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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We use a spectral theory perspective to reconsider properties of the Riemann zeta function. In particular, new integral representations are derived and used to present its value at odd positive integers.