20 ( S$_{d}$Gβmρ$_{0}$ ) 1 / $^{2}$R , we obtain which does not involve new integrals. Using the previous results, this expression relates the entropy S/Nk$_{B}$ to α . 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 α . In Figs. 31, 32, 33 and 34, we have plotted the free energy J/Nk$_{B}$ as a function of the inverse temperature η (parameterized by α ) in d = 1 , 2 , 3. In Figs. 35, 36, 37 and 38), we have plotted the entropy S/Nk$_{B}$ as a function of the energy Λ (parameterized by α ) in d = 1 , 2 , 3. µ > µ$_{c}$ . This interesting transition was first evidenced by Childress & Percus [54] using a different approach. In our thermodynamical approach, this implies the existence of a canonical tricritical point at µ$_{c}$ = √$_{2}$$_{π}$$_{.}$ For µ < µ$_{c}$ the phase transition is second order and for µ > µ$_{c}$ the phase transition is first order. S Nk$_{B}$ = − d − 2 2 ln η − 2 ln α − 2Λ η + λα 2 µ 2 , (85) J Nk$_{B}$ = S Nk$_{B}$ + η Λ . (86) Λ = − d 2 η + d 2 µ $_{2}$. (87) Let us first consider µ < µ$_{c}$ (see Fig. 39). The discus- sion is similar to that given for the modified Newtonian model. The first branch n = 1 of inhomogeneous states exists only for η > η ∗ $_{c}$. It has a higher free energy J than the homogeneous phase (see Fig. 31) and it is fully sta- ble. Secondary branches appear for smaller values of the temperature but they have smaller values of free energy J (see Fig. 31) and they are unstable (saddle points of free energy). Therefore, the canonical caloric curve displays a second order phase transition between homogeneous and inhomogeneous states marked by the discontinuity of ∂E ∂β at β = β ∗ $_{c}$. We note that, for the inhomogeneous states, there exists two solutions with the same tempera- ture and the same free energy but with different density profiles corresponding to α$_{1}$ < α$_{c}$ and α$_{2}$ > α$_{c}$ , where α$_{c}$ is the value of α at the point of contact with in the homogeneous branch. Thus, the inhomogeneous branch is degenerate. These two states can be distinguished by their central density α . Since ρ/ρ$_{0}$ = dη/α $^{2}$, the solution α$_{1}$ < α$_{c}$ corresponds to ρ$_{0}$ < ρ and the solution α$_{2}$ > α$_{c}$ corresponds to ρ$_{0}$ > ρ . The density profiles are similar to those represented in Fig. 17 for the modified Newto- nian model. In conclusion: (i) for η < η ∗ $_{c}$, there is only one stable state (homogeneous); (ii) for η > η ∗ $_{c}$, there are two stable states α$_{1}$ < α$_{c}$ and α$_{2}$ > α$_{c}$ (inhomogeneous) with the same free energy and one unstable state (homo- geneous). Therefore, the central density α plays the role of an order parameter (see Fig. 22). We shall now determine the caloric curve β ( E ) corre- sponding to the screened Newtonian model. First of all, we note that, for the homogeneous phase, we have ρ = ρ$_{0}$ and Φ = Φ$_{0}$ with − k 2 $_{0}$Φ$_{0}$ = S$_{d}$Gρ$_{0}$ (or equivalently ψ = 0, λ = 1 and α 2 = dη ). Therefore, the relationship between the energy and the temperature can be written This shows that η → + ∞ for Λ → Λ$_{max}$ = d/ (2 µ $^{2}$). On the other hand, eliminating α between η$_{n}$ ( α ) and Λ$_{n}$( α ) given by Eqs. (73) and (80), we get the series of equilibria for the n -th inhomogeneous branch. The series of equilibria (critical points) and the caloric curves (stable states) in CE and MCE are described below for different dimensions of space. In Figs. 39 and 40, we plot the series of equilibria in d = 1 for different values of the screening parameter µ . F. Caloric curve 1. The dimension d = 1 0.4 0.45 0.5 0.55 0.6 0.65 0 50 100 150 200 n=2 n=1 c → → d=1 c S S U U FIG. 39: Series of equilibria in d = 1 for µ = 1 < µ c . The caloric curve displays a second order phase transition in CE and MCE taking place at η = η ∗ c and Λ = Λ ∗ $_{c}$. It is marked by the discontinuity of ∂E/∂β in CE or ∂β/∂E in MCE. Note that the branches α < α c and α > α c coincide. For µ < µ c , the CE and MCE ensembles are equivalent. Let us now consider µ > µ$_{c}$ (see Fig. 40). The first branch n = 1 of inhomogeneous states exists only for Let us first describe the canonical ensemble (CE). The control parameter is the inverse temperature η and the stable states are maxima of free energy J at fixed mass M . The homogeneous phase exists for any value of η . It is stable for η < η ∗ c and unstable for η > η ∗ c (see Sec. V). Comparing Figs. 39 and 40, we see that the screened Newtonian model is characterized by a pitchfork bifurcation at η = η ∗ $_{c}$. The pitchfork bifurcation is super- critical if µ < µ$_{c}$ = √$_{2}$$_{π}$ ≃ 4 . 4428829 and sub-critical if