Pergamon Chaos, Solmnr & Fractals Vol. 5, No. 1, pp. 77-89. 1995 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved C960-077Y/Y5$9.50 + .OO 0960-0779(94)EOO145-6 Do Ergodic or Chaotic Properties of the Reflection Law Imply Ergodicity or Chaotic Behavior of a Particle’s Motion?* JANUSZ SZCZEPAfiSKI and ELIGIUSZ WAJNRYB Institute of Fundamental Technological Research, Polish Academy of Sciences, Swietokrzyska 21, 00-049 Warsaw, Poland (Received 11 June 1994) Abstract-The aim of this paper is to answer the question if such properties of reflection law as ergodicity, chaotic behavior and periodicity transfer directly to the motion of a particle in sufficiently large and commonly used classes of the containers. We present two examples. In the first, the above listed properties transfer directly, i.e. ergodicity, periodicity and chaos of the reflection law yield, respectively, ergodicity, periodicity and chaos of the motion but the second example exhibits an opposite relationship: ergodicity and chaotic behavior of the law each imply periodicity of the motion, while periodicity yields ergodicity. These examples show that the answer to the question is negative and the role of the shape of the container is very important even in the case when we assume very strong properties of the reflection laws. Some related macroscopic properties following from the microscopic dynamics are presented, e.g. the properties of the long-time behavior of the distribution function for the corresponding Knudsen gas. Conversely, it turns out that the dynamical systems obtained are closely related to some intensively studied dynamical systems, namely ‘standard maps’ (topologically conjugated) and one-dimensional (1D) systems. The reflection law correspond- ing to each standard map is given. INTRODUCTION The problem of what happens when the particle reaches the boundary is not only of theoretical interest but of practical importance. This question has been studied for a long time by a number of scientists; up to now there are only hypothesis in this subject, more or less confirmed by experiment. One of the approaches for studying the behavior of a fluid confined to a container is the approximation that consists of replacing the wall by a smooth surface and assuming that when a particle encounters the wall it ‘reflects’, that is, its velocity instantaneously changes from its incident value to another ‘reflected’ value, the latter being such as to take the particle back into the domain of the gas. We call any such transformation of incident velocities into reflected ones a reflection law. Reflection law models were first considered rigorously by Schnute and Shinbrot [l]. They showed that within a reasonable class of reflection laws (in particular, they assumed that a reflection law is a Cl, isotropic, planar, one-to-one map), there exist only two reflection laws: specular or reverse, such that the fluid does not slip at the boundary. Moreover, under the above assumptions these two reflection laws are the only two for which the entropy function cannot increase for any initial distribution function [2]. Specular reflection is a commonly investigated reflection laws: it states that the angle of incidence is equal to the angle of reflection. If we additionally assume that the gas is very o *A part of this paper has been presented at the SIAM Conference on Application of Dynamical Systems. Snowbird, Utah, USA, October 15-19, 1992. 77