Dynamics and Kinetics

提出的研究解决了动力系统理论和统计力学中一些长期存在的问题，以及一些从一般观点和应用角度来看很重要的新的自然问题。该项目开发了一种设计双曲台球的新方法，这将允许人们证明更多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种新的一般特征，绝对聚焦曲线是双曲台球在连分式中唯一允许的聚焦分量。一个长期存在的问题，是否可以平滑一个体育场台球的边界将得到解决。该项目还将揭示凸域中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在哪里的问题。由首席研究员提出的关于开放系统动力学的两个自然问题将被解决。第一个问题是通过洞的逃逸如何取决于洞在相空间中的位置。第二个问题是关于单孔逃逸和多孔逃逸的关系。这些问题揭示了组合学和数论之间的微妙联系。在非凸多边形中，有限大小的弹子粒子的动力学表现为双曲线。这将应用于统计力学中的经典Ehrenfest周期风树模型，并从自然物理的角度证明，该模型在动力学的丰富性方面超过了周期洛伦兹模型。该项目将提供新的视觉和相对简单的台球动力系统模型，包括混沌和混合（与混沌和规则动力学共存的区域）行为。(注意:“台球”是一个技术性的数学概念，与台球这个名字的室内游戏无关。)此外，首席研究员介绍的一些模型将（有些已经）被理论家和实验家用于物理学，他们实际上已经建造了这样的设备，因此将促进跨学科的合作。寻找最优（以确保最快/最慢的逃逸）孔位置的问题将在开放系统中具有潜在的各种应用。这个问题，以及通过多个洞逃逸的问题，是受到原子台球实验的启发。此外，这种方法开启了对动力学进行有限时间（而不是时间渐近）预测的可能性（例如，预测一个时刻之后，通过特定洞的逃逸比通过任何其他相同大小的洞逃逸的可能性更大）。对有限粒径粒子的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，预计本科生也会参与。该项目的成果将广泛传播，通过首席研究员（通常是全体会议）、他的合作者和他的学生在物理学家、生物学家和工程师广泛参与的跨学科会议上的参与，加强科学和技术的理解。

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.