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Cellular automata and statistical mechanical models

Abstract

We elaborate on the analogy between the transfer matrix of usual lattice models and the master equation describing the time development of cellular automata. Transient and stationary properties of probabilistic automata are linked to surface and bulk properties, respectively, of restricted statistical mechanical systems. It is demonstrated that methods of statistical physics can be successfully used to describe the dynamic and the stationary behavior of such automata. Some exact results are derived, including duality transformations, exact mappings, “disorder,” and “linear” solutions. Many examples are worked out in detail to demonstrate how to use statistical physics in order to construct cellular automata with desired properties. This approach is considered to be a first step toward the design of fully parallel,probabilistic systems whose computational abilities rely on the cooperative behavior of their components.

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On leave from Institute for Theoretical Physics, Roland Eötvös University, Budapest, Hungary.

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Rujàn, P. Cellular automata and statistical mechanical models. J Stat Phys 49, 139–222 (1987). https://doi.org/10.1007/BF01009958

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Key words

  • Cellular automata
  • multispin Ising models
  • static and dynamic critical phenomena
  • duality relations
  • exact solutions
  • stochastic processes
  • predictible behavior
  • topological invariants
  • microcanonical simulations