Cluster-approximation mean-field theory of a class of cellular automaton models

A cluster-approximation mean-field scheme, which may be used to describe some cellular automaton (CA) models, is introduced based on analyzing the evolution of cluster states. In this scheme the description of microscopic processes is exact within a cluster, and approximate for the interaction between clusters and the environment. The approach to building the theory is illustrated by a surface-reaction-like CA model very similar to the model of Ziff, Gulari, and Barshad. A set of equations of motion concerning the evolution of microscopic cluster states is derived, and some results from these equations are given.