Non-Poisson Renewal Events and Memory

TL;DR

This paper compares two types of fluctuation-dissipation processes leading to anomalous relaxation, highlighting how crucial events cause non-Poisson renewal processes with power-law waiting times, affecting the interpretation of relaxation phenomena.

Contribution

It introduces a detailed analysis of two fluctuation-dissipation processes, emphasizing the role of non-Poisson renewal events in anomalous relaxation dynamics.

Findings

Stationary processes have exponential waiting times.

Crucial events lead to power-law waiting times with diverging mean.

Both processes exhibit Mittag-Leffler type relaxation.

Abstract

We study two different forms of fluctuation-dissipation processes generating anomalous relaxations to equilibrium of an initial out of equilibrium condition, the former being based on a stationary although very slow correlation function and the latter characterized by the occurrence of crucial events, namely, non-Poisson renewal events, incompatible with the stationary condition. Both forms of regression to equilibrium have the same non-exponential Mittag-Leffler structure. We analyze the single trajectories of the two processes by recording the time distances between two consecutive origin re-crossings and establishing the corresponding waiting time probability density function (PDF), $\psi(t)$. In the former case, with no crucial events, $\psi(t)$ is exponential and in the latter case, with crucial events, $\psi(t)$ is an inverse power law PDF with a diverging first moment. We discuss the consequences that this result is expected to have for the correct interpretation of some anomalous relaxation processes.