Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 344 / 3 / c / Solution

Detailed balance says that each equilibrium transition is balanced by its time reverse:With the Boltzmann distribution $P_{\mathrm{eq}}\propto e^{-\beta H}$ and $\Delta H=H_2-H_1$, this requiresParts a and b instead give $\exp [-(2\zeta /{\sigma }^2)\Delta H]$. Equality for every pair of endpoints yields the fluctuation-dissipation relation for a Langevin particle