Non-Uniformly Hyperbolic Dynamical Systems with Singularities

具有奇点的非均匀双曲动力系统

• 批准号: 0098773
• 负责人: Nandor Simanyi
• 金额: $ 9.13万
• 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: 2001
• 资助国家: 美国
• 起止时间: 2001-06-01 至 2004-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0098773&HistoricalAwards=false
• 关键词: Non; Uniformly; Hyperbolic; Dynamical; Systems

This project is devoted to a special class of chaotic dynamical systems,namely hyperbolic mathematical billiards. They serve as the prototype examplesof non-uniformly hyperbolic dynamical systems with singularities. Such systemsplay an increasingly important role in the rigorous mathematical foundation ofstatistical physics, so that the study of their chaotic (i. e. mixing)properties is getting more and more physical relevance. The project mainlyfocuses on a fundamental conjecture regarding this family of dynamicalsystems, namely the celebrated "Boltzmann-Sinai Ergodic Conjecture", whichstates that any finite system of (totally elastic) hard spheres moving on aflat torus is fully hyperbolic and ergodic, of course, on the level set of itstrivial first integrals. The proof of this conjecture (in its full generality)has been so far notoriously withstanding any attack against it. The firstmajor part of the present proposal directly targets this conjecture. Thesecond and fourth parts are blueprints for further research in this directionby generalizing the original Boltzmann-Sinai Conjecture to cylindric billiards(mathematical billiards with cylindric scatterers) and billiards in physicallymore relevant containers, like rectangular boxes. The third part of the projectaims at the biggest open question in the topic of Wojtkowski's one-dimensionalfalling balls: Wojtkowski's still unsolved conjecture on the full hyperbolicityof the falling ball system with nonincreasing masses. (And such that not allmasses are the same, of course.) Beside these, the question of ergodicity(possibly, under the condition that a strictly concave potential acts) is alsoposed and targeted. The foundation of statistical physics (like heat theory, dynamical theoryof fluids and gases) took place in the last third of the 18th century, mainlyby the groundbreaking works of Boltzmann and Helmholz. That foundation was,however, based upon a strong hypothesis made by Boltzmann himself. Thathypothesis claims that any physical system with a huge number of interactingparticles (like molecules) has the property that for any fixed total energyand initial state, the system will evolve to any other state with the sameenergy. Although this conjecture, if taken literally, mathematically cannothappen, yet the precise mathematical formalism and its rigorous verificationfor different models of statistical physics bears a particular importance tothe understanding the physics of the surrounding world.

这个项目致力于一类特殊的混沌动力系统，即双曲数学台球。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严格的数学基础中扮演着越来越重要的角色，因此，对其混沌（即混沌）的研究也就越来越受到重视。e.混合）性质变得越来越物理相关。该项目主要集中在一个基本猜想关于这个家庭的dynamicalsystems，即著名的“玻尔兹曼-西奈遍历猜想”，其中指出，任何有限系统的（全弹性）硬球运动的flat环面是完全双曲和遍历，当然，在水平集上的strivial第一积分。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。第二部分和第四部分是在这个方向上进一步研究的蓝图，通过将原始的玻尔兹曼-西奈猜想推广到圆柱形台球（数学台球与圆柱形散射体）和物理上更相关的容器中的台球，如矩形盒子。第三部分是针对Wojtkowski一维落球问题中最大的一个悬而未决的问题：Wojtkowski关于质量不增的落球系统的全双曲性的猜想。(And当然，并不是所有的质量都是一样的。）除此之外，遍历性问题（可能，在严格凹势作用的条件下）也被提出和针对。统计物理学的基础（如热理论，流体和气体的动力学理论）发生在世纪的最后三分之一，主要是由玻尔兹曼和亥姆霍兹的开创性工作。然而，这个基础是基于玻尔兹曼自己提出的一个强有力的假设。该假说声称，任何具有大量相互作用粒子（如分子）的物理系统都具有这样的性质，即对于任何固定的总能量和初始状态，系统将演化到具有相同能量的任何其他状态。虽然这个猜想，如果从字面上理解，在数学上是不成立的，但是精确的数学形式及其对不同统计物理模型的严格验证，对理解周围世界的物理学具有特别重要的意义。