  We study a model of random colliding particles interacting with an infinite
reservoir at fixed temperature and chemical potential. Interaction between the
particles is modeled via a Kac master equation \cite{kac}. Moreover, particles
can leave the system toward the reservoir or enter the system from the
reservoir. The system admits a unique steady state given by the Grand Canonical
Ensemble at temperature $T=\beta^{-1}$ and chemical potential $\chi$. We show
that any initial state converges exponentially to equilibrium by computing the
spectral gap of the generator in a suitable $L^2$ space and by showing
exponential decrease of the relative entropy with respect to the steady state.
We also show propagation of chaos and thus the validity of a Boltzmann-Kac type
equation for the particle density in the infinite system limit.
