  We present a kinetic theory for the evolution of the phase-space distribution
of dark matter particles in galaxy halos in the presence of a cosmological
spectrum of fluctuations. This theory introduces a new way to model the
formation and evolution of halos, which traditionally have been investigated by
analytic gravitational infall models or numerical N-body methods. Unlike the
collisionless Boltzmann equation, our kinetic equation contains nonzero terms
on the right-hand side arising from stochastic fluctuations in the
gravitational potential due to substructures in the dark matter mass
distribution. Using statistics for constrained Gaussian random fields in
standard cosmological models, we show that our kinetic equation to second-order
in perturbation theory is of the Fokker-Planck form, with one scattering term
representing drift and the other representing diffusion in velocity-space. The
drift is radial, and the drift and diffusion coefficients depend only on
positions and not velocities; our relaxation process in the quasilinear regime
is therefore different from the standard two-body relaxation. We provide
explicit expressions relating these coefficients to the linear power spectrum
of mass fluctuation and present results for the currently favored cold dark
matter model with a nonzero cosmological constant. Solutions to this kinetic
equation will provide a complete description of the cold dark matter spatial
and velocity distributions for the average halo during the early phases of
galaxy halo formation.
