Past exam of the thermal and statistical physics course of the University of Cambridge / 2014 / Paper 1 / 2 / a

For one Cartesian velocity component of a free Brownian particle, the inertial Langevin equation isThe random force and viscous drag obey the fluctuation-dissipation relation. With a thermal equilibrium velocity distribution, the relaxation time is $\tau =m/\zeta$ andIntegrating velocity twice givesFor $t\gg \tau$, this is Einstein's diffusion law, $⟨\Delta x^2⟩=2Dt$, with $D=k_{B}T/\zeta$. In three dimensions $⟨|\Delta \mathbf{r}{|}^2⟩=6Dt$. At $t\ll \tau$ the motion is ballistic, $⟨\Delta x^2⟩\simeq (k_{B}T/m)t^2$. A sphere in the Stokes-drag regime has $\zeta =6\pi \eta a$, giving $D=k_{B}T/(6\pi \eta a)$.