For a gas trapped in a harmonic potential, the sloshing (or Kohn) mode is undamped and its frequency coincides with the trap frequency, independently of the statistics, interaction and temperature of the gas. However, experimental trap potentials have usually Gaussian shape and anharmonicity effects appear as the temperature and, in the case of Fermions, the filling of the trap are increased. We study the sloshing mode of a degenerate Fermi gas in an anharmonic trap within the Boltzmann equation, including in-medium effects in both the transport and collision terms. The calculated frequency shifts and damping rates of the sloshing mode due to the trap anharmonicity are in satisfactory agreement with the available experimental data. We also discuss higher-order dipole, octupole, and bending modes and show that the damping of the sloshing mode is caused by its coupling to these modes.

Using the test-particle method, we solve numerically the Boltzmann equation for an ultra-cold gas of trapped fermions with realistic particle number and trap geometry in the normal phase. We include a mean-field potential and in-medium modifications of the cross-section obtained within a T matrix formalism. After some tests showing the reliability of our procedure, we apply the method to realistic cases of practical interest, namely the anisotropic expansion of the cloud and the radial quadrupole mode oscillation. Our results are in good agreement with experimental data. Although the in-medium effects significantly increase the collision rate, we find that they have only a moderate effect on the anisotropic expansion and on frequency and damping rate of the quadrupole mode.

We consider a polarized Fermi gas in the BCS-BEC crossover region above the critical temperature within a T matrix formalism. By treating the mean-field like shift of the quasiparticle energies in a self-consistent manner, we avoid the known pathological behavior of the standard Nozières-Schmitt-Rink approach in the polarized case, i.e., the polarization has the right sign and the spin polarizability is positive. The momentum distributions of the correlated system are computed and it is shown that, in the zero-temperature limit, they satisfy the Luttinger theorem. Results for the phase diagram, the spin susceptibility, and the compressibility are discussed.

The phase-transition lines between the normal and the FFLO phase, shown as the blue dashed lines in Figs. 6 and 7 of our article, are not correct. Actually, it is not possible to calculate these lines within the present theoretical framework. This can be seen as follows. Consider first the transition from the normal to the BCS (q = 0) superfluid phase. In this case, the T matrix (,q) develops for T → T c (or δμ * → δμ * c) a pole at = 0, q = 0. In spite of this pole, the correction to the density ρ (1) and the self-energy σ remain well defined in the limit T → T c (or δμ * → δμ * c), because a pole at q = 0 has zero weight in the integral over q in the calculation of ρ (1) and in the integral over k = q − k in the calculation of σ. This is, however, no longer true for the phase transition towards the FFLO phase where the pole appears at a finite value q = 0. Then, ρ (1) and σ diverge in the limit T → T c (or δμ * → δμ * c). The phase-transition lines in Figs. 6 and 7 of our article were computed by extrapolating ρ (1) from two values of (δμ * ,T) close to the phase transition line but still in the normal phase to the point (δμ * c ,T c) on the phase-transition line. But as discussed above, this extrapolation cannot give a meaningful result in the case q = 0.

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