One needs to make sure that all regions of µ/T are physical, and not for instance metastable. In particular, the entropy must increase with temperature, otherwise the specific heat is negative. In Figure 5 we plot the entropy density s , and one sees that ∂s/∂T < 0 when µ/T > 11 . 7. One must bear in mind that our formalism, though non-perturbative in some respects, is still an approximation, and such a feature could be an artifact that disappears once the corrections are incorporated. Since it is beyond the scope of this work to systematically explore these corrections, we will instead try and interpret our results as they are, for this and subsequent cases. The above feature of the entropy density could signify a phase transition at µ/T = 11 . 7, where the density and temperature are related by: where T$_{F}$ is the Fermi temperature ϵ$_{F}$ /k$_{B}$ , where ϵ$_{F}$ is defined in section III, and T$_{c}$ is the critical temperature for this hypothetical phase transition. The above value is comparable to the prediction T$_{c}$ ≈ 0 . 1 T$_{F}$ in [25] for quasi 2d (trapped) systems. In all other regions, and for all other cases considered below, ∂s/∂T > 0. The scaling function b is shown in Figure 6. It has a well defined limit as T/µ → 0, however the limiting value depends on whether µ is positive or negative: Recall b = 1 is the free fermion value. The discontinuity at T/µ = 0 is traced to the fact that c ( ∞ ) = c ( −∞ ), and has no physical significance. Since the density approaches mµ/ 2 π , the equation of state is The energy per particle scaling function ˜ ξ is shown in Figure 7. It has the limiting values: n = 13 . 1 mT 2 π ⇐⇒ T$_{c}$ = 0 . 076 T$_{F}$ (63) lim T/µ → 0 − b = 0 , lim T/µ → 0 $_{+}$b = 1 (64) p = 2 πn $^{2}$/m, ( µ/T → ∞ ) (65) lim µ/T → 0 ˜ ξ = 0 . 907412 , lim µ/T →−∞ ˜ ξ = 0 . 842766 , lim µ/T →∞ ˜ ξ = ∞ (66) 21