Collective behaviors in spatially extended systems with local interactions and synchronous updating, Progr. Theoret. Phys

Assessing the extent to which dynamical systems with many degrees of freedom can be described within a thermodynamics formalism is a problem that currently attracts much attention. In this context, synchronously updated regular lattices of identical, chaotic elements with local interactions are promising models for which statistical mechanics may be hoped to provide some insights. This article presents a large class of cellular automata rules and coupled map lattices of the above type in space dimensions d=2 to 6. Such simple models can be approached by a mean-field approximation which usually reduces the dynamics to that of a map governing the evolution of some extensive density. While this approxima-tion is exact in the d= = limit, where macroscopic variables must display the time · dependent behavior of the mean·field map, basic intuition from equilibrium statistical mechanics rules out any such behavior in a low·dimensional system, since it would involve the collective motion of locally disordered elements. The models studied are chosen to be as close as possible to mean-field conditions, i.e., rather high space dimension, large connectivity, and equal·weight coupling between sites. While the mean·field evolution is never observed, a new type of non·trivial collective behavior is found, at odds with th