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 ${\hat{I}}^2$. The system admits the canonical distribution at inverse temperature ${\hat{I}}^2$ 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.

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