Three-level description of the domino cellular automaton

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.