Inhomogeneous sandpile model: Crossover from multifractal scaling to finite size scaling

TL;DR

This paper investigates an inhomogeneous sandpile model combining two toppling rules, revealing a crossover from multifractal to finite size scaling influenced by the density of stochastic sites, highlighting the importance of local rules in global behavior.

Contribution

The study introduces a mixed-rule sandpile model with a tunable parameter, demonstrating how local rule variations induce a transition in scaling behavior from multifractal to finite size scaling.

Findings

Crossover from multifractal to finite size scaling observed.

Critical density for crossover is between 0 and 0.01.

Local dynamical rules significantly affect global properties.

Abstract

We study an inhomogeneous sandpile model in which two different toppling rules are defined. For any site only one rule is applied corresponding to either the Bak, Tang and Wiesenfeld model {[}P.Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. \textbf{59}, 381 (1987){]} or the Manna two-state sandpile model {[}S. S. Manna, J. Phys. A \textbf{24}, L363 (1991){]}. A parameter $c$ is introduced which describes a density of sites which are randomly deployed and where the stochastic Manna rules are applied. The results show that the avalanche area exponent $\tau_{a}$, avalanche size exponent $\tau_{s}$, and capacity fractal dimension $D_{s}$ depend on the density $c$. A crossover from multifractal scaling of the Bak, Tang, and Wiesenfeld model ($c=0$) to finite size scaling was found. The critical density $c$ is found to be in the interval $0<c<0.01$. These results demonstrate that local dynamical rules are important and can change the global properties of the model.