On the meaning of Tsallis functional in astrophysics

Abstract

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.

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