Phase transition in an elementary probabilistic cellular automaton

Niels K. Petersen and Preben Alstrøm

Physica A: Statistical Mechanics and its Applications, 1997, vol. 235, issue 3, 473-485

Abstract: Cellular automata exhibit a large variety of dynamical behaviors, from fixed-point convergence and periodic motion to spatio-temporal chaos. By introducing probabilistic interactions, and regarding the asymptotic density Φ of non-quiescent cell states as an order parameter, phase transitions may be identified from a quiescent phase with Φ = 0 to a chaotic phase with non-zero Φ. We consider an elementary one-dimensional probabilistic cellular automaton (PCA) with deterministic limits given by the quiescent rule 0, the rule 72 that evolves into a non-trivial fixed point, and the chaotic rules 18 and 90. Despite the simplicity of the rules, the PCA shows a surprising number of transition phenomena. We identify ‘second-order’ phase transitions from Φ = 0 to Φ > 0 with static and dynamic exponents that differ from those of directed percolation. Moreover, we find that the non-trivial fixed-point rule 72 is a singular point in PCA space.

Date: 1997
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:235:y:1997:i:3:p:473-485

DOI: 10.1016/S0378-4371(96)00410-4

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