1 Random walks
In 1827 Robert Brown noticed that pollen grains suspended in water perform a chaotic and endless dance, but it took many years before it was realized that Brownian motion could reconcile an apparent paradox between thermodynamics and Newtonian mechanics. Einstein played a key role in understanding Brownian motion by predicting that the root mean square displacement of such a particle (green) with respect to its starting point (the centre of the box) increases with the square root of time. Before Einstein came along, experimentalists had assumed that this displacement varied linearly and had therefore been measuring the wrong quantity.