Tunneling of the Kawasaki dynamics at low temperatures in two dimensions

Beltrán, J.; Landim, C.

Beltrán, J. ; Landim, C.

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015), p. 59-88 / Harvested from Numdam

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Résumé
Abstract

On considère un gaz sur réseau évoluant selon la dynamique de Kawasaki à température inverse $β$ sur le tore bi-dimensionel $𝛬_{L} = {0, \dots, L - 1}^{2}$ . 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 $e^{2 β}$ le processus évolue comme une chaîne de Markov sur $𝛬_{L}$ qui saute d’un site $𝐱$ vers un site $𝐲 \neq 𝐱$ à un taux strictement positif qui peut-être exprimé en terme de probabilités d’atteinte de dynamiques markoviennes élémentaires.

Consider a lattice gas evolving according to the conservative Kawasaki dynamics at inverse temperature $β$ on a two dimensional torus $𝛬_{L} = {0, \dots, L - 1}^{2}$ . 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 $e^{2 β}$ the process evolves as a Markov process on $𝛬_{L}$ which jumps from any site $𝐱$ to any other site $𝐲 \neq 𝐱$ at a strictly positive rate which can be expressed in terms of the hitting probabilities of simple Markovian dynamics.

Publié le : 2015-01-01

MR 3300964 | Zbl 06412898

DOI : https://doi.org/10.1214/13-AIHP568
Classification: 82C44, 82C22, 60K35

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     author = {Beltr\'an, J. and Landim, Claudio},
     title = {Tunneling of the Kawasaki dynamics at low temperatures in two dimensions},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {51},
     year = {2015},
     pages = {59-88},
     doi = {10.1214/13-AIHP568},
     mrnumber = {3300964},
     zbl = {06412898},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2015__51_1_59_0}
}

Beltrán, J.; Landim, C. Tunneling of the Kawasaki dynamics at low temperatures in two dimensions. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) pp. 59-88. doi : 10.1214/13-AIHP568. http://gdmltest.u-ga.fr/item/AIHPB_2015__51_1_59_0/

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