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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L.E. Reichl, A Modern Course in Statistical Physics, 2nd edn. (Wiley, New York, 1998)
N.G. Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981)
H. Haken, Synergetics (Springer-Verlag, Berlin, 1977)
H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications (Springer-Verlag, Berlin, 1989)
T.M. Cover, J.A. Thomas, Elements of Information Theory (Wiley, New York, 1991)
Nonextensive Statistical Mechanics and Thermodynamics, edited by S.R.A. Salinas, C. Tsallis, Vol. 29, Braz. J. Phys. (1999)
Nonextensive Entropy - Interdisciplinary Applications edited by M. Gell-Mann, C. Tsallis (Oxford University Press., New York, 2004)
Nonextensive Statistical Mechanics: New Trends, New Perspectives, Vol. 36, Europhys. News (2005)
C. Tsallis, J. Stat. Phys. 52, 479 (1988)
S. Abe, Phys. Lett. A 224, 326 (1997)
C. Anteneodo, J. Phys. A 32, 1089 (1999)
E.P. Borges, I. Roditi, Phys. Lett. A 246, 399 (1998)
P.T. Landsberg, V. Vedral, Phys. Lett. A 247, 211 (1998)
E.M.F. Curado, Braz. J. Phys. 29, 36 (1999)
E.M.F. Curado, F.D. Nobre, Physica A 335, 94 2004
A. Renyi, Probability Theory (North-Holland, Amsterdam, 1970)
G. Kaniadakis, Physica A 296, 405 (2001)
G. Kaniadakis, Phys. Rev. E 66, 056125 (2002)
G. Kaniadakis, Phys. Rev. E 72, 036108 (2005)
T.D. Frank, Nonlinear Fokker-Planck equations: Fundamentals and Applications (Springer, Berlin, 2005)
A.R. Plastino, A. Plastino, Physica A 222, 347 (1995)
C. Tsallis, D.J. Bukman, Phys. Rev. E R2197, 54 (1996)
L. Borland, Phys. Rev. E 57, 6634 (1998)
L. Borland, F. Pennini, A.R. Plastino, A. Plastino, Eur. Phys. J. B 12, 285 (1999)
E.K. Lenzi, R.S. Mendes, C. Tsallis, Phys. Rev. E 67, 031104 (2003)
T.D. Frank, A. Daffertshofer, Physica A 272, 497 (1999)
T.D. Frank, A. Daffertshofer, Physica A 295, 455 (2001)
T.D. Frank, Physica A 301, 52 (2001)
L.C. Malacarne, R.S. Mendes, I.T. Pedron, E.K. Lenzi, Phys. Rev. E 63, 030101 (2001)
L.C. Malacarne, R.S. Mendes, I.T. Pedron, E.K. Lenzi, Phys. Rev. E 65, 052101 (2002)
P.H. Chavanis, Phys. Rev. E 68, 036108 (2003)
P.H. Chavanis, Physica A 340, 57 (2004)
A.R. Plastino, L. Borland, C. Tsallis, J. Math. Phys. 39, 6490 (1998)
A. Compte, D. Jou, J. Phys. A 29, 4321 (1996)
T.D. Frank, Phys. Lett. A 267, 298 (2000)
T.D. Frank, Physica A 292, 392 (2001)
S. Martinez, A.R. Plastino, A. Plastino, Physica A 259, 183 (1998)
E.M.F. Curado, F.D. Nobre, Phys. Rev. E 67, 021107 (2003)
F.D. Nobre, E.M.F. Curado, G. Rowlands, Physica A 334, 109 (2004)
M. Shiino, J. Math. Phys. 42, 2540 (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Schwämmle, V., Curado, E. & Nobre, F. A general nonlinear Fokker-Planck equation and its associated entropy. Eur. Phys. J. B 58, 159–165 (2007). https://doi.org/10.1140/epjb/e2007-00217-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjb/e2007-00217-1