Classical two-dimensional sandpile models

I consider sandpile models being open nonlinear systems demonstrating the phenomenon of avalanche-like response to a single disturbance of steady state. I study thoroughly the five most known variants of the two-dimensional rules referred as the models of Dhar–Ramaswamy, Pastor–Satorras–Vespignani, Feder–Feder, Manna and Bak–Tang–Wiesenfeld. The analytical solutions obtained in various ways are known for the first four models and the reasons preventing the construction of a solution are known for the fifth one. The generalization of these results allows to develop a common approach to the theoretical study of self-organized critical phenomena. Self-organization into the critical state gives rise to the scale-invariant properties. Theirs statistical description can not be generally obtained on the basis of the models’ rules. Intermediate level models mediates between the microlevel of the elements local behavior and the macrolevel of the entire system behavior. The rules of these models are not derived from the rules of original models, but are formulated on the ground of physical intuition, computer simulation results and common understanding of the dynamic processes that hold the system near the critical point. The collective dynamics of all of the models is reduced at the intermediate level to the processes familiar to mathematical physics, the first of them are asymmetric random walks. On this basis, I propose uniform methods of solution of models. The BTW model is solved for the first time. All the critical indices are analytically calculated for the models considered. The influence of the rules features of models on their common properties is analysed on theses ground. The most important for the rules is the aspect whether they are stochastic or deterministic. The former increases the number of avalanche characteristics with different properties, and the later helps large avalanches fit into a finite-size system and results in the system as a whole obtaining dynamic symmetries absent at the level of elements behaviour rules.