  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.
