Browna

Meet Browna, a pollen grain jiggling in water, kicked about by molecules too small to see.

In 1827 the botanist Robert Brown saw tiny particles from pollen jiggling endlessly in water. In 1905 Albert Einstein explained it: water molecules constantly hit the particle from all sides, and at each instant the kicks do not quite balance. The particle does a random walk. Einstein showed that its mean squared displacement along one axis grows in proportion to time, ⟨x²⟩ = 2Dt, with D set by the temperature, the liquid's viscosity and the particle's size.

Because the spread grows with the square root of time, wandering four times as far takes sixteen times as long.

Jean Perrin's measurements of Brownian motion around 1908 used Einstein's theory to estimate Avogadro's number, convincing remaining sceptics that atoms are real. Perrin received the 1926 Nobel Prize in Physics.

The same mathematics describes diffusion of molecules, noise in electrical circuits and models of fluctuating prices. Over very short times, the particle's motion is not a pure random walk, because it has inertia.