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

In the low-temperature regime, the particle rapidly equilibrates near a minimum and only rarely crosses a neighboring maximum. Applying Laplace's method to the exact current formula gives the Kramers escape ratesEach right or left escape changes position by $2\pi$, henceThe reduced continuous-time random walk on minima has off-diagonal transition ratesand diagonal generator entry $W(l|l)=-(k_{+}+k_{-})$. Its residence time in each well is exponentially distributed with rate $k_{+}+k_{-}$, and the next jump is right with probability $k_{+}/(k_{+}+k_{-})$.