  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.
