Non-Uniformly Hyperbolic Dynamical Systems with Singularities

本课题致力于研究一类特殊的混沌动力系统，即双曲数学台球。它们是具有奇点的非均匀双曲动力系统的典型例子。这类系统在统计物理严密的数学基础中发挥着越来越重要的作用，因此对其混沌（即混合）性质的研究越来越具有物理意义。该项目主要关注关于这类动力系统的一个基本猜想，即著名的“玻尔兹曼-西奈遍历猜想”，该猜想指出，在平面环面上运动的任何（完全弹性）硬球的有限系统都是完全双曲的和遍历的，当然，在平凡第一积分的水平集中。到目前为止，这一猜想的证明（在其全部普遍性方面）一直经受住了任何针对它的攻击。本提案的第一个主要部分直接针对这一猜想。第二和第四部分是该方向进一步研究的蓝图，将原始玻尔兹曼-西奈猜想推广到圆柱形台球（具有圆柱形散射体的数学台球）和物理上更相关的容器（如矩形盒子）中的台球。项目的第三部分是在Wojtkowski的一维落球主题中最大的开放问题：Wojtkowski关于非增加质量的落球系统的完全双曲性的尚未解决的猜想。（当然，并不是所有的质量都是一样的。）除此之外，遍历性的问题（可能，在严格凹的潜在行为的条件下）也被提出并有针对性。统计物理学（如热论、流体和气体的动力学理论）的基础是在18世纪的最后30年，主要是由玻尔兹曼和亥姆霍兹的开创性工作奠定的。然而，这个基础是建立在玻尔兹曼自己提出的一个强有力的假设之上的。该假说声称，任何具有大量相互作用的粒子（如分子）的物理系统都具有这样的性质：对于任何固定的总能量和初始状态，该系统将进化到具有相同能量的任何其他状态。虽然这个猜想，如果从字面上看，在数学上是不可能发生的，但是精确的数学形式主义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具有特别重要的意义。

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.