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future works. In order to find the stable states, we shall use physical considerations and exploit results obtained in related studies. 

In Fig. 16 we plot the series of equilibria in d = 1. 

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 fully stable for η < η ∗ c and unstable for η > η ∗ c (see Sec. V). 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. 9) and it is fully stable. Secondary branches of inhomogeneous states appear for smaller values of the temperature but they have smaller values of free energy J (see Fig. 9) and they are unstable (saddle points of free energy). Therefore, the canonical caloric curve displays a second order phase transition be- tween homogeneous and inhomogeneous states marked by the discontinuity of ∂E ∂β at β = β ∗ $_{c}$. For the inhomo- geneous states, there exists two solutions with the same temperature and the same free energy but with differ- ent density profiles corresponding to λ$_{1}$ < 1 and λ$_{2}$ > 1 (see Fig. 17). Thus, the inhomogeneous branch is de- generate. These two states can be distinguished by their central density 1 /λ . In conclusion: (i) for η < η ∗ $_{c}$, there is only one stable state λ = 1 (homogeneous); (ii) for η > η ∗ $_{c}$, there are two stable states λ$_{1}$ < 1 and λ$_{2}$ > 1 (inhomogeneous) with the same free energy and one un- stable state λ = 1 (homogeneous). Therefore, the central density 1 /λ plays the role of an order parameter (see Fig. 3). In d = 1, there exists a fully stable equilibrium state for any temperature. This is consistent with the usual Newtonian model in d = 1 [20, 29]. This is also consistent with results of chemotaxis since it has been rigorously proven that the Keller-Segel model does not blow up in d = 1 [8]. 

Let us now describe the microcanonical ensemble (MCE). The control parameter is the energy Λ and the stable states are maxima of entropy S at fixed mass M and energy E . The homogeneous phase exists for any value of energy Λ < 0. It is fully stable for Λ < Λ ∗ c and unstable for Λ > Λ ∗ c (see Sec. V). The first branch n = 1 of inhomogeneous states exists only for Λ ∗ c < Λ < Λ$_{max}$. It has a higher entropy S than the homogeneous phase (see Fig. 12) and it is fully stable. Secondary branches of inhomogeneous states appear for smaller values of the energy but they have smaller values of entropy S (see Fig. 12) and they are unstable (saddle points of en- tropy). Therefore, the microcanonical caloric curve dis- plays a second order phase transition marked by the dis- continuity of ∂β ∂E at E = E ∗ $_{c}$. For the inhomogeneous states, there exists two solutions with the same energy and the same entropy but with different density profiles corresponding to λ$_{1}$ < 1 and λ$_{2}$ > 1. Thus, the inhomo- 

## 1. The dimension d = 1 

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geneous branch is degenerate. These two states can be distinguished by their central density 1 /λ . In conclusion: (i) for Λ < Λ ∗ $_{c}$, there is only one stable state λ = 1 (homo- geneous); (ii) for Λ ∗ c < Λ < 0, there are two stable states λ$_{1}$ < 1 and λ$_{2}$ > 1 (inhomogeneous) with the same en- tropy and one unstable state λ = 1 (homogeneous). (iii) for 0 < Λ < Λ$_{max}$, there are two stable states λ$_{1}$ < 1 and λ$_{2}$ > 1 (inhomogeneous) with the same entropy. There- fore, the central density 1 /λ plays the role of an order parameter (see Fig. 6). In d = 1, there exists a fully stable equilibrium state for any accessible energy. This is consistent with the usual Newtonian model in d = 1 [20, 29]. 

The caloric curve, corresponding to the fully stable states in the series of equilibria, is denoted by (S) in Fig. 16. The branch (U) corresponds to unstable states. There exists a fully stable equilibrium state for any ac- 

