Abstract
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. in J. Stat. Phys. 149(4):722–737, 2012), 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.
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References
Kozlov, O., Vasilyev, N.: Reversible Markov chains with local interaction. In: Multicomponent Random Systems, p. 451 (1980)
Goldstein, S., Kuik, R., Lebowitz, J.L., Maes, C.: From pca’s to equilibrium systems and back. Commun. Math. Phys. 125(1), 71–79 (1989)
Lebowitz, J.L., Maes, C., Speer, E.R.: Statistical mechanics of probabilistic cellular automata. J. Stat. Phys. 59(1–2), 117–170 (1990)
Maes, C., Shlosman, S.B.: When is an interacting particle system ergodic? Commun. Math. Phys. 151(3), 447–466 (1993)
Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 74(14), 2694 (1995)
Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in stationary states. J. Stat. Phys. 80(5–6), 931–970 (1995)
Cirillo, E.N.M., Nardi, F.R.: Metastability for a stochastic dynamics with a parallel heat bath updating rule. J. Stat. Phys. 110(1–2), 183–217 (2003)
Iovanella, A., Scoppola, B., Scoppola, E.: Some spin glass ideas applied to the clique problem. J. Stat. Phys. 126(4–5), 895–915 (2007)
Jona-Lasinio, G.: From fluctuations in hydrodynamics to nonequilibrium thermodynamics. Prog. Theor. Phys. Suppl. 184, 262–275 (2010)
Scoppola, B.: Exact solution for a class of random walk on the hypercube. J. Stat. Phys. 143(3), 413–419 (2011)
Dai Pra, P., Scoppola, B., Scoppola, E.: Sampling from a gibbs measure with pair interaction by means of pca. J. Stat. Phys. 149(4), 722–737 (2012)
Bremaud, P.: Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues vol. 31. Springer, Berlin (1999)
Häggström, O.: Finite Markov Chains and Algorithmic Applications vol. 52. Cambridge University Press, Cambridge (2002)
Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. AMS, Providence (2009)
Acknowledgements
The authors want to thank Elisabetta Scoppola and Paolo Dai Pra for the useful comments and discussions. The first author would like to express appreciation to Alexander Agathos and Salvatore Filippone for the useful comments regarding the cuda implementation; he would also like to address a very special thank to Thomas L. Falch and Johannes Kvam for their gargantuan help in developing and running the code.
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To the memory of Francesco De Blasi.
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Lancia, C., Scoppola, B. Equilibrium and Non-equilibrium Ising Models by Means of PCA. J Stat Phys 153, 641–653 (2013). https://doi.org/10.1007/s10955-013-0847-0
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DOI: https://doi.org/10.1007/s10955-013-0847-0