Large Deviations in Large Non-equilibrium Systems

大型非平衡系统中的大偏差

• 批准号: 2153739
• 负责人: Li Cheng Tsai
• 金额: $ 22.5万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-15 至 2022-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2153739&HistoricalAwards=false
• 关键词: Large; Deviations; Non; equilibrium; Systems

Central to research in modern probability theory is the study of how complex random systems behave. Besides being of fundamental interest, studying the behavior of such systems improves understanding of how heat flows through a medium, how tumors grow, how water waves propagate, and how traffic jams form and dissolve. This project focuses on a particular aspect of the behavior known as large deviations, which is about small-probability events occurring in the systems of interest. The theory of large deviations is one of the pillars of probability theory; understanding how it manifests itself in different types of systems is a practical and important task. Further, the study of large deviations has applications to understanding excursions between meta-stable states, analyzing phase transitions, and benchmarking numerical algorithms. The research aims to advance understanding in these and other areas. The project also provides research training opportunities for graduate students.In concrete terms, this project will study the finite-dimensional and sample-path large deviations of asymmetric interacting particle systems and of nonlinear stochastic partial differential equations. They exhibit large deviations in various regimes, including the short-time, intermediately-long-time, and long-time regimes. The main goals of this project are: i) establishing the short-time large deviation principles of various systems; ii) studying how the large deviations in the short-time regime converge to the (mostly conjectural) large deviations in the long-time regime; and iii) investigating the large deviations in the intermediately-long-time regime. These goals involve the mathematical study of various physical phenomena, including the correspondences between soliton waves and shock waves, the spontaneous generation of soliton waves, and dynamical symmetry breaking. The goals of this project will be achieved by a combination of stochastic analysis and integrable structures using the Feynman-Kac formula, the connection to quantum-many-body systems, and the inverse scattering transform of integrable partial differential equations.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.

现代概率论研究的核心是研究复杂随机系统的行为。除了基本的兴趣之外，研究这种系统的行为可以提高对热量如何在介质中流动，肿瘤如何生长，水波如何传播以及交通堵塞如何形成和溶解的理解。这个项目关注的是被称为大偏差的行为的一个特定方面，它是关于在感兴趣的系统中发生的小概率事件。大偏差理论是概率论的支柱之一；了解它如何在不同类型的系统中表现出来是一项实际而重要的任务。此外，对大偏差的研究可以应用于理解亚稳定状态之间的漂移、分析相变和对数值算法进行基准测试。这项研究旨在促进对这些领域和其他领域的理解。该项目还为研究生提供研究培训机会。具体而言，本项目将研究非对称相互作用粒子系统和非线性随机偏微分方程的有限维和样本路径大偏差。它们在不同的制度下表现出很大的偏差，包括短期、中长期和长期制度。本项目的主要目标是：1)建立各系统的短时大偏差原理；Ii)研究短时状态下的大偏差如何收敛于长时间状态下的（主要是推测的）大偏差；iii)研究中长期体制的大偏差。这些目标涉及各种物理现象的数学研究，包括孤子波和激波之间的对应关系，孤子波的自发产生和动态对称性破缺。这个项目的目标将通过使用费曼-卡茨公式的随机分析和可积结构的结合，与量子多体系统的联系，以及可积偏微分方程的逆散射变换来实现。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。