Statistical mechanics of cellular automata

TLDR

Cellular automata are used as simple mathematical models to investigate self‑organization in statistical mechanics. The study analyzes elementary cellular automata with binary sites evolving deterministically by nearest‑neighbor rules, and briefly examines more complex automata, linking them to dynamical systems theory and computation. With simple initial states the automata evolve to homogeneous or self‑similar patterns with fractal dimensions ≈1.59 or ≈1.69, while random initial states produce irreversible self‑organization, and the resulting structures fall into two universality classes independent of initial conditions or rule details.

Abstract

Cellular automata are used as simple mathematical models to investigate self-organization in statistical mechanics. A detailed analysis is given of "elementary" cellular automata consisting of a sequence of sites with values 0 or 1 on a line, with each site evolving deterministically in discrete time steps according to definite rules involving the values of its nearest neighbors. With simple initial configurations, the cellular automata either tend to homogeneous states, or generate self-similar patterns with fractal dimensions \ensuremath{\simeq} 1.59 or \ensuremath{\simeq} 1.69. With "random" initial configurations, the irreversible character of the cellular automaton evolution leads to several self-organization phenomena. Statistical properties of the structures generated are found to lie in two universality classes, independent of the details of the initial state or the cellular automaton rules. More complicated cellular automata are briefly considered, and connections with dynamical systems theory and the formal theory of computation are discussed.