Deterministic Cellular Automata with Diffusive Behavior

Abstract

It is a classical result [1] that an ensemble of independent unbiased random walks on the one dimensional lattice, Z, and moving at discrete times, Z ⁺, has a continuum limit given by a diffusion equation. More recently, systems of randomly walking particles interacting via an exclusion principle have been studied [2] [3]. Another interesting problem is that of using deterministic dynamical systems for the same purpose. Of course, to the extent that the underlying microscopic dynamics of atoms in real diffusing media are deterministic, we know that this should be possible. In this work we describe two completely deterministic cellular automata that exhibit diffusive behavior in one dimension, possibly with spatial inhomogeneity. We analyze these automata both theoretically and experimentally to investigate their continuum limits. In the first of these, we experimentally find significant deviations from the Chapman-Enskog theory; these deviations are due to a buildup of correlations that invalidates the Boltzmann molecular chaos assumption. In the second, the relevant correlations have been suppressed, and good agreement with the Chapman-Enskog theory is obtained.