Chaotic motion and random matrix theories

  • Oriol Bohigas
  • Marie-Joya Giannoni
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 209)


“Cada ciència ha mester los vocables per los quals mills sia manifestada ; e car a aquesta c iencia ia demostrativa sien mester vocables s escurs e que los hòmens lecs no han en us, e cat nós facam aquest llibre als hómens lecs per açó breument e ab plans vocables parlarem d' esta ciènc cia” Ramon Llull (Libre de Gentil, 1273 ?)


Chaotic System Invariant Torus Riemann Zeta Function Random Matrix Theory Toda Lattice 


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  1. [AA-67]
    V.I. ARNOLD and A. AVEZ, Problèmes ezgodiques de to mécanique classique, Gauthier-Villars, Paris, 1967.Google Scholar
  2. [Ar-76]
    V.I. ARNOLD, Méthodes Mathématiques de la Mécanioue Ctassique, Mir, Moscou, 1976.Google Scholar
  3. [Ba-68]
    R. BALIAN, Nuov. Cim. 57 (1968) 183.Google Scholar
  4. [Ba-75]
    R. BALESCU, Equilibrium and Nonequilibrium Statistical Mechanics,Appendix A, Wiley-Interscience Publ., N.Y., 1975.Google Scholar
  5. [Be-1873]
    J. BERTRAND, Comptes Rendus Académie des Sciences LXXVII (1873) 849.Google Scholar
  6. [Be-78]
    M.V. BERRY, in Topics in Nonlinear Dynamics (ed.S. Jorna) Am.Inst.Conf. Proc. 46 (1978) 16.Google Scholar
  7. [Be-81a]
    M.V. BERRY, Eur.J.Phys. 2 (1981) 91.Google Scholar
  8. [Be-81b]
    M.V. BERRY, Ann.Phys., N.Y. 131 (1981) 163.Google Scholar
  9. [Be-83]
    M.V. BERRY, in Chaotic Behaviour of Deterministic Systems, Les Houches, Summer School Lectures XXXVI, R.H.G. Heileman and G. Joos (eds.), North-Holland, Amsterdam, 1983, p.171.Google Scholar
  10. [Be-84]
    M.V. BERRY in [Ca-84].Google Scholar
  11. [BF-71.
    O. BOHIGAS and J. FLORES, Phys. Lett. 34B (1971) 261.Google Scholar
  12. [BFF-81.
    T.A. BRODY, J. FLORES, J.B. FRENCH, P.A. MELLO, A. PANDEY and S.S.M. WONG, Rev.Mod.Phys. 53 (1981) 385.Google Scholar
  13. [BG-75]
    O. BOHIGAS and M.J. GIANNONI, Ann.Phys., N.Y. 89 (1975) 393.Google Scholar
  14. [BGP-84]
    O. BOHIGAS, M.J. GIANNONI and A. PANDEY, to be published.Google Scholar
  15. [BGS-83]
    O. BOHIGAS, M.J. GIANNONI and C. SCHMIT, in [Ca-84].Google Scholar
  16. [BGS-84a]
    O. BOHIGAS, M.J. GIANNONI and C. SCHMIT, Phys.Rev.Lett. 52 (1984) 1.Google Scholar
  17. [BGS-84b]
    O. BOHIGAS, M.J. GIANNONI and C. SCHMIT, preprint 1984.Google Scholar
  18. [BH-76]
    H.P. BALTES and E.R. HILF: Spectra of Finite Systems, Bibliographisches Institut, Mannheim 1976.Google Scholar
  19. [BHP-83]
    O. BOHIGAS, R.U. HAQ and A. PANDEY, in Nuclear Data for Science and Technology, K.H. Böchhoff (ed.), Reidel, Dordrecht, 1983, p. 809.Google Scholar
  20. [BHP-84]
    O. BOHIGAS, R.U. HAQ and A. PANDEY, preprint.Google Scholar
  21. [Bi-78]
    P. BILLINGSEY, Ezgodic Theory and Information, Robert E. Krieger Publishing Company, Huntington, New York, 1978.Google Scholar
  22. [BLM-76]
    E.G. BILPUCH, A.M. LANE, G.E. MITCHELL and J.D. MOSES, Phys.Rep. C28 (1976) 145.Google Scholar
  23. [BT-77]
    M.V. BERRY and M. TABOR, Proc.Roy.Soc.Lond. A356 (1977) 375.Google Scholar
  24. [Ca-84]
    G. CASATI (ed.), Proceedings of Conference on Quantum Chaos, Como 1983, Pergamon, to appear.Google Scholar
  25. [CFS-82]
    I.P. CORNFELD, S.V. FOMIN and Ya.G. SINAI, Ergodic Theory, Springer-Verlag, N.Y., 1982.Google Scholar
  26. [CG-83]
    H.S. CAMARDA and P.D. GEORGOPULOS, Phys.Rev.Lett. 50 (1983) 492.Google Scholar
  27. [CVG-80]
    G. CASATI, F. VALZ-CRIS and I. GUARNERI, Lett.Nuov.Cim. 28 (1980) 279.Google Scholar
  28. [DH-80]
    P.J. DAVIS and R. HERSH, The Mathematical Experience, Birkhduser, Boston, 1980.Google Scholar
  29. [DM-63]
    F.J. DYSON and M.L. MEHTA, J.Math.Phys. 4 (1963) 701, reprinted in [Po-65].Google Scholar
  30. [Dy-62a]
    F.J. DYSON, J.Math.Phys. 3 (1962) 140, reprinted in [Po-65].Google Scholar
  31. [Dy-62b]
    F.J. DYSON, J.Math.Phys. 3 (1962) 1199, reprinted in [Po-65].Google Scholar
  32. [Dy-70]
    F.J. DYSON, Comm.Math.Phys. 19 (1970) 235.Google Scholar
  33. [Dy-71]
    F.J. DYSON, Rev.Mex.Fis. 20 (1971) 231.Google Scholar
  34. [Dy-72]
    F.J. DYSON, J. Math.Phys. 13 (1972) 90.Google Scholar
  35. [Dy-83]
    F.J. DYSON, private communication 1983.Google Scholar
  36. [Ed-74]
    H.M. EDWARDS, Riemann's Zeta Function, Academic Press, 1974.Google Scholar
  37. [FK-82]
    J.B. FRENCH and V.K.B. KOTA, Ann. Rev.Nucl.Part.Sci.32 (1982) 35.Google Scholar
  38. [Fo-74]
    J. FORD in Lecture Notes in Physics n°28: Lectures in Statistical Physics, Springer-Verlag, Berlin-Heidelberg, 1974, p.204.Google Scholar
  39. [Fo-83]
    J. FORD, in Physics Today (April 1983)-See also J. FORD, in Long-Time Prediction in Dynamics, C.W. Horton, Jr., L.E. Reichl, V.G. Szebehely, (eds.) J. Wiley, 1983.Google Scholar
  40. [FST-73]
    J. FORD, S.D. STODDARD and J.S. TURNER, Prog.Theor.Phys. 50 (1973) 1547.Google Scholar
  41. [FW-70]
    J.B. FRENCH and S.S.M. WONG, Phys.Lett. 33B (1970) 449.Google Scholar
  42. [Ga-61]
    M. GAUDIN, Nucl.Phys. 25 (1961)447, reprinted in [Po-65].Google Scholar
  43. [GC-68]
    I.J. GOOD and R.F. CHURCHHOUSE, Math. of Computation 22 (1968) 857.Google Scholar
  44. [Ge-72]
    A. GERVOIS, Nucl.Phys. A184 (1972) 507.Google Scholar
  45. [Go-51]
    H. GOLDSTEIN, Classical Mechanics, Addison-Wesley, Reading, Massachusetts, 1951.Google Scholar
  46. [He-74]
    M. HENON, Phys.Rev. B9 (1974) 1921.Google Scholar
  47. [He-80]
    R.H.G. HELLEMAN in Fundamental Problems in Statistical Mechanics, Vol. 5, E.G.D. Cohen (ed.), North-Holland, Amsterdam,1980, p.165.Google Scholar
  48. [He-83]
    M. HENON in Chaotic Behaviour of Deterministic Systems, Les Houches, Summer School Lectures XXXVI, R.H.G. Heileman and G. Joos (eds.), North-Holland, Amsterdam 1983, p.53.Google Scholar
  49. [HH-64]
    M. HENON and C. HEILES, Astron.J. 69 (1964) 73.Google Scholar
  50. [HKC-83]
    E. HALLER, H. KOPPEL and L.S. CEDERBAUM, Chem.Phys.Lett. 101 (1983)215.Google Scholar
  51. [HM-63]
    C.B. HASELGROVE and J.C.P. MILLER, Tables of the Riemann Zeta Function, Royal Society Mathematical Tables _6, Cambridge University Press 1963.Google Scholar
  52. [Ho-75]
    A. HOBSON, Journ. of Math.Phys. 16 (1975) 2210.Google Scholar
  53. [HPB-82]
    R.U. HAQ, A. PANDEY and O. BOHIGAS, Phys.Rev.Lett. 48 (1982) 1086.Google Scholar
  54. [HWR-78]
    G. HACKEN, R. WERBIN and J. RAINWATER, Phys.Rev. C17 (1978) 43. This paper contains a complete set of references to earlier papers by the Columbia group.Google Scholar
  55. [HYK-84]
    H. HIROOKA, Y. YOTSUYA, Y. KOBAYASHI and N. SAITO, Phys.Lett. 1O1A (1984) 115.Google Scholar
  56. [Ka-59]
    M. KAC, Statistical Independence in Probability, Analysis and Number Theory, Math. Ass. America, 1959.Google Scholar
  57. [Ka-66]
    M. KAC, Am.Math.Monthly 73 (1966) 1.Google Scholar
  58. [Kh-64]
    A.Y. KHINCHINE, Continued Fractions, University of Chicago Press, Chicago, 1964.Google Scholar
  59. [La-70]
    C. LANCZOS, The Variational Principles of Mechanics, fourth edition, University of Toronto Press, Toronto, 1970.Google Scholar
  60. [La-73]
    V.F. LAZUTKIN, Izv.A.N.SSSR. ser matem 37 (1973); english translation in Math. USSR Izv. Vol. 7 (1973) 185.Google Scholar
  61. [Le-54]
    P. LEVY, Théorie de Vaddition des variables aléatoires, Gauthier-Villars, Paris, 1954.Google Scholar
  62. [LL-83]
    A.J. LICHTENBERG andM.A. LIEBERMAN, Regular and Stochastic Motion, Springer-Verlag, New-York, 1983.Google Scholar
  63. [LP-73]
    J.L. LEBOWITZ and O. PENROSE, in Physics Today 26 (February 1973) 23.Google Scholar
  64. [Ly-68]
    J.E. LYNN, The Theory of Neutron Resonance Reactions, Clarendon Press, Oxford 1968.Google Scholar
  65. [Ma-74]
    S.V. MANAKOV,Zh.Eksp. Teor.Fiz. 67 (1974) 543; english translation in Sov.Phys. JETP 40 (1974) 269.Google Scholar
  66. [Ma-83]
    V.R. MANFREDI, private communication, 1983.Google Scholar
  67. [MdC-72]
    M.L. MEHTA and J. des CLOIZEAUX, Ind.J.Pure Appl.Math. 3 (1972) 329.Google Scholar
  68. [Me-60]
    M.L. MEHTA, Nucl.Phys. 18 (1960) 395, reprinted in [Po-65].Google Scholar
  69. [Me-67]
    M.L. MEHTA, Random Matrices and the Statistical Theory of Energy Levels, Academic Press, 1967.Google Scholar
  70. [Me-71]
    M.L. MEHTA, Comm.Math.Phys. 20 (1971) 245.Google Scholar
  71. [ME-81]
    N.F.G. MARTIN and J.W. ENGLAND, Mathematical Theory of Entropy, Encyclopedia of Mathematics and its Applications, vol.12, Gian-Carlo Rota (ed.), Addison-Wesley, 1981.Google Scholar
  72. [MF-75]
    K.K. MON and J.B. FRENCH, Ann.Phys., N.Y. 95 (1975) 90.Google Scholar
  73. [MK-79]
    S.W. Mc DONALD and A.N. KAUFMAN, Phys.Rev.Lett. 42 (1979) 1189.Google Scholar
  74. [Mo-81]
    P.H. MORSE, Vibrations and Sound, 5th edition, American Inst. of Physics, 1981.Google Scholar
  75. [Mo-73a]
    H.L. MONTGOMERY, Proc.Symp.Pure Maths. 24 (1973) 181.Google Scholar
  76. [Mo-73b]
    J. MORER, Stable and Random Motions in Dynamical Systems, Princeton University Press, Princeton, 1973.Google Scholar
  77. [Mo-74]
    H.L. MONTGOMERY, Proc. of Int. Congr. of Mathematicians, Vancouver, 1974.Google Scholar
  78. [Mo-76]
    H.L. MONTGOMERY, Proc. Symp. Pure Maths. 28 (1976) 307.Google Scholar
  79. [MP-83]
    M.L. MEHTA and A. PANDEY, J.Phys. A16 (1983) L601.Google Scholar
  80. [MSP-84]
    S. MUKAMEL, J. SUE and A. PANDEY, Chem.Phys.Lett. 105 (1984) 134.Google Scholar
  81. [MT-84]
    T. MATSUSHITA and T. TERASAKA, Chem.Phys.Lett., in press.Google Scholar
  82. [MW-79]
    C. MAHAUX and H. WEIDENMÜLLER, Ann.Rev.Nucl.Part.Sci. 29 (1979) 1.Google Scholar
  83. [Od-82]
    A. ODLYZKO, private communication, 1982.Google Scholar
  84. [Or-74]
    D.S. ORNSTEIN, Ergodic Theory, Randomness, and Dynamical Systems, New Haven and London, Yale University Press, 1974.Google Scholar
  85. [Pa-78]
    A. PANDEY, Doctoral dissertation, University of Rochester, 1978, unpublished.Google Scholar
  86. [Pa-79]
    A. PANDEY, Ann.Phys., N.Y. 119 (1979) 170.Google Scholar
  87. [Pa-81]
    W. PARRY, Topics in Ergodic Theory, Cambridge University Press, 1981.Google Scholar
  88. [Pa-81a]
    A. PANDEY, Ann.Phys. 134 (1981) 110.Google Scholar
  89. [Pe-83]
    P. PECHUKAS, Phys. Rev.Lett. 51 (1983) 943.Google Scholar
  90. [Po-65]
    C.E. PORTER (ed.), Statistical theories of spectra:: fluctuations, Academic Press, 1965.Google Scholar
  91. [Po-65a]
    C.E. PORTER, Introductory article in [Po-65].Google Scholar
  92. [PS-83]
    J.C. PARIKH and V.B. SHEOREY, preprint 1983.Google Scholar
  93. [RB-81]
    P.J. RICHENS and M.V. BERRY, Physica 2D (1981) 495.Google Scholar
  94. [Ro-61]
    V.A. ROHLIN, Izv.Akad.Nauk.SSSR Ser.Mat., 25 (1961) 499; english translation, Amer.Math.Soc.Transl.Ser. 2, 39 (1964) 1.Google Scholar
  95. [Ro-84]
    A. ROBNIK, J.Phys. A17 (1984) 1049.Google Scholar
  96. [RP-60]
    N. ROSENZWEIG and C.E. PORTER, Phys.Rev. 120 (1960) 1698, reprinted in [Po-65].Google Scholar
  97. [Sc-84]
    C. SCHMIT, Contribution to Workshop on Semi-Clauical Methods, Grenoble 1984.Google Scholar
  98. [Sh-73]
    P. SHIELDS, The Theory of Bernouilli Shifts, University of Chicago Press, 1973.Google Scholar
  99. [Si-70]
    Ya.G. SINAI, Usp.Mat.Nauk 25 (1970) 141; english translation in Russ.Math.Surv. 25 (1970) 137.Google Scholar
  100. [Si-79]
    C. SIMO, But.Soc.Cat. C.F.Q.iM. I (1979) 145.Google Scholar
  101. [SVZ-84]
    T.H. Seligman, J.J.M. VERBAARSCHOT and M.R. ZIRNBAUER, preprint Heidelberg MPI H-V13,1984.Google Scholar
  102. [SW-49]
    C.E. SHANNON and W. WEAVER, The Mathematical Theory of Communication, University of Illinois, Urbana, Illinois, 1949.Google Scholar
  103. [SZ-72]
    M. SOYEUR and A. ZUKER, Phys.Lett. 41B (1972) 135.Google Scholar
  104. [TKS-83]
    M. TODA, R. KUBO and N. SAITO, Statistical Physics 1, Springer Series in SolidState Sciences, Peter Fulde (ed.), vol. 30, Springer-Verlag, BerlinHeidelberg, 1983.Google Scholar
  105. [To-67]
    M. TODA, J.Phys.Soc. Japan 22 (1967) 431.Google Scholar
  106. [To-70]
    M. TODA, Prog.Theor.Phys.Suppl. 45 (1970) 174.Google Scholar
  107. [VZ-84]
    J.J.M. VERBAARSCHOT and M.R. ZIRNBAUER, submitted to Ann.Phys. (N.Y.).Google Scholar
  108. [We-84]
    H. WEIDENMÜLLER, preprint Heidelberg MPI-H-V10, 1984.Google Scholar
  109. [Wh-77]
    K.J. WHITEMAN, Rep.Prog.Phys. 40 (1977) 1033.Google Scholar
  110. [Wi-56]
    E.P. Wigner, Contribution to Conference on neutron physics by time-of-flight, Gatlinburg, Tennessee 1956, reprinted in [Po-65].Google Scholar
  111. [Wi-67a]
    E.P. WIGNER, SIAM Rev. 9 (1967) 1.Google Scholar
  112. [Wi-67b]
    E.P. WIGNER, Symmetries and Reflections, Indiana University Press, Bloomington, 1967.Google Scholar
  113. [Wi-72]
    E.P. WIGNER, in Statistical Properties of Nuclei, J.B. Garg (ed.), Plenum, New York, 1972.Google Scholar
  114. [Wi-79]
    K.G. WILSON, Scientific American 241, no 2 p. 158.Google Scholar
  115. [Wo-70]
    S.S.M. WONG, Nucl.Phys. A159 (1970) 235.Google Scholar
  116. [Za-77]
    D. ZAGIER, The Math.Intelligencer 0 (1977) 7.Google Scholar
  117. [Za-81]
    G.M. ZASLAVSKY, Phys.Rep. 80 (1981) 157.Google Scholar
  118. [ZK-75]
    A.N. ZEMLYAKOV and A.B. KATOK, Math.Zametki 18 (1975) 291; english translation in Math. Notes 18 (1976) 760.Google Scholar
  119. [ZVW-83]
    M.R. ZIRNBAUER, J.J.M. VERBAARSCHOT and H.A. WEIDENMÜLLER, Nucl.Phys. A411 (1983) 161.Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Oriol Bohigas
    • 1
  • Marie-Joya Giannoni
    • 1
  1. 1.Division de Physique ThéoriqueInstitut de Physique NucléaireORSAY CEDEXFrance

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