Fractals in the Critical Attractor of the Classical Sandpile Model

Abstract

The classical sandpile model for self-organized criticality is analyzed with deterministic perturbations. This allows us to explore underlying long-range correlations in the critical attractor and quantify these as the fractal dimension of excitable sites and the fractal dimension of avalanche termination sites. The fractal of highly excitable sites, characterized by a dimension of 1.3, is associated with the avalanches responsible for most of the relaxations in the system. Furthermore, the fractal dimension of these highly excitable sites suggests a scaling exponent for avalanche sizes of 1.26, consistent with earlier literature.