\[cor2\] Assuming our perturbative solution is the global optimizer, there is a phase transition as one raises $t$, for any fixed $0< e < 1/2$.

Recall from Corollary \[cor1\] the connection between $t$ and $\epsilon$:

$$t= \epsilon^3 - (e-\epsilon)^3.$$

Note that $t$ is an increasing function of $\epsilon$ and reaches the value $2e^3$ when $\epsilon = 2e$. From equation (\[entropy\]) for the entropy we see that it cannot be extended analytically to $t >
2e^3$, yet for $e< 1/2$ we have $2e^3< e^2< e^{3/2}$ so $(e,t)$ is in the interior of the phase space (Figure 1).

Conclusion
==========

Our goal was to analyze possible phase transitions between multipartite phases of complex networks, analogous to solid/solid transitions in materials. To this end we adapted the Strauss model [@St], defined in the grand canonical ensemble, to a microcanonical ensemble. It is appropriate at this point to review the ‘equivalence of ensembles’ in statistical physics.

In thermodynamics the concavity of the entropy $S(E,N,V)$, as a function of internal energy $E$, particle number (or mass) $N$ and volume $V$, and the interpretation of equilibrium states as states maximizing the entropy, are both fundamental; see for instance [@Ca; @Ma]. Lagrange multipliers can be used to 