-0.4 -0.39 -0.38 -0.37 -2.15 -2.125 -2.1 -2.075 s -0.4 -0.3 -0.2 -0.1 0 -6 -5 -4 -3 -2 s → → n=1 n=2 d=3 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 -6 -4 -2 0 s ↑ n=1 n=2 d=1 11 We shall now determine the caloric curve β ( E ) corre- sponding to the modified Newtonian model. First of all, we note that for the homogeneous phase, the potential energy W = 0 so that the energy reduces to the kinetic energy. Therefore, the series of equilibria of the homoge- neous phase is simply FIG. 15: Entropy S Nk$_{B}$ as a function of energy Λ in d = 3. FIG. 12: Entropy S Nk$_{B}$ as a function of energy Λ in d = 1. Note that the branches λ < 1 and λ > 1 coincide. FIG. 13: Entropy S Nk$_{B}$ as a function of energy Λ in d = 2. -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -6 -5 -4 -3 -2 -1 0 s → → n=1 n=2 d=2 F. Caloric curves and phase transitions η = − d 2Λ . (63) On the other hand, eliminating λ between η$_{n}$ ( λ ) and Λ$_{n}$( λ ) given by Eqs. (52) and (58), we get the series of equilibria η$_{n}$ (Λ) for the n -th inhomogeneous branch. The series of equilibria contain all the critical points of the optimization problems (3) and (17). The series of equilibria are the same in the canonical and microcanon- ical ensembles because the critical points are the same. They contain fully stable states (global maxima of S or J ), metastable states (local maxima of S or J ) and un- stable states (saddle points of S or J ). The stable parts of the series of equilibria form the caloric curves in the canonical and microcanonical ensembles. We shall dis- tinguish the strict caloric curves formed by fully stable states and the physical caloric curves containing fully sta- ble and metastable states [66]. Metastable states are im- portant because they can be long-lived in systems with long-range interaction [55, 56]. The caloric curves may differ in CE and MCE in case of ensembles inequiva- lence. They are described below in different dimensions of space. Remark: In order to determine the stable branch, we shall compare the entropy (in MCE) or the free energy (in CE) of the different solutions in competition (with the same values of energy or temperature). However, this is not sufficient because a distribution could have a high entropy and be an unstable saddle point. A more rigorous study should therefore investigate the sign of the second order variations of entropy or free energy for each critical point. But this is a difficult task that is left for -0.1462 -0.1461 -0.1460 -0.1459 -0.1458 -0.1457 -1.925 -1.925 -1.924 -1.923 -1.923 s t d=2 n=1 n=1 n=1 FIG. 14: Enlargement of Fig. 13. The entropies of the ho- mogeneous phase and inhomogeneous phase become equal at Λ = Λ t ≃ − 0 . 146. This corresponds to a first-order phase transition in the microcanonical ensemble marked by the dis- continuity of the slope S $^{′}$( E ) = 1 /T .