Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 353 / 4 / c

Let $Q=(\sigma {\sigma }^T{)}^{-1}$ and use its induced inner product. The autonomous Lagrangian isIts conserved Hamiltonian isOn an infinite-duration fluctuation path $H=0$, so $\Vert \dot{\varphi }\Vert =\Vert a\Vert$. Expanding the action then givesSince metric arclength satisfies $ds=\Vert d\varphi \Vert =\Vert \dot{\varphi }\Vert dt=\Vert a\Vert dt$,This is the geometric minimum action representation and is independent of the speed used to parametrize the path.

For constant isotropic diffusion and detailed balance, the drift has gradient formfor a positive scalar mobility $M$ after absorbing temperature and diffusion constants into $V$. Deterministic relaxation follows $a$ downhill. The least-action escape trajectory is its time reverse,Its tangent is therefore parallel to $\nabla V$. Since the gradient is normal to every level set $V=\text{constant}$, the escape path from a local minimum crosses all equipotentials orthogonally.