0.5 1 1.5 2 0 10 20 30 40 n=2 n=1 c d=2 1 1.1 1.2 1.3 1.4 1.5 1.6 0 10 20 30 n=1 n=2 c → → d=3 23 are two stable states (inhomogeneous) and one unstable state (homogeneous). The pairs of inhomogeneous states have the same entropy. Therefore, the central density α plays the role of an order parameter (see Fig. 28). The microcanonical phase diagram is represented in Figs. 45 and 46 where we have plotted Λ ∗ $_{c}$, Λ$_{∗}$ and Λ$_{t}$ as a function of µ . The three energies coincide at the microcanonical tricritical point µ = µ$_{m}$ . At that point, the phase transition goes from second order ( µ < µ$_{m}$ ) to first order ( µ > µ$_{m}$ ). We have also represented the region of negative specific heats which appears at the canonical tricritical point µ = µ$_{c}$ . For µ$_{c}$ < µ < µ$_{m}$ , it is delimited by Λ ∗ c and Λ ′ and for µ > µ$_{m}$ , it is delimited by Λ$_{∗}$ and Λ $^{′}$. This region of negative specific heats also defines the physical region of ensembles inequivalence, i.e. the states that are stable in MCE but unstable in CE (metastable states are considered here as stable states). Finally, we have represented the strict region of ensembles inequiva- lence, i.e. the states that are stable in MCE but unstable or metastable in CE. It is delimited by Λ$_{1}$ and Λ$_{2}$ and, of course, contains the negative specific heats region. The strict caloric curve (see Figs. 39, 43 and 44), cor- responding to the fully stable states, is denoted (S). The states (U) are unstable. The states (M) are metastable but they are long-lived. We see that there exists a fully stable equilibrium state for any accessible energy and any screening length. This is consistent with the usual New- tonian model in d = 1 [20, 29]. In conclusion, for µ < µ$_{c}$ , the system displays canon- ical and microcanonical second order phase transitions. For µ$_{c}$ < µ < µ$_{m}$ (canonical tricritical point), the system displays canonical first order phase transitions and micro- canonical second order phase transitions. For µ > µ$_{m}$ (microcanonical tricritical point), the system displays canonical and microcanonical first order phase transi- tions. Note that the canonical and microcanonical tri- critcal points do not coincide as also observed in other models [28, 38, 40]. FIG. 47: Caloric curve in d = 2 for µ = 1. FIG. 48: Caloric curve in d = 3 for µ = 1. equation (64). We first consider the spectral stability of the homogeneous phase with respect to the Smoluchowski equation or, equivalently, with respect to the Keller-Segel model. This will allow us to determine the growth rate (unstable case) or the damping rate (stable case) of the perturbation. Then, we investigate the dynamical and thermodynamical stability of a larger class of systems by determining whether the homogeneous phase is a maxi- mum of entropy at fixed mass and energy in MCE or a minimum of free energy at fixed mass in CE. In Figs. 47 and 48, we plot the series of equilibria in d = 2 and d = 3. We have considered different values of µ but only the case µ = 1 is shown. We have observed that the shape of the diagrams does not significantly depend on the value of the screening parameter µ . Therefore, the description of these diagrams is similar to the one given in Secs. III F 2 and III F 3 for the modified Newtonian model. We consider the mean field Smoluchowski equation In this section, we study the stability of the homoge- neous phase in the case where the potential satisfies the modified Poisson equation (43) or the screened Poisson coupled to the modified Poisson equation (43) or to the screened Poisson equation (64). The boundary condi- tions are given by Eq. (45). Up to a change of notation, these equations also describe the Keller-Segel model (31)- (37) or (31)-(39). In the modified Newtonian model, the ∂ρ ∂t = ∇ · [ 1 ξ ( k$_{B}$T m ∇ ρ + ρ ∇ Φ )] , (88) A. Spectral stability 2. The dimensions d = 2 and d = 3 V. STABILITY OF THE HOMOGENEOUS PHASE