Macroscopic control parameter for avalanche models for bursty transport

TL;DR

This paper identifies a control parameter $R_A$ that governs the transition to Self-Organized Criticality in avalanching systems, highlighting fundamental differences from turbulence and implications for physical systems.

Contribution

It introduces a macroscopic control parameter $R_A$ for avalanche models and analyzes its role in the transition to SOC, contrasting it with turbulence.

Findings

SOC occurs as $R_A o 0$, indicating maximal excited degrees of freedom.

Power law scaling persists at finite $R_A$ if a large range of lengthscales exists.

SOC phenomenology can be observed in systems with finite $R_A$, given sufficient scale range.

Abstract

Similarity analysis is used to identify the control parameter $R_A$ for the subset of avalanching systems that can exhibit Self- Organized Criticality (SOC). This parameter expresses the ratio of driving to dissipation. The transition to SOC, when the number of excited degrees of freedom is maximal, is found to occur when $R_A \to 0$. This is in the opposite sense to (Kolmogorov) turbulence, thus identifying a deep distinction between turbulence and SOC and suggesting an observable property that could distinguish them. A corollary of this similarity analysis is that SOC phenomenology, that is, power law scaling of avalanches, can persist for finite $R_A$, with the same $R_A \to 0$ exponent, if the system supports a sufficiently large range of lengthscales; necessary for SOC to be a candidate for physical ($R_A$ finite) systems.