Abstract
Consequences of the connection between nonlinear Fokker-Planck equations and entropic forms are investigated. A particular emphasis is given to the feature that different nonlinear Fokker-Planck equations can be arranged into classes associated with the same entropic form and its corresponding stationary state. Through numerical integration, the time evolution of the solution of nonlinear Fokker-Planck equations related to the Boltzmann-Gibbs and Tsallis entropies are analyzed. The time behavior in both stages, in a time much smaller than the one required for reaching the stationary state, as well as towards the relaxation to the stationary state, are of particular interest. In the former case, by using the concept of classes of nonlinear Fokker-Planck equations, a rich variety of physical behavior may be found, with some curious situations, like an anomalous diffusion within the class related to the Boltzmann-Gibbs entropy, as well as a normal diffusion within the class of equations related to Tsallis’ entropy. In addition to that, the relaxation towards the stationary state may present a behavior different from most of the systems studied in the literature.
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References
M. Muskat, The Flow of Homogeneous Fluids Through Porous Media (McGraw–Hill, New York, 1937)
B. Berkowitz, H. Scher, Phys. Rev. Lett. 87, 4038 (1997)
A. Klemm, A.P. Müller, R. Kimmich, Phys. Rev. E 55, 4413 (1997)
G. Ford, J. Lewis, R. O’Connell, Phys. Rev. A 37, 4419 (1988)
H. Spohn, J. Phys. I 3, 69 (1993)
C.-K. Peng, J. Mietus, J. Hausdorff, S. Havlin, H. Stanley, A. Goldberger, Phys. Rev. Lett. 70, 1343 (1993)
R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)
T.D. Frank, Nonlinear Fokker-Planck Equations: Fundamentals and Applications (Springer, Berlin, 2005)
A. Renyi, Probability Theory (North-Holland, Amsterdam, 1970)
C. Tsallis, J. Stat. Phys. 52, 479 (1988)
E.M.F. Curado, Braz. J. Phys. 29, 36 (1999)
E.M.F. Curado, F.D. Nobre, Physica A 335, 94 (2004)
G. Kaniadakis, Physica A 296, 405 (2001)
G. Kaniadakis, Phys. Rev. E 66, 056125 (2002)
Nonextensive Entropy – Interdisciplinary Applications, edited by M. Gell-Mann, C. Tsallis (Oxford University Press, New York, 2004)
Nonextensive Statistical Mechanics: New Trends, New Perspectives, edited by J.P. Boon, C. Tsallis, Europhys. News 36 (2005)
A.R. Plastino, A. Plastino, Physica A 222, 347 (1995)
S. Kamin, Israel J. Math. 14, 76 (1973)
S. Kamin, Arch. Rat. Mech. Anal. 60, 171 (1976)
T.D. Frank, A. Daffertshofer, Physica A 272, 497 (1999)
J.A. Carrillo, G. Toscani, Indiana Univ. Math. J. 49, 113 (2000)
J.A. Carrillo, A. Jüngel, P.A. Markowich, G. Toscani, A. Unterreiter, Monatschefte für Mathematik 133, 1 (2001)
M. Shiino, J. Math. Phys. 42, 2540 (2001)
G. Kaniadakis, Phys. Lett. A 288, 283 (2001)
T.D. Frank, A. Daffertshofer, Physica A 295, 455 (2001)
L. Borland, A.R. Plastino, C. Tsallis, J. Math. Phys. 39, 6490 (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)
V. Schwämmle, F.D. Nobre, E.M.F. Curado, Phys. Rev. E 76, 041123 (2007)
P.H. Chavanis, P. Laurencot, M. Lemou, Physica A 341, 145 (2004)
P.H. Chavanis, Eur. Phys. J. B 54, 525 (2006)
V. Schwämmle, E.M.F. Curado, F.D. Nobre, Eur. Phys. J. B 58, 159 (2007)
M. Shiino, Phys. Rev. A 36, 2393 (1987)
T.D. Frank, Physica A 310, 397 (2002)
P.H. Chavanis, Phys. Rev. E 68, 036108 (2003)
P.H. Chavanis, Eur. Phys. J. B 62, 179 (2008)
C. Tsallis, D. Bukman, Phys. Rev. E R2197, 54 (1996)
T.D. Frank, Phys. Lett. A 305, 150 (2002)
V. Schwämmle, E.M.F. Curado, F.D. Nobre, AIP Conf. Proc. 965, 152 (2007)
D.S. Zhang, G.W. Wei, D.J. Kouri, D.K. Hoffman, Phys. Rev. E 56, 1197 (1997)
L.E. Reichl, A Modern Course in Statistical Physics, 2nd edn. (Wiley, New York, 1998)
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Schwämmle, V., Curado, E. & Nobre, F. Dynamics of normal and anomalous diffusion in nonlinear Fokker-Planck equations. Eur. Phys. J. B 70, 107–116 (2009). https://doi.org/10.1140/epjb/e2009-00172-9
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DOI: https://doi.org/10.1140/epjb/e2009-00172-9