  We study the evolution of a system of M particles in contact with a large
reservoir of N>>M particles. The reservoir is initially in equilibrium at
temperature T=1/\beta. The evolution of the system and reservoir is described
via a suitable Kac-style collision process. We show that for large N, this
evolution can be effectively described by replacing the reservoir with a
Maxwellian thermostat at temperature T. This description provides an
approximation that is uniform in time both in a suitable L^2 norm and in the
Gabetta-Toscani-Wennberg (GTW) distance.
