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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]

Underdamped Langevin dynamics retains both position $q$ and momentum $p$. For mass $m>0$, temperature $T>0$, friction rate $\gamma >0$ and smooth potential energy $U$, one common normalization isHere $W$ is vector Brownian motion. The noise amplitude is tied to friction by the fluctuation-dissipation relation for a Langevin particle. If the density is normalizable and the dynamics is nonexplosive, its invariant distribution is the Boltzmann distributionThe Hamiltonian flow preserves the total energy and phase-space volume; the momentum Ornstein-Uhlenbeck process preserves a centered normal distribution with coordinate variance $m{k}_{B}T$. These two facts verify the invariant density. In dimensionless units with $k_{B}T=1$ and momentum variance $\eta$, it is $\exp [-U(q)-|p{|}^2/(2\eta )]$, with the corresponding choice $m=\eta$.

Retaining inertia resolves the short-time ballistic-to-diffusive Brownian displacement. The overdamped Langevin dynamics approximation removes momentum relaxation when only much longer times are relevant.