Dynamic glass transition: bridging the gap between mode-coupling theory and the replica approach

TL;DR

This paper explores the connection between mode-coupling theory and the replica approach in describing the glass transition, showing their equations coincide under certain approximations, thus bridging static and dynamic perspectives.

Contribution

It demonstrates that the replica off-diagonal Ornstein-Zernicke equation aligns with mode-coupling theory equations through a specific factorization approximation.

Findings

The static replica approach reproduces the mode-coupling non-ergodicity parameter.

A factorization approximation links static correlations to dynamic transition predictions.

The derived equations unify static and dynamic descriptions of the glass transition.

Abstract

We clarify the relation between the ergodicity breaking transition predicted by mode-coupling theory and the so-called dynamic transition predicted by the static replica approach. Following Franz and Parisi [Phys. Rev. Lett. 79, 2486 (1997)], we consider a system of particles in a metastable state characterized by non-trivial correlations with a quenched configuration. We show that the assumption that in a metastable state particle currents vanish leads to an expression for the replica off-diagonal direct correlation function in terms of a replica off-diagonal static four-point correlation function. A factorization approximation for this function results in an approximate closure for the replica off-diagonal direct correlation function. The replica off-diagonal Ornstein-Zernicke equation combined with this closure coincides with the equation for the non-ergodicity parameter derived using the mode-coupling theory.