Do ergodic or chaotic properties of the reflection law imply ergodicity or chaotic behavior of a particleʹs motion?

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 corresponding to each standard map is given.