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

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.