Relaxation and overlap probability function in the spherical and mean spherical model

TL;DR

This paper investigates the dynamical differences between the spherical and mean spherical models after a quench, revealing a crossover time where their behaviors diverge and highlighting stability differences.

Contribution

It provides a dynamical analysis of the equivalence between spherical and mean spherical models, identifying a crossover time and stability properties.

Findings

Existence of a crossover time $t^* \,\sim\, V^{2/d}$ separating equivalent and divergent behaviors.

The response function relation holds for the spherical model but not for the mean spherical model.

The mean spherical model is an example of a system that is not stochastically stable.

Abstract

The problem of the equivalence of the spherical and mean spherical models, which has been thoroughly studied and understood in equilibrium, is considered anew from the dynamical point of view during the time evolution following a quench from above to below the critical temperature. It is found that there exists a crossover time $t^* \sim V^{2/d}$ such that for $t < t^*$ the two models are equivalent, while for $t > t^*$ macroscopic discrepancies arise. The relation between the off equilibrium response function and the structure of the equilibrium state, which usually holds for phase ordering systems, is found to hold for the spherical model but not for the mean spherical one. The latter model offers an explicit example of a system which is not stochastically stable.