Past exam of the thermal and statistical physics course of the University of Cambridge / 2016 / Paper 1 / 2 / b

Brownian motion is the irregular motion of a mesoscopic particle caused by thermal molecular impacts. A colloidal sphere in a fluid provides a physical example. The random force competes with dissipative drag; averaging many trajectories produces diffusion even though each trajectory fluctuates.

A free-particle Langevin equation with thermal Langevin force normalization isThe noise has zero mean, and its strength is linked to drag by the fluctuation-dissipation theorem. The velocity relaxes on the time scale $\tau =m/\zeta$. At long times $t\gg \tau$, the Einstein diffusion relation isFor a slowly moving no-slip sphere of radius $a$ in a continuum fluid of dynamic viscosity $\eta$, Stokes drag law gives $\zeta =6\pi \eta a$. The particle probability density obeys ${\partial }_{t}P=D{\nabla }^{2}P$; a localized initial position develops a Gaussian profile. The ballistic-to-diffusive Brownian displacement changes character: at $t\ll \tau$, equilibrium velocities instead give ballistic mean-square displacement $3(k_{B}T/m)t^2$.

The drag-noise model assumes a classical bath with a sufficiently short correlation time; a mathematical white-noise trajectory does not resolve individual molecular collisions. Tracking nonliving colloidal particles measures diffusion and tests the connection between thermal fluctuations and dissipation.^[ref]