On the interpretations of Tsallis functional in connection with Vlasov–Poisson and related systems: Dynamics vs thermodynamics

Abstract

We discuss different interpretations of Tsallis functional in astrophysics. In principle, for t → + ∞ , a self-gravitating system should reach a statistical equilibrium state described by the Boltzmann distribution. However, this tendency is hampered by the escape of stars and the gravothermal catastrophe. Furthermore, the relaxation time increases almost linearly with the number of particles N so that most stellar systems are in a collisionless regime described by the Vlasov equation. This equation admits an infinite number of stationary solutions. The system can be trapped in one of them as a result of phase mixing and incomplete violent relaxation and remains frozen in this quasi-stationary state for a very long time until collisional effects finally come into play. Tsallis distribution functions form a particular class of stationary solutions of the Vlasov equation named stellar polytropes. We interpret Tsallis functional as a particular H-function in the sense of Tremaine, Hénon and Lynden-Bell [Mon. Not. R. astr. Soc. 219 (1986) 285]. Furthermore, we show that the criterion of nonlinear dynamical stability for spherical stellar systems described by the Vlasov–Poisson system resembles a criterion of thermodynamical stability in the microcanonical ensemble and that the criterion of nonlinear dynamical stability for barotropic stars described by the Euler–Poisson system resembles a criterion of thermodynamical stability in the canonical ensemble. Accordingly, a thermodynamical analogy can be developed to investigate the nonlinear dynamical stability of barotropic stars and spherical galaxies but the notions of entropy, free energy and temperature are essentially effective. This analogy provides an interpretation of the nonlinear Antonov first law in terms of ensemble inequivalence. Similar ideas apply to other systems with long-range interactions such as two-dimensional vortices and the HMF model. We propose a general scenario to understand the emergence of coherent structures in long-range systems and discuss the dynamical/thermodynamical “duality” of their description. We stress that the thermodynamical analogy that we develop is only valid for systems whose distribution function depends only on energy. We discuss two other, independent, interpretations of Tsallis functional in astrophysics, in relation with generalized kinetic equations and quasi-equilibrium states of collisional stellar systems.