  An origin is sought for the ubiquity of cusps, both in computer simulations
of halo formation in hierarchical clustering cosmogonies and in observations of
galactic nuclei by the Hubble Space Telescope (HST). The encounters 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-state, self-consistent, cusped solution of the
collisional Boltzmann equation corresponding to $\rho \sim r^{-4/3}$. This
equilibrium is both stable and an attractor. It is the natural end-point 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 luminosity
density can be accentuated or softened, depending on the form of the
mass-luminosity relation. Possible applications to the cusped nuclei of
early-type galaxies are discussed.
