  Scalar dark matter (DM), and axions in particular, have an irreducible
abundance of particles produced by freeze-in due to portal interactions with
the Standard Model plasma in the early Universe. In addition, vacuum
misalignment and other mechanisms can lead to the presence of a cold,
oscillating condensate. Therefore, generically, the evolution of the DM in both
forms, condensate and particles, needs to be studied simultaneously. In
non-equilibrium quantum field theory, the condensate and particles are
described by one- and two-point functions, respectively. The fundamental
coupled equations of motion (EoMs) of these objects are non-local. To simplify
the EoMs and bring them into a familiar form for relic abundance calculations,
we perform a Markovianization process for a quasi-harmonically oscillating
homogeneous condensate, leading to local EoMs for the particle distribution
function and the envelope function of condensate oscillation. This reduces the
dynamics to a pair of coupled Boltzmann equations, and we derive explicitly the
form of the collision operators for all particle and condensate interactions.
