The proposed research is divided into two categories. The first pertains to the hydrodynamic description of self-organizing systems. Self-organizing automata or particle systems are open driven stochastic systems which evolve to a stationary distribution characterized by the nontrivial scaling behavior (as a function of system size) known as self-organized criticality. We have previously established that many of these systems near equilibrium can be described in certain limits by nonlinear diffusion equations, with a diffusion coefficient which has a singularity at a critical value of the local density. Scaling arguments suggest that the validity of this hydrodynamic description for the open (nonequilibrium) self-organizing automata depends on the size of the fluctuations in the local density in comparison to the rate at which the system converges to the critical density as the system size diverges. The purpose of this research is to rigorously investigate the validity of the singular diffusion description for the open systems, with the goal being general conditions for when fluctuations destroy the diffusion description. In the past several years there has been a considerable amount of activity in a number of scientific communities regarding self-organized criticality. Recent activity has focused on relatively simple systems, which, when simulated on the computer show broad distributions which change in a nontrivial way with the size of the system. The purpose of this research is to look at mathematical models which describe these systems.
