Sensitivity to initial conditions in self-organized critical systems

TL;DR

This paper investigates how small initial differences affect avalanche dynamics in self-organized critical systems, revealing that damage remains localized and scale-free due to the Abelian property of the models.

Contribution

It demonstrates that damage does not spread in Abelian models, providing a theoretical explanation for localized damage in self-organized critical systems.

Findings

Damage does not spread in the model.

Damage remains statistically time-invariant and scale-free.

Ensemble averaging shows apparent damage spreading due to avalanche timing.

Abstract

We discuss sensitivity to initial conditions in a model for avalanches in granular media displaying self-organized criticality. We show that damage, due to a small perturbation in initial conditions, does not spread. The damage persists in a statistically time-invariant and scale-free form. We argue that the origin of this behavior is the Abelian nature of the model, which generalizes our results to all Abelian models, including the BTW model and the Manna model. An ensemble average of the damage leads to seemingly time dependent damage spreading. Scaling arguments show that this numerical result is due to the time lag before avalanches reach the initial perturbation.