Dynamics and Kinetics

动力学和动力学

• 批准号: 0900945
• 负责人: Leonid Bunimovich
• 金额: $ 22.5万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-15 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0900945&HistoricalAwards=false
• 关键词: Dynamics; Kinetics

The proposed research addresses some long-standing problems in the theory of dynamical systems and statistical mechanics, as well as some new natural questions that are important from a general point of view and for applications. The project develops a new approach to the design of hyperbolic billiards that will allow one to prove hyperbolicity for more a general class of billiards. This approach is based on a new general characterization of absolutely focusing curves, which are the only admissible focusing components of hyperbolic billiards, in terms of continued fractions. A long-standing problem on whether one can smooth the boundary of a stadium billiard will be resolved. The project will also shed new light on the question of where the border lies between completely chaotic billiards in convex domains and billiards with divided phase spaces into chaotic and regular components. Two natural questions raised by the principal investigator on the dynamics of open systems will be addressed. The first one asks how the escape through a hole depends on the position of a hole in phase space. The second question is about the relationship between escape through one hole and escape through multiple holes. These questions reveal some subtle connections between combinatorics and number theory. The dynamics of a finite-size billiard particle in nonconvex polygon will be shown to be hyperbolic. This will be applied to the classical Ehrenfest periodic wind-tree model in statistical mechanics and demonstrate that, from a natural physical point of view, this model surpasses the periodic Lorentz model in the richness of its dynamics.The project will provide new visual and relatively simple models of billiard dynamical systems with chaotic as well as with mixed (coexisting regions with chaotic and with regular dynamics) behavior. (N.B."Billiards" is a technical mathematical concept that does not refer to the parlor game of that name.) Moreover, some of the models introduced by the principal investigator will be (and some already have been) used in physics by both theoreticians and experimentalists, who have actually built such devices, and therefore will foster interdisciplinary collaborations. A problem of finding an optimal (to ensure the fastest/slowest escape) placement of a hole will have a potentially large variety of applications for open systems. This question, as well as the one on escape through multiple holes, was inspired by experiments on atomic billiards. Moreover, this approach opens up the possibility of making finite-time (rather than asymptotic in time) predictions of dynamics (e.g., predicting a moment after which escape through a specific hole is more likely than escape through any other hole of the same size). The analysis of the wind-tree model with a finite-size particle will have applications in statistical mechanics. The project will enhance the infrastructure for research and education through collaborations with researchers in the US, Mexico, Canada, and Europe. Graduate students are already involved in this research, and the involvement of undergraduates is anticipated. The results of the project will be broadly disseminated to enhance scientific and technological understanding via participation of the principal investigator (often with plenary talks), his collaborators, and his students in interdisciplinary conferences with a broad participation of physicists, biologists, and engineers.

这项研究解决了动力系统理论和统计力学中的一些长期存在的问题，以及一些新的自然问题，这些问题从总体上看是重要的，对于应用来说也是重要的。该项目开发了一种设计双曲线台球的新方法，这将使人们能够证明更一般类型的台球的双曲性。这种方法是基于绝对聚焦曲线的一个新的一般刻画，绝对聚焦曲线是双曲台球中唯一允许的连续分数聚焦分量。一个长期存在的问题--能否使体育场台球的边界变得光滑--将得到解决。该项目还将为凸域中的完全混沌台球和将相空间划分为混沌和规则分量的台球之间的边界问题提供新的线索。将讨论首席研究员提出的关于开放系统动力学的两个自然问题。第一个问题是，通过孔的逃逸如何依赖于相空间中孔的位置。第二个问题是关于从一个洞逃生和通过多个洞逃生的关系。这些问题揭示了组合学和数论之间的一些微妙联系。有限大小的台球粒子在非凸多边形中的动力学将表现为双曲线。这将应用于统计力学中经典的Ehrenfest周期风树模型，并证明从自然物理的角度来看，该模型在动力学方面的丰富性超过了周期Lorentz模型，该项目将为具有混沌和混合(混沌和规则动态共存区域)行为的台球动力系统提供新的直观和相对简单的模型。(注：“台球”是一个技术性的数学概念，并不是指同名的室内游戏。)此外，首席研究员介绍的一些模型将被理论家和实验者用于物理学，他们实际上已经建造了这样的设备，因此将促进跨学科合作。对于开放系统来说，寻找一个洞的最佳位置(以确保最快/最慢的逃逸)的问题将具有潜在的各种应用。这个问题，以及关于通过多个洞逃生的问题，都是受到原子台球实验的启发。此外，这种方法提供了对动力学进行有限时间(而不是时间上的渐近)预测的可能性(例如，预测某个时刻之后，从特定的洞中逃脱的可能性比从任何其他相同大小的洞中逃脱的可能性更大)。有限大小粒子风树模型的分析将在统计力学中得到应用。该项目将通过与美国、墨西哥、加拿大和欧洲的研究人员合作，加强研究和教育的基础设施。研究生已经参与了这项研究，本科生也有望参与。该项目的成果将通过主要研究人员(通常是全体会议)、合作者和学生参加有物理学家、生物学家和工程师广泛参与的跨学科会议来广泛传播，以增进对科学和技术的了解。