Quantitative Methods for Modeling Properties of Random Media

随机介质属性建模的定量方法

基本信息

批准号：
1700329
负责人：
Scott Armstrong
金额：
$ 18万
依托单位：
New York University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2020-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700329&HistoricalAwards=false
关键词：
Quantitative Methods Modeling Properties Random

项目摘要

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.

最好将许多物理系统理解为许多较小的违规行为的集体行为。一个例子是一团气体，由大量的单个粒子相互碰撞。在较大（宏观的）尺度上，由于较小的较小相互作用的影响不可估量的影响，这种看似复杂的系统可能具有更简单但非常精确的描述（例如，理想的气体定律）。 “其他示例包括通过多孔岩石移动的水或复合材料的物理特性进行建模。复杂的小规模不规则和相互作用的大规模“平均”定律的推导在于统计物理学领域。该项目的目的是开发数学工具，以严格理解此类问题，假设小规模不规则发生了。我们想回答以下问题：如何通过小规模互动的行为准确预测大规模的物理定律，包括比例性常数？大规模定律的错误是什么，换句话说，什么长度尺度足够大，以至于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，而且还具有更广泛的应用，例如设计最佳复合材料或地热发电厂的建设。部分微分方程用于对许多重要的物理系统进行建模，该项目的重点是在该系数表现出小规模的随机振荡的假设下，了解解决方案的大规模行为。在数学文献中，获得描述原始方程的大规模行为的“平均”部分微分方程称为“均质化”。如果在模型中纳入了随机性，则通常称为“随机同质化”。近年来，研究人员为最简单的情况开发了一个相当完整的随机均质化理论：均匀的椭圆方程。这些方程式模型复合材料的静电性能。本项目的目的是通过获得“退化”方程式的均质化定量理论以及多孔介质中的方程。特定的目标包括严格地证明在具有定量误差范围的多孔岩石中的Darcy-Brinkman定律；得出液体中颗粒稀释液的有效粘度；获得在渗透群集上随机移动的粒子扩散性的边界，以及在随机培养基中扩散的长期行为（中间渐近性）的更精确估计；周期性和随机均质化中边界层的分析；并获得形状优化问题的均质化结果。对于这些问题，目标是开发一种定量数学理论，该理论提供明确的误差界限和鲁棒的分析技术。