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Thermodynamics and dynamics of a 1-D gravitational system

Abstract
Aims.The dynamics of large-scale structure formation in the universe by gravitational instability still presents many open issues and is mostly studied through numerical simulations. This motivates the study of simpler models which can be investigated by analytical means in order to understand the main processes at work. Thus, we describe here a one-dimensional self-gravitating system derived from the cosmological context, which leads to an effective external potential in addition to the standard gravitational self-interaction. As a first step we consider small times so that the expansion can be neglected. Then we present a thermodynamical analysis of this system as well as the stability properties of the associated hydrodynamical and collisionless systems.
Methods.We consider the mean field limit (i.e. continuum limit) to perform an analytical study.
Results.We find a second-order phase transition at T_c1 from an homogeneous equilibrium at high temperature to a clustered phase (with a density peak at one of the boundaries of the system) at low temperature. There also exists an infinite series of unstable equilibria which appear at lower temperatures T_cn, reflecting the scale-free nature of the gravitational interaction and the usual Jeans instability. We find that, as for the similar HMF (Hamiltonian mean field) model, all three micro-canonical, canonical and grand-canonical ensembles agree with each other, as well as with the stability properties associated with a hydrodynamical approach. On the other hand, the collisionless dynamics governed by the Vlasov equation yields the same results except that at low T the equilibrium associated with two density peaks (one at each boundary) becomes stable.