34 research outputs found
Best-fit quasi-equilibrium ensembles: a general approach to statistical closure of underresolved Hamiltonian dynamics
A new method of deriving reduced models of Hamiltonian dynamical systems is
developed using techniques from optimization and statistical estimation. Given
a set of resolved variables that define a model reduction, the
quasi-equilibrium ensembles associated with the resolved variables are employed
as a family of trial probability densities on phase space. The residual that
results from submitting these trial densities to the Liouville equation is
quantified by an ensemble-averaged cost function related to the information
loss rate of the reduction. From an initial nonequilibrium state, the
statistical state of the system at any later time is estimated by minimizing
the time integral of the cost function over paths of trial densities.
Statistical closure of the underresolved dynamics is obtained at the level of
the value function, which equals the optimal cost of reduction with respect to
the resolved variables, and the evolution of the estimated statistical state is
deduced from the Hamilton-Jacobi equation satisfied by the value function. In
the near-equilibrium regime, or under a local quadratic approximation in the
far-from-equilibrium regime, this best-fit closure is governed by a
differential equation for the estimated state vector coupled to a Riccati
differential equation for the Hessian matrix of the value function. Since
memory effects are not explicitly included in the trial densities, a single
adjustable parameter is introduced into the cost function to capture a
time-scale ratio between resolved and unresolved motions. Apart from this
parameter, the closed equations for the resolved variables are completely
determined by the underlying deterministic dynamics
A Mean-field statistical theory for the nonlinear Schrodinger equation
A statistical model of self-organization in a generic class of
one-dimensional nonlinear Schrodinger (NLS) equations on a bounded interval is
developed. The main prediction of this model is that the statistically
preferred state for such equations consists of a deterministic coherent
structure coupled with fine-scale, random fluctuations, or radiation. The model
is derived from equilibrium statistical mechanics by using a mean-field
approximation of the conserved Hamiltonian and particle number for
finite-dimensional spectral truncations of the NLS dynamics. The continuum
limits of these approximated statistical equilibrium ensembles on
finite-dimensional phase spaces are analyzed, holding the energy and particle
number at fixed, finite values. The analysis shows that the coherent structure
minimizes total energy for a given value of particle number and hence is a
solution to the NLS ground state equation, and that the remaining energy
resides in Gaussian fluctuations equipartitioned over wavenumbers. Some results
of direct numerical integration of the NLS equation are included to validate
empirically these properties of the most probable states for the statistical
model. Moreover, a theoretical justification of the mean-field approximation is
given, in which the approximate ensembles are shown to concentrate on the
associated microcanonical ensemble in the continuum limit.Comment: 24 pages, 2 figure
Reduced Models of Point Vortex Systems
Nonequilibrium statistical models of point vortex systems are constructed using an optimal closure method, and these models are employed to approximate the relaxation toward equilibrium of systems governed by the two-dimensional Euler equations, as well as the quasi-geostrophic equations for either single-layer or two-layer flows. Optimal closure refers to a general method of reduction for Hamiltonian systems, in which macroscopic states are required to belong to a parametric family of distributions on phase space. In the case of point vortex ensembles, the macroscopic variables describe the spatially coarse-grained vorticity. Dynamical closure in terms of those macrostates is obtained by optimizing over paths in the parameter space of the reduced model subject to the constraints imposed by conserved quantities. This optimization minimizes a cost functional that quantifies the rate of information loss due to model reduction, meaning that an optimal path represents a macroscopic evolution that is most compatible with the microscopic dynamics in an information-theoretic sense. A near-equilibrium linearization of this method is used to derive dissipative equations for the low-order spatial moments of ensembles of point vortices in the plane. These severely reduced models describe the late-stage evolution of isolated coherent structures in two-dimensional and geostrophic turbulence. For single-layer dynamics, they approximate the relaxation of initially distorted structures toward axisymmetric equilibrium states. For two-layer dynamics, they predict the rate of energy transfer in baroclinically perturbed structures returning to stable barotropic states. Comparisons against direct numerical simulations of the fully-resolved many-vortex dynamics validate the predictive capacity of these reduced model
Analysis of statistical equilibrium models of geostrophic turbulence
Statistical equilibrium lattice models of coherent structures in geostrophic turbulence, formulated by discretizing the governing Hamiltonian continuum dynamics, are analyzed. The first set of results concern large deviation principles (LDP's) for a spatially coarse-grained process with respect to either the canonical and/or the microcanonical formulation of the model. These principles are derived from a basic LDP for the coarse-grained process with respect to product measure, which in turn depends on Cramer's Theorem. The rate functions for the LDP'Ĺ› give rise to variational principles that determine the equilibrium solutions of the Hamiltonian equations. The second set of results addresses the equivalence or nonequivalence of the microcanonical and canonical ensembles. In particular, necessary and sufficient conditions for a correspondence between microcanonical equilibria and canonical equilibria are established in terms of the concavity of the microcanonical entropy. A complete characterization of equivalence of ensembles is deduced by elementary methods of convex analysis
An Introduction to the Thermodynamic and Macrostate Levels of Nonequivalent Ensembles
This short paper presents a nontechnical introduction to the problem of
nonequivalent microcanonical and canonical ensembles. Both the thermodynamic
and the macrostate levels of definition of nonequivalent ensembles are
introduced. The many relationships that exist between these two levels are also
explained in simple physical terms.Comment: Revtex4, 5 pages, 1 figur
Thermodynamic versus statistical nonequivalence of ensembles for the mean-field Blume-Emery-Griffiths model
We illustrate a novel characterization of nonequivalent statistical
mechanical ensembles using the mean-field Blume-Emery-Griffiths (BEG) model as
a test model. The novel characterization takes effect at the level of the
microcanonical and canonical equilibrium distributions of states. For this
reason it may be viewed as a statistical characterization of nonequivalent
ensembles which extends and complements the common thermodynamic
characterization of nonequivalent ensembles based on nonconcave anomalies of
the microcanonical entropy. By computing numerically both the microcanonical
and canonical sets of equilibrium distributions of states of the BEG model, we
show that for values of the mean energy where the microcanonical entropy is
nonconcave, the microcanonical distributions of states are nowhere realized in
the canonical ensemble. Moreover, we show that for values of the mean energy
where the microcanonical entropy is strictly concave, the equilibrium
microcanonical distributions of states can be put in one-to-one correspondence
with equivalent canonical equilibrium distributions of states. Our numerical
computations illustrate general results relating thermodynamic and statistical
equivalence and nonequivalence of ensembles proved by Ellis, Haven, and
Turkington [J. Stat. Phys. 101, 999 (2000)].Comment: 13 pages, 4 figures, minor typos corrected and one reference adde
An optimization principle for deriving nonequilibrium statistical models of Hamiltonian dynamics
A general method for deriving closed reduced models of Hamiltonian dynamical
systems is developed using techniques from optimization and statistical
estimation. As in standard projection operator methods, a set of resolved
variables is selected to capture the slow, macroscopic behavior of the system,
and the family of quasi-equilibrium probability densities on phase space
corresponding to these resolved variables is employed as a statistical model.
The macroscopic dynamics of the mean resolved variables is determined by
optimizing over paths of these probability densities. Specifically, a cost
function is introduced that quantifies the lack-of-fit of such paths to the
underlying microscopic dynamics; it is an ensemble-averaged, squared-norm of
the residual that results from submitting a path of trial densities to the
Liouville equation. The evolution of the macrostate is estimated by minimizing
the time integral of the cost function. The value function for this
optimization satisfies the associated Hamilton-Jacobi equation, and it
determines the optimal relation between the statistical parameters and the
irreversible fluxes of the resolved variables, thereby closing the reduced
dynamics. The resulting equations for the macroscopic variables have the
generic form of governing equations for nonequilibrium thermodynamics, and they
furnish a rational extension of the classical equations of linear irreversible
thermodynamics beyond the near-equilibrium regime. In particular, the value
function is a thermodynamic potential that extends the classical dissipation
function and supplies the nonlinear relation between thermodynamics forces and
fluxes
Optimal thermalization in a shell model of homogeneous turbulence.
We investigate the turbulence-induced dissipation of the large scales in a statistically homogeneous flow using an “optimal closure,” which one of us (BT) has recently exposed in the context of Hamiltonian dynamics. This statistical closure employs a Gaussian model for the turbulent scales, with corresponding vanishing third cumulant, and yet it captures an intrinsic damping. The key to this apparent paradox lies in a clear distinction between true ensemble averages and their proxies, most easily grasped when one works directly with the Liouville equation rather than the cumulant hierarchy. We focus on a simple problem for which the optimal closure can be fully and exactly worked out: the relaxation arbitrarily far-from-equilibrium of a single energy shell towards Gibbs equilibrium in an inviscid shell model of 3D turbulence. The predictions of the optimal closure are validated against DNS and contrasted with those derived from EDQNM closure