Cellular Automata: From Structural Principles to Transport and Correlation Methods

Abstract: Cellular automata (CA) are discrete-time dynamical systems with local update rules on a lattice. Despite their elementary definition, CA support a wide spectrum of macroscopic phenomena central to statistical physics: equilibrium and nonequilibrium phase transitions, transport and hydrodynamic limits, kinetic roughening, self-organized criticality, and complex spatiotemporal correlations. This survey focuses on three tightly connected themes. \emph{(i)} We present a structural view of CA as shift-commuting maps on configuration spaces, emphasizing rule complexity, reversibility, and conservation laws (including discrete continuity equations). \emph{(ii)} We organize transport in CA into ballistic, diffusive, and anomalous regimes, and connect microscopic currents to macroscopic laws through Green--Kubo formulas, scaling theory, and universality classes. \emph{(iii)} We develop correlation-based methods -- from structure factors and response formulas to computational mechanics and data-driven inference -- that diagnose regimes and enable coarse-graining.