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The European Physical Journal B

, Volume 58, Issue 2, pp 159–165 | Cite as

A general nonlinear Fokker-Planck equation and its associated entropy

  • V. Schwämmle
  • E. M.F. Curado
  • F. D. NobreEmail author
Statistical and Nonlinear Physics

Abstract.

A recently introduced nonlinear Fokker-Planck equation, derived directly from a master equation, comes out as a very general tool to describe phenomenologically systems presenting complex behavior, like anomalous diffusion, in the presence of external forces. Such an equation is characterized by a nonlinear diffusion term that may present, in general, two distinct powers of the probability distribution. Herein, we calculate the stationary-state distributions of this equation in some special cases, and introduce associated classes of generalized entropies in order to satisfy the H-theorem. Within this approach, the parameters associated with the transition rates of the original master-equation are related to such generalized entropies, and are shown to obey some restrictions. Some particular cases are discussed.

PACS.

05.40.Fb Random walks and Levy flights 05.20.-y Classical statistical mechanics 05.40.Jc Brownian motion 66.10.Cb Diffusion and thermal diffusion 

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References

  1. L.E. Reichl, A Modern Course in Statistical Physics, 2nd edn. (Wiley, New York, 1998) Google Scholar
  2. N.G. Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981) Google Scholar
  3. H. Haken, Synergetics (Springer-Verlag, Berlin, 1977) Google Scholar
  4. H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications (Springer-Verlag, Berlin, 1989) Google Scholar
  5. T.M. Cover, J.A. Thomas, Elements of Information Theory (Wiley, New York, 1991) Google Scholar
  6. Nonextensive Statistical Mechanics and Thermodynamics, edited by S.R.A. Salinas, C. Tsallis, Vol. 29, Braz. J. Phys. (1999) Google Scholar
  7. Nonextensive Entropy - Interdisciplinary Applications edited by M. Gell-Mann, C. Tsallis (Oxford University Press., New York, 2004) Google Scholar
  8. Nonextensive Statistical Mechanics: New Trends, New Perspectives, Vol. 36, Europhys. News (2005) Google Scholar
  9. C. Tsallis, J. Stat. Phys. 52, 479 (1988) zbMATHCrossRefGoogle Scholar
  10. S. Abe, Phys. Lett. A 224, 326 (1997) CrossRefADSGoogle Scholar
  11. C. Anteneodo, J. Phys. A 32, 1089 (1999) zbMATHCrossRefADSGoogle Scholar
  12. E.P. Borges, I. Roditi, Phys. Lett. A 246, 399 (1998) zbMATHCrossRefADSGoogle Scholar
  13. P.T. Landsberg, V. Vedral, Phys. Lett. A 247, 211 (1998) zbMATHCrossRefADSGoogle Scholar
  14. E.M.F. Curado, Braz. J. Phys. 29, 36 (1999) CrossRefGoogle Scholar
  15. E.M.F. Curado, F.D. Nobre, Physica A 335, 94 2004 Google Scholar
  16. A. Renyi, Probability Theory (North-Holland, Amsterdam, 1970) Google Scholar
  17. G. Kaniadakis, Physica A 296, 405 (2001) zbMATHCrossRefADSGoogle Scholar
  18. G. Kaniadakis, Phys. Rev. E 66, 056125 (2002) CrossRefADSGoogle Scholar
  19. G. Kaniadakis, Phys. Rev. E 72, 036108 (2005) CrossRefADSGoogle Scholar
  20. T.D. Frank, Nonlinear Fokker-Planck equations: Fundamentals and Applications (Springer, Berlin, 2005) Google Scholar
  21. A.R. Plastino, A. Plastino, Physica A 222, 347 (1995) CrossRefADSGoogle Scholar
  22. C. Tsallis, D.J. Bukman, Phys. Rev. E R2197, 54 (1996) Google Scholar
  23. L. Borland, Phys. Rev. E 57, 6634 (1998) CrossRefADSGoogle Scholar
  24. L. Borland, F. Pennini, A.R. Plastino, A. Plastino, Eur. Phys. J. B 12, 285 (1999) CrossRefADSGoogle Scholar
  25. E.K. Lenzi, R.S. Mendes, C. Tsallis, Phys. Rev. E 67, 031104 (2003) CrossRefADSGoogle Scholar
  26. T.D. Frank, A. Daffertshofer, Physica A 272, 497 (1999) CrossRefADSGoogle Scholar
  27. T.D. Frank, A. Daffertshofer, Physica A 295, 455 (2001) zbMATHCrossRefADSGoogle Scholar
  28. T.D. Frank, Physica A 301, 52 (2001) zbMATHCrossRefADSGoogle Scholar
  29. L.C. Malacarne, R.S. Mendes, I.T. Pedron, E.K. Lenzi, Phys. Rev. E 63, 030101 (2001) CrossRefADSGoogle Scholar
  30. L.C. Malacarne, R.S. Mendes, I.T. Pedron, E.K. Lenzi, Phys. Rev. E 65, 052101 (2002) CrossRefADSGoogle Scholar
  31. P.H. Chavanis, Phys. Rev. E 68, 036108 (2003) CrossRefADSGoogle Scholar
  32. P.H. Chavanis, Physica A 340, 57 (2004) CrossRefADSGoogle Scholar
  33. A.R. Plastino, L. Borland, C. Tsallis, J. Math. Phys. 39, 6490 (1998) zbMATHCrossRefADSGoogle Scholar
  34. A. Compte, D. Jou, J. Phys. A 29, 4321 (1996) zbMATHCrossRefADSGoogle Scholar
  35. T.D. Frank, Phys. Lett. A 267, 298 (2000) zbMATHCrossRefADSGoogle Scholar
  36. T.D. Frank, Physica A 292, 392 (2001) zbMATHCrossRefADSGoogle Scholar
  37. S. Martinez, A.R. Plastino, A. Plastino, Physica A 259, 183 (1998) CrossRefGoogle Scholar
  38. E.M.F. Curado, F.D. Nobre, Phys. Rev. E 67, 021107 (2003) CrossRefADSGoogle Scholar
  39. F.D. Nobre, E.M.F. Curado, G. Rowlands, Physica A 334, 109 (2004) CrossRefADSGoogle Scholar
  40. M. Shiino, J. Math. Phys. 42, 2540 (2001) zbMATHCrossRefADSGoogle Scholar

Copyright information

© EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007

Authors and Affiliations

  1. 1.Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, Rio de JaneiroRJBrazil

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