Scale Invariant Avalanches: A Critical Confusion

TL;DR

This paper clarifies the role of the power-law exponent in scale invariant avalanches, addressing confusion in the field by analyzing its impact on criticality, predictability, and system dynamics across various phenomena.

Contribution

It introduces a criticality condition based on the exponent value, explaining how smaller exponents lead to more predictable and less critical systems, and reviews prediction methods and common misconceptions.

Findings

Smaller exponents reduce critical properties and increase predictability.

The exponent controls the energy balance and size ratio of avalanches.

Logarithmic scales and dissipation influence avalanche analysis.

Abstract

The "Self-organized criticality" (SOC), introduced in 1987 by Bak, Tang and Wiesenfeld, was an attempt to explain the 1/f noise, but it rapidly evolved towards a more ambitious scope: explaining scale invariant avalanches. In two decades, phenomena as diverse as earthquakes, granular piles, snow avalanches, solar flares, superconducting vortices, sub-critical fracture, evolution, and even stock market crashes have been reported to evolve through scale invariant avalanches. The theory, based on the key axiom that a critical state is an attractor of the dynamics, presented an exponent close to -1 (in two dimensions) for the power-law distribution of avalanche sizes. However, the majority of real phenomena classified as SOC present smaller exponents, i.e., larger absolute values of negative exponents, a situation that has provoked a lot of confusion in the field of scale invariant avalanches. The main goal of this chapter is to shed light on this issue. The essential role of the exponent value of the power-law distribution of avalanche sizes is discussed. The exponent value controls the ratio of small and large events, the energy balance --required for stationary systems-- and the critical properties of the dynamics. A condition of criticality is introduced. As the exponent value decreases, there is a decrease of the critical properties, and consequently the system becomes, in principle, predictable. Prediction of scale invariant avalanches in both experiments and simulations are presented. Other sources of confusion as the use of logarithmic scales, and the avalanche dynamics in well established critical systems, are also revised; as well as the influence of dissipation and disorder in the "self-organization" of scale invariant dynamics.