A simple analytical description of the non-stationary dynamics in Ising spin systems

TL;DR

This paper extends a variational approach to analytically describe the non-stationary dynamics of Ising spin systems, demonstrating high accuracy across various models and proposing the 'diamond' approximation as a minimal standard.

Contribution

It introduces the application of the cluster variational method to non-stationary Ising dynamics and validates the 'diamond' approximation against numerical Glauber dynamics.

Findings

The 'diamond' approximation accurately describes non-stationary dynamics in various Ising models.

Discrepancies appear in spin glass models at very low temperatures due to metastable states.

The method offers a simple yet effective analytical tool for studying Ising dynamics.

Abstract

The analytical description of the dynamics in models with discrete variables (e.g. Ising spins) is a notoriously difficult problem, that can be tackled only under some approximation. Recently a novel variational approach to solve the stationary dynamical regime has been introduced by Pelizzola [Eur. Phys. J. B, 86 (2013) 120], where simple closed equations are derived under mean-field approximations based on the cluster variational method. Here we propose to use the same approximation based on the cluster variational method also for the non-stationary regime, which has not been considered up to now within this framework. We check the validity of this approximation in describing the non-stationary dynamical regime of several Ising models defined on Erdos-R\'enyi random graphs: we study ferromagnetic models with symmetric and partially asymmetric couplings, models with random fields and also spin glass models. A comparison with the actual Glauber dynamics, solved numerically, shows that one of the two studied approximations (the so-called 'diamond' approximation) provides very accurate results in all the systems studied. Only for the spin glass models we find some small discrepancies in the very low temperature phase, probably due to the existence of a large number of metastable states. Given the simplicity of the equations to be solved, we believe the diamond approximation should be considered as the 'minimal standard' in the description of the non-stationary regime of Ising-like models: any new method pretending to provide a better approximate description to the dynamics of Ising-like models should perform at least as good as the diamond approximation.