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.

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

In two dimensions, the mean squared displacement is

where $D$ is the diffusion coefficient. (In 1-D the prefactor is $2$; in 3-D it is $6$. The factor is twice the dimension.) Einstein further showed that $D$ is set by the thermal energy and the drag the particle feels from the bath:

the Einstein relation, where $\gamma$ is the friction coefficient. For a spherical particle of radius $a$ in a fluid of viscosity $\eta$, Stokes’s law gives $\gamma = 6\pi\eta a$, so

Larger temperature → bigger $D$ (more energetic kicks). Larger particle → smaller $D$ (more inertia, 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.