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An Investigation Of The Phase Transitions Of A Family Of Probabilistic Automata

Author

Listed:
  • HEINZ MÜHLENBEIN

    (Fraunhofer Institute for Autonomous Intelligent Systems, D-53754 Sankt Augustin, Germany)

  • THOMAS AUS DER FÜNTEN

    (Fraunhofer Institute for Autonomous Intelligent Systems, D-53754 Sankt Augustin, Germany)

Abstract

We investigate a family of totalistic probabilistic cellular automata (PCA) which depend on three parameters. For the uniform random neighborhood and for the symmetric 1D PCA the exact stationary distribution is computed for all finiten. This result is used to evaluate approximations (uni-variate and bi-variate marginals). It is proven that the uni-variate approximation (also called mean-field) is exact for the uniform random neighborhood PCA. The exact results and the approximations are used to investigate phase transitions. We compare the results of two order parameters, the uni-variate marginal and the normalized entropy. Sometimes different transitions are indicated by the Ehrenfest classification scheme. This result shows the limitations of using just one or two order parameters for detecting and classifying major transitions of the stationary distribution. Furthermore, finite size scaling is investigated. We show that extrapolations ton=∞from numerical calculations of finitencan be misleading in difficult parameter regions. Here, exact analytical estimates are necessary.

Suggested Citation

  • Heinz Mühlenbein & Thomas Aus Der Fünten, 2004. "An Investigation Of The Phase Transitions Of A Family Of Probabilistic Automata," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 7(01), pages 93-123.
    • Handle: RePEc:wsi:acsxxx:v:07:y:2004:i:01:n:s0219525904000081
    DOI: 10.1142/S0219525904000081
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