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III. Tsallis Theory, the Maximum Entropy Principle, and Evolution Equations

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

Abstract

The generalized thermostatistics advanced by Tsallis in order to treat nonextensive systems has greatly increased the range of possible applications of statistical mechanics to the description of natural phenomena. Here we consider some aspects of the relationship between Tsallis’ theory and Jaynes’ maximum entropy (MaxEnt) principle. We review some universal properties of general thermostatistical formalisms based on an entropy extremalization principle. We explain how Tsallis formalism provides a useful tool in order to obtain MaxEnt solutions of nonlinear partial differential equations describing nonextensive physical systems. In particular, we consider the case of the Vlasov-Poisson equations, where Tsallis MaxEnt principle leads in a natural way to the stellar polytrope solutions. We pay special attention to the “P-picture” formulation of Tsallis generalized thermostatistics based on the entropic functional Sq and standard linear constraints.

Keywords

Vlasov Equation Maximum Entropy Principle Tsallis Entropy Tsallis Distribution Nonextensive System 
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, Physica A 221, 277 (1995), and references therein.CrossRefMathSciNetADSGoogle Scholar
  2. 2.
    C. Tsallis, Fractals 6, 539 (1995), and references therein; an updated bibliography can be found in http://tsallis.cat.cbpf.br/biblio.htm zbMATHMathSciNetGoogle Scholar
  3. 3.
    C. Tsallis, Physics World 10, 42 (July 1997).Google Scholar
  4. 4.
    J. Binney and S. Tremaine, Galactic Dynamics (Princeton University Press, Princeton, 1987).zbMATHGoogle Scholar
  5. 5.
    A. R. Plastino and A. Plastino, in Condensed Matter Theories, Volume 11, ed. E. Ludeña (Nova Science Publishers, New York, 1996), p. 341.Google Scholar
  6. 6.
    D. Lynden-Bell, Mon. Not. R. Astron. Soc. 124, 279 (1967).ADSGoogle Scholar
  7. 7.
    W. C. Saslaw, Gravitational Physics of Stellar and Galactic Systems, (Cambrideg University Press, Cambridge, 1985).Google Scholar
  8. 8.
    S. Tremaine, M. Henon, and D. Lynden-Bell, Mon. Not. R. Astron. Soc. 219, 285 (1986).zbMATHADSGoogle Scholar
  9. 9.
    C. Tsallis, J. Stat. Phys. 52, 479 (1988).zbMATHCrossRefMathSciNetADSGoogle Scholar
  10. 10.
    E. M. F. Curado and C. Tsallis, J. Phys. A 24, L69 (1991); Corrigenda 24, 3187 (1991) and 25, 1019 (1992).CrossRefMathSciNetADSGoogle Scholar
  11. 11.
    A. R. Plastino and A. Plastino, in Condensed Matter Theories, Volume 11, ed. E. Ludeña (Nova Science Publishers, New York, 1996), p. 327.Google Scholar
  12. 12.
    D. H. Zanette and P. A. Alemany, Phys. Rev. Lett. 75, 366 (1995); M. O. Caceres and C. E. Budde, ibid 77, 2589 (1996); D. H. Zannette and P. A. Alemany, ibid 77, 2590 (1996).CrossRefADSGoogle Scholar
  13. 13.
    A. K. Rajagopal, Phys. Rev. Lett. 76, 3469 (1996).zbMATHCrossRefMathSciNetADSGoogle Scholar
  14. 14.
    E. K. Lenzi, L. C. Malacarne, and R. S. Mendes, Phys. Rev. Lett. 80, 218 (1998).CrossRefADSGoogle Scholar
  15. 15.
    A. R. Plastino, A. Plastino, and C. Tsallis, J. Phys. A, 27, 5707, (1994).zbMATHCrossRefMathSciNetADSGoogle Scholar
  16. 16.
    G. A. Raggio, J. Math. Phys. 36, 4785 (1995).zbMATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    A. M. Meson and F. Vericat, J. Math. Phys. 37, 4480 (1996).zbMATHCrossRefMathSciNetADSGoogle Scholar
  18. 18.
    R. J. V. dos Santos, J. Math. Phys. 38, 4104 (1997).zbMATHCrossRefMathSciNetADSGoogle Scholar
  19. 19.
    R. Silva Jr., A. R. Plastino, and J. A. S. Lima, Phys. Lett. A 249, 401 (1998).zbMATHCrossRefMathSciNetADSGoogle Scholar
  20. 20.
    S. Abe, Eur. Phys. J. B 9, 679 (1999).CrossRefADSGoogle Scholar
  21. 21.
    A. R. Plastino and A. Plastino, Phys. Lett. A177, 177 (1993).CrossRefMathSciNetADSGoogle Scholar
  22. 22.
    A. R. Plastino and C. Anteneodo, Ann. Phys. 255, 250 (1997).zbMATHCrossRefMathSciNetADSGoogle Scholar
  23. 23.
    A. R. Plastino and A. Plastino, Phys. Lett. A 174, 384 (1993); J. J. Aly, in NBody Problems and Gravitational Dynamics, oc. of the Meeting held at Aussois-France (21–25 March 1993), eds. F. Combes and E. Athanassoula (Publications de l’Observatoire de Paris, 1993).CrossRefMathSciNetADSGoogle Scholar
  24. 24.
    A. R. Plastino and A. Plastino, Phys. Lett. A 193, 251 (1994).zbMATHCrossRefMathSciNetADSGoogle Scholar
  25. 25.
    B. M. R. Boghosian, Phys. Rev. E 53, 4754 (1995); C. Anteneodo and C. Tsallis, J. Mol. Liq. 71, 255 (1997).CrossRefADSGoogle Scholar
  26. 26.
    G. Kaniadakis, A. Lavagno, and P. Quarati, Phys. Lett. B 369, 308 (1996); P. Quarati, A. Carbone, G. Gervino, G. Kaniadakis, A. Lavagno, and E. Miraldi, Nucl. Phys. A 621, 345c (1996); G. Kaniadakis, A. Lavagno, M. Lissia, and P. Quarati, Physica A 261, 359 (1998).CrossRefADSGoogle Scholar
  27. 27.
    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).Google Scholar
  28. 28.
    C. Tsallis, A. R. Plastino and W.-M. Zheng, Chaos, Solitons and Fractals 8, 885 (1997); M. L. Lyra and C. Tsallis, Phys. Rev. Lett. 80, 53 (1998).zbMATHCrossRefMathSciNetADSGoogle Scholar
  29. 29.
    U. M. S. Costa, M. L. Lyra, A. R. Plastino, and C. Tsallis, Phys. Rev. E 56, 245 (1997).CrossRefADSGoogle Scholar
  30. 30.
    C. Anteneodo and C. Tsallis, Phys. Rev. Lett. 80, 5313 (1998).CrossRefADSGoogle Scholar
  31. 31.
    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).CrossRefADSGoogle Scholar
  32. 32.
    P. A. Alemany and D. H. Zanette, Phys. Rev. E 49, 956 (1994); D. H. Zanette and P. A. Alemany, Phys. Rev. Lett. 75, 366 (1995); M. O. Caceres and C. E. Budde, ibid 77, 2589 (1996); D. H. Zannette and P. A. Alemany, ibid 77, 2590 (1996); C. Tsallis, S. V. F. Levy, A. M. C. Souza and R. Maynard, Phys. Rev. Lett. 75, 3589 (1995); Erratum 77, 5442 (1996).CrossRefADSGoogle Scholar
  33. 33.
    A. R. Plastino and A. Plastino, Physica A 222, 347 (1995).CrossRefMathSciNetADSGoogle Scholar
  34. 34.
    C. Tsallis and D. J. Bukman, Phys. Rev. E 54, R2197 (1996).CrossRefADSGoogle Scholar
  35. 35.
    L. Borland, F. Pennini, A. R. Plastino, and A. Plastino, Eur. Phys. J. B 12, 285 (1999).CrossRefADSGoogle Scholar
  36. 36.
    L. Borland, Phys. Rev E 57, 6634 (1998).CrossRefADSGoogle Scholar
  37. 37.
    D. A. Stariolo, Phys. Rev. E 55, 4806 (1997).CrossRefADSGoogle Scholar
  38. 38.
    A. Compte and D. Jou, J. Phys. A 29, 4321 (1996).zbMATHCrossRefMathSciNetADSGoogle Scholar
  39. 39.
    A. Compte, D. Jou, and Y. Katayama, J. Phys. A 30, 1023 (1997).zbMATHCrossRefADSGoogle Scholar
  40. 40.
    A. M. Mariz, Phys. Lett. A 165, 409 (1992).CrossRefMathSciNetADSGoogle Scholar
  41. 41.
    J. D. Ramshaw, Phys. Lett. A 175, 169 (1993).CrossRefMathSciNetADSGoogle Scholar
  42. 42.
    L. Borland, A. R. Plastino, and C. Tsallis, J. Math. Phys. 39, 6490 (1998); Erratum 40, 2196 (1999).zbMATHCrossRefMathSciNetADSGoogle Scholar
  43. 43.
    J.W. Gibbs, Elementary Principles in Statistical Mechanics (Yale University Press, New Haven, 1902).zbMATHGoogle Scholar
  44. 44.
    E.T. Jaynes, Phys. Rev. 106, 620 (1957).CrossRefMathSciNetADSGoogle Scholar
  45. 45.
    E.T. Jaynes, Phys. Rev. 108, 171 (1957).CrossRefMathSciNetADSGoogle Scholar
  46. 46.
    A. Katz, Statistical Mechanics (Freeman, San Francisco, 1967).Google Scholar
  47. 47.
    W. Grandy, Foundations of Statistical Mechanics (Reidel, Dordrecht, 1987).zbMATHGoogle Scholar
  48. 48.
    R. Balian, From Microphysics to Macrophysics (Springer-Verlag, Berlin, 1992).zbMATHGoogle Scholar
  49. 49.
    C. Beck and F. Schlögl, Thermodynamics ofChaotic Systems, (Cambridge University Press, Cambridge, 1993).CrossRefGoogle Scholar
  50. 50.
    J. Skilling and R. K. Bryan, Mon. Not. R. Astron. Soc. 211, 111 (1984).zbMATHADSGoogle Scholar
  51. 51.
    V. J. Cornwell and K. F. Evans, Astron. Astrophys. 143, 77 (1985).ADSGoogle Scholar
  52. 52.
    L. Borland, Physica D 99, 175 (1996).zbMATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    A. Plastino and A. R. Plastino, Phys. Lett. A 226, 257 (1997).zbMATHCrossRefMathSciNetADSGoogle Scholar
  54. 54.
    R. S. Mendes, Physica A 242, 299 (1997).CrossRefADSGoogle Scholar
  55. 55.
    E. M. F. Curado, Braz. J. Phys. 29, 36 (1999).CrossRefGoogle Scholar
  56. 56.
    R. P. Di Sisto, S. Martinez, R. B. Orellana, A. R. Plastino, and A. Plastino, Physica A 265, 590 (1999).CrossRefGoogle Scholar
  57. 57.
    A. R. R. Papa, J. Phys. A: Math. Gen. 31, 5271 (1998).zbMATHCrossRefMathSciNetADSGoogle Scholar
  58. 58.
    S. Abe, Phys. Lett. A 224, 326 (1997).CrossRefMathSciNetADSGoogle Scholar
  59. 59.
    E. P. Borges and I. Roditi, Phys. Lett. A 246, 399 (1998).zbMATHCrossRefMathSciNetADSGoogle Scholar
  60. 60.
    C. Anteneodo and A.R. Plastino, J. Phys. A: Math. Gen. 32, 1089 (1999).zbMATHCrossRefMathSciNetADSGoogle Scholar
  61. 61.
    C. Tsallis, R. S. Mendes and A. R. Plastino, Physica A 261, 534 (1998).CrossRefGoogle Scholar
  62. 62.
    A. R. Plastino and J. A. S. Lima, Equipartition and Virial Theorems within General Thermostatistical Formalisms, (1999) submitted for publication.Google Scholar
  63. 63.
    D. Otero, A. Plastino, A. N. Proto, and G. Zannoli, Phys. Rev. A 26, 1209 (1982).CrossRefADSGoogle Scholar
  64. 64.
    D. Otero, A. Plastino, A. N. Proto, and G. Zannoli, Z. Phys. A 316, 323 (1984).CrossRefADSGoogle Scholar
  65. 65.
    E. Duering, D. Otero, A. Plastino, and A. N. Proto, Phys. Rev. A 32, 2455 (1985).CrossRefADSGoogle Scholar
  66. 66.
    E. Duering, D. Otero, A. Plastino, and A. N. Proto, Phys. Rev. A 35, 2314 (1985).CrossRefMathSciNetADSGoogle Scholar
  67. 67.
    P. Hick and G. Stevens, Astron. Astrophys. 172, 350 (1987).ADSGoogle Scholar
  68. 68.
    A. R. Plastino and A. Plastino, Physica A 232, 458 (1996).CrossRefMathSciNetADSGoogle Scholar
  69. 69.
    A. R. Plastino, H. G. Miller, and A. Plastino, Phys. Rev. E56, 3927 (1997).CrossRefADSGoogle Scholar
  70. 70.
    A. Plastino, A. R. Plastino, and H. G. Miller, Phys. Lett. A 232, 349 (1997).zbMATHCrossRefMathSciNetADSGoogle Scholar
  71. 71.
    E. D. Malaza, H. G. Miller, A. R. Plastino, and F. Solms, Physica A 265, 224 (1999).CrossRefGoogle Scholar
  72. 72.
    M. Muskat, The Flow ofHomo geneous Fluids Through Porous Media (McGraw-Hill, New York, 1937).Google Scholar
  73. 73.
    J. Buckmaster, J. Fluid Mech. 81, 735 (1977).zbMATHCrossRefMathSciNetADSGoogle Scholar
  74. 74.
    H. Spohn, J. Phys. (France) I 3, 69 (1993).CrossRefADSGoogle Scholar
  75. 75.
    J. M. Carlson, E. R. Grannan, C. Singh, and G.H. Swindle, Phys. Rev. E 48 688 (1993).CrossRefADSGoogle Scholar
  76. 76.
    H. Risken, The Fokker-Planck Equation (Springer-Verlag, New York, 1989).zbMATHGoogle Scholar
  77. 77.
    A. R. Plastino and A. Plastino, Braz. J. Phys. 29, 79 (1999).Google Scholar
  78. 78.
    L. S. Lucena, L. R. da Silva, and C. Tsallis, Phys. Rev. E 51, 6247 (1995).CrossRefADSGoogle Scholar
  79. 79.
    J. R. Grigera, Phys. Lett. A 217, 47 (1996).CrossRefADSGoogle Scholar
  80. 80.
    P. Jund, S.G. Kim, and C. Tsallis, Phys. Rev. B 52, 50 (1995).CrossRefADSGoogle Scholar
  81. 81.
    S. Curilef and C. Tsallis, Critical behaviour in Lennard-Jones like fluids: Longrange interactions and nonextensivity, preprint (1998).Google Scholar
  82. 82.
    J. D. Brown and J. W. York Jr., in Physiscal Origins ofTime Assymetry, eds. J.J. Halliwell, J. Perez-Mercader, and W.H. Zurek (Cambrideg University Press, Cambridge, 1996), p. 465–474.Google Scholar
  83. 83.
    P. Morse and H. Feshbach, Methods ofThe oretical Physics (McGraw-Hill, New York, 1953).Google Scholar
  84. 84.
    W. Thomson (Lord Kelvin), in: Mathematical and Physical Papers, Vol.3 (Cambridge University Press, Cambridge, 1911), p. 255–360.Google Scholar
  85. 85.
    S. Chandrasekhar, An Introduction to the Theory of Stellar Structure(University of Chicago Press, Chicago, 1958).Google Scholar
  86. 86.
    A. Lavagno, G. Kaniadakis, M. Rego-Monteiro, P. Quarati, and C. Tsallis, Astroph. Lett. Comm. 35, 449 (1998).ADSGoogle Scholar
  87. 87.
    G. Rybicki, Astrophys. and Sapce Sci. 14, 56 (1971).CrossRefADSGoogle Scholar
  88. 88.
    A. R. Plastino and A. Plastino, unpublished.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • A.R. Plastino
    • 1
  1. 1.Faculty of Astronomy and GeophysicsNational University La PlataLa PlataArgentina

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