The particle model. A gas is made of tiny particles, very far apart compared with their size, flying about in straight lines and bouncing off each other and off the walls. The faster they move, the hotter the gas. The temperature is a measure of the average kinetic energy of the particles: for a gas in two dimensions the average kinetic energy of a particle is kT (and (3/2)kT in three dimensions), where k is the Boltzmann constant. So the typical speed goes up as the square root of the temperature:

v_typical ∝ √T

Brownian motion. In 1827 the botanist Robert Brown looked at pollen grains in water through a microscope, and saw them jiggling about for ever. At first people thought they were alive. In 1905 Einstein showed that the grains are being hit by the water molecules, which are too small to see. At any moment, more molecules hit one side than the other by chance, and so the grain moves a little. Over time the grain does a random walk: the average of the square of the distance from the start increases in proportion to the time,

⟨r²⟩ = 4Dt (in two dimensions)

and D is the diffusion constant. This was the first strong evidence that atoms and molecules really exist, and Jean Perrin got a Nobel Prize for the measurements which gave the size of atoms. The graph shows that ⟨r²⟩ grows roughly in a straight line. (Very close to the start the grain moves in a straight line, so the distance grows faster than that. In two dimensions the collisions also leave a little memory of the motion, so the growth is a little faster than proportional over the first few seconds.) A hotter gas makes the grain move more quickly, and a heavier grain moves more slowly.

What the model leaves out. The particles are hard discs in two dimensions, with perfectly elastic collisions and no forces between them. Real particles are much smaller, much more numerous and move in three dimensions. The units of the time and the speed are chosen so that you can see them. The “temperature” sets the average kinetic energy in the model in the same way as for a real gas.

Things to try. (1) Make the big particle lighter and heavier. How does its path change? (2) Raise and lower the temperature. What happens to the motion? (3) Why does the path of the big particle look like a random walk? (4) Why is the green line roughly straight? (5) Compare the values of D: does doubling the temperature double D? Does doubling the mass halve it?

Reading the graphs. Both graphs have fixed scales, so runs at different settings can be compared. The top graph is one particle’s r², so it is very jumpy. The bottom graph averages over many time intervals, so it is smoother; its slope is 4D, and D (the diffusion constant) is worked out from the first 3 seconds. A hotter gas or a smaller, lighter particle gives a steeper line and a bigger D. After a long time the line bends over because the particle cannot leave the box.