Hyperbolic Dynamics in Physical Systems and Ergodic Theory

物理系统中的双曲动力学和遍历理论

• 批准号: 2154725
• 负责人: Marian Gidea
• 金额: $ 16.42万
• 依托单位: Yeshiva University
• 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: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154725&HistoricalAwards=false
• 关键词: Hyperbolic; Dynamics; Physical; Systems; Ergodic

This project concerns mathematical research in the fields of dynamical systems and ergodic theory as well as applications to mathematical physics. The research focuses on hyperbolic dynamical systems, where a small perturbation causes exponential uncertainty in both negative and positive time. Such systems appear in many real-world dynamics, for example in interacting particle systems. Much is known for hyperbolic systems of particles with low degrees of freedom and with hard ball interactions by models of mathematical billiards. An important goal of this research is to investigate the case of high degrees of freedom, which is more relevant for physical systems - ergodic theory provides an abstract point of view on dynamical systems by studying time and space averages instead of the local geometry. Another research goal is to extend the understanding of infinite ergodic theory and partial chaos by hyperbolic dynamics examples. Moreover, a summer school will be offered to advanced high school or first year undergraduate students and several research projects will involve both undergraduate and Ph.D. students.The research is comprised of three main parts. The first part concerns the study high dimensional billiards and deterministic walks. Topics that are well understood in two-dimensional billiards (complexity bounds, long flight times) will be investigated in higher dimensions. The second part is devoted to hyperbolic dynamics in physical systems. Based on earlier results by the investigator on smaller building blocks, larger systems of mass and energy transport will be studied. Both these parts make significant advances towards better mathematical models of high dimensional real-life dynamical systems. The third part is to study the role of hyperbolic dynamics in infinite ergodic theory: a field which is more suitable for some large physical systems than traditional finite ergodic theory. New examples of partial chaos will also be constructed.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.

该项目涉及动力学系统和千古理论领域的数学研究以及对数学物理学的应用。该研究的重点是双曲动力学系统，其中少量的扰动会导致负时间和正时间的指数不确定性。这样的系统出现在许多现实世界动力学中，例如在交互的粒子系统中。以低自由度的颗粒的双曲系统以及数学台球模型的硬球相互作用而闻名。这项研究的一个重要目的是研究高度自由度的情况，这与物理系统更相关 - 奇异理论通过研究时间和空间平均而不是局部几何形状来对动态系统提供抽象的观点。另一个研究目的是通过双曲动力学示例扩展对无限的千古理论和部分混乱的理解。此外，将提供一所暑期学校的高级高中或第一年的本科生，几个研究项目将涉及本科生和博士学位。学生。研究由三个主要部分组成。第一部分涉及研究高维台球和确定性步行的研究。将在较高的维度上研究在二维台球（复杂性范围，漫长的飞行时间）中得到充分了解的主题。第二部分致力于物理系统中的双曲动力学。根据研究人员对较小构建块的较早结果，将研究较大的质量和能量运输系统。这两个部分都取得了重大进展，朝着更好的数学模型的高维实际动力学系统方面的进步。第三部分是研究双曲动力学在无限的厄运理论中的作用：比传统的有限麦片理论更适合某些大型物理系统的领域。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和更广泛的影响标准通过评估来支持的，这奖项也将被认为值得支持。