Relationship between long time scales and the static free-energy in the Hopfield model

TL;DR

This paper analytically links the long relaxation time scales in the Hopfield model's Glauber dynamics to the stationary points of its static mean-field free-energy, revealing a structured relationship between dynamics and energy landscape.

Contribution

It provides an analytical derivation of the relaxation time spectrum in the Hopfield model, connecting time scales to stationary points of the free-energy landscape.

Findings

Long relaxation times are grouped into families associated with stationary points.

Time scales within a family are inversely related to the eigenvalues of the free-energy Hessian.

The spectrum of relaxation times is derived for large system sizes.

Abstract

The Glauber dynamics of the Hopfield model at low storage level is considered. We analytically derive the spectrum of relaxation times for large system sizes. The longest time scales are gathered in families, each family being in one to one correspondence with a stationary (not necessarily stable) point of the static mean-field free-energy. Inside a family, the time scales are given by the reciprocals (of the absolute values) of the eigenvalues of the free-energy Hessian matrix.