Communications in Mathematical Physics

, Volume 125, Issue 1, pp 71–79 | Cite as

From PCA's to equilibrium systems and back

  • Sheldon Goldstein
  • Roelof Kuik
  • Joel L. Lebowitz
  • Christian Maes


Stationary measures for probabilistic cellular automata (PCA's) ind dimensions give rise to space-time histories whose statistics may naturally be described by Gibbs states ind+1 dimensions for an interaction energy ℋ obtained from the PCA. In this note we study the converse question: Do all Gibbs states for this ℋ correspond to statistical space-time histories for the PCA? Our main result states that the answer is yes, at least for translation invariant or periodic Gibbs states. Thus ergodicity questions for PCA's can, at least partially, be formulated as questions of uniqueness of Gibbs states.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Interaction Energy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Sheldon Goldstein
    • 1
  • Roelof Kuik
    • 1
  • Joel L. Lebowitz
    • 1
  • Christian Maes
    • 1
    • 2
  1. 1.Department of Mathematics and PhysicsRutgers UniversityNew BrunswickUSA
  2. 2.Aspirant N.F.W.O., Instituut voor Theoretische Fysika, K.U. Leuven, Celestijnenlaan 200 DLeuvenBelgium

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