  Short-range repulsion governs the dynamics of matter from atoms to animals.
Using theory, simulations, and experiments, we find that an ensemble of
repulsive particles spreads compactly with a sharp boundary, in contrast to the
diffusive spreading of Brownian particles. Starting from the pair interactions,
at high densities, the many-body dynamics follow non-linear diffusion with a
self-similar expansion, growing as $t^{1/4}$; At longer times, thermal motion
dominates with the classic $t^{1/2}$ expansion. A logarithmic growth controlled
by nearest-neighbor interactions connects the two self-similar regimes.
