On Ergodicity in 3D Closed Billiards

概要

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We discuss ergodic properties of 3D closed billiards with an emphasis on static statistical properties calculated according to a Liouville measure written in terms of Birkhoff coordinates. It is possible to calculate the Liouville measure in terms of four angle variables determining a corresponding 4D map independently of details of the boundary which may be in general defined piece-wisely. We elucidate how to obtain the Liouville measure in an elementary way. The final form of the Liouville measure leads to the definitions of the Birkhoff coordinates in 3D billiards. We next find statistical formulas of long time averages such as segment length, angular momentum, and pressure distribution in ergodic billiards. These formulas can be utilized as necessary conditions to investigate which 3D billiard systems are ergodic. We finally investigate a concrete example called a 3D oval billiard which is expected to be ergodic by comparing these three formulas with numerical results.
理論物理学刊行会の論文
1995-01-25