Fluctuations in self-organizing systems

Self-organized criticality (SOC) in a wide variety of systems is seen to arise as a consequence of a singularity in the diffusion coefficient of the hydrodynamic limit. We demonstrate that this description is valid for several models on a closed system and observe that it can break down if the driving is sufficiently strong on the open systems where SOC is observed. In this case fluctuations play an important role, and if fluctuations are large enough then pure power laws in event-size distributions are observed. In contrast, when diffusion holds on SOC systems the characteristic event size diverges sublinearly in the system size. We derive an exponent inequality which provides a necessary condition for the singular-diffusion description to hold on the open driven system. The inequality involves the order of the diffusion singularity, the driving rate, and standard critical exponents