manu/mldr-zoomed-100char-total-queries-documents

chunk_id stringlengths 3 9	chunk stringlengths 1 100
5_45	(Note: this equation is not equivalent to the classical one given in the French version of the
5_46	article.)
5_47	This is just Newton's second law for the j-th particle. The first factor is just the usual Hooke's
5_48	law form for the force. The factor with is the nonlinear force. We can rewrite this in terms of
5_49	continuum quantities by defining to be the wave speed, where is the Young's modulus for the
5_50	string, and is the density:
5_51	Connection to the KdV equation
5_52	The continuum limit of the governing equations for the string (with the quadratic force term) is
5_53	the Korteweg–de Vries equation (KdV equation.) The discovery of this relationship and of the
5_54	soliton solutions of the KdV equation by Martin David Kruskal and Norman Zabusky in 1965 was an
5_55	important step forward in nonlinear system research. We reproduce below a derivation of this limit,
5_56	which is rather tricky, as found in Palais's article. Beginning from the "continuum form" of the
5_57	lattice equations above, we first define u(x, t) to be the displacement of the string at position x
5_58	and time t. We'll then want a correspondence so that is .
5_59	We can use Taylor's theorem to rewrite the second factor for small (subscripts of u denote partial
5_60	derivatives):
5_61	Similarly, the second term in the third factor is Thus, the FPUT system is
5_62	If one were to keep terms up to O(h) only and assume that approaches a limit, the resulting
5_63	equation is one which develops shocks, which is not observed. Thus one keeps the O(h2) term as
5_64	well:
5_65	We now make the following substitutions, motivated by the decomposition of traveling-wave solutions
5_66	(of the ordinary wave equation, to which this reduces when vanish) into left- and right-moving
5_67	waves, so that we only consider a right-moving wave. Let . Under this change of coordinates, the
5_68	equation becomes
5_69	To take the continuum limit, assume that tends to a constant, and tend to zero. If we take , then
5_70	Taking results in the KdV equation:
5_71	Zabusky and Kruskal argued that it was the fact that soliton solutions of the KdV equation can pass
5_72	through one another without affecting the asymptotic shapes that explained the quasi-periodicity of
5_73	the waves in the FPUT experiment. In short, thermalization could not occur because of a certain
5_74	"soliton symmetry" in the system, which broke ergodicity.
5_75	A similar set of manipulations (and approximations) lead to the Toda lattice, which is also famous
5_76	for being a completely integrable system. It, too, has soliton solutions, the Lax pairs, and so
5_77	also can be used to argue for the lack of ergodicity in the FPUT model.
5_78	Routes to thermalization
5_79	In 1966, Izrailev and Chirikov proposed that the system will thermalize, if a sufficient amount of
5_80	initial energy is provided. The idea here is that the non-linearity changes the dispersion
5_81	relation, allowing resonant interactions to take place that will bleed energy from one mode to
5_82	another. A review of such models can be found in Livi et al. Yet, in 1970, Ford and Lunsford insist
5_83	that mixing can be observed even with arbitrarily small initial energies. There is a long and
5_84	complex history of approaches to the problem, see Dauxois (2008) for a (partial) survey.
5_85	Recent work by Onorato et al. demonstrates a very interesting route to thermalization. Rewriting
5_86	the FPUT model in terms of normal modes, the non-linear term expresses itself as a three-mode
5_87	interaction (using the language of statistical mechanics, this could be called a "three-phonon
5_88	interaction".) It is, however, not a resonant interaction, and is thus not able to spread energy
5_89	from one mode to another; it can only generate the FPUT recurrence. The three-phonon interaction
5_90	cannot thermalize the system.
5_91	A key insight, however, is that these modes are combinations of "free" and "bound" modes. That is,
5_92	higher harmonics are "bound" to the fundamental, much in the same way that the higher harmonics in
5_93	solutions to the KdV equation are bound to the fundamental. They do not have any dynamics of their
5_94	own, and are instead phase-locked to the fundamental. Thermalization, if present, can only be among
5_95	the free modes.
5_96	To obtain the free modes, a canonical transformation can be applied that removes all modes that are
5_97	not free (that do not engage in resonant interactions). Doing so for the FPUT system results in
5_98	oscillator modes that have a four-wave interaction (the three-wave interaction has been removed).
5_99	These quartets do interact resonantly, i.e. do mix together four modes at a time. Oddly, though,
5_100	when the FPUT chain has only 16, 32 or 64 nodes in it, these quartets are isolated from
5_101	one-another. Any given mode belongs to only one quartet, and energy cannot bleed from one quartet
5_102	to another. Continuing on to higher orders of interaction, there is a six-wave interaction that is
5_103	resonant; furthermore, every mode participates in at least two different six-wave interactions. In
5_104	other words, all of the modes become interconnected, and energy will transfer between all of the
5_105	different modes.
5_106	The three-wave interaction is of strength (the same as in prior sections, above). The four-wave
5_107	interaction is of strength and the six-wave interaction is of strength . Based on general
5_108	principles from correlation of interactions (stemming from the BBGKY hierarchy) one expects the
5_109	thermalization time to run as the square of the interaction. Thus, the original FPUT lattice (of
5_110	size 16, 32 or 64) will eventually thermalize, on a time scale of order : clearly, this becomes a
5_111	very long time for weak interactions ; meanwhile, the FPUT recurrence will appear to run unabated.
5_112	This particular result holds for these particular lattice sizes; the resonant four-wave or six-wave
5_113	interactions for different lattice sizes may or may not mix together modes (because the Brillouin
5_114	zones are of a different size, and so the combinatorics of which wave-vectors can sum to zero is
5_115	altered.) Generic procedures for obtaining canonical transformations that linearize away the bound
5_116	modes remain a topic of active research.
5_117	References
5_118	Further reading
5_119	Grant, Virginia (2020). "We thank Miss Mary Tsingou". National Security Science. Winter 2020:
5_120	36-43.
5_121	External links Nonlinear systems Ergodic theory History of physics Computational physics
6_0	Sharp County is a county located in the U.S. state of Arkansas. As of the 2010 census, the
6_1	population was 17,264. The county seat is Ash Flat. The county was formed on July 18, 1868, and
6_2	named for Ephraim Sharp, a state legislator from the area.
6_3	Sharp County was featured on the PBS program Independent Lens for its 1906 "banishment" of all of
6_4	its Black residents. A local newspaper at the time was quoted as saying that "The community is
6_5	better off without them."
6_6	Geography
6_7	According to the U.S. Census Bureau, the county has a total area of , of which is land and (0.3%)
6_8	is water.
6_9	Major highways U.S. Highway 62 U.S. Highway 63 U.S. Highway 167 U.S. Highway 412 Highway 56
6_10	Highway 58 Highway 175
6_11	Adjacent counties Oregon County, Missouri (north) Randolph County (northeast)
6_12	Lawrence County (southeast) Independence County (south) Izard County (southwest)
6_13	Fulton County (northwest)
6_14	Demographics 2020 census
6_15	As of the 2020 United States census, there were 17,271 people, 7,447 households, and 4,420 families
6_16	residing in the county.
6_17	2000 census
6_18	As of the 2000 census, there were 17,119 people, 7,211 households, and 5,141 families residing in
6_19	the county. The population density was 28 people per square mile (11/km2). There were 9,342
6_20	housing units at an average density of 16 per square mile (6/km2). The racial makeup of the county
6_21	was 97.14% White, 0.49% Black or African American, 0.68% Native American, 0.12% Asian, 0.02%
6_22	Pacific Islander, 0.16% from other races, and 1.39% from two or more races. 0.98% of the

5_45

(Note: this equation is not equivalent to the classical one given in the French version of the

5_46

article.)

5_47

This is just Newton's second law for the j-th particle. The first factor is just the usual Hooke's

5_48

law form for the force. The factor with is the nonlinear force. We can rewrite this in terms of

5_49

continuum quantities by defining to be the wave speed, where is the Young's modulus for the

5_50

string, and is the density:

5_51

Connection to the KdV equation

5_52

The continuum limit of the governing equations for the string (with the quadratic force term) is

5_53

the Korteweg–de Vries equation (KdV equation.) The discovery of this relationship and of the

5_54

soliton solutions of the KdV equation by Martin David Kruskal and Norman Zabusky in 1965 was an

5_55

important step forward in nonlinear system research. We reproduce below a derivation of this limit,

5_56

which is rather tricky, as found in Palais's article. Beginning from the "continuum form" of the

5_57

lattice equations above, we first define u(x, t) to be the displacement of the string at position x

5_58

and time t. We'll then want a correspondence so that is .

5_59

We can use Taylor's theorem to rewrite the second factor for small (subscripts of u denote partial

5_60

derivatives):

5_61

Similarly, the second term in the third factor is Thus, the FPUT system is

5_62

If one were to keep terms up to O(h) only and assume that approaches a limit, the resulting

5_63

equation is one which develops shocks, which is not observed. Thus one keeps the O(h2) term as

5_64

well:

5_65

We now make the following substitutions, motivated by the decomposition of traveling-wave solutions

5_66

(of the ordinary wave equation, to which this reduces when vanish) into left- and right-moving

5_67

waves, so that we only consider a right-moving wave. Let . Under this change of coordinates, the

5_68

equation becomes

5_69

To take the continuum limit, assume that tends to a constant, and tend to zero. If we take , then

5_70

Taking results in the KdV equation:

5_71

Zabusky and Kruskal argued that it was the fact that soliton solutions of the KdV equation can pass

5_72

through one another without affecting the asymptotic shapes that explained the quasi-periodicity of

5_73

the waves in the FPUT experiment. In short, thermalization could not occur because of a certain

5_74

"soliton symmetry" in the system, which broke ergodicity.

5_75

A similar set of manipulations (and approximations) lead to the Toda lattice, which is also famous

5_76

for being a completely integrable system. It, too, has soliton solutions, the Lax pairs, and so

5_77

also can be used to argue for the lack of ergodicity in the FPUT model.

5_78

Routes to thermalization

5_79

In 1966, Izrailev and Chirikov proposed that the system will thermalize, if a sufficient amount of

5_80

initial energy is provided. The idea here is that the non-linearity changes the dispersion

5_81

relation, allowing resonant interactions to take place that will bleed energy from one mode to

5_82

another. A review of such models can be found in Livi et al. Yet, in 1970, Ford and Lunsford insist

5_83

that mixing can be observed even with arbitrarily small initial energies. There is a long and

5_84

complex history of approaches to the problem, see Dauxois (2008) for a (partial) survey.

5_85

Recent work by Onorato et al. demonstrates a very interesting route to thermalization. Rewriting

5_86

the FPUT model in terms of normal modes, the non-linear term expresses itself as a three-mode

5_87

interaction (using the language of statistical mechanics, this could be called a "three-phonon

5_88

interaction".) It is, however, not a resonant interaction, and is thus not able to spread energy

5_89

from one mode to another; it can only generate the FPUT recurrence. The three-phonon interaction

5_90

cannot thermalize the system.

5_91

A key insight, however, is that these modes are combinations of "free" and "bound" modes. That is,

5_92

higher harmonics are "bound" to the fundamental, much in the same way that the higher harmonics in

5_93

solutions to the KdV equation are bound to the fundamental. They do not have any dynamics of their

5_94

own, and are instead phase-locked to the fundamental. Thermalization, if present, can only be among

5_95

the free modes.

5_96

To obtain the free modes, a canonical transformation can be applied that removes all modes that are

5_97

not free (that do not engage in resonant interactions). Doing so for the FPUT system results in

5_98

oscillator modes that have a four-wave interaction (the three-wave interaction has been removed).

5_99

These quartets do interact resonantly, i.e. do mix together four modes at a time. Oddly, though,

5_100

when the FPUT chain has only 16, 32 or 64 nodes in it, these quartets are isolated from

5_101

one-another. Any given mode belongs to only one quartet, and energy cannot bleed from one quartet

5_102

to another. Continuing on to higher orders of interaction, there is a six-wave interaction that is

5_103

resonant; furthermore, every mode participates in at least two different six-wave interactions. In

5_104

other words, all of the modes become interconnected, and energy will transfer between all of the

5_105

different modes.

5_106

The three-wave interaction is of strength (the same as in prior sections, above). The four-wave

5_107

interaction is of strength and the six-wave interaction is of strength . Based on general

5_108

principles from correlation of interactions (stemming from the BBGKY hierarchy) one expects the

5_109

thermalization time to run as the square of the interaction. Thus, the original FPUT lattice (of

5_110

size 16, 32 or 64) will eventually thermalize, on a time scale of order : clearly, this becomes a

5_111

very long time for weak interactions ; meanwhile, the FPUT recurrence will appear to run unabated.

5_112

This particular result holds for these particular lattice sizes; the resonant four-wave or six-wave

5_113

interactions for different lattice sizes may or may not mix together modes (because the Brillouin

5_114

zones are of a different size, and so the combinatorics of which wave-vectors can sum to zero is

5_115

altered.) Generic procedures for obtaining canonical transformations that linearize away the bound

5_116

modes remain a topic of active research.

5_117

References

5_118