From PCA's to equilibrium systems and back

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.

[1]  Bielefeld Encounters in Mathematics,et al.  Quantum Fields — Algebras, Processes , 1980 .

[2]  T. Liggett.  Interacting Particle Systems , 1985 .

[3]  P. Rujan,et al.  Cellular automata and statistical mechanical models , 1987 .

[4]  E. Domany.  Exact Results for Two- and Three-Dimensional Ising and Potts Models , 1984 .

[5]  Antoine Georges,et al.  From equilibrium spin models to probabilistic cellular automata , 1989 .

[6]  Benjamin Weiss,et al.  Commuting measure-preserving transformations , 1972 .

[7]  Markov and stability properties of equilibrium states for nearest-neighbor interactions , 1988 .

[8]  Bennett,et al.  Role of irreversibility in stabilizing complex and nonergodic behavior in locally interacting discrete systems. , 1985, Physical Review Letters.

[9]  David Griffeath,et al.  Multicomponent Random Systems , 1980 .

[10]  Eytan Domany,et al.  Equivalence of Cellular Automata to Ising Models and Directed Percolation , 1984 .

[11]  R. Israel.  Some examples concerning the global Markov property , 1986 .

[12]  R. Israel.  Convexity in the Theory of Lattice Gases , 1979 .

[13]  J. Lebowitz,et al.  Statistical mechanics of probabilistic cellular automata , 1990 .

[14]  F. Spitzer.  Infinite Systems with Locally Interacting Components , 1981 .

[15]  R. Kuik,et al.  Entropy and global Markov properties , 1990 .

[16]  Sheldon Goldstein,et al.  Remarks on the global Markov property , 1980 .

[17]  Hans-Otto Georgii,et al.  Gibbs Measures and Phase Transitions , 1988 .

[18]  H. Föllmer.  On the Global Markov Property , 1980 .