Cooperative Behavior of Kinetically Constrained Lattice Gas Models of Glassy Dynamics

TL;DR

This paper analyzes kinetically constrained lattice models of glasses, proving that their behavior on hypercubic lattices is limited to ergodic or trivial non-ergodic states, with cooperative dynamics and phase transitions observed on Bethe lattices.

Contribution

It demonstrates the dichotomy of behaviors in KA models depending on lattice type and introduces a detailed analysis of phase transitions and cooperativity in these models.

Findings

Ergodicity or trivial non-ergodicity on hypercubic lattices.

Cooperative dynamics at high densities in ergodic cases.

Dynamical phase transition with diverging scales on Bethe lattices.

Abstract

Kinetically constrained lattice models of glasses introduced by Kob and Andersen (KA) are analyzed. It is proved that only two behaviors are possible on hypercubic lattices: either ergodicity at all densities or trivial non-ergodicity, depending on the constraint parameter and the dimensionality. But in the ergodic cases, the dynamics is shown to be intrinsically cooperative at high densities giving rise to glassy dynamics as observed in simulations. The cooperativity is characterized by two length scales whose behavior controls finite-size effects: these are essential for interpreting simulations. In contrast to hypercubic lattices, on Bethe lattices KA models undergo a dynamical (jamming) phase transition at a critical density: this is characterized by diverging time and length scales and a discontinuous jump in the long-time limit of the density autocorrelation function. By analyzing generalized Bethe lattices (with loops) that interpolate between hypercubic lattices and standard Bethe lattices, the crossover between the dynamical transition that exists on these lattices and its absence in the hypercubic lattice limit is explored. Contact with earlier results are made via analysis of the related Fredrickson-Andersen models, followed by brief discussions of universality, of other approaches to glass transitions, and of some issues relevant for experiments.