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February 2015 Tunneling of the Kawasaki dynamics at low temperatures in two dimensions
J. Beltrán, C. Landim
Ann. Inst. H. Poincaré Probab. Statist. 51(1): 59-88 (February 2015). DOI: 10.1214/13-AIHP568

Abstract

Consider a lattice gas evolving according to the conservative Kawasaki dynamics at inverse temperature $\beta$ on a two dimensional torus $\varLambda _{L}=\{0,\dots,L-1\}^{2}$. We prove the tunneling behavior of the process among the states of minimal energy. More precisely, assume that there are $n^{2}$ particles, $n<L/2$, and that the initial state is the configuration in which all sites of the square $\{0,\dots,n-1\}^{2}$ are occupied. We show that in the time scale $\mathrm{e}^{2\beta}$ the process evolves as a Markov process on $\varLambda _{L}$ which jumps from any site $\mathbf{x}$ to any other site $\mathbf{y}\neq\mathbf{x}$ at a strictly positive rate which can be expressed in terms of the hitting probabilities of simple Markovian dynamics.

On considère un gaz sur réseau évoluant selon la dynamique de Kawasaki à température inverse $\beta$ sur le tore bi-dimensionel $\varLambda _{L}=\{0,\dots,L-1\}^{2}$. Nous étudions l’évolution du processus parmi les états d’énergie minimale.

Supposons la présence de $n^{2}$ particules, $n<L/2$ et qu’á l’état initial les sites du carré $\{0,\dots,n-1\}^{2}$ soient tous occupés. Nous montrons qu’á l’échelle de temps $\mathrm{e}^{2\beta}$ le processus évolue comme une chaîne de Markov sur $\varLambda _{L}$ qui saute d’un site $\mathbf{x}$ vers un site $\mathbf{y}\neq\mathbf{x}$ à un taux strictement positif qui peut-être exprimé en terme de probabilités d’atteinte de dynamiques markoviennes élémentaires.

Citation

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J. Beltrán. C. Landim. "Tunneling of the Kawasaki dynamics at low temperatures in two dimensions." Ann. Inst. H. Poincaré Probab. Statist. 51 (1) 59 - 88, February 2015. https://doi.org/10.1214/13-AIHP568

Information

Published: February 2015
First available in Project Euclid: 14 January 2015

zbMATH: 1314.82036
MathSciNet: MR3300964
Digital Object Identifier: 10.1214/13-AIHP568

Subjects:
Primary: 60K35, 82C22, 82C44

Rights: Copyright © 2015 Institut Henri Poincaré

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Vol.51 • No. 1 • February 2015
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