Local Rigidity and Self-Organized Criticality for Avalanches

The general conditions for a sandpile system to evolve spontaneously into a critical state characterized by a power law distribution of avalanches or bursts are identified as: a ) the existence of a stationary state with a global conservation law; b ) long-range correlations in the continuum limit ( i.e. Laplacian diffusive field); c ) the existence of a local rigidity for the microscopic dynamics. These conditions permit a classification of the models that have been considered up to now and the identification of the local rigidity as a new basic parameter that can lead to various possible scenarios ranging continuously from SOC behaviour to standard diffusion.