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 $${N \gg M}$$ particles within the framework of the Kac master equation modeling random collisions. The reservoir is initially in equilibrium at temperature $${T=\beta^{-1}}$$ . 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. [ABSTRACT FROM AUTHOR]