Uniform Approximation of a Maxwellian Thermostat by Finite Reservoirs

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 .