Stability of Continuous Cellular Automata

Abstract

We formally coarse-grain the microscopic equations of motion which determine the dynamical behavior of a class of cellular automata by projection-operator methods. Thus, we obtain a macroscopic description in terms of Langevin equations for the slow long-lifetime thermodynamic variables, such as order parameters. The dynamical rules for the automata are modeled by equations which are similar to Hamilton’s equations. For reversible automata, we obtain the standard results of nonequilibrium statistical physics, namely, fluctuation-dissipation relations, Onsager’s symmetry relations, and Green-Kubo relations. For dynamical rules which have both reversible and irreversible parts we find that none of these results apply. For irreversible rules, we obtain an exact result: the slow variable is linearly unstable due to a “fluctuation-enhancement” relation. This can imply that the structure in the irreversible cellular automata grows exponentially for early times. We also discuss the relationship of our results to the growth of ordered structure in physical systems.KeywordsCellular AutomatonDendritic GrowthLangevin EquationSlow VariableSpinodal DecompositionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.