An empirical method for modelling and simulating some complex macroscopic phenomena by cellular automata

Abstract

Novel parallel computing models sometime represent a valid alternative to standard differential equation methods in modelling complex phenomena. In particular, Cellular Automata (CA) provide such an alternative approach for some complex natural systems, whose behaviour can be described in terms of local interactions of their constituent parts. This paper illustrates an empirical method applied with interesting results in modelling and simulating some complex macroscopic phenomena. While classical CA are based upon elementary automata, with few states and a simple transition function, in order to deal with macroscopic phenomena it is often necessary to allow a large number of different states a more complicated transition. The notion of substate is introduced in the macroscopic case for decomposing the state of the cell. The values associated to substates can change in time either due to interactions among substates inside the cell (internal transformations) or to local interactions among neighbouring cells. The internal transformations are treated in a way similar to ordinary difference equations. The local interactions among cells can be often treated according to an algorithm for the minimisation of differences, which describes a tendency of conserved quantities to reach an equilibrium distribution. A large class of complex macroscopic phenomena seem to satisfy the applicability conditions of such an empirical method; some of them are briefly reviewed.