13 cessible values of energy in MCE and temperature in CE. The microcanonical and canonical ensembles are equiva- lent (like in the Newtonian case). In conclusion, the system displays second order phase transition in CE and MCE. This is similar to the HMF model [32–34]. In Fig. 18 we plot the series of equilibria in d = 2. Let us first describe the canonical ensemble (CE). The control parameter is the inverse temperature η . The ho- mogeneous phase exists for any η . It is stable for η < η ∗ c and unstable for η > η ∗ c (see Sec. V). The first branch n = 1 of inhomogeneous states exists for η > η$_{c}$ = 4 and it connects the homogeneous branch at η ∗ $_{c}$. For η < η ∗ $_{c}$, it has a lower free energy J than the homogeneous phase (see Fig. 10) and it is unstable. For η > η ∗ $_{c}$, it has a higher free energy J than the homogeneous phase (see Fig. 10). However, it is expected to be unstable or, pos- sibly, metastable (to settle this issue we have to study the sign of the second order variations of free energy as explained above). Secondary inhomogeneous branches appear for smaller values of the temperature but they have smaller values of the free energy (see Fig. 10) and they are unstable. The homogeneous branch is expected to be fully stable for η < η$_{c}$ = 4 and metastable for η$_{c}$ = 4 < η < η ∗ c (see Fig. 19). These conclusions are motivated by two arguments: (i) in the Newtonian model in d = 2, we know that there exists a fully sta- ble equilibrium state for η < η$_{c}$ = 4 and no equilibrium state for η > η$_{c}$ = 4. In that case, the system under- goes an isothermal collapse [19, 29]. For η > η$_{c}$ = 4, there is no global maximum of free energy J because we can make it diverge by creating a Dirac peak containing all the particles. In the modified Newtonian model, the same argument applies since it is independent of bound- ary conditions. Since we know that the homogeneous branch is stable for η < η ∗ $_{c}$, we conclude that it must be metastable in the range η$_{c}$ = 4 < η < η ∗ $_{c}$. There is there- fore a zeroth order phase transition at η$_{c}$ = 4 marked by the discontinuity of the free energy. (ii) In the chemo- tactic literature, it has been rigorously established that the Keller-Segel model in d = 2 does not blow up for η < η$_{c}$ = 4 while it can blow up for η > η$_{c}$ = 4 [8]. This is consistent with our stability results. 2. The dimension d = 2 -0.2 -0.1 0 0.1 0.2 0 20 40 60 n=3 n=2 n=1 → → d=2 -0.4 -0.3 -0.2 -0.1 0 0.1 2 4 6 8 10 12 U S U M d=2 $_{c}$=4 U or M * c FIG. 18: Series of equilibria in d = 2. The first inhomoge- neous branch n = 1 tends to a plateau η c = 4 for large central densities 1 /λ → $^{+}$∞ due to the formation of a Dirac peak. This is similar to the plateau appearing in the caloric curve of the classical self-gravitating gas [29]. FIG. 19: Caloric curve in the canonical ensemble in d = 2. The homogeneous branch is fully stable for η < η c = 4, metastable for η c < η < η ∗ c and unstable for η > η ∗ $_{c}$. The inho- mogeneous branch is always unstable (or, possibly, metastable for η > η ∗ $_{c}$). For sufficiently low temperatures, the system can experience an isothermal collapse. ity of the temperature (see Fig. 20). The energy of tran- sition has been determined by comparing the entropy of the homogeneous and inhomogeneous phases and looking at which point the curves S ( E ) intersect (see Fig. 14). Equivalently, it can be obtained by performing a vertical Maxwell construction [31]. The homogeneous phase is fully stable for Λ < Λ$_{t}$, metastable for Λ$_{t}$ < Λ < Λ ∗ c and unstable for Λ > Λ ∗ $_{c}$. The lower part of the first inhomo- geneous branch is fully stable for Λ > Λ$_{t}$ and metastable for Λ$_{∗}$ < Λ < Λ$_{t}$. The upper part of the first inhomoge- neous branch is unstable for Λ$_{∗}$ < Λ < Λ ∗ $_{c}$. For Λ > Λ ∗ $_{c}$, it is unstable or, possibly, metastable. Secondary inhomo- geneous branches appear for smaller values of the energy but they have smaller values of the entropy (see Fig. 13) and they are unstable. The stable states of the inhomo- geneous branch have 1 /λ > 1 indicating that the density Let us now describe the microcanonical ensemble (MCE). The control parameter is the energy Λ. The homogeneous phase exists for all values of Λ < 0. It is stable for Λ < Λ ∗ c and unstable for Λ > Λ ∗ c (see Sec. V). The first branch n = 1 of inhomogeneous states ex- ists for Λ > Λ$_{∗}$ and it connects the homogeneous branch at Λ ∗ $_{c}$. We see that the inhomogeneous branch β ( E ) is multi-valued. Considering the value of the entropy in the different phases (see Figs. 13 and 14), the caloric curve is expected to display a microcanonical first order phase transition at Λ = Λ$_{t}$ ≃ − 0 . 146 marked by the discontinu-