Scaling Limits via Stochastic Homogenization

该研究项目涉及被称为粒子系统的数学模型的统计特性，这些模型描述了重要物理系统的流体动力学和材料科学方面。更具体地说，该项目研究（1）团簇生长，（2）随机复合材料和（3）扩散气体的模型。 关于这些模型的许多基本和重要的问题仍然没有解决。 一个更突出的例子是，我们仍然缺乏一个完全严格的理解，流体方程是如何从分子相互作用中产生的。 研究人员将试图在这个问题上取得进展和其他应用最近发展的技术在随机均匀化理论。 这项工作将具有跨学科的特点，混合了概率和分析的结果。 该项目将利用计算机模拟，以经验检验的想法在理论的发展。技术在定量随机均匀化，特别是那些用来获得最佳速率的椭圆设置，现在已经足够成熟，攻击的问题，以前似乎不服从这种方法。 基本的均匀化思想保持不变，即把微观模型看作是它们光滑的宏观极限的扰动。 关键的新见解是，宏观极限的正则性理论通常有一个微观的类似物。 可能的应用的例子包括朗道方程的推导作为相互作用粒子的标度极限和格点上伯努利势的安德森局部化。 这一方法已被证明是成功的，研究人员预计它将继续产生新的结果。

This research project concerns the statistical properties of mathematical models known as particle systems that describe fluid-dynamical and materials-science aspects of important physical systems. More specifically, the project studies models of (1) cluster growth, (2) randomly composited materials, and (3) diffuse gases. Many basic and important questions about these models remain open. One of the more striking examples is that we still lack a fully rigorous understanding how fluid equations arise from molecular interactions. The investigator will attempt to make progress on this problem and others by applying recently-developed techniques in the theory of stochastic homogenization. The work will have an interdisciplinary character, mixing results from probability and analysis. The project will make use of computer simulation to empirically test ideas during development of the theory.Techniques in quantitative stochastic homogenization, particularly those used to obtain optimal rates in the elliptic setting, are now mature enough to attack problems that previously did not seem to be amenable to this approach. The basic homogenization idea remains the same, namely, to view the microscopic models as perturbations of their smooth macroscopic limits. The key new insight is that the regularity theory of the macroscopic limit often has a microscopic analogue. Examples of possible applications include the derivation of the Landau equation as a scaling limit of interacting particles and Anderson localization for a Bernoulli potential on the lattice. This approach has already proved successful, and the investigator expects it to continue to yield new results.