  We investigate a family of generalized Fokker-Planck equations that contains
Richardson and porous media equations as members. Considering a confining drift
term that is related to an effective potential, we show that each equation of
this family has a stationary solution that depends on this potential. This
stationary solution encompasses several well-known probability distributions.
Moreover, we verify an $H$ theorem for the generalized Fokker-Planck equations
using free-energy-like functionals. We show that the energy-like part of each
functional is based on the effective potential and the entropy-like part is a
generalized Tsallis entropic form, which has an unusual dependence on the
position and can be related to a generalization of the Kullback-Leibler
divergence. We also verify that the optimization of this entropic-like form
subjected to convenient constraints recovers the stationary solution. The
analysis presented here includes several studies about $H$ theorems for other
generalized Fokker-Planck equations as particular cases.
