RUI: Statistical Properties of Nonequilibrium and Extended Dynamical Systems

由这笔赠款资助的研究将在很大程度上集中于研究具有碰撞相互作用的粒子系统的数学模型，这是统计力学领域的核心，特别是对我们对混沌动力系统的理解。这样的系统在非平衡动力学的研究中扮演着重要的角色，非平衡动力学对粒子在外力或非弹性碰撞下的运动进行建模。非平衡动力学的其他例子包括允许质量或能量逃逸的系统，以及由较小的相互作用的组件交换质量或能量的大规模系统。这类系统已被用于理论物理和化学工程中，以模拟原子陷阱，探索固体中的热传导机制，并研究分子过程中的亚稳性。在这笔赠款过程中获得的严格的数学结果将有助于解释这些研究，并提出新的研究方向。这笔助学金还支持本科生参与数学研究。利用上述问题的高度直观性和物理动机，作者将在每个由助学金资助的暑假期间招募本科生从事这些主题的研究。将特别重视从研究数学中代表性不足的群体中招收学生。学生将在海报会议上传播他们的研究成果，并酌情通过在本科生或研究期刊上发表文章。通过激发人们对数学研究事业的兴趣，并创建一个支持这种兴趣的同龄人社区，这笔赠款将有助于实现研究和教育相结合的重要目标。动力系统中的许多研究都集中在动态自包含的封闭系统上。然而，在许多建模情况下，这样的全局视图是不可能的，因此有必要研究受其他未知系统影响的局部系统，可能在不同的尺度上。这些考虑促使许多系统在这一资助过程中被研究：其中质量或能量可以通过确定性或随机机制进入或离开的系统。这项资助是围绕三个具体项目组织的：第一个项目研究经典和非平衡粒子系统的统计性质，这是上述统计力学中的一类重要模型；第二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；第三个项目研究动力系统的行为，这些系统由(可能无限多)连接在一起的较小组件组成，允许轨道或能量在它们之间传递。当每次只关注一个组件时，这种系统通过允许进入和退出，以自然的方式概括了对开放系统的讨论。由赠款资助的研究活动的学术价值源于要开发的分析工具的深度以及正在审议的系统的复杂性和多样性。利用他最近关于弥散粒子系统转移算符的谱分解的工作，作者将从统计力学的角度研究平衡和非平衡模型。这种方法有望解决Bowen和Ruelle关于连续时间流的一个长期猜想，并提供对重要物理量的严格分析，如熵产生。作者将使用的第二个工具是构造马尔可夫扩张(有限的和可数的马尔可夫划分的推广)，它不对动力学做出马尔可夫假设，并且广泛适用于非一致双曲系统，包括Hénon映射和各种质点系统。这些工具在非平衡系统、开耦合映象格子和扩展系统中的应用将代表着这类系统研究的重大进展。在一个背景下理解这些工具的努力加强了它们，并有助于它们应用于数学的其他领域。通过应用这些想法来解决物理文献中正式提出和接近的问题，他们的智力兴趣得到了提高。

The research funded by this grant will focus in large part on the study of mathematical models of particle systems with collision interactions that are central to the field of statistical mechanics, and in particular to our understanding of chaotic dynamical systems. Such systems play an important role in the study of non-equilibrium dynamics, which model, for example, the motion of particles subjected to external forces or nonelastic collisions. Other examples of non-equilbrium dynamics include systems in which mass or energy is allowed to escape, and large-scale systems of smaller interacting components which exchange mass or energy. Such systems have been used in theoretical physics and chemical engineering to model atom traps, explore mechanisms for heat conduction in solids and investigate metastability in molecular processes. Rigorous mathematical results obtained during the course of this grant will aid in the interpretation of these studies as well as suggest new directions of inquiry. This grant also supports the involvement of undergraduates in mathematics research. Using the highly visual nature and physical motivation of the problems outlined above, the author will recruit undergraduate students to work on these topics during each summer funded by the grant. Special emphasis will be given to recruiting students from underrepresented groups in research mathematics. Students will disseminate 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 grant will contribute to the important goal of integrating research and education.Much research in dynamical systems focuses on closed systems in which the dynamics are self-contained. In many modeling situations, however, such a global view is not possible and it becomes necessary to study local systems that are influenced by other unknown systems, possibly on different scales. Such considerations motivate many of the systems to be studied during the course of this grant: systems in which mass or energy may enter or exit through deterministic or random mechanisms. The grant is organized around three specific projects: The first project investigates the statistical properties of both classical and non-equilibrium particle systems, which constitute an important class of models from statistical mechanics as described above; the second concerns open systems, which are inspired by physical models in which mass or energy is allowed to escape; 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 them. 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. The intellectual merit of the research activity funded by the grant stems from the depth of the analytical tools to be developed as well as the complexity and variety of the systems under consideration. Using his recent work concerning the spectral decomposition of the transfer operator for dispersing particle systems, the author will investigate both equilibrium and non-equilibrium models from statistical mechanics. This approach is expected to resolve a longstanding conjecture by Bowen and Ruelle regarding the continuous time flow and to provide rigorous analysis of physically important quantities such as entropy production. A second tool the author will use is the construction of Markov extensions (a generalization of finite and countable Markov partitions), which make no Markovian assumptions on the dynamics and are widely applicable to nonuniformly hyperbolic systems, including Hénon maps and a wide variety of particle systems. The application of such tools to systems out of equilibrium, open coupled map lattices and extended systems will represent significant advances in the study of such systems. Efforts to understand these tools in one context strengthens them and aids in their application to other areas of mathematics. Their intellectual interest is enhanced by the application of these ideas to resolve problems posed and approached formally in the physics literature.