ABSTRACT
The basic components of cellular automata are discrete. The investigation of cellular automaton models seems likely to provide some new insights. Based on master or transport equations, one can find partial differential equations satisfied by the densities of the extensive quantities conserved by the cellular automaton evolution. The mathematical origins of continuum behaviour in cellular automata are much the same as they are for many physical systems. A continuum system such as a fluid has the feature that its state can be described by just a few extensive quantities. A crucial issue, which relates to the foundations of thermodynamics, is the degree of randomness which is produced by a cellular automaton, or which, for that matter, is really necessary to reproduce macroscopic diffusion phenomena. If microscopic randomization is assumed, then overall continuum behaviour can be derived using statistical mechanics. The conservation laws necessary for macroscopic diffusion turn out to be quite straightforward to ensure in a class of cellular automata.
