Theory of Self-organized Criticality for Problems with Extremal Dynamics

TL;DR

This paper develops a theoretical framework for understanding self-organized criticality in systems with extremal dynamics and quenched disorder, enabling calculation of critical exponents and application to models like Invasion Percolation.

Contribution

It introduces a novel theoretical scheme transforming quenched dynamics into a stochastic model with memory, allowing direct computation of critical exponents.

Findings

Applicable to Invasion Percolation

Transforms quenched dynamics into stochastic models with memory

Enables direct calculation of critical exponents

Abstract

We introduce a general theoretical scheme for a class of phenomena characterized by an extremal dynamics and quenched disorder. The approach is based on a transformation of the quenched dynamics into a stochastic one with cognitive memory and on other concepts which permit a mathematical characterization of the self-organized nature of the avalanche type dynamics. In addition it is possible to compute the relevant critical exponents directly from the microscopic model. A specific application to Invasion Percolation is presented but the approach can be easily extended to various other problems.