Three-level description of the domino cellular automaton

TL;DR

This paper develops a three-level analytical framework for a domino cellular automaton with avalanches, deriving exact relations and an Ito equation to describe its dynamics, with implications for analyzing natural time series.

Contribution

It introduces a novel three-level description of cellular automata and derives an Ito equation from the automaton's properties, bridging microscopic and macroscopic analysis.

Findings

Exact relations for automaton parameters in equilibrium

Approximate validity of relations for deviations from equilibrium

Construction of an Ito equation for automaton dynamics

Abstract

Inspired by the approach of kinetic theory of gases, a three-level description (microscopic, mesoscopic and macroscopic) of cellular automaton is presented. To provide an analytical treatment a simple domino cellular automaton with avalanches was constructed. Formulas concerning exact relations for density, clusters, avalanches and other parameters in an equilibrium state were derived. It appears that some relations are approximately valid for deviations from the equilibrium, so the adequate Ito equation could be constructed. The equation provides the time evolution description of some variable on the macroscopic level. The results also suggest a motive for applying of the procedure of construction of the Ito equation (from time series data) to natural time series.