Systematic perturbation approach for a dynamical scaling law in a kinetically constrained spin model

TL;DR

This paper develops a systematic perturbation method to analyze the dynamical scaling law in a kinetically constrained spin model, revealing a universal relation between size and time scales near the nonergodic transition.

Contribution

It introduces an exact perturbation analysis approach that clarifies the dynamical scaling law in a kinetically constrained spin model on a Bethe lattice.

Findings

Perturbation solutions converge to Monte Carlo results with higher order.

Identifies a universal size-time scaling law near the transition.

Provides a systematic framework for analyzing nonergodic transitions.

Abstract

The dynamical behaviours of a kinetically constrained spin model (Fredrickson-Andersen model) on a Bethe lattice are investigated by a perturbation analysis that provides exact final states above the nonergodic transition point. It is observed that the time-dependent solutions of the derived dynamical systems obtained by the perturbation analysis become systematically closer to the results obtained by Monte Carlo simulations as the order of a perturbation series is increased. This systematic perturbation analysis also clarifies the existence of a dynamical scaling law, which provides a implication for a universal relation between a size scale and a time scale near the nonergodic transition.