Free energy and the Fokker-Planck equation

We establish a new and intriguing connection between the Fokker-Planck equation with gradient drift term and an associated free energy functional. Namely, we demonstrate that such a Fokker-Planck equation may be interpreted as a gradient flux, of a steepest descent, of a free energy functional with respect to a certain metric, This is accomplished through the construction of a time-discrete iterative variational scheme whose solutions converge to the solution of the Fokker-Planck equation. The time step in this scheme is governed by the Wasserstein metric on probability measures.