Point-occurrence self-similarity in crackling-noise systems and in other complex systems

TL;DR

This paper investigates the self-similarity in the timing of events in crackling noise systems and other complex systems, revealing unique scaling laws and their relation to renewal processes and correlations.

Contribution

It introduces a detailed analysis of the uncommon scaling law in crackling noise and non-correlated renewal processes, highlighting the role of finite and infinite mean waiting times.

Findings

Scaling law observed in crackling noise systems.

Processes without finite mean waiting time show double power-law distributions.

Short-range correlations do not alter the attraction to limit distributions.

Abstract

It has been recently found that a number of systems displaying crackling noise also show a remarkable behavior regarding the temporal occurrence of successive events versus their size: a scaling law for the probability distributions of waiting times as a function of a minimum size is fulfilled, signaling the existence on those systems of self-similarity in time-size. This property is also present in some non-crackling systems. Here, the uncommon character of the scaling law is illustrated with simple marked renewal processes, built by definition with no correlations. Whereas processes with a finite mean waiting time do not fulfill a scaling law in general and tend towards a Poisson process in the limit of very high sizes, processes without a finite mean tend to another class of distributions, characterized by double power-law waiting-time densities. This is somehow reminiscent of the generalized central limit theorem. A model with short-range correlations is not able to escape from the attraction of those limit distributions. A discussion on open problems in the modeling of these properties is provided.