Glauber Dynamics and Ageing

TL;DR

This paper investigates the Glauber dynamics of various models to understand the conditions under which ageing phenomena occur, analyzing eigenvalue spectra and relaxation times through numerical and analytical methods.

Contribution

It provides a detailed analysis of the eigenvalue density and relaxation behavior in Glauber dynamics for different models, linking spectral properties to ageing.

Findings

Ageing depends on the eigenvalue density at small eigenvalues.

Eigenvalue density relates to the spectral dimension of the associated random walk.

Relaxation time scales with system size in specific models.

Abstract

The Glauber dynamics of various models (REM-like trap models, Brownian motion, BM model, Ising chain and SK model) is analyzed in relation with the existence of ageing. From a finite size Glauber matrix, we calculate a time $\tau_w(N)$ after which the system has relaxed to the equilibrium state. The case of metastability is also discussed. If the only non zero overlaps between pure states are only self-overlaps (REM-like trap models, BM model), the existence or absence of ageing depends only on the behavior of the density of eigenvalues for small eigenvalues. We have carried out a detailed numerical and analytical analysis of the density of eigenvalues of the REM-like trap models. In this case, we show that the behavior of the density of eigenvalues for typical trap realizations is related to the spectral dimension of the equivalent random walk model.