 Fokker--Planck free-energy functional

For a normalized probability density $P$ and a dimensionless potential $V$, define the Fokker--Planck free-energy functionalAssume $Z=\int e^{-V}dx$ is finite and positive, and write $\pi =e^{-V}/Z$. Then $\mathcal{F}[P]=H(P\mid \pi )-\log Z$. Nonnegativity of relative entropy proves that the unique normalized minimizer is $\pi$, up to equality almost everywhere. The integrability assumption matters: a constant potential on the whole real line has no normalized density proportional to $e^{-V}$.

For a sufficiently regular solution of ${\partial }_{t}P=\nabla \cdot (PD\nabla (V+\log P))$ with symmetric positive-definite $D$ and no-flux boundaries, integration by parts givesThe free energy is thus a Lyapunov functional for the diffusion. Decrease alone is not a quantitative convergence rate; an inequality relating the energy gap to its dissipation, such as a logarithmic Sobolev inequality in the Gaussian case, supplies additional control.