Eur. Phys. J. Plus (2018) 133: 439
https://doi.org/10.1140/epjp/i2018-12330-5

Regular Article

General nonlinear Fokker-Planck equations with multiple potentials: H-theorem and constraints

Faculty of Science, Kanagawa University, 2946, 6-233 Tsuchiya, Hiratsuka, 259-1293, Kanagawa, Japan

^* e-mail: yamano@amy.hi-ho.ne.jp

Received: 27 August 2018
Accepted: 10 September 2018
Published online: 25 October 2018

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.