International Journal of Theoretical Physics 42:329-348 (2003).

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# Discrete, Amorphous Physical Models

**Erik Rauch**

## 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
are inherent in any discrete formulation. Is it possible to do without these
conditions and still have a useful physical model? 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
is be presented. Evidence is 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.

### Sections

## Table of Contents

*Erik Rauch*