  We discuss the meaning of Tsallis functional in astrophysics. The energy
functional of a polytropic star is similar to Tsallis free energy and the
H-function associated with a stellar polytrope is similar to Tsallis entropy.
More generally, the energy functional of a barotropic star is similar to a
generalized free energy and the H-function associated with a spherical stellar
system is similar to a generalized entropy. Their optimization under
appropriate constraints determines a condition of nonlinear dynamical
stability, not a condition of thermodynamical stability. The criterion of
dynamical stability for barotropic stars described by the Euler-Poisson system
is similar to a condition of canonical stability and the criterion of dynamical
stability for collisionless stellar systems described by the Vlasov-Poisson
system is similar to a condition of microcanonical stability. Hence, stellar
systems are stable whenever corresponding barotropic stars are stable but the
converse is wrong in general. This provides a new interpretation of Antonov
first law in terms of ensemble inequivalence. In this context, Tsallis
functional is not a true entropy or a true free energy. This is a useful
H-function making the bridge between polytropic and isothermal distributions. A
thermodynamical analogy can be developed to investigate the nonlinear dynamical
stability problem but the notions of entropy, free energy and temperature are
essentially effective. Similar ideas apply to other systems with long-range
interactions such as 2D vortices and the HMF model. We also discuss two other
interpretations of Tsallis functional in relation with generalized kinetic
equations and quasi-stationary states of collisional stellar systems.
