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I. Nonextensive Statistical Mechanics and Thermodynamics: Historical Background and Present Status

  • C. Tsallis
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 560)

Abstract

The domain of validity of standard thermodynamics and Boltzmann-Gibbs statistical mechanics is focused on along a historical perspective. It is then formally enlarged in order to hopefully cover a variety of anomalous systems. The generalization concerns nonextensive systems, where nonextensivity is understood in the thermodynamical sense. This generalization was first proposed in 1988 inspired by the probabilistic description of multifractal geometry, and has been intensively studied during this decade. In the present effort, we describe the formalism, discuss the main ideas, and then exhibit the present status in what concerns theoretical, experimental and computational evidences and connections, as well as some perspectives for the future. The whole review can be considered as an attempt to clarify our current understanding of the foundations of statistical mechanics and its thermodynamical implications.

Keywords

Generalize Entropy Point Vortex Entropic Form Solar Neutrino Problem Liapunov Exponent 
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.
    C. Tsallis, J. Stat. Phys. 52, 479 (1988).zbMATHMathSciNetCrossRefADSGoogle Scholar
  2. 3.
    C. Tsallis, in Nonextensive Statistical Mechanics and Thermodynamics, eds. S. R. A. Salinas and C. Tsallis, Braz. J. Phys. 29, 1 (1999) [accessible at http://sbf.if.usp.br/WWWpages/Journals/BJP/Vol29/Num1/index.htm]. The present review is an extended and updated version of the one just quoted.
  3. 5.
    F. Takens, Structures in dynamics—Finite dimensional deterministic studies, eds. H. W. Broer, F. Dumortier, S. J. van Strien, and F. Takens (North-Holland, Amsterdam, 1991), p. 253. [In his words: “The values of pi are determined by the following dogma: if the energy of the system in the ith state is Ei and if the temperature of the system is T then: pi = exp/kT/Z(T), here Z(T) =_i exp/kT, (this last constant is taken so that _i pi = 1). This choice of pi is called Gibbs distribution. We shall give no justification for this dogma; even a physicist like Ruelle disposes of this question as “deep and incompletely clarified”.”]Google Scholar
  4. 6.
    N. Krylov, Nature 153, 709 (1944). [In his words: “In the present investigation, the notion of ergodicity is ignored. I reject the ergodical hypothesis completely: it is both insufficient and unnecessary for statistics. I use, as starting point, the notion of motions of the mixing type, and show that the essential mechanical condition for the applicability of statistics consists in the requirement that in the phase space of the system all the regions with a sufficiently large size should vary in the course of time in such a way that while their volume remains constant—according to Liouville’s theorem—their parts should be distributed over the whole phase space (more exactly over the layer, corresponding to given values of the single-valued integrals of the motion) with a steadily increasing degree of uniformity. (...) The main condition of mixing, which ensures the fulfillment of this condition, is a sufficiently rapid divergence of the geodetic lines of this Riemann space (that is, of the paths of the system in the n-dimensional configuration space), namely, an exponential divergence (cf. Nopf1).”]. For full details on this pioneering approach see N.S. Krylov, Works on the Foundations of Statistical Physics, translated by A. B. Migdal, Ya. G. Sinai, and Yu. L. Zeeman, Princeton Series in Physics (Princeton University Press, Princeton, 1979).zbMATHMathSciNetCrossRefADSGoogle Scholar
  5. 7.
    R. Balescu, Equilibrium and Non-equilibrium Statistical Mechanics (Wiley, New York, 1975), p. 727. [In his words: “It therefore appears from the present discussion that the mixing property of a mechanical system is much more important for the understanding of statistical mechanics than the mere ergodicity. (...) A detailed rigorous study of the way in which the concepts of mixing and the concept of large numbers of degrees of freedom in.uence the macroscopic laws of motion is still lacking.”]Google Scholar
  6. 8.
    M. Courbage and D. Hamdan, Phys. Rev. Lett. 74, 5166 (1995).CrossRefADSGoogle Scholar
  7. 9.
    J.R. Dorfman, An Introduction to Chaos in Nonequilibrium Statistical Mechanics, Cambridge Lecture Notes in Physics 14, eds. P. Goddard and J. Yeomans (Cambridge University Press, Cambridge, 1999), footnote on p. 9. [In his words: “It is worth mentioning that there are examples of mixing systems with no non-zero Liapunov exponents. The concepts of ergodicity, mixing, and chaos can be quite subtle.”]Google Scholar
  8. 10.
    N. G. van Kampen, Braz. J. Phys. 28, 90 (1998) [accessible at http://www.sbf.if.usp.br/WWWpages/Journals/BJP/Vol28/Num2/index.htm].CrossRefADSGoogle Scholar
  9. 11.
    G. M. Zaslavsky, Physics Today 52, 39 (August 1999).Google Scholar
  10. 13.
    L. Tisza, Ann. Phys. 13, 1 (1961). [Or, in Generalized Thermodynamics (MIT Press, Cambridge, 1966), p. 123]. [In his words: “The situation is different for the additivity postulate P a2, the validity of which cannot be inferred from general principles.We have to require that the interaction energy between thermodynamic systems be negligible. This assumption is closely related to the homogeneity postulate P d1. From the molecular point of view, additivity and homogeneity can be expected to be reasonable approximations for systems containing many particles, provided that the intramolecular forces have a short range character.”]zbMATHMathSciNetCrossRefADSGoogle Scholar
  11. 14.
    H. Grad, in Rarefied Gas Dynamics, ed. J. A. Laurmann (Academic Press, New York, 1963), p. 26. [A variety of anomalous effects on the spectrum of the collision operator of the Boltzmann equation, due to the range of the forces, are discussed in this article.]Google Scholar
  12. 15.
    M. E. Fisher, Arch. Rat. Mech. Anal. 17, 377 (1964); J. Chem. Phys. 42, 3852 (1965); J. Math. Phys. 6, 1643 (1965); M. E. Fisher and D. Ruelle, J. Math. Phys. 7, 260 (1966); M. E. Fisher and J. L. Lebowitz, Commun. Math. Phys. 19, 251 (1970).CrossRefGoogle Scholar
  13. 16.
    P.T. Landsberg, Thermodynamics and Statistical Mechanics (Oxford University Press, Oxford, 1978; also Dover, New York, 1990), p. 102. [In his words: “The presence of long-range forces causes important amendments to thermodynamics, some of which are not fully investigated as yet.”]Google Scholar
  14. 17.
    W. Thirring, Foundations of Physics 20, 1103 (1990).MathSciNetCrossRefADSGoogle Scholar
  15. 18.
    N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981), footnote on p. 114. [In his words: “Actually an additional stability criterion is needed, see M. E. Fisher, Archives Rat. Mech. Anal. 17, 377 (1964); D. Ruelle, Statistical Mechanics, Rigorous Results (Benjamin, New York 1969). A collection of point particles with mutual gravitation is an example where this criterion is not satisfied, and for which therefore no statistical mechanics exists.”]zbMATHGoogle Scholar
  16. 19.
    P. T. Landsberg, J. Stat. Phys. 35, 159 (1984); L. G. Ta., Celestial Mechanics (John Wiley and Sons, New York, 1985), p. 437; W. C. Saslaw, Gravitational Physics of Stellar and Galactic Systems (Cambridge University Press, Cambridge, 1985), p. 217; J. Binney and S. Tremaine, Galactic Dynamics (Princeton University Press, Princeton, 1987), p. 267; D. Pavon, Gen. Rel. and Gravit. 19, 375 (1987); H. E. Kandrup, Phys. Rev. A 40, 7265 (1989); H. S. Robertson, Statistical Thermophysics (Prentice Hall, Englewood Cli.s, New Jersey, 1993), p. 96.MathSciNetCrossRefADSGoogle Scholar
  17. 20.
    B. J. Hiley and G. S. Joice, Proc. Phys. Soc. 85, 493 (1965).MathSciNetCrossRefADSGoogle Scholar
  18. 21.
    G. Kotliar, P. W. Anderson, and D. L. Stein, Phys. Rev. B 27, 602 (1983); see also K. Binder and A. P. Young, Rev. Mod. Phys. 58, 801 (1986), p. 924. [The system which is analyzed is a d = 1 Ising spin-glass whose random coupling constants follow a centered Gaussian distribution and decay with distance as 1/rá. It can be seen that the size scaling is still given by ~ N = N + 1, to be introduced later on, but with á replaced by 2á; in other words, the system is extensive if and only if 2α > d.]MathSciNetCrossRefADSGoogle Scholar
  19. 22.
    H. Risken, The Fokker-Planck Equation (Springer-Verlag, Berlin, 1984), p. 9.zbMATHGoogle Scholar
  20. 23.
    M. O. Caceres, in Nonextensive Statistical Mechanics and Thermodynamics, eds. S. R. A. Salinas and C. Tsallis, Braz. J. Phys. 29, 124 (1999) [accessible at http://sbf.if.usp.br/WWWpages/Journals/BJP/Vol29/Num1/index.htm].
  21. 24.
    X.-P. Huang and C. F. Driscoll, Phys. Rev. Lett. 72, 2187 (1994).CrossRefADSGoogle Scholar
  22. 25.
    E. M. Montroll and B. J. West, in Fluctuation Phenomena, eds. E. W. Montroll and J. L. Lebowitz (North-Holland, Amsterdam, 1979) [2nd edition: North-Holland Personal Library (1987)]; E. Montroll and M. F. Shlesinger, J. Stat. Phys. 32, 209 (1983); Lévy Flights and Related Topics in Physics, eds. M. F. Shlesinger, G. M. Zaslavsky, and U. Frisch (Springer-Verlag, Berlin, 1995); P. Allegrini, P. Grigolini, and B. J. West, Phys. Rev. E 54, 4760 (1996); B. J. West and P. Grigolini, Phys. Rev. E 55, 99 (1997); P. Grigolini, A. Rocco, and B. J. West, Phys. Rev. E 59, 2603 (1999); B. J. West, Physiology, Promiscuity and Prophecy at the Millenium: A Tale of Tails (World Scientific, Singapore, 1999).Google Scholar
  23. 26.
    Y.-H. Taguchi and H. Takayasu, Europhys. Lett. 30, 499 (1995).CrossRefADSGoogle Scholar
  24. 27.
    K. T. Waldeer and H. M. Urbassek, Physica A 176, 325 (1991).CrossRefADSGoogle Scholar
  25. 28.
    I. Koponen, Phys. Rev. E 55, 7759 (1997).CrossRefADSGoogle Scholar
  26. 29.
    D. C. Clayton, Nature 249, 131 (1974).CrossRefADSGoogle Scholar
  27. 30.
    N. A. Bahcall and S. P. Oh, Astrophys. J. 462, L49 (1996).CrossRefADSGoogle Scholar
  28. 31.
    J. M. Liu, J. S. De Groot, J. P. Matte, T. W. Johnston, and R. P. Drake, Phys. Rev. Lett. 72, 2717 (1994).CrossRefADSGoogle Scholar
  29. 32.
    J. Maddox, Nature 365, 103 (1993).CrossRefADSGoogle Scholar
  30. 33.
    H. P. de Oliveira, S. L. Sautu, I. D. Soares, and E. V. Tonini, Phys. Rev. D 60, 121301 (1999).CrossRefADSGoogle Scholar
  31. 34.
    G. Wilk and Z. Wlodarcsyk, Nucl. Phys. B (Proc. Suppl.) 75A, 191 (1999); G. Wilk and Z. Wlodarcsyk, Phys. Rev. D 50, 2318 (1994); G. Wilk and Z. Wlodarcsyk, Phys. Rev. Lett. 84, 2770 (2000); M.L.D. Ion and D.B. Ion, Phys. Lett. B 482, 57 (2000).CrossRefADSGoogle Scholar
  32. 35.
    I. Bediaga, E. M. F. Curado, and J. Miranda, Physica A 286, 156 (2000); C. Beck, Physica A 286, 164 (2000).CrossRefADSGoogle Scholar
  33. 36.
    O. V. Utyuzh, G. Wilk, and Z. Wlodarcsyk, preprint (1999) [hep-ph/9906442].Google Scholar
  34. 37.
    O. V. Utyuzh, G. Wilk, and Z. Wlodarcsyk, preprint (1999) [hep-ph/9906500].Google Scholar
  35. 38.
    T. Alber et al., Eur. Phys. J. C 2, 643 (1998).CrossRefADSGoogle Scholar
  36. 39.
    D. B. Walton and J. Rafelski, Phys. Rev. Lett. 84, 31 (2000).CrossRefADSGoogle Scholar
  37. 40.
    S. Abe and A. K. Rajagopal, Phys. Rev. A 60, 3461 (1999); A. Vidiella-Barranco, Phys. Lett. A 260, 335 (1999).CrossRefADSGoogle Scholar
  38. 41.
    E. M. F. Curado and C. Tsallis, J. Phys. A 24, L69 (1991); (Corrigenda) 24, 3187 (1991) and 25, 1019 (1992).MathSciNetCrossRefADSGoogle Scholar
  39. 42.
    C. Tsallis, R. S. Mendes, and A. R. Plastino, Physica A 261, 534 (1998).CrossRefGoogle Scholar
  40. 43.
    A. R. Plastino and A. Plastino, Phys. Lett. A 177, 177 (1993).MathSciNetCrossRefADSGoogle Scholar
  41. 44.
    J. L. Lebowitz, Physica A 194, 1 (1993); Physics Today 46, 32 (1993); Physica A 263, 516 (1999), and references therein.MathSciNetCrossRefADSGoogle Scholar
  42. 45.
    J. Harvda and F. Charvat, Kybernetica 3, 30 (1967).Google Scholar
  43. 46.
    I. Vajda, Kybernetika 4, 105 (1968) [in Czech].MathSciNetzbMATHGoogle Scholar
  44. 47.
    Z. Daroczy, Inf. and Control 16, 36 (1970).zbMATHMathSciNetCrossRefGoogle Scholar
  45. 48.
    A. Wehrl, Rev. Mod. Phys. 50, 221 (1978); I. J. Taneja, Advances in Electronics and Electron Physics 76, 327 (1989); M. Behara, Additive and Nonadditive Measures of Entropy (Wiley Eastern, New Delhi, 1990); M. Basseville, Institut de Recherche en Informatique et Systemes Aleatoires-IRISA (France), Report 1020 (May 1996).MathSciNetCrossRefADSGoogle Scholar
  46. 49.
    C. Tsallis, Chaos, Solitons and Fractals 6, 539 (1995).zbMATHMathSciNetCrossRefADSGoogle Scholar
  47. 50.
    C. Anteneodo and A. R. Plastino, J. Phys. A 32, 1089 (1999).zbMATHMathSciNetCrossRefADSGoogle Scholar
  48. 51.
    A. R. R. Papa, J. Phys. A 31, 5271 (1998).zbMATHMathSciNetCrossRefADSGoogle Scholar
  49. 52.
    E. P. Borges and I. Roditi, Phys. Lett. A 246, 399 (1998).zbMATHMathSciNetCrossRefADSGoogle Scholar
  50. 53.
    P. T. Landsberg and V. Vedral, Phys. Lett. A 247, 211 (1998).zbMATHMathSciNetCrossRefADSGoogle Scholar
  51. 54.
    P. T. Landsberg, in Nonextensive Statistical Mechanics and Thermodynamics, eds. S. R. A. Salinas and C. Tsallis, Braz. J. Phys. 29, 46 (1999) [accessible at http://sbf.if.usp.br/WWWpages/Journals/BJP/Vol29/Num1/index.htm].
  52. 55.
    E. M. F. Curado, in Nonextensive Statistical Mechanics and Thermodynamics, eds. S. R. A. Salinas and C. Tsallis, Braz. J. Phys. 29, 36 (1999) [accessible at http://sbf.if.usp.br/WWWpages/Journals/BJP/Vol29/Num1/index.htm].
  53. 56.
    R. P. Di Sisto, S. Martinez, A. R. Plastino, and A. Plastino, Physica A 265, 590 (1999).CrossRefGoogle Scholar
  54. 57.
    R. S. Johal, Phys. Rev. E 58, 4147 (1998).CrossRefADSGoogle Scholar
  55. 58.
    A. K. Rajagopal and S. Abe, Phys. Rev. Lett. 83, 1711 (1999).zbMATHMathSciNetCrossRefADSGoogle Scholar
  56. 59.
    R. S. Johal and R. Rai, Physica A 282, 525 (2000).CrossRefADSGoogle Scholar
  57. 60.
    R. Rossignoli and N. Canosa, Phys. Lett. A 264, 148 (1999).zbMATHMathSciNetCrossRefADSGoogle Scholar
  58. 61.
    A. Renyi, in Proc. Fourth Berkeley Symposium, 1960, Vol. 1 (University of California Press, Berkeley, Los Angeles, 1961), 547; Probability Theory (North-Holland, 1970) and references therein.Google Scholar
  59. 62.
    I. Csiszar, Information measures: A critical survey, in Transactions of the Seventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes and the European Meeting of Statisticians, 1974 (Reidel, Dordrecht, 1978), p. 73.Google Scholar
  60. 63.
    M. P. Schutzenberger, Contribution aux applications statistiques de la theorie de l’information, Publ. Inst. Statist. Univ. Paris 3, 3 (1954).MathSciNetGoogle Scholar
  61. 64.
    M. Hotta and I. Joichi, Phys. Lett. A 262, 302 (1999).zbMATHMathSciNetCrossRefADSGoogle Scholar
  62. 65.
    R. J. V. dos Santos, J. Math. Phys. 38, 4104 (1997); see also S. Abe, Phys. Lett. A 271, 74 (2000).zbMATHMathSciNetCrossRefADSGoogle Scholar
  63. 66.
    S. Abe, Phys. Lett. A 224, 326 (1997).MathSciNetCrossRefADSGoogle Scholar
  64. 67.
    F. Jackson, Mess. Math. 38, 57 (1909); Quart. J. Pure Appl. Math. 41, 193 (1910).Google Scholar
  65. 69.
    C. Tsallis, Quimica Nova 17, 468 (1994).Google Scholar
  66. 70.
    E. P. Borges, J. Phys. A 31, 5281 (1998).zbMATHMathSciNetCrossRefADSGoogle Scholar
  67. 71.
    P. Grassberger and I. Procaccia, Phys. Rev. Lett. 50, 346 (1983) and Phys. Rev. A 28, 2591 (1983); T. A. Halsey et al., Phys. Rev. A 33, 1141 (1986).MathSciNetCrossRefADSGoogle Scholar
  68. 72.
    S. Watanabe, Knowing and Guessing (Wiley, New York, 1969).zbMATHGoogle Scholar
  69. 73.
    H. Barlow, Vision. Res. 30, 1561 (1990); see also G. Toulouse, J. Phys. I (France) 3, 229 (1993).CrossRefGoogle Scholar
  70. 74.
    A. Plastino and A. R. Plastino, in Nonextensive Statistical Mechanics and Thermodynamics, eds. S. R. A. Salinas and C. Tsallis, Braz. J. Phys. 29, 50 (1999) [accessible at http://sbf.if.usp.br/WWWpages/Journals/BJP/Vol29/Num1/index.htm].
  71. 75.
    L. R. da Silva, E. K. Lenzi, J. S. Andrade, and J. Mendes Filho, Physica A 275, 396 (2000).CrossRefGoogle Scholar
  72. 76.
    S. Abe, Phys. Lett. A 263, 424 (1999); (Erratum) 267, 456 (2000).zbMATHMathSciNetCrossRefGoogle Scholar
  73. 77.
    A. R. Lima and T. J. P. Penna, Phys. Lett. A 256, 221 (1999).CrossRefGoogle Scholar
  74. 78.
    F. D. Nobre and C. Tsallis, Physica A 213, 337 (1995) [Erratum: 216, 369 (1995)]; F. D. Nobre and C. Tsallis, Phil. Mag. B 73, 545 (1996).MathSciNetCrossRefGoogle Scholar
  75. 79.
    C. Beck and F. Schlögl, Thermodynamics of Chaotic Systems (Cambridge University Press, Cambridge, 1993).Google Scholar
  76. 80.
    C. Tsallis and S. Abe, Physics Today 51, 114 (October, 1998).Google Scholar
  77. 81.
    M. C. S. Vieira and C. Tsallis, J. Stat. Phys. 48, 97 (1987).CrossRefADSGoogle Scholar
  78. 82.
    G. R. Guerbero. and G. A. Raggio, J. Math. Phys. 37, 1776 (1996); G. R. Guerberoff, P. A. Pury, and G. A. Raggio, J. Math. Phys. 37, 1790 (1996).MathSciNetCrossRefADSGoogle Scholar
  79. 83.
    A. Chame, Physica A 255, 423 (1998).CrossRefGoogle Scholar
  80. 84.
    S. Abe, Physica A 269, 403 (1999).Google Scholar
  81. 85.
    R. F. S. Andrade, Physica A 203, 486 (1994).CrossRefGoogle Scholar
  82. 86.
    A. K. Rajagopal, in the present volume.Google Scholar
  83. 88.
    K. Sasaki and M. Hotta, preprint (1999) [cond-mat/9912454].Google Scholar
  84. 89.
    V. Latora, A. Rapisarda, and S. Ruffo, Phys. Rev. Lett. 83, 2104 (1999).CrossRefADSGoogle Scholar
  85. 90.
    D. Prato and C. Tsallis, Phys. Rev. E 60, 2398 (1999).ADSGoogle Scholar
  86. 91.
    A. M. Mariz, Phys. Lett. A 165, 409 (1992); J. D. Ramshaw Phys. Lett. A 175, 169 and 171 (1993).MathSciNetCrossRefGoogle Scholar
  87. 92.
    L. Borland, A. R. Plastino, and C. Tsallis, J. Math. Phys. 39, 6490 (1998); (Erratum) 40, 2196 (1999).zbMATHMathSciNetCrossRefADSGoogle Scholar
  88. 93.
    M. O. Caceres, Phys. Lett. A 218, 471 (1995); A. Chame and E. V. L. de Mello, Phys. Lett. A 228, 159 (1997).MathSciNetGoogle Scholar
  89. 94.
    A. K. Rajagopal, Phys. Rev. Lett. 76, 3469 (1996).zbMATHMathSciNetCrossRefADSGoogle Scholar
  90. 95.
    E. K. Lenzi, L. C. Malacarne, and R. S. Mendes, Phys. Rev. Lett. 80, 218 (1998).CrossRefADSGoogle Scholar
  91. 96.
    C. Tsallis, Phys. Lett. A 206, 389 (1995).zbMATHMathSciNetCrossRefADSGoogle Scholar
  92. 97.
    A. R. Plastino, A. Plastino, and C. Tsallis, J. Phys. A 27, 5707 (1994).zbMATHMathSciNetCrossRefADSGoogle Scholar
  93. 98.
    T. Yamano, Phys. Lett. A 264, 276 (1999).zbMATHMathSciNetCrossRefADSGoogle Scholar
  94. 99.
    C. Tsallis, A. R. Plastino, and W.-M. Zheng, Chaos, Solitons and Fractals 8, 885 (1997).zbMATHMathSciNetCrossRefADSGoogle Scholar
  95. 100.
    E. P. da Silva, C. Tsallis, and E. M. F. Curado, Physica A 199, 137 (1993); (Erratum) 203, 160 (1994).MathSciNetCrossRefADSGoogle Scholar
  96. 101.
    C. Tsallis, in New Trends in Magnetism, Magnetic Materials and Their Applications, eds. J. L. Moran-Lopez and J. M. Sanchez (Plenum Press, New York, 1994); A. Chame and E. V. L. de Mello, J. Phys. A 27, 3663 (1994).Google Scholar
  97. 102.
    B. M. Boghosian, in Nonextensive Statistical Mechanics and Thermodynamics, eds. S. R. A. Salinas and C. Tsallis, Braz. J. Phys. 29, 90 (1999) [accessible at http://sbf.if.usp.br/WWWpages/Journals/BJP/Vol29/Num1/index.htm].
  98. 103.
    A. R. Plastino and A. Plastino, Phys. Lett. A 174, 384 (1993); J. J. Aly, Minimum energy / maximum entropy states ofself-gravitating systems, in N-body problems and gravitational dynamics, oceedings of the Meeting held at Aussois-France (21–25 March 1993), eds. F. Combes and E. Athanassoula (Publications de l’Observatoire de Paris, Paris, 1993), page 19; A. R. Plastino and A. Plastino, Phys. Lett. A 193, 251 (1994); A. R. Plastino and A. Plastino, in Nonextensive Statistical Mechanics and Thermodynamics, eds. S. R. A. Salinas and C. Tsallis, Braz. J. Phys. 29, 1 (1999) [accessible at http://sbf.if.usp.br/WWWpages/Journals/BJP/Vol29/Num1/index.htm].Google Scholar
  99. 104.
    B. M. Boghosian, Phys. Rev. E 53, 4754 (1996).CrossRefADSGoogle Scholar
  100. 105.
    D. A. Stariolo, Phys. Lett. A 185, 262 (1994); L. Borland, Phys. Lett. A 245, 67 (1998).zbMATHMathSciNetCrossRefADSGoogle Scholar
  101. 106.
    A. R. Plastino and A. Plastino, Phys. Lett. A 222, 347 (1995).MathSciNetGoogle Scholar
  102. 107.
    C. Tsallis and D. J. Bukman, Phys. Rev. E 54, R2197 (1996).CrossRefADSGoogle Scholar
  103. 108.
    A. Compte and D. Jou, J. Phys. A 29, 4321 (1996); D. A. Stariolo, Phys. Rev. E 55, 4806 (1997); L. Borland, Phys. Rev. E 57, 6634 (1998); L. Borland, Phys. Lett. A 245, 67 (1998).zbMATHMathSciNetCrossRefADSGoogle Scholar
  104. 109.
    A. K. Rajagopal, Physica A 253, 271 (1998).MathSciNetCrossRefGoogle Scholar
  105. 110.
    H. S. Wio and S. Bouzat, in Nonextensive Statistical Mechanics and Thermodynamics, eds. S. R. A. Salinas and C. Tsallis, Braz. J. Phys. 29, 135 (1999) [accessible at http://sbf.if.usp.br/WWWpages/Journals/BJP/Vol29/Num1/index.htm].
  106. 111.
    C. Tsallis, Phys. Rev. E 58, 1442 (1998).CrossRefADSGoogle Scholar
  107. 112.
    A. K. Rajagopal and C. Tsallis, Phys. Lett. A 257, 283 (1999).zbMATHMathSciNetCrossRefADSGoogle Scholar
  108. 113.
    E. K. Lenzi, L. C. Malacarne, and R. S. Mendes, Phys. Rev. Lett. 80, 218 (1998).CrossRefADSGoogle Scholar
  109. 114.
    A. K. Rajagopal, R. S. Mendes and E. K. Lenzi, Phys. Rev. Lett. 80, 3907 (1998); E. K. Lenzi, R. S. Mendes and A. K. Rajagopal, Phys. Rev. E 59, 1398 (1999); R. S. Mendes, Braz. J. Phys. 29, 66 (1999).CrossRefADSGoogle Scholar
  110. 115.
    R. A. Treumann, Phys. Rev. E 57, 5150 (1998); E. K. Lenzi, L. C. Malacarne, and R. S. Mendes, preprint (1998); J. E. Straub and T. Whitfield, preprint (1998).CrossRefADSGoogle Scholar
  111. 116.
    A. K. Rajagopal, in Nonextensive Statistical Mechanics and Thermodynamics, eds. S. R. A. Salinas and C. Tsallis, Braz. J. Phys. 29, 61 (1999) [accessible at http://sbf.if.usp.br/WWWpages/Journals/BJP/Vol29/Num1/index.htm].
  112. 117.
    A. R. Plastino and C. Anteneodo, Ann. Phys. 255, 250 (1997).zbMATHMathSciNetCrossRefADSGoogle Scholar
  113. 118.
    C. Tsallis and D. A. Stariolo, Notas de Fisica/CBPF (Brazil) 026 (June 1994) and Physica A 233, 395 (1996); D.A. Stariolo and C. Tsallis, Ann. Rev. Comput. Phys., Vol. II, ed. D. Stauffer (World Scientific, Singapore, 1995)), page 343.Google Scholar
  114. 119.
    T. J. P. Penna, Phys. Rev. E 51, R1 (1995).CrossRefADSGoogle Scholar
  115. 120.
    I. Andricioaei and J. E. Straub, Pys. Rev. E 53, R3055 (1996); Physica A 247, 553 (1997); J. E. Straub and I. Andricioaei, in Nonextensive Statistical Mechanics and Thermodynamics, eds. S. R. A. Salinas and C. Tsallis, Braz. J. Phys. 29, 179 (1999) [accessible at http://sbf.if.usp.br/WWWpages/Journals/BJP/Vol29/Num1/index.htm].CrossRefADSGoogle Scholar
  116. 121.
    J. Schulte, Phys. Rev. E 53, 1348 (1996).CrossRefADSGoogle Scholar
  117. 122.
    U. H. E. Hansmann, Physica A 242, 250 (1997); Chem. Phys. Lett. 281, 140 (1997); U. H. E. Hansmann and Y. Okamoto, Phys. Rev. E 56, 2228 (1997); U. H. E. Hansmann, M. Masuya, and Y. Okamoto, Proc. Natl. Acad. Sci. USA 94, 10652 (1997); U. H. E. Hansmann, F. Eisenmenger, and Y. Okamoto, Chem. Phys. Lett. 297, 374 (1998); U. H. E. Hansmann and Y. Okamoto, in Nonextensive Statistical Mechanics and Thermodynamics, eds. S. R. A. Salinas and C. Tsallis, Braz. J. Phys. 29, 187 (1999) [accessible at http://sbf.if.usp.br/WWWpages/Journals/BJP/Vol29/Num1/index.htm].CrossRefADSGoogle Scholar
  118. 123.
    P. Serra, A. F. Stanton, S. Kais, and R. E. Bleil, J. Chem. Phys. 106, 7170 (1997); P. Serra and S. Kais, Chem. Phys. Lett. 275, 211 (1997).CrossRefADSGoogle Scholar
  119. 124.
    Y. Xiang, D. Y. Sun, W. Fan, and X. G. Gong, Phys. Lett. A 233, 216 (1997).CrossRefADSGoogle Scholar
  120. 125.
    M. R. Lemes, C. R. Zacharias, and A. Dal Pino Jr., Phys. Rev. B 56, 9279 (1997).CrossRefADSGoogle Scholar
  121. 126.
    D. E. Ellis, K. Mundim, V. P. Dravid, and J. W. Rylander, Hybrid classical and quantum modeling of defects, interfaces and surfaces, in Computer Aided-Design ofHigh-T emperature Materials (Oxford University Press, Oxford, 1999), page 350; M. A. Moret, P. M. Bisch, and F. M. C. Vieira, Phys. Rev. E 57, R2535 (1998); K. C. Mundim, T. Lemaire, and A. Bassrei, Physica A 252, 405 (1998); M. A. Moret, P. G. Pascutti, P. M. Bisch, and K. C. Mundim, J. Comput. Chem. 19, 647 (1998); L. Guo, D. E. Ellis, and K. C. Mundim, J. Porphyrins and Phthalocyanines 3, 196 (1999); K. C. Mundim and D. E. Ellis, in Nonextensive Statistical Mechanics and Thermodynamics, eds. S. R. A. Salinas and C. Tsallis, Braz. J. Phys. 29, 197 (1999) [accessible at http://sbf.if.usp.br/WWWpages/Journals/BJP/Vol29/Num1/index.htm].Google Scholar
  122. 127.
    K. C. Mundim and C. Tsallis, Int. J. Quantum Chem. 58, 373 (1996).CrossRefGoogle Scholar
  123. 128.
    H. Nishimori and J. Inoue, J. Phys. A 31, 5661 (1998).zbMATHMathSciNetCrossRefADSGoogle Scholar
  124. 129.
    R. Salazar and R. Toral, Phys. Rev. Lett. 83, 4233 (1999); Computer Physics Communications 121/122, 40 (1999); A Monte Carlo method for the numerical simulation of Tsallis statistics, preprint (1999) [cond-mat/9911063]; A. R. Lima, J. S. Sa Martins, and T. J. P. Penna, Physica A 268, 553 (1999).CrossRefADSGoogle Scholar
  125. 130.
    A. R. Plastino, H. G. Miller, A. Plastino, and G. D. Yen, J. Math. Phys. 38, 6675 (1997).zbMATHMathSciNetCrossRefADSGoogle Scholar
  126. 131.
    S. Abe and A. K. Rajagopal, Physica A (2001), in press.Google Scholar
  127. 132.
    A. K. Rajagopal, Phys. Lett. A 205, 32 (1995).zbMATHMathSciNetCrossRefADSGoogle Scholar
  128. 133.
    M. Portesi and A. Plastino, Physica A 225, 412 (1996).MathSciNetCrossRefADSGoogle Scholar
  129. 134.
    D. B. Ion and M. L. D. Ion, Phys. Rev. Lett. 81, 5714 (1998); Phys. Rev. Lett. 83, 463 (1999); Phys. Rev. E 60, 5261 (1999).CrossRefADSGoogle Scholar
  130. 135.
    F. Buyukkilic and D. Demirhan, Phys. Lett. A 181, 24 (1993); F. Buyukkilic, D. Demirhan, and A. Gulec, Phys. Lett. A 197, 209 (1995); F. Pennini, A. Plastino, and A.R. Plastino, Phys. Lett. A 208, 309 (1995); F. Pennini, A.R. Plastino, and A. Plastino, Physica A 235, 388 (1997); U. Tirnakli, F. Buyukkilic, and D. Demirhan, Phys. Lett. A 245, 62 (1998).MathSciNetCrossRefADSGoogle Scholar
  131. 136.
    A. K. Rajagopal, Physica B 212, 309 (1995).CrossRefADSGoogle Scholar
  132. 137.
    A. K. Rajagopal, Phys. Lett. A 214, 127 (1996).zbMATHMathSciNetCrossRefADSGoogle Scholar
  133. 138.
    E. K. Lenzi, R. S. Mendes, and A. K. Rajagopal, Physica A 286, 503 (2000).zbMATHMathSciNetCrossRefADSGoogle Scholar
  134. 139.
    A. Ott, J. P. Bouchaud, D. Langevin, and W. Urbach, Phys. Rev. Lett. 65, 2201 (1990); J. P. Bouchaud, A. Ott, D. Langevin, and W. Urbach, J. Phys. II (France) 1, 1465 (1991).CrossRefADSGoogle Scholar
  135. 140.
    F. Bardou, J. P. Bouchaud, O. Emile, A. Aspect, and C. Cohen-Tannoudji, Phys. Rev. Lett. 72, 203 (1994).CrossRefADSGoogle Scholar
  136. 141.
    T. H. Solomon, E. R. Weeks, and H. L. Swinney, Phys. Rev. Lett. 71, 3975 (1993).CrossRefADSGoogle Scholar
  137. 142.
    G. M. Viswanathan, V. Afanasyev, S. V. Buldyrev, E. J. Murphy, P. A. Prince, and H. E. Stanley, Nature 381, 413 (1996).CrossRefADSGoogle Scholar
  138. 143.
    C.-K. Peng, J. Mietus, J. M. Hausdorff, S. Havlin, H. E. Stanley, and A. L. Goldberger, Phys. Rev. Lett. 70, 1343 (1993).CrossRefADSGoogle Scholar
  139. 144.
    R. A. Antonia, N. Phan-Thien, and B. R. Satyoparakash, Phys. Fluids 24, 554 (1981).CrossRefADSGoogle Scholar
  140. 145.
    C.-K. Peng, S. V. Buldyrev, A. L. Goldberger, S. Havlin, F. Scirotino, M. Simons, and H. E. Stanley, Nature 356, 168 (1992); H. E. Stanley, S. V. Buldyrev, A. L. Goldberger, S. Havlin, R. N. Mantegna, C.-K. Peng, M. Simons, and M. H. R. Stanley, in Lévy Flights and Related Topics, eds. M. F. Shlesinger, G. M. Zaslavsky, and U. Frisch (Springer, Berlin, 1995), p. 331.CrossRefADSGoogle Scholar
  141. 146.
    B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1983).Google Scholar
  142. 147.
    B. B. Mandelbrot, Fractals and Scaling in Finance: Discontinuity, Concentration, Risk (Springer-Verlag, New York, 1997), and references therein.zbMATHGoogle Scholar
  143. 148.
    R. N. Mantegna, and H. E. Stanley, Nature 376, 46 (1995).CrossRefADSGoogle Scholar
  144. 149.
    P. A. Alemany and D. H. Zanette, Phys. Rev E 49, R956 (1994).CrossRefADSGoogle Scholar
  145. 150.
    D. H. Zanette and P. A. Alemany, Phys. Rev. Lett. 75, 366 (1995); 77, 2590 (1996); M. O. Caceres and C. E. Budde, Phys. Rev. Lett. 77, 2589 (1996); D. H. Zanette, Braz. J. Phys. 29, 107 (1999) [accessible at http://sbf.if.usp.br/WWWpages/Journals/BJP/Vol29/Num1/index.htm]; A. Robledo, Phys. Rev. Lett. 83, 2289 (1999).CrossRefADSGoogle Scholar
  146. 151.
    C. Tsallis, S. V. F. Levy, A. M. C. de Souza, and R. Maynard, Phys. Rev. Lett. 77, 5422 (1996) [Erratum 77, 5442 (1996)]; C. Tsallis, Physics World 10, 42 (July 1997).CrossRefADSGoogle Scholar
  147. 152.
    V. V. Uchaikin and V. M. Zolotarev, Chance and Stability—Stable Distributions and Their Applications, Series Modern Probability and Statistics (VSP, The Netherlands, 1999).Google Scholar
  148. 153.
    E. W. Montroll and B. J. West, in Studies in Statistical Mechanics, Vol. VII, eds. E. W. Montroll and J. L. Lebowitz (North-Holland, Amsterdam, 1979), p. 62.Google Scholar
  149. 154.
    A. S. Chaves, Phys. Lett. A 239, 13 (1998); M. P. Almeida, Phys. Lett. A 249, 560 (1998).zbMATHMathSciNetCrossRefADSGoogle Scholar
  150. 155.
    G. Drazer, H. S. Wio and C. Tsallis, Phys. Rev. E 61, 1417 (2000).CrossRefADSGoogle Scholar
  151. 156.
    A. R. Plastino, M. Casas, and A. Plastino, Physica A (2000), in press.Google Scholar
  152. 157.
    P. Quarati, A. Carbone, G. Gervino, G. Kaniadakis, A. Lavagno, and E. Miraldi, Nucl. Phys. A 621, 345c (1997); G. Kaniadakis, A. Lavagno, M. Lissia, and P. Quarati, Physica A 261, 359 (1998); M. Coraddu, G. Kaniadakis, A. Lavagno, M. Lissia, G. Mezzorani, and P. Quarati, in Nonextensive Statistical Mechanics and Thermodynamics, eds. S. R. A. Salinas and C. Tsallis, Braz. J. Phys. 29, 153 (1999) [accessible at http://sbf.if.usp.br/WWWpages/Journals/BJP/Vol29/Num1/index.htm].CrossRefADSGoogle Scholar
  153. 158.
    V. H. Hamity and D. E. Barraco, Phys. Rev. Lett. 76, 4664 (1996); D. F. Torres, H. Vucetich, and A. Plastino, Phys. Rev. Lett. 79, 1588 (1997).CrossRefADSGoogle Scholar
  154. 159.
    G. K. Zipf, Human Behavior and the Principle of Least Effort (Addison-Wesley, Cambridge-MA, 1949); see also R. Gunther, L. Levitin, B. Schapiro, and P. Wagner, Intern. J. Theor. Phys. 35, 395 (1996).Google Scholar
  155. 160.
    S. Denisov, Phys. Lett. A 235, 447 (1997).CrossRefADSGoogle Scholar
  156. 161.
    D. Kahneman and A. Tversky, Econometrica 47, 263 (1979); A. Tverky and D. Kahneman, Journal of Risk and Uncertainty 5, 297 (1992).zbMATHCrossRefGoogle Scholar
  157. 162.
    C. Tsallis, Chaos, Solitons and Fractals 6, 539 (1995).zbMATHMathSciNetCrossRefADSGoogle Scholar
  158. 163.
    C. Tsallis, A. S. Martinez and R. Maynard, unpublished.Google Scholar
  159. 164.
    C. Tsallis, Physica A 221, 277 (1995).MathSciNetCrossRefADSGoogle Scholar
  160. 165.
    X.-P. Huang, F. Anderegg, E. M. Hollmann, C. F. Driscoll, and T. M. O’Neil, Phys. Rev. Lett. 78, 875 (1997); F. Anderegg, X.-P. Huang, C. F. Driscoll, E. M. Hollmann, T. M. O’Neil, and D. H. E. Dubin, Phys. Rev. Lett. 78, 2128 (1997).CrossRefADSGoogle Scholar
  161. 166.
    C. Anteneodo and C. Tsallis, J. Mol. Liq. 71, 255 (1997).CrossRefGoogle Scholar
  162. 167.
    C. Anteneodo, private communication. This theory was first presented in [3].Google Scholar
  163. 168.
    H. Brands, P. H. Chavanis, R. Pasmanter, and J. Sommeria, Phys. Fluids 11, 3465 (1999).MathSciNetCrossRefADSzbMATHGoogle Scholar
  164. 169.
    V. Berezinsky, Solar Neutrino Problem, 30eme Rencontre de Moriond (12–18 March, 1995) [Laboratori Nazionali del Gran Sasso, Report 14 (June 1995)].Google Scholar
  165. 170.
    G. Kaniadakis, A. Lavagno, and P. Quarati, Phys. Lett. B 369, 308 (1996).CrossRefADSGoogle Scholar
  166. 171.
    A. Lavagno, G. Kaniadakis, M. Rego-Monteiro, P. Quarati, and C. Tsallis, Astrophys. Lett. and Comm. 35, 449 (1998).ADSGoogle Scholar
  167. 172.
    C. Tsallis and A. M. C. de Souza, Phys. Lett. A 235, 444 (1997).CrossRefADSGoogle Scholar
  168. 173.
    J. C. Mather et al., Astrophys. J. 420, 439 (1994); D.J. Fixsen et al., Astrophys. J. 420, 457 (1994).CrossRefADSGoogle Scholar
  169. 174.
    C. Tsallis, F. C. Sa Barreto, and E. D. Loh, Phys. Rev. E 52, 1447 (1995).CrossRefADSGoogle Scholar
  170. 175.
    A. R. Plastino, A. Plastino, and H. Vucetich, Phys. Lett. A 207, 42 (1995).zbMATHMathSciNetCrossRefADSGoogle Scholar
  171. 176.
    A. Einstein, in Physics and Reality. [In his words: “To be sure, it has been pointed out that the introduction of a spacetime continuum may be considered as contrary to nature in view of the molecular structure of everything which happens on a small scale. It is maintained that perhaps the success of the Heisenberg method points to a purely algebraical method of description of nature, that is to the elimination of continuum functions from physics. Then, however, we must also give up, by principle, the spacetime continuum.”]; in The Meaning of Relativity. [In his words: “One can give good reasons why reality cannot at all be represented by a continuous field. From the quantum phenomena it appears to follow with certainty that a finite system of finite energy can be completely described by a finite set of numbers (quantum numbers). This does not seem to be in accordance with a continuum theory, and must lead to an attempt to find a purely algebraic theory for the description of reality. But nobody knows how to obtain the basis of such a theory.”]Google Scholar
  172. 177.
    R. Hagedorn, Suppl. Nuovo Cimento 3, 147 (1965).Google Scholar
  173. 178.
    W. M. Alberico, A. Lavagno, and P. Quarati, Eur. Phys. J. C 12, 499 (1999).ADSCrossRefGoogle Scholar
  174. 179.
    C. M. G. Lattes, Y. Fujimoto, and S. Hasegawa, Phys. Rep. 65, 151 (1980).CrossRefADSGoogle Scholar
  175. 180.
    R. H. Austin, K. Beeson, L. Eisenstein, H. Frauenfelder, I. C. Gunsalus, and V. P. Marshall, Phys. Rev. Lett. 32, 403 (1974); R. H. Austin, K. Beeson, L. Eisenstein, and H. Frauenfelder, Biochemistry 14, 5355 (1975); F. Parak and H. Frauenfelder, Physica A 201, 332 (1993); H. Frauenflder and P. G. Wolynes, Physics Today 58 (February 1994); P. G. Wolynes, J. N. Onuchic, and D.Thirumalai, Science 267, 1169 (1995).CrossRefADSGoogle Scholar
  176. 181.
    C. Tsallis, G. Bemski, and R. S. Mendes, Phys. Lett. A 257, 93 (1999); Erratum: see caption of Fig. 15.CrossRefADSGoogle Scholar
  177. 182.
    A. Upadhyaya, J. P. Rieu, J. A. Glazier, and Y. Sawada (1998), private communication.Google Scholar
  178. 183.
    S. Redner, Eur. Phys. J. B 4, 131 (1998).CrossRefADSGoogle Scholar
  179. 184.
    C. Tsallis and M. P. de Albuquerque, Eur. Phys. J. B 13, 777 (2000).CrossRefADSGoogle Scholar
  180. 185.
    L. G. Gamero, A. Plastino, and M. E. Torres, Physica A 246, 487 (1998); A. Capurro, L. Diambra, D. Lorenzo, O. Macadar, M. T. Martin, C. Mostaccio, A. Plastino, J. Perez, J. Perez, E. Rofman, M. E. Torres, and J. Velluti, Physica A 257, 149 (1998); A. Capurro, L. Diambra, D. Lorenzo, O. Macadar, M. T. Martin, C. Mostaccio, A. Plastino, J. Perez, E. Rofman, M. E. Torres, and J. Velluti, Physica A 265, 235 (1999); M. T. Martin, A. R. Plastino, and A. Plastino, Physica A 275, 262 (2000).CrossRefADSGoogle Scholar
  181. 186.
    A. C. Tsallis, C. Tsallis, A. C. N. de Magalhaes, and F.A. Tamarit, to be published.Google Scholar
  182. 187.
    S. A. Cannas, D. A. Stariolo, and F. A. Tamarit, Network: Computation in neural sciences 7, 141 (1996).zbMATHCrossRefGoogle Scholar
  183. 188.
    S. Ghashghaie, W. Breymann, J. Peinke, P. Talkner, and Y. Dodge, Nature 381, 767 (1996).CrossRefADSGoogle Scholar
  184. 189.
    F. M. Ramos, C. Rodrigues Neto, and R. R. Rosa, preprint (1999) [condmat/9907348].Google Scholar
  185. 190.
    B. Chabaud, A. Naert, J. Peinke, F. Chilla, B. Castaing, and B. Hebral, Phys. Rev. Lett. 73, 3227 (1994).CrossRefADSGoogle Scholar
  186. 191.
    T. Arimitsu and N. Arimitsu, Phys. Rev. E 61, 3237 (2000).CrossRefADSGoogle Scholar
  187. 192.
    F. Anselmet, Y. Cagne, E. J. Hopfinger, and R. A. Antonia, J. Fluid Mech. 140, 63 (1984); C. Meneveau and K. R. Sreenivasan, Nucl. Phys. B (Proc. Suppl.) 2, 49 (1987).CrossRefADSGoogle Scholar
  188. 193.
    C. Beck, Physica A 277, 115 (2000).CrossRefADSGoogle Scholar
  189. 194.
    B. Castaing, Y. Gagne, and E. J. Hopfinger, Physica D 46, 177 (1990).zbMATHCrossRefADSGoogle Scholar
  190. 195.
    T. Arimitsu and N. Arimitsu, J. Phys. A 33, L 235 (2000).MathSciNetCrossRefADSGoogle Scholar
  191. 196.
    C. Meneveau and K. R. Sreenivasan, J. Fluid Mech. 224, 429 (1991).zbMATHCrossRefADSGoogle Scholar
  192. 197.
    C. Tsallis, Fractals 3, 541 (1995).zbMATHCrossRefGoogle Scholar
  193. 198.
    P. Jund and S. G. Kim, Phys. Rev. B 52, 50 (1995); J. R. Grigera, Phys. Lett. A 217, 47 (1996); S. A. Cannas and F. A. Tamarit, Phys. Rev. B 54, R12661 (1996); S. A. Cannas and A. C. N. Magalhaes, J. Phys. A 30, 3345 (1997); L. C. Sampaio, M. P. de Albuquerque, and F. S. de Menezes, Phys. Rev. B 55, 5611 (1997); S. E. Curilef, Ph.D. thesis (CBPF, Rio de Janeiro, 1997); S. Curilef and C. Tsallis, Phys. Lett. A 264, 270 (1999); H. H. A. Rego, L. S. Lucena, L. R. da Silva, and C. Tsallis, Physica A 266, 30 (1999); F. Tamarit and C. Anteneodo, Phys. Rev. Lett. 84, 208 (2000).CrossRefADSGoogle Scholar
  194. 199.
    H. N. Nazareno and P. E. de Brito, Phys. Rev. B 60, 4629 (1999).CrossRefADSGoogle Scholar
  195. 200.
    L. Borland and J. G. Menchero, Braz. J. Phys. 29, 169 (1999); L. Borland, J. G. Menchero, and C. Tsallis, Phys. Rev. B 61, 1650 (2000).CrossRefGoogle Scholar
  196. 201.
    I. Goldhirsch and G. Zanetti, Phys. Rev. Lett. 70, 1619 (1993).CrossRefADSGoogle Scholar
  197. 202.
    U. M. S. Costa, M. L. Lyra, A. R. Plastino, and C. Tsallis, Phys. Rev. E 56, 245 (1997).CrossRefADSGoogle Scholar
  198. 203.
    M. L. Lyra and C. Tsallis, Phys. Rev. Lett. 80, 53 (1998); U. Tirnakli, C. Tsallis, and M. L. Lyra, Eur. Phys. J. B 10, 309 (1999); C. R. da Silva, H. R. da Cruz, and M. L. Lyra, in Nonextensive Statistical Mechanics and Thermodynamics, eds. S. R. A. Salinas and C. Tsallis, Braz. J. Phys. 29, 144 (1999) [accessible at http://sbf.if.usp.br/WWWpages/Journals/BJP/Vol29/Num1/index.htm].CrossRefADSGoogle Scholar
  199. 204.
    P. Grassberger and M. Scheunert, J. Stat. Phys. 26, 697 (1981); T. Schneider, A. Politi, and D. Wurtz, Z. Phys. B 66, 469 (1987); G. Anania and A. Politi, Europhys. Lett. 7, 119 (1988) [It can be shown that from its Eq. (2) can be derived the present Eq. (156)]; H. Hata, T. Horita, and H. Mori, Progr. Theor. Phys. 82, 897 (1989).MathSciNetCrossRefADSGoogle Scholar
  200. 205.
    V. Latora and M. Baranger, Phys. Rev. Lett. 82, 520 (1999).CrossRefADSGoogle Scholar
  201. 206.
    R. C. Hilborn, Chaos and Nonlinear Dynamics (Oxford University Press, New York, 1994), page 390.zbMATHGoogle Scholar
  202. 207.
    G. Guerbero., private communication (1998).Google Scholar
  203. 208.
    V. Latora, M. Baranger, A. Rapisarda, and C. Tsallis, Phys. Lett. A 273, 97 (2000).MathSciNetCrossRefADSzbMATHGoogle Scholar
  204. 209.
    S. Montangero, L. Fronzoni, and P. Grigolini, preprint (1999) [cond-mat/9911412].Google Scholar
  205. 211.
    P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987).MathSciNetCrossRefADSGoogle Scholar
  206. 212.
    F. A. Tamarit, S. A. Cannas, and C. Tsallis, Eur. Phys. J. B 1, 545 (1998); A. R. R. Papa and C. Tsallis, Phys. Rev. E 57, 3923 (1998); see also A. Bhowal, Physica A 247, 327 (1997).CrossRefADSGoogle Scholar
  207. 213.
    V. Latora, A. Rapisarda, and S. Ruffo, Phys. Rev. Lett. 80, 692 (1998).CrossRefADSGoogle Scholar
  208. 214.
    C. Anteneodo and C. Tsallis, Phys. Rev. Lett. 80, 5313 (1998).CrossRefADSGoogle Scholar
  209. 215.
    L. Borland and C. Tsallis, in preparation (1999).Google Scholar
  210. 216.
    E. P. Borges and C. Tsallis, in preparation (1999).Google Scholar
  211. 217.
    A. Campa, A. Giansanti, D. Moroni, and C. Tsallis, cond-mat/0002168Google Scholar
  212. 218.
    M.-C. Firpo, Phys. Rev. E 57, 6599 (1998).CrossRefADSGoogle Scholar
  213. 219.
    M. Antoni and A. Torcini, Phys. Rev. E 57, R6233 (1998).CrossRefADSGoogle Scholar
  214. 220.
    M. Buiatti, P. Grigolini, and L. Palatella, Physica A 268, 214 (1999).CrossRefGoogle Scholar
  215. 221.
    S. Kirkpatrick, C.D. Gelatt, and M.P. Vecchi, Science 220, 671 (1983).MathSciNetCrossRefADSGoogle Scholar
  216. 222.
    A. M. C. de Souza and C. Tsallis, Physica A 236, 52 (1997).CrossRefADSGoogle Scholar
  217. 223.
    S. Lloyd, Phys. Rev. A 61, R010301 (2000).MathSciNetCrossRefADSGoogle Scholar
  218. 224.
    P. Gaspard and X.-J. Wang, Proc. Natl. Acad. Sci. USA 85, 4591 (1988); P. A. Alemany, Phys. Lett. A 235, 452 (1997); C. Tsallis, L. R. da Silva, R. S. Mendes, R. O. Vallejos, and A. M. Mariz, Phys. Rev. E 56, R4922 (1997).zbMATHMathSciNetCrossRefADSGoogle Scholar
  219. 225.
    A. Wiles, Annals of Mathematics 142, 443 (1995); R. Taylor and A. Wiles, Annals of Mathematics 142, 553 (1995).MathSciNetCrossRefGoogle Scholar
  220. 226.
    R. S. Mendes and C. Tsallis, Renormalization group approach to nonextensive statistical mechanics, preprint (2000) [cond-mat/0003365].Google Scholar
  221. 227.
    E. K. Lenzi, E. P. Borges, and R. S. Mendes, J. Phys. A 32, 8551 (1999).zbMATHMathSciNetCrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • C. Tsallis
    • 1
    • 2
  1. 1.Department of PhysicsUniversity of North TexasTexasUSA
  2. 2.Centro Brasileiro de Pesquisas FísicasRio de Janeiro-RJBrazil

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