Astrophysics and Space Science

, Volume 290, Issue 3–4, pp 259–274

Nonextensive Statistical Mechanics: Some Links with Astronomical Phenomena

  • Constantino Tsallis
  • Constantino Tsallis
  • Domingo Prato
  • Angel R. Plastino
  • Angel R. Plastino
Article

Abstract

A variety of astronomical phenomena appear to not satisfy the ergodic hypothesis in the relevant stationary state, if any. As such, there is no reason for expecting the applicability of Boltzmann–Gibbs (BG) statistical mechanics. Some of these phenomena appear to follow, instead, nonextensive statistical mechanics. In the same manner that the BG formalism is based on the entropy SBG=−kipi ln pi, the nonextensive one is based on the form Sq=k(1 −∑ipiq)/(q− 1) (with S1=SBG). The stationary states of the former are characterized by an exponential dependence on the energy, whereas those of the latter are characterized by an (asymptotic) power law. A brief review of this theory is given here, as well as of some of its applications, such as the solar neutrino problem, polytropic self-gravitating systems, galactic peculiar velocities, cosmic rays and some cosmological aspects. In addition to these, an analogy with the Keplerian elliptic orbits versus the Ptolemaic epicycles is developed, where we show that optimizing Sq with a few constraints is equivalent to optimizing SBG with an infinite number of constraints.

nonextensive statistical mechanics Tsallis entropy solar neutrino problem cosmic rays polytropes 

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References

  1. Fermi, E.: 1956, Thermodynamics, Dover, New York.Google Scholar
  2. Tisza, L.: 1966, Generalized Thermodynamics, MIT Press, Cambridge, MA, p. 123.Google Scholar
  3. Landsberg, P.T.: 1978, Thermodynamics and Statistical Mechanics, Oxford university Press, Oxford, p. 102 (also Dover, New york, 1990).Google Scholar
  4. Cohen, E.G.D.: 2002, Physica A 305, 19.MATHMathSciNetCrossRefADSGoogle Scholar
  5. Baranger, M.: 2002, Physica A 305, 27.MATHMathSciNetCrossRefADSGoogle Scholar
  6. Beck, C., Lewis, G.S. and Swinney, H.L.: 2001, Phys. Rev. E 63, 035303; Beck, C.: 2001, Phys. Rev.Lett. 87, 180601.CrossRefADSGoogle Scholar
  7. Arimitsu, T. and Arimitsu, N.: 2002, Physica A 305, 218.MATHMathSciNetCrossRefADSGoogle Scholar
  8. Bediaga, I., Curado, E.M.F. and Miranda, J.: 2000, Physica A 286, 156.CrossRefADSGoogle Scholar
  9. Walton, D.B. and Rafelski, J.: 2000, Phys. Rev. Lett. 84, 31.CrossRefADSGoogle Scholar
  10. Anteneodo, C., Tsallis, C. and Martinez, A.S.: 2002, Europhys. Lett. 59, 635.CrossRefADSGoogle Scholar
  11. Borland, L.: 2002, Phys. Rev. Lett. 89, 098701; Quantitative Finance 2, 415.CrossRefADSGoogle Scholar
  12. Osorio, R., Borland, L. and Tsallis, C.: in preparation, in: M. Gell-Mann and C. Tsallis (eds.), Nonextensive Entropy–Interdisciplinary Applications, Oxford University Press, NewYork, 2004; Michael, F. and Johnson, M.D.: Physica A 320, 525 (2003), and Physica A 324, 359 (2003).Google Scholar
  13. Upadhyaya, A., Rieu, J.-P., Glazier, J.A. and Sawada, Y.: 2001, Physica A 293, 549.Google Scholar
  14. de Lima, J.A.S., Silva, R. and Plastino, A.R.: 2001, Phys. Rev. Lett. 86, 2938.CrossRefADSGoogle Scholar
  15. Tsallis, C., Plastino, A.R. and Zheng, W.-M.: 1997, Chaos, Solitons Fractals 8, 885; Costa, U.M.S., Lyra, M.L., Plastino, A.R. and Tsallis, C.: 1997, Phys. Rev. E 56, 245; Lyra, M.L. and Tsallis, C.: 1998, Phys. Rev. Lett. 80, 53; Tirnakli, U., Tsallis, C. and Lyra, M.L.: 1999, Eur. Phys. J. B 11, 309; Latora, V., Baranger, M., Rapisarda, A., Tsallis, C.: 2000, Phys. Lett. A 273, 97; de Moura, F.A.B.F., Tirnakli, U., Lyra, M.L.: 2000, Phys. Rev. E 62, 6361; Tirnakli, U., Ananos, G.F.J., Tsallis, C.: 2001, Phys. Lett. A 289, 51; de Oliveira, H.P., Soares, I.D. and Tonini, E.V.: 2001, Physica A 295, 348; Baldovin, F. and Robledo, A.: 2002, Europhys. Lett. 60, 518; Baldovin, F. and Robledo, A.: 2002, Phys. Rev. E 66, 045104(R); Borges, E.P., Tsallis, C., Ananos, G.F.J. and Oliveira, P.M.C.: 2002, Phys. Rev. Lett. 89, 254103; Tirnakli, U.: 2002, Physica A 305, 119; Tirnakli, U.: 2002, Phys. Rev. E 66, 066212.MATHMathSciNetCrossRefGoogle Scholar
  16. Weinstein, Y., Lloyd, S. and Tsallis, C.: 2002, Phys. Rev. Lett. 89, 214101.CrossRefADSGoogle Scholar
  17. Abe, S. and Rajagopal, A.K.: 2001, Physica A 289, 157, Tsallis, C., Lloyd, S. and Baranger, M.: 2001, Phys. Rev. A 63, 042104; Tsallis, C., Lamberti, P.W. and Prato, D.: 2001, Physica A 295, 158; Alcaraz, F.C. and Tsallis, C.: 2002, Phys. Lett. A 301, 105; Tsallis, C., Prato, D. and Anteneodo, C.: 2002, Eur. Phys. J. B 29, 605; Batle, J., Plastino, A.R., Casas, M. and Plastino, A.: J. Phys. A 35 10311 (2002).MATHMathSciNetCrossRefADSGoogle Scholar
  18. Plastino, A.R. and Plastino, A.: 1995, Physica A 222, 347; Tsallis, C. and Bukman, D.J.: 1996, Phys.Rev. E 54, R2197; Giordano, C., Plastino, A.R., Casas, M. and Plastino, A.: 2001, Eur. Phys. J. B 22, 361; Compte, A. and Jou, D.: 1996, J. Phys. A 29, 4321; Plastino, A.R., Casas, M. and Plastino, A.: 2000, Physica A 280, 289; Bologna, M., Tsallis, C. and Grigolini, P.: 2000, Phys. Rev. E 62, 2213; Tsallis, C. and Lenzi, E.K.: 2002. Chem. Phys. 284, 341; Lenzi, E.K., Malacarne, L.C., Mendes, R.S. and Pedron, I.T.: 319, 245 (2003), Physica A; Lenzi, E.K., Anteneodo, C. and Borland, L.: 2001, Phys. Rev. E 63, 051109; Curado, E.M.F. and Nobre, F.D.: Phys. Rev. E 67, 021107 (2003); F.D. Nobre, E.M.F. Curado and G. Rowlands, Physica A 334, 109 (2004); Anteneodo, C. and Tsallis, C. J. Math. Phys. 44, 5194 (2003).MathSciNetCrossRefADSGoogle Scholar
  19. Anteneodo, C. and Tsallis, C.: 1998, Phys. Rev. Lett 80, 5313; Latora, V., Rapisarda, A. and Tsallis, C.: 2001, Phys. Rev. E 64, 056134; Campa, A., Giansanti, A. and Moroni, D.: 2002, Physica A 305, 137; Cabral, B.J.C. and Tsallis, C.: 2002, Phys. Rev. E 66, 065101(R);. Borges, E.P and Tsallis, C.: 2002, Physica A 305, 148; Campa, A., Giansanti, A., Moroni, D. and Tsallis, C.: 2001, Phys. Lett. A 286, 251; Firpo, M.-C. and Ruffo, S.: 2001, J. Phys. A 34, L511; Anteneodo, C. and Vallejos, R.O.: 2002 Phys. Rev. E 65, 016210; Vallejos, R.O. and Anteneodo, C.: 2002, Phys. Rev. E 66, 021110; Montemurro, M.A., Tamarit, F. and Anteneodo, C, Phys. Rev. E 67, 031106 (2003).CrossRefADSGoogle Scholar
  20. Abe, S. and Suzuki, N.: 2003, Phys. Rev. E 67, 016106.CrossRefADSGoogle Scholar
  21. Tsallis, C.: 1988, J. Stat. Phys. 52, 479.MATHMathSciNetCrossRefGoogle Scholar
  22. Curado, E.M.F. and Tsallis, C.: 1991, J. Phys. A: Math. Gen. 24, L69 [Corrigenda: 24, 3187 (1991) and 25, 1019 (1992)]; Tsallis, C., Mendes, R.S. and Plastino, A.R.: 1998, Physica A 261, 534.MathSciNetCrossRefADSGoogle Scholar
  23. Salinas, S.R.A. and Tsallis, C. (eds.): 1999, Braz. J. Phys. 29(1). Abe, S. and Okamoto, Y. (eds.): 2001, Nonextensive Statistical Mechanics and its Applications, Series Lecture Notes in Physics, Springer-Verlag, Berlin; Kaniadakis, G., Lissia, M. and Rapisarda, A. (eds.): 2002, Physica A 305(1/2); Gell-Mann, M. and Tsallis, C. (eds.) Nonextensive Entropy–Interdisciplinary Applica-tions, Oxford University Press, New York, 2004; Swinney, H.L. and Tsallis, C. (eds.); anomalous distributions, nonlinear dynamics and nonextensivity, special volume of Physica D (Elsevier, Amsterdam, 2004).Google Scholar
  24. Henry, J.: 2001, Moving Heavens and Earth: Copernicus and the Solar System, Icon Books, Cam-bridge,UK.Google Scholar
  25. Hoyle, F.: 1973, Nicolaus Copernicus–An Essay on His Life and Work, Harper and Row, New York, p. 14.Google Scholar
  26. Danby, J.M.A.: 1989, Fundamentals of Celestial Mechanics, (2nd edn.), Willmann-Bell Inc., Richmond, VA.Google Scholar
  27. de Moura, F.A.B.F., Tirnakli, U. and Lyra, M.L.: 2000, Phys. Rev. E 62, 6361.CrossRefADSGoogle Scholar
  28. P.H. Chavanis, arXiv: Phys. Rev. E 68, 036108 (2003) see also Chavanis, P.-H., Physica A 332,89 (2004).CrossRefADSGoogle Scholar
  29. Kaniadakis, G., Lavagno, A. and Quarati, P.: 1996, Phys. Lett. B 369, 308; Quarati, P., Carbone, A., Gervino, G., Kaniadakis, G., Lavagno, A. and Miraldi, E.: 1997, Nucl. Phys. A 621, 345c; Kaniadakis, G., Lavagno, A. and Quarati, P.: 1998, Astrophysics and Space Science 258, 145; Kaniadakis, G., Lavagno, A., Lissia, M. and Quarati, P.: 1989, in: T. Kodama, C.E. Aguiar, S.B. Duarte, Y. Hama, G. Odyniec, and H. Strobele, (eds.), Proc. 5th International Workshop on Relativistic Aspects of Nuclear Physics, Rio de Janeiro-Brazil, 1997, World Scientific, Singapore, p. 193. Gervino, G., Kaniadakis, G., Lavagno, A., Lissia, M. and Quarati, P.: 1998, in: Nuclei in the Cosmos, Volos-Greece, July 1998; Coraddu, M., Kaniadakis, G., Lavagno, A., Lissia, M.,. Mezzorani, G. and Quarti, P.: 1999, Braz. J. Phys. 29, 153; Lavagno, A. and Quarati, P.: 2000, Nucl. Phys. B, Proc. Suppl. 87, 209; Cossu, C.M.: 2000, Masters Thesis, Universita degli Studi di Cagliari.CrossRefADSGoogle Scholar
  30. Plastino, A.R. and Plastino, A.: 1993, Physics Letters A 174, 384.MathSciNetCrossRefADSGoogle Scholar
  31. Plastino, A.R.: 2001, in: S. Abe, and Y. Okamoto, (eds.), Nonextensive Statistical Mechanics and Its Applications, Springer, Berlin.Google Scholar
  32. Aly, J.-J. and Perez, J.: 1999, Phys. Rev. E 60, 5185; Taruya, A. and Sakagami, M.: 2002, Physica A 307, 185; Taruya, A. and Sakagami, M.: Physica A 318, 387 (2003), Chavanis, P.-H., 2003, Astro. Astrophys.: 386, 732; Chavanis, P.H.: Astro. Astrophys. 401,15(2003); Taruya A. and Sakagami, M. Physica A 322, 285 (2003) and Phys. Rev. Lett. 90 181101 (2003).CrossRefADSGoogle Scholar
  33. Lavagno, A., Kaniadakis, G., Rego-Monteiro, M., Quarati, P. and Tsallis, C.: 1998, Astrophysical Letters and Communications 35, 449.ADSGoogle Scholar
  34. Tsallis, C., Anjos, J.C. and Borges, E.P.: Phys. Lett. A 310, 372 (2003).CrossRefADSGoogle Scholar
  35. Hamity, V.H. and Barraco, D.E.: 1996, Phys. Rev. Lett. 76, 4664; Hamity, V.H. and Barraco, D.E.: 2000, Physica A 282, 203; Chimento, L.P.: 1997, J. Math. Phys. 38, 2565; Torres, D.F., Vucetich, H. and Plastino, A.: 1997 Phys. Rev. Lett. 79, 1588 [Erratum: 80, 3889 (1998)]; Tirnakli, U. and Torres, D.F.: 1999, Physica A 268, 225; Chimento, L.P., Pennini, F. and Plastino, A.: 1998, Physica A 256, 197; Chimento, L.P., Pennini, F. and Plastino, A.: 1999 Phys. Lett. A 257, 275; Torres, D.F. and Vucetich, H.: 1998, Physica A 259, 397; Torres, D.F.: 1998, Physica A 261, 512; de Oliveira, H.P., Sautu, S.L., Soares, I.D. and Tonini, E.V.: 1999, Phys. Rev. D 60, 121301–1; Pessah, M.E., Torres, D.F. and Vucetich, H.: 2001, Physica A 297, 164; Pessah, M.E. and Torres, D.F.: 2001, Physica A 297, 201; Hanyu, C. and Habe, A.: 2001, Astrophys. J. 554, 1268; Tonini, E.V.: 2002, Doctoral Thesis, Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro.CrossRefADSGoogle Scholar
  36. Binney, J. and Tremaine, S.: 1987, Galactic Dynamics, Princeton University Press, Princeton.Google Scholar
  37. Tremaine, S., Henon, M. and Lynden-Bell, D.: 1986, Mon. Not. R. Astron. Soc. 219, 285.MATHADSGoogle Scholar
  38. Boghosian, B.M.: 1996, Phys. Rev. E, 53, 4754.CrossRefADSGoogle Scholar
  39. Anteneodo, C. and Tsallis, C.: 1997, J. Mol. Liq. 71, 255.CrossRefGoogle Scholar
  40. Bahcall, N.A. and Oh, S.P.: 1996, Ap. J. Lett. 462, L49.ADSGoogle Scholar
  41. Nauenberg, M.: 2002, cond-mat/0210561 [version 2]; see also Nauenberg, M., Phys. Rev. E 67 036114 (2003), Tsallis, C., Phys. Rev. E (March 2004), in press, and Tsallis, C., cond-mat/ 0304696.CrossRefADSGoogle Scholar
  42. Guerberoff, G.R. and Raggio, G.A.: 1996, J. Math. Phys. 37, 1776; Guerberoff, Pury, P.A. and Raggio, G.A.: 1996, J. Math. Phys. 37, 1790.MATHMathSciNetCrossRefADSGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Constantino Tsallis
    • 1
  • Constantino Tsallis
    • 2
  • Domingo Prato
    • 3
  • Angel R. Plastino
    • 4
  • Angel R. Plastino
    • 5
  1. 1.Centro Brasileiro de Pesquisas FísicasBrazil
  2. 2.Santa Fe InstituteUSA
  3. 3.Facultad de Matematica, Astronomia y FisicaUniversidad Nacional de CordobaArgentina
  4. 4.Facultad de Ciencias Astronomicas y GeofisicasUniversidad Nacional de La Plata and CONICETArgentina
  5. 5.Departament de FisicaUniversitat de les Illes BalearsPalma de MallorcaSpain

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