Brownian motion and random walks.

Step 1 of 5

A speck that never rests

The Scene has a drop of water under a microscope, with a speck in it a micrometre across, a thousandth of a millimetre. It moves exactly as fast as such a speck really does in water. The blue line marks where it has been over the last half minute.

In 1827 the botanist Robert Brown saw motion like this in the particles that came out of pollen grains, and could not explain it. It has been called Brownian motion ever since. Watch the speck for a few moments.

Try it: after answering, step Zoom to the molecules to Molecules. The gold arrow is the net push over the last moment.

Back to the experiment ↑

Step 2 of 5

The speck’s size

Each molecule that hits the speck gives it a tiny push, and a vast number of them hit it every second. If the pushes from all sides balanced exactly, the speck would stand still. They never quite do. The Water temperature is fixed now. Predict what a bigger speck will change.

Step 3 of 5

Four times as long

A path made of steps in random directions, each forgetting the last, is called a random walk. A hundred steps in a straight line take you a hundred steps from the start. A hundred random steps undo each other a lot, and take you on average only about ten steps: after $N$⁠ steps the typical distance is one step’s length times $\sqrt{N}$⁠.

Zoom to the molecules is at Many specks now: sixty specks set off together from the centre, and the red ring marks their typical distance from the start. The chart beside the Scene plots it against time. In symbols, the average of the squared distance grows in step with time:

$\langle r^2 \rangle = 4Dt$$\langle r^2 \rangle =$$4Dt$⁠

$D$⁠ is the diffusion coefficient, which says how quickly something spreads.

Step 4 of 5

Einstein’s formula

In 1905 Albert Einstein worked out what sets $D$⁠ for a ball in a liquid:

$D = \frac{kT}{6\pi\eta r}$$D =$$\frac{kT}{6\pi\eta r}$⁠

The temperature $T$⁠, in kelvin, is on top: warmer molecules move faster and hit harder. Underneath are the speck’s radius $r$⁠ and the viscosity $\eta$⁠, how thick the liquid is. $k$⁠ is Boltzmann’s constant. In 1908 Jean Perrin measured how specks like these spread, found exactly what Einstein had predicted, and convinced the last doubters that atoms and molecules really exist. The water is boiling now, at 373 K, and both Parameters are fixed.

Step 5 of 5

Faster

Water also gets much runnier as it warms: its viscosity falls from about 1.8 × 10⁻³ Pa·s at 0 °C to about 0.28 × 10⁻³ Pa·s near boiling. So heating the water raises $D$⁠ more than the temperature alone would.