Volume 107, Number 3, August 2014
|Number of page(s)||6|
|Section||Condensed Matter: Structural, Mechanical and Thermal Properties|
|Published online||24 July 2014|
The influence of dimension on the relaxation process of East-like models: Rigorous results
1 Mathematics Institute, University of Warwick - Coventry CV4 7AL, UK
2 Department of Mathematics, La Sapienza University - P.le Aldo Moro 2, 00185 Rome, Italy
3 Department of Mathematics and Physics, Roma Tre University - Largo S.L. Murialdo 1, 00146 Rome, Italy
Received: 8 July 2014
Accepted: 11 July 2014
We study facilitated models which extend to arbitrary dimensions the one-dimensional East process and which are supposed to catch some of the main features of the complex dynamics of fragile glasses. We focus on the low-temperature regime (small density of the facilitating sites). In the literature the relaxation process has been assumed to be quasi–one-dimensional and the equilibration time has been computed using the relaxation time of the East model on the equilibrium length scale in d-dimension. This led to a super-Arrhenius scaling for the relaxation time of the form . In a companion paper, using renormalization group ideas and electrical networks methods, we rigorously establish that instead , contradicting the quasi–one-dimensional assumption. The above scaling confirms previous MCAMC simulations. Next we compute the relaxation time at finite and mesoscopic length scales, and show a dramatic dependence on the boundary conditions. Our final result is related to the out-of-equilibrium dynamics. Starting with a single facilitating site at the origin we show that, up to length scales , its influence propagates much faster (on a logarithmic scale) along the diagonal direction than along the axes directions.
PACS: 64.70.Q- – Theory and modeling of the glass transition / 61.43.Fs – Glasses / 64.60.De – Statistical mechanics of model systems (Ising model, Potts model, field-theory models, Monte Carlo techniques, etc.)
© EPLA, 2014
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