14 Let us first describe the canonical ensemble (CE). The control parameter is η . The homogeneous phase exists for all 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 > 2 . 64 and it connects the homoge- neous branch at η = η ∗ $_{c}$. For large central densities 1 /λ , it forms a spiral towards a singular solution. For η < η ∗ $_{c}$, it has a lower free energy J than the homogeneous phase (see Fig. 11) and it is unstable. For η > η ∗ $_{c}$, it has a higher free energy J than the homogeneous phase (see Fig. 11). However, it is expected to be unstable or, pos- sibly, metastable. Secondary inhomogeneous branches appear for smaller values of the temperature but they have a higher value of free energy J and they are unsta- ble. The homogeneous branch is metastable for η < η ∗ $_{c}$. These conclusions are motivated by two arguments: (i) in the Newtonian model in d = 3, we know that there is no fully stable equilibrium state in CE. The system can undergo an isothermal collapse [29]. There is no global maximum of free energy J because we can make it diverge by creating a Dirac peak containing all the particles [21]. In the modified Newtonian model, the same argument applies since it is independent on boundary conditions. Since we know that the homogeneous branch is stable for η < η ∗ $_{c}$, then it can only be metastable. (ii) In the chemotactic literature, it has been rigorously established that the Keller-Segel model in d = 3 can blow up for any η [8]. This is consistent with our stability results. -0.16 -0.15 -0.14 -0.13 -0.12 -0.11 -0.1 6 7 8 9 U M S U or M U M S t d=2 U C < 0 * c FIG. 20: Caloric curve in the microcanonical ensemble in d = 2. A first-order phase transition is expected to take place in the microcanonical ensemble at Λ = Λ t . Note that the lower branch has negative specific heats. Λ$_{∗}$ and possibly Λ ∗ c represent microcanonical spinodal points marking the end of the metastable phase. FIG. 21: Series of equilibria in d = 3. The inhomogeneous branch forms a spiral for large central densities 1 /λ → $^{+}$∞ due to the damped oscillations of the inverse temperature η ( λ ) and energy Λ( λ ). This is similar to the spiral appearing in the series of equilibria of the classical self-gravitating gas as we approach the singular isothermal sphere [29]. -0.4 -0.3 -0.2 -0.1 0 0 10 20 30 40 -0.4 -0.35 -0.3 -0.25 -0.2 2 4 6 8 n=1 n=2 → ← U M U U or M d=3 is concentrated at the center. The possibly metastable states for Λ > Λ ∗ c have 1 /λ < 1 indicating that the den- sity is concentrated near the box. In conclusion, there exists a fully stable equilibrium state for any value of en- ergy. This is similar to the Newtonian model in d = 2 [24, 29]. However, in the modified Newtonian model, we expect a first order phase transition at Λ$_{t}$ that is not present in the Newtonian model. The strict caloric curve, corresponding to the fully sta- ble states (global maxima) in the series of equilibria, is denoted (S) in Figs. 19 and 20. The unstable states (sad- dle points) are denoted (U) and the metastable states (local maxima) are denoted (M). There exists a fully sta- ble equilibrium state for any accessible value of energy in MCE and for sufficiently high values of the temperature in CE ( η < η$_{c}$ = 4). Here, the microcanonical and canon- ical ensembles are inequivalent (unlike in the Newtonian case). In particular, the lower part of the first inhomo- geneous branch is stable in MCE while it is unstable in CE. This branch has negative specific heats C < 0 (see Fig. 20) which is not possible in the canonical ensemble. In conclusion, the system displays a zeroth order phase transition in CE (associated with an isothermal collapse) and a first order phase transition in MCE. Note also that the energy E ( β ) and its first derivative E $^{′}$( β ) are contin- uous at the critical point β ∗ c but its second derivative E $^{′′}$( β ) is discontinuous. Provided that the inhomoge- neous branch for η > η ∗ c is metastable, this would corre- spond to a third order canonical phase transition between a homogeneous metastable state and an inhomogeneous metastable state. Let us now describe the microcanonical ensemble (MCE). The control parameter is the energy Λ. The homogeneous phase exists for all 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 exists for Λ > − 0 . 405 and it connects the homogeneous branch at Λ = Λ ∗ $_{c}$. For large central densities 1 /λ , it forms a spiral towards a singular solution. For Λ < Λ ∗ $_{c}$, it has a lower In Fig. 21 we plot the series of equilibria in d = 3. 3. The dimension d = 3