  We generalize previously obtained results for the (all orders in the 't Hooft
coupling) thermal free energy of bosonic and fermionic large $N$ Chern-Simons
theories with fundamental matter, to values of the chemical potential larger
than quasiparticle thermal masses. Building on an analysis by Geracie, Goykhman
and Son, we present a simple explicit formula for the occupation number for a
quasiparticle state of any given energy and charge as a function of the
temperature and chemical potential. This formula is a generalization to finite
't Hooft coupling of the famous occupation number formula of Bose-Einstein
statistics, and implies an exclusion principle for Chern-Simons coupled bosons:
the total number of bosons occupying any particular state cannot exceed the
Chern-Simons level. Specializing our results to zero temperature we construct
the phase diagrams of these theories as a function of chemical potential and
the UV parameters. At large enough chemical potential, all the bosonic theories
we study transit into a compressible Bose condensed phase in which the runaway
instability of free Bose condensates is stabilized by the bosonic exclusion
principle. This novel Bose condensate is dual to - and reproduces the
thermodynamics of - the fermionic Fermi sea.
