Mathematical Sciences: Rigorous Results on Statistical Mechanics of Interacting Particle Systems

拟议的研究分为两类。第一个是关于自组织系统的流体动力学描述。自组织自动机或粒子系统是开放驱动的随机系统，其演化为以称为自组织临界性的非平凡标度行为（作为系统大小的函数）为特征的平稳分布。我们以前已经建立了许多这些系统接近平衡，可以描述在一定的限制下，由非线性扩散方程，与扩散系数具有奇异性的临界值的局部密度。标度参数表明，这种流体动力学描述的有效性的开放（非平衡）自组织自动机依赖于局部密度的波动的大小相比，系统收敛到临界密度的系统大小发散的速率。本研究的目的是严格调查的有效性的奇异扩散描述的开放系统，目标是一般条件时，波动破坏的扩散描述。在过去几年中，许多科学界就自组织临界性开展了大量的活动。最近的活动集中在相对简单的系统，其中，当在计算机上模拟时，显示出广泛的分布，这些分布随着系统的大小以非平凡的方式变化。本研究的目的是研究描述这些系统的数学模型。

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.