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stable equilibrium state for any temperature and any screening length. This is consistent with the usual New- tonian model in d = 1 [20, 29]. This is also consistent with the results of chemotaxis since it has been estab- lished rigorously that there is no blow up in d = 1 [8]. 

Let us finally 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 Λ < Λ$_{max}$ = d/ (2 µ $^{2}$). It is stable for Λ < Λ ∗ c and unsta- ble for Λ > Λ ∗ c (see Sec. V). Comparing Figs. 43 and 44, we see that there exists a microcanonical tricritical point at µ$_{m}$ ≃ 11 . 8 and Λ ≃ 2 . 37 10 − 4 (corresponding to η ≃ 149 . 1096). For µ < µ$_{m}$ the phase transition is second order and for µ > µ$_{m}$ the phase transition is first 

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order. 

Let us first consider µ < µ$_{m}$ (see Figs. 39 and 43). The first branch n = 1 of inhomogeneous states exists for Λ ∗ c < Λ < Λ (1) max = 1 / (2 µ tanh( µ )) (see Appendix D). It has a higher entropy S than the homogeneous phase and it is stable. Secondary branches appear for smaller values of the energy but they have smaller values of en- tropy and are unstable. The microcanonical caloric curve displays a second order phase transition marked by the discontinuity ∂β ∂E at E = E ∗ $_{c}$. For µ < µ$_{c}$ , the specific heat is always positive. In that case, the microcanonical and canonical ensembles are equivalent. For µ > µ$_{c}$ , a region of negative specific heats appears. This leads to a convex dip in the entropic curve S ( E ) (see Fig. 36). In that case, the microcanonical and canonical ensembles are inequivalent: the states with negative specific heats are stable in MCE while they are unstable in CE (com- pare Figs. 41 and 43). Therefore, these energies cannot be achieved in a canonical description. 

Let us now consider µ > µ$_{m}$ (see Fig. 44). The first branch n = 1 of inhomogeneous states exists only for Λ > Λ$_{∗}$( µ ). The caloric curve displays a microcanoni- cal first order phase transition at Λ$_{t}$( µ ) marked by the discontinuity of the temperature T and the existence of metastable states. The energy of transition Λ$_{t}$( µ ) can be obtained by plotting the entropy of the two phases as a function of the energy and determining at which en- ergy they become equal. Equivalently, it can be obtained by performing a vertical Maxwell construction [31]. The discussion is similar to that given in the canonical en- semble except that the axis are reversed. In conclusion: (i) for Λ < Λ$_{∗}$, there is only one stable state (homoge- neous); (ii) for Λ$_{∗}$ < Λ < Λ ∗ $_{c}$, there are three stable states (one homogeneous and two inhomogeneous) and two un- stable states (inhomogeneous); (iii) for Λ > Λ ∗ $_{c}$, there 

