  Dark energy and dark matter are two of the biggest mysteries of modern
cosmology, and our understanding of their fundamental nature is incomplete.
Many parameterizations of couplings between the two in the continuity equation
have been studied in the literature, and observational data from the growth of
perturbations can constrain these parameterizations. Assuming standard general
relativity with a simple Yukawa-type coupling between dark energy and dark
matter fields in the Lagrangian, we use the Boltzmann equation to analytically
express and calculate the interaction kernel $Q$ in the continuity equation and
compare it to that of a typical parametrization. We arrive at a comparably very
small result, as expected. Since the interaction is a function of the dark
matter mass, other observational data sets can be used to constrain the mass.
This calculation can be modified to account for other couplings of the dark
energy and dark matter fields. This calculation required obtaining a
distribution function for dark energy that leads to an equation of state
parameter that is negative, which neither Bose-Einstein nor Fermi-Dirac
statistics can supply, and this is the main result of this paper. Treating dark
energy as a quantum scalar field, we use adiabatic subtraction to obtain a
finite analytic approximation for its distribution function that assumes the
FLRW metric and nothing more.
