  There is no agreement in the literature on the rate of diffusion of a
particle in a cooling granular gas. Predictions and model assumptions range
from the conventional to very exotic dependence of the mean square distance
(MSD) on time. This problem is addressed here by calculating the MSD from
first-principles. The calculation is based on random-walking and it circumvents
the common use of continuum equations and equations of states, which involve
approximations that erode at low gas particle densities. The MSD is found to
increase logarithmically with time -- slower than even in anomalous diffusion.
This result is consistent with the well-established Haff's law for the decay of
the kinetic energy, which is also derived along the way from the same first
principles. This derivation also pins down Haff's time constant, alleviating
the usual need for a fitting parameter. The diffusion theory is then used to
calculate explicitly the time evolution of any initial particle velocity
distribution, yielding an unusual functional form. The limitations of the
theory are discussed and extensions to it are outlined.
