= Solution
\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> is
$$
m\dot{\mathbf v}=-\zeta\mathbf v+\boldsymbol\xi(t),\qquad\langle\xi_i(t)\xi_j(t')\rangle=2\zeta k_BT\,\delta_{ij}\delta(t-t').
$$
The 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> is
$$
\boxed{D=\frac{k_BT}{\zeta},\qquad\langle|\mathbf r(t)-\mathbf r(0)|^2\rangle=6Dt\quad\hbox{in three dimensions}.}
$$
For 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 $\partial_tP=D\nabla^2P$; 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 $3(k_BT/m)t^2$.
\Image[/past-exam-of-the-thermal-and-statistical-physics-course-of-the-university-of-cambridge/2016/paper-1-brownian-motion.png]
{title=An illustrative random trajectory and the inertial crossover from ballistic to diffusive mean-square displacement}
{source=paper-1-brownian-motion.py}
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.<Stochastic Processes by David Tong>[Sections 3.1–3.2: Brownian motion, Langevin equation, diffusion and the Einstein relation]{ref}
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