Analytical description of the diffusion in a cellular automaton with the Margolus neighbourhood in terms of the two-dimensional Markov chain

TL;DR

This paper provides an exact analytical solution for diffusion in a two-dimensional cellular automaton with Margolus neighbourhood, using Markov chain analysis and Jacobi polynomials, refining previous empirical results.

Contribution

It introduces a Markov chain approach to analytically describe diffusion in cellular automata with Margolus neighbourhood, deriving explicit formulas for the diffusion coefficient.

Findings

Analytical expression for the probability distribution using Jacobi polynomials

Exact formula for the diffusion coefficient's dependence on parameters

Results agree with and refine previous empirical findings

Abstract

The one-parameter two-dimensional cellular automaton with the Margolus neighbourhood is analyzed based on the considering the projection of the stochastic movements of a single particle. Introducing the auxiliary random variable associated with the direction of the movement, we reduce the problem under consideration to the study of a two-dimensional Markov chain. The master equation for the probability distribution is derived and solved exactly using the probability generating function method. The probability distribution is expressed analytically in terms of Jacobi polynomials. The moments of the obtained solution allowed us to derive the exact analytical formula for the parametric dependence of the diffusion coefficient in the two-dimensional cellular automaton with the Margolus neighbourhood. Our analytic results agree with earlier empirical results of other authors and refine them. The results are of interest for the modelling two-dimensional diffusion using cellular automata especially for the multicomponent problem.