The Kac Model Coupled to a Thermostat

TL;DR

This paper analyzes a particle system interacting with a thermal bath using the Kac model, demonstrating exponential convergence to equilibrium, propagation of chaos, and deriving an effective Boltzmann equation in the large-system limit.

Contribution

It provides rigorous proofs of exponential convergence to equilibrium and chaos propagation for the Kac model coupled to a thermostat, with explicit spectral gap analysis.

Findings

Exponential decay to equilibrium proven via spectral gap

Propagation of chaos established in the large-system limit

Effective Boltzmann equation derived for the one-particle marginal

Abstract

In this paper we study a model of randomly colliding particles interacting with a thermal bath. Collisions between particles are modeled via the Kac master equation while the thermostat is seen as an infinite gas at thermal equilibrium at inverse temperature $\beta$. The system admits the canonical distribution at inverse temperature $\beta$ as the unique equilibrium state. We prove that any initial distribution approaches the equilibrium distribution exponentially fast both by computing the gap of the generator of the evolution, in a proper function space, as well as by proving exponential decay in relative entropy. We also show that the evolution propagates chaos and that the one-particle marginal, in the large-system limit, satisfies an effective Boltzmann-type equation.