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

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