A stochastic theory for temporal fluctuations in self-organized critical systems

TL;DR

This paper develops a stochastic mean-field theory modeling the temporal fluctuations of activity in sandpile models, capturing their anti-persistent behavior and aligning well with numerical simulations.

Contribution

It introduces a novel stochastic differential equation framework with fractional Gaussian noise to describe sandpile activity dynamics.

Findings

The theory accurately reproduces temporal features observed in simulations.

It models activity as an anti-persistent Gaussian walk with activity-dependent diffusion.

The approach generalizes existing models by incorporating fractional Gaussian noise.

Abstract

A stochastic theory for the toppling activity in sandpile models is developed, based on a simple mean-field assumption about the toppling process. The theory describes the process as an anti-persistent Gaussian walk, where the diffusion coefficient is proportional to the activity. It is formulated as a generalization of the It\^{o} stochastic differential equation with an anti-persistent fractional Gaussian noise source. An essential element of the theory is re-scaling to obtain a proper thermodynamic limit, and it captures all temporal features of the toppling process obtained by numerical simulation of the Bak-Tang-Wiesenfeld sandpile in this limit.