Mode-Coupling Theory as a Mean-Field Description of the Glass Transition

TL;DR

This paper compares mode-coupling theory and replica theory in describing the glass transition, revealing significant discrepancies especially in high dimensions, and highlights limitations of MCT due to pathological behaviors.

Contribution

It provides a detailed comparison of MCT and replica theory across dimensions, exposing fundamental differences and limitations of MCT in high-dimensional systems.

Findings

MCT better predicts nonergodic parameters in 3D than replica theory.

Discrepancies increase with dimension, especially in the predicted transition points.

MCT exhibits pathological negative tails in nonergodic parameters at high dimensions.

Abstract

Mode-coupling theory (MCT) is conjectured to be a mean-field description of dynamics of the structural glass transition and the replica theory to be its thermodynamic counterpart. However, the relationship between the two theories remains controversial and quantitative comparison is lacking. In this Letter, we investigate MCT for monatomic hard sphere fluids at arbitrary dimensions above three and compare the results with replica theory. We find grave discrepancies between the predictions of two theories. While MCT describes the nonergodic parameter quantitatively better than the replica theory in three dimension, it predicts a completely different dimension dependence of the dynamical transition point. We find it to be due to the pathological behavior of the nonergodic parameters derived from MCT, which exhibit negative tails in real space at high dimensions.