A Hierarchical Model of Slow Constrained Dynamics

TL;DR

This paper presents a hierarchical constrained model demonstrating extremely slow relaxation dynamics, with relaxation behavior depending on temperature change, and provides analytical and numerical insights into the underlying mechanisms.

Contribution

It introduces a new hierarchical model with simple energy structure and hierarchical constraints, capturing slow relaxation phenomena and analytical descriptions of the dynamics.

Findings

Relaxation is extremely slow when cooling from T0 to Tf, fitting a stretched exponential.

Relaxation is exponential when Tf > T0, with times obeying an Arrhenius law.

A simple equation effectively describes the slow relaxation behavior.

Abstract

We introduce a new simple hierarchically constrained model of slow relaxation. The configurational energy has a simple form as there is no coupling among the spins defining the system; the associated stationary distribution is an equilibrium, Gibbsian one. However, due to the presence of hierarchical constraints in the dynamics the system is found to relax to its equilibrium distribution in an extremely slow fashion when suddenly cooled from an initial temperature, $T_0$, to a final one $T_f$. The relaxation curve in that case can be fit by an stretched exponential curve. On the other hand the relaxation function is found to be exponential when $T_f >T_0$, with characteristic times depending on both $T_f$ and $T_0$, with characteristic times obeying an Arrhenius law. Numerical results as well as some analytical studies are presented. In particular we introduce a simple equation that captures the essence of the slow relaxation.