Complex Systems

Lattice Models of the Lorentz Gas: Physical and Dynamical Properties Download PDF

Philippe M. Binder
Center For Nonlinear Studies, Los Alamos National Laboratory,
Los Alamos, NM 87545, USA
Applied Physics Section, Yale University,
New Haven, CT 06520, USA


This paper examines the validity and usefulness of cellular automaton models of fluid motion by means of a simple problem in kinetic theory.

We formulate three lattice models of the motion of a particle in a two-dimensional matrix of fixed, randomly placed, non-overlapping scatterers (the Lorentz gas). We measure several macroscopic and microscopic properties of this system, such as diffusion coefficients and mean-free paths. The results agree with analytical predictions, except at a high density of scatterers, where the models break down.

We also study these models as discrete dynamical systems. The properties of their state-transition diagrams, which give the number of all possible trajectories of the particle and their lengths, are similar to those of chaotic and random discrete maps. This agrees with analytical predictions that this gas exhibits chaotic behavior.

For this problem, we conclude that agreement between cellular automaton simulations analytical results is very good.