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Prog. Theor. Phys. Vol. 93 No. 1 (1995) pp. 19-46

[ Full Text PDF : FREE ACCESS (851K) ]

On Ergodicity in 3D Closed Billiards

Shinji Koga

Department of Physics, Osaka Kyoiku University, Osaka 582

(Received August 31, 1994)

Abstract:

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.


URL : http://ptp.ipap.jp/link?PTP/93/19/
DOI : 10.1143/PTP.93.19

[ Full Text PDF : FREE ACCESS (851K) ] Citation:

References:

  1. G. D. Birkhoff, Dynamical Systems (American Mathematical Society, 1972).
  2. V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics (Addison-Wesley, 1968).
  3. V. V. Kozlov and D. V. Treshchëv, Billiards-A Genetic Introduction to the Dynamics of Systems with Impacts (American Mathematical Society, 1991).
  4. L. A. Bunimovich, Commun. Math. Phys. 65 (1979), 295[CrossRef]; Sov. Phys.-JETP 62 (1985), 842.
  5. See for linear stability analysis
    M. P. Wojtkowski, Commun. Math. Phys. 129 (1990), 319[CrossRef].
    V. V. Kozlov and I. I. Chigur, Appl. Math. Mech. 55 (1991), 576.
  6. J. Moser and A. P. Veselov, Commun. Math. Phys. 139 (1991), 217.
  7. G. Benettin, L. Galgani and J. M. Strelchn, Phys. Rev. 14A (1976), 2338[APS].

Citing Article(s) :

  1. Journal of the Physical Society of Japan 69 (2000) pp. 2011-2033 :
    Numerical Evidence of Universal Scaling Laws of Moments in 2D Sinai's Billiard Systems
    Shinji Koga
  2. Journal of the Physical Society of Japan 70 (2001) pp. 1260-1299 :
    Unified Treatment of Birkhoff Coordinates in Billiard Systems of Particles Moving under an Influence of a Potential Including Particle-Particle Interaction
    Shinji Koga
  3. Progress of Theoretical Physics Vol. 96 No. 6 (1996) pp. 1301-1305 :
    Global Scaling Laws in 3D Sinai's Billiards
    Shinji Koga
  4. Progress of Theoretical Physics Vol. 97 No. 3 (1997) pp. 363-378 :
    Scaling Laws of Moments in Sinai's Billiard Systems. I
    Shinji Koga

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