Convergence of independent particle systems

We consider a system of particles moving independently on a countable state space, according to a general (non-space-homogeneous) Markov process. Under mild conditions, the number of particles at each site will converge to a product of independent Poisson distributions; this corresponds to settling to an ideal gas. We derive bounds on the rate of this convergence. In particular, we prove that the variation distance to stationarity decreases proportionally to the sum of squares of the probabilities of each particle to be at a given site. We then apply these bounds to some examples. Our methods include a simple use of the Chen-Stein lemma about Poisson convergence. Our results require certain strong hypotheses, which further work might be able to eliminate.