  We study the phase diagram of a two component Fermi system with a weak
attractive interaction. Our analysis includes the leading order Hartree energy
shifts and pairing correlations at finite temperature and chemical potential
difference between the two fermion species. We show that in an asymmetric
system, the Hartree shift to the single particle energies are important for the
phase competition between normal and superfluid phase and can change the phase
transition curve qualitatively. At large asymmetry we find that a novel but
somewhat fragile superfluid state can be favored due to finite temperature
effects. We also investigate the transition between the normal phase and an
inhomogeneous superfluid phase to study how gradient instabilities evolve with
temperature and asymmetry. Finally, we adopt our analysis to study the density
profiles of similar asymmetric Fermi systems that are being observed in cold
atom experiments.
