Brownian motion and the Stokes–Einstein relation

A pollen grain in water jitters visibly under the microscope because it is struck by molecules from every side and the strikes never quite balance — Brownian motion, the visible end of a random walk driven by thermal energy. Einstein's insight in 1905 was that the same molecular collisions cause both the kicking and the drag, so the two must be linked: the Stokes–Einstein relation gives the diffusion coefficient as thermal energy divided by the drag coefficient of the particle, tying a random microscopic quantity to two things anyone can measure. It was the argument that made atoms hard to deny, and it is still how particle sizes are measured in a dynamic light scattering instrument.

Practise this concept

Robert Brown watched pollen grains in water in 1827 and found them in ceaseless irregular motion. The grains were not alive and nothing was stirring the water: they were being hit by water molecules, roughly a million million million times a second, from every direction at once. The hits nearly cancel, and the small residue left over at each instant is what shakes the grain. What you see under the microscope is therefore a random walk with its steps too small and too frequent to resolve individually.

Einstein's 1905 paper turned that picture into something measurable. The trick was to notice that the molecular collisions do two jobs at the same time: they kick the particle around, and they resist it when it moves. Those cannot be independent, and the connection between them — the first fluctuation-dissipation relation — fixes the diffusion coefficient as the thermal energy divided by the drag coefficient. For a sphere in a viscous fluid the drag is Stokes', six pi times viscosity times radius, which gives the Stokes-Einstein relation: warmer means faster diffusion, thicker or bigger means slower, and no free parameters are left over.

Two consequences are worth carrying away. First, the relation contains Boltzmann's constant, so measuring the jitter of a visible particle gives a value for a molecular quantity — Perrin did exactly that in 1908, extracted Avogadro's number, and effectively ended the argument about whether atoms were real. Second, run it backwards: measure a diffusion coefficient by watching how quickly a speckle pattern of scattered laser light fluctuates, and the same equation returns the particle's radius. That is what a dynamic light scattering instrument does, and it is routine for sizing proteins, polymers and nanoparticles today.

Key ideas

Requires: Random walks and diffusion, Viscosity and Poiseuille flow

Formulas

D = \frac{k_\mathrm{B} T}{6 \pi \eta r}

Symbol	Name	Unit
$D$	diffusion coefficient	m^2/s (square metre per second)
$T$	temperature	K (kelvin)
$eta$	viscosity	Pa*s (pascal-second)
$r$	radius	m (metre)

Common misconceptions

Thinking a Brownian particle is pushed by a current, a temperature gradient or some other external influence. Nothing external is needed: a fluid in perfect equilibrium still delivers an uneven hail of molecular collisions, and the imbalance at any instant is what moves the grain. Stir the fluid and you add drift on top, but the jitter is there in still, uniform water and does not stop.
Expecting a thicker, more viscous fluid to hold a particle still but leave its diffusion coefficient alone, as if drag and random kicking were separate effects. They are the same molecular collisions counted two ways, which is exactly what the Stokes–Einstein relation says: raise the viscosity and the diffusion coefficient falls in proportion, so a particle in glycerol wanders far more slowly than the same particle in water at the same temperature.
Assuming every particle in a fluid jitters visibly, so failing to see it means the fluid is too still. The kicks are the same size whatever the particle, but the response is not: a grain a micrometre across is displaced measurably, while a sand grain of the same material is millions of times heavier and better damped, so its Brownian displacement is far below anything a microscope resolves. Brownian motion does not stop at large sizes; it becomes unobservable.

Resources

Articles

Part of these UT courses

This concept is mapped as covering part of the following course(s) in the University of Tartu curriculum.

Stochastics for physicistsLOFY.04.052