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.Comment: 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 comment
We study a system of M particles in contact with a large but finite reservoir of N >> M particles within the framework of the Kac master equation modeling random collisions. The reservoir is initially in equilibrium at temperature T = β −1 . We show that for large N, this evolution can be approximated by an effective equation in which the reservoir is described by a Maxwellian thermostat at temperature T . This approximation is proven for a suitable L 2 norm as well as for the Gabetta-Toscani-Wennberg (GTW) distance and is uniform in time.
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 1 β . 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 m N for large N ii) the relative entropy approaches its equilibrium value (at least) at an eventually exponential rate ∼
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