Statistical Mechanics of Violent Relaxation in Stellar Systems
We discuss the statistical mechanics of violent relaxation in stellar systems following the pioneering work of (1967 ). The solutions of the gravitational Vlasov-Poisson system develop finer and finer filaments so that a statistical description is appropr
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Pierre-Henri Chavanis Laboratoire de Physique Quantique, Universite Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France
Abstract. We discuss the statistical mechanics of violent relaxation in stellar systems following the pioneering work of Lynden-Bell (1967). The solutions of the gravitational Vlasov-Poisson system develop finer and finer filaments so that a statistical description is appropriate to smooth out the small-scales and describe the "coarse-grained" dynamics. In a coarse-grained sense, the system is expected to reach an equilibrium state of a Fermi-Dirac type within a few dynamical times. We describe in detail the equilibrium phase diagram and the nature of phase transitions which occur in self-gravitating systems. Then, we introduce a small-scale parametrization of the Vlasov equation and propose a set of relaxation equations for the coarse-grained dynamics. These relaxation equations, of a generalized FokkerPlanck type, are derived from a Maximum Entropy Production Principle (MEPP). We make a link with the quasilinear theory of the Vlasov-Poisson system and derive a truncated model appropriate to collisionless systems subject to tidal forces. With the aid of this kinetic theory, we qualitatively discuss the concept of "incomplete relaxation" and the limitations of Lynden-Bell's theory.
Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2 The gravitational Vlasov-Poisson system ... . . . . . . . . . . . . . . . . . . .. 3 Lynden-Bell's approach of violent relaxation. . . . . . . . . . . . . . . . . . .. 4 Computation of Fermi-Dirac spheres. . . . . . . . . . . . . . . . . . . . . . . . . .. 5 The maximum entropy production principle 6 The quasilinear theory 7 Truncated models for collisionless stellar systems . . . . . . . . . . . . . . . . 8 Conclusion References
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Introduction
It has long been realized that galaxies, and self-gravitating systems in gen-
eral, follow a kind of organization despite the diversity of their initial conditions and their environment. This organization is illustrated by morphological classification schemes such as the Hubble sequence and by simple rules which N. Antonic et al.(eds.), Multiscale Problems in Science and Technology © Springer-Verlag Berlin Heidelberg 2002
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govern the structure of individual self-gravitating systems. For example, elliptical galaxies display a quasi-universal luminosity profile described by de Vaucouleurs' R 1 / 4 law and most of globular clusters are well fitted by the Michie-King model [8]. The question that naturally emerges is, what determines the particular configuration to which a self-gravitating system settles. It is possible that their present configuration crucially depends on the conditions that prevail at their birth and on the details of their evolution. However, in view of their apparent regularity, it is tempting to investigate whether their organization can be favoured by some fundament
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