= Fokker--Planck free-energy functional
For a normalized <probability density function>[probability density] $P$ and a dimensionless potential $V$, define the <Fokker--Planck free-energy functional>
$$
\mathcal F[P]=\int(P\log P+VP)\,dx.
$$
Assume $Z=\int e^{-V}\,dx$ is finite and positive, and write $\pi=e^{-V}/Z$. Then $\mathcal F[P]=H(P\mid\pi)-\log Z$. Nonnegativity of <Kullback-Leibler divergence>[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 $\partial_tP=\nabla\cdot(PD\nabla(V+\log P))$ with symmetric <positive-definite matrix>[positive-definite] $D$ and no-flux boundaries, <integration by parts> gives
$$
\frac d{dt}\mathcal F[P_t]=-\int P_t\nabla(V+\log P_t)^{\mathsf T}D\nabla(V+\log P_t)\,dx\leq0.
$$
The free <energy> is thus a <Lyapunov function>[Lyapunov functional] for the <diffusion process>[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.
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