Discrete, amorphous physical models

Abstract

Discrete models of physical phenomena are an attractive alternative to continuous models such as partial differential equations. In discrete models, such as cellular automata, space is treated as having finitely many locations per unit volume. Physical processes are modelled by rules that typically depend on a small number of nearby locations. Such models have the feature that they depend critically on a regular (crystalline) lattice, as well as the global synchronization of all sites. We might well ask, on the grounds of minimalism, whether the global synchronization and crystalline lattice arc inherent in any discrete formulation. Is it possible to do without these conditions and still model physics? Or are they somehow fundamental? We will answer this question by presenting a class of models that are "extremely local" in the sense that the update rule does not depend on synchronization with other sites, or on detailed knowledge of the lattice geometry. All interactions involve only a single pair of sites. The models have the further advantage that they exactly conserve the analog of quantities such as momentum and energy which are conserved in physics. A framework for simulating the asynchronous, parallel model with irregular geometry on a sequential computer will be presented. Evidence will be given that the models agree well qualitatively and quantitatively with continuous differential equations. We will draw parallels between the various kinds of physical models and various computing architectures, and show that the class of models presented corresponds to a new parallel computing architecture known as an amorphous computer.