I really feel better saying that with the jargon, so:

Complete uniformity is an unstable equilibrium. Assume that the initial distribution is *nearly* uniform, so we can use linear stability analysis. Said distribution can be broken down into sine waves by Fourier decomposition, and their rate of growth depends on their frequency. In reasonable models, the growth rate has a maximum, so that component of the initial distribution grows exponentially faster than the others, and dominates the resulting form.