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Equilibrium and non-equilibrium Ising models by means of PCA. (English) Zbl 1302.82066
Summary: We propose a unified approach to reversible and irreversible pca dynamics, and we show that in the case of 1D and 2D nearest neighbor Ising systems with periodic boundary conditions we are able to compute the stationary measure of the dynamics also when the latter is irreversible. We also show how, according to P. Dai Pra et al. [J. Stat. Phys. 149, No. 4, 722–737 (2012; Zbl 1264.82080)], the stationary measure is very close to the Gibbs for a suitable choice of the parameters of the PCA dynamics, both in the reversible and in the irreversible cases. We discuss some numerical aspects regarding this topic, including a possible parallel implementation.
Reviewer: Reviewer (Berlin)

MSC:
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
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References:
[1] Kozlov, O.; Vasilyev, N., Reversible Markov chains with local interaction, 451, (1980) · Zbl 0444.60099
[2] Goldstein, S.; Kuik, R.; Lebowitz, J.L.; Maes, C., From pca’s to equilibrium systems and back, Commun. Math. Phys., 125, 71-79, (1989) · Zbl 0683.68045
[3] Lebowitz, J.L.; Maes, C.; Speer, E.R., Statistical mechanics of probabilistic cellular automata, J. Stat. Phys., 59, 117-170, (1990) · Zbl 1083.82522
[4] Maes, C.; Shlosman, S.B., When is an interacting particle system ergodic?, Commun. Math. Phys., 151, 447-466, (1993) · Zbl 0765.60102
[5] Gallavotti, G.; Cohen, E.G.D., Dynamical ensembles in nonequilibrium statistical mechanics, Phys. Rev. Lett., 74, 2694, (1995)
[6] Gallavotti, G.; Cohen, E.G.D., Dynamical ensembles in stationary states, J. Stat. Phys., 80, 931-970, (1995) · Zbl 1081.82580
[7] Cirillo, E.N.M.; Nardi, F.R., Metastability for a stochastic dynamics with a parallel heat Bath updating rule, J. Stat. Phys., 110, 183-217, (2003) · Zbl 1035.82029
[8] Iovanella, A.; Scoppola, B.; Scoppola, E., Some spin Glass ideas applied to the clique problem, J. Stat. Phys., 126, 895-915, (2007) · Zbl 1153.82026
[9] Jona-Lasinio, G., From fluctuations in hydrodynamics to nonequilibrium thermodynamics, Prog. Theor. Phys. Suppl., 184, 262-275, (2010) · Zbl 1201.82040
[10] Scoppola, B., Exact solution for a class of random walk on the hypercube, J. Stat. Phys., 143, 413-419, (2011) · Zbl 1219.82097
[11] Dai Pra, P.; Scoppola, B.; Scoppola, E., Sampling from a Gibbs measure with pair interaction by means of pca, J. Stat. Phys., 149, 722-737, (2012) · Zbl 1264.82080
[12] Bremaud, P.: Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues vol. 31. Springer, Berlin (1999) · Zbl 0949.60009
[13] Häggström, O.: Finite Markov Chains and Algorithmic Applications vol. 52. Cambridge University Press, Cambridge (2002) · Zbl 0999.60001
[14] Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. AMS, Providence (2009) · Zbl 1160.60001
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