Continuum versus Discrete: A Physically Interpretable General Rule For Cellular Automata By Means of Modular Arithmetic

Describing complex phenomena by means of cellular automata (CA) has shown to be a very effective approach in pure and applied sciences. In fact, the number of published papers concerning this topic has tremendously increased over the last twenty years. Most of the applications, notwithstanding, use cellular automata to qualitatively describe the phenomena, which is surely a consequence of the way the automata rules have been defined. In the present paper a general rule which describes every of Wolfram's cellular automata is derived. The new representation is given in terms of a new function hereby defined, the iota-delta function. The latter function is further generalized in order to provide a general rule for not only Wolfram's but also to every CA rule which depends on the sum and products of the values of cells in the automaton mesh. By means of a parallel between the finite difference method and the iota-delta function, the new representation provides a straightforward physical interpretation of CA, which gives, for the first time, a quantitative interpretation of the generating rule itself. By means of the new formulation, advective-diffusive phenomena are analyzed. In particular, the relation between CA automata and anomalous diffusion is briefly discussed.