Theory of Extremal Dynamics with Quenched Disorder: Self-Organization, Avalanche Dynamics and Critical Exponents

In this paper we discuss a general theoretical scheme, that we have recently proposed, for a class of phenomena characterized by extremal dynamics with quenched disorder. The approach is based on a transformation of the quenched dynamics into a stochastic dynamics with cognitive memory. This transformation, together with other concepts, permits a mathematical characterization of the self-organized nature of the avalanche type dynamics. By combining the mapping with real space methods, like the fixed scale transformation (FST), it is also 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 with quenched disorder.