrix specifying how flows in the transmission network interact with the supplydemand balance for the goods at each market. For example, for the DCOPF model considered below, this matrix indicates how current that is injected at a node will distribute over the transmission lines, and is determined by the line susceptances. We are concerned with supporting an optimal allocation with prices. An optimal allocation is a solution (z * ℓ ) ℓ∈L of the welfare maximization problem
max x,y,f b∈B v b (x b ) − s∈S c s (y s ) − r∈R d r (f r ) subject to (1).(2)
Example 1. Consider a network of coupled electricity markets for a single time period, described by a directed graph G = (M, F ) where M is the set of nodes (electricity at each node corresponds to goods) and F is the set of pairs of nodes (transmission parameters) that are connected by an edge. Electricity at each node v ∈ M is encoded as a distinct good. As there are only capacity constraints on the flow, we do not require any additional flow parameters on top of the set of edges. Each buyer b ∈ B and seller s ∈ S participates respectively in market ν(b) and ν(s). Meanwhile, for each directed edge (v, w), there exists a transmission operator which may operate the line to transmit power between v and w. Therefore, we parameterize the feasible bundles of transmission operators via the directional flow f (v,w) . A feasible bundle provides a quantity f (v,w) of good w and a quantity −f (v,w) of good v, and the cost for this flow is denoted d (v,w) (f (v,w) ). We assume that line flows are only subject to capacity constraints,
F (v,w) ≤ f (v,w) ≤ F (v,w) ∀ (v, w) ∈ F. We therefore set d (v,w) (f (v,w) ) = 0 if f (v,w) ∈ [F (v,w) , F (v,w) ] and d (v,w) (f (v,w) ) = +∞ otherwise.
In this market the set of supply-demand constraints is given by
s∈S|ν(s)=v y s − b∈B|ν(b)=v x b + (w,v)∈F f (w,v) − (v,w)∈F f (v,w) = 0 ∀ v ∈ M,
and the matrix B r in (1) is simply the oriented node-incidence matrix of G. An optimal allocation then prescribes buyers' consumption, sellers' supply and the flows on power lines such that the gains from trade are maximized.
Then, in a coupled market, prices p ∈ R M ∪F correspond to the per-unit cost of purchase of each unit of a good or flow. Utilities are assumed to be quasilinear in payment -thus each market participant has utility
u b (x|p) = v b (x) − p T x ∀ b ∈ B,(3)u s (y|p) = p T y − c s (y) ∀ s ∈ S, u r (f |p) = p T B r f − d r (f ) ∀ r ∈ R.
By optimizing over x, y, f in (3), each market participant has an indirect utility function, denoting the utility they would have from consuming / providing the bundle that maximizes their utility given prices,û
b (p) = max x v b (x) − p T x ∀ b ∈ B,(4)u s (p) = max y p T y − c s (y) ∀ s ∈ S, u r (p) = max f p T B r f − d r (f ) ∀ r ∈ R.
By comparison with Definition 3, we see that the indirect utility function is simply the convex conjugate of the preferences of buyers and sellers,
û b (p) = (−v b ) * (−p) andû s (p) = c * s (p)
for any price vector p and any buyer b or seller s. Similarly for transmission operators,û r (p) = d * r (B T r p). An optimal allocation (z * ℓ ) ℓ∈L and prices p together are then said to form a Walrasian equilibrium if the allocation of each market participant is utility-maximizing at the given prices.
Definition 5 (Walrasian equilibrium in coupled markets). A price vector p and a feasible allocation (z * ℓ ) ℓ∈L in R M ∪F form a Walrasian equilibrium if:
1. (Market clearing) The supply-demand equivalence constraints (1) are satisfied.
(Envy-freeness)
The allocation of every agent maximizes their utility at the prices -i.e. for any market participant ℓ, u ℓ (z * ℓ |p) =û ℓ (p).
(Budget balance) The sum of payments equals zero
, p T s∈S y * s − b∈B x * ℓ + r∈R B r f r = 0.
At a Walrasian equilibrium, envy-freeness implies that no agent's utility can be less than that for consuming / supplying zero goods. Therefore, it is also individually rational for each buyer and seller to participate in the market, as no agent earns a negative payoff as a result of their market participation. A market is convex if −v b , c s , d r are all closed convex functions. This allows us to derive a version of the First and Second Welfare Theorem for coupled and convex markets.
Theorem 3.3 (The Welfare Theorems for Coupled and Convex Markets). Let price vector p * ∈ R M ∪F and the allocation (z ℓ ) * ℓ∈L be a Walrasian equilibrium, then this allocation maximizes social welfare. Conversely, if (z ℓ ) * ℓ∈L is a welfare-maximizing allocation, then it can be supported by a Walrasian price vector p * that forms a Walrasian equilibrium.
Proof: The theorem follows by first considering the welfare function, defined as the value of the welfare maximization problem as its linear constraints are perturbed (cf. Rockafellar (2015) for a detailed discussion). Assuming strong supply-demand equivalence is required, 4 the social welfare ω : R M ∪F → R is defined as a function of excess supply such that
ω(σ) = max x,y,f b∈B v b (x b ) − s∈S c s (y s ) − r∈R d r (f r ) subject to s∈S y s − b∈B x b + r∈R B r f r = σ.
Thus ω is a convolution, and by Proposition 3.1.4 the convex conjugate of −ω is given by
(−ω) * (p) = b∈B (−v b ) * (p) + s∈S c * s (p) + r∈R d * r (B T r p) = ℓ∈Lû ℓ (p).
Given that valuations are concave and costs are convex, ω is concave in all arguments and thus −ω is convex. As all constraints are linear, constraint qualification is satisfied and −ω is closed. Therefore, −ω admits a subdifferential ∂(−ω)(0) at σ = 0. Any element of the subdifferential provides prices that correspond to the per-unit value of the provision of an additional constraint violation. As the constraint fixes excess supply to 0, this is precisely the value of the provision of an additional unit of each good to the market.
By the Fenchel-Young inequality, in general we have (−ω) * (p) − ω(0) ≥ 0, with equality holding if and only if p ∈ ∂(−ω)(0). Therefore, for an optimal solution p of the subgradient problem
min p ℓ∈Lû ℓ (p) − ω(0),(5)we note that p T s∈S y * s − b∈B x * b + r∈R B r f r = 0 and ω(0) = b∈B v b (x * b ) − s∈S c s (y * s ) − r∈R d r (f * r )
. Then by the Definition (3) of utilities and by re-arranging terms, this subgradient problem may be rewritten
min p ℓ∈Lû ℓ (p) − ω(0) = min p b∈Bû b (p) + p T x * b − v b (x * b ) + s∈Sû s (p) + c s (y * s ) − p T y * s + r∈Rû r (p) + d r (f * r ) − p T B T r f * r = min p ℓ∈L (û ℓ (p) − u ℓ (z * ℓ |p)).(6)
For any market participant ℓ and for any price vector p,û ℓ (p) ≥ u ℓ (z * ℓ |p). Therefore as the value of the problem equals zero, at an optimal solution p these must all hold with equality, which implies that an optimal solution to (5) p together with a welfare-maximizing allocation (z * ℓ ) ℓ∈L form a Walrasian equilibrium. Likewise, if p, (z * ℓ ) ℓ∈L form a Walrasian equilibrium then the value of (6) is zero, which implies that (z * ℓ ) ℓ∈L maximizes social welfare. We do not need differentiability, and convexity of −v b , c s , d r is sufficient for this proof. As convex optimization is also computationally efficient so long as each valuation and cost function is tractable to compute, the subgradient provides a natural way of computing prices. This motivates us to define dual pricing functions for an optimal outcome ((
x * b ) b∈B , (y * s ) s∈S , (f * r ) r∈R ), λ b (p|x * b ) =û b (p) − u b (x * b |p), λ s (p|y * s ) =û s (p) − u s (y * s |p), λ r (p|f * r ) =û r (p) − u r (f * r |p). (7)
Furthermore, we call (5) the dual pricing problem associated with the welfare maximization problem (2). Although the connections between convexity and the existence of a Walrasian equilibrium are well-known Mamer 1997, Liberopoulos andAndrianesis 2016), the version of the welfare theorems for coupled markets clearly delineates when we can expect Walrasian prices in coupled markets. This provides a foundation for our discussion of pricing rules in non-convex electricity markets. The Lagrangian dual and subgradients can also be used to derive prices in nonconvex markets. Due to a positive duality gap, however, such prices no longer imply a Walrasian equilibrium.
Note that in this paper, we do not consider how different pricing rules incentivize participants to bid truthfully. We assume large markets as complete-information games and argue that individual participants have little market power and act as price takers.
Pricing Non-Convex and Coupled Markets
In the absence of convexity, the negative welfare function −ω is non-convex in general and the subgradient problem (5) has value > 0, pointing to a duality gap. The representation (6) of the convex pricing problem then suggests that for any welfare-maximizing outcome (z * ℓ ) ℓ∈L and any price vector p * , Definition 5.2 is not satisfied and market participants incur lost opportunity costs (LOCs). This raises the question of how to price these markets.
In this section, we review some proposals for pricing optimal outcomes of such non-convex markets via the formalism through which we established the existence of Walrasian equilibria in coupled and convex markets. We will see that these pricing rules consist of convex models for the dual pricing problem (5). Moreover, they assert a convexified model of the original welfare maximization problem corresponding to minimization of a class of LOCs, given prices, as a central design goal: global LOCs (GLOCs) which correspond to unrealized profit as participants deviate to any feasible outcome, local LOCs (LLOCs) which correspond to unrealized profit as participants deviate to another outcome under fixed commitment, and make-whole payments (MWPs) as the required amount of compensation to market participants to ensure they do not make a loss.
We conclude the section by illustrating that the minimization of each class of LOCs results in prices causing market participants to incur large LOCs in other classes. This motivates framing pricing in non-convex markets as a multi-objective optimization problem.
Minimizing Global Lost Opportunity Costs
Global lost opportunity costs quantify the incentives of market participants to deviate from the optimal allocation. In particular, they are defined as the difference between the maximum profit a market participant can achieve (obtained from the indirect utility function) and the actual profit they yield under the optimal allocation (obtained from the direct utility function), given some prices p.
Definition 6. Let ℓ ∈ L be a market participant with allocation z * ℓ , facing prices p ∈ R M . Then the (global) lost opportunity cost (GLOC) of ℓ is given
GLOC ℓ (p|z * ℓ ) =û ℓ (p) − u ℓ (z * ℓ |p).(8)
Zero GLOCs for every market participant would imply no incentives to deviate from the optimal solution and therefore a Walrasian equilibrium. Minimizing GLOCs aims to minimize envy and thus to approximate a Walrasian equilibrium as much as possible. As mentioned earlier, in highly regulated electricity markets, GLOCs are typically not compensated by side-payments. Instead, market operators impose penalties of at least GLOC ℓ on each participant ℓ if they deviate from the optimal allocation. Thus envy-freeness of prices might be less of a concern in such transparent and regulated markets.
The respective pricing rule that minimizes GLOCs is known as Convex Hull pricing because the associated welfare maximization problem is obtained by convexifying the preferences of market participants in the original welfare maximization problem Ring 2003, Gribik et al. 2007).
max x,y,f b∈B conc[v b ](x b ) − s∈S conv[c s ](y s ) − r∈R conv[d r ](f r ) subject to (1). (Primal CH)
Given the optimal allocation (z * ℓ ) ℓ∈L , we thus relabel the contribution λ ℓ (p|z * ℓ ) of market participant ℓ ∈ L to the dual pricing problem as λ CH ℓ (p|z * ℓ ) and call this contribution the Convex Hull pricing function of ℓ at z * ℓ . From Definition (2), we infer that if p is a solution to (5) with objective value 0, then
−p ∈ ∂(−v b ) * * (x * b ) ∀ b ∈ B, p ∈ ∂c * * s (y * s ) ∀ s ∈ S, B T r p ∈ ∂d * * r (f * r ) ∀ r ∈ R.
In other words, when the objective of the CH pricing problem equals zero, prices form supragradients of the concave closures of buyers' valuation functions and subgradients of the convex closure of sellers' and transmission operators' cost functions. Furthermore, in this case the duality gap equals zero, the concave closures at the optimal allocation equal exactly the valuation functions and the convex closures equal the cost functions. This implies that prices reflect the marginal valuations and costs each market participant has for any extra unit of a good or flow. In general, CH pricing provides a supragradient of the concave closure of the welfare function at σ = 0, though these prices might not necessarily belong to the subdifferential of market participants at an optimal outcome when there exists a positive duality gap due to non-convexities. One issue with CH prices is that they are generally intractable to compute (Schiro et al. 2016). There have been efforts to establish conditions under which CH prices can be computed by simple linear programs (Hua and Baldick 2017) or to design more efficient algorithms (Andrianesis et al. 2021, Knueven et al. 2022), yet as of today CH pricing cannot be applied for practical problems. Certain approximations of the dual pricing problem have been proposed as simple heuristics, for example taking the continuous relaxation when the welfare maximization problem (2) is a mixedinteger welfare maximization problem. Such an approximation expands the feasible region of the dual pricing problem, providing tractability by accounting for exaggerated lost opportunity costs during price computation. This is referred to as the ELMP pricing rule (MISO 2019). Note that these approaches are only used to derive prices, i.e. after the allocation problem has been solved to optimality.
Besides, CH prices have come under scrutiny due to their signaling properties. CH prices do not signal network congestion properly and they allow offline units to set prices (Schiro et al. 2016). We demonstrate in our experimental results that CH prices lead to high MWPs and LLOCs, and thus there might be price differences among adjacent network nodes, even though there is no congestion on the link connecting both.
Minimizing Local Lost Opportunity Costs
Local lost opportunity costs quantify the incentives of a market participants to deviate from the optimal allocation locally. While we provide a more formal definition below, intuitively local lost opportunity costs consider only deviations which keep fixed the commitment status of the optimal allocation. For example, if a generator is committed in the optimal allocation, we do not consider lost opportunity costs that result from uncommitting at a certain time period, or leaving the market altogether. We will also show that minimizing LLOCs coincides with the widely used IP pricing rule (O'Neill et al. 2005).
To discuss local lost opportunity costs formally, we first present the notion of a piecewise convex function. The disjoint convex sets which form the domain of each market participant's cost or valuation functions may be interpreted as different commitment levels / operating ranges.
Definition 7. We say that a function f : R n → R is piecewise convex if there exist disjoint convex sets X 1 , X 2 , .., X K ⊆ R n and closed convex functions f 1 , f 2 , .., f k such that for any 1 ≤ k ≤ K, f k :
X k → R and f = min 1≤k≤K f k . For each k ∈ K, we let dom(f k ) = X k denote the domain of f k . For any x ∈ R n , f k ′ is called active at x if k ′ ∈ arg min 1≤k≤K f k (x).
In particular, most U.S. and European electricity markets implement bidding languages which allow the expression of piecewise-concave valuations for buyers and piecewise-convex costs for sellers. This is because, in electricity markets, whether a market participant is committed or not is encoded via binary variables.
Example 2. Consider, in a market with a single good, a generator s with a cost function c s (y) = min u cy + hu subject to P · u ≤ y ≤ P · u, u ∈ {0, 1}
= 0 y = 0, cy + h P ≤ y ≤ P , +∞ else.
This is in fact the cost function of a generator with marginal cost c and fixed cost h for being committed. The cost function is in fact the minimum of two convex functions c ′ s , c ′′ s and is thus piecewise convex,
c ′ s (y) = 0 y = 0 +∞ y ̸ = 0 , c ′′ s (y) = cy + hu P ≤ y ≤ P +∞ else .
The definition of piecewise convex functions thus motivates us to define local lost opportunity costs as the difference between the maximum profit a market participant may achieve amongst allocations in the domain of their active cost function, and the actual profit they obtain under the optimal allocation, given prices p.
Definition 8. Let ℓ ∈ L be a market participant with a piecewise-convex cost function, who obtains allocation z * ℓ and faces prices p ∈ R M . Suppose that c ′ is the active cost function of ℓ, and writê
u ′ ℓ (p) = max z p T z − c ′ ℓ (z)(9)
for the indirect utility function of ℓ with respect to their active cost function. Then the local lost opportunity cost (LLOC) of ℓ is given
LLOC ℓ (p|z * ℓ ) =û ′ ℓ (p) − u ℓ (z * ℓ |p).(10)
Prices that imply zero LLOCs are often considered to generalize marginal pricing for non-convex markets as prices equal the marginal value of electricity. This is because when we restrict attention to only active cost / valuation functions, we obtain a convex market for which Walrasian prices are marginal costs. In fact, the widely used IP pricing rule (O'Neill et al. 2005) minimizes LLOCs. As in Example 2, piecewise-convex preferences may be modeled via the addition of binary variables that represent the choice of active valuation / cost functions. When each v b , c s , d r is also piecewise linear, this allows casting the welfare maximization problem as a mixed-integer linear program. It is shown in O'Neill et al. (2005) that marginal prices which eliminate LLOCs can be obtained by fixing binary variables to their optimal values and solving the resulting dual problem. This provides what is known as IP pricing in electricity markets.
Formally, IP pricing is given by the dual pricing problem associated with the welfare maximization problem
max x bv b (x b ) − p T x b − u b (x * b |p) + s∈S max ys p T y s −ĉ s (y s ) − u s (y * s |p) (IP Pricing) + r∈R max fr p T B r f r −d r (f r ) − u r (f * r |p) subject to x b ∈ dom(v b ) ∀ b ∈ B, y s ∈ dom(ĉ s ) ∀ s ∈ S, f r ∈ dom(d r ) ∀ r ∈ R,
minimizing the total LLOCs incurred by market participants for deviations under fixed commitment. We then denote the contribution of market participant ℓ to the dual pricing problem as λ IP ℓ (p|z * ℓ ) and call this contribution the IP dual pricing function of ℓ at z * ℓ . It is important to note that LLOCs are an important indicator for inadequate congestion signals. With fixed commitment, prices also reflect the marginal value of additional transmission and price differences across nodes only occur when the network is congested, i.e. transmission network cannot allow for further profitable trades between the nodes. Therefore IP pricing provides adequate signaling of congestion in the network as prices reflect the marginal value of additional transmission capacity (Yang et al. 2019), mitigating a shortfall of congestion income that was identified for CH pricing. On the other hand, in the presence of non-convexities CH pricing may result in large LLOCs (cf. Section 4.4). In this case, market participants have high incentives for just small deviations from the welfare-maximizing allocation. Furthermore, the resulting incorrect congestion signals might also set flawed incentives for market participants, e.g., for demand response.
Minimizing Make-Whole Payments
Make-whole payments quantify the losses that a market participant incurs under the optimal allocation, given some prices p. This is equivalent to lost opportunity costs that only consider the deviation to non-participation, i.e. leaving the market entirely.
Definition 9. Let ℓ ∈ L be a market participant with allocation z * ℓ , facing prices p ∈ R M . Then the make-whole payment required for ℓ is given,
MWP ℓ (p|z * ℓ ) = max{−u ℓ (z * ℓ |p), 0}.(11)
Market participants are compensated for their MWPs, resulting in a revenue imbalance for the market operator. Usually, MWPs are funded by additional charges imposed on consumers. As discussed in the introduction, increasing MWPs pose a concern in electricity markets, implying discriminatory pricing and diminishing the signaling value of prices. For IP prices, some market participants can incur substantial losses, as we find in our numerical experiments.
To rectify this issue, Bichler et al. (2022) introduced the following optimization problem, where MWPs are minimized directly. min λ≥0,p,γ b∈B
λ M W P b (p|x * b ) + s∈S λ M W P s (p|y * s ) + r∈R λ M W P r (p|f * r ) (Min-MWP) subject to −v b (x * b ) + p T x * b − λ M W P b (p|x * b ) ≤ 0 ∀ b ∈ B, −p T y * s + c s (y * s ) − λ M W P s (p|y * s ) ≤ 0 ∀ s ∈ S, −p T B r f * r + d r (f * r ) − λ M W P r (p|f * r ) ≤ 0 ∀ r ∈ R.
The associated dual pricing functions for this problem are λ M W P ℓ (p|z * ℓ ) = max{−u ℓ (z * ℓ |p), 0} for each market participant ℓ, and we call this contribution the Min-MWP dual pricing function of ℓ. By definition the Min-MWP dual pricing function accounts only for the lost opportunity cost with respect to non-participation, thus the value of (Min-MWP) is precisely the minimum MWPs required to compensate participants' losses.
Conflicting Design Goals
We established in the preceding discussion that CH pricing, IP pricing, and Min-MWP pricing each optimize a corresponding class of LOCs. In practice, typically only MWPs are paid out to market participants, while all remaining LOCs are enforced by penalizing market participants for deviating from the optimal dispatch. However, regardless of the payment rule, the minimization of all LOC classes is evidently desirable. Yet as we demonstrate in the following, focusing on only one such design goal by using the pricing rules introduced above can lead to large LOCs in other classes. This finding follows from the different objective functions, but we argue that trade-offs can be substantial and that a pricing rule should thus consider all classes of LOCs, as the following example shows.
Example 3. Consider a single-item market with two sellers in a single hour and two coupled locations. The first (second) seller is located at the first (second) node and has a minimum sales quantity of 2 (8) units, a maximum quantity of 15 (15) units, variable per-unit costs of $10 ($1) and fixed costs of $1000 ($10) when supplying a positive amount. At the first (second) node a fixed demand of 3 (1) units needs to be satisfied, and the line capacity is 2 in either direction. The optimal solution is for the first seller to supply the entire demand. In this case, 1 unit is transmitted from the first node to the second node, and there is no congestion in the network. We record prices, GLOCs, MWPs, LLOCs for the pricing rules under consideration in Table 1. While CH pricing minimizes GLOCs, it may very well imply large MWPs and LLOCs and a false congestion signal (by the price difference across the uncongested line). To illustrate the reason for this discrepancy, we compute participants' preferences and the optimal allocation for (Primal CH). By convexification the seller's minimum loads are relaxed and no longer binding. As a result, the first (second) seller can supply up to 15 units with variable per-unit costs of $10 + 1000 15 ($1 + 10 15 ), while the fixed demand and line capacity are unchanged. Therefore, the (hypothetical) optimal allocation is for the second seller to provide 3 units while the first seller provides 1 unit, and 2 units are transmitted from the second node to the first. The first seller's marginal costs sets the price at the first node, and the second seller's marginal costs sets the price at the second node. However, in the true optimal allocation the second seller could not be committed (due to its minimum quantity of 8), and 1 unit was transmitted in the opposite direction.
In contrast to CH pricing, IP prices fix LLOCs to zero via local convexity, but discard any information on changing commitment. As a result, IP prices may fail to adequately account for GLOCs or MWPs.
Example 4. Consider a single-item market with two sellers in a single hour and two coupled locations. The first (second) seller is located at the first (second) node and has a minimum sales quantity of 2 (8) units, a maximum quantity of 8 (15) units, variable per-unit costs of $1 ($10) and fixed costs of $100 ($100) when supplying a positive amount. At the first (second) node a fixed demand of 6 (1) units needs to be satisfied, and the line capacity is 4 in either direction. The optimal solution is for the first seller to supply the entire demand, and there is no congestion in the network. We record prices, GLOCs, MWPs, LLOCs for different pricing rules in Table 2 Finally, Min-MWP directly minimizes the make-whole payments but loses all information on subgradients in the process. As a result, while MWPs may be reduced to insignificant amounts, the solution set will be large and the computed prices will in general incur both significant LLOCs and GLOCs.
Example 5. Consider a single-item market with two sellers in a single hour and two coupled locations. The first (second) seller is located at the first (second) node and has a minimum sales quantity of 2 (8) units, a maximum quantity of 50 (15) units, variable per-unit costs of $10 ($10) and fixed costs of $1000 ($10) when supplying a positive amount. At the first node there is a fixed demand of 4 units, at the second node a buyer is willing to pay $50 per unit to consume up to 3 units, and the line capacity is 2 in either direction. The optimal solution is for the first seller to supply the entire demand at the first node and 2 units to the second node, and as a consequence there is congestion in the network. We record prices, GLOCs, MWPs, LLOCs for different pricing rules in Table 3. While Min-MWP pricing minimizes MWPs, it may very well imply large GLOCs and LLOCs and the congestion is not reflected in the prices as the price is identical at both nodes. In contrast, with IP pricing the price on the second node is higher than on the first node, indicating correctly directed congestion. The examples in this section might appear artificial to the reader. However, we remark that our numerical results verify that this contradiction between design objectives persists when we consider larger and more realistic market data in Section 6.
Pricing as Multi-Objective Optimization
In the previous section, we discussed dual pricing problems that minimize either GLOCs (CH pricing), LLOCs (IP pricing), or MWPs (Min-MWP pricing). We provided examples illustrating that a focus on one of these objectives can lead to high costs in the others. Ultimately, we are faced with trade-offs between the three conflicting objectives. We note that the market designer can also have design goals that go beyond minimization of deviation incentives, e.g. (approximate) budget-balance. However, we argue that a pricing rule that finds a good trade-off between GLOCs, LLOCs, and MWPs already satisfies crucial requirements for electricity markets: low incentives to deviate from the optimal allocation, adequate congestion signals, as well as low side-payments that distort the price signal. We therefore infer that the pricing problem in non-convex markets is one of multi-objective optimization rather than single-objective optimization, which deviates from prior literature on the subject. While many techniques for multi-objective optimization have been proposed (see, e.g., Deb (2001), Miettinen (2012) or Emmerich and Deutz (2018)), not all of them are suitable to be used as pricing rules in electricity markets. A poor choice of a multi-objective (dual pricing) formulation can impose a highly distorted (primal) convex model for the welfare maximization problem, resulting in a divergence between its implied optimal allocation and the true optimal allocation. For example, the primal convex model implied by the multi-objective formulation might possess fictitious goods and agents, unrealistically expand the feasible sets of market participants, or consolidate market participants to a single entity. This is a subtle issue that we refer to as economic interpretability problem, and discuss in detail in Appendix A. Observe that the introduced dual pricing problems possess an additively separable objective function with linear prices for each good, i.e. they are of the form min p∈R M ℓ∈L λ ℓ (p|z * ℓ ). We require the same from alternative pricing rules and respective multi-objective optimization problems, in order to prevent distortions in the number of market participants and goods, and thus to allow for economic interpretability. Each dual pricing function is a representation of some lost opportunity cost terms. Thus, we consider multi-objective problems which account for trade-offs between GLOCs, LLOCs, and MWPs. Linear scalarization is the most well-known multi-objective optimization technique and it satisfies the above requirement. Therefore, we first discuss linear scalarization before we introduce a new multi-objective optimization technique that also satisfies separability but is parameter-free: the join.
Linear Scalarization
Linear scalarization optimizes a weighted sum of individual objective functions. In particular, given dual pricing functions for market participants ℓ∈L λ i ℓ (·|z * ℓ ) i∈{1,..,m} associated with m dual pricing problems and a non-negative weight vector w ∈ R m ≥0 , the linear scalarization problem is given by
min p ℓ∈L m i=1 w i · λ i ℓ (p|z * ℓ )(12)
As each dual pricing objective λ i is separable over the market participants, linear scalarization is also equivalent to picking certain convex approximations to each participant's valuation or cost function, as in the case for CH, IP, and Min-MWP pricing. In fact, the resulting pricing problem is that of a convex market where each participant is replaced by an admixture of participants, where preferences from each dual pricing problem i are present in proportion to the weight w i . Specifically, suppose the weights are non-negative and normalized such that m i=1 w i = 1. Then, if buyer b has allocation x * b and valuation function v i b in the welfare maximization problem corresponding to dual pricing objective i, by Proposition 3.1 the weighted dual pricing function
m i=1 w i λ i (p|x * b ) corresponds to the valuation function v b (x) = max m i=1 w i v i b (χ i b ) subject to m i=1 w i χ i b = x.(13)
The prices computed via linear scalarization then equal the marginal value of electricity for this mixed market. In a multi-objective optimization problem, it is generally not possible for a solution to be optimal for each objective. Therefore, the optimality criterion is, in general, Pareto optimality. For m dual pricing problems, a price vector p is said to be Pareto optimal with respect to objectives if there does not exist another price vector q such that for any pricing problem i,
ℓ∈L λ i ℓ (q|z * ℓ ) ≤ ℓ∈L λ i ℓ (p|z * ℓ )
, with one inequality holding strictly. It is known that solutions to the linear scalarization problem yield the Pareto frontier of the multi-objective (Emmerich and Deutz 2018). We thus infer that if the dual pricing functions under consideration correspond to CH, IP, and Min-MWP pricing, linear scalarization allows to obtain other Pareto-efficient solutions with respect to total GLOCs, MWPs, and LLOCs.
However, linear scalarization comes with certain disadvantages for practical applications despite providing a Pareto optimal solution with respect to GLOCs, MWPs, and LLOCs. In particular, it requires preference information in order to set weights that produce a desirable outcome. In practice, however, preferences might be difficult to define in an environment with various stakeholders and without being able to study the impact of weights on the outcomes. Market operators might want to re-calibrate and fine-tune the weights after the Pareto frontier has been computed. For this reason, linear scalarization cannot be considered a full a priori multi-objective technique as weights and therefore the pricing rule might be set only after bids have been elicited. Therefore, linear scalarization would be difficult to implement in practice.
Join
In what follows, we focus on a pricing rule that treats low LLOCs and low MWPs as first-order objectives. Together, they imply that the market allocation is as locally stable as possible against considerations of non-participation or small deviations by the market participants. While GLOCs are not accounted for, we will see as a result in our experiments that GLOCs remain comparable to IP pricing.
Given prices p and an optimal allocation (z * ℓ ) ℓ∈L , market participant ℓ's opportunity cost under consideration of local deviations in an allocation is given by λ IP ℓ (p|z * ℓ ) while their opportunity cost against exiting the market is given by λ M W P ℓ (p|z * ℓ ). Therefore, the amount of compensation required to disincentivize such deviations from ℓ is given by max{λ IP ℓ (p|z * ℓ ), λ M W P ℓ (p|z * ℓ )}. This motivates us to consider the following dual pricing problem.
Definition 10. For each market participant ℓ, let λ IP ℓ (p|z * ℓ ) and λ M W P ℓ (p|z * ℓ ) denote the LLOCs and MWPs of ℓ at prices p and allocation z * , respectively. Then the join IP ∨ MWP of IP and Min-MWP pricing is the dual pricing problem
min p ℓ∈L max{λ IP ℓ , λ M W P ℓ }(p|z * ℓ ). (IP ∨ MWP)
By Proposition 3.1.5, if f, g are convex functions, then the convex conjugate of max{f, g} is the function conv min{f * , g * }. Thus (IP ∨ MWP) utilizes minimal concave closures of valuation functions and convex closures of cost functions that account for both LLOCs and MWPs. Note that if we considered λ CH ℓ (p|z * ℓ ) as well, the resulting pricing rule would correspond to the minimization of GLOCs and thus to CH pricing. This is because GLOCs capture all incentives to deviate, including local deviations and non-participation, and therefore are always greater or equal to both MWPs and LLOCs. We also remark that when the welfare maximization problem is a MILP, the join prices may be found via a linear program. For instance, for the formulation of IP pricing in Appendix D and Min-MWP pricing in Appendix E, the linear programming formulation of the join pricing problem is as shown in Appendix F.
The main motivation to use join prices is based on the joint minimization of LLOCs and MWPs. In fact, the join IP ∨ MWP is guaranteed to achieve lower MWPs than IP pricing.
Proposition 5.1. Suppose that p ∨ is an optimal solution of (IP ∨ MWP) and p IP is an optimal solution of (IP Pricing).
Then ℓ∈L λ M W P ℓ (p ∨ |z * ℓ ) ≤ ℓ∈L λ M W P ℓ (p IP |z * ℓ ).
Proof: The result follows since
ℓ∈L λ M W P ℓ (p ∨ |z * ℓ ) ≤ ℓ∈L max{λ M W P ℓ , λ IP ℓ }(p ∨ |z * ℓ ) ≤ ℓ∈L max{λ M W P ℓ , λ IP ℓ }(p IP |z * ℓ ) = ℓ∈L λ M W P ℓ (p IP |z * ℓ ).
Here, the first inequality holds since the terms of the second sum are element-wise no less than the terms of the first sum, the second inequality holds by the minimum property of p ∨ . The final equality then holds since for any market participant ℓ, prices p IP achieve zero LLOCs due to convexity. While Proposition 5.1 might appear to immediately follow from its definition, this is only guaranteed since IP pricing achieves zero LLOCs for each market participant. Trying to modify the proof to show that a solution of (IP ∨ MWP) achieves lower LLOCs than any solution of (Min-MWP), we obtain the following guarantee.
Corollary 5.2. Suppose that p ∨ is an optimal solution of (IP ∨ MWP) and p M W P is an optimal solution of (Min-MWP) such that ℓ∈L λ M W P
ℓ (p M W P |z * ℓ ) = 0. Then ℓ∈L λ IP ℓ (p ∨ |z * ℓ ) ≤ ℓ∈L λ IP ℓ (p M W P |z * ℓ ).
Therefore, a solution of (IP ∨ MWP) achieves lower LLOCs than any solution of (Min-MWP) in settings where zero MWPs may be achieved with linear and anonymous prices. In general, the presence of non-convexities might render this impossible. Indeed, the following example demonstrates that the zero MWP condition in Corollary 5.2 is necessary.
Example 6. Consider a market with one good, one seller s and two buyers b 1 and b 2 . Buyer b 1 has a block bid of $20 for 1 MWh and buyer b 2 has a block bid of $10 for 1 MWh, both of which must be fulfilled completely or not at all. Seller s has a cost function for generation
c s (y) = 0 y = 0, 30 ϵ · (y − 2 + ϵ) y ∈ [2 − ϵ, 2], +∞ otherwise,
with some small ϵ > 0. In this case, note that only the seller s can have strictly positive LLOCs, thus she determines IP prices and LLOCs are minimized for p IP ≥ 30/ϵ. Meanwhile, setting the price p M W P equal to 15 minimizes MWPs. However, since p ∨ ∈ [0, 10], seller s incurs a greater LLOC than for p M W P .
A sufficient condition to satisfy the zero MWP condition with linear and anonymous prices is the presence of purely inelastic demand (Bichler et al. 2022). In our numerical analysis in Section 6 we also find that the set of optimal solutions to (Min-MWP) is large and that a solver typically picks solutions with very high GLOCs and LLOCs. Then, the solution of (IP ∨ MWP) achieves lower LLOCs than the solution of (Min-MWP) picked by the solver. The join does not necessarily exhibit Pareto optimality with respect to the objectives of total LLOCs and total MWPs. However, the join satisfies a participant-wise Pareto optimality criterion.
Proposition 5.3. For an optimal outcome (z * ℓ ) of a market, there exists some optimal solution p ∨ of (IP ∨ MWP) such that deviations cannot jointly reduce MWPs and LLOCs of all market participants. Formally, there is no other price vector q, such that for any market participant ℓ and
any i ∈ {IP, M W P }, λ i ℓ (q|z * ℓ ) ≤ λ i ℓ (p ∨ |z * ℓ )
, with the inequality holding strictly for some ℓ, i.
CH pricing dominates the literature on electricity market pricing. By minimization of GLOCs, CH prices naturally satisfy Pareto optimality. However, Schiro et al. (2016) show that CH pricing allows offline sellers to distort both locational prices and congestion signals, as we have also demonstrated. The join avoids such distortions to the welfare maximization problem exhibited by CH pricing. It only modifies the dual pricing function of online units and thus disallows offline units from having any effect on the price. Unlike CH pricing, the join can also be implemented efficiently (in poly-time) in practical electricity markets. Overall, the join provides a straightforward, parameter-free method of jointly minimizing LLOCs and make-whole payments, eliminating the need for fine-tuning or preference elicitation.
Numerical Results
To analyze the proposed pricing rules in an exemplary electricity market, we consider a simplified multi-period direct current optimal power flow (DCOPF) model (Frank et al. 2012, Molzahn andHiskens 2019). This model describes a nodal electricity market with linearized transmission flows. 5 We refer to Appendix B for the notation and problem formulation. The electricity market consists of a set of nodes V , with a specified reference node R * ∈ V . Each buyer b (seller s) has an associated node ν(b)(ν(s)), and we denote by N (v) the set of neighboring nodes to node v. With T as the set of considered time periods and B vw as the line susceptances, we formulate the DCOPF as follows.
max x,y,u,ϕ,f,α b∈B v b (x b ) − s∈S c s (y s , u s , ϕ s ) − v∈V,w∈N (v) d vw (f vw ) − v∈V d v (α v ) (DCOPF) subject to s∈S|ν(s)=v y st − b∈B|ν(b)=v x bt + w∈N (v) B vw (α wt − α vt ) = 0 ∀ v ∈ V, t ∈ T, f vwt + B vw α wt − B vw α vt = 0 ∀ v ∈ V, w ∈ N (v), t ∈ T,
where x are variables corresponding to buyers' consumption at each time frame, y are variables corresponding to the generation of sellers at each time frame, u are binary variables that model non-convexities in sellers' cost functions, ϕ are binary variables that indicate generation start for sellers, f are variables corresponding to flows on the grid, and α are variables that correspond to voltage angles.
To proceed with our analysis, we will impose an explicit form on our valuation and cost functions. We assume that the valuation and cost functions are given as follows:
1. For each buyer b, we have concave, piecewise linear valuation functions, represented by hourly bids β t b for each time period t. Moreover, buyer b has inelastic demand P bt , and has maximum power consumption P bt ,
v b (x b ) = max x b t∈T,ℓ∈β t b x btℓ v btℓ subject to x btℓ ∈ [0, q btℓ ] ∀t ∈ T, ℓ ∈ β t b , x bt − ℓ∈β t b x btℓ = P bt ∀t ∈ T,
x bt ≤ P bt ∀t ∈ T.
2. For each seller s, when u st = 1, seller s has some convex cost function for producing a positive amount at time t on some closed interval -modeled again by hourly bids β t s . Moreover, u st ∈ {0, 1} is a binary variable that denotes whether the unit of seller s is active at time t (associated with fixed costs h s ), and the variable ϕ st = u st − u s(t−1) represents whether a unit is turned on or off at time 1 < t ≤ T . There potentially exists a set of minimum uptime constraints that are linear for u s , ϕ s , c s (y s , u s , ϕ s ) = min ys,us,ϕs t∈T,ℓ∈β t
s y stℓ c stℓ + t∈T h s u st subject to y stℓ ∈ [0, q stℓ ] ∀ t ∈ T, ℓ ∈ β t s , y st − P st u st ≥ 0 ∀ t ∈ T, u st ∈ {0, 1} ∀ t ∈ T, y st − P st u st ≤ 0 ∀ t ∈ T, ϕ st ≥ 0 ∀ t ∈ T, ϕ st − u st + u s(t−1) ≥ 0 ∀ 1 < t ≤ T, y st − ℓ∈β t s y stℓ = 0 ∀ t ∈ T, t i=t−Rs+1 ϕ st − u st ≤ 0 ∀ 1 < t ≤ T.
3. For the DCOPF, transmission operators decide about line flows and phase angles. First, each flow between two nodes v and w, f vwt lies on a closed interval [F vw , F vw ],
d vw (f vw ) = ∞ ∃1 ≤ t ≤ T such that f vwt / ∈ [F vw , F vw ], 0 else.
Second, phase angles α vt at node v can be set at zero cost. The phase angle at the reference node, however, is fixed to 0 at each time period,
d v (α v ) = ∞ v = R * and ∃1 ≤ t ≤ T, α vt ̸ = 0, 0 else.
With this explicit welfare maximization problem at hand, we implement the pricing rules discussed in Section 4. While CH pricing is intractable in general, a common heuristic in practical electricity markets is to employ ELMP (MISO 2019). ELMP relaxes the binary constraints in DCOPF for u st , {0, 1} → [0, 1], and prices are obtained from the resulting dual problem. It has been shown by Hua and Baldick (2017) that with the valuation / cost functions in our setting, the dual of this relaxation of DCOPF is in fact equivalent to CH pricing. Therefore, we have an explicit and tractable dual formulation of CH prices, provided in Appendix C.
In order to compute IP prices, we follow the approach suggested by O'Neill et al. (2005). In particular, we restrict each integer variable of the welfare maximization problem to the value it takes in the optimal allocation and solve the dual of the resulting problem. More explicitly, we set u st as {0, 1} → [0, 1] and u st = u * st in DCOPF and solve the dual problem provided in Appendix D.
For our implementation of Min-MWP, we note that given some prices the transmission operators have either infinite or zero LLOCs (and hence GLOCs) associated with phase angles. To be able to obtain a solution to Min-MWP with finite LLOCs / GLOCs, we thus opt to replace the Min-MWP dual pricing functions for phase angles with their actual CH dual pricing functions in our implementation, thus accounting for their GLOCS. This implementation, provided in Appendix E, minimizes MWPs of buyers, sellers and transmission operators under the constraint that no GLOCs occur with respect to phase angles.
For linear scalarization, we combine these pricing problems and optimize a weighted average of the individual objective functions. The formulation for the join (IP ∨ MWP) is provided in Appendix F and considers each participant's maximum of the IP and Min-MWP dual pricing function.
We parameterize the DCOPF for the IEEE RTS system (Section 6.1) as well as for the German bidding zone (Section 6.2) based on data that was recently used for the bid zone review in the European Union. In Appendix G, we further report results for the ARPA-E Grid Optimization Competition.
IEEE RTS System
First, we report results based on the IEEE RTS System, originally introduced by IEEE RTS Task Force of APM Subcommittee (1979) and used in a variety of studies on electricity markets (Garcia-Bertrand et al. 2006, Morales et al. 2009, Zoltowska 2016, Hytowitz et al. 2020, Zocca and Zwart 2021. Grigg et al. (1999) provide the stylized system topology, transmission network parameters, hourly (nodal) demand data as well as characteristics of generating units. In accordance with Zoltowska (2016), we select the single-area, 24-node topology by Grigg et al. (1999) for a representative 24-hour winter day with 32 generators (total capacity: 6.81 GW) and 17 consumers (average hourly demand: 2.60 GWh). Generators exhibit several non-convexities, such as no-load costs, minimum loads, or minimum runtimes. For data on generation costs or demand valuations we rely on the bid and offer curves provided by the case studies of Garcia-Bertrand et al. (2006) and Zoltowska (2016) on this system. Generators and consumers are embedded in a DC power flow model with 24 nodes. The optimal dispatch is computed by solving the mixed-integer DCOPF problem. All applied pricing rules constitute linear programs, with negligible computational effort compared to the initial DCOPF MILP. To begin with, the results underpin the finding that the optimization of individual objectives via CH, IP, and Min-MWP pricing is undesirable in at least one of the other objectives.
Result 1. The LOC classes under consideration (GLOCs, LLOCs, MWPs) are conflicting. Pricing rules that optimize one class of LOCs lead to undesirable outcome in other classes of LOCs.
Applying linear scalarization to IP and Min-MWP pricing leads to alternative results on the Pareto frontier (e.g., Table 4 displays a linear scalarization with 90% weight on IP and 10% weight on Min-MWP 7 ), yet setting the weights is arbitrary. It should be noted that the set of outcomes implied by linear scalarization is not continuous. That is, there is only a limited number of outcomes and different weight vectors might produce the same result. The join IP ∨ Min-MWP requires no MWPs and LLOCs are reduced by 85% compared to CH pricing (and close to 100% compared to Min-MWP pricing). Moreover, GLOCs are significantly reduced compared to Min-MWP pricing and are only 16% higher than the GLOCs of IP pricing. Figure 1 illustrates the GLOCs, MWPs, and LLOCs for the tested pricing rules and the Pareto frontier. Note that Min-MWP possesses very high GLOCs and LLOCs, and can therefore not be meaningfully depicted. The Pareto frontier is obtained by applying linear scalarizations to the three pricing rules and is presented in 1a. The lines in Figures 1b -1d correspond to pairwise linear scalarizations, i.e. projections of the Pareto frontier on the respective two-dimensional space. Those lines connect the discrete number of Pareto-efficient outcomes implied by linear scalarization.
Result 2. The Pareto frontier of GLOCs, LLOCs, and MWPs possesses a high curvature. This suggests that there are significant merits to balancing the trade-offs between different classes of LOCs by applying multi-objective optimization. For example, with an equal-weight linear scalarization of IP and CH pricing, we obtain prices that have less GLOCs and LLOCs than the average over IP and CH. This effect is particularly prominent for Min-MWP pricing: although the original price profile is fairly extreme, a linear scalarization causes the GLOCs and LLOCs to collapse very fast. 8 However, finding the correct weight vector for an a priori pricing rule requires information on preferences, which may require some degree of exploration and thus poses an impediment in practice. In contrast, the proposed join IP ∨ Min-MWP does not require such preference elicitation. However, joining it with standard IP pricing retains the congestion signals almost perfectly and at the same time reduces the substantial MWPs that IP pricing implies. The join further improves the congestion signals compared to CH pricing. For example, in hours 4-6 the single join price correctly signals that no congestion occurs in the network (as is also signaled by the single IP price), while CH pricing exhibits price differences and thus wrongly implies congestion.
European Bid Data
Next, we report results for a DCOPF for the German bidding zone. The model is based on data published for the ongoing bidding zone review (BZR) in the European Union (EU). The BZR describes the process of evaluating the current bidding zone configuration as a result of structural congestion and efficiency losses in the EU day-ahead market. In this context, the European Network of Transmission System Operators for Electricity (ENTSO-E) conducted a nodal pricing study and published the related data in September 2022. 9 We calibrate our DCOPF model based on these data.
The data sets contain hourly aggregated demand data as well as generator characteristics for 24 representative weeks of the years 1989, 1995, and 2009, as well as a network grid model for the German transmission system. We utilize these data to construct the DCOPF as provided above. We obtain a system with 1687 nodes and 3232 transmission lines as well as bids for 24 × 7 = 168 days.
We exemplarily report the result for three days: February 18, 2009, as a representative uncongested day according to the ENTSO-E study; July 30, 2009, as a representative congested day;
and January 23, 2009 as a representative average day. The resulting MILPs of the DCOPF are large with 4,538 generators, 1444 consumers, and more than 850,000 continuous and 200,000 binary variables for January 23, 2009. To reduce the computational effort 10 and memory requirements, we therefore split the 24-hour interval into four consecutive 6-hour intervals. In particular, instead of the regular 24-hour time horizon in European day-ahead markets, we use six-hour intervals and aggregate the respective lost opportunity costs. Technically, this means that intertemporal constraints are not enforced between the different six-hour intervals, but the impact on total lost opportunity costs is minor. Tables 5, 6, and 7 provide an overview of the results. The results confirm what was found for the IEEE RTS system. Due to the large size of this test system, the trade-offs between GLOCs, MWPs, and LLOCs are even more evident. In particular, as demand is price-inelastic, Min-MWP provides a solution with zero MWPs (Bichler et al. 2022). The GLOCs and LLOCs implied by Min-MWP pricing are very high, yet adding a small weight of make-whole payments to IP can already reduce MWPs significantly. IP prices can lead to large MWPs, which are reduced by linear scalarization and the join, leading to only a small increase in LLOCs. As discussed earlier, linear scalarization is sensitive to the weights which are difficult to set a priori. In contrast, IP ∨ Min-MWP requires no parameterization and trades off MWPs and congestion signals (by means of LLOCs) in a meaningful way. The join also reduces GLOCs compared to IP and Min-MWP.
Result 3. Pure minimization of MWPs leads to very high GLOCs and LLOCs. Market participants have high incentives to deviate from the optimal outcome and congestion signals are flawed. The join IP ∨ Min-MWP significantly reduces LLOCs compared to Min-MWP pricing and MWPs compared to IP pricing. It produces a better congestion signal than CH pricing, while exhibiting lower MWPs at the same time. The differences in the lost opportunity costs between the pricing rules are more pronounced for the large instances of the German bidding zone compared to the small IEEE RTS data set.
Compared to IP ∨ Min-MWP, CH prices compromise on MWPs and LLOCs (and thus congestion signals) to reduce GLOCs. Typically, CH prices are intractable. As indicated earlier, we chose a model formulation that allows to compute CH prices in polynomial time. For more complex valuation or cost functions, convex hull formulations are not readily available, rendering CH pricing intractable (Schiro et al. 2016). In contrast, IP ∨ Min-MWP can always be computed in polynomial time and it is practically tractable even for large problem sizes.
Conclusions
The welfare theorems provide a solid foundation for the analysis of electricity market prices as they introduce the design desiderata one can hope for with a market-based allocation. A version of the welfare theorems for coupled markets delineates those environments for which we can expect Walrasian equilibria to exist from those where we do not. These theorems neither need linearity nor differentiability, but convexity of the cost and value functions. In particular, Walrasian prices cease to exist in the presence of non-convexities. Electricity spot markets are coupled and nonconvex, and consequently there is no linear price vector that would be budget-balanced and envyfree. With linear and anonymous market prices on non-convex power markets, some participants require an uplift to compensate their loss, which incurs a budget deficit for the seller. In addition, the congestion signals on lines are biased, which results in non-zero LLOCs. This means, prices between nodes might differ even if no congestion occurs and the price difference does not reflect the marginal value of transmission capacity.
The extended welfare theorems provide a foundation to classify different pricing rules for electricity markets as they have been suggested in the literature. We show that existing pricing rules minimize certain classes of lost opportunity costs, which can cause substantial increases in other relevant lost opportunity costs. Based on this trade-off, we propose to view the design of pricing rules in non-convex electricity markets as a multi-objective optimization problem.
We analyze linear scalarization as a weighted sum of individual objective functions to derive a Pareto frontier of the different design goals. As this comes with certain practical limitations, i.e. the need for preference elicitation, we propose a novel and scalable pricing rule that requires no parametrization or fine-tuning. This join of the IP and Min-MWP pricing rule addresses current policy issues in U.S. and European electricity markets. In particular, the join minimizes side-payments, maintains adequate congestion signals, and can be computed efficiently. Based on realistic data sets, we demonstrate the possibilities of the join to capitalize on the upsides of several pricing rules. In view of recent concerns by regulators regarding increasing levels of MWPs with the IP pricing rule and the desire to maintain good congestion signals, the join provides a straightforward and easy-to-implement alternative for regulators. This implies that, given a convex dual pricing functionλ b (p|x * b ) for a buyer b, denoting byṽ b the corresponding concave valuation function for buyer b,
(−ṽ b ) * (p) =λ b (−p|x * b ) + p T x * b + v b (x * b ),(14)
where v b (x * b ) is the value buyer b has for allocation x * b in the original welfare maximization problem (2). Noting that for a closed convex function f the biconjugate equals the function itself, i.e. f = f * * , we conclude that
v b (x) = −(λ b (−p|x * b ) + p T x * b + v b (x * b )) * (x) (15) = −(max p p T (x − x * b ) − v b (x * b ) −λ b (−p|x * b )) = −λ * b (x * b − x) + v b (x * b ).
By a similar analysis, the corresponding cost function for a seller s is given bỹ
c s (y) =λ * c (y − y * s ) + c s (y * s ).(16)
The induced cost functions of transmission operators, in turn, need to be more carefully defined due to the presence of matrices B r which specify the interaction of flows with the supply-demand constraints. In this case, as a first attempt one may consider a modified conjugate functioñ
d r (f ) = max p p T B r (f − f * r ) + d r (f * r ) −λ r (p|f * r ).
However, this allows for transmission operator r a feasible flow of the form f = f ′ + δ, where B r δ = 0 (i.e. δ is in the null-space null(B r ) of B r ) and f ′ is orthogonal to δ. Such flows δ do not have any effect on the supply-demand balance, and can be considered to be infeasible. Therefore, to rectify this, we instead set
d r (f ) = max p p T B r (f − f * r ) + d r (f * r ) −λ r (p|f * r ) if f T δ = 0, ∀ δ ∈ null(B r ), +∞ else.(17)
Then by Proposition 3.1.4, we note for the welfare functioñ
ω(σ) = max x,y,f b∈Bṽ b (x b ) − s∈Sc s (y s ) − r∈Rd r (f r ) subject to s∈S y s − b∈B x b + r∈R B r f r = σ,
the conjugate of the negative welfare function (−ω) * equals
(−ω) * (p) = ℓ∈Lλ ℓ (p|z * ℓ ) + ω(0).(18)
The pricing problem for this convex market, given by the subgradient problem (5), is therefore precisely min p ℓ∈Lλ ℓ (p|z * ℓ ). To illustrate how the calculations work, we derive the convex model corresponding to (Min-MWP):
Example 7. In (Min-MWP), each market participant ℓ has a dual pricing function λ M W P ℓ (p|z * ℓ ) = max{−u ℓ (z * ℓ |p), 0}. For any buyer b,
u b (x * b ) = v b (x * b ) − p T x * b .
We know that the convex modelṽ b here satisfies (15), and by Proposition 3.1.5,
[λ M W P b (·)] * (x) = conv min{(−u b (x * b |·)) * , 0 * }(x)
. The convex conjugate of the identical zero function is χ {0} (x), the indicator function for the singleton set containing 0. Meanwhile, (−u b (x * b |·)) * is given by
(−u b (x * b |·)) * (x) = max p p T x + u b (x * b |p) = max p p T (x − x * b ) + v b (x * b ) = χ {x * b } (x) + v b (x * b )
. Therefore, substituting for the expression (15), we get
−ṽ b (x) = conv min{χ {0} (x) + v b (x * b ), χ {x * b } } − v b (x * b ), which then implies thatṽ b (x) = conc max{−χ {0} , −χ {x * b } + v b (x * b )}.
Specifically, buyer b's valuations are modelled as the convexification of the valuation they have for their allocation, and the valuation they have for not participating in the market. Likewise, for a seller s and a transmission operator r, we havẽ
c s (y) = conv min{χ {0} , χ {y * s } + c s (y * s )}, d r (f ) = conv min{χ {0} , χ {f * r } + d r (f * r )}.
Thus it is indeed possible to extract a convex welfare maximization problem from a given dual pricing problem, and the dual pricing problem prices the optimal outcome of this welfare maximization problem. The set of participants and goods of this welfare maximization problem might in fact greatly differ for this problem; we provide two examples.
Example 8 (Penalty functions: Fictitious goods and agents). As mentioned previously, in nonconvex markets a Walrasian equilibrium need not exist. In this case, there is no price vector p such that the conditions of Walrasian equilibria (5) are all satisfied. While we consider budget balance in linear payments, market participants' losses still need to be compensated. To finance these uplift payments, O'Neill et al. (2016) consider imposing personalized price vectors to participants. Specifically, their approach involves finding a set of participants who require make-whole payments, and then imposing personalized prices so that their uplifts are financed via the payments of other market participants. As a measure of fairness, they minimize the magnitude of these transfers.
Here, we analyze the distortionary effect of a similar pricing problem, stylized for simplicity of analysis. We impose personalized prices to the dual pricing problem (6), with a quadratic penalty term for the magnitude of differences in prices. Furthermore, we add a budget balance constraint. Then the resulting dual pricing problem is given by
min (p ℓ ) ℓ∈L ℓ∈L λ CH ℓ (p ℓ |z * ℓ ) + 1 2 (ℓ,ℓ ′ )∈L 2 ∥p ℓ − p ℓ ′ ∥ 2 2 subject to ℓ∈L p T ℓ z * ℓ = 0.
By the examples above, we see that the addition of constraints and penalty functions lead to an addition of new agents in the primal welfare maximization problem, while combining participants' lost opportunity costs under a single term consolidates them into a single entity. We expect such drastic changes in the welfare maximization problem to widen the gap between the true optimal allocation and the allocation priced by the dual pricing problem, as in the case of CH pricing. Therefore, we would like to avoid such distortions if possible.
We note that the dimension M of the price vector p ∈ R M is precisely the number of priced goods. This is because the welfare function ω takes as its argument supply-demand constraint violations in real goods, and prices are in the corresponding dual space. To maintain the number of goods, we must thus impose linear prices -as modifying the size of the price vector (e.g. by imposing personalized prices) modifies the number of goods in the corresponding welfare maximization problem.
To maintain the number of participants, note that by Proposition 3.1.4 a welfare maximization problem with a set of agents L necessarily has an associated subgradient problem (5) that is additively separable over the participants. That is, for a welfare maximization problem of the form
max x,y,f b∈Bṽ b (x b ) − s∈Sc s (y s ) − r∈Rd r (f r ) subject to s∈S y s − b∈B x b + r∈R B r f r = 0,
the dual pricing problem is of the form min p ℓ∈Lλ ℓ (p|z * ℓ ). The converse implication also holds and an additively separable dual pricing function leads to a welfare maximization problem with set of participants L, as shown by our discussion in A.1. This leads us to observe that, to obtain a dual pricing problem with minimal distortion, we should restrict attention to dual pricing problems with an additively separable objective function (over L) and linear prices for each good.
B DCOPF Problem
Sets B Buyers S Sellers T = {1, .., T } Time Periods V Nodes (with some R * ∈ V as reference node) N (v) Neighboring nodes of a node v ∈ V β t b Bids of buyer b ∈ B in period t ∈ T β t s Bids of seller s ∈ S in period t ∈ T Mappings ν(b) B → V Mapping of buyer b ∈ B to its node v ∈ V ν(s) S → V Mapping of seller s ∈ S to its node v ∈ V Parameters B vw [pu] Susceptance of the line connecting v, w ∈ V v btℓ [$/MWh] Value of bid ℓ ∈ β t b of buyer b ∈ B in period t ∈ T q btℓ [MWh] Maximum quantity of bid ℓ ∈ β t b of buyer b ∈ B in period t ∈ T P bt [MWh] Price-inelastic demand of buyer b ∈ B in period t ∈ T P bt [MWh] Maximum demand of buyer b ∈ B in period t ∈ T c stℓ [$/MWh] Cost of bid ℓ ∈ β t s of seller s ∈ S in period t ∈ T h s [$] No-load costs of seller s ∈ S q stℓ [MWh] Maximum quantity of bid ℓ ∈ β t s of seller s ∈ S in period t ∈ T P st [MWh] Minimum output of seller s ∈ S in period t ∈ T P st [MWh]
Maximum output of seller s ∈ S in period t ∈ T R s Minimum uptime of seller s ∈ S F vw [MWh] Minimum flow on the line connecting v, w ∈ V F vw [MWh] Maximum flow on the line connecting v, w ∈ V Primal Variables
x bt ≥ 0 [MWh] Consumption of buyer b ∈ B in period t ∈ T x btℓ ≥ 0 [MWh] Consumption of buyer b ∈ B in period t ∈ T regarding bid ℓ ∈ β t b y st ≥ 0 [MWh] Generation of seller s ∈ S in period t ∈ T y stℓ ≥ 0 [MWh] Generation of seller s ∈ S in period t ∈ T regarding bid ℓ ∈ β t s u st ∈ {0, 1} Commitment of seller s ∈ S in period t ∈ T ϕ st ≥ 0 Start-up indicator for seller s ∈ S in period t ∈ T α vt ∈ R [rad] Voltage angle at node v ∈ V in period t ∈ T f vwt ∈ R [MWh] Flow on the line connecting v, w ∈ V in period t ∈ T Dual Variables p vt ∈ R [$/MWh] Price at node v ∈ V in period t ∈ T γ vwt ∈ R [$/MWh] Congestion price for the line connecting v, w ∈ V in period t ∈ T r t ∈ R
Dual of the reference node voltage angle constraint in period t ∈ T
h s u st (DCOPF-MILP) subject to x btℓ ∈ [0, q btℓ ] ∀b ∈ B, t ∈ T, ℓ ∈ β t b x bt − ℓ∈β t b x btℓ = P bt ∀b ∈ B, t ∈ T x bt ≤ P bt ∀b ∈ B, t ∈ T y stℓ ≥ 0 ∀ s ∈ S, t ∈ T, ℓ ∈ β t s y stℓ − u st q stℓ ≤ 0 ∀ s ∈ S, t ∈ T, ℓ ∈ β t s y st − ℓ∈β t s y stℓ = 0 ∀ s ∈ S, t ∈ T y st − P st u st ≥ 0 ∀ s ∈ S, t ∈ T y st − P st u st ≤ 0 ∀ s ∈ S, t ∈ T ϕ st − u st + u s(t−1) ≥ 0 ∀ s ∈ S, 1 < t ≤ T ϕ si − u st ≤ 0 ∀ s ∈ S, 1 < t ≤ T f vwt ∈ [F vw , F vw ] ∀ v ∈ V, w ∈ N (v), t ∈ T f vwt − B vw (α vt − α wt ) = 0 ∀ v ∈ V, w ∈ N (v), t ∈ T s:ν(s)=v y st − b:ν(b)=v x bt − w∈N (v) f vwt = 0 ∀ v ∈ V, t ∈ T α R * t = 0 ∀ t ∈ T
C ELMP Pricing Problem
The continuous relaxation of the binary integer constraints in the welfare maximization problem provide an LP. The associated dual pricing problem with this relaxation of the welfare maximization problem is the pricing rule known as ELMP (MISO 2019). Furthermore, the class of valuation / cost functions we consider are such that the optimal solutions to the dual LP provide CH prices (Hua and Baldick 2017). Therefore, to formulate the Convex Hull pricing problem in our setting we consider the dual LP to (DCOPF) where for each seller s and each time period t, the binary integer constraints are relaxed u st ∈ [0, 1]. Furthermore, we add constants to the dual objective function which do not alter the solution sets. However, the addition of these constants allows the objective function value to be sum of GLOCs. The resulting dual LP is given as: min ϵ,ψ,χ≥0,ϵ,ψ,χ,χ≤0,p,γ,r,λ,ϵ b∈B
λ b + s∈S λ s + v∈V,w∈N (v),t∈T λ vwt (ELMP-LP) subject to λ b − t∈T ϵ bt P bt + ϵ bt P bt + ℓ∈ν t b ϵ btℓ q btℓ + v b (x * b ) − p T ν(b) x * b ≥ 0 ∀ b ∈ B λ s − t∈T ψ st + p T ν(s) y * s − c s (y * s , u * s ) ≥ 0 ∀ s ∈ S λ vwt − ϵ vwt F vw − ϵ vwt F vw + γ vwt f * vwt ≥ 0 ∀ v ∈ V, w ∈ N (v), t ∈ T w|v∈N (w) B wv (p wt + γ wvt ) − w∈N (v) B vw (p vt + γ vwt ) = 0 ∀ v ∈ V \ {R * }, t ∈ T r t + w|R * ∈N (w) B wR * (p wt + γ wR * t ) − w∈N (R * ) B R * w (p R * t + γ R * wt ) = 0 ∀ t ∈ T − γ vwt + ϵ vwt + ϵ vwt = 0 ∀ v ∈ V, w ∈ N (v), t ∈ T ϵ btℓ + ϵ btℓ − ϵ bt = v btℓ ∀ b ∈ B, t ∈ T, ℓ ∈ ν t b ϵ bt + ϵ bt + p ν(b)t = 0 ∀ b ∈ B, t ∈ T ϵ stℓ + ϵ stℓ − ϵ st = −c stℓ ∀ s ∈ S, t ∈ T, ℓ ∈ ν t s ϵ st + ϵ st + ϵ st − p ν(s)t = 0 ∀ s ∈ S, t ∈ T − ℓ∈ν 1 s q s1ℓ ε s1ℓ + ψ s1 + ψ s1 − P s1 ϵ s1 − P s1 ϵ s1 + χ s2 = −h s1 , ∀ s ∈ S
D IP Pricing Problem
Note that in our setting, an optimal allocation ((x * b ) b∈B , (y * s ) s∈S , (f * r ) r∈R ) fixes the binary integer variables u * st which indicate whether seller s is generating at period t. IP pricing then restricts the primal welfare maximization problem (DCOPF) by adding the constraints u st = u * st ∀ s ∈ S, t ∈ T . The resulting welfare maximization problem is an LP, and the dual LP provides prices which minimize LLOCs. Again, we modify the dual objective function to be the sum of LLOCs by adding constants that do not affect the optimal solution. In our setting, this dual LP is given as: min ϵ≥0,ϵ≤0,p,γ,r,λ,ϵ b∈B
λ b + s∈S λ s + v∈V,w∈N (v),t∈T λ vwt (IP-LP) subject to λ b − t∈T ϵ bt P bt + ϵ bt P bt + ℓ∈ν t b ϵ btℓ q btℓ + v b (x * b ) − p T ν(b) x * b ≥ 0 ∀ b ∈ B λ s − t∈T ϵ st P st u * st + ϵ st P st u * st + ℓ∈ν t b ϵ stℓ q stℓ u * st + p T ν(s) y * s − c s (y * s , u * s ) ≥ −h T u * s ∀ s ∈ S λ vwt − ϵ vwt F vw − ϵ vwt F vw + γ vwt f * vwt ≥ 0 ∀ v ∈ V, w ∈ N (v), t ∈ T w|v∈N (w) B wv (p wt + γ wvt ) − w∈N (v) B vw (p vt + γ vwt ) = 0 ∀ v ∈ V \ {R * }, t ∈ T r t + w|R * ∈N (w) B wR * (p wt + γ wR * t ) − w∈N (R * ) B R * w (p R * t + γ R * wt ) = 0 ∀ t ∈ T − γ vwt + ϵ vwt + ϵ vwt = 0 ∀ v ∈ V, w ∈ N (v), t ∈ T ϵ btℓ + ϵ btℓ − ϵ bt = v btℓ ∀ b ∈ B, t ∈ T, ℓ ∈ ν t b ϵ bt + ϵ bt + p ν(b)t = 0 ∀ b ∈ B, t ∈ T ϵ stℓ + ϵ stℓ − ϵ st = −c stℓ ∀ s ∈ S, t ∈ T, ℓ ∈ ν t s ϵ st + ϵ st + ϵ st − p ν(s)t = 0 ∀ s ∈ S, t ∈ T.
E Min-MWP Pricing Problem
As mentioned before, while a direct implementation of (Min-MWP) is possible, solutions to the resulting dual pricing problem may result in phase angle operators having infinite LLOCs / GLOCs. To rectify this issue, in our implementation of (Min-MWP) we still account for GLOCs of transmission operators with respect to phase angles. This is achieved by including the dual constraints associated with primal variables α vt for v ∈ V, t ∈ T , and the resulting dual pricing problem is given as: min p,γ,r,λ b∈B
λ b + s∈S λ s + v∈V,w∈N (v),t∈T λ vwt (Min-MWP-LP) subject to −v b (x * b ) + p T ν(b) x * b − λ b ≤ 0 ∀ b ∈ B −p T ν(s) y * s + c s (y * s , u * s ) − λ s ≤ 0 ∀ s ∈ S −γ vwt f * vwt − λ vwt ≤ 0 ∀ v ∈ V, w ∈ N (v), t ∈ T w|v∈N (w) B wv (p wt + γ wvt ) − w∈N (v) B vw (p vt + γ vwt ) = 0 ∀ v ∈ V \ {R * }, t ∈ T r t + w|R * ∈N (w) B wR * (p wt + γ wR * t ) − w∈N (R * ) B R * w (p R * t + γ R * wt ) = 0 ∀ t ∈ T.
F IP ∨ Min-MWP Pricing Problem
As discussed, the join of IP and Min-MWP pricing considers each participant's maximum of the IP and Min-MWP dual pricing function. For DCOPF, we need not introduce new decision variables and the addition of only |B ∪S| constraints to IP-LP is sufficient. Then, by choice of the constraints, each participant's contribution to the objective function λ ℓ will equal the maximum of their MWPs and LLOCs. The dual LP is given as: min ϵ≥0,ϵ≤0,p,γ,r,λ,ϵ b∈B
λ b + s∈S λ s + v∈V,w∈N (v),t∈T λ vwt ((IP ∨ Min-MWP)-LP) subject to λ b − t∈T ϵ bt P bt + ϵ bt P bt + ℓ∈ν t b ϵ btℓ q btℓ + v b (x * b ) − p T ν(b) x * b ≥ 0 ∀ b ∈ B v b (x * b ) − p T ν(b) x * b + λ b ≥ 0 ∀ b ∈ B λ s − t∈T ϵ st P st u * st + ϵ st P st u * st + ℓ∈ν t b ϵ stℓ q stℓ u * st + p T ν(s) y * s − c s (y * s , u * s ) ≥ −h T u * s ∀ s ∈ S p T ν(s) y * s − c s (y * s , u * s ) + λ s ≥ 0 ∀ s ∈ S λ vwt − ϵ vwt F vw − ϵ vwt F vw + γ vwt f * vwt ≥ 0 ∀ v ∈ V, w ∈ N (v), t ∈ T w|v∈N (w) B wv (p wt + γ wvt ) − w∈N (v) B vw (p vt + γ vwt ) = 0 ∀v ∈ V \ {R * }, dt ∈ T r t + w|R * ∈N (w) B wR * (p wt + γ wR * t ) − w∈N (R * ) B R * w (p R * t + γ R * wt ) = 0 ∀ t ∈ T − γ vwt + ϵ vwt + ϵ vwt = 0 ∀ v ∈ V, w ∈ N (v), t ∈ T ϵ btℓ + ϵ btℓ − ϵ bt = v btℓ ∀ b ∈ B, t ∈ T, ℓ ∈ ν t b
ϵ bt + ϵ bt + p ν(b)t = 0 ∀ b ∈ B, t ∈ T ϵ stℓ + ϵ stℓ − ϵ st = −c stℓ ∀ s ∈ S, t ∈ T, ℓ ∈ ν t s ϵ st + ϵ st + ϵ st − p ν(s)t = 0 ∀ s ∈ S, t ∈ T.
G ARPA-E Grid Optimization Competition Data
Apart from the IEEE RTS System and the BZR data, we test our pricing rules on the network and bid data provided for the ARPA-E Grid Optimization Competition Challenge 2. 11 This competition seeks the development of modern and scalable optimization techniques for solving complex power flow problems. To that end, they provide large-scale and realistic test sets of single-period power flow problems. For our purposes, we test the five different scenarios provided for an exemplary 617-node grid. We parameterize DCOPF with the available data (with capped susceptance values to avoid numerical instability) and report the results of exemplary linear scalarizations and the join in the table below. The results confirm the observations made for the other test systems. 12 Min-MWP requires small make-whole payments, but incurs very large GLOCs and LLOCs. CH pricing minimizes GLOCs but incurs very large MWPs and LLOCs. Finally, IP pricing provides correct congestion signals, but requires large side-payments and penalties.
3. (Multiplication by a number) Rescaling f by some positive constant rescales both the magnitude and the argument of its conjugate, (αf ) * = αf (·) α . 4. (Convolution) Suppose the function ω : R n → R is defined via a convolution of f and g, i.e.
b ,ĉ s ,d r are the corresponding active valuation and cost functions given the optimal allocation. The dual pricing problem can be equivalently stated as the minimization of LLOCs of all
Figure 1 :
1RTS System -GLOCs, MWPs, LLOCs [$]
Figure 2 :
2RTS System -Price Heatmaps [$/MWh] Figure 2 contains price heatmaps (in $/MWh) along the different nodes and periods. The Min-MWP pricing without additional constraints leads to high price volatility and inadequate congestion signals.
ε stℓ + ψ st + ψ st − P st ϵ st − P st ϵ st − χ st − χ st + χ s(t+1) = −h st ∀s ∈ S, ℓ ε sT ℓ + ψ sT + ψ sT − P sT ϵ sT − P sT ϵ sT − χ sT − χ sT = −h sT ∀ s ∈ Ŝ χ st + χ st + min{T ,t+R s −1} t ′ =tχ st = 0 ∀ s ∈ S, 1 < t ≤ T.
Price 1 [$/Unit] Price 2 [$/Unit] GLOCs [$] MWPs [$] LLOCs [$] Table 1: Example for CH pricing with large MWPs and LLOCsCH
76.67
1.67
733.33
733.33
733.33
IP
10.00
10.00
1125.00
1000.00
0
Min-MWP
260.00
260.00
6625.00
0
2750.00
. While IP pricing minimizes LLOCs, it may very well imply large GLOCs and MWPs.Price 1 [$/Unit] Price 2 [$/Unit] GLOCs [$] MWPs [$] LLOCs [$]
CH
13.5
13.5
12.5
12.5
12.5
IP
1
1
100
100
0
Min-MWP
15.29
15.29
14.29
0
14.29
Table 2 :
2Example for IP pricing with large GLOCs and MWPs
Price 1 [$/Unit] Price 2 [$/Unit] GLOCs [$] MWPs [$] LLOCs [$]CH
30.00
10.67
996.67
918.67
996.67
IP
10.00
50.00
1590.00
1000.00
0
Min-MWP
176.67
176.67
10076.67
253.33
7586.67
Table 3 :
3Example for Min-MWP pricing with large GLOCs and LLOCs
Table 4
4provides a first high-level overview of the results. 6GLOCs [$] MWPs [$] LLOCs [$]
CH
1436.21
202.57
1272.12
IP
10364.24
3387.00
0
Min-MWP
2.1 ×10 11
0
1.1 ×10 11
0.9 IP + 0.1 Min-MWP
11025.03
693.00
111.79
0.9 CH + 0.1 Min-MWP 1432.93
199.27
1119.85
0.5 IP + 0.5 CH
1570.13
606.21
729.29
IP ∨ Min-MWP
12022.38
0
191.14
Table 4 :
4RTS System -GLOCs, MWPs, LLOCs
Table 5 :
5BZR Data -February 18, 2009 -GLOCs, MWPs, LLOCs
GLOCs [$] MWPs [$] LLOCs [$]
CH
68392.90
2735.06
33562.90
IP
184097.08
3982.19
0.00
Min-MWP
6736749.48 0.00
9955661.19
0.9 IP + 0.1 Min-MWP 233327.03
854.46
128.50
IP ∨ Min-MWP
220355.71
200.27
230.73
Table 6 :
6BZR Data -July 30, 2009 -GLOCs, MWPs, LLOCs
GLOCs [$]
MWPs [$] LLOCs [$]
CH
79158.72
24577.01
44726.50
IP
514294.88
22487.21
0.00
Min-MWP
11166195.10 0.00
8933860.68
0.9 IP + 0.1 Min-MWP 532385.17
2907.31
541.85
IP ∨ Min-MWP
523313.00
325.51
1292.38
Table 7 :
7BZR Data -January 23, 2009 -GLOCs, MWPs, LLOCs
Table 8 :
8DCOPF Notation max b∈B t∈T ℓ∈β t b v btℓ x btℓ − s∈S t∈T ℓ∈β t s c stℓ y stℓ −s∈S t∈T
Table 9 :
9ARPA Grid Optimization Competition -GLOCs, MWPs, LLOCs [$]
https://www.ferc.gov/industries-data/electric/electric-power-markets/energy-price-formation
The remaining GLOCs are typically enforced by imposing penalties on market participants that deviate from the optimal allocation.3 Commitment decisions on spot markets determine whether a generator is scheduled to produce electricity during a market time unit (a binary decision variable in the allocation problem), but not the production quantity (in Megawatt hours (MWh)). Commitment decisions occur because many types of generators require a long time to turn on/off.
The discussion generalizes easily to the case of weak supply-demand equivalence
Tighter power flow relaxations exist, but are currently not applied in practical electricity markets for computational reasons. We refer toMolzahn and Hiskens (2019) for a comprehensive overview of power flow problems.
Note that Min-MWP has multiple optimal solutions, i.e. there exist multiple sets of prices (forming a polyhedron) that lead to zero MWPs. We report the solution that our solver provides. 7 A simple 50:50 weight assignment would lead to a high emphasis on MWPs and outcomes that are much worse than that of the join.
We also tested linear scalarizations with a non-exact ELMP approximation of CH pricing, in order to simulate a situation when CH prices are not readily available. This provided similar results, albeit with slightly increased GLOCs due to the non-exactness of the approximation.
The data sets are available here: https://www.entsoe.eu/network_codes/bzr/.
Note that in our experiments, the computational effort of the six-hour MILPs was negligible with runtime of only a few minutes on a standard computer, yet the memory requirements of computing all lost opportunity costs were high.
t i=t−Rs+1
See https://gocompetition.energy.gov/challenges/challenge-2 for further details.
Note that the ARPA-E test systems are much bigger in size, but only consider a single hour. For both the IEEE RTS system and the German bidding zone, we consider 24 consecutive hours. Therefore, the results reported inTable 9has different orders of magnitude.
AcknowledgmentsThe financial support from the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) (BI 1057/9-1) is gratefully acknowledged.A Economic Interpretability ProblemIn Section 5, we had mentioned that a dual pricing problem induces a convexified model for the underlying welfare maximization problem (2), and that a poor choice of multi-objective formulation for the dual pricing may expand the feasible region of the underlying welfare maximization problem, exaggerating the gap between the dual pricing and the primal allocation problems. This, in turn, may lead to a loss of semantic meaning, disallowing any meaningful economic interpretation. In this section of the electronic companion, we elaborate on this phenomenon. In order to do so, we first explain how to derive the corresponding convex model of a dual pricing problem by application of results from Section 3.1. We then illustrate how severe distortions in the welfare maximization problem may occur by considering two commonly used multi-objective solution concepts.A.1 Derivation of convex market modelsTo derive the welfare maximization problem corresponding to some dual pricing problem min p ℓ∈L λ ℓ (p|z * ℓ ), we note that for a convex market the dual pricing functions are defined as in(7). Using convex conjugates, we can rewrite this as follows:We can then compute the associated welfare maximization problem for this dual pricing problem by following the procedure in Section A.1. The corresponding market has set of personalized goods and flow parameters M × L, F × L. We also add a fictitious exchange operator (ℓ, ℓ ′ ) ∈ L 2 for each pair of participants, who can exchange goods and flows in {ℓ} × (M ∪ F ) with the corresponding goods and flows in {ℓ ′ } × (M ∪ F ) at a one-to-one ratio. Finally, we add an auctioneer A who can provide any multiple of (z * ℓ ) ℓ∈L to the market at cost 0, clearing personalized goods in proportion to the priced optimal outcome. The resulting welfare maximization problem is thusThe additional exchange operators may be thought of as participants who can take advantage of personalized price differences for arbitrage, while the auctioneer attempts to enforce optimal market clearing.Example 9 (Chebyshev scalarization: Consolidation of agents). To obtain a more balanced outcome with respect to the GLOCs of market participants, one might consider minimizing the (weighted) maximum of participants' GLOCs instead. Such a method is known as Chebyshev scalarization in the literature, where for objectives to f 1 , f 2 , ..., f n to be jointly minimized, one seeks a solution ofHere, (w i ) 1≤i≤n are the weights on each objective f i , signifying their relative importance, and F is the feasible region of the problem. Let us consider implementing Chebyshev scalarization as a dual pricing problem, where each participant's GLOCs are scaled by the 2-norm of the participant's allocation. Intuitively, this minimizes the GLOCs incurred for each unit of good purchased. This leads us to consider the dual pricing problemHowever, this dual pricing problem is not additively separable over the market participants. Assuming the market does not have transmission operators for simplicity and by evaluating the convex conjugate of the dual pricing problem equalsIt does not seem possible, however, to transform this expression into a utilitarian welfare function for the original set of market participants. In fact, the expression does not appear to admit a straightforward interpretation as any meaningful welfare function.
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