Smoluchowski dynamics and the ergodic-nonergodic transition

TL;DR

This paper investigates the ergodic-nonergodic transition in systems with Smoluchowski dynamics using a self-consistent perturbation theory, finding a transition at an unphysically high packing fraction.

Contribution

It develops a second-order perturbation theory linking static structure factors to effective potentials for Smoluchowski systems near the glass transition.

Findings

Identifies an ergodic-nonergodic transition at eta* > 0.76

Establishes a linear fluctuation-dissipation theorem in the context

Uses Percus-Yevick approximation for hard spheres

Abstract

We use the recently introduced theory for the kinetics of systems of classical particles to investigate systems driven by Smoluchowski dynamics. We investigate the existence of ergodic-nonergodic (ENE) transitions near the liquid-glass transition. We develop a self-consistent perturbation theory in terms of an effective two-body potential. We work to second order in this potential. At second order we have an explicit relationship between the static structure factor and the effective potential. We choose the static structure factor in the case of hard spheres to be given by the solution of the Percus-Yevick approximation for hard spheres. Then using the analytically determined ENE equation for the ergodicity function we find an ENE transition for packing fraction, eta, greater than a critical value eta*=0.76 which is physically unaccessible. The existence of a linear fluctuation-dissipation theorem in the problem is shown and used to great advantage.