Transitions in non-conserving models of Self-Organized Criticality

TL;DR

This paper analyzes a non-conserving earthquake model, showing that criticality persists despite dissipation, and identifies the average branching ratio as a key parameter controlling the transition to self-organized criticality.

Contribution

It provides analytical and numerical evidence that criticality can occur in non-conserving models and introduces the average branching ratio as an order parameter for the transition.

Findings

Criticality persists even with dissipation.

The avalanche size cutoff scales as ( ext{alpha}_c - ext{alpha})^{-3/2}.

The average branching ratio acts as an order parameter.

Abstract

We investigate a random--neighbours version of the two dimensional non-conserving earthquake model of Olami, Feder and Christensen [Phys. Rev. Lett. {\bf 68}, 1244 (1992)]. We show both analytically and numerically that criticality can be expected even in the presence of dissipation. As the critical level of conservation, $\alpha_c$, is approached, the cut--off of the avalanche size distribution scales as $\xi\sim(\alpha_c-\alpha)^{-3/2}$. The transition from non-SOC to SOC behaviour is controlled by the average branching ratio $\sigma$ of an avalanche, which can thus be regarded as an order parameter of the system. The relevance of the results are discussed in connection to the nearest-neighbours OFC model (in particular we analyse the relevance of synchronization in the latter).