Non-Equilibrium Steady States in Kac's Model Coupled to a Thermostat

TL;DR

This paper investigates the existence, uniqueness, and convergence to non-equilibrium steady states in Kac's model with external coupling, providing new insights into non-Maxwellian steady states and their behavior as particle number grows.

Contribution

It introduces methods to analyze non-equilibrium steady states in Kac's model with non-Maxwellian external coupling, including convergence rates and large particle limits.

Findings

Existence and uniqueness of non-equilibrium steady states.

Quantitative convergence rates to steady states.

Behavior of the model as the number of particles increases.

Abstract

This paper studies the existence, uniqueness and convergence to non-equilibrium steady states in Kac's model with an external coupling. We work in both Fourier distances and Wasserstein distances. Our methods work in the case where the external coupling is not a Maxwellian equilibrium. This provides an example of a non-equilibrium steady state. We also study the behaviour as the number of particles goes to infinity and show quantitative estimates on the convergence rate of the first marginal.