Quantitative Methods for Modeling Properties of Random Media

许多物理系统最好理解为许多较小的不规则性的集体行为。一个例子是由大量相互碰撞的单个粒子组成的气体云。在更大的(宏观)尺度上，这个看似非常复杂的系统可能有一个简单得多，但仍然非常精确的描述(例如，理想气体定律)，这是由于无数小尺度相互作用的影响。其他例子包括水在多孔岩石中流动的模拟，或复合材料的物理性质。从复杂的小尺度不规则性和相互作用中推导出大尺度的“平均”定律属于统计物理领域。这个项目的目标是开发数学工具，在假设小规模的违规行为是随机发生的情况下，严格理解这类问题。我们想回答这样的问题：如何从小尺度相互作用的行为中准确地预测包括比例常数在内的大尺度物理定律？大尺度定律的误差是什么，换句话说，多大的长度尺度才足以让我们观察到平均行为，而不是看到复杂的随机波动？这些问题不仅对统计物理和概率很重要，而且也有更广泛的应用，例如设计最优复合材料或建造地热发电厂。偏微分方程被用来模拟许多重要的物理系统，本项目的重点是在假设系数呈现小规模随机振荡的情况下了解此类方程的解的大尺度行为。在数学文献中，得到一个描述原始的、更复杂的方程的大规模行为的“平均”偏微分方程式被称为“齐次化”。如果模型中包含了随机性，则通常将其称为“随机同质化”。近年来，对于最简单的情形：一致椭圆型方程，研究人员发展了一种比较完整的随机齐次化的定量理论。例如，这些方程对复合材料的静电特性进行了建模。本项目的目标是通过获得“退化”方程以及多孔介质中的方程的齐次化的量化理论来更进一步。具体目标包括用定量误差界严格证明流体在多孔岩石中流动的达西-布林克曼定律；推导流体中颗粒稀释悬浮液的有效粘度；获得在渗流簇上随机运动的颗粒的扩散系数的界限以及对随机介质中扩散的长期行为(中间渐近性)的更精确估计；周期性和随机均质中的边界层分析；以及获得形状优化问题的均化结果。对于这些问题中的每一个，目标都是开发一种量化的数学理论，提供明确的误差界和稳健的分析技术。

Many physical systems are best understood as the collective behavior of many smaller irregularities. An example is a cloud of gas comprised of a huge number of individual particles colliding with each other. On a larger (macroscopic) scale, this very seemingly complicated system may have a much simpler, but still very precise, description (the ideal gas law, for instance) due to the effects of the incalculable number of smaller scale interactions "averaging out." Other examples include the modeling of water moving through a porous rock, or the physical properties of composite materials. The derivation of large-scale, "averaged" laws from complicated small-scale irregularities and interactions lies in the realm of statistical physics. The goal of this project is to develop mathematical tools for a rigorous understanding of such problems under the assumption that the small-scale irregularities are occurring randomly. We would like to answer such questions as: How can the large-scale physical law, including the proportionality constants, be accurately predicted from the behavior of the small-scale interactions? What is the error in the large-scale law, in other words, what length scale is large enough that we observe the averaged behavior rather than seeing the complicated random fluctuations? Such questions are not just important to statistical physics and probability, but they also have wider applications, such as to the design of optimal composite materials or the construction of geothermal power plants. Partial differential equations are used to model many important physical systems, and the focus of the project is to understand the large-scale behavior of solutions to such equations under the assumption that the coefficients exhibit small-scale random oscillations. Obtaining an "averaged" partial differential equation that describes the large-scale behavior of the original, more complicated equation is called "homogenization" in the mathematical literature. If randomness is incorporated in the model, it is usually referred to as "stochastic homogenization." In recent years, researchers have developed a rather complete quantitative theory of stochastic homogenization for the simplest situation: uniformly elliptic equations. These equations model electrostatic properties of composite materials, for instance. The present project aims to go further by obtaining a quantitative theory of homogenization for "degenerate" equations as well as equations in porous media. Particular goals include rigorously justifying the Darcy-Brinkman law for fluid flow in a porous rock with quantitative error bounds; deriving the effective viscosity of a dilute suspension of particles in a fluid; obtaining bounds for the diffusivity of a particle moving randomly on a percolation cluster as well as more precise estimates for the long-time behavior (intermediate asymptotics) of a diffusion in a random medium; the analysis of boundary layers in periodic and stochastic homogenization; and obtaining homogenization results for shape optimization problems. For each of these problems, the goal is to develop a quantitative mathematical theory that provides explicit error bounds and robust analytic techniques.