Abstract.
In one of the most general possible forms of nonlinear Fokker-Planck equations in one-dimensional space, we consider systems under multiple external potentials with a single diffusion term. We prove the associated H-theorem: that is, the free-energy functional decreases in time under a suitable condition, which relates the entropic form and the two nonlinear functionals in the equations. We consider the constraint on the entropic form and provide a constraint on the functional contained in the diffusion term. Moreover, we see that the gauge field associated with this type of general nonlinear Fokker-Planck equations represents an indicator for fluctuations of the probability flux velocity.
Similar content being viewed by others
References
T.D. Frank, Nonlinear Fokker-Planck Equations: Fundamentals and Applications (Springer, Berlin, 2005)
A.D. Fokker, Ann. Phys. 43, 810 (1914)
M. Planck, Sitzungsber. Preuss. Akad. Wissens. 24, 324 (1917)
G.A. Casas, F.D. Nobre, E.M.F. Curado, Phys. Rev. E 86, 061136 (2012)
E.M.F. Curado, F.D. Nobre, Entropy 20, 183 (2018)
V. Schwämmle, F.D. Nobre, E.M.F. Curado, Phys. Rev. E 76, 041123 (2007)
V. Schwämmle, E.M.F. Curado, F.D. Nobre, Eur. Phys. J. B 58, 159 (2007)
M.S. Ribeiro, G.A. Casas, F.D. Nobre, J. Phys. A 50, 065001 (2017)
C. Tsallis, Introduction to Nonextensive Statistical Mechanics (Springer, New York, NY, USA, 2009)
A.R. Plastino, A. Plastino, Physica A 222, 347 (1995)
T. Yamano, Phys. Lett. A. 374, 3116 (2010)
A. Plastino, A.R. Plastino, Phys. Lett. A 226, 257 (1997)
T. Yamano, Eur. Phys. J. B 18, 103 (2000)
A.M. Scarforne, H. Matsuzoe, T. Wada, Phys. Lett. A 380, 3022 (2016)
H. Feng, J. Wang, J. Chem. Phys. 135, 234511 (2011)
J. Wang, L. Xu, E. Wang, Proc. Natl. Acad. Sci. 105, 12271 (2008)
M.E. Peskin, D.V. Schröder, An Introduction to Quantum Field Theory (Perseus Books Publishing, Cambridge, Massachusetts, 1995) Chapt. 15
T. Yamano, Cent. Eur. J. Phys. 11, 910 (2013)
B.R. Frieden, Science from Fisher Information - A Unification (Cambridge University Press, Cambridge, 2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yamano, T. General nonlinear Fokker-Planck equations with multiple potentials: H-theorem and constraints. Eur. Phys. J. Plus 133, 439 (2018). https://doi.org/10.1140/epjp/i2018-12330-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/i2018-12330-5