Four state deterministic cellular automaton rule emulating random diffusion

Abstract

We show how to construct a deterministic nearest-neighbour cellular automaton (CA) with four states which emulates diffusion on a one-dimensional lattice. The pseudo-random numbers needed for directing random walkers in the diffusion process are generated with the help of rule 30. This CA produces density profiles which agree very well with solutions of the diffusion equation, and we discuss this agreement for two different boundary and initial conditions. We also show how our construction can be generalized to higher dimensions.