Renewal, Modulation and Superstatistics

TL;DR

This paper compares renewal and modulation methods for generating non-Poisson waiting time distributions, revealing their distinct physical effects such as aging and diffusion behaviors, and emphasizing the role of critical events in modulation.

Contribution

It demonstrates that renewal and modulation, despite similar statistical distributions, produce different physical phenomena, highlighting the importance of critical events in modulation.

Findings

Renewal produces aging and anomalous scaling.

Modulation results in no aging and variable diffusion behaviors.

Critical events are key to understanding modulation effects.

Abstract

We consider two different proposals to generate a time series with the same non-Poisson distribution of waiting times, to which we refer to as renewal and modulation. We show that, in spite of the apparent statistical equivalence, the two time series generate different physical effects. Renewal generates aging and anomalous scaling, while modulation yields no aging and either ordinary or anomalous diffusion, according to the prescription used for its generation. We argue, in fact, that the physical realization of modulation involves critical events, responsible for scaling. In conclusion, modulation rather than ruling out the action of critical events, sets the challenge for their identification.