Sub-exponential mixing of random billiards driven by thermostats

TL;DR

This paper proves that certain open mechanical particle systems driven by thermostats have a unique steady state and exhibit sub-exponential mixing, with convergence affected by slow particles.

Contribution

It extends discrete-time results to continuous time, establishing sub-exponential mixing and convergence to the steady state for these systems.

Findings

Unique steady state exists in continuous-time systems.

Convergence to the steady state is sub-exponential.

Slow particles hinder exponential mixing.

Abstract

We study the class of open continuous-time mechanical particle systems introduced in the paper by Khanin and Yarmola [Ergodic Properties of Random Billiards Driven by Thermostats. Commun. Math. Phys. 320, no. 1, 121-147 (2013)]. Using the discrete-time results from that paper we demonstrate rigorously that, in continuous time, a unique steady state exists and is sub-exponentially mixing. Moreover, all initial distributions converge to the steady state and, for a large class of initial distributions, convergence to the steady state is sub-exponential. The main obstacle to exponential convergence is the existence of slow particles in the system.