DO ERGODIC OR CHAOTIC PROPERTIES OF THE REFLECTION LAW IMPLY ERGODICITY OR CHAOTIC BEHAVIOR OF A PARTICLES MOTION

The aim of this paper is to answer the question if such properties of reflection law as ergodicity, chaotic behavior and periodicity transfe r directly to the motion of a particle in sufficiently large and commo nly used classes of the containers. We present two examples. In the fi rst, the above listed properties transfer directly, i.e. ergodicity, p eriodicity and chaos of the reflection law yield, respectively, ergodi city, periodicity and chaos of the motion but the second example exhib its an opposite relationship: ergodicity and chaotic behavior of the l aw each imply periodicity of the motion, while periodicity yields ergo dicity. These examples show that the answer to the question is negativ e 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 d ynamics are presented, e.g. the properties of the long-time behavior o f the distribution function for the corresponding Knudsen gas. Convers ely, it turns out that the dynamical systems obtained are closely rela ted to some intensively studied dynamical systems, namely 'standard ma ps' (topologically conjugated) and one-dimensional (1D) systems. The r eflection law corresponding to each standard map is given.