  We study the maximization of the Tsallis functional at fixed mass and energy
in the HMF model. We give a thermodynamical and a dynamical interpretation of
this variational principle. This leads to q-distributions known as stellar
polytropes in astrophysics. We study phase transitions between spatially
homogeneous and spatially inhomogeneous equilibrium states. We show that there
exists a particular index q_c=3 playing the role of a canonical tricritical
point separating first and second order phase transitions in the canonical
ensemble and marking the occurence of a negative specific heat region in the
microcanonical ensemble. We apply our results to the situation considered by
Antoni & Ruffo [Phys. Rev. E 52, 2361 (1995)] and show that the anomaly
displayed on their caloric curve can be explained naturally by assuming that,
in this region, the QSSs are polytropes with critical index q_c=3. We
qualitatively justify the occurrence of polytropic (Tsallis) distributions with
compact support in terms of incomplete relaxation and inefficient mixing
(non-ergodicity). Our paper provides an exhaustive study of polytropic
distributions in the HMF model and the first plausible explanation of the
surprising result observed numerically by Antoni & Ruffo (1995). In the course
of our analysis, we also report an interesting situation where the caloric
curve presents both microcanonical first and second order phase transitions.
