On the polynomial convergence rate to nonequilibrium steady-states

TL;DR

This paper proves the existence, uniqueness, and polynomial convergence rate to the nonequilibrium steady state in a stochastic 1D heat conduction model, using a novel induced chain method.

Contribution

It introduces a new technique called the induced chain method to analyze convergence rates in nonequilibrium stochastic energy exchange models.

Findings

Proves existence and uniqueness of NESS in the model.

Establishes polynomial speed of convergence to NESS.

Demonstrates consistency with deterministic dynamical system properties.

Abstract

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 steady state (NESS) and, furthermore, the polynomial speed of convergence to the NESS. Our result shows that the asymptotic properties of this model and its deterministic dynamical system origin are consistent. The proof uses a new technique called the induced chain method. We partition the state space and work on both the Markov chain induced by an "active set" and the tail of return time to this "active set".