1.3 Brownian motion as fluctuation

In 1827 the botanist Robert Brown watched, under a microscope, pollen grains suspended in water. The grains jittered erratically, as if alive — but the same jitter was visible in inanimate dust particles, in soot, in anything small enough. Brown could not explain it. Three-quarters of a century later, Einstein could: the grains were being kicked by the unseen molecules of the surrounding fluid, themselves in vigorous thermal motion of the kind we just visualised. Einstein’s 1905 paper on Brownian motion was the first concrete evidence that molecules are real. The probabilistic structure underlying this jitter — random walks, mean-square displacement growing as $\sqrt{N}$ , the diffusion equation as continuum limit — is developed in Foundations 11.3.

The setup

The simulation below sets the stage Brown actually saw. One large red particle of mass $M$ sits in a bath of about 200 small black particles, each of mass $m \ll M$ . Every collision between the bath and the big particle is perfectly elastic and time-reversible — same physics as the kinetic-theory gas. But the mass asymmetry means each hit barely perturbs the big particle, and the motion you observe is the sum of many small random kicks.

Each 120-second run of the heavy particle's r²(t) is noisy (light blue). Averaged over many runs, the noise washes out (black). If the particle moved in a straight line at the thermal speed ⟨V²⟩ = 2kT/M, the average would grow as t² — the green parabola. It does not. Because each collision randomises direction, the average grows linearly in t, with slope 4D. Linearity is the signature of randomness.

The plot on the right is the squared displacement $|\Delta \mathbf{r}(t)|^2$ of the big particle from its starting position, as a function of elapsed time. Watch what it does over a few tens of seconds: it grows, on average, linearly in $t$ .

Einstein’s relation

Einstein’s 1905 analysis ties together three things the chapter has now met: the random thermal kicks of the bath, the drag a fluid exerts on a moving body, and the thermal energy $k_B T$ . The result, the Einstein relation, is built in three steps — define how fast the particle spreads, define how hard the fluid resists being pushed through, and show that the two are the same physics.

The diffusion coefficient

The squared-displacement plot grows linearly in time, and that linear growth is what defines the diffusion coefficient. Averaged over many runs (the angle brackets $\langle\,\cdot\,\rangle$ denote that average), in $d$ dimensions

\bigl\langle |\Delta \mathbf{r}(t)|^2 \bigr\rangle \;=\; 2 d\, D\, t,

so $1$ -D gives $2Dt$ , $2$ -D gives $4Dt$ (the case plotted), and $3$ -D gives $6Dt$ . The diffusion coefficient $D$ (units $\text{m}^2/\text{s}$ ) measures how fast the particle’s probability cloud spreads: the typical distance wandered grows as $\sqrt{\langle|\Delta\mathbf r|^2\rangle} \sim \sqrt{Dt}$ — the square-root-of-time signature of a random walk, slower than the linear-in-time progress of directed motion.

▶ Why the mean-square displacement grows as 2dDt Derivation

The big particle performs a random walk: each molecular kick nudges it a small step in a random direction, uncorrelated with the previous one. For a sum of independent steps the variances add, so the mean-square displacement grows in proportion to the number of steps — and therefore in proportion to time. Defining the coefficient of that growth to be $2D$ in one dimension,

\langle \Delta x^2\rangle \;=\; 2 D t.

The full random-walk derivation of this linear law, and the diffusion equation it becomes in the continuum limit, are in Foundations 11.3. The motions along $x$ , $y$ , $z$ are independent and each contributes $2Dt$ , so in $d$ dimensions the squared displacements add:

\bigl\langle |\Delta\mathbf r|^2\bigr\rangle \;=\; \langle\Delta x^2\rangle + \langle\Delta y^2\rangle + \cdots \;=\; 2 d\, D\, t.

The prefactor $2, 4, 6$ is twice the number of dimensions. ✓

Drag: the same collisions, biased by motion

When the big particle sits still, the kicks are random and cancel on average; their leftover is the wander measured by $D$ . Push the particle through the fluid at velocity $\mathbf v$ , though, and the kicks stop cancelling: it runs into more molecules on its leading face than its trailing one, and the imbalance is a net drag force opposing the motion,

\mathbf{F}_\text{drag} \;=\; -\gamma\,\mathbf{v},

where $\gamma$ is the friction (drag) coefficient, units $\text{kg/s}$ . (Its inverse $1/\gamma$ is the mobility, the drift speed a unit force produces.) For a sphere of radius $a$ in a fluid of dynamic viscosity $\eta$ , the slow-motion drag is Stokes’s law $\gamma = 6\pi\eta a$ .

▶ Why the drag scales as γ = 6πηa Derivation

At low speed a small sphere’s drag can depend only on its radius $a$ , the fluid’s dynamic viscosity $\eta$ (units $\text{Pa·s} = \text{kg}/(\text{m}\,\text{s})$ ), and its speed $v$ ; the fluid’s inertia — its density — drops out when motion is slow enough. The drag must be linear in $v$ (doubling the speed doubles the rate at which momentum is handed to the fluid), so $F_\text{drag} = \gamma v$ with $\gamma$ assembled from $\eta$ and $a$ alone. Only one combination has the units of $\gamma$ :

[\eta\,a] \;=\; \frac{\text{kg}}{\text{m}\,\text{s}}\cdot\text{m} \;=\; \frac{\text{kg}}{\text{s}} \;=\; [\gamma],

so $\gamma \propto \eta a$ . Solving the fluid equations at low Reynolds number fixes the constant at $6\pi$ , giving $\gamma = 6\pi\eta a$ ; the solution is in Physics → Viscosity & diffusion. ✓

The relation

Diffusion ( $D$ ) and drag ( $\gamma$ ) are the same molecular collisions seen two ways: random and self-cancelling, they make the particle fluctuate; biased by the particle’s motion, they dissipate it. Einstein’s result is that the two are locked together by the thermal energy $k_B T$ — with $k_B$ Boltzmann’s constant, introduced in 1.2:

\boxed{\;D \;=\; \frac{k_B T}{\gamma}.\;}

This is the first instance of the fluctuation–dissipation principle: the size of a system’s spontaneous fluctuations ( $D$ ) is fixed by the friction that damps its driven motion ( $\gamma$ ), with temperature setting the scale. A fluid that resists motion strongly lets its particles wander slowly, and vice versa.

▶ The Einstein relation from a balance of drift and diffusion Derivation

Einstein’s argument is an equilibrium balance. Subject the Brownian particles to a weak external force — gravity on a colloid, say — derived from a potential $U(x)$ , so the force is $F = -\,dU/dx$ . Two particle currents result, and in equilibrium they must cancel everywhere.

Drift. The force drives each particle through the fluid, which resists with the drag $-\gamma v$ . The particle quickly reaches the terminal velocity where force balances drag, $F = \gamma v_\text{drift}$ , so $v_\text{drift} = F/\gamma$ . A population of number density $n(x)$ drifting at this speed is a flux

J_\text{drift} \;=\; n\,v_\text{drift} \;=\; \frac{nF}{\gamma}.

Diffusion. Random kicks carry particles from high to low concentration — Fick’s law — a flux

J_\text{diff} \;=\; -D\,\frac{dn}{dx}.

Equilibrium. With nothing flowing on average, the particles settle into the Boltzmann distribution in the potential (the equilibrium law, Physics → Kinetic theory),

n(x) \;=\; n_0\, e^{-U(x)/k_B T},

and the total flux vanishes, $J_\text{drift} + J_\text{diff} = 0$ . Differentiating the Boltzmann distribution,

\frac{dn}{dx} \;=\; -\frac{1}{k_B T}\frac{dU}{dx}\,n \;=\; \frac{F}{k_B T}\,n,

using $F = -dU/dx$ . Put the two fluxes equal and opposite:

\frac{nF}{\gamma} \;=\; D\,\frac{dn}{dx} \;=\; D\,\frac{nF}{k_B T}.

The common factor $nF$ cancels, leaving

D \;=\; \frac{k_B T}{\gamma}. \;✓

The external force was only a scaffold to set up the balance; it has dropped out, so the relation is a property of the free particle in the bath.

For the spherical particle, substituting Stokes’s law $\gamma = 6\pi\eta a$ gives the form Einstein used to read molecular sizes off a microscope:

D \;=\; \frac{k_B T}{6\pi\eta a}.

Higher temperature raises $D$ (more energetic kicks); a larger or more viscously-embedded particle lowers it (more drag). In the simulation, slide $T$ up and watch the squared-displacement plot grow faster; slide the mass ratio up and watch it grow slower.

Why this matters for sound

Brownian motion is not sound. It is the equilibrium fluctuation of the medium itself — the residual jitter that remains in a fluid even when nothing macroscopic is happening. The bath kicks the big particle, but the kicks are incoherent: positive and negative, in random directions, uncorrelated from one moment to the next. They do not propagate; they do not carry information.

A sound, by contrast, is a coherent deviation from equilibrium. The kicks are organised: the molecules at one location all push in the same direction at the same time, then return, and then push the other direction, in sync. The disturbance has structure both in space and in time. It carries something — energy, information, the contents of a sentence — from where it was generated to wherever the air arrives.

We are about to spend a chapter learning to derive the equation that governs such coherent deviations. Before we do, the next lesson states clearly what a “sound” is, so we know what we are deriving an equation for.