Open Problems in the Theory of Mathematical Billiards

数学台球理论中的未决问题

• 批准号: 0457168
• 负责人: Nandor Simanyi
• 金额: --
• 依托单位: University of Alabama at Birmingham
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0457168&HistoricalAwards=false
• 关键词: Open; Problems; Theory; Mathematical; Billiards

This project is devoted to a special class of chaotic dynamical systems, namely hyperbolic (and focusing, or `elliptic') mathematical billiards. In the hyperbolic case they are the prototype examples of non-uniformly hyperbolic dynamical systems with singularities. Such systems have been playing an important role in the rigorous mathematical foundation of statistical physics, so that the study and establishing their strong mixing properties is getting more and more physical relevance. The first part of the proposal focuses on a fundamental conjecture regarding this family of dynamical systems, namely the so called `Boltzmann-Sinai Ergodic Hypothesis', which states that any finite system of (totally elastic) hard spheres moving on a flat torus is fully hyperbolic and ergodic, of course, on the level set of its trivial first integrals. The proof of this conjecture (in its full generality) has been so far notoriously withstanding any attack against it. The first major part of the present proposal directly targets this conjecture. The second part is a blueprint for further research in this direction by generalizing the original Boltzmann-Sinai Hypothesis to cylindric billiards, and billiards in physically more relevant containers, i.e. rectangular boxes, convex domains, etc. The third part addresses a question posed by M. Herman, which asks if the scattering (hyperbolic) effect of the hard ball dynamics eventually prevails over the focusing effect of the convex boundary, if the motion takes place in a compact, convex domain. In the fourth part the fundamental complexity problems are targeted for the $n$-step singularity sets of higher dimensional, strictly dispersive, non-uniformly hyperbolic billiards. The answers to those questions are pivotal in further studies of the fine statistical properties of such systems. Part five poses some basic questions and problems concerning the ergodic properties of high-dimensional billiards, in which the hard core interaction potential is replaced by a smooth, rotational symmetric one. Finally, the closing part aims at some open problems in the theory of planar billiards.The theory of dynamical systems studies the time-evolution of complicated,multi-component systems, like particle systems in statistical physics, reaction kinetics from chemistry, the behavior of the atmosphere (hence the relevance in weather forecasting), population dynamics, developments on the stock market, etc. By nature, this theory is closely related to - and is partly arising from - the theory of differential equations and stochastic processes. An interesting feature of the theory of dynamical systems is that it helps us better understand such crucial phenomena in the time evolution of the above mentioned systems, as the high sensitivity of the solution to the initial conditions, sometimes referred to as chaos, or chaotic behavior. My present proposal targets the investigation and better understanding of a popular and important class of mostly chaotically behaving mathematical models, namely the so called mathematical billiards. They got their name after the fact that they model the physical motion of ball shaped particles interacting with each other via elastic collisions.

这个项目致力于一类特殊的混沌动力系统，即双曲（和聚焦，或“椭圆”）数学台球。在双曲的情况下，它们是具有奇异性的非一致双曲动力系统的原型例子。这类系统在统计物理的严格数学基础中一直扮演着重要的角色，因此研究和建立它们的强混合性质越来越具有物理意义。该提案的第一部分重点关注有关这类动力系统的基本猜想，即所谓的“波尔兹曼-西奈遍历假说”，该假说指出，任何在平坦环面上运动的（全弹性）硬球体的有限系统都是完全双曲的和遍历的，当然，在其平凡的第一积分的水平集上。这个猜想的证明（在其充分的一般性）到目前为止已经臭名昭著地经受住了任何攻击。本建议的第一个主要部分直接针对这个猜想。第二部分是一个蓝图，在这个方向上进一步研究，通过推广原来的玻尔兹曼西奈假说圆柱台球，台球在物理上更相关的容器，即矩形盒，凸域等第三部分解决了M提出的问题。赫尔曼，其中询问如果运动发生在紧凑的凸域中，硬球动力学的散射（双曲线）效应最终是否胜过凸边界的聚焦效应。第四部分主要研究高维、严格色散、非一致双曲台球的n阶奇异集的基本复杂性问题。这些问题的答案是关键，在进一步研究这些系统的精细统计特性。第五部分提出了一些基本的问题和高维台球遍历性的问题，其中硬核相互作用势被替换为一个光滑的，旋转对称的。最后，本文针对平面台球理论中的一些开放性问题，动力系统理论研究复杂的多组分系统的时间演化，如统计物理学中的粒子系统，化学中的反应动力学，大气行为等（因此在天气预报的相关性），人口动态，股票市场的发展，等等。这一理论与微分方程和随机过程理论密切相关，部分源于微分方程和随机过程理论。动力系统理论的一个有趣的特点是，它有助于我们更好地理解上述系统时间演化中的关键现象，如解对初始条件的高灵敏度，有时称为混沌或混沌行为。我目前的建议的目标是调查和更好地了解一个流行的和重要的一类主要是混沌行为的数学模型，即所谓的数学台球。他们因模拟球形粒子通过弹性碰撞相互作用的物理运动而得名。