Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 353 / 2 / a

SetThe stated curl-free condition says that $C_j=({\kappa }^{-1}{)}_{ji}{a}_i$ is a gradient, $C=\nabla \Psi$. DefineThen $a=-D\nabla V$, and the Fokker-Planck probability current becomesThis is the potential condition for a Fokker--Planck equation.

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 givesSubstituting the gradient current,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.

Let $Z={\int }_X{e}^{-V}dx$ and $P_B=e^{-V}/Z$. ThenBy nonnegativity of Kullback-Leibler divergence,with equality exactly when $P=P_B$ almost everywhere. Hence a normalizable $P_B$ is the unique minimizer and, because $\dot{F}<0$ elsewhere, the unique steady density compatible with the boundary condition.