Recently there has been significant interest in the claim that dark matter
axions gravitationally thermalize and form a Bose-Einstein condensate with
cosmologically long-range correlation. This has potential consequences for
galactic scale observations. Here we critically examine this claim. We point
out that there is an essential difference between the thermalization and
formation of a condensate due to repulsive interactions, which can indeed drive
long-range order, and that due to attractive interactions, which can lead to
localized Bose clumps (stars or solitons) that only exhibit short range
correlation. While the difference between repulsion and attraction is not
present in the standard collisional Boltzmann equation, we argue that it is
essential to the field theory dynamics, and we explain why the latter analysis
is appropriate for a condensate. Since the axion is primarily governed by
attractive interactions -- gravitation and scalar-scalar contact interactions
-- we conclude that while a Bose-Einstein condensate is formed, the claim of
long-range correlation is unjustified.
