  We use the recently developed kinetic theory with Berry curvature to describe
the fermions and antifermions of a chiral relativistic plasma. We check that
this transport approach allows to reproduce the chiral anomaly equation of
relativistic quantum field theory at finite temperature. We also check that it
allows to describe the anomalous gauge polarization tensor that appears in the
Hard Thermal (and/or Dense) effective field theory. We also construct an energy
density associated to the gauge collective modes of the chiral relativistic
plasma, valid in the case of small couplings or weak fields, which can be the
basis for the study of their dynamical evolution.
