随机力学系统中的多重散射与扩散近似
Multiple scattering in random mechanical systems and diffusion approximation
摘要：
本文关注的是在经典力学系统的台球型，定义如下模型的多重（或迭代）散射的随机过程。从一个给定的（确定性）系统的台球类型，一个随机过程的转移概率算子P引入的假设，一些动态变量是随机的，具有规定的概率分布。特别令人感兴趣的是系统弱散射，这是与参数家庭的运营商P_h，取决于几何或力学参数h，接近身份h去0。证明了（P_h-I）/h对小h收敛于紧支撑函数上的二阶椭圆微分算子L，并且与P_h相关联的马氏链过程收敛于具有无穷小生成元L的扩散. P_h和L都（稠密）定义在R^m中（下）半空间H上的平方可积函数空间L2（H，η）上，其中η是平稳测度。该测度的密度是（碰撞后）Maxwell-Boltzmann分布或Knudsen余弦分布，生成元为无穷小的随机过程分别对应于MB扩散和（广义）Legendre扩散。给出了简单力学系统的具体例子，并通过对随机过程的数值模拟进行了说明。
英文摘要：
This paper is concerned with stochastic processes that model multiple (or iterated) scattering in classical mechanical systems of billiard type, defined below. From a given (deterministic) system of billiard type, a random process with transition probabilities operator P is introduced by assuming that some of the dynamical variables are random with prescribed probability distributions. Of particular interest are systems with weak scattering, which are associated to parametric families of operators P_h, depending on a geometric or mechanical parameter h, that approaches the identity as h goes to 0. It is shown that (P_h -I)/h converges for small h to a second order elliptic differential operator L on compactly supported functions and that the Markov chain process associated to P_h converges to a diffusion with infinitesimal generator L. Both P_h and L are selfadjoint (densely) defined on the space L2(H,η) of square-integrable functions over the (lower) half-space H in R^m, where η is a stationary measure. This measure's density is either (post-collision) Maxwell-Boltzmann distribution or Knudsen cosine law, and the random processes with infinitesimal generator L respectively correspond to what we call MB diffusion and (generalized) Legendre diffusion. Concrete examples of simple mechanical systems are given and illustrated by numerically simulating the random processes.
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