  It is proposed that the cooling of a thermalized SU($N$) gauge theory can be
formulated in terms of a cascade involving three effective theories with
successively reduced (and spontaneously broken) gauge symmetries, SU($N$) $\to$
U(1)$^{N-1}$ $\to$ Z$_N$. The approach is based on the assumption that away
from a phase transition the bulk of the quantum interaction inherent to the
system is implicitly encoded in the (incomplete) classical dynamics of a
collective part made of low-energy condensed degrees of freedom. The properties
of (some of the) statistically fluctuating fields are determined by these
condensate(s). This leads to a quasi-particle description at tree-level. It
appears that radiative corrections, which are sizable at large gauge coupling,
do not change the tree-level picture qualitatively. The thermodynamic
self-consistency of the quasi-particle approach implies nonperturbative
evolution equations for the associated masses. The temperature dependence of
these masses, in turn, determine the evolution of the gauge coupling(s). The
hot gauge system approaches the behavior of an ideal gas of massless gluons at
asymptotically large temperature. A negative equation of state is possible at a
stage where the system is about to settle into the phase of the (spontaneously
broken) Z$_N$ symmetry.
