Self-organizing systems at finite driving rates

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Abstract
We consider finite driving-rate perturbations of models which were previously seen to exhibit self-organized criticality (SOC). These perturbations lead to more realistic models which we expect will have applications to a broader class of systems. At infinitesimal driving rates the separation of time scales between the driving mechanism (addition of grains) and the relaxation mechanism (avalanches) is infinite, while at finite driving rates what were once individual relaxation events may now overlap. For the unperturbed models, the hydrodynamic limits are singular diffusion equations, through which much of the scaling behavior can be explained. For these perturbations we find that the hydrodynamic limits are nonlinear diffusion equations, with diffusion coefficients which converge to singular diffusion coefficients as the driving rate approaches zero. The separation of time scales determines a range of densities, and, therefore, of system sizes over which scaling reminiscent of SOC is observed. At high densities the nature of the nonlinear diffusion coefficient is sensitive to the form of the perturbation, and in a sandpile model it is seen to have novel structure.

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