Abstract
We consider the problem of metastability for a stochastic dynamics with a parallel updating rule with single spin rates equal to those of the heat bath for the Ising nearest neighbors interaction. We study the exit from the metastable phase, we describe the typical exit path and evaluate the exit time. We prove that the phenomenology of metastability is different from the one observed in the case of the serial implementation of the heat bath dynamics. In particular we prove that an intermediate chessboard phase appears during the excursion from the minus metastable phase toward the plus stable phase.
Similar content being viewed by others
REFERENCES
O. Penrose and J. L. Lebowitz, Molecular theory of metastability: An update, appendix to the reprinted edition of the article Toward a rigorous molecular theory of metastability, by the same authors, in Fluctuation Phenomena, 2nd edn., E. W. Montroll and J. L. Lebowitz, eds. (North-Holland Physics Publishing, Amsterdam, 1987).
M. Cassandro, A. Galves, E. Olivieri, and M. E. Vares, Metastable behavior of stochastic dynamics: A pathwise approach, J. Statist. Phys. 35:603-634 (1984).
R. Kotecky and E. Olivieri, Droplet dynamics for asymmetric Ising model, J. Statist. Phys. 70:1121-1148 (1993).
E. J. Neves and R. H. Schonmann, Critical droplets and metastability for a Glauber dynamics at very low temperatures, Commun. Math. Phys. 137:209-230 (1991).
R. H. Schonmann, The pattern of escape from metastability of a stochastic Ising model, Commun. Math. Phys. 147:231-240 (1992).
S. Bigelis, E. N. M. Cirillo, J. L. Lebowitz, and E. R. Speer, Critical droplets in metastable probabilistic cellular automata, Phys. Rev. E 59:3935(1999).
E. N. M. Cirillo, A note on the metastability of the Ising model: the alternate updating case, J. Statist. Phys. 106:335-390 (2002).
P. Rujan, Cellular automata and statistical mechanical models, J. Statist. Phys 49:139-222 (1987). A. Georges and P. Le Doussal, From equilibrium spin models to probabilistic cellular automata, J. Statist. Phys. 54:1989.
O. N. Stavskaja, Gibbs invariant measures for Markov chains on finite lattices with local interactions, Math. USSR Sobrnik 21:395-411 (1973). A. L. Toom, N. B. Vasilyev, O. N. Stavskaja, L. G. Mitjushin, G. L. Kurdomov, and S. A. Pirogov, Discrete local Markov systems, Preprint 1989.
B. Derrida, Dynamical phase transition in spin model and automata, in Fundamental Problem in Statistical Mechanics VII, H. van Beijeren, ed. (Elsier Science Publisher B.V., 1990).
V. Kozlov and Vasiljev, Reversible Markov chain with local interactions, in Multicomponent Random System, Adv. in Prob. and Rel. Topics, 1980.
Vasiljev, Bernoulli and Markov stationary measures in discrete local interactions, Lect. Notes in Math. 653 (1978).
E. Olivieri and E. Scoppola, Markov chains with exponentially small transition probabilities: First exit problem from a general domain. I. The reversible case, J. Statist. Phys. 79:613-647 (1995).
F. R. Nardi and E. Olivieri, Low temperature stochastic dynamics for an ising model with alternating field, Markov Proc. and Rel. Fields 2:117-166 (1996).
J. L. Lebowitz, C. Maes, and E. Speer, Statistical mechanics of probabilistic cellular automata, J. Statist. Phys. 59:117-170 (1990); J. L. Lebowitz, C. Maes, and E. Speer Probabilistic cellular automata: Some statistical mechanics considerations, in Lectures in Complex Systems, SFI Studies in the Sciences of Complexity, Lecture Volume II, E. Jen, ed. (Addison–Wesley, New York, (1990).
E. J. Neves and R. H. Schonmann, Behavior of droplets for a class of Glauber dynamics at very low temperatures, Prob. Theor. Rel. Fields 91:331-354 (1992).
E. Olivieri, private communication.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cirillo, E.N.M., Nardi, F.R. Metastability for a Stochastic Dynamics with a Parallel Heat Bath Updating Rule. Journal of Statistical Physics 110, 183–217 (2003). https://doi.org/10.1023/A:1021070712382
Issue Date:
DOI: https://doi.org/10.1023/A:1021070712382