An origin is sought for the ubiquity of cusps, both in computer simula
tions of halo formation in hierarchical clustering cosmogonies and in
observations of galactic nuclei by the Hubble Space Telescope. The enc
ounters of merging clumps that built the galaxies can be described by
the collisional Boltzmann equation. Using insights gained by studying
the simpler Fokker-Planck equation, we show that there is a steady-sta
te, self-consistent, cusped solution of the collisional Boltzmann equa
tion corresponding to rho similar to r(-4/3). This equilibrium is both
stable and an attractor. It is the natural endpoint of the diffusive
encounters of an ensemble of equal mass clumps. The introduction of a
mass spectrum weakens the mass density cusp. The spike in the luminosi
ty density can be accentuated or softened, depending on the form of th
e mass-luminosity relation. Possible applications to the cusped nuclei
of early-type galaxies are discussed.