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

Equation (1) is Underdamped Langevin dynamics for a unit-mass particle in potential $U$, coupled to a heat bath of temperature $T$. The coefficient $\gamma >0$ is viscous friction, and the noise amplitude is fixed by the Fluctuation-dissipation theorem. Its Fokker-Planck equation isFor $\gamma \gg 1$, velocity relaxes rapidly, so formallyThus the Overdamped Langevin dynamics isand its density obeys

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:

Each net crossing of a periodic cell advances the unwrapped particle by $2\pi$. The stationary mean velocity is thereforeFor $F_0>0$ the numerator is positive, so motion is on average toward increasing $x$, down the tilted potential.

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 $k_{+}/(k_{+}+k_{-})$.

An exact Gillespie algorithm for one particle is:

For $N\ll K$, simulate particle identities independently and maintain a priority queue of their next event times. For $N\gg K$, store occupation numbers $n_k$ and use aggregate event rates $n_k{k}_{+}$ and $n_k{k}_{-}$ for each well; one population-level Gillespie event then decrements one $n_k$ and increments its neighbor. This replaces work proportional to particle number by work proportional to the number of occupied wells.