Mathematical Sciences: Large Scale Properties of Interacting Systems

9504791夸斯特摘要研究人员正在研究相互作用粒子系统的大尺度行为。在这些模型中，大的粒子系统根据一个简单的局域规则进行统计演化，对于这个局域规则，存在一定的守恒量。其目的是证明在宏观尺度上，守恒量是按照确定性守恒定律演化的，而确定性守恒定律通常采用非线性偏微分方程组的形式。特别令人感兴趣的是输运系数以及它们如何从微观动力学中产生，特别是在存在涨落净效应的非梯度模型中。我们将研究几种类型的非梯度模型：无序系统模型、表面和界面模型以及不可压缩的Navier-Stokes方程的格子气体模型。一个基本但在很大程度上仍然悬而未决的问题是，理解微观粒子的大系统是如何自我组织起来，产生我们在周围世界中看到的集体宏观行为的。研究人员在微观动力学具有一定随机性的模型上追求这一目标。这些研究包括凝聚态物理中的无序介质输运模型、表面和界面模型以及不可压缩流体流动的格子气体模型。除了为宏观守恒定律提供严格的理由外，这些研究的其他重要结果是输运系数的内在公式，以及对这些微观系统偏离其宏观极限的精确估计。

9504791 Quastel Abstract The investigator is studying the large scale behavior of interacting particle systems. In these models large systems of particles evolve statistically according to a simple local rule for which there are certain conserved quantities. The goal is to show that on a macroscopic scale the conserved quantities evolve according to deterministic conservation laws which usually take the form of nonlinear partial differential equations. Of particular interest are the transport coefficients and how they arise from the microscopic dynamics, in particular in the nongradient models where there is a net effect of fluctuations. Several types of nongradient models will be studied: Models of disordered systems, models of surfaces and interfaces, and lattice gas models for the incompressible Navier-Stokes equation. A fundamental, and still largely open, problem is to understand how large systems of microscopic particles organize themselves to produce the collective macroscopic behavior we see in the world around us. The investigator is pursuing this goal on models in which the microscopic dynamics has some degree of randomness. The research includes models of transport in disordered media which come from condensed matter physics, models of surfaces and interfaces, and lattice gas models for incompressible fluid flow. Besides providing a rigorous justification for the macroscopic conservation laws, other important consequences of these investigations are intrinsic formulas for transport coeffiaients, and precise estimates of the deviations of these microscopic systems from their macroscopic limits.