The Kac Model Coupled to a Thermostat
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 . The system admits the canonical distribution at inverse temperature 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.
Cite
@article{arxiv.1309.2715,
title = {The Kac Model Coupled to a Thermostat},
author = {Federico Bonetto and Michael Loss and Ranjini Vaidyanathan},
journal= {arXiv preprint arXiv:1309.2715},
year = {2014}
}
Comments
21 pages; obtained a stronger entropy decay result and hence modified the statement and proof of Theorem 1.3 accordingly; rewrote proof of Theorem 1.2 to improve clarity; and made other changes to address referee comments