10 0 10 20 30 40 50 60 -5 -4 -3 -2 -1 j → → n=1 n=2 $_{c}$=4 d=2 0 20 40 60 80 100 120 -3 -2.5 -2 -1.5 -1 -0.5 0 j → ← n=1 n=2 d=1 2.7 2.8 2.9 3 3.1 3.2 -3.35 -3.3 -3.25 -3.2 -3.15 j 0 5 10 15 20 25 30 35 -7 -6 -5 -4 -3 -2 j → → n=1 n=2 d=3 FIG. 10: Free energy J Nk$_{B}$ as a function of the inverse tem- perature η in d = 2. FIG. 9: Free energy J Nk$_{B}$ as a function of the inverse temper- ature η in d = 1. Note that the branches λ < 1 and λ > 1 coincide. FIG. 11: Free energy J Nk$_{B}$ as a function of the inverse tem- perature η in d = 3. Substituting Eq. (46) in Eq. (59), and introducing the dimensionless variables defined previously, we get 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 curves will be helpful in the next section to analyze the phase transitions in the canonical and microcanonical en- sembles respectively. Using the previous results, this expression relates the free energy J/Nk$_{B}$ to the central density 1 /λ . S Nk$_{B}$ = − d 2 ln β − ln ρ$_{0}$ + ρ$_{0}$ Nm S$_{d}$ ( R α ) $^{d}$∫ α 0 ψe − $^{ψ}$ξd − $^{1}$dξ, (60) S Nk$_{B}$ = − d − 2 2 ln η − 2 ln α + 1 η 1 α d − 2 ∫ α 0 ψe − $^{ψ}$ξd − 1 dξ, (61) J Nk$_{B}$ = S Nk$_{B}$ + η Λ . (62) 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]. 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