Introduction

Self-organized criticality was introduced in 1987 by Bak, Tang, and Wiesenfeld (BTW), who employed some appealing yet heuristic handwaving arguments. To substantiate the hypothesis, computer simulations of a simple algorithm – inspired by the avalanches induced when one plays with a pile of sand – were presented. Substantial analytic understanding was lacking for some time. Soon, however, standard mathematical tools were being applied to the new set of models. Statistical mechanics traditionally makes use of a combination of approaches, one of which consists of defining models that have a structure that allows an exact calculation of specific quantities. The art is to formulate a model of the right degree of complexity. One wants a model with sufficient structure to contain nonobvious behavior, but the model should not be so complicated that analytic approaches cannot be carried through. This last property obviously depends strongly on who is going to perform the analysis. The mathematical power of Deepak Dhar and his co-workers made it possible for them to solve an only slightly altered version of the original BTW cellular automata (Dhar and Ramaswamy 1989; Dhar 1990). After Dhar's work it was clear that, at least in some cases, the observed critical behavior was not merely an artefact of simulations on too-small systems. We shall outline the approach developed by Dhar and co-workers in Section 5.3.

Despite their undeniable beauty, the exact solutions have one drawback: the specific mathematics tends to be tailored to the details of the solved model. This means that generalization to other, similar models is often not possible.