Generalized mode-coupling theory of the glass transition: schematic results at finite and infinite order

TL;DR

This paper develops an extensive generalized mode-coupling theory for the glass transition, demonstrating convergence of schematic models and exploring how high-order contributions influence the nature of the transition.

Contribution

It introduces schematic GMCT models, derives an exact solution for a specific case, and shows convergence and control of transition types without relying on small parameters.

Findings

Infinite hierarchy of GMCT equations converges as closure level increases.

Numerical results show robust convergence patterns in schematic models.

High-order contributions determine whether the transition is continuous, discontinuous, or avoided.

Abstract

We present an extensive treatment of the generalized mode-coupling theory (GMCT) of the glass transition, which seeks to describe the dynamics of glass-forming liquids using only static structural information as input. This theory amounts to an infinite hierarchy of coupled equations for multi-point density correlations, the lowest-order closure of which is equivalent to standard mode-coupling theory. Here we focus on simplified schematic GMCT hierarchies, which lack any explicit wavevector-dependence and therefore allow for greater analytical and numerical tractability. For one particular schematic model, we derive the unique analytic solution of the infinite hierarchy, and demonstrate that closing the hierarchy at finite order leads to uniform convergence as the closure level increases. We also show numerically that a similarly robust convergence pattern emerges for more generic schematic GMCT models, suggesting that the GMCT framework is generally convergent, even though no small parameter exists in the theory. Finally, we discuss how different effective weights on the high-order contributions ultimately control whether the transition is continuous, discontinuous, or strictly avoided, providing new means to relate structure to dynamics in glass-forming systems.