Quasi equilibrium construction for the long time limit of glassy dynamics

Abstract

In this paper we review a recent proposal to understand the long time limit of glassy dynamics in terms of an appropriate Markov Chain. [1]. The advantages of the resulting construction are many. The first one is that it gives a quasi equilibrium description on how glassy systems explore the phase space in the slow relaxation part of their dynamics. The second one is that it gives an alternative way to obtain dynamical equations starting from a dynamical rule that is static in spirit. This provides a way to overcome the difficulties encountered in the short time part of the dynamics where current conservation must be enforced. We study this approach in detail in a prototypical mean field disordered spin system, namely the p-spin spherical model, showing how we can obtain the well known equations that describes its dynamics. Then we apply the same approach to structural glasses. We first derive a set of dynamical Ornstein-Zernike equations which are very general in nature. Finally we consider two possible closure schemes for them, namely the Hypernetted Chain approximation of liquid theory and a closure of the BBGKY hierarchy that has been recently introduced by G. Szamel. From both approaches we finally find a set of dynamical Mode-Coupling like equations that are supposed to describe the system in the long time/slow dynamics regime.

On weak ergodicity breaking in mean-field spin glasses

“…How to do that remains, however, an open problem. A possible suggestion could come from exploring the quasi-equilibrium dynamics [58,61].…”

An alternative, dynamic density functional-like theory for time-dependent density fluctuations in glass-forming fluids

“…In the context of the non-local in time relaxation processes we would like to comment on the relation of our approximation and the quasi-equilibrium construction of Franz et al [46,47] We assumed that to calculate the time derivative of the density correlation function we can approximate the exact probability distribution ( 9) by an equilibrium distribution in an external potential that assures that the correlation between the density of the system at time t and its initial initial density is equal to F (k; t). In contrast, the starting assumption of Franz et al is a quasi-equilibrium condition for the transition probability.…”

“…We propose the name quasi-equilibrium approximation for formula (11) to emphasize that the distribution of particle positions at time t is the same as in an equilibrium state that satisfies conditions (13)(14). The additional motivation for this name is the conceptual similarity of our approximation with quasi-equilibrium construction for the long-time limit of glassy dynamics proposed by Franz et al [46,47]. We further comment on this point in the Discussion section.…”