Prog. Theor. Phys. Vol. 87 No. 1 (1992) pp. 1-60
Collective Behaviors in Spatially Extended Systems with Local Interactions and Synchronous Updating
Service de Physique de l'Etat Condensé, Centre d'Etudes de Saclay,
91191 Gif-sur-Yvette cedex
(Received December 3, 1991)
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 approximation 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 systems, 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 the predictions of equilibrium statistical mechanics. Both in the cellular automata models and in the coupled map lattices, macroscopic variables frequently display a non-transient, time-dependent, low-dimensional dynamics emerging out of local disorder. Striking examples are period 3 cycles in two-state cellular automata and a Hopf bifurcation for a d = 5 lattice of coupled logistic maps. An extensive account of the phenomenology is given, including a catalog of behaviors, classification tables for the celular automata rules, and bifurcation diagrams for the coupled map lattices.
The observed underlying dynamics is accompanied by an intrinsic quasi-Gaussian noise (stemming from the local disorder) which disappears in the infinite-size limit. The collective behaviors constitute a robust phenomenon, resisting external noise, small changes in the local dynamics, and modifications of the initial and boundary conditions. Synchronous updating, high space dimension and the regularity of connections are shown to be crucial ingredients in the subtle build-up of correlations giving rise to the collective motion. The discussion stresses the need for a theoretical understanding that neither equilibrium statistical mechanics nor higher-order mean-field approximations are able to provide.
DOI : 10.1143/PTP.87.1
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Citing Article(s) :
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Progress of Theoretical Physics Supplement No.161 (2006) pp. 244-250
Critical Properties of Phase Transitions in Lattices of Coupled Logistic Maps
Philippe Marcq, Hugues Chaté and Paul Manneville
Progress of Theoretical Physics Supplement No.162 (2006) pp. 104-111
Collective Motion as a Transient Structure in a Hamiltonian Dynamical System
Hidetoshi Morita and Kunihiko Kaneko