We consider the non-equilibrium dynamics of the East model, a linear chain of 0-1 spins evolving under a simple Glauber dynamics in the presence of a kinetic constraint which forbids flips of those spins whose left neighbor is 1. We focus on the glassy effects caused by the kinetic constraint as , where q is the equilibrium density of the 0s. In the physical literature this limit is equivalent to the zero temperature limit. We first prove that, for any given L = O(1/q), the divergence as of three basic characteristic time scales of the East process of length L is the same. Then we examine the problem of dynamic heterogeneity, i.e., non-trivial spatio-temporal fluctuations of the local relaxation to equilibrium, one of the central aspects of glassy dynamics. For any mesoscopic length scale L = O(q (-gamma) ), gamma < 1, we show that the characteristic time scale of two East processes of length L and lambda L respectively are indeed separated by a factor q (-alpha) , alpha = alpha(gamma) > 0, provided that lambda a parts per thousand yen 2 is large enough (independent of q, lambda = 2 for gamma < 1/2). In particular, the evolution of mesoscopic domains, i.e., maximal blocks of the form 111..10, occurs on a time scale which depends sharply on the size of the domain, a clear signature of dynamic heterogeneity. A key result for this part is a very precise computation of the relaxation time of the chain as a function of (q, L), well beyond the current knowledge, which uses induction on length scales on one hand and a novel algorithmic lower bound on the other. Finally we show that no form of time scale separation occurs for gamma = 1, i.e., at the equilibrium scale L = 1/q, contrary to what was assumed in the physical literature based on numerical simulations.

Time Scale Separation and Dynamic Heterogeneity in the Low Temperature East Model / P., Chleboun; Faggionato, Alessandra; F., Martinelli. - In: COMMUNICATIONS IN MATHEMATICAL PHYSICS. - ISSN 0010-3616. - STAMPA. - 328:3(2014), pp. 955-993. [10.1007/s00220-014-1985-1]

Time Scale Separation and Dynamic Heterogeneity in the Low Temperature East Model

FAGGIONATO, ALESSANDRA;
2014

Abstract

We consider the non-equilibrium dynamics of the East model, a linear chain of 0-1 spins evolving under a simple Glauber dynamics in the presence of a kinetic constraint which forbids flips of those spins whose left neighbor is 1. We focus on the glassy effects caused by the kinetic constraint as , where q is the equilibrium density of the 0s. In the physical literature this limit is equivalent to the zero temperature limit. We first prove that, for any given L = O(1/q), the divergence as of three basic characteristic time scales of the East process of length L is the same. Then we examine the problem of dynamic heterogeneity, i.e., non-trivial spatio-temporal fluctuations of the local relaxation to equilibrium, one of the central aspects of glassy dynamics. For any mesoscopic length scale L = O(q (-gamma) ), gamma < 1, we show that the characteristic time scale of two East processes of length L and lambda L respectively are indeed separated by a factor q (-alpha) , alpha = alpha(gamma) > 0, provided that lambda a parts per thousand yen 2 is large enough (independent of q, lambda = 2 for gamma < 1/2). In particular, the evolution of mesoscopic domains, i.e., maximal blocks of the form 111..10, occurs on a time scale which depends sharply on the size of the domain, a clear signature of dynamic heterogeneity. A key result for this part is a very precise computation of the relaxation time of the chain as a function of (q, L), well beyond the current knowledge, which uses induction on length scales on one hand and a novel algorithmic lower bound on the other. Finally we show that no form of time scale separation occurs for gamma = 1, i.e., at the equilibrium scale L = 1/q, contrary to what was assumed in the physical literature based on numerical simulations.
2014
01 Pubblicazione su rivista::01a Articolo in rivista
Time Scale Separation and Dynamic Heterogeneity in the Low Temperature East Model / P., Chleboun; Faggionato, Alessandra; F., Martinelli. - In: COMMUNICATIONS IN MATHEMATICAL PHYSICS. - ISSN 0010-3616. - STAMPA. - 328:3(2014), pp. 955-993. [10.1007/s00220-014-1985-1]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/558885
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