Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 356 / 2 / b / Solution

For $0\le F_0<1$, stationary points satisfy $\cos x=F_0$. There is one minimum $m_k=2\pi -\arccos F_0+2\pi k$ and one maximum $M_k=\arccos F_0+2\pi k$ per period. The potential is a sinusoidal washboard tilted downward to the right by $2\pi F_0$ per period.

Periodization partitions the real line into translated cells, soTranslation by $2\pi$ merely reindexes the sum, proving periodic boundary conditions; summing the Fokker--Planck equation proves that $\hat{p}$ obeys it.

At stationarity the current $J_0=-D{\hat{p}}_s^{\prime }-{\varphi }^{\prime }{\hat{p}}_s$ is constant. Solving this first-order equation and imposing periodicity givesHere $J_0$ is the stationary probability crossing any point per unit time. Normalization determines it: