Partially Thermostated Kac Model

We study a system of $N$ particles interacting through the Kac collision, with $m$ of them interacting, in addition, with a Maxwellian thermostat at temperature $frac{1}{beta}$. We use two indicators to understand the approach to the equilibrium Gaussian state. We prove that i) the spectral gap of the evolution operator behaves as $frac{m}{N}$ for large $N$ ii) the relative entropy approaches its equilibrium value (at least) at an eventually exponential rate $sim frac{m}{N^2}$ for large $N$. The question of having non-zero entropy production at time $0$ remains open. A relationship between the Maxwellian thermostat and the thermostat used in Bonetto, Loss, Vaidyanathan (J. Stat. Phys. 156(4):647-667, 2014) is established through a van Hove limit.

The Kac Model Coupled to a Thermostat

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.