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Time scale separation and dynamic heterogeneity in the low temperature East model | | Paul Chleboun
; Alessandra Faggionato
; Fabio Martinelli
; | | Date: |
11 Dec 2012 | | 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 $qdownarrow
0$, where $q$ is the equilibrium density of the 0’s. 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 $qdownarrow 0$ 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 $q^{-a}$, $a=a(gamma)>0$, provided that $lambda geq 2$
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. | | Source: | arXiv, 1212.2399 | | Services: | Forum | Review | PDF | Favorites | |
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