Quasi equilibrium construction for the long time limit of glassy dynamics

TL;DR

This paper reviews a Markov Chain approach to understand the long-time behavior of glassy systems, providing a quasi-equilibrium perspective and deriving dynamical equations for both spin glasses and structural glasses.

Contribution

It introduces a novel Markov Chain framework for long-time glassy dynamics, deriving dynamical equations and applying them to spin and structural glasses.

Findings

Derivation of dynamical Ornstein-Zernike equations for glasses

Application of Hypernetted Chain and BBGKY closures

Recovery of Mode-Coupling equations in the long-time limit

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.