Grand Canonical Evolution for the Kac Model

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 fast 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.