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Regular and Irregular Motion in Classical And Quantum Systems

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Photophysics and Photochemistry in the Vacuum Ultraviolet

Part of the book series: NATO ASI Series ((ASIC,volume 142))

Abstract

It is now widely known that simple, nonlinear, deterministic dynamical systems can exhibit chaotic motion, which is unpredictable even in principle [1–4]. By simple, we mean that the system has only a few degrees of freedom N, say N = 2. By deterministic, we mean that the laws of motion are perfectly known locally; they are given by some velocity vector field in phase space. Time evolution of the system is governed by some ordinary differential equation, so that the future and the past are completely determined by the initial state of the system. The motion in time is obtained by finding the integral curves of the vector field (i.e., by solving the differential equation). This can always be done (at least numerically) for small times; and yet, the motion can still be unpredictable for large times. The reason for such a chaotic motion lies in the instability of trajectories, which can separate exponentially as time goes on. The motion thus displays a sensitive dependence on initial conditions. In such a case, the number of accurately calculated digits will decrease in proportion to the time elapsed from the initial condition, so that after a finite time our prediction will be completely wrong, no matter how good the measurement of the initial state was. An exponential instability can result in a type of motion which is as random as the outcome of tossing a coin. In fact, there is a whole hierarchy of chaotic systems, ranging from Bernoulli systems (most chaotic), through mixing and ergodic systems, down to almost-integrable (KAM) systems. An ergodic (or more chaotic) system will pass arbitrarily close to any point on the energy surface in phase space.

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References

  1. R. H. G. Helleman, in: Fundamental Problems in Statistical Mechanics, Vol. 5, E. G. D. Cohen, ed. (North-Holland, Amsterdam, 1980 )

    Google Scholar 

  2. B. V. Chirikov, Phys. Rep. 52, 263 (1979)

    Article  Google Scholar 

  3. M. V. Berry, in: Proceedings of the July 1981 ‘Les Houches’ Summer School on Chaotic Behaviour of Deterministic Systems, R. H. G. Helleman and G. looss, eds. ( North-Hoiland, Amster¬dam, 1982 )

    Google Scholar 

  4. G. M. Zaslavsky, Phys. Rep. 80, 158 (1981)

    Article  Google Scholar 

  5. L. J. Lebowitz and O. Penrose, Physics Today, 23 (Feb., 1973 )

    Google Scholar 

  6. M. I. Rabinovich, Sov. Phys. Usp. 21, 443 (1978)

    Article  Google Scholar 

  7. A. Libchaber and J. Maurer, J. Physique Lett. 39, L-369 (1978)

    Article  Google Scholar 

  8. For applications to magnetic, confinement in fusion and plasma physics see: Intrinsic Stochasticity in Plasmas, G. Laval and D. Gresillon, eds. ( Les Editions de Physique Publ., Orsay, 1980 )

    Google Scholar 

  9. F. Doveil, Phys. Rev. Lett. 46, 532 (1981)

    Article  Google Scholar 

  10. J. B. Delos and R. T. Swimm, Chem. Phys. Lett. 47, 76 (1977)

    Article  CAS  Google Scholar 

  11. R. A. Marcus, Ann. N.Y. Acad. Sci. 357, 169 (1980)

    Article  CAS  Google Scholar 

  12. M. Robnik, J. Phys. A 14, 3195 (1981)

    Article  CAS  Google Scholar 

  13. M. Robnik, in: Proceedings of the Colloque International CNRS on Atomic and Molecular Physics near Ionization Thresholds(Aussois, France, 1982), J. de Physique Colloque C2, Supplement 11 (1982)

    Google Scholar 

  14. W. P. Reinhardt and D. Farrelly, in: Proceedings of the Colloque International CNRS on Atomic and Molecular Physics near Ionization Thresholds(Aussois, France, 1982); J. de Physique Colloque C2, Supplement 11 (1982)

    Google Scholar 

  15. A. R. Edmonds and R. A. Pullen, Preprints Imperial College, London (1980)

    Google Scholar 

  16. D. Richards, J. Phys. B., submitted

    Google Scholar 

  17. R. D. Williams and S. E. Koonin, “Semiclassical Quantization of the Shell Model,” Preprint Caltech (1982)

    Google Scholar 

  18. H. A. Weidenmüller, Phys. Blätter 38, 176 (1982)

    Google Scholar 

  19. See the contributions in: Nonlinear Dynamics and the Beam- Beam Interaction, M. Month and J. C. Herrera, eds. (AIP Conf. Proc. 57, 1979)

    Google Scholar 

  20. Life in the Universe, J. Billingham, ed. (MIT Press, Cambridge, 1981)

    Google Scholar 

  21. H. Haken, Synergetics( Springer, Berlin, 1978 )

    Google Scholar 

  22. M. A. Olshanetsky and A. M. Perelomov, Phys. Rep. 71, 313 (1981)

    Article  Google Scholar 

  23. V. I. Arnold, Mathematical Methods of Classical Mechanics( Springer, New York, 1980 )

    Google Scholar 

  24. H. Goldstein, Classical Mechanics( Addison-Wesley, Reading, (1950)

    Google Scholar 

  25. For the moment “ergodic” means just that there are no inte-grals of the motion except for the energy. Later, we give the precise definitions

    Google Scholar 

  26. R. J. Whiteman, Rep. Prog. Phys. 40, 1033 (1977)

    Article  Google Scholar 

  27. V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics( Benjamin, New York, 1968 )

    Google Scholar 

  28. M. V. Berry, in: AIP Conf. Proc. 46, S. Jorna, ed. ( AIP, New York, 1978 )

    Google Scholar 

  29. M. Henon and C. Heiles, Astron. J. 69, 73 (1964)

    Article  Google Scholar 

  30. V. I. Arnold, Ordinary Differential Equations( MIT Press, Cambridge, 1980 )

    Google Scholar 

  31. J. M. Greene, R. S. MacKay, F. Vivaldi and M. J. Feigenbaum, Physica 3D, 468 (1981)

    Google Scholar 

  32. J. Guckenheimer, Lect. in Appl. Math. 17, 187 (1979)

    Google Scholar 

  33. C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics( Springer, New York, 1971 )

    Google Scholar 

  34. J. K. Moser, Stable and Random Motions in Dynamical Systems(Princeton Univ. Press, Princeton, 1973 )

    Google Scholar 

  35. Ya. G. Sinai, Introduction to Ergodic Theory(Princeton Univ. Press, Princeton, 1976 )

    Google Scholar 

  36. M. C. Gutzwiller, Phys. Rev. Lett. 45, 150 (1980)

    Article  CAS  Google Scholar 

  37. F. G. Gustavson, Astron. J. 71, 670 (1966)

    Article  Google Scholar 

  38. E. Schrüfer and M. Robnik, in preparation

    Google Scholar 

  39. M. Robnik, Preprint Bonn (1982)

    Google Scholar 

  40. D. F. Escande and F. Doveil, J. Stat. Phys. 26, 257 (1981)

    Article  Google Scholar 

  41. J. M. Greene, J. Math. Phys. 20, 1183 (1979)

    Article  Google Scholar 

  42. R. C. Churchill, G. Pecelli and D. L. Rod, in: Lecture Notes in Physics 93( Springer, Berlin, 1978 )

    Google Scholar 

  43. P. Collet and J. -P. Eckmann, Iterated Maps on the Interval as Dynamical Systems( Birkhäauser, Basel, 1980 )

    Google Scholar 

  44. B. V. Chirikov, F. M. Izrailev and D. L. Shepelyansky, “Transient Stochasticity in Quantum Mechanics,” Preprint 80–210 Novosibirsk (1980)

    Google Scholar 

  45. I. C. Percival, J. Phys. B 6, L229 (1973)

    Article  Google Scholar 

  46. N. Pomphrey, J. Phys. B 7, 1909 (1974)

    Article  CAS  Google Scholar 

  47. E. Vock and W. Hunziker, “Stability of Schrödinger Eigenvalue Problems,” Preprint Zürich (1982)

    Google Scholar 

  48. D. W. Noid, M. L. Koszykowski and R. A. Marcus, J. Chem. Phys. 67, 404 (1977)

    Article  CAS  Google Scholar 

  49. See [11], and references therein

    Google Scholar 

  50. R. Ramaswamy and R. A. Marcus, J. Chem. Phys. 74, 1379 (1981)

    CAS  Google Scholar 

  51. J. Chem. Phys. 74, 1385 (1981)

    Article  CAS  Google Scholar 

  52. A. Einstein, Verh. d. Deutsch. Phys. Ges. 19, 82 (1917)

    Google Scholar 

  53. V. P. Maslov and M. V. Fedoryuk, Quasi classical Approximation for Equations of Quantum Mechanics (Nauka, Moscow, 1976); in Russian

    Google Scholar 

  54. The rule is that the phase of a wave decreases by p/2 upon crossing a caustic with the classical allowed region on the right; similarly, it increases by p/2 in the case of opposite orientation. For an irreducible cycle, the orientation is the same for all caustics encountered during such a closed path

    Google Scholar 

  55. M. V. Berry, Preprint Utrecht (1981)

    Google Scholar 

  56. M. C. Gutzwiller, J. Math. Phys. 12, 343 (1971); in: Path Integrals, J. T. Devreese, ed. ( Plenum, New York, 1978 ), p. 163

    Google Scholar 

  57. R. Balian and C. Bloch, Ann. Phys. (N.Y.) 69, 76 (1972)

    Google Scholar 

  58. G. M. Zaslavsky, Sov. Phys. Usp. 22, 788 (1979)

    Article  Google Scholar 

  59. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals( McGraw-Hill, New York, 1965 )

    Google Scholar 

  60. M. V. Berry and M. Tabor, Proc. Roy. Soc. London A 349, 101 (1976)

    Article  Google Scholar 

  61. C. DeWitt-Morette, A. Maheshwari and B. Nelson, Phys. Rep. 50, 255 (1979)

    Article  Google Scholar 

  62. M. V. Berry, Ann. Phys. (N.Y.) 131, 163 (1981)

    Article  Google Scholar 

  63. J. von Neumann and E. Wigner, Phys. Z. 30, 467 (1929)

    Google Scholar 

  64. G. M. Zaslavsky and N. N. Filonenko, Sov. Phys. JETP 38, 317 (1974)

    Google Scholar 

  65. M. V. Berry and M. Tabor, Proc. Roy. Soc. London A 356, 375 (1977)

    Article  Google Scholar 

  66. R. T. Swimm and J. B. Delos, J. Chem. Phys. 71, 1706 (1979)

    Article  CAS  Google Scholar 

  67. I. M. Gelfand and B. M. Levitan, Amer. Math. Soc. Transl. 1, 253 ( 1955 )

    Google Scholar 

  68. F. Calogero and A. Degasperis, Preprint Rome (1978)

    Google Scholar 

  69. W. P. Reinhardt, private discussion; I should like to thank him for bringing this fact to my attention

    Google Scholar 

  70. H. J. Korsch, Phys. Lett. 90A, 113 (1982)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Astronomische Institute, Universität Bonn, Auf dem Hügel 71, D-5300, Bonn, West Germany

    Marko Robnik

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  1. Marko Robnik
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Editors and Affiliations

  1. Department of Chemistry, Louisiana State University, Baton Rouge, Louisiana, USA

    S. P. McGlynn

  2. Department of Chemistry, New York University, New York, NY, USA

    G. L. Findley

  3. Argonne National Laboratory, Argonne, Illinois, USA

    R. H. Huebner

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© 1985 D. Reidel Publishing Company, Dordrecht, Holland

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Robnik, M. (1985). Regular and Irregular Motion in Classical And Quantum Systems. In: McGlynn, S.P., Findley, G.L., Huebner, R.H. (eds) Photophysics and Photochemistry in the Vacuum Ultraviolet. NATO ASI Series, vol 142. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-5269-0_15

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  • DOI: https://doi.org/10.1007/978-94-009-5269-0_15

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-8827-5

  • Online ISBN: 978-94-009-5269-0

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