Improved mean-field dynamical equations are able to detect the two-steps relaxation in glassy dynamics at low temperatures

TL;DR

This paper develops improved mean-field dynamical equations for the Ising p-spin model, successfully capturing the two-step relaxation characteristic of glassy dynamics at low temperatures, with adjustments for entropic barriers.

Contribution

It introduces a new closure scheme for the master equation and a renormalized timescale to accurately model low-temperature glassy dynamics.

Findings

The dynamical mean-field equations match Monte Carlo simulations.

The two-step relaxation is correctly reproduced with the renormalized timescale.

The approach captures out-of-equilibrium dynamics across temperature regimes.

Abstract

We study the stochastic relaxation dynamics of the Ising p-spin model on a random graph, a well-known model with glassy dynamics at low temperatures. We introduce and discuss a new closure scheme for the master equation governing the continuous-time relaxation of the system, that translates into a set of differential equations for the evolution of local probabilities. The solution to these dynamical mean-field equations describes very well the out-of-equilibrium dynamics at high temperatures, notwithstanding the key observation that the off-equilibrium probability measure contains higher-order interaction terms, not present in the equilibrium measure. In the low-temperature regime, the solution to the dynamical mean-field equations shows the correct two-step relaxation (a typical feature of the glassy dynamics), but with a relaxation timescale too short. We propose a solution to this problem by identifying the range of energies where entropic barriers play a key role and defining a renormalized microscopic timescale for the dynamical mean-field solution. The final result perfectly matches the complex out-of-equilibrium dynamics computed through extensive Monte Carlo simulations.