Theory of the $\beta$-Relaxation Beyond Mode-Coupling Theory: A Microscopic Treatment

TL;DR

This paper extends mode-coupling theory to include critical fluctuations, deriving a microscopic stochastic model for $eta$-relaxation that predicts a smooth crossover replacing the mean-field transition.

Contribution

It develops a microscopic, fluctuation-inclusive extension of MCT, linking it to stochastic processes and providing explicit calculations for hard-sphere systems.

Findings

Fluctuations restore ergodicity in the system.

The theory predicts a smooth crossover instead of a sharp transition.

Explicit microscopic coupling constants are computed for hard spheres.

Abstract

We develop a systematic extension of mode-coupling theory (MCT) that incorporates critical dynamical fluctuations. Starting from a microscopic diagrammatic theory, we identify dominant classes of divergent diagrams near the mode-coupling transition and show that the corresponding asymptotic series dominates the mean-field below an upper critical dimension $d_c=8$. To resum these divergences, we construct a mapping to a stochastic dynamical process in which the order parameter evolves under random spatiotemporal fields. This reformulation provides a controlled, fully dynamical derivation of an effective theory for the $\beta$-relaxation which remarkably coincides with stochastic beta-relaxation theory [T. Rizzo, EPL 106, 56003 (2014)]. All coupling constants of the latter theory are expressed microscopically in terms of the liquid static structure factor and are computed for the paradigmatic hard-sphere system. The analysis demonstrates that fluctuations alone restore ergodicity and replace the putative mean-field transition by a smooth crossover. Our results establish a predictive framework for structural relaxation beyond mean-field.