1 9 0.25 0.5 0.75 1 1.25 1.5 1.75 -4 -3 -2 -1 0 s c → ← d=2 0 5 10 15 20 25 0 10 20 30 j 2.8 2.9 3 3.1 1 1.1 1.2 1.3 1.4 j c → → d = 3  n=1 FIG. 37: Entropy S Nk$_{B}$ as a function of the energy Λ for d = 2. We have taken µ = 1. FIG. 34: Free energy J Nk$_{B}$ as a function of the inverse tem- perature η for d = 3. We have taken µ = 1. FIG. 38: Entropy S Nk$_{B}$ as a function of the energy Λ for d = 3. We have taken µ = 1. FIG. 35: Entropy S Nk$_{B}$ as a function of the energy Λ for d = 1. We have taken µ = 1 < µ c . 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 -6 -5 -4 -3 -2 -1 s 1.11 1.12 1.13 -2.18 -2.16 -2.14 -2.12 -2.1 s c ← → d = 3 0.2 0.3 0.4 0.5 0.6 -6 -4 -2 0 s c n=1 n=2 d = 1 c -0.06 -0.04 -0.02 0 0.02 0.04 0.06 -6 -5 -4 -3 -2 -1 0 s c → ← n=1 n=2 d=1 c Convex dip We can proceed exactly as in Sec. III E and obtain up to unimportant constants. However, we can also obtain a simpler expression. Substituting ρ = ρ$_{0}$e − βm (Φ − Φ 0 ) in Eq. (81), we obtain This can be rewritten up to unimportant constants. Finally, using Eqs. (79), (51), (57) and the relations κ = µ/α and α = S Nk$_{B}$ = − d − 2 2 ln η − 2 ln α + 1 η 1 α d − 2 ∫ α 0 ψe − $^{ψ}$ξd − 1 dξ, (82) S = d 2 Nk$_{B}$ ln T − k$_{B}$ ∫ ρ m ln ρ$_{0}$ m d r + k$_{B}$β ∫ ρ (Φ − Φ$_{0}$) d r . (83) S Nk$_{B}$ = − d 2 ln β − ln ρ$_{0}$ + 2 βE N − βm Φ$_{0}$ , (84) FIG. 36: Entropy S Nk$_{B}$ as a function of the energy Λ for d = 1. We have taken µ = 10 > µ c . There is a (small) convex dip associated with the region of negative specific heats in the microcanonical ensemble.