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Phase Transitions in Cellular Automata

  • Reference work entry
Computational Complexity

Article Outline

Glossary

Definition of the Subject

Introduction

The Domany–Kinzel Cellular Automaton

Car Traffic Models

Epidemic Models

Future Directions

Bibliography

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Abbreviations

Phase transition :

In statistical physics, a phase transition is the transformation of a macroscopic system from one phase to another. Phase transitions are divided into two types. First-order phase transitions are characterized by discontinuities of first‐order derivatives of the Gibbs free energy, such as the entropy or the volume whereas second‐order phase transitions are characterized by power‐law behaviors of second‐order derivatives of the Gibbs free energy such as the specific heat or the magnetic susceptibility. These singular behaviors occur for specific values of intensive variables such as the temperature or the magnetic field. Similar behaviors have also been discovered in many‐component systems.

Critical behavior :

Critical behavior manifests itself in many‐component systems and is characteristic of a cooperative behavior of the various components. This notion has been introduced in equilibrium statistical physics for many‐body systems that exhibit second‐order phase transitions. In the vicinity of the transition temperature T c, a singular physical quantity Q has a power‐law behavior, that is, it behaves as \( { |T-T_\mathrm{c}|^\varepsilon } \), where ε is the critical exponent characterizing the critical behavior of Q.

Universality :

Despite the great variety of physical systems that exhibit equilibrium second‐order phase transitions, their critical behaviors, characterized by a set of critical exponents, fall into a small number of universality class es that only depend on the symmetry of the order parameter and the space dimension. Critical behavior is universal in the sense that it does not depend upon details whose characteristic size is much less than the correlation length, such as lattice structure, range of interactions (as long as this range is finite), spin length, and so on. Since time is involved in nonequilibrium critical phenomena, universality classes are expected to offer a richer variety.

Cellular automaton :

A cellular automaton is a fully discrete dynamical system. It consists of a regular finite‐dimensional lattice of cells, each one in a state belonging to a finite set of states. The state of each cell evolves in discrete time steps. At time t the state of a given cell is a function of the states of a finite number of neighboring cells at time \( { t-1 } \). The neighborhood of a given cell may or may not include the cell itself. All cells evolve according to the same evolution rule which may be either deterministic or probabilistic.

Model :

A model is a simplified mathematical representation of a system. In the actual system, although many features are likely to be important, not all of them, however, should be included in the model. Only a few relevant features which are thought to play an essential role in the interpretation of the observed phenomena should be retained. If it captures the key elements of a complex system, a simple model, may elicit highly relevant questions.

Mean‐field approximation :

In a many‐agent system the mean‐field approximation is a first attempt to understand the behavior of the system. Although rather crude, it is often useful. Ignoring space correlations between the agents and replacing local interactions by uniform long‐range ones, the mean‐field approximation only deals with average quantities. Historically, under the name “molecular field theory”, it was first used by Pierre Weiss (1869–1940) in 1907 to build up the first simple theory of the para‐ferromagnetic second‐order phase transition.

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Books and Reviews

  1. Here are a few articles on different problems of phase transitions in cellular automata that might be of interest to readers wishing to go in more details.

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Author information

Authors and Affiliations

  1. Department of Physics, University of Illinois, Chicago, USA

    Nino Boccara

  2. CE Saclay, Gif-sur-Yvette, France

    Nino Boccara

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  1. Nino Boccara
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Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Boccara, N. (2012). Phase Transitions in Cellular Automata. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_136

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