Mathematical Sciences: Large Scale Properties of Interacting Systems

9504791 Quastel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，粒子的大系统根据一个简单的局部规则进行统计演化，其中存在某些守恒量。 我们的目标是表明，在宏观尺度上的保守量的发展，根据确定性的守恒定律，通常采取非线性偏微分方程的形式。 特别感兴趣的是输运系数，以及它们是如何从微观动力学中产生的，特别是在非梯度模型中，其中存在波动的净效应。 几种类型的非梯度模型将被研究：无序系统的模型，表面和界面的模型，以及不可压缩的Navier-Stokes方程的格子气模型。 一个基本的，而且仍然很大程度上是开放的问题是理解微观粒子的大系统如何组织自己，以产生我们在周围世界中看到的集体宏观行为。 研究人员正在追求这一目标的模型，其中微观动力学具有一定程度的随机性。 研究内容包括来自凝聚态物理学的无序介质中的输运模型、表面和界面模型以及不可压缩流体流动的格子气模型。 除了为宏观守恒定律提供严格的证明外，这些研究的其他重要结果是输运系数的内在公式，以及这些微观系统与其宏观极限的偏差的精确估计。

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.