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Quasi-stationary states in the self-gravitating sheet model

We study quasi-stationary states (QSS) resulting from violent relaxation in the one-dimensional self-gravitating "sheet model", revisiting in particular the question of the adequacy of the theory of Lynden-Bell (LB) to describe them. For "waterbag" initial conditions characterized by a single phase space density, the prediction of this theory is, in this model, a function of only one parameter, which can conveniently be chosen to be the ratio of the energy to that in the degenerate limit. Studying a class of such initial conditions in which the shape of the initial waterbag is varied, we find that the LB predictions are reasonably good always in the low energy region, while at higher energies (i.e. in the non-degenerate limit) they are generally not even qualitatively correct, although certain initial conditions can still be found where they are as good as at low energy. We find notably that, in line with what has been observed by Levin et al. in some other models, when LB theory does not work the QSS are always characterized by the presence of a {\it degenerate} core, which these authors explain as the result of dynamical resonances. In short LB theory appears to be a good approximation only when violent relaxation is sufficiently "gentle", and otherwise a degenerate core-halo structure results.