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

From QFT to Boltzmann: Freeze-in in the presence of oscillating condensates