We study quasistationary states (QSSs) 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 nondegenerate 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 QSSs are always characterized by the presence of a 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.

We revisit the issue of relaxation to thermal equilibrium in the socalled "sheet model", i.e., particles in one dimension interacting by attractive forces independent of their separation. We show that this relaxation may be very clearly detected and characterized by following the evolution of order parameters defined by appropriately normalized moments of the phase space distribution which probe its entanglement in space and velocity coordinates. For a class of quasi-stationary states which result from the violent relaxation of rectangular waterbag initial conditions, characterized by their virial ratio R 0 , we show that relaxation occurs on a time scale which (i) scales approximately linearly in the particle number N , and (ii) shows also a strong dependence on R 0 , with quasi-stationary states from colder initial conditions relaxing much more rapidly. The temporal evolution of the order parameter may be well described by a stretched exponential function. We study finally the correlation of the relaxation times with the amplitude of fluctuations in the relaxing quasi-stationary states, as well as the relation between temporal and ensemble averages.