  We analyse the dynamics leading to radiative cooling of an atomic ensemble
confined inside an optical cavity when the atomic dipolar transitions are
incoherently pumped and can synchronize. Our study is performed in the
semiclassical regime and assumes that cavity decay is the largest rate in the
system dynamics. We identify three regimes characterising the cooling. At first
hot atoms are individually cooled by the cavity friction forces. After this
stage, the atoms' center-of-mass motion is further cooled by the coupling to
the internal degrees of freedom while the dipoles synchronize. In the latest
stage dipole-dipole correlations are stationary and the center-of-mass motion
is determined by the interplay between friction and dispersive forces due to
the coupling with the collective dipole. We analyse this asymptotic regime by
means of a mean-field model and show that the width of the momentum
distribution can be of the order of the photon recoil. Furthermore, the
internal excitations oscillate spatially with the cavity standing wave forming
an antiferromagnetic-like order.
