Journal of Statistical Physics, Vol. 81, Nos. I/2, 1995 Dynamical Chaos in the Lorentz Lattice Gas J. R. Dorfman, l"2 M. H. Ernst,~ and D. Jacobs 1"3 Received September 10, 1994; final December 8, 1994 This paper provides an introduction to the applications of dynamical systems theory to nonequilibrium statistical mechanics, in particular to a study of non- equilibrium 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. Expres- sions 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. KEY WORDS: Lorentz lattice gas; Lyapunov exponents; dynamical chaos; KS entropy; diffusion, kinetic theory of gases, random walks. 1. INTRODUCTION Over the past several years a number of workers have begun to explore the connections between the statistical mechanics of irreversible processes in fluids and the ergodic behavior of dynamical systems. Studies by Posch and Hoover I1~ and Evans et al. ~'-~ of fluid systems subjected to external forces producing shear flows or electric currents with internal Gaussian thermo- stats maintaining a constant kinetic or total energy led to very interesting connections between the transport coefficients and the Lyapunov exponents for the thermostatted systems. A good example of the work is the study of shear flows by Evans et al/2~. They were able to relate the coefficient of shear viscosity r/(),) of a thermostatted system subjected to an external force Institute for Theoretical Physics University of Utrecht, Utrecht, The Netherlands. 2 Permanent address: Department of Physics and IPST, University of Maryland, College Park, Maryland 20742. 3 Department of Physics, Michigan State University, East Lansing, Michigan 48824. 497 0022-4715/95/1000-0497507.50/0 (d) 1995 Plenum Publishing Corporation