Continuum Theory with Memory for Avalanches in Self-Organized Criticality

TL;DR

This paper develops a continuum theory incorporating memory effects to describe avalanche dynamics in self-organized criticality, revealing anomalous spreading behavior and verifying the theory through exact and numerical models.

Contribution

It introduces a novel continuum framework with memory for avalanche propagation in self-organized critical systems, extending traditional models with nonlocal, history-dependent potentials.

Findings

Derived an anomalous tail for avalanche spread probability.

Verified the theory with an exactly solvable model (D=4).

Numerically tested the theory on the Bak-Sneppen model.

Abstract

The propagator for the activity in a broad class of self-organized critical models obeys an imaginary-time Schr\"odinger equation with a nonlocal, history-dependent potential representing memory. Consequently, the probability for an avalanche to spread beyond a distance $r$ in time $t$ has an anomalous tail $\exp{[-C\,x^{1/(D-1)}]}$ for $x=r^D/t \gg 1$ and $D>2$, indicative of glassy dynamics. The theory is verified for an exactly solvable model, where $D=4$ and $C=3/4$, and for the Bak-Sneppen model where it is tested numerically.