Lattice Glass Models

Abstract

Motivated by the concept of geometrical frustration, we introduce a class of statistical mechanics lattice models for the glass transition. Monte Carlo simulations in three dimensions show that they display a dynamical glass transition which is very similar to that observed in other off-lattice systems and which does not depend on a specific dynamical rule. A mean-field study shows the existence of a discontinuous glass transition, in agreement with the numerical observations.Understanding the glass transition, and the glass phase, is one of the present major challenges in condensed matter physics. The experimental glass transition is related to a dramatic dynamical slowing down in which the structural relaxation time changes of 14 orders of magnitudes in a relatively small window of temperatures. As a matter of fact, the glass transition temperature T g is empirically defined as that where the structural relaxation time becomes of the order of an hour. In the last fifty years there have been many efforts to try to understand whether this phenomenon is just a dramatic crossover, it is related to a pure dynamical transition, or it is the signature of a true thermodynamic glass transition, called the "ideal glass transition", which would take place at temperature T K below T g but which is kinetically avoided [1]. The possibility of this last scenario in fragile glasses is supported by the closeness of the Vogel-Fulcher temperature (where the extrapolated relaxation time has a divergence) and of the Kauzmann temperature (where the extrapolated excess entropy of the supercooled liquid vanishes) [2], and has been widely explored theoretically [1,[3][4][5][6][7]. The analogy with mean field discontinuous spin glasses [6] has given recently a new boost to this line of research [7]. These models can be solved analytically and display striking similarities with the experimental fragile glasses. However, they predict (because of their mean field character) a dynamic freezing transition at a temperature T c > T g , equal to the mode coupling transition temperature, where real systems still have a finite relaxation time. This difference is generally attributed to the existence in finite dimensions of activated processes which would transform the 'mean field' dynamical transition into a crossover in such a way that the relaxation time increase very fast in the temperature region T K < T < T c and eventually diverges at T K .In this paper we introduce new lattice models of glasses. We study them numerically in three dimensions, and show (1) that they display a dynamic glass transition similar to the one seen in the simulation of glass formers like binary Lennard-Jones [8], (2) that their mean field solution, obtained using the Bethe approximation, predicts the same physical scenario found for mean field discontinuous spin glasses (in particular a dynamic freezing transition preceding the "ideal glass transition").Our lattice glass models are defined as follows. On each node of the lattice (e.g. a cubic lattice), there can be...

Pure Glass in Finite Dimensions

“…The condition provides a precise definition of pure glass in this Letter. It should be noted that pure glass presented thus far are limited to models in a random graph [8][9][10] or with long-range interactions in the Kac limit [11,12]. One of the main results in this Letter is the presentation of a pure glass model with short-range interactions in finite dimensions.…”

Lattice models of disorder with order

“…Models with geometrical frustration and only two body interaction have been employed as coarse grained approximation of hard sphere models in the continuum limit. A dynamical glass transition has been observed in threedimensional models with a constraint on the local equilibrium density [29] together with a first order transition. Similar results have also been found in [30] and in [31] although in the latter model the crystalline phase does not seem accessible by numerical simulation even at very low cooling rate.…”

Lattice Boltzmann model with hierarchical interactions

“…These models reach out larger scales but the corresponding outcomes have to be critically weighted against the simplifying assumptions they are based upon. At the extreme of this line one finds spin-glasses [3,4] and their modern lattice glass variants [5,6].…”