具有壁温的随机台球和相关的马尔可夫链
Random billiards with wall temperature and associated Markov chains
摘要：
我们所说的随机台球是指一个台球系统，其中标准的镜面反射规则被马尔可夫转移概率算子P所取代，该算子P在台球粒子与台球域边界的每次碰撞时，给出了给定碰撞前速度的碰撞后速度的概率分布。具有微结构的随机台球（RBM）是一种随机台球，其中P是从台球域边界上的几何/机械结构的选择中导出的。RBM提供了简单而明确的颗粒-表面相互作用的力学模型，可以将热效应，并允许从一般状态空间上的马尔可夫链的标准理论的角度详细研究粒子作用。我们专注于运营商P本身，以及它如何涉及到的微观结构的机械/几何特征，如质量比，曲率和潜力。主要结果如下：（1）刻画了平稳概率（平衡态）的P和显示如何标准的平衡分布研究在经典的统计力学，如麦克斯韦玻尔兹曼分布和克努森余弦定律，自然产生的广义不变台球措施;（2）得到了P的一些基本的泛函理论性质，在很一般的条件下，证明了P是适当的Hilbert空间上范数为1的自伴算子。在一个简单但说明性的例子中，我们证明了P是一个紧（希尔伯特-施密特）算子。这导致的问题有关的频谱的特征值的微观结构的P;（3）我们探讨后者的问题，分析和数值在几个有代表性的例子;（4）我们提出了一个通用的算法来模拟这些马尔可夫链的基础上的几何描述的不变体积的经典统计力学。
英文摘要：
By a random billiard we mean a billiard system in which the standard specular reflection rule is replaced with a Markov transition probabilities operator P that, at each collision of the billiard particle with the boundary of the billiard domain, gives the probability distribution of the post-collision velocity for a given pre-collision velocity. A random billiard with microstructure (RBM) is a random billiard for which P is derived from a choice of geometric/mechanical structure on the boundary of the billiard domain. RBMs provide simple and explicit mechanical models of particle-surface interaction that can incorporate thermal effects and permit a detailed study of thermostatic action from the perspective of the standard theory of Markov chains on general state spaces. We focus on the operator P itself and how it relates to the mechanical/geometric features of the microstructure, such as mass ratios, curvatures, and potentials. The main results are as follows: (1) we characterize the stationary probabilities (equilibrium states) of P and show how standard equilibrium distributions studied in classical statistical mechanics, such as the Maxwell-Boltzmann distribution and the Knudsen cosine law, arise naturally as generalized invariant billiard measures; (2) we obtain some basic functional theoretic properties of P. Under very general conditions, we show that P is a self-adjoint operator of norm 1 on an appropriate Hilbert space. In a simple but illustrative example, we show that P is a compact (Hilbert-Schmidt) operator. This leads to the issue of relating the spectrum of eigenvalues of P to the features of the microstructure;(3) we explore the latter issue both analytically and numerically in a few representative examples;(4) we present a general algorithm for simulating these Markov chains based on a geometric description of the invariant volumes of classical statistical mechanics.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
