  We examine in detail the mechanisms behind thermalization and Bose-Einstein
condensation of a gas of photons in a dye-filled microcavity. We derive a
microscopic quantum model, based on that of a standard laser, and show how this
model can reproduce the behavior of recent experiments. Using the rate equation
approximation of this model, we show how a thermal distribution of photons
arises. We go on to describe how the non-equilibrium effects in our model can
cause thermalization to break down as one moves away from the experimental
parameter values. In particular, we examine the effects of changing cavity
length, and of altering the vibrational spectrum of the dye molecules. We are
able to identify two measures which quantify whether the system is in thermal
equilibrium. Using these we plot "phase diagrams" distinguishing BEC and
standard lasing regimes. Going beyond the rate equation approximation, our
quantum model allows us to investigate both the second order coherence,
$g^{(2)}$, and the linewidth of the emission from the cavity. We show how the
linewidth collapses as the system transitions to a Bose condensed state, and
compare the results to the Schawlow--Townes linewidth.
