# Asymptotic Behavior of Thermal Nonequilibrium Steady States for a Driven Chain of Anharmonic Oscillators

@article{ReyBellet2000AsymptoticBO,
title={Asymptotic Behavior of Thermal Nonequilibrium Steady States for a Driven Chain of Anharmonic Oscillators},
author={Luc Rey-Bellet and Lawrence E. Thomas},
journal={Communications in Mathematical Physics},
year={2000},
volume={215},
pages={1-24}
}
• Published 7 January 2000
• Mathematics
• Communications in Mathematical Physics
Abstract: We consider a model of heat conduction introduced in [6], which consists of a finite nonlinear chain coupled to two heat reservoirs at different temperatures. We study the low temperature asymptotic behavior of the invariant measure. We show that, in this limit, the invariant measure is characterized by a variational principle. The main technical ingredients are some control theoretic arguments to extend the Freidlin–Wentzell theory of large deviations to a class of degenerate…
Exponential Convergence to Non-Equilibrium Stationary States in Classical Statistical Mechanics
• Mathematics, Physics
• 2002
Abstract: We continue the study of a model for heat conduction [6] consisting of a chain of non-linear oscillators coupled to two Hamiltonian heat reservoirs at different temperatures. We establish
Nonequilibrium statistical mechanics of open classical systems
We describe the ergodic and thermodynamical properties of chains of anharmonic oscillators coupled, at the boundaries, to heat reservoirs at positive and dierent temperatures. We discuss existence
On the derivation of Fourier's law for coupled anharmonic oscillators
• Physics, Mathematics
• 2006
We study the Hamiltonian system made of weakly coupled anharmonic oscillators arranged on a three dimensional lattice and subjected to a stochastic forcing mimicking heat baths of temperatures T_1
Existence of Nonequilibrium Steady State for a Simple Model of Heat Conduction
• Physics
• 2013
This paper contains rigorous results for a simple stochastic model of heat conduction similar to the KMP (Knipnis–Marchiori–Presutti) model but with possibly energy-dependent interaction rates. We
On the polynomial convergence rate to nonequilibrium steady states
• Yao Li
• Mathematics
The Annals of Applied Probability
• 2018
We consider a stochastic energy exchange model that models the 1D microscopic heat conduction in the nonequilibrium setting. In this paper, we prove the existence and uniqueness of the nonequilibrium
Non-equilibrium steady states for networks of oscillators
• Mathematics
• 2018
Non-equilibrium steady states for chains of oscillators (masses) connected by harmonic and anharmonic springs and interacting with heat baths at different temperatures have been the subject of
Towards a Derivation of Fourier’s Law for Coupled Anharmonic Oscillators
• Physics
• 2007
We consider a Hamiltonian system made of weakly coupled anharmonic oscillators arranged on a three dimensional lattice $${\mathbb{Z}}_{2N} \times \mathbb{Z}^2$$, and subjected to stochastic forcing
Persistent energy flow for a stochastic wave equation model in nonequilibrium statistical mechanics
We consider a one-dimensional partial differential equation system modeling heat flow around a ring. The system includes a Klein-Gordon wave equation for a field satisfying spatial periodic boundary
Diffusive limit and Fourier's law for the discrete Schroedinger equation
We consider the one-dimensional discrete linear Schrodinger (DLS) equation perturbed by a conservative stochastic dynamics, that changes the phase of each particles, conserving the total norm (or

## References

SHOWING 1-10 OF 37 REFERENCES
Non-Equilibrium Statistical Mechanics¶of Strongly Anharmonic Chains of Oscillators
• Mathematics
• 1999
Abstract: We study the model of a strongly non-linear chain of particles coupled to two heat baths at different temperatures. Our main result is the existence and uniqueness of a stationary state at
Weak Noise Limit and Nonequilibrium Potentials of Dissipative Dynamical Systems
The possibility of generalizing thermodynamic potentials to non-equilibrium steady states of dissipative dynamical systems subject to weak stochastic perturbations is discussed. In the first part the
Non-Equilibrium Statistical Mechanics of Anharmonic Chains Coupled to Two Heat Baths at Different Temperatures
• Mathematics
• 1999
Abstract:We study the statistical mechanics of a finite-dimensional non-linear Hamiltonian system (a chain of anharmonic oscillators) coupled to two heat baths (described by wave equations). Assuming
Entropy Production in Nonlinear, Thermally Driven Hamiltonian Systems
• Mathematics
• 1999
We consider a finite chain of nonlinear oscillators coupled at its ends to two infinite heat baths which are at different temperatures. Using our earlier results about the existence of a stationary
Ergodic properties of classical dissipative systems I
• Mathematics
• 1998
We consider a class of models in which a Hamiltonian system A, with a finite number of degrees of freedom, is brought into contact with an infinite heat reservoir B. We develop the formalism required
Hamiltonian Derivation of a Detailed Fluctuation Theorem
We analyze the microscopic evolution of a system undergoing a far-from-equilibrium thermodynamic process. Explicitly accounting for the degrees of freedom of participating heat reservoirs, we derive
Dynamical ensembles in stationary states
• Physics
• 1995
We propose, as a generalization of an idea of Ruelle's to describe turbulent fluid flow, a chaotic hypothesis for reversible dissipative many-particle systems in nonequilibrium stationary states in
A Gallavotti–Cohen-Type Symmetry in the Large Deviation Functional for Stochastic Dynamics
• Mathematics
• 1999
We extend the work of Kurchan on the Gallavotti–Cohen fluctuation theorem, which yields a symmetry property of the large deviation function, to general Markov processes. These include jump processes
Fluctuation theorem for stochastic dynamics
The fluctuation theorem of Gallavotti and Cohen holds for finite systems undergoing Langevin dynamics. In such a context all non-trivial ergodic theory issues are bypassed, and the theorem takes a
Probability of second law violations in shearing steady states.
• Mathematics
Physical review letters
• 1993
An expression for the probability of fluctuations in the shear stress of a fluid in a nonequilibrium steady state far from equilibrium is given and a formula for the ratio that, for a finite time, theShear stress reverse sign is violating the second law of thermodynamics.