Exact stability results in stochastic lattice gas cellular automata

Abstract

In this paper we consider a lattice gas as a discrete Markov process, with a Markov operator\(\mathfrak{L}\) acting on the phase space of the lattice gas cellular automata. We are interested in the asymptotic properties of the sequences of densities in both Liouville and Boltzmann descriptions. We show that under appropriate hypotheses, in both descriptions, the sequence of densities are asymptotically periodic. It is then possible, by introducing a slight modification in the transition process, to avoid the existence of cycles and to ensure the stability of the stationary densities. We point out the particular part played by the regular global linear invariants that define the asymptotic Gibbs states in a one-to-one way for most models.