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

建议的研究分为两类。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，它演化到平稳分布，其特征是被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)。我们以前已经证明，在一定的极限下，许多接近平衡的系统可以用非线性扩散方程来描述，扩散系数在局部密度的临界值处具有奇异性。标度论证表明，这种对开放(非平衡)自组织自动机的流体动力学描述的有效性取决于局部密度涨落的大小，相对于系统在系统大小发散时收敛到临界密度的速率。本研究的目的是严格考察奇异扩散描述对于开放系统的有效性，目标是当波动破坏扩散描述时的一般条件。在过去的几年里，在一些科学界开展了大量关于自组织临界性的活动。最近的活动集中在相对简单的系统上，当在计算机上模拟时，这些系统显示出广泛的分布，这些分布随着系统的大小以一种不平凡的方式变化。这项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.