Axion dark matter has the unique property of having extremely high quantum occupancy. This may lead to think that axion dark matter can be described in terms of a classical field.

The usual equations for the linear growth of density perturbations in the cold dark matter fluid can be obtained from the classical description of a self-gravitating scalar field in the non-relativistic regime, with differences appearing only on scales smaller than a critical length.

However, axion dark matter is expected to thermalize through gravitational self-interactions and to form a Bose-Einstein condensate. I will argue that the classical field approximation is in general not valid for thermalizing quantum fields, even in the large occupancy regime, and discuss the different evolution of a homogeneous condensate in classical and quantum field theory.