Relaxation times of kinetically constrained spin models with glassy dynamics

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 $d\ge 2$, we also prove a super-Arrhenius scaling of the form $\exp (1/(1-\rho ))\le \tau \le \exp (1/(1-\rho {)}^2)$. For FA1f in $d$=$1,2$ we rigorously prove the power law scalings recently derived in \cite{JMS} while in $d\ge 3$ 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.

4 pages

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• Time scale separation in the low temperature East model: Rigorous results
• Space-time phase transitions in the East model with a softened kinetic constraint
• Relaxation to equilibrium of generalized East processes on ${\mathbb{Z}}^d$: Renormalization group analysis and energy-entropy competition
• Is there a fractional breakdown of the Stokes-Einstein relation in Kinetically Constrained Models at low temperature?
• The relation between the structure of blocked clusters and the relaxation dynamics in kinetically-constrained models
• The influence of dimension on the relaxation process of East-like models
• Exponential convergence to equilibrium for the d-dimensional East model
• Counting metastable states in a kinetically constrained model using a patch repetition analysis
• Time scale separation and dynamic heterogeneity in the low temperature East model
• Systematic perturbation approach for a dynamical scaling law in a kinetically constrained spin model
• Fredrickson--Andersen model in two dimensions