Asymptotic laws for tagged-particle motion in glassy systems

TL;DR

This paper derives and tests asymptotic laws for the motion of a tagged particle near a glass transition, showing how corrections significantly affect the interpretation of experimental and simulation data.

Contribution

The paper provides explicit asymptotic laws and correction formulas for tagged-particle dynamics within mode-coupling theory, validated by numerical simulations.

Findings

Long-time relaxation follows alpha-scaling and von Schweidler decay.

Corrections to asymptotic laws significantly alter the non-Gaussian parameter.

Results qualitatively agree with molecular dynamics simulations.

Abstract

Within the mode-coupling theory for structural relaxation in simple systems the asymptotic laws and their leading-asymptotic correction formulas are derived for the motion of a tagged particle near a glass-transition singularity. These analytic results are compared with numerical ones of the equations of motion evaluated for a tagged hard sphere moving in a hard-sphere system. It is found that the long-time part of the two-step relaxation process for the mean-squared displacement can be characterized by the $\alpha $-relaxation-scaling law and von Schweidler's power-law decay while the critical-decay regime is dominated by the corrections to the leading power-law behavior. For parameters of interest for the interpretations of experimental data, the corrections to the leading asymptotic laws for the non-Gaussian parameter are found to be so large that the leading asymptotic results are altered qualitatively by the corrections. Results for the non-Gaussian parameter are shown to follow qualitatively the findings reported in the molecular-dynamics-simulations work by Kob and Andersen [Phys. Rev. E 51, 4626 (1995)].