Dynamical and statistical aspects of Self-organized criticality

In 1987 physicists P. Bak, C. Tang and K. Wiesenfeld (BTW) discovered that there exists a broad class of non-linear dynamical systems, with many degrees of freedom, that exhibit self-affine statistical characteristics similar to what is observed in eq uilibrium systems at critical points. Contrary to equilibrium systems, these reach their critical states with out fine tuning of external parameters, and hence they can be used to explain generic statistical features of complex behavior in nature. Thi s class of dynamical behavior was named Self- Organized Criticality (SOC), and in the last twenty years SOC has been suggested as the mechanism behind the observed complexity in many natural systems. Examples include earthquakes, forest fires, stock ma rket fluctuations, biological evolution and much more. Parallel to the search of SOC in nature, several mathematical models have been constructed to demonstrate SOC-dynamics. A majority of these models, for instance the Zhang model, are generalizations of the original BTW sand pile model. This means that their dynamical rules are inspired by the toppling of sand in a pile at a critical steepness. Despite extensive numerical investigation of these models, sufficient understanding of the relation betwee n the dynamical rules and the large-scale behavior is still far from achieved, and hence there does not yet exist precise formulation of SOC. As a consequence, a natural system with some scaling properties is often erroneously considered to exhibit SO C if one can draw some superficial analogues between its dynamical behavior and the avalanching one observes in sand pile models. To resolve this unfortunate situation we will combine recently developed mathematical results with recent advances on stoc hastic description of SOC to give a consistent and sufficient characterization of SOC. This will enable us to exploit the full depth of the concept and its relation to other classes of complex dynamics.