Sand as Maxwell's Demon

We consider a dilute gas of granular material inside a box, kept in a stationary state by vertical vibrations. A wall separates the box into two identical compartments, save for a small hole at some finite height $h$. As the gas is cooled, a second order phase transition occurs, in which the particles preferentially occupy one side of the box. We develop a quantitative theory of this clustering phenomenon and find good agreement with numerical simulations.