Dynamical chaos in the Lorentz lattice gas

This paper provides an introduction to the applications of dynamical systems theory to nonequilibrium statistical mechanics, in particular to a study of nonequilibrium phenomena in Lorentz lattice gases with stochastic collision rules. Using simple arguments, based upon discussions in the mathematical literature, we show that such lattice gases belong to the category of dynamical systems with positive Lyapunov exponents. This is accomplished by showing how such systems can be expressed in terms of continuous phase space variables. Expressions for the Lyapunov exponent of a one-dimensional Lorentz lattice gas with periodic boundaries are derived. Other quantities of interest for the theory of irreversible processes are discussed.