21 1 2 3 4 5 6 7 8 9 0 20 40 60 80 100 * c t * c Inhomogeneous phase Homogeneous phase * c -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0 50 100 150 200 n=2 n=1 c → → d=1 c 0 0.01 0.02 0.03 0.04 60 80 100 120 U S S M U M t d=1 c n=1 U * * c 1 2 FIG. 42: Canonical phase diagram in d = 1 exhibiting a tri- critical point at µ c = √$_{2}$$_{π}$ ≃ 4 . 44 and η ≃ 29 . 6. We have represented η c , η$_{∗}$ and η t as a function of µ . The region be- tween η$_{∗}$ and η ∗ c contains stable and metastable states. FIG. 40: Series of equilibria in d = 1 for µ = 10 > µ c . FIG. 41: Canonical caloric curve in d = 1 for µ = 10 > µ c . It displays a canonical first-order phase transition marked by the discontinuity of the energy at η = η t ( µ ). The region of negative specific heats is unstable in the canonical ensemble and replaced by a phase transition (Maxwell plateau). The temperatures η ∗ c and η$_{∗}$ represent canonical spinodal points marking the end of the metastable phase. FIG. 43: Microcanonical caloric curve in d = 1 for µ = 10 > µ c . It displays a microcanonical second order phase transition marked by the discontinuity of ∂β ∂E at E = E ∗ $_{c}$. For µ > µ c , there exists a region of negative specific heats that is stable in the microcanonical ensemble. -0.01 0 0.01 0.02 0.03 0.04 60 80 100 120 140 U S S d=1 c * c ’ C < 0 n=1 η < η$_{∗}$ , there is only one stable state (homogeneous); (ii) for η$_{∗}$ < η < η ∗ $_{c}$, there are three stable states (one homo- geneous and two inhomogeneous) and two unstable states (inhomogeneous); (iii) for η > η ∗ $_{c}$, there are two stable states (inhomogeneous) and one unstable state (homo- geneous). The pairs of inhomogeneous states have the same free energy. Therefore, the central density α plays the role of an order parameter (see Fig. 23). The canonical phase diagram is represented in Fig. 42 where we have plotted η ∗ $_{c}$, η$_{∗}$ and η$_{t}$ as a function of µ . The three temperatures coincide at the tricritical point µ = µ$_{c}$ . At that point, the phase transition goes from second order ( µ < µ$_{c}$ ) to first order ( µ > µ$_{c}$ ). The strict caloric curve (see Figs. 39 and 41), cor- responding to the fully stable states, is denoted (S). The physical caloric curve should take into account the metastable states (M) because they are long-lived. The states (U) are unstable. We see that there exists a fully η > η$_{∗}$ ( µ ). The caloric curve displays a canonical first order phase transition at η$_{t}$ ( µ ) marked by the disconti- nuity of the energy E (see Fig. 41). The temperature of transition η$_{t}$ ( µ ) can be obtained by plotting the free energy of the two phases as a function of the tempera- ture and determining at which temperature they become equal (see Fig. 32). Equivalently, it can be obtained by performing a horizontal Maxwell construction [31]. The homogeneous phase is fully stable for η < η$_{t}$ , metastable for η$_{t}$ < η < η ∗ c and unstable for η > η ∗ $_{c}$. The right branch of the inhomogeneous phase is fully stable for η > η$_{t}$ and metastable for η$_{∗}$ < η < η$_{t}$ . The left branch is unstable. Note that this branch has negative specific heats which is not permitted in the canonical ensemble. Secondary branches appear for smaller values of the tem- perature but they have smaller values of free energy J and they are unstable. We also note that the branch of inhomogeneous states is degenerate since the curves α < α$_{c}$ and α > α$_{c}$ coincide. In conclusion: (i) for