  We examine the thermal evolution of particle number densities in the early
universe when the particles have a finite diffusion length. Assuming that
annihilations are impossible when the mean separation of the particles is
larger than their diffusion length, we derive a version of the Boltzmann
equation for freeze out in this scenario and an approximate solution, accurate
to better than 2\%. The effect of a finite diffusion length is to increase the
final relic freeze-out abundance over its corresponding value when diffusion
effects are ignored. When diffusion is limited only by scattering off of the
thermal background, and the annihilation cross section is bounded by unitarity,
a significant effect on the freeze-out abundance requires a scattering cross
section much larger than the annihilation cross section. A similar effect is
demonstrated when the relic particles are produced via the freeze-in mechanism,
but in this case the finite diffusion length is due to the scattering of
particles that annihilate into the relic particle of interest. For freeze in,
the effect of a finite diffusion length is to reduce the final relic particle
abundance. The effects of a finite diffusion length are most important when the
scattering cross section or the relic mass are very large. While we have not
found a particularly compelling example where this would affect previous
results, with the current interest in new dark matter candidates it could
become an important consideration.
