周期台球模型中的确定性扩散
Deterministic Diffusion in Periodic Billiard Models
摘要：
研究了几类具有扩散的周期台球模型的统计性质。第二章介绍了动力系统的统计性质，并对动力系统的统计性质进行了综述。在第三章中，我们研究了二维两参数周期洛仑兹气体模型中扩散系数的几何依赖性，包括如何从数据中估计扩散系数的讨论。在第四章中，我们研究了中心极限定理中出现的位置分布和位移分布的形状。我们表明，有一个振荡的精细结构和它的起源是什么。这使我们能够猜想在这些系统中的中心极限定理的改进。一个非麦克斯韦的速度分布，导致非高斯极限分布。第5章处理多边形台球通道，开发正常和异常扩散时发生的图片，后者是由于平行的散射体在台球造成沟道效应。我们还描述了从正常到异常扩散的交叉。在第六章中，我们将我们的方法推广到三维周期性Lorentz气体模型，证明了在一定条件下正常扩散发生。特别地，我们构造了一个显式的有限视界模型，并讨论了位形空间中的空穴对系统扩散性质的影响。最后，我们得出结论和未来研究的方向。
英文摘要：
We investigate statistical properties of several classes of periodic billiard models which are diffusive. An introductory chapter gives motivation, and then a review of statistical properties of dynamical systems is given in chapter 2. In chapter 3, we study the geometry dependence of diffusion coefficients in a two-parameter 2D periodic Lorentz gas model, including a discussion of how to estimate them from data. In chapter 4, we study the shape of position and displacement distributions, which occur in the central limit theorem. We show that there is an oscillatory fine structure and what its origin is. This allows us to conjecture a refinement of the central limit theorem in these systems. A non-Maxwellian velocity distribution is shown to lead to a non-Gaussian limit distribution. Chapter 5 treats polygonal billiard channels, developing a picture of when normal and anomalous diffusion occur, the latter being due to parallel scatterers in the billiard causing a channelling effect. We also characterize the crossover from normal to anomalous diffusion. In chapter 6, we extend our methods to a 3D periodic Lorentz gas model, showing that normal diffusion occurs under certain conditions. In particular, we construct an explicit finite-horizon model, and we discuss the effect that holes in configuration space have on the diffusive properties of the system. We finish with conclusions and directions for future research.
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