content_image/402663.jpg content_image/402664.jpg content_image/402662.jpg content_image/402669.jpg
<img src='content_image/402663.jpg'>

<img src='content_image/402664.jpg'>

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 

<img src='content_image/402662.jpg'>

<img src='content_image/402669.jpg'>

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 

