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abstract: 'We establish the global well-posedness of overdamped dynamical density functional theory (DDFT): a nonlinear, nonlocal integro-partial differential equation used in statistical mechanical models of colloidal flow and other applications including nonlinear reaction-diffusion systems and opinion dynamics. With no-flux boundary conditions, we determine the well-posedness of the full nonlocal equations including two-body hydrodynamic interactions (HI) through the theory of Fredolm operators. Principally, this is done by rewriting the dynamics for the density $\varrho$ as a nonlocal Smoluchowski equation with a non-constant diffusion tensor $\bm{D}$ dependent on the diagonal part ($\bm{Z}_1$) of the HI tensor, and an effective drift $\bm{A}[\vec{a}]$ dependent on the off-diagonal part ($\bm{Z}_2$). We derive a scheme to uniquely construct the mean colloid flux $\vec{a}(\vec{r},t)$ in terms of eigenvectors of $\bm{D}$, show that the stationary density $\varrho(\vec{r})$ is independent of the HI tensors, as well as proving exponentially fast convergence to equilibrium. The stability of the equilibria $\varrho(\vec{r})$ is studied by considering the bounded (nonlocal) perturbation of the differential (local) part of the linearised operator. We show that the spectral properties of the full nonlocal operator with no-flux boundary conditions can differ considerably from those with periodic boundary conditions. We showcase our results by using the numerical | 1 | member_48 |
methods available in the pseudo-spectral collocation scheme 2DChebClass.'
author:
- '$^\dagger$,'
- '[^1] ,'
- '[^2]'
bibliography:
- 'bibYear3.bib'
title: '****'
---
dynamic density functional theory (DDFT), colloids, overdamped limit, hydrodynamic interactions, nonlocal-differential PDEs, interacting particle systems, McKean-Vlasov equation, phase transitions, bifurcation theory.
60F10; 60J75; 62P10; 92C37
[^3]
Introduction {#intro}
============
For suspended particles in a viscous fluid, the Navier-Stokes equations are not sufficient to model flows on a spatial scale comparable with the size of the individual particles. Instead, one requires a computationally tractable model that captures meso/macro-scale dynamics whilst also including physical effects driven by particle-level interactions. Dynamic density functional theories (DDFTs) are excellent candidates for modelling such systems [@marconi1999dynamic; @ArcherEvans04]. They are typically applied in condensed matter physics in the colloidal particle regime with particles of typical diameters 1nm$-$1$\mu$m. Recent advances have allowed the inclusion of inertia [@MarconiTarazonaCecconiMelchionna07; @archer2009dynamical], multiple species [@Archer05; @RothRauscherArcher09; @GNK13; @LichtnerArcherKlapp12], hydrodynamic interactions (HI) [@rex2009dynamical; @Rauscher10; @goddard2012general; @goddard2012unification], background flows [@RauscherDominguezKrugerPenna07], temperature gradients [@WittkowskiLowenBrand12; @anero2013functional], hard spheres [@Rosenfeld89; @RothEvansLangKahl02; @roth2010fundamental; @StopperMaroltRothHansen-Goos15], confined geometries [@goddard2016dynamical; @zimmermann2016flow], arbitrary shaped particles [@WittkowskiLowen11], and active microswimmers [@menzel2016dynamical; @hoell2017dynamical].
For equilibrium fluids, there is a rigorous mathematical framework proving the existence of nontrivial fluid densities, different from those | 1 | member_48 |
found by classical fluid dynamical formalisms, by taking into account both many body effects and external force fields. This is commonly known as (classical) density functional theory (DFT) [@Mermin:1965lo]. It is able to predict effects driven by the microscale, e.g., the non-smooth droplet profiles which are formed at the gas-liquid-solid trijunction in contact line problems [@berim2009simple] and the coexistence of multiple fluid films at critical values of the chemical potential energy in droplet spreading [@pereira2012equilibrium]. It has been used to resolve the paradox of stress and pressure singularities normally found in classical moving contact line problems [@sibley2013contact]. What is more, DFT agrees well with molecular dynamics simulations; see, e.g., [@Lutsko10] and references therein. These advancements motivate more mathematical analysis, in particular, on the well-posedness of the underlying equations being used and on the number and structure of equilibrium states.
As a non-equilibrium extension to DFT for classical fluids, dynamic DFT (DDFT) has been applied to a wide range of problems: polymeric solutions [@PennaDzubiellaTarazona03], spinodal decomposition [@ArcherEvans04], phase separation [@Archer05], granular dynamics [@MarconiMelchionna07; @MarconiTarazonaCecconi07], nucleation [@vanTeeffelenLikosLowen08], liquid crystals [@WittkowskiLowenBrand10], and evaporating films [@ArcherRobbinsThiele10]. Recently, a stochastic version of DDFT has been derived [@Lutsko12], which allows the study of energy barrier crossings, | 1 | member_48 |
such as in nucleation.
A crucial point is that the computational complexity of DDFT is (essentially) constant in the number of particles, which allows the treatment of macroscopically large systems, whilst retaining microscopic information. Furthermore, due to the universality of the underlying nonlinear, nonlocal partial differential equations, DDFT may be considered as a generalisation of a wider class of such models used in the continuum modelling of many natural phenomena consisting of complex, many body, multi-agent interparticle effects including: pattern formation [@camazine2003self], the flocking of birds, cell proliferation, the self organising of morphogenetic and bacterial species [@canizo2010collective; @carrillo2009double], nonlocal reaction-diffusion equations [@al2018dynamical] and even consensus modelling in opinion dynamics[@chazelle2017well]. Many of these applications are often described as systems of interacting (Brownian) particles and, in the case of hard particle viscous suspensions, bath-mediated HI effects may be included.
The HI are forces on the colloids mediated by the bath flow, generated by the motion of the colloidal particles. This in turn produces a nontrivial particle–fluid–particle hydrodynamic phenomenon, the inclusion of which has been shown to have substantial effects on the physics of many systems; for example, they have been found to be the underlying mechanism for the increased viscosity of suspensions | 1 | member_48 |
compared to a pure bath [@Einstein06], the blurring of laning that arises in driven flow [@WysockiLowen11], the migration of molecules away from a wall [@HodaKumar07], and are particularly complex in confined systems [@happel2012low; @LaugaSquires05], and for active particles and microswimmers, which result in additional HI [@HuberKoehlerYang11].
Mathematically, the inter-particle forces and HI can be described through the hydrodynamic fields $\varrho$ and $\vec{v}$, the one-body density and one-body velocity fields, respectively. These fields, inherent to a continuum description of a collection of particles, are derived by considering successive moments (density, velocity, heat flux, …) of the underlying kinetic system [@gorban2014hilbert]. In particular, for systems of interacting Newtonian particles, when the momenta are non-negligible, the evolution of the phase space density $f(\vec{r}^N,\vec{p}^N, t)$ for a system of $N$ colloids determining the probability of finding the system in the state $(\vec{r}^N,\vec{p}^N)$ at time $t$ is described by the $N$-body Fokker-Planck equation and the dynamics of the hydrodynamic fields are defined by obtaining closed equations for $\{\varrho, \varrho\times \vec{v}\} := \int \mathrm{d}\vec{r}^{N-1}\,\mathrm{d}\vec{p}^N\, \{1, \vec{p}/m\}f(\vec{r}^N,\vec{p}^N, t)$, where $m$ is the particle mass. Here, $\vec{r}^N$ and $\vec{p}^N$ denote the $3N$-dimensional position and momentum vectors of all $N$ particles.
The inclusion of HI leads to a much | 1 | member_48 |
richer hierachy of fluid equations compared to systems without HI; compare e.g. [@goddard2012unification] and [@archer2009dynamical]. In particular, see e.g. [@goddard2012unification], by integration over all but one particle position, the one-body Fokker-Planck equation may be obtained. If, in addition, two-body HI and interparticle interactions are assumed and the inertia of the colloids is considered small, a high friction limit $\gamma\to \infty$ may be taken [@goddard2012overdamped]. The result is that the velocity distribution converges to a Maxwellian, and one can eliminate the momentum variable through an adiabatic elimination process that is based on multiscale analysis [@pavliotis2008multiscale]. The final one-body Smoluchowski equation for $\varrho$ is a novel, nonlinear, nonlocal PDE shown to be independent of the unknown kinetic pressure term $\int \mathrm{d}\vec{r}\,\mathrm{d}\vec{p}\, m^{-2} \vec{p}\otimes\vec{p} f(\vec{r},\vec{p}, t)$, which normally persists at $\gamma = O(1)$ (see[@goddard2012overdamped], Theorem 4.1).
Existence, uniqueness and global asymptotic stability of the novel Smoluchowski equation in this overdamped limit has, until this work, remained unproven. It is the inclusion of HI that provides richness through additional nonlinearities in both the dissipation and convection terms. The inclusion of HI is interesting from both physical and mathematical standpoints. Physically, as above, the HI give rise to a much more complex evolution in the | 1 | member_48 |
density. Mathematically, the convergence to equilibrium will depend inherently on the spectral properties of the effective diffusion tensor and effective drift vector arising from the HI. What is more, since the full $N$-body Fokker-Planck equation is a PDE in a very high dimensional phase space, well-posed nonlinear, nonlocal PDEs governing the evolution of the one-particle distribution function, valid in the mean field limit, describing the flow of nonhomogeneous fluids are desirable for computational reasons.
The equations studied in this paper are related to the McKean-Vlasov equation [@chayes2010mckean], a nonlinear nonlocal PDE of Fokker-Planck type that arises in the meanfield limit of weakly interacting diffusions. The novelty of the present problem lies in the space dependent diffusion tensor and nonlinear, nonlocal boundary conditions. Additionally, the problem that we study in this paper may in general not be written as a gradient flow, with the exception of the modelling assumption that the off-diagonal elements of the friction tensor $\bm{\Gamma}$ are zero. This choice is equivalent to setting $\bm{Z}_2$ to zero, and would be physically relevant for a diffuse system of particles with a strong hydrodynamic interaction with a wall but weak inter-particle hydrodynamic interactions [@goddard2016dynamical].
Description of the Model.
-------------------------
In this | 1 | member_48 |
work we analyse the overdamped partial differential equation (PDE) associated to a system of interacting stochastic differential equations (SDEs) on $U$ an open, bounded subset of $\mathbb{R}^d$ of the following form, governing the positions $\vec{r}_i$ and momenta $\vec{p}_i$ of $i = 1,\dots, N$ colloidal particles immersed in a bath of many more, much smaller and much lighter particles:
$$\begin{aligned}
\frac{\mathrm{d}\vec{r}_i}{\mathrm{d}t} &= \frac{1}{m}\vec{p}_i,\label{eq:SDE_overdamped_pos}\\
\frac{\mathrm{d}\vec{p}_i}{\mathrm{d}t} &= -\nabla_{\vec{r}_i} V(\vec{r}^N,t)-\sum_{j=1}^{N}\boldsymbol{\Gamma}_{ij}(\vec{r}^N)\vec{p}_j + \sum_{j=1}^N\boldsymbol{B}_{ij}(\vec{r}^N)\vec{f}_{j}(t) \label{eq:SDE_overdamped_mom}\end{aligned}$$
where $\vec{r}^N = (\vec{r}_1,\cdots,\vec{r}_N)$, $\boldsymbol{B} = \left( mk_BT\boldsymbol{\Gamma}\right)^{1/2}$, $\boldsymbol{\Gamma} = \gamma (\bm{1} + \tilde{\boldsymbol{\Gamma}})$ (where the tilde denotes the nondimensional tensor and $\bm{1}$ is the $3N\times 3N$ identity matrix), $V$ is a potential, $k_B$, $\mathrm{T}$, $\gamma$ are Boltzmann’s constant, temperature and friction, respectively, and $\vec{f}_i(t) = (\zeta^x_i(t),\zeta^y_i(t),\zeta^z_i(t))^\top$ is a Gaussian white noise term with mean and correlation given by $\langle \zeta_i^a(t)\rangle = 0$ and $\langle\zeta_i^a(t),\zeta_j^b(t) \rangle = 2\delta_{ij}\delta^{ab}\delta(t-t')$.
In $d = 3$ dimensions, the friction tensor $\boldsymbol{\Gamma}$ comprises $N^2$ positive definite $3\times 3$ mobility matrices $\boldsymbol{\Gamma}_{ij}$ for the colloidal particles. These couple the momenta of the colloidal particles to HI forces on the same particles, mediated by fluid flows in the bath. Typically, in the underdamped limit with dense suspensions, the HI may be short range lubrication forces, whereas in disperse | 1 | member_48 |
systems in the overdamped limit, the HI are taken to be the long range forces given by the Rotne-Prager-Yamakawa tensor [@rotne1969variational]. However, we do not make any such assumptions on the form of the tensors here.
We have described a general set of coupled Langevin equations with spatially-dependent friction tensor $\boldsymbol{\Gamma}(\vec{r}^N)$. As we will see, the dynamics – tend towards an equilibrium given by the Gibbs probability measure, which we will show to be independent of the friction tensor. Instead of computing the trajectories of individual particles we consider the evolution of the density of particles $\varrho(\vec{r},t)$ given by the Smoluchowski equation in the high friction limit $\gamma\to \infty$, $$\begin{aligned}
\label{eq:mk-eq}
\qquad \partial_{t}\varrho(\vec{r},t) =-\tfrac{k_B\mathrm{T}}{m\gamma} \nabla_{\vec{r}}\cdot\vec{a}(\vec{r},[\varrho], t) \qquad \text{ for } \vec{r} \in U,\,t\in [0,T]\end{aligned}$$ where $\vec{a}(\vec{r},[\varrho], t)$ is the flux, $[\varrho]$ denotes functional dependence, $U\subseteq\mathbb{R}^d$ and $T<\infty$. Equation was derived rigorously as a solvability condition of the corresponding Vlasov-Fokker-Planck equation for the one-body density in position and momentum space $f(\vec{r},\vec{p},t)$ by writing $f$ as a Hilbert expansion in a small nondimensional parameter $\epsilon\propto\gamma^{-1}$ [@goddard2012overdamped]. Therein, $\epsilon$ has units length, and therefore a problem specific length scale must be introduced to make it truly nondimensional.
We are interested in global existence, | 1 | member_48 |
uniqueness, positivity and regularity of the weak solution to when $\vec{a}(\vec{r},t)$ is given by the integral equation
$$\begin{aligned}
\vec{a}(\vec{r},t) + \boldsymbol{H}[\vec{a},\varrho](\vec{r},t)+\frac{\varrho(\vec{r},t)}{k_BT}\bm{D}(\vec{r},[\varrho],t)\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho](\vec{r},t)
=0,\label{eq:eqn_for_a}\end{aligned}$$
$$\begin{aligned}
\boldsymbol{H}[\vec{a},\varrho](\vec{r},t):=
\varrho(\vec{r},t)\bm{D}(\vec{r},[\varrho],t)\int_U\mathrm{d}\vec{r}'\, g(\vec{r},\vec{r}')\boldsymbol{Z}_2(\vec{r},\vec{r}')\vec{a}(\vec{r}',t),\label{eq:eqn_for_H}\end{aligned}$$
$$\begin{aligned}
&\frac{\varrho(\vec{r},t)}{k_BT}\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho](\vec{r},t) := [\nabla_{\vec{r}}+\tfrac{1}{k_B\mathrm{T}}\Big(\nabla_{\vec{r}}V_1(\vec{r},t)\nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad+\int_U\mathrm{d}\vec{r}'\varrho(\vec{r}',t)g(\vec{r},\vec{r}')\nabla_{\vec{r}}V_{2}(\vec{r},\vec{r}')\Big) ]\varrho(\vec{r},t),\label{eq:eqn_for_J}\end{aligned}$$
where to ease notation we have suppressed $[\varrho]$ in the argument of $\vec{a}$ and $\mathcal{F}$ is the free energy functional which will be defined in Section \[subsec:free\_energy\_framework\]. The functions $V_1$ and $V_2$ are the external and (two body) interparticle potentials respectively. Additionally, the non-constant diffusion tensor $$\begin{aligned}
\label{eq:def_diffusion_tensor}
\boldsymbol{D}(\vec{r},[\varrho],t):=\frac{k_{\text{B} }\mathrm{T}}{m \gamma}\Big[\boldsymbol{1}+\int\mathrm{d}\vec{r}'g(\vec{r},\vec{r}')\varrho(\vec{r}',t)\boldsymbol{Z}_1(\vec{r},\vec{r}')\Big]^{-1}\end{aligned}$$ will be considered; this is interesting from a physical point of view. It has been previously shown (see [@goddard2012overdamped]) that for $\boldsymbol{Z}_1$ being positive definite, $\boldsymbol{D}$ is also positive definite and therefore has positive, finite eigenvalues. The term $g(\vec{r},\vec{r}')$ (regarded as known) is the correlation function defined by the two-body density $\varrho^{(2)}(\vec{r},\vec{r}',t) =g(\vec{r},\vec{r}')\varrho(\vec{r},t)\varrho(\vec{r}',t)$ and the operator $\boldsymbol{H}[\cdot]$ describes terms corresponding to HI.
We note that if $\boldsymbol{D}$ were positive semidefinite, a zero eigenvalue of $\boldsymbol{D}$ is permitted, which physically-speaking would amount to the colloidal system possessing a zero diffusion rate in some subset of $U$ with nonzero measure. Such systems are interesting (for example, in many biological systems the physical domain $U$ could be a substrate including | 1 | member_48 |
cuts, voids or interior walls) but are not considered in this paper. Throughout this work the largest and smallest eigenvalues of $\boldsymbol{D}$ will be denoted $\mu_{\max}$ and $\mu_{\min}$, respectively.
Furthermore, for two-body HI, $\boldsymbol{Z}_1$, $\boldsymbol{Z}_2$ are the diagonal and off-diagonal blocks respectively of the translational component of the grand resistance matrix originating in the classical theory of low Reynolds number hydrodynamics between suspended particles [@happel2012low], [@jeffrey1984calculation], related to the friction tensor by $$\begin{aligned}
\tilde{\boldsymbol{\Gamma}}_{ij}(\vec{r}^N) = \delta_{ij}\sum_{l\neq i}\boldsymbol{Z}_{1}(\vec{r}_i,\vec{r}_l)+(1-\delta_{ij})\boldsymbol{Z}_{2}(\vec{r}_i,\vec{r}_j).\end{aligned}$$
In $d = 3$ dimensions, and for the particular case $N=2$ (where $N$ is the number of particles in the system), $\bm{\Gamma}\in \mathbb{R}^{6\times 6}$ and $\bm{\Gamma}_{ij}$ may be seen as equivalent to the second-rank tensor of the translational part of the resistance matrix as found in [@jeffrey1984calculation] used to model lubrication forces. It should be noted however that the definition of those resistance matrices are formalism dependent, that is, the individual entries are scalar functions arising from the solution of Stokes equations for two-body lubrication interactions using multipole methods. Conversely, $\bm{\Gamma}_{ij}$ are general tensors, independent of the type of HI under consideration, and are therefore a more general representation of hydrodynamic phenomena of colloidal suspensions. Additionally, $\bm{\Gamma}_{ij}$ may be used to model not | 1 | member_48 |
just lubrication forces between particles but also long range forces, wall effects and more. In the case of inter-particle HI, the diagonal blocks $\bm{\Gamma}_{ii}$ each represent the force exerted on the fluid due to the motion of particle $i$, which is simply the sum of all the pairwise HI from the perspective of particle $i$. The off-diagonal blocks $\bm{\Gamma}_{ij}$ represent the force on particle $i$ due to the motion of particle $j$.
The stationary equations for the equilibrium density $\varrho(\vec{r})$ and equilibrium flux $\vec{a}(\vec{r})$ are given by
$$\begin{aligned}
&\qquad\qquad\qquad\qquad\nabla_{\vec{r}}\cdot \vec{a}(\vec{r}) = 0,\label{eq:div_a_0}\\
&\vec{a}(\vec{r}) + \boldsymbol{H}[\vec{a},\varrho](\vec{r})+\frac{\varrho(\vec{r},t)}{k_BT}\bm{D}(\vec{r},[\varrho],t)\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho](\vec{r})
=0.\label{eq:eqn_for_a_in_equilibrium}
\end{aligned}$$
Note that given a finite flux vector $\vec{a}$ solving -, it is not obvious that $\varrho$ is necessarily a minimiser of the free energy $\mathcal{F}-\int_U\mathrm{d}\vec{r}\,\mu_{c}\varrho$ (where $\mu_{c}$ is the chemical potential of the species). However, for the particular choice $\vec{a}\equiv \vec{0}$ (which is a natural and physically realistic solution), $\varrho$ is necessarily a minimiser of $\mathcal{F}-\int_U\mathrm{d}\vec{r}\,\mu_{c}\varrho$, and we will show that under reasonable assumptions these are indeed the only fixed points of the system.
Previous well-posedness studies of similar nonlinear, nonlocal PDEs focused on periodic boundary conditions; see, e.g., [@chazelle2017well; @greg_mckean_vlasov_torus]. In contrast, we are interested in the well-posedness of , - subject | 1 | member_48 |
to no-flux boundary conditions. This choice admits the nontrivial effect of the two body forces generated by the potential $V_2$ interacting with density on the boundary of the physical domain. We also seek to understand the asymptotic stability of stationary states. The motivation for this choice of boundary condition is physical; it corresponds to a closed system of particles in which the particle number is conserved over time. It is clear that most applications of such equations will be in confined systems, rather than a periodic domain and, as such, no-flux boundary conditions are natural. We note that the choice of boundary condition is expected to have significant effects on the dynamics, including the form of the bifurcation diagram.
Free Energy Framework. {#subsec:free_energy_framework}
----------------------
Related to the system -, we define the free energy functional $\mathcal{F}:P^+_{\text{ac}}(U)\to \mathbb{R}$ where $P^+_{\text{ac}}$ is the set of strictly positive definite absolutely continuous probability measures on $U$. We define $$\begin{aligned}
\label{eq:def_of_F}
\mathcal{F}[\varrho]&:=\int_U\mathrm{d}\vec{r}\,\varrho(\vec{r},t)\log\varrho(\vec{r},t)+\int_U\mathrm{d}\vec{r}\,\varrho(\vec{r},t)\,\Big[V_1(\vec{r},t)+\tfrac{1}{2}(gV_2)\star\varrho \Big],\end{aligned}$$ where $\star$ denotes convolution in space. Here we assume the probability measure $\varrho$ has density with respect to the Lebesgue measure. Additionally we define the probability measure on $U$ $$\begin{aligned}
\label{eq:def_of_measure}
\mu(\mathrm{d}\vec{r}) = \mathrm{d}\vec{r}\,Z^{-1}e^{-\tfrac{(V_1+(gV_2)\star\varrho)}{k_BT}}\end{aligned}$$ where $Z = \int_U \mathrm{d}\vec{r}\,e^{-\tfrac{(V_1+(gV_2)\star\varrho)}{k_BT}}$ and $\varrho$ (when | 1 | member_48 |
it exists) satisfies the nonlinear equation $$\varrho = Z^{-1}e^{-\tfrac{(V_1+(gV_2)\star\varrho)}{k_BT}}.$$ The existence of a probability density $\varrho$, and therefore a probability measure $\mu$ in , is obtained by Lemma \[thm:exis\_fix\_point\]. The functional $\mathcal{F}$ gives rise to the density minimising the free energy associated to the system - as $\gamma \to \infty$, which will be shown in Theorem \[thm:association \_of\_free\_energy\].
To make the connection between the free energy functional $\mathcal{F}$ in and the theory of non-uniform classical fluids, one may consider the Helmholtz free energy functional, which is the central energy functional of DFT [@evans1979nature] $$\begin{aligned}
\label{eq:ddft-helmholtz_func}
\mathcal{F}_{H}[\varrho] = \int_U\mathrm{d}\vec{r}\,\varrho(\vec{r},t)V_1(\vec{r},t)+ k_BT\int_U\mathrm{d}\vec{r}\,\varrho(\vec{r},t)[\log(\Lambda^3\varrho(\vec{r},t))-1]+\mathcal{F}_\text{ex}[\varrho]\end{aligned}$$ where $\mathcal{F}_{\text{ex}}$ is the excess over ideal gas term and $\Lambda$ the de Broglie wavelength, which turns out to be superfluous. The term $\mathcal{F}_{\text{ex}}$ is not in general known, the exception being for one dimensional hard rods [@percus1976equilibrium]. Using the free energy functional $\mathcal{F}_H$, the corresponding Euler-Lagrange equation is $$\begin{aligned}
\label{eq:chemical_potential_euler_lagrange_eqn}
\mu_{c}=V_1(\vec{r}) + k_BT [\log (\Lambda^3\varrho(\vec{r}))-1]+\tfrac{\delta\mathcal{F}_{\text{ex}}}{\delta\rho}[\varrho]\end{aligned}$$ where $\mu_{c}$ is the chemical potential which is constant at equilibrium. Note that $\mu_c$ should not be confused with the measure $\mu$ defined in . After taking the gradient of and multiplying by $\varrho$ we obtain $$\begin{aligned}
0=\varrho(\vec{r})\nabla_{\vec{r}}\tfrac{\delta\mathcal{F}}{\delta\rho}[\varrho]=k_BT\nabla_{\vec{r}}\varrho+\varrho(\vec{r})\nabla_{\vec{r}}\Big(V_1(\vec{r})+\tfrac{\delta\mathcal{F}_{\text{ex}}}{\delta\rho}[\varrho]\Big).\end{aligned}$$ At equilibrium, the sum rule holds (see, | 1 | member_48 |
e.g. [@goddard2012unification]) $$\begin{aligned}
\label{eq:excess_free_energy_equilibrium_sum}
\varrho(\vec{r})\nabla_{\vec{r}}\tfrac{\delta\mathcal{F}_{\text{ex}}}{\delta\varrho}[\varrho]=\sum_{n=2}^N\int\mathrm{d}\vec{r}^{n}\nabla_{\vec{r}}V_n(\vec{r}^n)\varrho_n(\vec{r}^n).\end{aligned}$$ where $\varrho_n(\vec{r}^n)$ is the standard $n-$particle configuration distribution function in equilibrium. Limiting the particle interactions to two-body, for example with the approximation $\varrho_2(\vec{r},\vec{r}') = \varrho(\vec{r})\varrho(\vec{r}')g(\vec{r},\vec{r}',[\varrho])$, we take the first term in the above series to obtain the equality $\nabla_{\vec{r}}\mathcal{F}_H[\varrho]= \nabla_{\vec{r}}\mathcal{F}[\varrho]$. In this way wee see that the density minimising $\mathcal{F}_H$ will minimise $\mathcal{F}$.
When $\boldsymbol{Z}_2\equiv 0 $, and by using the adiabatic approximation that holds out of equilibrium, we note that PDE simplifies to (cf. [@rex2009dynamical]) $$\begin{aligned}
\label{eq:ddft-eq}
\partial_{t}\varrho = \nabla_{\vec{r}}\cdot \left[\boldsymbol{D}(\vec{r},t)\varrho(\vec{r},t)\,\nabla_{\vec{r}}\tfrac{\delta \mathcal{F}}{\delta \varrho}[\varrho]\right].\end{aligned}$$ From we conclude that the dynamics under the choice $\bm{Z}_2 \equiv 0$ has a gradient flow structure. When $\bm{Z}_2$ is not necessarily zero, one cannot in general write the full dynamics as a gradient flow and, hence, the inclusion of HI introduces a novel perturbation away from the classical theory of gradient flow structure. Additionally, one sees how the free energy functional gives rise to the concept of a local pressure variation by the term inside the divergence of . In particular, the term $\tfrac{k_\text{B}\mathrm{T}}{m}\varrho(\vec{r},t)\,\nabla_{\vec{r}}\tfrac{\delta \mathcal{F}}{\delta \varrho}[\varrho]$ represents the spatial variation of the energy available to change particle configurations per unit volume at fixed particle number, in other words, it is | 1 | member_48 |
an analogue of a local pressure gradient for the particle density. We will show that $\mathcal{F}[\varrho]$ is associated to the PDE even when $\bm{Z}_2 \neq 0$, that is $\partial_t \varrho = 0$ implies $\varrho$ is a critical point of $\mathcal{F}$.
[0.48]{} ![(a). The bifurcation diagram for (a). $V_2(x,y) = x\cdot y$ and (b). $V_2(x,y) = -\cos\left(\frac{2\pi(x-y)}{\mathrm{L}}\right)$ in Section \[subsec:numericalexperiments\]: the solid [blue]{} line denotes the stable branch of solutions while the dotted [red]{} line denotes the unstable branch of solutions. In (a) the stationary density $e^{-x^2}/Z$ changes stability at the critical interaction energy $\kappa_2 = \kappa_{2\sharp}= -2.4$ and the new stable density is asymmetric adhering to one wall (Figure \[fig:bif\_fig\_right\]). In (b), in the absence of a confining potential, the uniform density becomes unstable at the critical interaction energy $\kappa_2 = \kappa_{2\sharp} = 0.4$ and the density may become multi-modal (Figure \[fig:bif\_stable\_equilibria\_V2\_cosine\]).[]{data-label="fig:bifurcation_diagram"}](bifurcation_diagram_for_Phi2.pdf "fig:"){width="\textwidth"}
[0.48]{} ![(a). The bifurcation diagram for (a). $V_2(x,y) = x\cdot y$ and (b). $V_2(x,y) = -\cos\left(\frac{2\pi(x-y)}{\mathrm{L}}\right)$ in Section \[subsec:numericalexperiments\]: the solid [blue]{} line denotes the stable branch of solutions while the dotted [red]{} line denotes the unstable branch of solutions. In (a) the stationary density $e^{-x^2}/Z$ changes stability at the critical interaction energy $\kappa_2 = \kappa_{2\sharp}= -2.4$ | 1 | member_48 |
and the new stable density is asymmetric adhering to one wall (Figure \[fig:bif\_fig\_right\]). In (b), in the absence of a confining potential, the uniform density becomes unstable at the critical interaction energy $\kappa_2 = \kappa_{2\sharp} = 0.4$ and the density may become multi-modal (Figure \[fig:bif\_stable\_equilibria\_V2\_cosine\]).[]{data-label="fig:bifurcation_diagram"}](bifurcation_diagram_for_Phi2_cosine.pdf "fig:"){width="\textwidth"}
Description of Main Results and Organisation of the Paper.
----------------------------------------------------------
### Main Results {#main-results .unnumbered}
The main results of this work are threefold.
1. We establish existence and uniqueness of weak solutions to DDFTs including two-body HI governed by equations , - with no-flux boundary conditions.
2. We derive *a priori* convergence estimates of the density $\varrho(\vec{r},t)$ to equilibrium in $L^2$ and relative entropy.
3. We study the stability of equilibrium states and construct bifurcation diagrams for two numerical applications.
These results are of particular interest for physical applications of colloidal systems where conservation of mass is either a desirable or necessary property of the system. Additionally, the stability theorem contrasts with simpler linear stability analyses of similar systems of gradient flow structure with periodic boundary conditions [@martzel2001mean], [@greg_mckean_vlasov_torus] which may be tackled by means of Fourier analysis.
### Organisation of the Paper {#organisation-of-the-paper .unnumbered}
The paper is organised as follows: in Section \[sec:preliminaries\] | 1 | member_48 |
we present the boundary and initial conditions, introduce the main notation, nondimensionalise the main equations, state the stationary equation for the density, define the weak formulation of the Smoluchowski equation including full HI and provide a list of assumptions. In Section \[sec:statement\_of\_main\_results\] we state the main results of the present work in a precise manner. In Section \[sec:ex\_uni\_full\_HI\] we provide an existence and uniqueness theorem for the flux $\vec{a}$ when full HI are included. In Section \[sec:char\_stationary\_sol\] we characterise solutions of the stationary problem and convergence to equilibrium in $L^2$ as $t \to \infty$. In Section \[sec:global\_asymptotic\_stability\] we obtain results on the global asymptotic stability of the stationary densities by showing that the free energy is a continuous functional for all two-body interaction strengths. Additionally we prove an H- theorem for the equilibria, provide *a priori* convergence estimates in relative entropy, derive an asymptotic expansion of the equilibria for small interaction energy and perform a spectral analysis of the linearised nonlocal Smoluchowski operator. In Section \[sec:bifurcation\_theory\] we provide necessary and sufficient conditions for phase transitions in generalised DDFT-like systems with no-flux boundary conditions. In Section \[sec:manufactured\_bif\] we construct the bifurcation diagram for some example problems. In Section \[sec:existence\_uniqueness\_with\_partial\_HI\] we obtain | 1 | member_48 |
an existence and uniqueness theorem for the Smoluchowski equation with non-constant diffusion tensor and effective drift vector dependent on the two-body HI tensors $\bm{Z}_1$ and $\bm{Z}_2$. In Section \[sec:discussion\] we present our concluding remarks and state some open problems. In Appendix \[sec:classical\_paraboliv\_pde\] we provide some technical results that are used in the proof of Theorem \[thm:exis\_uniq\_weak\_sol\_rho\]. Finally in Appendix \[app:nomenclature\] we provide a list of nomenclature.
Preliminaries {#sec:preliminaries}
=============
In this section we specify the nonlinear boundary conditions and initial data for the DDFT . We also nondimensionalise the governing equations and provide the assumptions on the regularity of the potentials, correlation function, diffusion tensor and initial data.
Boundary Conditions. {#subsec:boundary_conditions}
--------------------
When $U = \mathbb{R}^d$ we take $$\begin{aligned}
\label{bc:density_and_flux_decaying}
\left\{\begin{aligned}
\varrho(\vec{r},t)\to 0 \\
\vec{a}(\vec{r},t)\to \vec{0}
\end{aligned}\right. \quad \text{ as } \quad |\vec{r}|\to \infty,\end{aligned}$$ where we require $V_1$ to be growing at least quadratically as $\vec{r}\to \infty$. Physically-speaking this prevents the density from running out to infinity. When $U\subset \mathbb{R}^d$ is open and bounded we impose that the total mass of the system $M$ remains constant, in particular we have $$\begin{aligned}
\label{bc:mass_preserving}
\vec{a}(\vec{r},t)\cdot\vec{n}\bigg|_{\partial U\times [0,T]} = 0.\end{aligned}$$ The boundary condition may be viewed as a [*nonlinear*]{} Robin condition imposing the | 1 | member_48 |
flux through the boundary $\partial U$ is zero for all time $t\in [0,T]$. If $\varrho$ is a number density then $\int \mathrm{d}\vec{r}\, \varrho = N$ for all time, however for the analysis in Section \[sec:ex\_uni\_full\_HI\] and onwards we will assume $\varrho$ is a probability density so that $\int \mathrm{d}\vec{r}\, \varrho = 1$. The rescaling between number and probability densities is discussed in the following section.
Initial Conditions.
-------------------
We will assume that the initial data has finite free energy and is consistent with the imposed boundary conditions. For example, one could prescribe initial data $(\varrho_0,\vec{a}_0)^\top$ such that $$\begin{aligned}
\label{eq:initial_data_for_a_and_rho}
\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho_0](\vec{r}) = \mu_{c},\qquad
\vec{a}_0 = \vec{0}.\end{aligned}$$ where $\mu_{c}$ is the chemical potential, constant at equilibrium. It is straightforward to check that $(\varrho_0,\vec{a}_0)^\top$ is an equilibrium point of the system . Commonly, one then drives the system out of equilibrium via a time-dependent external potential. In principle $\mu_{c}$ may be given and the equations , are well defined. In practice, for complicated particle configurations, $\mu_{c}$ is not known but can be computed by minimising the free energy along with the additional constraint $\int_U\mathrm{d}\vec{r}\, \varrho_0(\vec{r})=N$, where $N$ is the (expected) number of particles for a finite system and $\varrho_0$ is a number density. | 1 | member_48 |
Note that $\mu_{c}$ is a potential, so by raising it one may force more particles into the system. We will assume that $\mu_c$ is constant to fix the number of particles. To ensure $\varrho$ (and $\varrho_0$) is a probability density one may rescale $\varrho/N = \tilde{\varrho}$, $N g = \tilde{g}$ and $\vec{a}/N^2 = \tilde{\vec{a}}$, where the tilde denotes the new variable, so that $\int_U\mathrm{d}\vec{r}\, \varrho_0(\vec{r})=1$ and equations - become independent of $N$.
This provides a method of converting back to the number density which is typically used in numerical modelling of finite colloidal systems [@goddard2016dynamical], [@goddard2012unification], [@goddard2012general]. Throughout however, since we will frequently use the integral of the density, we will assume $\varrho$ and $\varrho_0$ are probability densities to ease notation. With this, one has three equations for three unknowns $\mu_{c}$, $\varrho_0$, $\vec{a}_0$ and the initial density $\varrho_0$ can be computed. For the rest of paper it is convenient to work in dimensionless units. We now nondimensionalise the governing equations.
Evolution Equations.
--------------------
We now nondimensionalise our equations. Let $\mathrm{L}$, $\tau$, $\text{U}$ be characteristic length, time and velocity scales respectively, then by nondimensionalising $$\begin{aligned}
\vec{r}\sim \mathrm{L} \tilde{\vec{r}},\quad t\sim\tau\tilde{t}, \quad \mathrm{U} = \tfrac{\mathrm{L}}{\tau}, \quad \varrho\sim\tfrac{1}{\mathrm{L}^d}\tilde{\varrho},\quad \mathcal{F}\sim k_BT \tilde{\mathcal{F}},\quad \vec{a}\sim \mathrm{A}\tilde{ \vec{a}}.\end{aligned}$$ | 1 | member_48 |
where $d$ is the physical dimension and $\mathrm{A}$ is a characteristic flux scale. The system becomes (after dropping tildes) $$\begin{aligned}
\partial_t\varrho(\vec{r},t) = -\tfrac{1}{Fr}\times \tfrac{\tau^{-1}}{\gamma}\times \mathrm{A}\times \mathrm{L}^{d+1} \nabla_{\vec{r}}\cdot \vec{a}(\vec{r},t),\end{aligned}$$ where we have defined the Froude number $Fr = m\mathrm{U}^2/(k_BT)$. By choosing $Fr = 1$, $\tau = \gamma^{-1}$ and $\mathrm{A} = 1/\mathrm{L}^d$ we simplify the system of equations to the following boundary value problem.\
\[prop:non\_dimensional\_time\_evolving\_flux\_eqn\]
The non-dimensional one-body density $\varrho(\vec{r},t)$ and flux $\vec{a}(\vec{r},t)$ evolve according the the boundary value problem $$\begin{aligned}
\label{eq:evolution_eqn_for_a_dimensionless}
\begin{cases}
&\qquad\qquad\partial_t\varrho = -\nabla_{\vec{r}}\cdot \vec{a}(\vec{r},t), \\
&\vec{a}(\vec{r},t) + \bm{H}[\vec{a},\varrho] +\varrho(\vec{r},t)\bm{D}(\vec{r},[\varrho],t)\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho]=0,
\\
&\qquad [\bm{H}[\vec{a},\varrho] +\varrho(\vec{r},t)\bm{D}(\vec{r},[\varrho],t)\nabla_{\vec{r}}\frac{\delta\mathcal{F}}{\delta\varrho}[\varrho]]\cdot\vec{n}\big|_{\partial U}=0.
\end{cases}\end{aligned}$$
We note that when the off-diagonal HI tensor $\bm{Z}_{2}=0$, by using the definitions of $\mathcal{F}$ and $\bm{D}$ , the evolution equations in may be written as a nonlinear Smoluchowski equation (such as ) with non-constant diffusion coefficient. However we observe that even when $\bm{Z}_2 \neq 0$ the dynamics may be recast into a Smoluchowski equation for $\varrho$ under an effective drift vector dependent on $\bm{Z}_2$.\
\[prop:non\_dimensional\_time\_evolving\_rho\_eqn\]
The non-dimensional one-body density $\varrho(\vec{r},t)$ evolves according the the boundary value problem $$\begin{aligned}
\label{eq:evolution_eqn_for_rho_dimensionless}
\begin{cases}
&\qquad\partial_t\varrho=\,\nabla_{\vec{r}}\cdot\left[Pe^{-1}\bm{D}\nabla_{\vec{r}}\varrho+\varrho\,\bm{D}\left(\nabla_{\vec{r}} (\kappa _1V_1+\kappa _2\,(gV_2)\star\varrho) + \bm{A}[\bm{a}]\right)\right],
\\
&\qquad \qquad \qquad \qquad \qquad \Pi [\varrho]\cdot\vec{n}\big|_{\partial U} = 0,\\
&\qquad\Pi[\varrho]:= \bm{D}\,\left(\nabla_{\vec{r}}\varrho +
\varrho\, \nabla_{\vec{r}}(\kappa_1 V_1(\vec{r},t)+ | 1 | member_48 |
\kappa_2 (gV_2)\star \varrho)+\bm{A}[\bm{a}]\right),
\end{cases}\end{aligned}$$ where $\bm{A}[\bm{a}]$ is an effective background flow induced by the hydrodynamic interactions defined by $$\begin{aligned}
\label{eq:def_of_V1_eff}
\bm{A}[\vec{a}]:=\int_U\mathrm{d}\vec{r}'\, g(\vec{r},\vec{r}') \bm{Z}_2(\vec{r},\vec{r}')\vec{a}(\vec{r'},t),\end{aligned}$$ $\kappa_1$, $\kappa_2$ are non-dimensional constants measuring the strength of confining and interaction potentials respectively, $Pe=\mathrm{L}\text{U}/\alpha$ is the P[é]{}clet number measuring the ratio of advection rates to diffusive rates and $\alpha = k_B\mathrm{T}/(m\gamma)$.
Corollary \[prop:non\_dimensional\_time\_evolving\_rho\_eqn\] is the general formulation of the nondimensional equations - when $\bm{Z}_2\neq0$, including a non-constant diffusion coefficient and an effective drift. Throughout this paper, to study the intermediate regime of equally strong advection and diffusion, we set $Pe = 1$. Additionally, we redefine the two-body potential to absorb the correlation function $g$ to ease notation, $V_2(\vec{r},\vec{r}'):= g(\vec{r},\vec{r}') V_2(\vec{r},\vec{r}')$. In practice, there are many choices for $g$, for example the hard sphere approximation takes $g(|\vec{r}- \vec{r}'|) = 0$ for $|\vec{r}-\vec{r}'| < 1$ and unity otherwise. Alternatively $g$ may be obtained numerically from microscopic dynamics. We consolidate the choices for equations , in Section \[subsec:assumptions\_definitions\].
The effective drift $\bm{A}[\vec{a}]$, dependent on $\bm{Z}_2$ and $\vec{a}$ may be determined once $\vec{a}(\vec{r},t)$ is solved from the second equation in . Note that the evolution equation in may be viewed as a generalised McKean-Vlasov equation with a non-constant diffusion tensor | 1 | member_48 |
and confining potential. In particular the McKean-Vlasov equation may be recovered in the special case $\bm{Z}_1 = \bm{Z}_2 = V_1 = 0$, see for example [@greg_mckean_vlasov_torus], [@chayes2010mckean]. We will use Corollary \[prop:non\_dimensional\_time\_evolving\_rho\_eqn\], to write the full dynamics including full HI, to obtain our results on weak solutions for $\varrho(\vec{r},t)$ (see Theorem \[thm:eigenfn\_expansion\_of\_flux\], Section \[sec:ex\_uni\_full\_HI\] and Theorem \[thm:existence\_and\_uniqueness\], Section \[sec:existence\_uniqueness\_with\_partial\_HI\]). We continue to the next section by stating the stationary boundary value problem for equilibrium states $\varrho(\vec{r})$.
Stationary Equations.
---------------------
For general $\bm{Z}_2$ we will show in Theorem \[thm:association \_of\_free\_energy\] that the stationary density $\varrho(\vec{r})$ satisfies $$\begin{aligned}
\label{eq:stationary_eqn_for_rho_dimensionless}
\begin{cases}
&\qquad 0=\,\nabla_{\vec{r}}\cdot[\bm{D}\nabla_{\vec{r}}\varrho+\varrho\,\bm{D}\nabla_{\vec{r}}(\kappa _1V_1+\kappa _2\,V_2\star\varrho)],
\\
&\qquad \qquad \qquad \qquad \qquad \Pi [\varrho]\cdot\vec{n}\big|_{\partial U} = 0,\\
&\qquad\Pi[\varrho]:= \bm{D}\,(\nabla_{\vec{r}}\varrho +
\varrho\, \nabla_{\vec{r}}(\kappa_1 V_1(\vec{r},t)+ \kappa_2 \,V_2\star \varrho)).
\end{cases}\end{aligned}$$ We now discuss regularity on the potentials and diffusion tensor.
Assumptions & Definitions. {#subsec:assumptions_definitions}
--------------------------
Typically for long range HI the $\boldsymbol{Z}_i$ exhibit singularities at the origin (particle centres) so the correlation function $g$ is a necessary inclusion and provides a way of smoothing $\boldsymbol{D}$ and we assume $g \in L^\infty(U)$. For $\varrho \geq 0$ the diffusion tensor $\boldsymbol{D}$ as a convolution with the density will then be a weakly differentiable function. For the existence and uniqueness | 1 | member_48 |
theory in Appendix \[sec:classical\_paraboliv\_pde\] and Section \[sec:existence\_uniqueness\_with\_partial\_HI\] we require that first derivatives of $\bm{D}_{ij}$ to be bounded in $L^\infty(U)$ so that all coeeffiecients of the PDE are uniformly bounded.
Out of equilibirum, we will suppress the time dependence on $\bm{D}$, $V_1$ simply to ease notation. However at equilibrium $\bm{D}$, $V_1$ are assumed to be independent of time, indeed in order for equilibrium states of the density and flux to be well defined. We note that is $\bm{D}$ positive definite and symmetric, as it has been rigorously shown to be [@goddard2012overdamped]. In summary we have the following notational choices and assumptions for the evolution problem .
#### **[Notation]{}**
Throughout we ease notation on the two-body interaction potential.
- The two-body interaction potential is redefined to absorb the correlation function $g$ $$\begin{aligned}
\label{ass:V2_redef}
V_2\stackrel{\text{redef}}{:=} g V_2. \tag{N1}\end{aligned}$$
For the dynamics we assume:
#### **[Assumptions D]{}**
- The diffusion tensor $\bm{D}$ is symmetric, positive definite, and the first derivatives of $\bm{D}_{ij}$ are bounded in $L^\infty(U)$ $$\begin{aligned}
\label{ass:D_pos_def_weak_diffable}
\bm{D}_{ij}\in W^{1,\infty}(U). \tag{D1}\end{aligned}$$
- The diagonal and off-diagonal blocks of the HI tensors are uniformly bounded in the sense $$\begin{aligned}
\label{ass:Z2_uniformly_bd}
\|gZ_2\|_{L^\infty(U)}<\infty, \quad \|gZ_1\|_{L^\infty(U)}<\infty\tag{D2}\end{aligned}$$
- The initial data $\varrho_0$ is a non-negative, square-integrable, absolutely continuous probability | 1 | member_48 |
---
abstract: 'We present a sequence of high resolution (R$\sim$20,000 or 15 km s$^{-1}$) infrared spectra of stars and brown dwarfs spanning spectral types M2.5 to T6. Observations of 16 objects were obtained using eight echelle orders to cover part of the $J$-band from 1.165-1.323 $\mu$m with NIRSPEC on the Keck II telescope. By comparing opacity plots and line lists, over 200 weak features in the $J$-band are identified with either FeH or H$_{2}$O transitions. Absorption by FeH attains maximum strength in the mid-L dwarfs, while H$_{2}$O absorption becomes systematically stronger towards later spectral types. Narrow resolved features broaden markedly after the M to L transition. Our high resolution spectra also reveal that the disappearance of neutral Al lines at the boundary between M and L dwarfs is remarkably abrupt, presumably because of the formation of grains. Neutral Fe lines can be traced to mid-L dwarfs before Fe is removed by condensation. The neutral potassium (K I) doublets that dominate the $J$-band have pressure broadened wings that continue to broaden from $\sim$ 50 km s$^{-1}$ (FWHM) at mid-M to $\sim$ 500 km s$^{-1}$ at mid-T. In contrast however, the measured pseudo-equivalent widths of these same lines reach a maximum in | 1 | member_49 |
the mid-L dwarfs. The young L2 dwarf, G196-3B, exhibits narrow potassium lines without extensive pressure-broadened wings, indicative of a lower gravity atmosphere. Kelu-1AB, another L2, has exceptionally broad infrared lines, including FeH and H$_{2}$O features, confirming its status as a rapid rotator. In contrast to other late T objects, the peculiar T6 dwarf 2MASS 0937+29 displays a complete absence of potassium even at high resolution, which may be a metallicity effect or a result of a cooler, higher-gravity atmosphere.'
author:
- '[IAN S. MCLEAN, L. PRATO, MARK R. MCGOVERN, ADAM J. BURGASSER, J. DAVY KIRKPATRICK, EMILY L. RICE AND SUNGSOO S. KIM]{}'
title: 'THE NIRSPEC BROWN DWARF SPECTROSCOPIC SURVEY II: HIGH-RESOLUTION J-BAND SPECTRA OF M, L and T DWARFS[^1]'
---
Introduction
============
With effective temperatures $\la$ 2200 K, the cool atmospheres of L and T dwarfs generate complex spectra that are rich in molecular features, especially at near-infrared (NIR) wavelengths where ro-vibrational transitions of many molecules dominate. Fortunately, these cool, very low luminosity objects are also brightest in the NIR. Until recently, most infrared spectroscopic investigations of L and T dwarfs have concentrated on the identification of strong, broad spectral features, useful for the establishment of spectral classification, and have | 1 | member_49 |
employed resolving powers of $R = \lambda/\Delta\lambda \la 2000$ (Burgasser et al. 2002, 2004, 2006; Cushing et al. 2003, 2005; Geballe et al. 1996, 2002; Jones et al. 1994; Leggett et al. 2000, 2001; McLean et al. 2000a, 2001, 2003a; Reid et al. 2000; Testi et al. 2001). Observations with significantly higher spectral resolution are potentially very important because line-blending from molecular transitions is reduced and weak features are resolved. Higher resolution spectra are more useful for constraining models of the complex molecular chemistry of brown dwarf atmospheres and for characterizing properties such as gravity and metallicity (Mohanty et al. 2004). For example, less massive brown dwarfs and younger brown dwarfs have smaller surface gravities which results in less pressure broadening and a different line shape. Furthermore, spectra with $R\ga$ 20,000 ($\la$15 km s$^{-1}$) are required for the measurement of radial and rotational velocities, and to search for radial velocity variability associated with brown dwarf spectroscopic binaries.
Obtaining high signal-to-noise observations with an increase in spectral resolution of a factor of ten is difficult because brown dwarfs are so faint. Basri et al. (2000) successfully resolved the resonance absorption lines of Cs and Rb in the far-red visible regime for | 1 | member_49 |
a sample of M and L dwarfs using the HIRES echelle spectrograph on the Keck 10-m telescope and derived effective temperatures through comparison with available model atmospheres. Reid et al. (2002) also used high-resolution optical echelle spectroscopy to study 39 dwarfs with spectral types between M6.5 and L0.5. However, because brown dwarf fluxes are significantly less in visible light, high-resolution observations of fainter L dwarfs and of the even dimmer T dwarfs are not tenable in the optical and require infrared observations. With the advent and development of sensitive large-format IR array detectors, IR spectroscopy with the requisite spectral resolution is now possible (McLean et al. 1998, 2000b).
In this paper we present the first well-sampled spectral sequence of late M, L and T dwarfs observed at high resolution (R $\sim$ 20,000) in the NIR. This work is part of the NIRSPEC Brown Dwarf Spectroscopic Survey (BDSS) being carried out at the Keck Observatory; preliminary results were presented in McLean et al. (2003b). The goals of the BDSS are to obtain a significant sample of NIR spectra of low-mass stars and brown dwarfs of differing ages, surface gravities, and metallicities at both medium (R $\sim$ 2,000) and high spectral resolution | 1 | member_49 |
for spectral classification studies and comparisons with model atmospheres. McLean et al. (2003a), hereafter M03, describes the lower resolution part of the survey; spectra from that study are available online.[^2] Here, we investigate the $J$-band using ten times higher spectral resolution than in M03. The $J$-band (defined as 1.15-1.36 $\mu$m in this paper) is important because this region contains four strong lines of neutral potassium (K I) that are both temperature and gravity-sensitive, and which persist throughout the M, L, and T dwarf sequence. In §2 we describe our observations and data reduction procedures. §3 provides a discussion of the rich, spectral morphology. In addition to atomic K I, there are lines of Al I, Fe I, Mn I, Na I and Ti I, and transitions of molecular species such as CrH, FeH, and H$_{2}$O that can provide a unique resource for improving model atmospheres at these low temperatures. We show that the sudden disappearance of the Al I lines critically defines the M-L boundary at these resolutions. Concentrating on the strongly pressure-broadened K I lines, we look for correlations between spectral type and equivalent widths, velocity widths (FWHM), and residual intensity. The relation between molecular line strengths and spectral | 1 | member_49 |
type is also investigated. The effects of rotation, surface gravity and metallicity are explored in §4. A summary of the overall results and concluding remarks is given in §5.
Observations and Data Reduction
===============================
Targets and Instrumentation
---------------------------
Targets for the initial survey, the BDSS (M03), were selected primarily from well-known M dwarfs and from L and T dwarfs identified in the Two Micron All Sky Survey (2MASS; Kirkpatrick et al. 1997, 1999, 2000, 2001; Burgasser et al. 2000, 2002; Reid et al. 2000; Wilson et al. 2001), augmented with discoveries from the Deep Near-Infrared Survey of the Southern Sky (DENIS; Delfosse et al. 1997, 1999), the Sloan Digital Sky Survey (SDSS; Leggett et al. 2000; Geballe et al. 2002), and other investigations (Becklin & Zuckerman 1988; Ruiz, Leggett and Allard 1997). To ensure high signal-to-noise spectra for the high-resolution part of the survey, a subset of 12 of the brightest objects ($J=7-15$), spanning the spectral type range from M6 to T6, was selected. Of these, only 2MASS 0140+27 was not part of the initial survey. The M2.5 star G196-3A was also observed along with its L2 companion G196-3B, both examples of objects substantially younger than 1 Gyr (Rebolo et | 1 | member_49 |
al. 1998) and therefore most likely to exhibit gravity effects (McGovern et al. 2004). In addition, the peculiar T6 dwarf 2MASS 0937+29 (Burgasser et al. 2002) was added to the list because of its apparent lack of K I features in lower resolution spectra. Another late T dwarf (2MASS 2356$-$15, T5.5) was observed after completion of the initial set for comparison to the 2MASS 0937+29. Although the signal-to-noise ratio was sufficient to establish the presence of K I absorption in this T5.5 dwarf, the fainter magnitude ($J=15.8$) and stronger 2o absorption made quantitative analysis too difficult so the spectrum is not shown.
Table 1 provides the complete list of 16 targets and the observing log. Shorthand names such as 2MASS 1507$-$16 are used in the text for simplicity, but the full designations are listed in Table 1. Two targets were known visual doubles at the time of observing, 2MASS 0746+20 (L0.5) and DENIS 0205$-$11 (L7), but in neither case did we have sufficient angular resolution to separate the components. Subsequent to making our observations, DENIS 0205-11 was reported as a possible triple brown dwarf system (Bouy et al. 2005) based on Hubble Telescope images. Burgasser et al. (2005) subsequently found | 1 | member_49 |
SDSS 0423$-$04 to be double, the average spectral type of T0 being due to an L6 and T2 combination. Even more recently, the binary nature of Kelu-1, a 0$\farcs$29 pair, was revealed using Laser Guide Star adaptive optics on the Keck telescope (Liu & Leggett 2005; Gelino et al. 2006). Again, in neither case were these targets resolved in our NIRSPEC observations.
All of the observations were made using the NIRSPEC cryogenic spectrometer on the Keck II 10-m telescope on Mauna Kea, Hawaii. Detailed descriptions of the design and performance of this UCLA-built instrument are given elsewhere (McLean et al. 1998; 2000b). For this study, NIRSPEC was used in its cross-dispersed echelle mode. High resolution spectra are dispersed across the 1024$\times$1024 InSb detector at 0$\farcs$143 per pixel while the spatial scale in the cross-dispersed direction is 0$\farcs$19 per pixel. An independent slit-viewing camera with a scale of 0$\farcs$18 per pixel is available for centering and guiding. With the gratings used in NIRSPEC, the relationship between the blaze wavelength ($\lambda_{b}$) and echelle order number (m) is $m\lambda_{b}$ = 76.56 $\mu$m; together with the free spectral range (see below), this equation gives the order location of a given wavelength. The spectrometer was | 1 | member_49 |
set up with the NIRSPEC-3 order-sorting filter and specific echelle and cross-dispersion grating angles to record 11 echelle orders ($m$ = 66 to $m$ = 56) covering the wavelength range from 1.15–1.36$\mu$m, corresponding approximately to the standard $J$-band. The free spectral range ($\lambda_{b}/m$) at 1.255 $\mu$m (order 61) is $\sim$206 Å, but the effective dispersion is 0.179 Å/pixel, allowing for only 183 Å (89%) of this order to be captured by the detector. In fact, the captured wavelength range varies from 171 Å (94%) in order 65 to 192 Å (84%) in order 58. Thus, because the spectral interval captured by the detector is slightly smaller than the free spectral range in each order, there are small gaps, increasing with wavelength, in the total spectral coverage. Table 2 summarizes the spectral range for each order used in the subsequent analysis. In practice, for an entrance slit 0$\farcs$43 (3 pixels) wide, the final spectral resolution in the reduced data is $R \sim 20,000$, (or 15 km s$^{-1}$), compared to the theoretical value of $R = 24,000$. The average value of one spectral resolution element is $\sim$0.625Å (equivalent to 3.6 pixels) over most of the $J$-band region.
Spectroscopic observations were made as | 1 | member_49 |
nodded pairs. Typically, integrations of 600 s each were taken with the object placed at two positions, designated A and B, separated by $\sim$7$''$ on the $\sim$12$''$ long entrance slit of NIRSPEC. Shorter exposure times were used for brighter objects. Exposures of 300 s per nod position were used for 2MASS 0746+2000AB, Kelu-1AB and 2MASS 1507-1627, 120 s for Wolf 359 and 60 s per nod position for G196-3A. Total integration times per object ranged from a few minutes to 1.5 hours depending on the apparent $J$ magnitude. Signal-to-noise ratios were typically greater than 20 (5%) per resolution element over most orders, and sometimes greater than 100 (1% noise). Seeing conditions were $\sim0\farcs5-0\farcs6$ and therefore a slit width of 0$\farcs$43 (3 pixels) was used for all observations, except in the case of 2MASS 1507$-$16, for which we used a $0\farcs576$ (4 pixels) slit because of poorer seeing. A0 V stars were observed at an airmass very close to that of the target object to calibrate for absorption features caused by the Earth’s atmosphere. Arc lamp spectra, taken immediately after each observation, and OH night sky lines in the observed spectra, were used for wavelength and dispersion calibration. A white-light spectrum | 1 | member_49 |
and a corresponding dark frame were obtained for flat-fielding.
Data Reduction Methods
----------------------
For the data reduction we used REDSPEC, an IDL-based software package developed at UCLA for NIRSPEC by S. Kim, L. Prato, and I. McLean[^3]. For each echelle order, REDSPEC uses the position of the two-dimensional spectra on the NIRSPEC array and the calibration line spectra to construct spatial and spectral maps necessary to transform the raw data onto a uniform grid. If the target spectrum itself is too faint to provide the spatial rectification, then the A0V star observed with the same set up was used instead. Although four arc lamps are available, it is often the case that there are too few well-distributed lines per echelle order for good spectral rectification. Consequently, OH night sky lines were also used. The dispersion was more than adequately fit by a second order polynomial of the form $\lambda = c_{0} + c_{1} x + c_{2} x^{2}$ where $c_{1} \sim$ 0.17$\pm$0.01 Å/pixel and $c_{2} \sim$ 7 x 10$^{-6}$ Å/pixel$^2$. To extract spectra free from atmospheric background and uneven detector response, the difference of an A/B image pair was formed and flat-fielded. The flat-fielded difference frame was then rectified using the | 1 | member_49 |
spatial and spectral maps and the raw spectrum produced by summing 5$-$10 rows from each trace in the rectified image. The extracted traces (one positive, one negative) are subtracted again to produce a positive spectrum with residual night sky emission line features removed, unless a line was saturated. In the $J$-band, none of the night sky emission lines are saturated. A0 V star spectra were reduced in the same way, interpolating over the intrinsic Pa$\beta$ hydrogen absorption line at 1.28$\mu$m in the $J$-band spectra. The raw target spectrum was then divided by the raw A0 V star spectrum to remove telluric features. The true slope of the target spectrum was restored by multiplication with a blackbody spectrum of T$_{eff}=9500$ K for an A0 V star (Tokunaga 2000). Finally, the spectra reduced from multiple A/B pairs were averaged together to improve the signal-to-noise ratio.
J-band Spectral morphology at R$\sim$20,000
===========================================
Overview
--------
For each of the 16 targets we have extracted 8 echelle orders (see Table 2) yielding a total of 128 spectra. Before examining and interpreting the new spectra in detail, it is very useful to have a broad overview of the basic spectral features present and an awareness of | 1 | member_49 |
the general trends that occur in the high-resolution $J$-band data as a function of spectral type. A convenient way of doing this is to select a representative source for a few spectral types and present all eight echelle orders on the same plot, thus enabling the entire $J$-band to be viewed at a glance. Figures 1$-$6 show the reduced spectra of Wolf 359 (M6), 2MASS 0140+27 (M9), 2MASS 0345+25 (L0), 2MASS 1507$-$16 (L5), SDSS 0423$-$04AB (T0) and 2MASS 0559$-$14 (T4.5). The double nature of SDSS 0423$-$04AB means that we do not have a true T0 spectrum, but lower resolution studies (M03) show that J-band spectral variations are relatively weak from L6 to T2 and therefore this binary remains a useful proxy for a T0 dwarf. In these plots, echelle orders 58 through 65 are shown together; the remaining orders at the edges of the $J$-band are too contaminated by strong atmospheric absorption to be useful. For ease of comparison, all spectra are shown in the laboratory reference frame and vacuum wavelengths are used throughout; radial velocities and searches for radial velocity variations will be reported and discussed in a separate forthcoming paper (Prato et al. 2006 in prep.). Each order | 1 | member_49 |
is normalized to unity at the same wavelength. Comparison of the spectra in these six figures, all of which have excellent signal to noise ratios (at least 20:1 per pixel), shows that the region is densely populated with numerous weak absorption features and a few stronger lines. We will show that the fine-scale spectral structure is real and repeatable, and that it is mainly attributable to FeH or H$_{2}$O. The strongest atomic features are the doublets of K I that occur in orders 61 and 65. These lines persist from M6-T4.5 but clearly change their character with spectral type. In later sections the K I lines will be singled out for closer inspection. For reference, Table 3 summarizes the main spectral transitions observed in the $J$-band over the spectral type range from M6-T4.5, including the energy levels of the atomic transitions.
As shown in Figure 1, the M6 dwarf Wolf 359 has at least one distinguishing feature in each order. Atomic lines of Al I at 1.31270 and 1.31543 $\mu$m appear in order 58. There is a moderately strong line of Mn I at 1.29033 $\mu$m in order 59, plus some weaker lines of Ti I at 1.28349 and 1.28505 | 1 | member_49 |
$\mu$m. A weak unresolved Na I doublet is seen at 1.26826 $\mu$m in order 60. The first pair of strong K I lines at 1.24357 and 1.25256 $\mu$m appears in order 61. Multiple weak absorption features occur in both orders 62 and 63, the most notable grouping being the set of lines around 1.222 $\mu$m. Several of the stronger features have been identified with FeH from lower resolution studies (Jones et al. 1996; Cushing et al. 2003; M03). Note however, that a major FeH band head at 1.24 $\mu$m is just off the detector at the short wavelength edge of this order. In order 64 there is a pair of strong Fe I lines at 1.18861 and 1.18873 $\mu$m and another Fe I line at 1.19763 $\mu$m. Order 65 contains the second set of strong K I lines, one at 1.169342 $\mu$m and the close pair at 1.177286, 1.177606 $\mu$m. In general, the lines are relatively sharp and well-resolved. Wolf 359 is a bright source and therefore the signal-to-noise ratio in this spectrum is at least 100:1.
Following these spectral features order by order through Figures 1-6 reveals certain general trends as a function of spectral type. Comparing the M9 | 1 | member_49 |
object (Figure 2) with the M6 source (Figure 1) we see that the Al I lines at 1.3127 and 1.3154 $\mu$m in order 58 are somewhat weaker at M9 and then suddenly they are no longer present at L0 (Figure 3), or in any later spectral types (Figures 4-6). This is an important observation that relates to the M-L transition and we will discuss the Al I lines in the next section. Throughout order 58 there are other weaker spectral features, the so-called fine-scale spectral structure. This spectral structure becomes more pronounced at M9 (Figure 2), seems broader in the L0 and L5 objects (Figures 3 and 4), weakens at T0 or more accurately from L6 to T2 (Figure 5) and then completely changes character by T4.5 (Figure 6). The most likely interpretation of this spectral sequence is that it represents changes in the physical structure of these cool atmospheres (temperature, pressure, chemistry). In subsequent sections we compare opacity data for different molecular species to identify the primary absorbers at each spectral class.
In order 59 the sharp Mn I line at 1.2903 $\mu$m seen at M6 and M9 (Figures 1 and 2) broadens and disappears after L0 (Figure 3). | 1 | member_49 |
The fine-scale spectral structure in this order is dominant until T0 composite type (Figure 5) when the spectrum becomes remarkably smooth. Here again the high resolution data reveal a striking effect, this time at the transition from L to T dwarfs. New spectral structure develops in this order between types T0 at T4.5 (Figure 6) but, as was the case for order 58, the pattern is different, indicating different atmospheric conditions.
The weak Na I line at 1.2683 $\mu$m detected in order 60 in the M6 object (Figure 1) is already absent in the M9 object (Figure 2). Otherwise, the behavior of the fine-scale structure follows a pattern similar to order 59 becoming remarkably weak at T0 (Figure 5) and leading to a smoother appearance for these spectra near the L-T transition.
Order 61 contains one of the pairs of strong K I doublets located at 1.2436 and 1.2525 $\mu$m. These lines deepen and widen slightly from M6 to M9, and then become increasingly broader and shallower from L0 to T4.5. NIRSPEC spectral order 61 also has many fine-scale features attributable to molecular transitions. Two features, one at 1.24637 and the other at 1.24825 $\mu$m have been identified previously with | 1 | member_49 |
FeH (Cushing et al. 2003). These FeH lines strengthen slightly from M6 to M9 (Figures 1 and 2), become much broader in the L dwarfs (Figures 3 and 4) and then vanish completely in the T dwarfs (Figures 5 and 6) to leave, once again, a remarkably smooth continuum between the K I lines.
Comparing orders 62 and 63 in Figures 1-6, the known FeH features in these spectral bands strengthen from M6 to M9, broaden markedly at L0 and remain strong and broad through L5 before becoming weaker in the T0 and T4.5. As with the atomic lines, the individual FeH lines seem to broaden significantly at the transition from spectral types M9 to L0. Evidence of weak FeH absorption is still present around 1.222 $\mu$m at spectral type T0, and possibly even at T4.5, as shown in Figures 5 and 6, but this molecular species is clearly not dominant in T dwarfs.
For the M6 dwarf (Figure 1), order 64 is characterized by a pair of strong lines of Fe I at 1.18861 and 1.18873 $\mu$m that are easily resolved, and another Fe I line at 1.19763 $\mu$m that is blended with FeH. These features remain strong at | 1 | member_49 |
M9 (Figure 2) and persist into the L-dwarf range, becoming broader at L0 (Figure 3), and then undetectable by L5 (Figure 4). From T0 to T4.5 (Figures 5 and 6), order 64 becomes increasingly chopped up by new spectral features, some of which are quite sharp and deep. In section §3.3 we show that these features are caused by absorption by 2o.
Finally, there is order 65, which contains the second K I doublet and exhibits some of the largest changes with spectral type. The slightly weaker K I companion line at 1.17728 $\mu$m, only 3.3Å from the longer wavelength member of the doublet is easily resolved in the M9 object (Figure 2), already blended from line broadening in the L0 (Figure 3), barely discernable at L5 (Figure 4), and completely washed out by line broadening and numerous molecular features at T4.5 (Figure 6). Order 65, being close to the short wavelength edge of the $J$-band where terrestrial water vapor absorption is expected, also contains many strong intrinsic transitions of hot H$_{2}$O, for example, the feature at 1.175 $\mu$m.
Al I, Fe I and Mn I; indicators of the M-L transition
-----------------------------------------------------
Figure 7 provides a more detailed view of | 1 | member_49 |
the Al I doublet in order 58. In this plot, the spectra for the M6, M9 and L0 objects shown in Figures 1-3 are expanded and overlaid. Evidently, there is significant spectral structure in this part of the spectrum making it difficult to identify a true continuum level. All three spectral types show consistent features, in particular the wide depression containing the shorter wavelength Al I line. The equivalent width of the Al I lines clearly decreases from M6 to M9, but the change over these three spectral types is only about 25%. Because this region of spectrum is contaminated by 2o absorption, it is difficult to obtain accurate equivalent widths. A pseudo-equivalent width over a 4.2Å interval centered on each line was obtained relative to the local continuum in the troughs where the Al I lines are found. For the stronger line of the pair at 1.3127 $\mu$m, the measured values of equivalent width for the M6 and M9 dwarfs respectively are 420$\pm$20 mÅ and 300$\pm$40 mÅ. At L0, however, the pseudo-equivalent width of this line is $\le$40 mÅ. Clearly, at the transition from M9 to L0, both Al I lines vanish completely. Although only three objects bridging this | 1 | member_49 |
transition were observed at high resolution, the conclusions given here are supported fully by the results of our low resolution BD spectroscopic survey (M03) where two objects of every spectral type from M6 to L5 was included.
As shown in Table 3, these lines arise from absorption from an energy level at 3.14 eV. Interestingly, the Na I line at 1.268 $\mu$m in order 60 is already absent in the M9, and careful inspection shows that a somewhat broadened Mn I line at 1.290 $\mu$m in order 59 persists through L0. The Na I line is excited from a high energy state at 3.6 eV whereas the Mn I line comes from a state at only 2.1 eV. Thus, the sequence in which the lines disappear is at least qualitatively consistent with thermal excitation. But the abrupt loss of Al I lines at the classical M-L boundary is too great to be explained by Boltzmann factors alone. For example, for a temperature change from 2850 K from the M6 to about 2400 K for the M9 say, the population of excited atoms in the upper level would drop by 51% and the equivalent width of the line might change from | 1 | member_49 |
420 mÅ to about 210 mÅ if all other factors remain the same. From M9 to L0 the change would be a further 13% assuming a change in effective temperature of 150 K. Thus, the line should still be measurable with an equivalent width of 140 $\pm$40 mÅ, or about one-third its value at M6. Yet, both lines disappear abruptly. It is likely, from the models of Lodders (2002), that aluminum has been sequestered in compounds such as hibonite (CaAl$_{12}$O$_{19}$) and that this abrupt change in absorption line strength is really caused by the sudden depletion of aluminum as an absorber due to a significant change in atmospheric chemistry, rather than simply a drop in effective temperature. Gas temperatures typical of this transition are near 2000 K (Lodders 2002).
It is also curious that the intensity ratio of the components of the Al I doublet is closer to 3:2 than the expected 2:1 ratio based on their statistical weights. However, as shown in Figure 7, this spectral region is highly complex with many overlapping transitions which makes it difficult to determine the true continuum level for each line. Alternatively, the peculiar line ratios may be a non-LTE effect, or the | 1 | member_49 |
result of line blending.
Another element that is also important for understanding the temperature structure of these cool, dust-forming objects is iron. As previously mentioned, order 64 contains a remarkably strong pair of Fe I lines at 1.1886 and 1.1976 $\mu$m. The shorter wavelength Fe I line is a resolved double with a separation of 1.2Å in the M6 and M9 objects, but appears as a single broad feature at L0. By L5 the Fe lines are completely absent. Lower resolution studies (M03) also suggested that Fe disappeared around L2 or L3. These Fe I lines arise from low-lying energy levels near 2.2 eV, and gas phase iron requires temperatures above about 1700 K (Burrows et al. 2001). Combining the results that Al disappears at L0 and Fe is no longer present by L3, and using the chemistry temperature scale, suggests that there is about a 300 K temperature change from L0 to L3, which is shallower but still consistent with the interval of about 140 K per spectral type derived by Burgasser et al. (2002) as well as the effective temperature scale of Golimowski et al. (2004). As noted by Burgasser et al. (2002), temperatures derived from condensation chemistry | 1 | member_49 |
tend to be systematically cooler by about 500 K than those derived from empirical determinations of T$_{eff}$ using objects with known parallax. These conclusions are not necessarily inconsistent if different spectral features probe a range of optical depths in the atmosphere.
Finally, we note the presence of several weak lines of Ti I that arise from energy states near 1.4 eV, even lower than those of the strong potassium lines. A strong Ti multiplet at 0.97 has also been seen in the spectra of M dwarfs up to at least M9 (Cushing et al. 2005). Unfortunately, these weak lines are impossible to trace after M9.
Fine-scale structure; the role of FeH and 2o
--------------------------------------------
The astronomical $J$-band is bounded by 2o absorption bands from terrestrial water vapor. It is therefore no surprise that high-temperature 2o (hot steam) transitions intrinsic to M, L and T dwarfs encroach far into the $J$-band from both the short and long wavelength ends. These so-called infrared water bands are difficult to model because millions of transitions are needed (Partridge & Schwenke 1997). Typically, models over-estimate the depth of the infrared water bands. In addition to 2o, some of the stronger non-atomic transition features are known | 1 | member_49 |
to be attributable to FeH from lower resolution studies (Jones et al.1996; McLean et al. 2000). These features occur at 1.2091, 1.2113, 1.2135 and 1.2221 $\mu$m. Cushing et al. (2003) verified the features at 1.1939 and 1.2389 $\mu$m as the 0–1 and 1–2 band heads of the F$^{4}\Delta$ – X$^{4}\Delta$ system of FeH, and attributed a blended feature described by McLean et al. (2000) at 1.2221 $\mu$m as the F$^{4}\Delta_{7/2}$ – X$^{4}\Delta_{7/2}$ Q-branch. These authors also listed 24 other relatively strong features lying within the $J$-band. In the Cushing et al. list, no FeH features were identified in the wavelength interval covered by our order 65, which includes the strong shorter-wavelength doublet of K I, and only one feature (at 1.2464 $\mu$m) was tabulated for order 61, where the other K I doublet dominates.
To identify many more of the complex fine-scale features seen in the spectral sequences of Figures 1–6, we analyzed opacity (cross-section) data for both FeH and H$_{2}$O, (R. Freedman 2003, private communication) and utilized the FeH line list and transition catalog by Phillips et al. (1987). We are also grateful to Adam Burrows who provided CrH opacity data (Burrows et al. 2002) and Linda Brown who | 1 | member_49 |
---
abstract: 'We study the gravitational Dirichlet problem in AdS spacetimes with a view to understanding the boundary CFT interpretation. We define the problem as bulk Einstein’s equations with Dirichlet boundary conditions on fixed timelike cut-off hypersurface. Using the fluid/gravity correspondence, we argue that one can determine non-linear solutions to this problem in the long wavelength regime. On the boundary we find a conformal fluid with Dirichlet constitutive relations, viz., the fluid propagates on a ‘dynamical’ background metric which depends on the local fluid velocities and temperature. This boundary fluid can be re-expressed as an emergent hypersurface fluid which is non-conformal but has the same value of the shear viscosity as the boundary fluid. The hypersurface dynamics arises as a collective effect, wherein effects of the background are transmuted into the fluid degrees of freedom. Furthermore, we demonstrate that this collective fluid is forced to be non-relativistic below a critical cut-off radius in AdS to avoid acausal sound propagation with respect to the hypersurface metric. We further go on to show how one can use this set-up to embed the recent constructions of flat spacetime duals to non-relativistic fluid dynamics into the AdS/CFT correspondence, arguing that a version of the | 1 | member_50 |
membrane paradigm arises naturally when the boundary fluid lives on a background Galilean manifold.'
author:
- |
Daniel Brattan$^a$[^1], Joan Camps$^a$[^2], R. Loganayagam$^b$[^3], Mukund Rangamani$^a$[^4]\
\
,\
\
,\
title: |
[**CFT dual of the AdS Dirichlet problem:\
Fluid/Gravity on cut-off surfaces**]{}
---
(0,0)(0,0) (380, 330)[DCPT-11/25]{}
Introduction {#s:intro}
============
The AdS/CFT correspondence [@Maldacena:1997re] which postulates a remarkable duality between large $N$ quantum field theories and gravitational dynamics, provides a useful theoretical laboratory to address questions underlying the dynamics of these systems. Not only has it proven useful to obtain quantitative information about the dynamics of strongly coupled field theories, but it also provides a unique perspective into the geometrization of field theoretic concepts.
Since the early days of the AdS/CFT correspondence it has been known that the radial direction of the bulk spacetime encodes in some sense the energy scale of the dual field theory [@Susskind:1998dq]. While the nature of this map is not terribly precise outside of the simple example of pure geometry (dual to the vacuum state of the field theory), it nevertheless provides valuable intuition about certain basic aspects of effective field theory dynamics [@Banks:1998dd; @Peet:1998wn], and has led to the idea of the holographic renormalisation group | 1 | member_50 |
[@deBoer:1999xf], which relates the radial ‘evolution’ in AdS to RG flows in field theories. More recently this idea has been exploited to geometrize Wilson’s concept of integrating out momentum shells to generate field theory effective actions, in terms of integrating out regions of the bulk geometry which in turn lead to effective multi-trace boundary conditions on the cut-off surface, a fixed radial slice (in some preferred foliation) in AdS [@Heemskerk:2010hk; @Faulkner:2010jy]. One of the key features of this holographic Wilsonian approach was the emergence of multi-trace deformations of the field theory even in the planar limit, consistent with field theory expectation.
A natural question in this context is what does this RG flow mean for the gravitational equations of motion? More precisely, consider the problem of integrating out radial geometric shells in Einstein gravity with negative cosmological constant (which is a consistent truncation of string theory/supergravity). One anticipates based on the standard dictionary which relates the bulk metric to the boundary energy momentum tensor to obtain a scale dependent effective action for the energy momentum tensor, containing arbitrarily high multi-traces of the stress tensor. The reason for the generation of these multi-traces is clear, once one factors in the intrinsic | 1 | member_50 |
non-linearity of gravity. The basic equations in this context are of course easy to write down; as explained in [@Heemskerk:2010hk; @Faulkner:2010jy] the flow is driven by the radial ADM Hamiltonian and one can in principle solve the resulting Hamilton-Jacobi like equation for the effective action on the cut-off hypersurface. Despite the conceptual simplicity of the formulation of Wilsonian RG in terms of geometric effective actions, the point still remains that gravity’s intrinsic non-linearity makes explicit solutions hard to come by.
One can ask whether there is a tractable sector of the gravitational flow equations which leads to new insight. A natural avenue for exploration is suggested by the long-wavelength regime where we restrict attention to fluctuations of low frequency in the field theory directions. As evidenced by the fluid/gravity correspondence [@Bhattacharyya:2008jc] there is an essential simplification in this regime; bulk Einstein’s equations can be explicitly solved order by order in a long-wavelength expansion along the boundary.[^5] As such one should be able to use this framework in conjunction with the fluid dynamical expansion to derive an effective action for the low frequency degrees of freedom which live on a cut-off surface in the interior of the AdS spacetime.[^6] Rather than | 1 | member_50 |
tackle this problem directly we will take a slightly different tack in this paper, one which we believe clarifies some aspects of evolution in the radial direction and its possible connection to RG flows. One of our main conclusions will be that imposing rigid cut-offs in AdS is more naturally viewed in terms of perturbing the CFT by some non-local deformation or equivalently by introducing explicit state-dependent sources in the boundary theory.
A second motivation is the recent work [@Bredberg:2010ky; @Bredberg:2011jq] which derives an explicit map between solutions of vacuum Einstein equations (with no cosmological constant) and those of incompressible Navier-Stokes equations, thereby making direct contact with some of the ideas of the black hole membrane paradigm in asymptotically flat spacetime [@Damour:1978cg; @Thorne:1986iy]. This problem, which has been further generalized in [@Compere:2011dx], is the zero cosmological constant analog of the problem we consider (see also [@Eling:2009pb; @Eling:2009sj] for another approach and [@Cai:2011xv] for related work). The idea is to consider a fixed timelike hypersurface with Dirichlet data enforcing a flat metric on the slice. Given these boundary conditions one wants to solve vacuum Einstein equations so as to obtain a solution which has a regular future horizon.[^7] By explicit construction | 1 | member_50 |
which involves long wavelength fluctuations around flat space in a Rindler patch the authors of [@Bredberg:2011jq; @Compere:2011dx] construct solutions to vacuum Einstein’s equations order by order in a perturbation expansion in gradients along the hypersurface directions. The resulting geometry has a regular Rindler horizon, and one obtains a regular solution to Einstein’s equations contingent on the fact that dynamics of the induced stress tensor on the hypersurface satisfies the incompressible Navier-Stokes equations.
While this development is fascinating, one is hampered from a first principles understanding of the physics from a holographic viewpoint, owing to the rather poorly understood concepts of flat space holography. Moreover, given the connection between fluid dynamics (albeit relativistic and conformal) and Einstein’s equations with negative cosmological constant as described by the aforementioned fluid/gravity correspondence [@Bhattacharyya:2008jc](and its non-relativistic extension in [@Bhattacharyya:2008kq] ), it is interesting to ask whether the construction in [@Bredberg:2011jq] can be obtained as a limit of the fluid/gravity map. If this is possible, one can then look for the field theoretic interpretation of the flat space problem.
Motivated by these issues, we consider a region of the AdS spacetime bounded by a timelike hypersurface $\Sigma_D$ at some radial position, say $r = r_D$ in | 1 | member_50 |
the supergravity limit of AdS/CFT. We are interested in solving for the bulk dynamics where will give ourselves the freedom to specify boundary conditions on $\Sigma_D$. The second boundary condition (which is necessary to zero-in onto a unique solution) will be specified by demanding regularity in the interior of the spacetime. We have schematically depicted the set-up in . In the large $N$ limit, the specification of the problem thus is tantamount to solving classical partial differential equations (PDEs) in AdS with a Dirichlet boundary condition imposed on various fields at the hypersurface $\Sigma_D$. The question we would like to know the answer to is simply: “What is the problem that we are solving in the CFT language?”
![Schematic representation of the Dirichlet problem we consider in this paper. The Dirichlet surface is taken to be at some value $r = r_D$ where we impose boundary conditions on the fields. The solutions will further be constrained by requiring that they be regular on any putative horizon ${\mathcal H}^+$ (shown in the figure) or the origin. The question we are after is what is the boundary image of this Dirichlet data?[]{data-label="f:setup"}](Dir-schema)
(0,0) (-4.8,1.36)[$\Sigma_D$]{} (-8.4,2.3)[${\mathcal H}^+$]{} (-5.24,-0,36)[data = $\hat{\mathfrak X}$]{} (-1.7,-0.36)[data = | 1 | member_50 |
${\mathfrak X}$]{}
As we have reviewed above, various results exist in literature that suggest that solving such a Dirichlet problem is analogous to some kind of RG from the CFT point of view. Despite the strongly suggestive nature of this holographic RG point of view, it is also not very clear what kind of an RG is one speaking of within a CFT. A-priori, for one, it does not seem like the RG flow that arises from cutting off a CFT a la Wilson is the correct way to dualize the Dirichlet problem. Hence, our question - what is the CFT dual of a bulk Dirichlet problem?[^8]
As a warm-up we first consider the bulk Dirichlet problem for linear PDEs, using the simple setting of a Klein-Gordon field propagating in a cut-off AdS spacetime. In this case it is not hard to see that one is deforming the field theory by a non-local double-trace operator, whose precise form, we argue, can be extracted by suitable convolution of appropriate bulk-to-boundary propagators.
We then turn to the issue of setting up the problem in a gravitational setting, outlining it in general before moving on to the tractable setting of the fluid/gravity regime. | 1 | member_50 |
In the long wavelength regime we will argue that the bulk Dirichlet problem reduces to a particular forcing of the fluid on the boundary of the asymptotically AdS spacetime. The fluid/gravity correspondence was generalized to fluids propagating on curved backgrounds with slowly varying curvatures in [@Bhattacharyya:2008ji] and the most general solutions which will prove to be of interest to us were presented in [@Bhattacharyya:2008mz]. Using these results it transpires that we can immediately write down the solution to the bulk Dirichlet problem in the long wavelength regime.
The logic is the following: we wish to prescribe on the hypersurface $r =r_D$ a Lorentzian metric which we denote as ${\hat g}_{\mu\nu}$. This is arbitrary subject to the requirement that its curvatures be slowly varying so that we can treat it with the fluid/gravity perturbation scheme. We then solve Einstein’s equations demanding regularity in the interior of the spacetime. Using standard intuition from the AdS/CFT correspondence it can be argued that the seed geometry which we need to set up the gradient expansion should simply be a black hole geometry which has a regular future event horizon, which furthermore satisfies the prescribed Dirichlet boundary condition.[^9] It is not hard to see that | 1 | member_50 |
such a seed solution is obtained by simply performing a coordinate transformation of the well known planar Schwarzschild-AdS black hole.
But this is precisely the set-up of [@Bhattacharyya:2008ji; @Bhattacharyya:2008mz], the only difference being the fact that in these works the Dirichlet data is imposed on the boundary at $r =\infty$. Let’s call this boundary metric $g_{\mu\nu}$, which is also by definition slowly varying etc.. The solution to the asymptotic Dirichlet problem is characterized by the boundary metric $g_{\mu\nu}$, a distinguished velocity field $u^\mu$ (which is unit normalized) and a scalar function $b$ (determining the temperature or equivalently the local energy density). Let us denote these variables collectively as ${\mathfrak X}$. The boundary Brown-York stress tensor (up to counter-terms) takes the fluid dynamical form and is built out of the data contained in ${\mathfrak X}$.
Now given the space of solutions to the asymptotic Dirichlet problem, we can reparametrize that space of solutions appropriately to obtain the solutions of the new bulk Dirichlet problem. The only condition we have to satisfy is that the induced metric on $\Sigma_D$ in the solutions obtained this way[^10] be equal to ${\hat g}_{\mu\nu}$. Furthermore, we can extract the stress tensor on $\Sigma_D$[^11]. We will argue | 1 | member_50 |
that there is a corresponding velocity field ${\hat u}^\mu$ (normalized with respect to the hypersurface metric) and a scalar function (the hypersurface temperature), which parameterize the stress tensor of the hypersurface, which not surprisingly takes the fluid dynamical form. The main novelty is that the stress tensor does not however correspond to that of a conformal fluid. The introduction of an explicit scale by way of the Dirichlet surface’s location engenders a non-vanishing trace, which curiously evolves in a highly suggestive manner under change of cut-off surface position, see .
Calling the totality of the data on the hypersurface ${\hat {\mathfrak X}}$ we further show that within the gradient expansion there is a one-to-one correspondence between the hypersurface data and the boundary data; $\varphi_D: {\mathfrak X} \to {\hat {\mathfrak X}}$ is bijective. This then has the advantage that we can immediately understand the boundary dual of the bulk Dirichlet problem as a conformal fluid which is placed on a[^12] ‘dynamical background’ whose metric depends on the same set of variables that characterizes the fluid itself (in addition to the prescribed hypersurface metric). So from the boundary viewpoint there is a complete mixing between intrinsic and extrinsic data, which is the | 1 | member_50 |
long-wavelength non-linear analog of the double trace deformation seen for the scalar toy model. Moreover, this solution allows us to see that the dynamics of the fluid on the Dirichlet surface, as given by the conservation equation on $\Sigma_D$, ‘emerges’ as collective dynamics of the boundary CFT. In particular, the boundary fluid lives on a ‘dynamical background’, and the effects of the background can be suitably subsumed into the fluid description. This suggests that the correct way to think about the hypersurface physics is in terms of a ‘dressed fluid’ living on an inert geometry. Thus, the effective description of a fluid on this dynamical background is geometrically encapsulated in terms of the Dirichlet hypersurface dynamics.
Examining the resulting dynamics on $\Sigma_D$ we find that the hypersurface or effective fluid suffers from a possible pathology for $r_D$ smaller than some critical value $r_{D,snd}$. At $r_{D,snd}$ the sound mode of the effective fluid starts to propagate outside the inert background $\Sigma_D$’s light-cone. We suggest that in the CFT, this effect is due to the extreme forcing of the fluid on the boundary by the ‘dynamical’ metric, and moreover propose that one can obtain sensible dynamics by projecting out the sound mode. | 1 | member_50 |
This involves looking at the fluid at a scaling limit and this can be formalized as taking the incompressible non-relativistic limit of the fluid on the hypersurface in a manner entirely analogous to the scaling limit described in for generic relativistic fluids in [@Bhattacharyya:2008kq; @Fouxon:2008tb].[^13]
Having understood the Dirichlet problem for generic $\Sigma_D$ away from the horizon ${\mathcal H}^+$, we then proceed to push this surface deeper into the spacetime and ask what happens as we approach the horizon. In this regime $\Sigma_D$ dynamics continues to be described by incompressible Navier-Stokes equations in the limit, though with some slight differences from the BMW limit mentioned above. Zooming in onto the region between $\Sigma_D$ and the horizon, we provide an embedding of the construction of [@Bredberg:2011jq; @Compere:2011dx] into the fluid/gravity correspondence [@Bhattacharyya:2008jc]. Further, we demonstrate that in this limit both the bulk metric in the region between $\Sigma_D$ and the boundary, and the boundary metric degenerate from metrics on a Lorentzian manifold to Newton-Cartan like structures. This raises interesting questions about the natural emergence of the Galilean structures in the AdS/CFT correspondence which we postpone for future work.
The plan of this paper is as follows: In we first address the | 1 | member_50 |
Dirichlet problem for a scalar field propagating in using this linear problem to build intuition. In we pose the bulk Dirichlet problem for gravity in spacetime and solve it in the long wavelength approximation borrowing heavily on the results from the fluid/gravity correspondence. The remainder of the paper is then devoted to understanding the physics of our construction in various regimes: demonstrates how the Dirichlet surface dynamics, as governed by the conservation equation, arises from the boundary physics. Aided by this we argue that the Dirichlet dynamics is probably pathological past a critical radius and propose a non-relativistic scaling of the resulting fluid a la BMW in to cure this possible pathology. Finally, in we study the near-horizon Dirichlet problem and make contact with the recent work on the flat space Dirichlet problem (and its connection with Navier-Stokes equations). We end with a discussion in . Various appendices contain useful technical results. In particular, to aid the reader we provide a comprehensive glossary of our conventions and key formulae in . This is followed by a complete ‘Dirichlet dictionary’ relating hypersurface variables to boundary variables in for ready reference.
[*Note added:*]{} While this work being completed we received [@Kuperstein:2011fn] which | 1 | member_50 |
has partial overlap with the results presented in . These authors also attempt to solve for bulk geometries with prescribed boundary conditions on $\Sigma_D$ in the long wavelength regime and interpret their results in terms of a RG flow of fluid dynamics.
[*Note added in v2:*]{} In the first version of the paper the non-relativistic metrics quoted in and in were incorrect; the metrics as presented do not solve the bulk Einstein’s equations to the desired order. These are now corrected in the current version. However, the full set of terms that we need to include in order to see the Naiver-Stokes equation on the boundary is quite large. Hence in the main text we only report the results for the case where the non-relativistic fluid moves on a Ricci flat spatial manifold in and present the general results in a new appendix . We note that the results of also correct the expressions originally derived in [@Bhattacharyya:2008kq].
Dirichlet problem for probe fields {#s:dscalar}
==================================
To set the stage for the discussion let us consider setting up the bulk Dirichlet problem for linear PDEs in an asymptotically spacetime. As a canonical example we will consider the dynamics of a probe | 1 | member_50 |
scalar field $\Phi(r,x^\mu)$ of mass $m$. Generalizations to other linear wave equations such as the free Maxwell equation are straightforward.
We will let this scalar field propagate on a background asymptotically geometry with spatio-temporal translational symmetries so that the background metric can be brought to the form $$ds^2 =r^2\, g_{\mu\nu}(r) \, dx^\mu \, dx^\nu + \frac{g_{rr}(r)}{r^2} \, dr^2
\label{bggen}$$ The boundary of the spacetime is at $r \to \infty$ and we will assume that the boundary metric is the Minkowski metric on ${\mathbb R}^{d-1,1}$ for simplicity, so that asymptotically $g_{\mu\nu} \to \eta_{\mu\nu}$ and $g_{rr} \to 1$ (as $r\to \infty$).
The dynamical equation of motion for the scalar is the free Klein-Gordon equation which can be written as an ODE in the radial direction for the Fourier modes $\Phi_{k}(r)$ of $\Phi(r,x^\mu)$ $$\Phi(r,x^\mu)=\int \frac{d^dk}{(2\pi)^d} \,e^{i\, k\cdot x} \, \Phi_k(r)\ ,$$ and takes the form $$\frac{1}{\sqrt{-g\, g_{rr}}}\, \partial_r \, \left(\sqrt{-g\, g^{rr}} \, \partial_r \Phi_{k}(r) \right) - \left(g_{\mu\nu} \,k^\mu\,k^\nu +m^2 \right) \Phi_{k} = 0$$ As a second order equation we need to specify two boundary conditions. We are going to restrict attention to the finite part of the geometry as illustrated in and impose Dirichlet boundary conditions for the field at some hypersurface $\Sigma_D$ | 1 | member_50 |
at $r = r_D$. The second boundary condition in general can take the form of a regularity boundary condition in the interior of the spacetime. If we were working in a spacetime with a horizon this would demand that the mode functions of interest are purely ingoing at the future horizon.
The question we wish to pose is the following: usually in an asymptotically spacetime we know that the solution to the scalar wave equation above has two linearly independent modes with power-law fall-off characterized by the source $J_\phi$ and vev $\phi$ of the dual boundary operator ${\cal O}_\Phi$ (which we recall is a conformal primary). We wish to ask what is the characterization of the boundary data as a functional of the Dirichlet hypersurface data. In this simple linear problem it is easy to see that there is a one-one map between the two sets of data. Essentially we are asking how to tune $J_\phi$ and $\phi$ so that the value of the scalar field on the Dirichlet hypersurface at $r=r_D$ takes on its given value.
While it is possible to derive a formal answer to the above question, it is useful to first visit the simple setting of | 1 | member_50 |
pure spacetime where we have the luxury of being able to solve the scalar wave-equation explicitly to see an explicit answer to the question.
Probe scalar in {#s:adss}
----------------
We specialize our consideration to the pure geometry where $g_{\mu\nu} = \eta_{\mu\nu} $ and $g_{rr} = 1$ and one has enhanced Lorentz symmetry on the constant $r$ slices. The wave equation simplifies to $$\frac{1}{r^{d-1} }\, \frac{d}{dr} \, \left(r^{d+1} \, \frac{d}{dr} \Phi_{k}(r) \right) - \left(k^2 +m^2 \right) \Phi_{ k} = 0
\label{mkgads}$$ This is well known to have solutions in terms of Bessel functions, but we will proceed to examine the behavior in a gradient expansion to set the stage for the real problem of interest later.
### The $k=0$ case
The translationally invariant solution of the massive Klein-Gordon equation in the bulk is[^14] $$\Phi(r) = \frac{\phi}{(2\nu)\,r^\Delta} + r^{\Delta-d} J_\phi$$ where the dual primary has a scaling dimension $\Delta$ obeying $\Delta(\Delta-d)=m^2$ along with a source $J_\phi$ and a normalized vev[^15] $\phi$ defined via $$J_\phi \equiv \left[r^{d-\Delta}\, \Phi(r)\right]_{r\to\infty}$$ $$\phi \equiv \left[-r^{2\nu}\times r\partial_r \left( r^{d-\Delta}\Phi \right)\right]_{r\to\infty} = \left[-r^\Delta\left( r\partial_r \Phi -(\Delta-d)\Phi\right)\right]_{r\to\infty}$$ with $$\nu \equiv \Delta - \frac{d}{2} = \sqrt{\frac{d^2}{4} + m^2}$$ for convenience. For simplicity, we will assume $\Delta > \frac{d}{2} $ and choose | 1 | member_50 |
$m$ such that $\nu \notin {\mathbb Z}$ to avoid complications with logarithms. Extension to $\Delta \in [\frac{d}{2}-1, \frac{d}{2}]$ with the lower end of the interval saturating the unitary bound is possible with the added complication of taking proper account of the necessary boundary terms.
We will begin by rewriting this solution in terms of the quantities on the Dirichlet surface which we denote with a hat to distinguish them from the boundary data: $$\hat{J}_\phi \equiv \left[r^{d-\Delta}\,\Phi(r)\right]_{r\to r_D}=J_\phi+ \frac{\phi}{(2\nu)\,r_D^{2\nu}} \ ,
\label{j0}$$ $$\hat{\phi} \equiv \left[-r^{2\nu}\times r\partial_r \left( r^{d-\Delta}\Phi \right)\right]_{r\to r_D} =\phi \ .
\label{ph0}$$
Since the transformation between the data on the boundary $\{J_\phi,\phi\}$ and that on the hypersurface $\{\hat{J}_\phi, \hat{\phi}\}$ is linear it is a simple matter to write the bulk solution in terms of the hypersurface variables. One simply has $$\Phi(r) =\frac{\hat{\phi}}{(2\nu)\, r^\Delta} + r^{\Delta-d} \left(\hat{J}_\phi- \frac{\hat{\phi}}{(2\nu)r_D^{2\nu}}\right) .
\label{pDsol}$$ This is the answer we seek and all that remains is to interpret this result.
It is now easy to notice that the imposition of the Dirichlet condition on a hypersurface inside the bulk is equivalent to making the boundary source a specific function of the vev. From we can read off the specific deformation of the boundary CFT action | 1 | member_50 |
to be given by $$\delta \mathcal{L}_{CFT} = -\frac{1}{16\pi \,G_{d+1}\, } \, \left(\hat{J}_\phi\,\hat{\phi}- \frac{1}{2(2\nu)\, r_D^{2\nu}}\, \hat{\phi}^2 \right) \propto \hat{J}_\phi \, {\cal O}_\Phi - \frac{(16\pi \,G_{d+1})}{2(2\nu)\, r_D^{2\nu}}\, {\cal O}_\Phi^2$$ which happens to be an irrelevant double-trace deformation [@Witten:2001ua; @Berkooz:2002ug] of the boundary CFT. Hence, at least in this simple setup the dual of the Dirichlet problem is to make the source of a primary ${\cal O}_\Phi$ a particular joint function of the vev of the primary in the given state and another fixed (state-independent) auxiliary source.
### The $k\neq0$ case : Derivative expansion up to $k^2$
Having seen the result for the translationally invariant case $k=0$, we now proceed with $k \neq 0$. It is well known that general solution to the wave equation is given in terms of Bessel functions which we parameterize as[^16] $$\Phi_k(r) = \frac{\phi_k}{r^\Delta}\times\frac{\Gamma(\nu)}{2(k/2r)^{\nu}}I_{\nu}(k/r) + r^{\Delta-d} (J_\phi)_k \times \frac{2(k/2r)^{\nu}}{\Gamma(\nu)}K_{\nu}(k/r)$$ Note that our previous result for $k=0$ follows from just keeping the leading $x^0$ terms in the expansions $$\begin{split}
\frac{2x^{\nu}}{\Gamma(\nu)}\;K_{\nu}(2x) &= \sum_{j=0}^{\infty} \frac{\Gamma(\nu-j)}{\Gamma(\nu)} \frac{(-x^2)^j}{j!} +x^{2\nu}\sum_{j=0}^{\infty} \frac{\Gamma(-\nu-j)}{\Gamma(\nu)} \frac{(-x^2)^j}{j!}\\
\frac{\Gamma(\nu)}{2x^{\nu}}I_{\nu}(2x) &=\sum_{j=0}^{\infty} \frac{\Gamma(\nu)}{(2\nu+2j)\Gamma(\nu+j)} \frac{x^{2j}}{j!}
\end{split}
\label{eq:expn}$$ For a general $k$, we can repeat the analysis of the previous section. While this can be done generally at all orders in $k$ with some | 1 | member_50 |
work, for simplicity we will resort to derivative expansion keeping terms upto order $k^2$. Not only will this allow us to see some of the structures emerging explicitly, but it also sets the stage for our gravitational computation in later sections.
Using the expansion above, we have $$\begin{split}
\Phi_k(r) &= \frac{\phi_k}{(2\nu)\, r^\Delta}\left(1 +\frac{2\nu}{(2\nu+2)^2} \frac{k^2}{2r^2}+\ldots \right) + r^{\Delta-d} (J_\phi)_k \left( 1-\frac{1}{(2\nu-2)} \frac{k^2}{2r^2}+\ldots\right.\\
&\qquad\qquad \quad \left. +
\frac{\Gamma(-\nu)}{\Gamma(\nu)}\left(\frac{k}{2r}\right)^{2\nu}\left\{1 +\frac{1}{(2\nu+2)} \frac{k^2}{2r^2}+\ldots\right\}\right)
\end{split}$$ with the ellipses representing order $k^4$ terms and higher.
The source at the intermediate surface is as before easily determined $$\begin{split}
(\hat{J}_\phi)_k &\equiv \left[r^{d-\Delta}\Phi_k(r)\right]_{r\to r_D}\\
&= (J_\phi)_k \left(1-\frac{1}{(2\nu-2)} \frac{k^2}{2\,r_D^2}+\ldots + \frac{\Gamma(-\nu)}{\Gamma(\nu)}\left(\frac{k}{2r_D}\right)^{2\nu}\left\{1 +\frac{1}{(2\nu+2)} \frac{k^2}{2\,r_D^2}+\ldots\right\}\right)\\
&\qquad \qquad +\frac{\phi_k}{(2\nu)\, r_D^{2\nu}}\left(1 +\frac{2\nu}{(2\nu+2)^2} \frac{k^2}{2\,r_D^2}+\ldots \right)
\end{split}
\label{j2}$$ while the normalized vev of the primary to this order in derivative expansion can be determined after subtracting an appropriate counter-term as[^17] $$\begin{split}
\hat{\phi}_k &=\left[-r^{2\nu}\times r\partial_r \left( r^{d-\Delta}\Phi_k \right)+\frac{r^\Delta}{2\nu-2}\frac{k^2}{r^2}\Phi_k +\ldots \right]_{r\to r_D}\\
&=\phi_k\left(1 +\frac{2\nu-2}{(2\nu)^2} \frac{k^2}{2\,r_D^2}+\ldots \right) +\frac{4(J_\phi)_k}{(2\nu-2)} \times\frac{\Gamma(-\nu)}{\Gamma(\nu+2)}\left(\frac{k}{2}\right)^{2\nu}+\ldots \\
\end{split}
\label{ph2}$$ To solve the Dirichlet problem,we need to solve for $\phi_k,(J_\phi)_k $ in terms of the hatted variables from and which can be inverted to get $$\begin{split}
(J_\phi)_k &= \frac{(\hat{J}_\phi)_k}{\mathfrak{D}}\left(1 +\frac{2\nu-2}{(2\nu)^2} \frac{k^2}{2r_D^2}+\ldots \right) -\frac{\hat{\phi}_k}{\mathfrak{D}\, (2\nu) \, r_D^{2\nu}}\left(1 +\frac{2\nu}{(2\nu+2)^2} \frac{k^2}{2\,r_D^2}+\ldots \right) \\
\phi_k &=\frac{\hat{\phi}_k}{\mathfrak{D}} \left( 1-\frac{1}{(2\nu-2)} \frac{k^2}{2\,r_D^2}+\ldots +
\frac{\Gamma(-\nu)}{\Gamma(\nu)}\left(\frac{k}{2\,r_D}\right)^{2\nu}\left\{1 +\frac{1}{(2\nu+2)} | 1 | member_50 |
\frac{k^2}{2\,r_D^2}+\ldots\right\}\right)\\
&\qquad \qquad -\frac{4(\hat{J}_\phi)_k}{\mathfrak{D}(2\nu-2)} \times\frac{\Gamma(-\nu)}{\Gamma(\nu+2)}\left(\frac{k}{2}\right)^{2\nu}+\ldots \\
\end{split}$$ where the momentum dependent coefficient ${\mathfrak D}$ is $$\begin{split}
\mathfrak{D}&\equiv 1-\frac{4(2\nu-1)}{(2\nu)^2(2\nu-2)} \frac{k^2}{2\,r_D^2}+
\frac{\Gamma(-\nu)}{\Gamma(\nu)}\left(\frac{k}{2\,r_D}\right)^{2\nu}\left\{1-\frac{1}{\nu^2(\nu-1)} \right.\\
&\left.\qquad \qquad \qquad+\; \frac{(2 \nu +1)^4-4 (2 \nu +2)^2+7}{(2 \nu)^2
(2 \nu +2)^3}\;\frac{k^2}{r_D^2}+\ldots\right\}\\
\end{split}$$ As we saw in the $k=0$ case, we have yet again determined a state dependent source on the boundary for the primary operator ${\cal O}_\Phi$. The key feature to note from the above analysis, is that the expression for the boundary source $J_\phi$ is non-analytic in $k$ and hence non-local when Fourier-transformed back to position space. Hence, we see that in general we have a map between the non-local double trace deformation on the boundary and the Dirichlet data on $\Sigma_D$ (similar non-local double-trace deformations were explored earlier in [@Marolf:2007in]).
A general proposal for linear systems
-------------------------------------
From the analysis of the free scalar wave equation in the picture is rather clear. In the CFT, in general one can make the source a non-local functional of the vev of the primary operator. Usually such a function can be fed into the holographic dictionary via a ‘state-dependent’ boundary condition, which whilst somewhat unnatural from a field theory is a perfectly sensible boundary condition to consider. For | 1 | member_50 |
some special classes of functionals, this state-dependent boundary condition has a very simple bulk interpretation as a Dirichlet boundary condition imposed on an intermediate surface, implying that we can trade the non-locality of the boundary sources into local behavior at some lower radius.
We just have one further question to answer before we declare victory: how do we in practice determine this special set of sources in various holographic setups? For the general backgrounds we can formally write the solution to the wave equations in terms of integrals over the Dirichlet data convolved with suitable ‘Dirichlet bulk to boundary propagators’, ${\cal K}_\text{source}$ and ${\cal K}_\text{vev}$. The former propagates the information contained in $\hat{J}_\phi$ to the boundary source, while the latter allows determination of the contribution from the vev $\hat{\phi}$ on $\Sigma_D$, i.e., formally $$J_\phi(x) = \int d^dx' \, \left\{{\cal K}_\text{source}(r_D, x; x') \, \hat{J}_\phi(x') + {\cal K}_\text{vev}(r_D, x; x') \, \hat{\phi}(x')\right\}$$ Note that implicit in our definition of these Dirichlet bulk to boundary propagators is the information of the boundary condition in the interior of the geometry and the necessary counter-terms. While it is possible to work this out in more specific geometries, such as a Schwarzschild-AdS$_{d+1}$ spacetime to see | 1 | member_50 |
the interplay of these IR boundary conditions, we will leave this toy problem for now, and proceed to analyze the more interesting case of gravitational dynamics in wherein we do have to face-up with non-linearities of the equations of motion.[^18]
Before proceeding to the gravitational setting, however, let us make a few pertinent observations relevant to the motivation mentioned at the beginning of . The result we have obtained is quite intuitive; demanding that our fields take on the desired value at $\Sigma_D$ entails a linear relation between the two pieces of data at infinity, thereby leading to the observation about the source depending on the vev. We also see that despite some superficial resemblance to the Wilsonian RG flow where too one encounters multi-trace operators there is a crucial distinction in the physics. In the formulation of [@Heemskerk:2010hk; @Faulkner:2010jy] one finds that for fixed asymptotic data, upon integrating out the region of the geometry between the boundary and a cut-off surface (which we can for simplicity take to be $\Sigma_D$ for the sake of discussion) one obtains an effective action for a cut-off field theory living on $\Sigma_D$ with scale dependent sources. These are irrelevant double traces (which are | 1 | member_50 |
the only terms generated in a Gaussian theory which the linear models under discussion are), and one obtains the $\beta$-functions for the double trace couplings along the flow. In the present context however what we have is a situation wherein we are forced to engineer a specific double trace deformation on the boundary so as to ensure that we satisfy the Dirichlet boundary conditions on $\Sigma_D$. This is conceptually different from usual notions of RG, where one does not conventionally consider state dependent boundary conditions in the UV. However, there is a sense in which renormalisation of sources takes place which will become quite clear when we look at the gravitational problem.
The Dirichlet problem for gravity {#s:dgrav}
=================================
Having understood the boundary meaning of the Dirichlet problem for probe fields in a fixed background, we now turn to the situation where we consider dynamical gravity in the bulk. While we could consider other matter degrees of freedom in the bulk whose backreaction we now have to take into account, we choose for simplicity to restrict attention to the dynamics in the pure gravity sector which, as is well known, is a consistent truncation of the supergravity equations of motion. | 1 | member_50 |
---
abstract: 'We extended a previous qualitative study of the intermittent behaviour of a chaotical nucleonic system, by adding a few quantitative analyses: of the configuration and kinetic energy spaces, power spectra, Shannon entropies, and Lyapunov exponents. The system is regarded as a classical “nuclear billiard” with an oscillating surface of a 2D Woods-Saxon potential well. For the monopole and dipole vibrational modes we bring new arguments in favour of the idea that the degree of chaoticity increases when shifting the oscillation frequency from the adiabatic to the resonance stage of the interaction. The order-chaos-order-chaos sequence is also thoroughly investigated and we find that, for the monopole deformation case, an intermittency pattern is again found. Moreover, coupling between one-nucleon and collective degrees of freedom is proved to be essential in obtaining chaotic states.'
author:
- Daniel Felea
- Cristian Constantin Bordeianu
- Ion Valeriu Grossu
- Călin Beşliu
- Alexandru Jipa
- 'Aurelian-Andrei Radu'
- Emil Stan
title: 'Intermittency route to chaos for the nuclear billiard - a quantitative study'
---
\[intro\]Introduction
=====================
We begin by briefly reminding that a conjugated continuous effort has been made to relate the emergence of the collective energy dissipation through one and two-body nuclear | 1 | member_51 |
processes with the chaotical behaviour of nuclear systems .
A few options in choosing the collective oscillation frequencies have come into focus in the past years, in connection with the onset of chaoticity for “nuclear billiards”. First of all, the issue of dissipation into thermal motion of the adiabatic collective vibrational energy of the potential well was treated for several multipolarities by Burgio, Baldo *et al.* . On the other hand, when trying to associate different vibration frequencies to various nuclear processes, the path to chaos was found to be changed with the order of multipole [@felea-01; @felea-02; @bordeianu-08a; @bordeianu-08b; @bordeianu-08c; @felea-09a].
This paper was intended to bring a quantitative argumentation, based on a systematic study of the configuration and kinetic energy spaces, power spectra, informational entropies, and largest Lyapunov exponents. The study was done in completion of a few qualitative types of analysis previously presented [@felea-09a]: sensitive dependence on the initial conditions, single-particle phase space maps, fractal dimensions of Poincare maps, and autocorrelation functions.
In short, we remind that, by studying the nucleonic dynamics in a Woods-Saxon potential, one can find an increase of the chaotical degree of the system behaviour as raising the frequency of 2D wall oscillation.
| 1 | member_51 |
The main result of [@felea-09a] was reported in relation with an intermittent route to chaos for the monopole vibrations close to the resonance phase of a nuclear interaction. Still, we mention that the purpose of these two coupled etudes was only to emphasize the detection of such intermission for the “nuclear billiards” and not to establish its type according to [@pomeau-80], nor to compare it with other intermittency patterns from known experimental results .
\[sec:1\]Toy model
==================
We continue the study on a classical dynamical system proposed by Burgio, Baldo *et al.* , system composed of a number of $A$ nucleons with no charge, spin, or internal structure. A two-dimensional deep Woods-Saxon potential well, regarded as “nuclear billiard”, is periodically hit by the nucleons. The Bohr Hamiltonian in polar coordinates is a sum of two components: kinetic ($E_{kin.}$) and potential ($E_{pot.}$), the kinetic one decoupling into radial ($E_r$), centrifugal ($E_L$), and collective terms ($E_{coll.}$):
$$E_{kin.} = E_r + E_L + E_{coll.} = {\sum_{j=1}^{A}}\left( \frac{p_{r_{j}}^{2}}{2m}+\frac{p_{\theta _{j}}^{2}}{2mr_{j}^{2}} \right)%
+\frac{p_{\alpha }^{2}}{2M},$$
$$E_{pot.} = {\sum_{j=1}^{A}} V\left(r_{j},R\left( \theta _{j}\right) \right)+\frac{M\Omega ^{2}\alpha ^{2}}{2}.$$
The phase space is defined by particle and collective momenta and their conjugate coordinates: $\left( r,p_{r}\right)$, $\left( \theta,p_{\theta}\right)$ and $\left( \alpha,p_{\alpha}\right)$. The collective | 1 | member_51 |
coordinate $\alpha $ oscillates with $\Omega $ frequency, the Inglis mass $M$ is equal to $mAR_{0}^{2}$, and the nucleon mass: $m=938\ \rm{MeV}$.
For the time being we are not interested in studying the nucleon dynamics beyond the Woods-Saxon barrier:
$$V\left( r_j,R\left( \theta _j\right) \right) =\frac{V_0}{1+\exp \left[ \frac{r_j-R\left( \theta _j,\alpha \right) }a\right] },$$
and therefore we choose a deep well: $V_{0}=-1500\ \rm{MeV}$ and accordingly, a low value for the diffusivity coefficient: $a=0.01\ \rm{fm}$.
When considering the two-dimensional case, the frontier of the collective motion is described as a function of the collective variable $\alpha $ and of the Legendre polynomials $P_{L}\left( \cos \theta_{j}\right)$ [@burgio-95; @baldo-96; @baldo-98; @felea-09a]:
$$R_j = R\left( \theta _j,\alpha \right) =R_0\left[ 1+\alpha P_L\left( \cos \theta_j\right) \right].$$
The oscillation degree of the potential well $L$ is considered for the monopole $\left(0\right)$, dipole $\left(1\right)$ and quadrupole case $\left(2\right)$.
If the surface has a stationary behaviour, or whenever one takes into account the uncoupled Hamilton equations ([UCE]{}) for the particle:
$$\stackrel{\cdot }{r_j}=\frac{p_{r_j}}m,\
\stackrel{\cdot }{\theta _j}=\frac{p_{\theta _j}}{mr_j^2},\
\stackrel{\cdot }{p_{r_j}}=\frac{p_{\theta _j}^2}{mr_j^3}-\frac{\partial V}{\partial r_j},\
\stackrel{\cdot }{p_{\theta _j}}=-\frac{\partial V}{\partial R_j}\cdot \frac{\partial R_j}{\partial \theta _j},$$
and collective degrees of freedom (*d.o.f.*):
$$\stackrel{\cdot }{\alpha }=\frac{p_\alpha }M,\
\stackrel{\cdot }{p_\alpha}=-M\Omega ^2\alpha -\sum_{j=1}^{A}\left(\frac{\partial V}{\partial R_j} \cdot \frac{\partial R_j}{\partial \alpha }\right),$$
| 1 | member_51 |
$R_{0}$ has a fix value, chosen for consistency with previous papers [@burgio-95; @baldo-96; @baldo-98; @felea-09a] as $6\ \rm{fm}$.
A Runge-Kutta type algorithm (order 2-3) with an optimized step size was used for solving the system of differential equations, while keeping the absolute errors for the phase space variables under $10^{-6}$ and conserving the total energy with relative error: $\Delta E/E \approx 10^{-8}$ (Fig. \[fig:1\]).
We imposed the equilibrium condition between the pressure exerted by the particles and the mechanical pressure of the wall [@burgio-95; @baldo-96; @baldo-98; @felea-09a] and thus obtained for the initial equilibrium value of the collective variable, perturbed with a small value:
$$\alpha_{0}=\frac{-1+\sqrt{1+8T/{mR_0^2\Omega ^2}}}{2}+0.15,$$
where $T=36\ \rm{MeV}$ [@burgio-95; @baldo-96; @baldo-98; @felea-09a] is the two-dimensional kinetic energy and also the temperature of the nuclear system, when considering the natural system of units ($\hbar=c=k_{B}=1$).
\[sec:2\]The quantitative analysis of the route to chaos
========================================================
We carry on the study begun in [@felea-09a] by gradually changing the degree of vibration of the potential wall from a slow motion (adiabatic state) to a rapid one (the so-called resonance state of the interaction). In the first case, the collective motion is described by a radian frequency smaller than $0.05\ c/\rm{fm}$. The latter is dominated | 1 | member_51 |
by frequencies close to the one-nucleon collisional frequency:
$$\omega _{part}=\frac{\pi}{R_0}\cdot\sqrt{\frac{2T}{m}}\approx 0.145\ c/\rm{fm}.$$
The physical motivation for studying the one-nucleon chaotical dynamics in the “nuclear billiard”, ranging the frequencies from the adiabatic to the resonance regime of a nuclear interaction, is explained in some detail in [@felea-09a].
We briefly remind that the process of nuclear multifragmentation can be viewed as a resonance process and that for smaller excitation energies, nuclear evaporation or breakup of a projectile nucleus occurs when the energy is shared between the collective and one-nucleon degrees of freedom.
By using a few types of analyses: sensitive dependence on the initial conditions, single-particle phase space maps, fractal dimensions of Poincare maps and autocorrelation functions, we emphasized that an intermittent route to chaos is observed in the monopole case when increasing the vibrational frequency to $\Omega = 0.1\ c/\rm{fm}$ [@felea-09a]. In the resonance phase of the interaction the onset of chaotical behaviour was found to be earlier than at any other adiabatic oscillations of the Woods-Saxon potential well.
We present here other methods [@schuster-84] promoting the idea that the degree of chaoticity increases when moving from the adiabatic to the resonance regime: analyses of the configuration and kinetic energy spaces, | 1 | member_51 |
power spectra, generalized informational entropies and Lyapunov exponents. Furthermore, we try to identify possible pathways to chaos, including the intermittent one, previously put in evidence [@felea-09a].
Configuration and kinetic energy spaces
---------------------------------------
In order to establish if a specific physical system presents a chaotic dynamics and to identify possible routes to chaos we analyzed the behaviour of a small bunch of trajectories in the configuration and in the kinetic energy space, respectively. For example, we took five trajectories separated by an $\epsilon = \Delta r=0.01\ \rm{fm}$ aperture, while keeping the rest of the initial phase space variables constant and let the system evolve over a given time $\left(\Delta t = 1,600\ \rm{fm}/c\right)$.
For the transient stages from adiabatic to resonance, the temporal evolution of an initially confined trajectory bundle was studied for the monopole and dipole oscillation modes of the potential wall and also for the limit situation, in which the individual and collective degrees of freedom remain uncoupled (Figs. \[fig:2\] and \[fig:3\]).
The configuration space revealed a high degree of symmetry in $\left(r,\theta\right)$ plane in both cases, $\left( 4+2\right)$ uncoupled nonlinear differential equations and monopole (left and middle panels). Also, that the central zone remained uncovered, reflecting the conservation | 1 | member_51 |
of the nucleon angular momentum.
Another important conclusion was issued from the definition of the stability concept of a dynamical system. For the aforementioned cases the dispersed trajectory pack periodically regroups on the frontier that delimits the forbidden zone of the phase space. This type of behaviour corresponds with the definition of stability given by Poisson (for e.g., in [@holmes-96]). We will herewith remind that the Poisson stability defines as steady the movement of a particle system of which configuration comes close, from time to time, to the initial position.
At a first glance, on the simple [UCE]{} case one can distinguish two extremities of the radius of the particle periodic motion in the 2D potential well: $r_{min}(UCE) \approx 1.42\ \rm{fm}$, and $r_{Max}(UCE) \approx 5.96\ \rm{fm}$. These values correspond to the roots of the boundary equation:
$$E = \frac{p_{\theta}^2}{2mr^2}+V\left(r \right),$$
for a given one-nucleon energy $E$, when the radial component of velocity vanishes ($\stackrel{\cdot }{r}=0$).
The analysis of the particle motion in the configuration space is similar to that applied to any system with bound unclosed trajectories. The nucleonic motion takes place within a circular crown (the so-called *annulus*) determined by the concentric circles of $r_{min}$ and $r_{Max}$ radii. The | 1 | member_51 |
trajectory is symmetric about any turning point.
For $\Delta t = 1,600\ \rm{fm}/c$ we descry in the [UCE]{} case as much as $27$ distinct apocenters (at $r=r_{Max}$). These are correlated with a number of $13$ complete and one incomplete revolutions about the center of the force field (*i.e.* $27$ straight lines before $r(t)$ changes its sense of variation).
The particle completely sweeps over twice the 2D configuration space after $1,537\ \rm{fm}/c$. However, we notice that the bound trajectories are open, which means that the orbits never pass twice through a given point (see Fig. \[fig:2\]), which is in concordance with Bertrand’s theorem. We briefly remind that for a bound orbit to be closed, the angle between two consecutive apocenters must be:
$$\Delta \theta = 2\pi \cdot \frac{n_{1}}{n_{2}},$$
*i.e.* after $n_{2}$ revolutions about the center, the radius vector should sweep out a multiple $n_{1}$ of $2\pi$ [radians]{}. For the [UCE]{} case above considered we consequently obtain: $\Delta \theta_{UCE} \approx 4\pi/13$ [radians]{} (Fig. \[fig:2\] - left column).
The kinetic energy points are displayed in right isosceles triangular shaped patterns (Fig. \[fig:3\]), whose hypotenuses are described by Eqs. (11) and (12), for the two specific non-chaotical situations: [UCE]{} and the intermittent monopolar “window” | 1 | member_51 |
emerged at $\Omega = 0.1\ c/\rm{fm}$:
$$E_{L_{UCE}} = 18.05 - E_r,$$
$$E_{L_{\Omega=0.1\: c/\rm{fm}}} = 18.09 - E_r.$$
Moreover, for the intermittency frequency of monopole oscillations, one can notice a second smaller segment with the same negative slope:
$$E_{L_{\Omega=0.1\: c/\rm{fm}}} = 5.87 - E_r.$$
For the uncoupled differential equations there are a couple of extreme values for the centrifugal kinetic energy: $E_{L_{min}}(UCE) = 1.02\ \rm{MeV}$, and $E_{L_{Max}}(UCE) = 18.05\ \rm{MeV}$, associated with $r_{Max}(UCE)$ and respectively, with $r_{min}(UCE)$ (left column of Fig. \[fig:3\] and Eq. (1)).
As for the monopolar intermittency, we can distinguish just five distinct values for the $E_{L}$: $1.32\ \rm{MeV}$, $2.13\ \rm{MeV}$, $3.01\ \rm{MeV}$, $5.87\ \rm{MeV}$, and $18.09\ \rm{MeV}$ (central plot of Fig. \[fig:3\]), correlated with stationary radii: $r_{1} \approx 5.26\ \rm{fm}$, $r_{2} %
\approx 4.14\ \rm{fm}$, $r_{3} \approx 3.49\ \rm{fm}$, $r_{4} \approx 2.51\ \rm{fm}$, and $r_{5} \approx %
1.43\ \rm{fm}$ (Eq. (1), Fig. 2 of [@felea-09a], and central plot of Fig. \[fig:2\]). Thus, the nucleonic motion for the intermittent case is composed of alternated revolutions about the force field centre, forming a cyclic symmetrical structure, for e.g., $r_{1}$, $r_{5}$, $r_{2}$, $r_{4}$, $r_{3}$, $r_{4}$, $r_{2}$, $r_{5}$, $r_{1}$, and so on (Fig. \[fig:2\]). This behaviour can be easily verified through | 1 | member_51 |
the sensitivity dependence on the initial conditions analysis, previously presented (third column of Fig. 2 - [@felea-09a]). It should also be mentioned that, following this radius alternation, the nucleon covers in 2D configuration space $\approx 2\pi$ radians after $6$ full revolutions in almost $590\ \rm{fm}/c$.
Concluding, we highlight once more, that ordered, non-chaotical events, exhibit periodical symmetrical patterns in the configuration and kinetic energy spaces. This was shown to be a characteristic feature of the uncoupled nonlinear Hamilton equations case and also, of the steady, intermittent behaviour arisen in the monopole case at $0.1\ c/\rm{fm}$ vibrational radian frequency.
A tendency to compactly fill the kinetic energy space when increasing the monopolar vibrations (from $0.05\ c/\rm{fm}$ to $0.145\ c/\rm{fm}$) was observed, except for the intermittency situation above described. For the $L = 1$ oscillation mode of the potential well, it seems that at the same frequency ($\Omega = 0.1\ c/\rm{fm}$) a somewhat intermittent behaviour could also come out, but this was proved to be elusive, as verified when reverting to this issue with the help of informational entropies and Lyapunov exponents and analyzing the system on longer time periods.
Power spectra
-------------
In order to better distinguish between a multiple periodical | 1 | member_51 |
behaviour that can also exhibit an erratic pattern and chaos we used the Fourier transform of the analyzed signals:
$$x\left( \omega \right) = \lim_{T\rightarrow \infty} \int^{T}_{0} e^{i\omega t} \cdot x\left( t\right) \ dt\ ,$$
$$x\left( \omega \right) = \lim_{N\rightarrow \infty} \sum^{N}_{n=0} e^{i\omega t_{n}} \cdot x\left( t_{n}\right).$$
For a multiple periodical movement the power spectrum:
$$P\left( \omega \right) = \left|x\left( \omega \right)\right|^{2},$$
will only contain a number of discrete lines: the fundamental frequencies of the system and their associated sets of harmonics, while the chaotical behaviour is completely aperiodical and is represented by a continuous or quasi-continuous broadband.
The obtained results are presented in Figures \[fig:4\]-\[fig:7\]. As a persistent feature of the physical system analyzed one should mention that for the monopole and dipole deformation degrees of the potential well (Figures \[fig:5\] and \[fig:6\]) the chaotic behaviour increases in time, thus confirming previous results.
The transition towards a chaotic regime was put again in evidence once passing from the adiabatic to the resonance stage of the interaction. The power spectra reveal, as expected, the intermittent feature of the transition in the monopolar case at $\Omega=0.1\ c/\rm{fm}$.
This can be detected for periods of time large enough ($\Delta t \geq 1,600\ \rm{fm}/c$) | 1 | member_51 |
to positively identify chaotic patterns, by transition from a quasi-continuous spectrum of the one-nucleon radial coordinate ($\Omega _{ad}=0.02\ c/\rm{fm}$) to a discrete periodical one, containing fundamental frequencies of the system and its harmonics ($\Omega=0.1\ c/\rm{fm}$) and again to a continuous spectrum at the resonance vibrational frequency ($\Omega _{res}=0.145\ c/\rm{fm}$) (Fig. \[fig:5\] - right panels). The order-chaos-order-chaos sequence can be also spotted out for the monopole oscillations in the power spectra of the collective degree of freedom (Fig. \[fig:7\] - second column).
The temporal series of the radius variable show a symmetrical sawtooth waveform for the uncoupled situation at any chosen vibration frequency, and also for the monopole case at adiabatic collective oscillations. For the rest, in general an asymmetrical sawtooth form defines the series, but sometimes, more complicated patterns appear at higher multipole orders (Figures 1-4 - [@felea-09a]).
The difference between two successive maxima in the temporal series of the radius variable for the [UCE]{} case is: $T_{0_{UCE}} \approx 1,537/26 = 59.1\ \rm{fm}/c$ (left column of Fig. \[fig:2\]) and can be also obtained from the sensitive dependence on the initial conditions analysis (see Fig. 1 - [@felea-09a]). The fundamental frequency for the radial sawtooth temporal series: $\omega_{0_{UCE}}=2\pi/T_{0_{UCE}}=0.106\ c/\rm{fm}$ and its | 1 | member_51 |
first three harmonics: $\omega_{1_{UCE}}=0.212\ c/\rm{fm}$, $\omega_{2_{UCE}}=0.318\ c/\rm{fm}$, and $\omega_{3_{UCE}}=0.424\ c/\rm{fm}$, can be easily traced down in Figure \[fig:4\].
As for the “window” of intermittency at $L = 0$, we obtained: $T_{0_{int}} \approx 590/12 = 49.2\ \rm{fm}/c$ (Fig. 2 - [@felea-09a] and central plot of current Fig. \[fig:2\]). This gives the corresponding fundamental frequency: $\omega_{0_{int}}=0.128\ c/\rm{fm}$ and its associated harmonics: $\omega_{1_{int}}=0.256\ c/\rm{fm}$, $\omega_{2_{int}}=0.384\ c/\rm{fm}$, and $\omega_{3_{int}}=0.512\ c/\rm{fm}$ (Fig. \[fig:5\]).
Shannon entropies
-----------------
In order to further investigate route to chaos, we paid attention to the time evolution of the generalized informational entropy (or Shannon entropy), introduced as usually :
$$S_{Shannon}\left(t\right)=-{\sum^{N\left(t\right)}_{k=1}} p_k \cdot \ln p_k,$$
$N\left(t\right)$ being the number of gradually occupied cells until the time $t$.
This type of entropy is actually a number which quantifies the time rate of information production for a chaotic trajectory [@ott-93]. We consider in the first place the case of a particle that at every moment occupies a cell of the two-dimensional lattice phase space with a $p_k$ probability:
$$p_k=1/N_{total\ cells},$$
where:
$$N_{total\ cells}=N_{r} \cdot N_{p_{r}} \cdot N_{\theta},$$
$N_{r}$, $N_{p_{r}}$, and $N_{\theta}$ are the number of bins of the $\left( r,p_{r},\theta\right)$ lattice. For $p_{\theta}$ is a constant of motion for the monopole and the [UCE]{} | 1 | member_51 |
cases, we use for comparisons only these three phase space variables.
As an alternative measure for the above defined entropy we also used the cumulative filling percentage of the one-nucleon phase space:
$$\eta\left(t\right)=\frac{N\left(t\right)}{N_{total\ cells}} \cdot 100\ \left(\%\right).$$
In the first place, for a given wall frequency of vibration and for a certain multipolarity (here, for $\Omega _{res}=0.145\ c/\rm{fm}$ and $L = 0$), we studied the dependence of the Shannon entropy with the number of bins. A clear tendency for smoothing the entropy curve was found when decreasing the bin. A reduced number of cells ($N_{b}=2^3$) is characterized by an entropy formed from a small number of high-amplitude Heaviside functions. As the number of bins increases (for e.g., here to $12^3$), the entropy gets a more realistic representation, being composed of a superior number of low-amplitude step functions (Fig. \[fig:8\]).
Moreover, the filling percentage $\eta$ of the one-nucleon phase space maps can drastically differ with the size of the bin. Thus, after the system evolved over $400\ \rm{fm}/c$, a phase space with $8$ bins is entirely covered, $64$ bins can be filled in with $0.8594$ probability, a $26.95$ filling percentage for $512$ cells can be found, and we counted only as | 1 | member_51 |
much as $226$ bins occupied out of a total of $1,728$ (*i.e.* $\eta = 13.08\ \%$).
At a first glance one can identify a series of entropy plateaus, which could be put in correspondence with stationary or quasi-stationary thermodynamic values of the system if a large number of particles would be under study. Some of them will vanish when considering a large number of bins. However, those surviving for $N_{b} \rightarrow \infty$ could be associated with stationary nucleonic states in the chosen potential well in the limit of a large number of degrees of freedom.
For a given 2D phase space lattice formed of $N_{b}=4^3$ bins we present in Figures \[fig:9\]-\[fig:12\] a comparison between the informational entropies of the physical system in study, starting from the adiabatic stage of interaction and gradually increasing the vibrational wall frequency towards the dipole resonance value, $\Omega _{res}=0.145\ c/\rm{fm}$. The slopes for the resonance frequency case were found to be significantly higher than for the adiabatic one ($\Omega _{ad}=0.02\ c/\rm{fm}$) for all multipolarities involved.
Another comparison revealed significant differences between the onset times of the quasi-constant Shannon entropy values for all cases taken into consideration. Thus, for four vibrational radian frequencies and for four | 1 | member_51 |
coupling modes of the Hamilton equations we show the informational entropy values after $800\: \rm{fm}/c$ (Table \[tab:1\]) and the associated phase space filling degrees (Table \[tab:2\]). Also, in Table \[tab:3\], are presented the periods of time after which the filling percentages $\eta$ equal unity.
----------------------- ------------------------------------------------------------------------------------- -- -- --
Oscillation frequency [UCE]{} & $L=0$ & $L=1$ & $L=2$\
$\Omega \:_{ad} = 0.020\: c/\rm{fm}$ & $3.6889$ & $3.7136$ & $3.7842$ & $3.4340$\
$\Omega \:_{ad} = 0.050\: c/\rm{fm}$ & $3.6889$ & $4.0775$ & $4.0775$ & $3.2958$\
$\Omega \:\;\ \ = 0.100\: c/\rm{fm}$ & $3.6889$ & $3.6889$ & $3.8501$ & $4.0073$\
$\Omega _{res} = 0.145\: c/\rm{fm}$ & $3.6889$ & $4.1589$ & $4.0431$ & $4.0254$\
----------------------- ------------------------------------------------------------------------------------- -- -- --
: \[tab:1\]The computed $S_{Shannon}\left(t=800\: \rm{fm}/c\right)$ of the $\left( r\leftrightarrow p_{r}\leftrightarrow\theta\right)$ one-particle phase space maps at several multipolarities and frequencies of wall vibration
----------------------- --------------------------------------------------------------------------------- -- -- --
Oscillation frequency [UCE]{} & $L=0$ & $L=1$ & $L=2$\
$\Omega \:_{ad} = 0.020\: c/\rm{fm}$ & $62.50$ & $64.06$ & $68.75$ & $48.44$\
$\Omega \:_{ad} = 0.050\: c/\rm{fm}$ & $62.50$ & $92.19$ & $92.19$ & $42.19$\
$\Omega \:\;\ \ = 0.100\: c/\rm{fm}$ & $62.50$ & $62.50$ & $73.44$ & $85.94$\
$\Omega _{res} = 0.145\: c/\rm{fm}$ & $62.50$ & | 1 | member_51 |
$100.00$ & $89.06$ & $87.50$\
----------------------- --------------------------------------------------------------------------------- -- -- --
: \[tab:2\]The filling percentage $\eta$ of the $\left( r\leftrightarrow p_{r}\leftrightarrow\theta\right)$ one-particle phase space maps at several multipolarities and frequencies of wall vibration
----------------------- ------------------------------------------------------------------------------------ -- -- --
Oscillation frequency [UCE]{} & $L=0$ & $L=1$ & $L=2$\
$\Omega \:_{ad} = 0.020\: c/\rm{fm}$ & $>10^{5}$ & $6,023$ & $6,359$ & $5,356$\
$\Omega \:_{ad} = 0.050\: c/\rm{fm}$ & $>10^{5}$ & $1,618$ & $4,223$ & $>10^{5}$\
$\Omega \:\;\ \ = 0.100\: c/\rm{fm}$ & $>10^{5}$ & $11,442$ & $3,241$ & $2,758$\
$\Omega _{res} = 0.145\: c/\rm{fm}$ & $>10^{5}$ & $729$ & $1,887$ & $10,571$\
----------------------- ------------------------------------------------------------------------------------ -- -- --
: \[tab:3\]The time (in $\rm{fm}/c$) at which the informational entropies of the $\left( r\leftrightarrow p_{r}\leftrightarrow\theta\right)$ one-particle phase space maps at several multipolarities and frequencies of wall vibration have the maximum value (*i.e.* $\eta = 100\ \%$)
We continue the analysis by further defining the Shannon entropy for a group of $w$ nearby orbits:
$$S_{traject.\ pack}\left(t\right)=ln\ N_{w}\left(t\right),$$
so that the number of occupied cells is:
$$1 \leq N_w(t) \leq w,$$
thus describing the spread of the trajectories at each moment of time $t$ (Figs. \[fig:13\]-\[fig:16\]). When reaching the maximum divergence, the entropy for five distinct phase space | 1 | member_51 |
paths gets its highest value (Table \[tab:4\]).
----------------------- ------------------------------------------------------------------------------- -- -- --
Oscillation frequency [UCE]{} & $L=0$ & $L=1$ & $L=2$\
$\Omega \:_{ad} = 0.020\: c/\rm{fm}$ & $>10^{4}$ & $1,095$ & $555$ & $688$\
$\Omega \:_{ad} = 0.050\: c/\rm{fm}$ & $>10^{4}$ & $855$ & $476$ & $122$\
$\Omega \:\;\ \ = 0.100\: c/\rm{fm}$ & $>10^{4}$ & $4,133$ & $333$ & $395$\
$\Omega _{res} = 0.145\: c/\rm{fm}$ & $>10^{4}$ & $279$ & $327$ & $469$\
----------------------- ------------------------------------------------------------------------------- -- -- --
: \[tab:4\]The time (in $\rm{fm}/c$) at which the one-particle Shannon entropies of a pack of $w=5$ close orbits begin having the maximum value (*i.e.* $S_{traject.\ pack} = 1.60944$) for various coupling degrees between the one-nucleon and the collective *d.o.f.* and for the standard chosen wall frequencies
We begin the analysis with the [UCE]{} case. The single and collective uncoupled *d.o.f.* give birth to a quasi-laminar behaviour with a weak development of chaotic states. The one-particle informational entropy shows an identical evolution, no matter the frequency chosen. The orbit covers, after $800\ \rm{fm}/c$, only $62.50\ \%$ of the entire lattice (Table \[tab:2\] and Figure \[fig:9\]) and does not reach $100\ \%$, even after $\Delta t = 100,000\ \rm{fm}/c$ (Table \[tab:3\]). Also, the | 1 | member_51 |
phase space is not covered up by all five trajectories for the whole range of $10,000\ \rm{fm}/c$ considered, when analyzing $S_{traject.\ pack}$ (Fig. \[fig:13\] and Table \[tab:4\]).
For the dipole oscillations mode, at $\Omega_{ad} = 0.05\ c/\rm{fm}$, it appears that, after only $800\ \rm{fm}/c$, the entropy closes in upon its maximum value: $S_{Max} = ln\ N_{total\ cells} = 4.1589$ (Fig. \[fig:11\] and Table \[tab:1\]). However, on long periods of time, the real tendency is towards filling up the nucleonic phase space as rapid as the vibrational frequency is increased (Table \[tab:3\]). The exact pattern is repeated when studying the Shannon entropy for closeby nucleonic trajectories (Fig. \[fig:15\] and Table \[tab:4\]).
We found quite the same feature for the monopole case, with exception for the intermittent “window” at $\Omega=0.1\ c/\rm{fm}$ (Tables \[tab:1\], \[tab:2\] and Fig. \[fig:10\]). The occupying rate is so small in the intermittent zone, that just at $11,442\ \rm{fm}/c$, the particle would have covered the whole phase space (see Table \[tab:3\]). A similar conclusion can be drawn from Table \[tab:4\] and Fig. \[fig:14\] (with a double temporal scale scanned for the intermittent frequency). The trajectory pack informational entropy reaches its highest value after the longest one-particle evolution time of | 1 | member_51 |
all: $4,133\ \rm{fm}/c$.
The quadrupole oscillation also reveals an apparent intermittent pattern, this time at $\Omega_{ad} = %
0.05\ c/\rm{fm}$. We call it intermittent because after $800\ \rm{fm}/c$ the nucleon fills in only $42.19\ \%$ of the total number of bins (Figure \[fig:12\] and Table \[tab:2\]), and a longer time than $100,000\ \rm{fm}/c$ is required to get to $\eta = 100\ \%$ (Table \[tab:3\]). However, this behaviour can be a misleading one, the Shannon entropy for a trajectory bunch showing exactly the opposite (see Fig. \[fig:16\] and Table \[tab:4\]), after $122\ \rm{fm}/c$ the orbits being completely dispersed.
Lyapunov exponents
------------------
We furthermore presented another quantitative analysis: the temporal evolution of the Lyapunov exponents, $\lambda\left(t\right)$. As previously shown, initial adjacent points in the phase space $\Delta x_{0} (t=0)$, can generate in time separated trajectories $\Delta x\left(t\right)$. When studying the evolution of a single phase space parameter, the one-dimensional Lyapunov exponent takes the form:
$$\lambda\left(t\right)=\lim_{\left|\Delta x_{0}\right|\rightarrow 0} ln\left|\frac{\Delta x\left(t\right)}{\Delta x_{0}}\right|.$$
The generalization for obtaining the multi-dimensional Lyapunov exponent is then straightforward:
$$\lambda\left(t\right)=ln\frac{\left(\sum_{k=1}^{m}\left[x_{k}\left(t\right)-x_{k0}\right]^{2}\right)^{\frac{1}{2}}}{0.01},$$
where the sum is taken over all $m = 4$ squared differences between final $x_{k}\left(t\right)$ and initial $x_{k0}$ one-nucleon phase space variables. Integration times of the order of $10^{3}$ [fm]{}/c | 1 | member_51 |
exclude errors when computing the Lyapunov exponents.
In short, we here remind that the trajectories can be classified as function of the Lyapunov exponents. Thus, one can distinguish periodical behaviours, for $\lambda=0$, dissipative movements with a fixed point or a basin of attraction $\left(\lambda<0\right)$, and aperiodical chaotic states $\left(\lambda>0\right)$, when the iterative discrete evolution of the solution series (Eqs. 5 and 6) leads to a chaotic pattern.
Another way of measuring the system sensitivity to initial conditions is to compute the largest Lyapunov exponent ([LLE]{}). Usually a couple of methods can be employed, one based on the time dependence of the multi-dimensional Lyapunov exponent, the other on Wolf’s standard method that uses a Gram-Schmidt Reorthonormalization of the tangent vectors [@wolf-85]. In the latter, the [LLE]{} is obtained by taking the asymptotic value of the multi-dimensional Lyapunov exponent:
$$\lambda_{Max}=\lim_{t \rightarrow \infty} \frac{\lambda\left(t\right)}{t}.$$
Still, this method has the disadvantage that the integration times have to be at least an order of magnitude larger than those here considered. Other methods are slightly less efficacious, being more CPU time-consuming when simulating strong chaotic systems [@ramasubramanian-00].
We consequently used the first method and noticed the saturation behaviour, *i.e.* the arising of a plateau after a | 1 | member_51 |
certain time $t_{c}$ (Fig. \[fig:17\]). The straight lines represent fits whose slopes match the [LLE]{} (Table \[tab:5\]). They are in inverse proportion with the onset times of chaoticity $(\tau = 1/\lambda_{Max})$, being a measure of the trajectory decoupling at a microscopic level.
[cccc]{} Oscillation frequency & $L=0$ & $L=1$ & $L=2$\
$\Omega \:_{ad} = 0.020\: c/\rm{fm}$ & $0.003939$ & $0.008689$ & $0.004306$\
$\Omega \:_{ad} = 0.050\: c/\rm{fm}$ & $0.004432$ & $0.009829$ & $0.023203$\
$\Omega \:\;\ \ = 0.100\: c/\rm{fm}$ & $0.000761$ & $0.015454$ & $0.014402$\
$\Omega _{res} = 0.145\: c/\rm{fm}$ & $0.008739$ & $0.016662$ & $0.010086$\
When the single-particle and collective *d.o.f.* remain uncoupled, the Lyapunov exponents basically oscillate between two quasi-stationary regimes. This happens for all vibrational frequencies involved, reflecting a periodical regrouping of orbits in two basins of attraction. The phase space not being covered, even after a hundred of thousand of $\rm{fm}/c$, computing the [LLE]{} becomes futile for this case.
One can remark for dipole oscillations (Fig. \[fig:17\] - middle column) a faster evolution towards reaching saturation states of the 4-dimensional Lypaunov exponents, once passing from the adiabatic $(\tau _{ad}=115\ \rm{fm}/c)$ to the resonance phase of the interaction $(\tau _{res}=60\ \rm{fm}/c)$.
In the monopolar case the intermittency | 1 | member_51 |
can be easily traced at $0.1\ c/\rm{fm}$ vibrational frequency (Fig. \[fig:17\] - left panels). During the intermittent stage, independent nearby orbits microscopically diverge with the slowest rate of all: $\tau =1,314\ \rm{fm}/c$ (Table \[tab:5\]). In order to catch the heaving in sight of the stationary plateau at $\approx 4,917\ \rm{fm}/c$, the temporal scale was scanned over $6,400\ \rm{fm}/c$.
The study of the quadrupole collective oscillation case confirms the results obtained with all previous analyses. Namely, the neighbouring trajectories deviate one from each other after just $43\ \rm{fm}/c$ at an adiabatic frequency: $0.05\ c/\rm{fm}$. Also, when increasing $\Omega$, the [LLE]{} evolution pattern exactly matches that found with informational entropy measured for a group of orbits (Tables \[tab:4\] and \[tab:5\]).
\[sec:3\]Conclusions
====================
We investigated the chaotic nucleonic behaviour in a two-dimensional deep Woods-Saxon potential well for specific phases of the nuclear interaction. By comparing the order-to-chaos transition for these cases of interest, from adiabatic to resonance regime, it was shown that the couplings between the one-particle dynamics and high multipole vibrational modes significantly decrease the onset of the chaotic nucleonic motion towards realistic nuclear interaction time scale.
The quantitative study enfolded a plethora of analyses, pointing out that the paths to chaos | 1 | member_51 |
for the “nuclear billiard” are dissimilar for the studied multipolarities. For the first two multipole degrees we noticed a more rapid emergence of chaotic states as moving on towards higher radian frequencies of oscillation. When analyzing the system with quadrupole collective deformations of the potential well, an order-strong chaos-weak chaos-order sequence is revealed. Still, as emphasized in the “Shannon entropies” subsection, the quadrupole case represents an intricate one, and further analysis would be required before concluding it.
Every type of quantitative analysis strengthened previous results regarding the monopolar intermittency route to chaos for the “nuclear billiard”. The collective oscillation frequency for the intermittent behaviour was located prior to the resonance state of interaction (at $\Omega =0.1\ c/\rm{fm}$).
Further studies along the above issues are currently in progress. The used formalism can be improved by adding spin and charge to the nucleons. A semi-quantal treatment of this problem, including Pauli blocking effect, is hoped to shed more light on the discussed issue in the near future.
We wish to thank to R.I. Nanciu, I.S. Zgură, A.Ş. Cârstea, G. Păvălaş, S. Zaharia, A. Gheaţă, M. Rujoiu, A. Mitruţ, and R. Mărginean for fruitful discussions on this paper.
G.F. Burgio, M. Baldo, A. | 1 | member_51 |