Exponential Convergence to Non-Equilibrium Stationary States in Classical Statistical Mechanics

TL;DR

This paper proves that a chain of nonlinear oscillators coupled to heat reservoirs converges exponentially fast to a unique non-equilibrium stationary state, using Lyapunov functions and energy propagation bounds.

Contribution

It introduces a Lyapunov function for the chain dynamics and demonstrates exponential convergence to the stationary state in a nonlinear heat conduction model.

Findings

Existence of a Lyapunov function for the model.

Exponential convergence to a unique stationary state.

Bound on energy propagation in anharmonic chains.

Abstract

We continue the study of a model for heat conduction consisting of a chain of non-linear oscillators coupled to two Hamiltonian heat reservoirs at different temperatures. We establish existence of a Liapunov function for the chain dynamics and use it to show exponentially fast convergence of the dynamics to a unique stationary state. Ingredients of the proof are the reduction of the infinite dimensional dynamics to a finite-dimensional stochastic process as well as a bound on the propagation of energy in chains of anharmonic oscillators.