Relaxation times of kinetically constrained spin models with glassy dynamics

TL;DR

This paper rigorously analyzes the relaxation times of kinetically constrained spin models relevant to glassy dynamics, revealing exponential decay of correlations and super-Arrhenius scaling, challenging previous numerical findings.

Contribution

It provides the first rigorous proofs of relaxation time scalings and decay behaviors in kinetically constrained models, introducing a novel multi-scale analytical approach.

Findings

Exponential decay of persistence and auto-correlation functions.

Super-Arrhenius scaling of relaxation times in FA2f models.

Power law scalings in FA1f models for dimensions 1 and 2.

Abstract

We analyze the density and size dependence of the relaxation time $\tau$ for kinetically constrained spin systems. These have been proposed as models for strong or fragile glasses and for systems undergoing jamming transitions. For the one (FA1f) or two (FA2f) spin facilitated Fredrickson-Andersen model at any density $\rho<1$ and for the Knight model below the critical density at which the glass transition occurs, we show that the persistence and the spin-spin time auto-correlation functions decay exponentially. This excludes the stretched exponential relaxation which was derived by numerical simulations. For FA2f in $d\geq 2$, we also prove a super-Arrhenius scaling of the form $\exp(1/(1-\rho))\leq \tau\leq\exp(1/(1-\rho)^2)$. For FA1f in $d$=$1,2$ we rigorously prove the power law scalings recently derived in \cite{JMS} while in $d\geq 3$ we obtain upper and lower bounds consistent with findings therein. Our results are based on a novel multi-scale approach which allows to analyze $\tau$ in presence of kinetic constraints and to connect time-scales and dynamical heterogeneities. The techniques are flexible enough to allow a variety of constraints and can also be applied to conservative stochastic lattice gases in presence of kinetic constraints.