Modeling temporal fluctuations in avalanching systems

TL;DR

This paper introduces a stochastic differential equation model for avalanching systems' activity, capturing fluctuations and avalanche statistics, and compares it with numerical simulations of sandpile models.

Contribution

It generalizes mean field sandpile dynamics using Itoh's SDE with fractional Gaussian noise, enabling analytical computation of avalanche exponents and activity fluctuations.

Findings

Model agrees with numerical sandpile simulations

Sandpiles do not exhibit universal non-Gaussian activity PDFs

Differences found between sandpile activity and turbulence energy fluctuations

Abstract

We demonstrate how to model the toppling activity in avalanching systems by stochastic differential equations (SDEs). The theory is developed as a generalization of the classical mean field approach to sandpile dynamics by formulating it as a generalization of Itoh's SDE. This equation contains a fractional Gaussian noise term representing the branching of an avalanche into small active clusters, and a drift term reflecting the tendency for small avalanches to grow and large avalanches to be constricted by the finite system size. If one defines avalanching to take place when the toppling activity exceeds a certain threshold the stochastic model allows us to compute the avalanche exponents in the continum limit as functions of the Hurst exponent of the noise. The results are found to agree well with numerical simulations in the Bak-Tang-Wiesenfeld and Zhang sandpile models. The stochastic model also provides a method for computing the probability density functions of the fluctuations in the toppling activity itself. We show that the sandpiles do not belong to the class of phenomena giving rise to universal non-Gaussian probability density functions for the global activity. Moreover, we demonstrate essential differences between the fluctuations of total kinetic energy in a two-dimensional turbulence simulation and the toppling activity in sandpiles.