Construction of an isotropic cellular automaton for a reaction-diffusion equation by means of a random walk

TL;DR

This paper introduces a novel method to create isotropic cellular automata for reaction-diffusion equations using microscopic particle random walks and vector fields, successfully reproducing complex patterns like the Belousov-Zhabotinsky reaction.

Contribution

It presents a new approach to construct isotropic cellular automata that accurately model reaction-diffusion systems through microscopic particle dynamics.

Findings

The automaton retains isotropy and pattern formation.

Successfully applied to model Belousov-Zhabotinsky reaction.

Reproduces numerical solutions of reaction-diffusion equations.

Abstract

We propose a new method to construct an isotropic cellular automaton corresponding to a reaction-diffusion equation. The method consists of replacing the diffusion term and the reaction term of the reaction-diffusion equation with a random walk of microscopic particles and a discrete vector field which defines the time evolution of the particles. The cellular automaton thus obtained can retain isotropy and therefore reproduces the patterns found in the numerical solutions of the reaction-diffusion equation. As a specific example, we apply the method to the Belousov-Zhabotinsky reaction in excitable media.