  We study diffusion in systems of classical particles whose dynamics conserves
the total center of mass. This conservation law leads to several interesting
consequences. In finite systems, it allows for equilibrium distributions that
are exponentially localized near system boundaries. It also yields an unusual
approach to equilibrium, which in $d$ dimensions exhibits scaling with
dynamical exponent $z = 4+d$. Similar phenomena occur for dynamics that
conserves higher moments of the density, which we systematically classify using
a family of nonlinear diffusion equations. In the quantum setting, analogous
fermionic systems are shown to form real-space Fermi surfaces, while bosonic
versions display a real-space analog of Bose-Einstein condensation.
