Criticality in cellular automata

Abstract

Using recent results obtained for the transition to turbulence via spatiotemporal intermittency in extended dynamical systems, critical cellular automata rules are built. Thanks to a systematic procedure, the continuous phase transition observed in a coupled map lattice is translated into a sequence of cellular automata rules for which the critical properties of the original system are reproduced as precisely as desired. It is shown that criticality, as understood from the point of view of statistical mechanics, is intimately related to the various characteristics of Wolfram's class IV rules. This suggests in turn the picture of a “critical surface” in the space of rules and provides the basis for a discussion of the problem of classification schemes. We review recent cellular automata studies where questions linked to criticality arise and argue that they are unified in light of our results, leaving the relationship between computational and statistical characterizations of critical rules as the central problem for future studies on this subject.