Advertisement

Random Matrix Theory

  • Linda E. Reichl
Part of the Institute for Nonlinear Science book series (INLS)

Abstract

Classical conservative systems that undergo a transition to chaos have very complex dynamical behavior, as we have seen in previous chapters. How much of this complex behavior remains in the corresponding quantum systems? That is the question we address in much of the remainder of this book. An essential new result has emerged: quantum systems, whose classical counterpart is chaotic, have spectra whose statistical properties are similar to those of random matrices that extremize information. Thus, any study of the quantum manifestations of chaos requires an analysis of information content of quantum systems using concepts from random matrix theory (RMT). We have attempted to give a complete grounding on random matrix theory in this book. Much of our discussion of random matrix theory is in the appendices, but we give an overview of key results in this chapter. Our analysis of quantum dynamics, the behavior of solutions of the Schrödinger equation, will actually begin in Chapter 6.

Keywords

Random Matrix Theory Wigner Distribution Cluster Function Eigenvalue Density Gaussian Ensemble 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bateman, H. (1953): Higher Trancendental Functions, Vol. 2, edited by A. Erdelyi (McGraw-Hill, New York).Google Scholar
  2. Brody, T.A. (1974): Lett. Nuovo Cimento 7 482.CrossRefGoogle Scholar
  3. Brody, T.A., Flores, J., French, J.B., Mello, P.A., Pandey, A., and Wong, S.S.M. (1981): Rev. Mod. Phys. 53 385.MathSciNetADSCrossRefGoogle Scholar
  4. Brookes, B.C. and Dick, W.F.L. (1969): Introduction to Statistical Methods (Heinemann, London).Google Scholar
  5. Dyson, F.J. (1962a): J. Math. Phys. 3 140.MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. Dyson, F. J. (1962b): J. Math. Phys. 3 166.MathSciNetADSCrossRefGoogle Scholar
  7. Dyson, F.J. (1962c): J. Math. Phys. 3 1191.MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. Dyson, F.J. and Mehta, M.L. (1963): J. Math. Phys. 4 489.CrossRefGoogle Scholar
  9. Gaudin, M. (1961): Nucl. Phys. 25 447.zbMATHCrossRefGoogle Scholar
  10. Guhr, T., Muller-Groeling, A., and Weidenmuller, H.A. (1998): Phys. Rept 299 Google Scholar
  11. 189.
    Haller, E., Koppel, H., and Cederbaum, L.S. (1983): Chem. Phys. Lett. 101 215.ADSCrossRefGoogle Scholar
  12. Li, W., Reichl, L.E., and Wu, B. (2002): Phys. Rev. E 65 56220.ADSCrossRefGoogle Scholar
  13. Mehta, M.L. (1960): Nucl. Phys. 18 420.CrossRefGoogle Scholar
  14. Mehta, M.L. (1991): Random Matrices and the Statistical Theory of Energy Levels, 2nd Edition (Academic Press, New York).Google Scholar
  15. Meyer, S.L. (1975): Data Analysis for Scientists and Engineers (John Wiley and Sons, Inc., New York).Google Scholar
  16. Porter, C.E. (1965): Statistical Theories of Spectra: Fluctuations (Academic Press, New York).Google Scholar
  17. Porter, C.E. and Thomas, R.G. (1956): Phys. Rev. 104 483.ADSCrossRefGoogle Scholar
  18. Reichl, L.E. (1998): A Modern Course in Statistical Physics, Second Edition (John Wiley and Sons, New York)zbMATHGoogle Scholar
  19. Terasaka, T. and Matsushita, T. (1985): Phys. Rev. A 32 538.ADSCrossRefGoogle Scholar
  20. Venkataraman, R. (1982): J. Phys. B 15 4293.ADSCrossRefGoogle Scholar
  21. Wigner, E.P. (1951): Ann. Math. 53 36.MathSciNetzbMATHCrossRefGoogle Scholar
  22. Wigner, E.P. (1955): Ann. Math. 62 548.MathSciNetzbMATHCrossRefGoogle Scholar
  23. Wigner, E.R (1957a): Ann. Math. 65 203.MathSciNetzbMATHCrossRefGoogle Scholar
  24. Wigner, E.P. (1957b): Can. Math. Congr. Proc. (Univ. of Toronto Press, Toronto, Canada), p. 174. Reprinted in [Porter 1965].Google Scholar
  25. Wigner, E.P. (1958): Ann. Math. 67 325.MathSciNetzbMATHCrossRefGoogle Scholar
  26. Wigner, E.P. (1959): Conference on Neutron Physics by Time of Flight, Gatlinburg, Tennessee, November 1956, Oak Ridge Natl. Lab. Rept. ORNL-2309, p.67 (1959). Reprinted in [Porter 1965], p. 188.Google Scholar
  27. Wilson, K. G. (1962): J. Math. Phys. 3 1040.ADSzbMATHCrossRefGoogle Scholar
  28. Zyczkowski, K. (1991): in Quantum Chaos, edited by H.A. Cerdeira, R. Ramaswamy, M.C. Gutzwiller, and G. Casati (World Scientific, Singapore).Google Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Linda E. Reichl
    • 1
  1. 1.Department of Physics and Center for Statistical Mechanics and Complex SystemsUniversity of Texas at AustinAustinUSA

Personalised recommendations