Universal Cellular Automata Based on the Collisions of Soft Spheres

TL;DR

This paper introduces universal cellular automata models based on elastic collisions of soft spheres, which are more physically realistic than traditional hard-sphere models, and explores their computational capabilities and physical interpretations.

Contribution

The paper presents 2D and 3D soft sphere models of universal cellular automata, connecting lattice gas automata with continuous dynamics and discussing momentum conservation and physical interpretations.

Findings

Soft sphere models correspond to lattice gas automata with extended impact periods.

Universal cellular automata can be constructed using elastic collisions of soft spheres.

Models can be interpreted as relativistic or semi-classical systems.

Abstract

Fredkin's Billiard Ball Model (BBM) is a continuous classical mechanical model of computation based on the elastic collisions of identical finite-diameter hard spheres. When the BBM is initialized appropriately, the sequence of states that appear at successive integer time-steps is equivalent to a discrete digital dynamics. Here we discuss some models of computation that are based on the elastic collisions of identical finite-diameter soft spheres: spheres which are very compressible and hence take an appreciable amount of time to bounce off each other. Because of this extended impact period, these Soft Sphere Models (SSM's) correspond directly to simple lattice gas automata--unlike the fast-impact BBM. Successive time-steps of an SSM lattice gas dynamics can be viewed as integer-time snapshots of a continuous physical dynamics with a finite-range soft-potential interaction. We present both 2D and 3D models of universal CA's of this type, and then discuss spatially-efficient computation using momentum conserving versions of these models (i.e., without fixed mirrors). Finally, we discuss the interpretation of these models as relativistic and as semi-classical systems, and extensions of these models motivated by these interpretations.