Eur. Phys. J. B 76, 21-30 (2010)
https://doi.org/10.1140/epjb/e2010-00164-8

Scaling and complex avalanche dynamics in the Abelian sandpile model

¹ Physics Department, College of Sciences, Shiraz University, 71454, Shiraz, Iran
² Max-Planck Institute for the Science of Light, Guenther-Scharowsky Str. 1/24, 91058 Erlangen, Germany

Corresponding author: ^a montakhab@shirazu.ac.ir

Received: 1 December 2009
Revised: 11 April 2010
Published online: 4 June 2010

Abstract

We study the two-dimensional Abelian Sandpile Model on a square lattice of linear size L. We introduce the notion of avalanche's fine structure and compare the behavior of avalanches and waves of toppling. We show that according to the degree of complexity in the fine structure of avalanches, which is a direct consequence of the intricate superposition of the boundaries of successive waves, avalanches fall into two different categories. We propose scaling ansätz for these avalanche types and verify them numerically. We find that while the first type of avalanches (α) has a simple scaling behavior, the second complex type (β) is characterized by an avalanche-size dependent scaling exponent. In particular, we define an exponent γ to characterize the conditional probability distribution functions for these types of avalanches and show that γ_α = 0.42, while 0.7 ≤ γ_β ≤ 1.0 depending on the avalanche size. This distinction provides a framework within which one can understand the lack of a consistent scaling behavior in this model, and directly addresses the long-standing puzzle of finite-size scaling in the Abelian sandpile model.