RUI: Open, coupled and extended dynamical systems with nonuniform hyperbolicity

RUI：具有非均匀双曲性的开放、耦合和扩展动力系统

• 批准号: 1101572
• 负责人: Mark Demers
• 金额: $ 13万
• 依托单位: Fairfield University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101572&HistoricalAwards=false
• 关键词: RUI; Open; coupled; extended; dynamical

The author will undertake several projects in the area of dynamical systems and ergodic theory. The first project concerns open systems, which are inspired by physical models in which mass or energy is allowed to escape. The project will study various mechanisms that facilitate or hinder escape in nonuniformly hyperbolic systems and use the relation between entropy and positive Lyapunov exponents to quantify the escape rate. The second project develops a powerful approach, the spectral decomposition of the transfer operator, to study the statistical properties of particle systems, which are an important class of models from mathematical physics. The third project investigates the behavior of dynamical systems which are comprised of (possibly infinitely many) smaller components linked together, with orbits or energy allowed to pass between components. When focused on one component at a time, such systems generalize the discussion of open systems in a natural way by allowing both entry and escape. All three projects involve a detailed analysis of the spectral properties of the transfer operator associated with the corresponding closed system without relying on restrictive Markovian assumptions on the dynamics.Much research in dynamical systems has focused on closed systems in which the dynamics are self-contained. In many modeling situations, however, it is not possible to obtain such a global view so that it becomes necessary to study local systems that are influenced by other unknown systems, possibly on different scales. Such considerations motivate the study of the types of systems considered in this project: systems in which mass or energy may enter or exit through deterministic or random mechanisms. Many of these problems are motivated by models from mathematical physics. For example, open particle systems are used to model atom traps; extended and linked particle systems are used to create mechanical models of heat conduction in solids and to investigate metastability in molecular processes. The research will provide analytical tools to solve problems posed and approached formally in the physics literature. The project will both promote and be informed by this interdisciplinary dialogue. In addition, the project will support undergraduate research in mathematics. Using the highly visual nature and physical motivation of the problems described, the PI will recruit undergraduate students to work on projects related to these topics during the summers. Students will disseminate the results of their research at poster sessions and through publication in undergraduate or research journals, as appropriate. By stimulating interest in research careers in mathematics and creating a peer community supportive of that interest, the project will contribute to the important goal of integrating research and education.

作者将在动力系统和遍历理论领域进行几个项目。 第一个项目涉及开放系统，它受到允许质量或能量逃逸的物理模型的启发。 该项目将研究促进或阻碍非均匀双曲系统逃逸的各种机制，并利用熵和正李雅普诺夫指数之间的关系来量化逃逸率。 第二个项目开发了一种强大的方法，转移算子的谱分解，以研究粒子系统的统计特性，这是数学物理中的一类重要模型。第三个项目研究动力系统的行为，这些系统由（可能是无限多个）连接在一起的较小组件组成，允许组件之间传递轨道或能量。 当一次只关注一个组件时，这样的系统通过允许进入和退出，以一种自然的方式概括了开放系统的讨论。所有这三个项目都涉及到与相应的封闭系统相关的转移算子的谱特性的详细分析，而不依赖于限制性的马尔可夫假设的dynamics. Most动力系统的研究集中在封闭系统中的动态是自包含的。 然而，在许多建模情况下，不可能获得这样的全局视图，因此有必要研究可能在不同尺度上受其他未知系统影响的局部系统。 这种考虑激发了本项目中所考虑的系统类型的研究：质量或能量可以通过确定性或随机机制进入或退出的系统。 这些问题中的许多问题都是由数学物理模型引起的。 例如，开放粒子系统用于模拟原子陷阱;扩展和链接粒子系统用于创建固体热传导的力学模型，并研究分子过程中的亚稳态。 这项研究将提供分析工具，以解决物理文献中提出的问题和正式处理。该项目将促进这一跨学科对话，并从中了解情况。 此外，该项目将支持数学本科生的研究。 利用所描述的问题的高度视觉性和物理动机，PI将招募本科生在夏季从事与这些主题相关的项目。 学生将在海报会议上传播他们的研究成果，并通过在本科或研究期刊上发表，视情况而定。 通过激发对数学研究职业的兴趣，并建立一个支持这种兴趣的同行社区，该项目将有助于实现研究与教育相结合的重要目标。