The influence of dimension on the relaxation process of East-like models

TL;DR

This paper investigates how the dimension affects the relaxation process of East-like models, revealing that higher dimensions alter the relaxation time scaling and influence the out-of-equilibrium dynamics significantly.

Contribution

The study rigorously establishes the correct super-Arrhenius scaling of relaxation time in higher dimensions and highlights the impact of boundary conditions and dimensional effects on dynamics.

Findings

Relaxation time scales as (eta^2/2d\log 2) instead of previous estimates.

Boundary conditions significantly affect relaxation times at finite scales.

Diagonal propagation of influence is faster than along axes due to entropic effects.

Abstract

We consider the relaxation process and the out-of-equilibrium dynamics of natural generalizations to arbitrary dimensions of the well known one dimensional East process. These facilitated models are supposed to catch some of the main features of the complex dynamics of fragile glasses. We focus on the low temperature regime (small density $c \approx e^{-\beta}$ of the facilitating sites). In the literature the relaxation process for the above models has been assumed to be quasi-one dimensional and, in particular, their equilibration time has been computed using the relaxation time of the East model ($d=1$) on the equilibrium length scale $L_c=(1/c)^{1/d}$ in $d$-dimension. This led to the derivation of a super-Arrhenius scaling for the relaxation time of the form $T_{\rm rel} \asymp \exp(\beta^2/d\log 2)$. In a companion paper, using mainly renormalization group ideas and electrical networks methods, we rigorously establish that instead $T_{\rm rel} \asymp \exp(\beta^2/2d\log 2)$, a result showing that the relaxation process cannot be quasi-one-dimensional. The above scaling sharply confirms previous MCAMC (Monte Carlo with Absorbing Markov Chains) simulations. Next we compute the relaxation time at finite and mesoscopic length scales, and show a dramatic dependence on the boundary conditions, yet another indication of key dimensional effects. Our final result is related to the out-of-equilibrium dynamics. Starting with a single facilitating site at the origin we show that, up to length scales $L=O(L_c)$, its influence propagates much faster (on a logarithmic scale) along the diagonal direction than along the axes directions. Such unexpected result is due to a rather delicate balance between dynamical energy barriers and entropic effects in the constrained dynamics.