  We study the time evolution of the quantum fluctuations of the axion field
for both the QCD axion as well as axions arising in the context of supergravity
and string theories. We explicitly keep track not only of the coherently
oscillating zero momentum mode of the axion but also of the higher non-zero
momentum modes using the full axion potential. The full axion potential makes
possible two kinds of instabilities: spinodal instabilities and parametric
resonance instabilities. The presence of either of these instabilities can lead
to a quasi-exponential increase in the occupation of non-zero momentum modes
and the build-up of the quantum fluctuations of the axions. If either of these
becomes a significant effect then axions would no longer be a suitable cold
dark matter candidate. Our results confirm the conventional wisdom that these
effects are not significant in the setting of an expanding FRW universe and
hence axions are indeed cold dark matter candidates.
