Mean-Field Theory and Sandpile Models

TL;DR

This paper develops a detailed mean-field theory for sandpile models, clarifying how finite driving rates influence their critical behavior and universal properties, advancing understanding of self-organized criticality.

Contribution

It introduces a solvable mean-field framework based on a one-dimensional random walker to analyze sandpile models under various driving conditions.

Findings

Finite driving rate affects sandpile criticality.

Universal quantities are sensitive to driving conditions.

Complete mean-field solutions for different driving types.

Abstract

We review and refine the concept of a mean-field theory for the study of sandpile models, which are of central importance in the study of self-organized criticality. By considering the simple one-dimensional random walker with an absorbing and reflecting boundary we are able to construct a complete mean-field picture which we can solve in detail for different types of driving. Using this theory, we are able to clarify the effect of finite driving rate on sandpile models, as well as the observed sensitivity of certain universal quantities on the driving.