We present a generalization of the fluid-dynamical scheme developed for nuclear physics to the case of two trapped fermion species with pairing interactions. To establish a macroscopic description of the mass and momentum conservation laws, we adopt a generalization of the usual Thomas-Fermi approach that includes the pairing energy. We analyze the equilibrium density and gap profiles for an equal population mixture of harmonically trapped
6Li atoms for different choices of the local equation of state. We examine slight departures from equilibrium within our formulation, finding that density oscillations can propagate as first sound coupled to pairing vibrations, that in a homogeneous fermion system exhibit a Bogoliubov-like quasiparticle spectrum. In this case, the dispersion relation for the coupled modes displays a rich scenario of stable, unstable, and damped regimes.
An upper bound is derived for $\ensuremath{\Delta}$ for a cold dilute fluid of equal amounts of two species of fermion in the unitary limit ${k}_{f}a\ensuremath{\rightarrow}\ensuremath{\infty}$ (where
kf is the Fermi momentum,
a is the scattering length, and $\ensuremath{\Delta}$ is a pairing energy: the difference in energy per particle between adding to the system a macroscopic number (but infinitesimal fraction) of particles of one species compared to adding equal numbers of both. The bound is $\ensuremath{\delta}\ensuremath{\le}\frac{5}{3}[2(2\ensuremath{\xi}{)}^{2/5}\ensuremath{-}(2\ensuremath{\xi})]$ where $\ensuremath{\xi}=ϵ/{ϵ}_{\mathrm{FG}}$, $\ensuremath{\delta}=2\ensuremath{\Delta}/{ϵ}_{\mathrm{FG}}$;
ϵ is the energy per particle and
ϵFG is the energy per particle of a noninteracting Fermi gas. If the bound is saturated, then systems with unequal densities of the two species will separate spatially into a superfluid phase with equal numbers of the two species and a normal phase with the excess. If the bound is not saturated, then $\ensuremath{\Delta}$ is the usual superfluid gap. If the superfluid gap exceeds the maximum allowed by the inequality, phase separation occurs.