Approach to the steady state in kinetic models with thermal reservoirs at different temperatures

TL;DR

This paper investigates kinetic models with multiple thermal reservoirs at different temperatures, explicitly computes the unique non-equilibrium steady state, and proves exponential convergence to this state using probabilistic techniques.

Contribution

It introduces a new approach to analyze kinetic models with multiple reservoirs, explicitly computes the NESS, and proves exponential convergence with probabilistic methods.

Findings

Explicitly computed the unique non-equilibrium steady state (NESS).

Proved exponential approach to the NESS from any initial state.

Established non-existence of non-uniform NESS under simplified models.

Abstract

We continue the investigation of kinetic models of a system in contact via stochastic interactions with several spatially homogeneous thermal reservoirs at different temperatures. Considering models different from those investigated in earlier work, we explicitly compute the unique spatially uniform non-equilibrium steady state (NESS) and prove that it is approached exponentially fast from any uniform initial state. This leaves open the question of whether there exist NESS that are not spatially uniform. Making a further simplification of our models, we then prove non-existence of such NESS and exponential approach to the unique spatially uniform NESS (with a computably boundable rate). The method of proof relies on refined Doeblin estimates and other probabilisitic techniques, and is quite different form the analysis in earlier work that was based on contraction mapping methods.