Condensation of hard spheres under gravity

Starting from the Enskog equation of hard spheres of mass m and diameter D under the gravity g, we first derive the exact equation of motion for the equilibrium density profile at a temperature T and examine its solutions via the gradient expansion. The solutions exist only when βμ μ0, where μ is the dimensionless initial layer thickness and β=mgD/T, and the precise value of the upper bound μ0 depends on the underlying packing. When this inequality breaks down, a fraction of particles condenses from the bottom toward the surface.