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

Solved by gpt-5.6-sol high.