  We investigate how to describe the dissipative spin dynamics of the
driven-dissipative Dicke model, describing $N$ two-level atoms coupled to a
cavity mode, after adiabatic elimination of the cavity mode. To this end, we
derive a Redfield master equation which goes beyond the standard secular
approximation and large detuning limits. We show that the secular (or rotating
wave) approximation and the large detuning approximation both lead to
inadequate master equations, that fail to predict the Dicke transition or the
damping rates of the atomic dynamics. In contrast, the full Redfield theory
correctly predicts the phase transition and the effective atomic damping rates.
Our work provides a reliable framework to study the full quantum dynamics of
atoms in a multimode cavity, where a quantum description of the full model
becomes intractable.
