Tagged-particle motion of Percus-Yevick hard spheres from first principles

TL;DR

This paper develops a first-principles generalized mode-coupling theory to analyze the tagged-particle dynamics in glassy systems, providing improved quantitative predictions and insights into relaxation behaviors near the glass transition.

Contribution

It introduces a hierarchical GMCT framework for self-particle motion in glassy systems, extending traditional mode-coupling theory to higher levels for better accuracy.

Findings

GMCT results qualitatively match standard MCT predictions

Quantitative results improve with higher GMCT closure levels

Current GMCT cannot explain Stokes-Einstein violation

Abstract

We develop a first-principles-based generalized mode-coupling theory (GMCT) for the tagged-particle motion of glassy systems. This theory establishes a hierarchy of coupled integro-differential equations for self-multi-point density correlation functions, which can formally be extended up to infinite order. We use our GMCT framework to calculate the self-nonergodicity parameters and the self-intermediate scattering function for the Percus-Yevick hard sphere system, based on the first few levels of the GMCT hierarchy. We also test the scaling laws in the $\alpha$- and $\beta$-relaxation regimes near the glass-transition singularity. Furthermore, we study the mean-square displacement and the Stoke-Einstein relation in the supercooled regime. We find that qualitatively our GMCT results share many similarities with the well-established predictions from standard mode-coupling theory, but the quantitative results change, and typically improve, by increasing the GMCT closure level. However, we also demonstrate on general theoretical grounds that the current GMCT framework is unable to account for violation of the Stokes-Einstein relation, underlining the need for further improvements in the first-principles description of glassy dynamics.