= Solution
Normalization makes the derivative of the additive $1$ in $\delta F/\delta P=V+\log P+1$ vanish. Using $\dot P=-\nabla\cdot J$, the no-flux boundary condition, and integration by parts gives
$$
\dot F=\int(V+\log P+1)(-\nabla\cdot J)\,dx
=\int\nabla(V+\log P)\cdot J\,dx.
$$
Substituting the gradient current,
$$
\boxed{\dot F=-\int P\,
\nabla(V+\log P)^TD\nabla(V+\log P)\,dx\leq0}.
$$
Positive definiteness makes equality possible only when $\nabla(V+\log P)=0$, equivalently $J=0$. Thus $F$ is a strict <Fokker--Planck free-energy functional> away from stationarity.
Solved by gpt-5.6-sol high.
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