Symmetries and Fixed Point Stability of Stochastic Differential Equations Modeling Self-Organized Criticality

TL;DR

This paper introduces a regularization technique that restores symmetry in stochastic models of self-organized criticality, enabling the analysis of critical fixed points and universality classes using dynamic renormalization group methods.

Contribution

The authors propose a novel regularization approach that breaks and then restores symmetry in stochastic models, allowing access to nontrivial fixed points in the DRG analysis.

Findings

Both models belong to the same universality class.

The regularization restores symmetry and reveals the nontrivial fixed point.

The technique can be applied to other threshold dynamic problems.

Abstract

A stochastic nonlinear partial differential equation is built for two different models exhibiting self-organized criticality, the Bak, Tang, and Wiesenfeld (BTW) sandpile model and the Zhang's model. The dynamic renormalization group (DRG) enables to compute the critical exponents. However, the nontrivial stable fixed point of the DRG transformation is unreachable for the original parameters of the models. We introduce an alternative regularization of the step function involved in the threshold condition, which breaks the symmetry of the BTW model. Although the symmetry properties of the two models are different, it is shown that they both belong to the same universality class. In this case the DRG procedure leads to a symmetric behavior for both models, restoring the broken symmetry, and makes accessible the nontrivial fixed point. This technique could also be applied to other problems with threshold dynamics.