Skip to main content
SpringerLink
Log in

Relativistic entropy and related Boltzmann kinetics

  • Regular Article - Theoretical Physics
  • Published:
The European Physical Journal A Aims and scope Submit manuscript

Abstract

It is well known that the particular form of the two-particle correlation function, in the collisional integral of the classical Boltzmman equation, fixes univocally the entropy of the system, which turns out to be the Boltzmann-Gibbs-Shannon entropy. In the ordinary relativistic Boltzmann equation, some standard generalizations, with respect to its classical version, imposed by the special relativity, are customarily performed. The only ingredient of the equation, which tacitly remains in its original classical form, is the two-particle correlation function, and this fact imposes that also the relativistic kinetics is governed by the Boltzmann-Gibbs-Shannon entropy. Indeed the ordinary relativistic Boltzmann equation admits as stationary stable distribution, the exponential Juttner distribution. Here, we show that the special relativity laws and the maximum entropy principle suggest a relativistic generalization also of the two-particle correlation function and then of the entropy. The so obtained, fully relativistic Boltzmann equation, obeys the H-theorem and predicts a stationary stable distribution, presenting power law tails in the high-energy region. The ensued relativistic kinetic theory preserves the main features of the classical kinetics, which recovers in the c \( \rightarrow\) ∞ limit.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. Kaniadakis, to be published in Eur. Phys. J. B 69 (2009) DOI: 10.1140/epjb/e2009-00161-0.

  2. A. Hasegawa, A.M. Kunioki, M. Duong-van, Phys. Rev. Lett. 54, 2608 (1985).

    Google Scholar 

  3. V.M. Vasyliunas, J. Geophys. Res. 73, 2839 (1968).

    Google Scholar 

  4. P.L. Biermann, G. Sigl, Physics and Astrophysics of Ultra-Hight-Energy Cosmic Rays, Lect. Notes Phys., Vol. 576 (Spring-Verlag, Berlin, 2001).

  5. G. Wilk, Z. Wlodarczyk, Phys. Rev. D 50, 2318 (1994).

    Google Scholar 

  6. D.B. Walton, J. Rafelski, Phys. Rev. Lett. 84, 31 (2000).

    Google Scholar 

  7. G. Kaniadakis, P. Quarati, Physica A 192, 677 (1993).

  8. G. Kaniadakis, P. Quarati, Physica A 237, 229 (1997).

  9. G. Kaniadakis, A. Lavagno, P. Quarati, Nucl. Phys. B 466, 527 (1996).

    Google Scholar 

  10. G. Kaniadakis, A. Lavagno, P. Quarati, Phys. Lett. A 227, 227 (1997).

    Google Scholar 

  11. S. Abe, J. Phys. A: Math. Gen. 36, 8733 (2003).

    Google Scholar 

  12. T.D. Frank, Phys. Lett. A 299, 153 (2002).

    Google Scholar 

  13. P.-H. Chavanis, Physica A 332, 89 (2004).

  14. T.D. Frank, Phys. Lett. A 305, 150 (2002).

    Google Scholar 

  15. V. Schwammle, E.M.F. Curado, F.D. Nobre, Eur. Phys. J. B 58, 159 (2007).

    Google Scholar 

  16. G. Kaniadakis, Physica A 296, 405 (2001).

  17. G. Kaniadakis, Phys. Lett. A 288, 283 (2001).

    Google Scholar 

  18. G. Kaniadakis, Phys. Rev. E 66, 056125 (2002).

    Google Scholar 

  19. G. Kaniadakis, Phys. Rev. E 72, 036108 (2005).

    Google Scholar 

  20. T.S. Biro, G. Kaniadakis, Eur. Phys. J. B 50, 3 (2006).

    Google Scholar 

  21. S.R. de Groot, W.A. van Leeuwen, Ch.G. van Weert, Relativistic Kinetic Theory (North-Holland Publishing Company, Amsterdam, 1980).

  22. R. Silva, Eur. Phys. J. B 54, 499 (2006).

    Google Scholar 

  23. R. Silva, Phys. Lett. A 352 17 (2006).

  24. T. Wada, Phisica A 340, 126 (2004).

  25. T. Wada, Contin. Mech. Thermodyn. 16, 263 (2004).

    Google Scholar 

  26. G. Kaniadakis, A.M. Scarfone, Physica A 340, 102 (2004).

  27. S. Abe, G. Kaniadakis, A.M. Scarfone, J. Phys. A: Math. Gen. 37, 10513 (2004).

    Google Scholar 

  28. J. Naudts, Physica A 316, 323 (2002).

  29. J. Naudts, Rev. Math. Phys. 16, 809 (2004).

    Google Scholar 

  30. A.M. Scarfone, T. Wada, Prog. Theor. Phys. Suppl. 162, 45 (2006).

    Google Scholar 

  31. Guo Lina, Du Jiulin, Liu Zhipeng, Phys. Lett. A 367, 431 (2007).

  32. Guo Lina, Du Jiulin, Phys. Lett. A 362, 368 (2007).

  33. G. Lapenta, S. Markidis, A. Marocchino, G. Kaniadakis, Astrophys. J. 666, 949 (2007).

    Google Scholar 

  34. G. Lapenta, S. Markidis, G. Kaniadakis, J. Stat. Mech. P02024 (2009).

  35. J.C. Carvalho, R. Silva, J.D. do Nascimento jr., J.R. De Medeiros, EPL 84, 59001 (2008).

  36. J.C. Carvalho, J.D. do Nascimento jr., R. Silva, J.R. De Medeiros, Astrophys. J. Lett. 696, L48 (2009).

  37. A. Rossani, A.M. Scarfone, J. Phys. A 37, 4955 (2004).

    Google Scholar 

  38. J.M. Silva, R. Silva, J.A.S. Lima, Phys. Lett. A 372, 5754 (2008).

    Google Scholar 

  39. A.M. Teweldeberhan, H.G. Miller, R. Tegen, Int. J. Mod. Phys. E 12, 669 (2003).

    Google Scholar 

  40. M. Coraddu, M. Lissia, R. Tonelli, Physica A 365, 252 (2006).

    Google Scholar 

  41. A. Celikoglu, U. Tirnakli, Physica A 372, 238 (2006).

  42. A.I. Olemskoi, V.O. Kharchenko, V.N. Borisyuk, Physica A 387, 1895 (2008).

    Google Scholar 

  43. A.Y. Abul-Magd, Phys. Lett. A 361, 450 (2007).

    Google Scholar 

  44. T. Wada, H. Suyari, Phys. Lett. A 348, 89 (2006).

    Google Scholar 

  45. F. Topsoe, Physica A 340, 11 (2004).

  46. T. Wada, H. Suyari, Phys. Lett. A 368, 199 (2007).

    Google Scholar 

  47. F. Clementi, M. Gallegati, G. Kaniadakis, Eur. Phys. J. B 57, 187 (2007).

    Google Scholar 

  48. F. Clementi, T. Di Matteo, M. Gallegati, G. Kaniadakis, Physica A 387, 3201 (2008).

    Google Scholar 

  49. D. Rajaonarison, D. Bolduc, H. Jayet, Econ. Lett. 86, 13 (2005).

    Google Scholar 

  50. D. Rajaonarison, Econ. Lett. 100, 396 (2008).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Fisica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy

    G. Kaniadakis

Authors
  1. G. Kaniadakis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Kaniadakis.

Additional information

Communicated by U.-G. Meißner

This paper is part of the Topical Issue Statistical Power Law Tails in High-Energy Phenomena.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kaniadakis, G. Relativistic entropy and related Boltzmann kinetics. Eur. Phys. J. A 40, 275 (2009). https://doi.org/10.1140/epja/i2009-10793-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epja/i2009-10793-6

PACS

Search

Navigation