SELF-ORGANIZING SYSTEMS AT FINITE DRIVING RATES

We consider finite driving-rate perturbations of models which were pre viously seen to exhibit self-organized criticality (SOC). These pertur bations lead to more realistic models which we expect will have applic ations to a broader class of systems. At infinitesimal driving rates t he separation of time scales between the driving mechanism (addition o f grains) and the relaxation mechanism (avalanches) is infinite, while at finite driving rates what were once individual relaxation events m ay now overlap. For the unperturbed models, the hydrodynamic limits ar e singular diffusion equations, through which much of the scaling beha vior can be explained. For these perturbations we find that the hydrod ynamic limits are nonlinear diffusion equations, with diffusion coeffi cients which converge to singular diffusion coefficients as the drivin g rate approaches zero. The separation of time scales determines a ran ge of densities, and, therefore, of system sizes over which scaling re miniscent of SOC is observed. At high densities the nature of the nonl inear diffusion coefficient is sensitive to the form of the perturbati on, and in a sandpile model it is seen to have novel structure.