Local thermal equilibrium for certain stochastic models of heat transport

TL;DR

This paper proves local thermal equilibrium and describes the energy profile in certain stochastic heat transport models out of equilibrium, using duality and random walk techniques.

Contribution

It establishes local thermal equilibrium and characterizes the energy profile in stochastic models of heat transport with nonconstant boundary temperatures.

Findings

Local marginal distributions tend to a Gibbs measure with position-dependent inverse temperature.

The mean energy profile satisfies Laplace's equation with boundary conditions.

The duality method links local energy distributions to hitting probabilities of random walks.

Abstract

This paper is about nonequilibrium steady states (NESS) of a class of stochastic models in which particles exchange energy with their "local environments" rather than directly with one another. The physical domain of the system can be a bounded region of $\mathbb R^d$ for any $d \ge 1$. We assume that the temperature at the boundary of the domain is prescribed and is nonconstant, so that the system is forced out of equilibrium. Our main result is local thermal equilibrium in the infinite volume limit. In the Hamiltonian context, this would mean that at any location $x$ in the domain, local marginal distributions of NESS tend to a probability with density $\frac{1}{Z} e^{-\beta (x) H}$, permitting one to define the local temperature at $x$ to be $\beta(x)^{-1}$. We prove also that in the infinite volume limit, the mean energy profile of NESS satisfies Laplace's equation for the prescribed boundary condition. Our method of proof is duality: by reversing the sample paths of particle movements, we convert the problem of studying local marginal energy distributions at $x$ to that of joint hitting distributions of certain random walks starting from $x$, and prove that the walks in question become increasingly independent as system size tends to infinity.