Brownian Motion and the Langevin Equation

A grain of pollen suspended in water jitters ceaselessly, a motion Brown observed under a microscope in 1827 and could not explain. The explanation, given by Einstein in 1905, is that the grain is bombarded from all sides by water molecules in thermal motion, and the imbalance between collisions on opposite faces at any instant drives it on a random walk.¹ The argument connects a visible, macroscopic motion to the invisible molecular one, and it does so quantitatively: the diffusion of the grain measures the thermal energy per molecule, hence Avogadro's number. This lesson builds the theory in two complementary forms. Einstein's diffusion picture tracks the probability distribution of the grain's position and yields the relation between diffusion and mobility. Langevin's picture writes a stochastic equation of motion, splits the molecular force into a drag and a random part, and fixes the random part by demanding thermal equilibrium — the first instance of a fluctuation–dissipation relation.

The random walk and diffusion

Model the grain's displacement as a sum of independent random steps. In one dimension, after $n$ steps of independent displacements ${\ell }_i$ with $⟨{\ell }_i⟩=0$ and $⟨{\ell }_i{\ell }_j⟩=⟨{\ell }^2⟩{\delta }_{ij}$, the total displacement $x={\sum }_i{\ell }_i$ has zero mean and variance

The mean-square displacement grows linearly in the number of steps, hence linearly in time if steps occur at a fixed rate. Writing $⟨x^2⟩=2Dt$ defines the diffusion constant $D$. The same content appears as a partial differential equation for the probability density $P(x,t)$ of finding the grain at $x$: the continuity of probability under a current proportional to the concentration gradient, $J=-D\partial P/\partial x$, gives the diffusion equation

whose solution from a point release at the origin is a spreading Gaussian,

In $d$ dimensions the independent Cartesian components add, giving $⟨r^2⟩=2dDt$. The linear growth of $⟨r^2⟩$ in time is the signature of diffusion, distinct from the $⟨r^2⟩\propto t^2$ of ballistic motion; it holds once many uncorrelated collisions have occurred.

The Einstein relation between diffusion and mobility

Diffusion and drag are two faces of the same molecular collisions, and Einstein tied them together with an equilibrium argument. Apply a weak steady force $F$ to each suspended particle — gravity on a colloidal grain, for instance. The force drives a steady drift at velocity $v_{\mathrm{drift}}={\mu }_{\mathrm{mob}}F$, where ${\mu }_{\mathrm{mob}}$ is the mobility (drift velocity per unit force). The drift carries a particle current $J_{\mathrm{drift}}=n{\mu }_{\mathrm{mob}}F$ down the force. Opposing it, the concentration gradient that builds up drives a diffusion current $J_{\mathrm{diff}}=-D\partial n/\partial x$ back.

In equilibrium the two currents cancel and the concentration is barometric, $n(x)\propto e^{-U(x)/k_{B}T}$ with $F=-\partial U/\partial x$. Then $\partial n/\partial x=-nF/k_{B}T$, and the balance $J_{\mathrm{drift}}+J_{\mathrm{diff}}=0$ reads

The relation ties a fluctuation (diffusion, the spreading driven by random kicks) to a dissipation (mobility, the response to a steady force). It is the first appearance of the fluctuation–dissipation principle: the same collisions that randomly push the particle also resist its steady drift, and the two effects are locked together by $k_{B}T$. Nothing in the derivation depends on the mechanism of the drag, only on the medium being at temperature $T$.

The Langevin equation

Langevin recast the problem as a stochastic equation of motion. The water exerts two forces on the grain: a systematic drag opposing its velocity, and a rapidly varying random force from individual collisions. For a grain of mass $m$ and velocity $v$,

where $\gamma$ is the friction coefficient — related to the mobility by ${\mu }_{\mathrm{mob}}=1/\gamma$, since a steady force $F$ balances the drag at $v=F/\gamma$ — and $\xi (t)$ is the random force. The split is a separation of time scales: the drag is the slow average of the collisions, and $\xi$ is the fast fluctuating remainder with zero mean, $⟨\xi (t)⟩=0$, so that $⟨v⟩$ obeys the deterministic decay $m\mathbf{d}⟨v⟩/\mathbf{d}t=-\gamma ⟨v⟩$.

Fixing the noise by the fluctuation–dissipation relation

The random force cannot have an arbitrary strength. If it were too weak, drag would drain the grain's kinetic energy and it would fall still; if too strong, the grain would heat without bound. Thermal equilibrium at temperature $T$ pins the strength exactly. Model $\xi (t)$ as Gaussian white noise: uncorrelated from one instant to the next,

with $B$ the noise strength to be determined. Solve the linear equation for the velocity,

square it, and average using the correlation of $\xi$. At long times the initial condition decays and the stationary variance is

Equipartition fixes the left side: a grain in equilibrium has $\frac{1}{2}m⟨v^2⟩=\frac{1}{2}{k}_{B}T$ per translational degree of freedom, so $⟨v^2⟩=k_{B}T/m$. Matching the two expressions,

The friction $\gamma$ appears twice: as the coefficient of the systematic drag (dissipation) and as the strength of the random force (fluctuation). They are not independent because both arise from the same molecular collisions. A medium that resists motion strongly also kicks strongly, and the ratio is fixed by $k_{B}T$. This is the microscopic content of the Einstein relation and the direct ancestor of the general fluctuation–dissipation theorem.

Velocity correlation and the mean-square displacement

The same solution gives the velocity autocorrelation function. For $t\ge 0$,

an exponential decay on the momentum relaxation time $\tau =m/\gamma$. The grain remembers its velocity for a time $\tau$, then the memory is erased by collisions. The diffusion constant is the time integral of this correlation, a relation of the Green–Kubo type,

recovering the Einstein relation from the dynamics. Integrating twice gives the mean-square displacement at all times,

with two regimes separated by $\tau =m/\gamma$:

• Ballistic, $t\ll \tau$: expanding the exponential gives $⟨x^2⟩\approx (k_{B}T/m)t^2$, the free-flight motion of a particle at the thermal speed $\sqrt{k_{B}T/m}$ before it has collided.
• Diffusive, $t\gg \tau$: the bracket is $\approx t$, so $⟨x^2⟩\approx 2Dt$ with $D=k_{B}T/\gamma$, the random walk.

The Stokes–Einstein relation and Avogadro's number

The friction on a sphere of radius $a$ moving slowly through a fluid of viscosity $\eta$ is given by Stokes' law, $\gamma =6\pi \eta a$. Substituting it into $D=k_{B}T/\gamma$ gives the Stokes–Einstein relation,

Every quantity on the right except $k_B$ is measurable: the temperature, the fluid's viscosity, and the grain's radius under the microscope. Every quantity on the left is measurable by tracking the grain, since $⟨r^2⟩=2dDt$ gives $D$ from the slope of the mean-square displacement against time. The relation therefore delivers $k_B$, and with the gas constant $R=N_A{k}_B$ known from the ideal-gas law it delivers Avogadro's number $N_A=R/k_B$.

Perrin's experiments settled the physical reality of molecules. A visible grain, too large to be a molecule and too small to ignore the molecular buffeting, translated the thermal energy $k_{B}T$ of the invisible medium into a measurable diffusion, and the number that came out matched Avogadro's number obtained from entirely independent methods.²

Summary

• A Brownian particle executes a random walk; independent steps make $⟨r^2⟩$ grow linearly in time, $⟨r^2⟩=2dDt$, and the probability density obeys the diffusion equation with a spreading-Gaussian solution.
• The Einstein relation $D={\mu }_{\mathrm{mob}}{k}_{B}T$ follows from balancing drift and diffusion currents in a barometric equilibrium; it ties a fluctuation (diffusion) to a dissipation (mobility) through $k_{B}T$.
• The Langevin equation $m\dot{v}=-\gamma v+\xi$ splits the molecular force into drag and noise; equilibrium forces the fluctuation–dissipation relation $⟨\xi (t)\xi (t^{\prime })⟩=2\gamma k_{B}T\delta (t-t^{\prime })$, so the same $\gamma$ sets both the drag and the noise strength.
• The velocity autocorrelation decays as $(k_{B}T/m)e^{-\gamma t/m}$; its time integral is $D=k_{B}T/\gamma$, and the mean-square displacement crosses over from ballistic $t^2$ at $t\ll m/\gamma$ to diffusive $2Dt$ at $t\gg m/\gamma$.
• The Stokes–Einstein relation $D=k_{B}T/6\pi \eta a$ made the grain's diffusion a measurement of $k_B$, hence of Avogadro's number; Perrin's confirmation established the molecular hypothesis.