Deterministic cellular automata resembling diffusion

TL;DR

This paper studies number conserving cellular automata to understand their diffusion-like behavior, introducing the decompression ratio concept and revealing phase transition phenomena in their dynamics.

Contribution

It introduces the decompression ratio to analyze CA dynamics and proves a phase transition in rule 184, highlighting rules suitable for diffusion modeling.

Findings

Many rules show abrupt changes in decompression ratio with initial density.

Rule 184 exhibits a formal phase transition in decompression ratio.

Some rules have infinite decompression ratio, useful for diffusion modeling.

Abstract

We investigate number conserving cellular automata with up to five inputs and two states with the goal of comparing their dynamics with diffusion. For this purpose, we introduce the concept of decompression ratio describing expansion of configurations with finite support. We find that a large number of number-conserving rules exhibit abrupt change in the decompression ratio when the density of the initial pattern is increasing, somewhat analogous to the second order phase transition. The existence of this transition is formally proved for rule 184. Small number of rules exhibit infinite decompression ratio, and such rules may be useful for "engineering" of CA rules which are good models of diffusion, although they will most likely require more than two states.