Exponential approach to, and properties of, a non-equilibrium steady state in a dilute gas

TL;DR

This paper analyzes a kinetic model of a dilute gas interacting with multiple thermal reservoirs, proving the existence, uniqueness, and exponential convergence to a non-equilibrium steady state using a specialized metric, and deriving positive entropy production.

Contribution

It establishes the existence and uniqueness of a non-equilibrium steady state in a dilute gas model with multiple reservoirs and demonstrates exponential convergence using the GTW metric.

Findings

Proves existence and uniqueness of NESS in the model

Shows exponential convergence to NESS in GTW metric

Derives positive entropy production expressions

Abstract

We investigate a kinetic model of a system in contact with several thermal reservoirs at different temperatures $T_\alpha$. Our system is a spatially uniform dilute gas whose internal dynamics is described by the nonlinear Boltzmann equation with Maxwellian collisions. Similarly, the interaction with reservoir $\alpha$ is represented by a Markovian process that has the Maxwellian $M_{T_\alpha}$ as its stationary state. We prove existence and uniqueness of a non-equilibrium steady state (NESS) and show exponential convergence to this NESS in a metric on probability measures introduced into the study of Maxwellian collisions by Gabetta, Toscani and Wenberg (GTW). This shows that the GTW distance between the current velocity distribution to the steady-state velocity distribution is a Lyapunov functional for the system. We also derive expressions for the entropy production in the system plus the reservoirs which is always positive.