Advertisement

The European Physical Journal B

, Volume 70, Issue 1, pp 39–48 | Cite as

Nonextensive and superstatistical generalizations of random-matrix theory

  • A. Y. Abul-Magd
Topical issue on Generalized Entropies and Non-Linear Kinetics

Abstract

Random matrix theory (RMT) is based on two assumptions: (1) matrix-element independence, and (2) base invariance. Most of the proposed generalizations keep the first assumption and violate the second. Recently, several authors presented other versions of the theory that keep base invariance on the expense of allowing correlations between matrix elements. This is achieved by starting from non-extensive entropies rather than the standard Shannon entropy, or following the basic prescription of the recently suggested concept of superstatistics. We review these generalizations of RMT and illustrate their value by calculating the nearest-neighbor-spacing distributions and comparing the results of calculation with experiments and numerical-experiments on systems in transition from order to chaos.

PACS

05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion 05.45.Mt Quantum chaos; semiclassical methods 03.65.-w Quantum mechanics 02.30.Mv Approximations and expansions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A.J. Lichtenberg, M.A. Lieberman, Regular and Stochastic Motion, Applied Mathematical Sciences (Springer, New York, 1983) Google Scholar
  2. C. Chandre, H.R. Jauslin, Phys. Rep. 365, 1 (2002) Google Scholar
  3. J.-P. Eckmann, D. Ruelle, Rev. Mod. Phys. 67, 617 (1985) Google Scholar
  4. S.S.E.H. Elnashaie, S.S. Elshishini, Dynamical Modelling, Bifurcation and Chaotic Behavior of Gas-Solid Catalytic Reactions (Gordon & Breach, Amsterdam, 1996) Google Scholar
  5. L.A. Bunimovich, S. Venkatuyiri, Phys. Rep. 290, 81 (1997) Google Scholar
  6. M.V. Berry, J. Phys. A 10, 2083 (1977) Google Scholar
  7. O. Bohigas, M.J. Giannoni, C. Schmit, Phys. Rev. Lett. 52, 1 (1984) Google Scholar
  8. M.L. Mehta, Random Matrices, 2nd edn. (Academic, New York, 1991) Google Scholar
  9. T. Guhr, A. Müller-Groeling, H.A. Weidenmüller, Phys. Rep. 299, 189 (1998) Google Scholar
  10. R. Balian, Nuovo Cim. 57, 183 (1958) Google Scholar
  11. N. Rosenzweig, C.E. Porter, Phys. Rev. 120, 1698 (1960) Google Scholar
  12. M.S. Hussein, M.P. Sato, Phys. Rev. Lett. 70, 1089 (1993); Phys. Rev. C 47, 2401 (1993) Google Scholar
  13. A. Casati, L. Molinari, F. Izrailev, Phys. Rev. Lett. 64, 1851 (1990) Google Scholar
  14. Y.V. Fyodorov, A.D. Mirlin, Phys. Rev. Lett. 67, 2405 (1991) Google Scholar
  15. A.D. Mirlin, Y.V. Fyodorov, F.M. Dittes, J. Quezada, T.H. Seligman, Phys. Rev. E 54, 3221 (1996) Google Scholar
  16. V.E. Kravtsov, K.A. Muttalib, Phys. Rev. Lett. 79, 1913 (1997). Google Scholar
  17. F. Evers, A.D. Mirlin, Phys. Rev. Lett. 84, 3690 (2000); Phys. Rev. B 62, 7920 (2000) Google Scholar
  18. J. Evans, F. Michael, e-prints arXiv:cond-mat/0207472 and /0208151 Google Scholar
  19. F. Toscano, R.O. Vallejos, C. Tsallis, Phys. Rev. E 69, 066131 (2004) Google Scholar
  20. F.D. Nobre, A.M.C. Souza, Physica A 339, 354 (2004) Google Scholar
  21. A.Y. Abul-Magd, Phys. Lett. A 333, 16 (2004) Google Scholar
  22. A.C. Bertuola, O Bohigas, M.P. Prato, Phys. Rev. E 70, 065102(R) (2004) Google Scholar
  23. A.Y. Abul-Magd, Phys. Rev. E 71, 066207 (2005) Google Scholar
  24. A.Y. Abul-Magd, Phys. Lett. A 361, 450 (2007) Google Scholar
  25. A.Y. Abul-Magd, Physica A 361, 41 (2006) Google Scholar
  26. A.Y. Abul-Magd, Phys. Rev. E 72, 066114 (2005) Google Scholar
  27. C. Beck, E.G.D. Cohen, Physica A 322, 267 (2003) Google Scholar
  28. E.G.D. Cohen, Physica D 193, 35 (2004) Google Scholar
  29. C. Beck, Physica D 193, 195 (2004) Google Scholar
  30. C. Beck, Europhys. Lett. 64, 151 (2003) Google Scholar
  31. F. Sattin, L. Salasnich, Phys. Rev. E 65, 035106(R) (2003) Google Scholar
  32. F. Sattin, Phys. Rev. E 68, 032102 (2003) Google Scholar
  33. A. Reynolds, Phys. Rev. Lett. 91, 084503 (2003) Google Scholar
  34. M. Ausloos, K. Ivanova, Phys. Rev. E 68, 046122 (2003) Google Scholar
  35. C. Beck, Physica A 331, 173 (2004) Google Scholar
  36. S. Abe, Phys. Rev. E 66, 046134 (2002) Google Scholar
  37. A.M.C. Souza, C. Tsallis, Phys. Lett. A 319, 273 (2003) Google Scholar
  38. C. Tsallis, A.M.C. Souza, Phys. Rev. E 67, 026106 (2003); C. Tsallis, A.M.C. Souza, J. Phys. Lett. A 319, 273 (2003) Google Scholar
  39. E.T. Jaynes, Phys. Rev. 106, 620 (1957); E.T. Jaynes, Phys. Rev. 108, 171 (1957) Google Scholar
  40. C. Tsallis, J. Stat. Phys. 52, 479 (1988) Google Scholar
  41. C. Tsallis, Lect. Notes Phys. 560, 3 (2001) Google Scholar
  42. Q.A. Wang, Eur. Phys. J. B 26, 357 (2002) Google Scholar
  43. C. Tsallis, R.S. Mendes, A.R. Plastino, Physica A 261, 534 (1998) Google Scholar
  44. G. Kaniadakis, Physica A 296, 405 (2001) Google Scholar
  45. G. Kaniadakis, Phys. Rev. E 66, 056125 (2002) Google Scholar
  46. G. Kaniadakis, Phys. Rev. E 72, 036108 (2005) Google Scholar
  47. Gradshteyn, I.M. Ryzhik, Table of integrals, series and products (Academic Press, New York, 1980) Google Scholar
  48. C. Beck, E.G.D. Cohen, H.L. Swinney, Phys. Rev. E 72, 026304 (2005) Google Scholar
  49. G. Le Caër, R. Delannay, Phys. Rev. E 59, 6281 (1999) Google Scholar
  50. K.A. Muttalib, J.R. Klauder, Phys. Rev. E 71, 055101(R) (2005) Google Scholar
  51. A.Y. Abul-Magd, B. Dietz, T. Friedrich, A. Richter, Phys. Rev. E 77, 046202 (2008). Google Scholar
  52. A. Relaño, J.M.G. Gómez, R.A. Molina, J. Retamosa, E. Faleiro, Phys. Rev. Lett. 89, 244102 (2002); E. Faleiro, J.M.G. Gómez, R.A. Molina, L. Muñoz, A. Relaño, J. Retamosa, Phys. Rev. Lett. 93, 244101 (2004); E. Faleiro, U. Kuhl, R.A. Molina, L. Muñoz, A. Relaño, J. Retamosa, Phys. Lett. A 358, 251 (2006); R.A. Molina, J. Retamosab, L. Muñoz, A. Relaño, E. Faleiro, Phys. Lett. B 644, 25 (2007) Google Scholar
  53. J.M.G. Gómez, A. Relaño, J. Retamosa, E. Faleiro, L. Salasnich, M. Vraničar, M. Robnik, Phys. Rev. Lett. 94, 084101 (2005) Google Scholar
  54. M.S. Santhanam, J.N. Bandyopadhyay, Phys. Rev. Lett. 95, 114101 (2005) Google Scholar
  55. P. Manimaran, P.A. Lakshmi, P.K. Panigrahi, Phys. Rev. E 72, 046120 (2005); J. Phys. A 39, L599 (2006) Google Scholar
  56. M.S. Santhanam, J.N. Bandyopadhyay, D. Angom, Phys. Rev. E 73, 015201 (2006) Google Scholar
  57. E. Bogomolny, U. Gerald, C. Schmit, Phys. Rev. E 59, R1315 (1999) Google Scholar
  58. P.J. Richens, M.V. Berry, Physica (Amsterdam) 2D, 495 (1981) Google Scholar
  59. L. Bunimovich, Funct. Anal. Appl. 8, 254 (1974) Google Scholar
  60. L.A. Bunimovich, Chaos 11, (2001) Google Scholar
  61. M.V. Berry, M. Robnik, J. Phys. A 17, 2413 (1984) Google Scholar
  62. T. Brody, Lett. Nuovo Cim. 7, 482 (1973) Google Scholar
  63. B. Dietz, T. Friedrich, M. Miski-Oglu, A. Richter, F. Schäfer, Phys. Rev. E 75, 035203(R) (2007) Google Scholar
  64. H.P. Baltes, E.R. Hilf, Spectra of Finite Systems (Bibliographisches Institut Mannheim, 1975) Google Scholar
  65. Y. Gu, K.W. Yu, Z.R. Yang, Phys. Rev. E 65, 046129 (2002) Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Faculty of Engineering Sciences, Sinai UniversityEl ArishEgypt

Personalised recommendations