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Substituting Eq. (46) in Eq. (59), and introducing the dimensionless variables defined previously, we get 

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up to some unimportant constants. Using α = ( S$_{d}$Gβmρ$_{0}$ ) 1 / $^{2}$R to express ρ$_{0}$ in terms of α and introduc- ing the normalized temperature (51), we finally obtain 

up to some unimportant constants. Using the previous results, this expression relates the entropy S/Nk$_{B}$ to the central density 1 /λ . The free energy is F = E − T S . In the following, it will be more convenient to work in terms of the Massieu function J = S − k$_{B}$βE (by an abuse of language, we shall often refer to J as the free energy). We have 

Using the previous results, this expression relates the free energy J/Nk$_{B}$ to the central density 1 /λ . 

In Figs. 9, 10 and 11, we have plotted the free en- ergy J/Nk$_{B}$ as a function of the inverse temperature η (parameterized by the central density 1 /λ ) in d = 1 , 2 , 3. In Figs. 12, 13, 14 and 15, we have plotted the entropy S/Nk$_{B}$ as a function of the energy Λ (parameterized by the central density 1 /λ ) in d = 1 , 2 , 3. In these figures, the solid lines without label refer to the homogeneous phase. The solid lines with label n = 1 refer to the first inhomogeneous branch. The dashed lines with label n = 2 refer to the second inhomogeneous branch. These 

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curves will be helpful in the next section to analyze the phase transitions in the canonical and microcanonical en- sembles respectively. 

Remark: Since δS = k$_{B}$βδE , the extrema of entropy S ( λ ) and energy E ( λ ) coincide. Since the series of equi- libria E ( λ ) exhibits damped oscillations for 1 /λ → + ∞ in d = 3 (see Fig. 8), this implies that the curve S ( λ ) will also exhibit damped oscillations at the same loca- tions. Correspondingly, S ( E ) will present some “spikes” for 1 /λ → + ∞ in d = 3 (see inset of Fig. 15). Similarly, since δJ = − Ek$_{B}$δβ , the extrema of free energy J ( λ ) and temperature β ( λ ) coincide. Since the series of equi- libria β ( λ ) undergoes damped oscillations for 1 /λ → + ∞ in d = 3 (see Fig. 5), this implies that the curve J ( λ ) will also exhibit damped oscillations at the same loca- tion, and that the curve J ( β ) will present some “spikes” for 1 /λ → + ∞ in d = 3 (see inset of Fig. 11). In ad- dition, the curve J ( β ) presents a minimum for η ≃ 24 . 7 corresponding to E = 0. Similar behaviors were previ- ously observed in the model of self-gravitating fermions [28, 31]. 

