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Disordered Systems and Quasi-Invariant Dynamics

无序系统和准不变动力学

基本信息

批准号：
1811093
负责人：
Philippe Sosoe
金额：
$ 16.2万
依托单位：
Cornell University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-06-01 至 2023-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811093&HistoricalAwards=false
关键词：
Disordered Systems Quasi Invariant Dynamics

项目摘要

Statistical physics provides an immensely useful set of tools to analyze natural phenomena involving complex interacting systems whose dynamics is difficult to describe exactly, but whose average behavior is amenable to analysis. It has found applications far beyond the realm of physics. To gain a deeper understanding of the mathematical underpinnings of this theory, it is necessary to study simplified models that nevertheless exhibit rich behavior similar to physical systems. This research project will study one class of such simplified discrete models, percolation models. In a separate topic area, probabilistic tools inspired by statistical physics will be applied to differential equations, to analyze the long-time behavior of their solutions.The project will study the behavior of metrics of physical significance, including the chemical distance and the effective resistance in critical random environments, such as Bernoulli percolation and oriented percolation in low dimensions. These metrics are of central importance for the study of random walks, for instance. Separately, the project will also study the behavior of nonlinear Hamiltonian partial differential and stochastic differential equations with random initial data, especially data chosen from invariant and quasi-invariant measures of Gibbs type. These objects allow for the construction of special solutions with improved long-time behavior compared to the generic case. The principal investigator will participate in the organization of summer schools, as well as the design and teaching of classes on statistical mechanics of both lattice systems like percolation, and continuous dynamical systems like partial differential equations with random data.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

统计物理学提供了一套非常有用的工具来分析涉及复杂相互作用系统的自然现象，这些系统的动力学很难准确描述，但其平均行为是可以分析的。它已经发现了远远超出物理学领域的应用。为了更深入地理解这一理论的数学基础，有必要研究简化的模型，这些模型仍然表现出与物理系统相似的丰富行为。本研究项目将研究一类这样的简化离散模型--渗流模型。在另一个单独的主题区域，受统计物理启发的概率工具将应用于微分方程，以分析其解的长期行为。该项目将研究具有物理意义的指标的行为，包括化学距离和临界随机环境中的有效阻力，如伯努利渗流和低维定向渗流。例如，这些指标对于随机行走的研究至关重要。另外，该项目还将研究具有随机初始数据的非线性哈密顿偏微分方程和随机微分方程的行为，特别是从Gibbs类型的不变和准不变测量中选取的数据。与一般情况相比，这些对象允许构造具有改进的长期行为的特殊解决方案。首席研究员将参与暑期学校的组织以及统计力学课程的设计和教学，这些课程既包括渗流等晶格系统，也包括具有随机数据的偏微分方程等连续动力系统。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。