0 1 2 3 4 5 0.05 0.10 0.15 0.20 f for repulsive fermions PSfrag replacements f β k $^{2}$/ 2 m n$_{c}$ FIG. 19: Filling fraction f for repulsive fermions for µ/T = − 3 , − 2 , − 0 . 55. The top curve corresponds to the critical density in eq. (95). a pseudo-energy ε$_{a}$ ( k ) for each type of particle and the filling fractions: As shown in Appendix B, in our approximation, the pseudo-energies satisfy the coupled integral equations: where ω$_{a}$ ( k ) = k $^{2}$/ 2 m$_{a}$ , and G$_{ab}$ is the 2-body scattering kernel. By Galilean invariance, the kernel is a function of | v$_{a}$ − v$_{b}$ | where v = k/m is a velocity. For any function g we defined: f$_{a}$ ( k ) = 1 e βε a ( k ) − s$_{a}$ (102) ε$_{a}$ ( k ) = ω$_{a}$ ( k ) − µ$_{a}$ − 1 β log ( 1 + β ∑ b G$_{ab}$ ∗ y − 1 b e βε b − s$_{b}$ ) (103) y$_{a}$ ≡ e − βδ a , δ$_{a}$ ≡ ε$_{a}$ − ω$_{a}$ + µ$_{a}$ (104) ( G$_{ab}$ ∗ g ) ( k ) ≡ ∫ ( d k $^{′}$) G$_{ab}$ ( k , k $^{′}$) g ( k $^{′}$) (105) 39