Complex Systems

Continuum versus Discrete: A Physically Interpretable General Rule for Cellular Automata by Means of Modular Arithmetic Download PDF

Luan C. de S. M. Ozelim *
André L. B. Cavalcante †
Lucas P. de F. Borges ‡
Department of Civil and Environmental Engineering, University of Brasília
Brasília, DF, 70910-900, Brazil
* luanoz@gmail.com
abrasil@unb.br Full address for mailing: Geotecnia—Departamento de Engenharia Civil e Ambiental/FT—UnB, 70910-900—Brasília/DF/Brazil
lucaspdfborges@gmail.com

Abstract

Describing complex phenomena by means of cellular automata (CAs) 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 20 years. Most of the applications use CAs to qualitatively describe the phenomena, which is surely a consequence of the way the automata rules are commonly defined. In this paper, a physical application of a general rule that describes each of Stephen Wolfram's CAs is discussed. The new representation is given in terms of the so-called 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 that 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, a straightforward physical interpretation of CAs is derived. Such an application regards advective-diffusive phenomena without a constant source. Finally, the relation between CAs and anomalous diffusion is briefly discussed.