Knudsen's cosine law and random billiards

Abstract

This article lays the groundwork for understanding the details of gas transport in the so-called Knudsen regime, particularly during the early, nonstationary stages of the transport process. A simple mathematical model of gas–surface interaction is introduced, which is both fairly general and sufficiently explicit to permit a detailed analysis. The model, which we call a random billiard, is an example of a random dynamical system based on the idea of a billiard system with geometric microstructure. Among the main results, we show that Knudsen's cosine law is a stationary probability distribution of post-collision velocities, although not always the unique one. Through numerical experiments with a variety of microgeometries we also provide some insight concerning the approach to stationarity.