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II. Quantum Density Matrix Description of Nonextensive Systems

  • A.K. Rajagopal
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 560)

Abstract

A description of nonextensive systems based on the Tsallis generalization of the Boltzmann-Gibbs (BG) formalism is given in terms of quantum density matrix. After developing reasons for using the density matrix description as well as generalizations of the BG-formalism in the first section, we formulate the theory of quantum entangled states and information theory in the second section. In the third section, the maximum Tsallis entropy principle with given normalized q-mean value constraints is developed in detail, leading to quantum statistical mechanics of nonextensive systems. In the fourth section, time-dependent unitary dynamics is given. Here, the Green function theory as well as linear response theory are described in detail. A brief description of nonunitary (Lindblad) dynamics is outlined in the fifth section. In the final section, open problems and possible resolution are discussed as concluding remarks. In eight Tables, the summaries of the various sections are given with a view to give intercomparison of the present developments with the familiar ones found in the literature. In Appendix, several forms of the q-Kullback-Leibler entropy are given in seeking a guide to introduce invariance principles in the theory of nonextensive Tsallis entropy.

Keywords

Density Matrix Entangle State Quantum Information Theory Tsallis Entropy Mutual Entropy 
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.

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References

  1. 1.
    L. Boltzmann, Wien Ber. 53, 195 (1866).Google Scholar
  2. 2.
    J. W. Gibbs, Tr. Conn. Acad. 3, 108 (1876); see also his Scientific Papers Vols. 1 and 2 (Dover, New York, 1961).Google Scholar
  3. 3.
    J. von Neumann, Gott. Nach., 273 (1927); see also his book, Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, 1955).Google Scholar
  4. 4.
    C. E. Shannon, Bell Syst. Tech. L., 27, 379 (1946).MathSciNetGoogle Scholar
  5. 5.
    E. T. Jaynes, Phys. Rev. 106, 620 (1957); see also his collection of papers, E. T. Jaynes: Papers on Probability, Statistics and Statistical Physics, ed. R. D. Rosenkrantz (Kluwer, Dordrecht, 1989).CrossRefMathSciNetADSGoogle Scholar
  6. 6.
    See for example,M. Ohya and D. Pertz, Quantum Entropy and Its Use, (Springer-Verlag, Berlin, 1993).zbMATHGoogle Scholar
  7. 7.
    C. Tsallis, J. Stat. Phys. 52, 479 (1988). Since this paper first appeared, a large number of papers based on this work have been and continues to be published in a wide variety of topics. A list of this vast literature which is continually being updated and enlarged may be obtained from http://tsallis.cat.cbpf.br/biblio.htm.zbMATHCrossRefMathSciNetADSGoogle Scholar
  8. 8.
    C. Tsallis, R. S. Mendes, and A. R. Plastino, Physica A 261, 534 (1998).CrossRefGoogle Scholar
  9. 9.
    M. L. Lyra and C. Tsallis, Phys. Rev. Lett. 80, 53 (1998).CrossRefADSGoogle Scholar
  10. 10.
    C. Anteneodo and C. Tsallis, Phys. Rev. Lett. 80, 5313 (1998).CrossRefADSGoogle Scholar
  11. 11.
    A. K. Rajagopal, Phys. Rev. Lett. 76, 3469 (1996).zbMATHCrossRefMathSciNetADSGoogle Scholar
  12. 12.
    N. L. Balazs and T. Bergeman, Phys. Rev. A 58, 2359 (1998).CrossRefADSGoogle Scholar
  13. 13.
    W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1990), p. 323–328.zbMATHGoogle Scholar
  14. 14.
    E. T. Jaynes and F. W. Cummings, Proc. I. R. E., 51, 89 (1963). See also H. Paul, Ann. der Phys. 11, 411 (1963), who also reported an exact solution to the JC model but did not explore in full detail the ramifications of the solution.Google Scholar
  15. 15.
    See for example various articles in Physics and Probability—Essays in honor of Edwin T. Jaynes, eds. W. T. Grandy, Jr. and P. W. Milonni (Cambridge University Press, Cambridge, 1993).Google Scholar
  16. 16.
    N. J. Cerf and C. Adami, Phys. Rev. Lett. 79, 5194 (1997).zbMATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    N. J. Cerf and C. Adami, Physica D 120, 62 (1998).zbMATHCrossRefMathSciNetADSGoogle Scholar
  18. 18.
    A. K. Rajagopal, K. L. Jensen, and F. W. Cummings, Phys. Lett. A 259, 285 (1999).zbMATHCrossRefMathSciNetADSGoogle Scholar
  19. 19.
    C. Tsallis, Phys. Rev. E 58, 1442 (1998).CrossRefADSGoogle Scholar
  20. 20.
    L. Borland, A. R. Plastino, and C. Tsallis, J. Math. Phys. 39, 6490 (1998); (E) 40, 2196 (1999).zbMATHCrossRefMathSciNetADSGoogle Scholar
  21. 22.
    R. Karplus and J. Schwinger, Phys. Rev. 73, 1020 (1948).zbMATHCrossRefADSGoogle Scholar
  22. 23.
    A. K. Rajagopal, Braz. J. Phys. 29, 61 (1998).Google Scholar
  23. 24.
    R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957).CrossRefMathSciNetADSGoogle Scholar
  24. 25.
    A. K. Rajagopal, R. S. Mendes, and E. K. Lenzi, Phys. Rev. Lett. 80, 3907 (1998). See also an expanded version of this work by E. K. Lenzi, R. S. Mendes, and A. K. Rajagopal, Phys. Rev. E 59, 1398 (1999).CrossRefADSGoogle Scholar
  25. 26.
    L. P. Kadano. and G. Baym, Quantum Statistical Mechanics (Benjamin, New York, 1962).Google Scholar
  26. 27.
    E. K. Lenzi, R. S. Mendes, and A. K. Rajagopal, cond-mat/9904100. See also S. Abe, Eur. Phys. J. B 9, 679 (1999).Google Scholar
  27. 28.
    K. Huang, Statistical Mechanics (Wiley, New York, 1963).Google Scholar
  28. 29.
    S. Abe, Phys. Lett. A 263, 424 (1999); (E) 267, 456 (2000).zbMATHCrossRefMathSciNetADSGoogle Scholar
  29. 30.
    A. R. Plastino, A. Plastino, and C. Tsallis, J. Phys. A: Math. Gen. 27, 5707 (1994).zbMATHCrossRefMathSciNetADSGoogle Scholar
  30. 31.
    A. K. Rajagopal, Phys. Rev. A 54, 3916 (1996).CrossRefADSGoogle Scholar
  31. 32.
    A. K. Rajagopal, Phys. Lett. A 228, 66 (1997).zbMATHCrossRefMathSciNetADSGoogle Scholar
  32. 33.
    A. K. Rajagopal, Physica A 253, 271 (1998).CrossRefMathSciNetGoogle Scholar
  33. 34.
    A. K. Rajagopal, Phys. Lett. A 246, 237 (1998).CrossRefADSGoogle Scholar
  34. 35.
    C. Kittel, Elementary Statistical Physics (Wiley, New York, 1958), Chapter 1.Google Scholar
  35. 36.
    P. T. Landsberg and V. Vedral, Phys. Lett. A 247, 211 (1998).zbMATHCrossRefMathSciNetADSGoogle Scholar
  36. 37.
    R. J. V. dos Santos, J. Math. Phys. 38, 4104 (1997).zbMATHCrossRefMathSciNetADSGoogle Scholar
  37. 38.
    S. Abe and A. K. Rajagopal (private ruminations, 1998).Google Scholar
  38. 39.
    P. Zeische, O. Gunnarsson, W. John, and H. Beck, Phys. Rev. B 55, 10 270 (1997).Google Scholar
  39. 40.
    D. B. Ion and M. L. D. Ion, Phys. Rev. Lett. 81, 5714 (1998).CrossRefADSGoogle Scholar
  40. 41.
    A. K. Rajagopal, Phys. Lett. A 205, 32 (1995).zbMATHCrossRefMathSciNetADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • A.K. Rajagopal
    • 1
  1. 1.Naval Research LaboratoryWashington D.C.USA

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