The nonlinear Fokker-Planck equation: Comparison of the classical and quantum (boson and fermion) characteristics

Abstract

A "free energy" functional F is studied for the one-dimensional Fokker-Planck equation (FPE). Using F, a notion of distance between a global Maxwellian function M, which is a stationary explicit solution of the FPE, and an arbitrary solution f (with the same total density ρ at the fixed moment t) is introduced. In this way we generalize the important Kullback-Leibler distance. A comparison between classical and quantum characteristics of the FPE solutions is made.