When an open system of classical point particles interacting by Newtonian gravity collapses and relaxes violently, an arbitrary amount of energy may, in principle, be carried away by particles which escape to infinity. We investigate here, using numerical simulations, how this released energy and other related quantities (notably the binding energy and size of the virialized structure) depend on the initial conditions, for the one-parameter family of starting configurations given by randomly distributing N cold particles in a spherical volume. Previous studies have established that the minimal size reached by the system scales approximately as N1/3, a behaviour which follows trivially when the growth of perturbations (which regularize the singularity of the cold collapse in the N -> ∞ limit) is assumed to be unaffected by the boundaries. Our study shows that the energy ejected grows approximately in proportion to N1/3, while the fraction of the initial mass ejected grows only very slowly with N, approximately logarithmically, in the range of N simulated. We examine in detail the mechanism of this mass and energy ejection, showing explicitly that it arises from the interplay of the growth of perturbations with the finite size of the system. A net lag of particles compared to their uniform spherical collapse trajectories develops first at the boundaries and then propagates into the volume during the collapse. Particles in the outer shells are then ejected as they scatter through the time-dependent potential of an already re-expanding central core. Using modified initial configurations, we explore the importance of fluctuations at different scales and discreteness (i.e. non-Vlasov) effects in the dynamics

Abstract"Quasi-stationary" states are approximately time-independent out of equilibrium states which have been observed in a variety of systems of particles interacting by long-range interactions. We investigate here the conditions of their occurrence for a generic pair interaction V (r → ∞) ∼ 1/r γ with γ > 0, in d > 1 dimensions. We generalize analytic calculations known for gravity in d = 3 to determine the scaling parametric dependences of their relaxation rates due to two body collisions, and report extensive numerical simulations testing their validity. Our results lead to the conclusion that, for γ < d − 1, the existence of quasi-stationary states is ensured by the large distance behavior of the interaction alone, while for γ > d − 1 it is conditioned on the short distance properties of the interaction, requiring the presence of a sufficiently large soft core in the interaction potential. In recent years there has been renewed interest in the statistical physics of long-range interactions (for a review, see e.g. [1]), a subject which has been treated otherwise mostly in the astrophysical literature for the specific case of gravity. The defining property of such interactions is the non-additivity of the potential energy of a uniform system, which corresponds to the non-integrability at large distances of the associated pair interaction, i.e., a pair interaction V (r → ∞) ∼ 1/r γ with γ < d in d space dimensions. The equilibrium thermodynamic analysis of these systems is very different to the canonical one for short-ranged interactions (with γ > d), leading notably to inhomogeneous equilibria as well as other unusual properties -e.g. non-equivalence of the statistical ensembles, negative specific heat in the microcanonical ensemble. Studies of simple toy models have shown that, like for gravity in d = 3, these equilibria (when defined) are reached only on time scales which are extremely long compared to those characteristic of the mean field dynamics. On the latter time scales one observes typically the formation, through "violent relaxation", of so-called "quasi-stationary" states (QSS), interpreted theoretically as stable stationary states of the Vlasov equation (which describes the kinetics in the mean field limit). In this letter we consider whether the occurrence of such QSS driven by mean-field dynamics can be considered as a behavior arising generically when there are long-range interactions in play. Using both simple analytical results and numerical simulations, we argue for the conclusion that it is only for γ < d − 1, i.e. when the pair force is absolutely integrable at large separations, that QSS can be expected to occur independently of the short distance properties of the interaction. For γ > d − 1, on the other hand, their occurrence will be conditioned strongly also on short distance properties, and thus cannot be considered to be a result simply of the long-range nature of the interaction. Our analysis shows the relevance of a classification of the range of interactions according to the conv...

We apply a simple linearisation, used standardly in solid state physics, to give an approximation describing the evolution under its self-gravity of an infinite perfect lattice perturbed from its equilibrium. In the limit that the initial perturbations are restricted to wavelengths much larger than the lattice spacing, the evolution corresponds exactly to that derived from an analagous linearisation of the Lagrangian formulation of the dynamics of a pressureless self-gravitating fluid, with the Zeldovich approximation as a sub-class of asymptotic solutions. Our less restricted approximation allows one to trace the evolution of the fully discrete distribution until the time when particles approach one another (i.e. "shell crossing"), with modifications of the fluid limit explicitly depending on the lattice spacing. We note that the simple cubic lattice presents both oscillating modes and modes which grow faster than in the fluid limit. In current cosmological theories the physics of structure formation in the universe reduces, over a large range of scales, to understanding the evolution of clustering under Newtonian gravity, with only a simple modification of the dynamical equations due to the expansion of the Universe. The primary instrument for solving this problem is numerical N -body simulation (NBS, see e.g.[1]). These simulations are most usually started from configurations which are simple cubic (sc) lattices perturbed in a manner prescribed by a theoretical cosmological model. In this letter we observe that, up to a change in sign in the force, the initial configuration is identical to the Coulomb lattice (or Wigner crystal) in solid state physics (see e.g.[2]), and we exploit this analogy to develop an approximation to the evolution of these simulations. We show that one obtains, for long wavelength perturbations, the evolution predicted by an analagous fluid description of the self-gravitating system, and in particular, as a special case, the Zeldovich approximation [3]. Further we can study precisely the deviations from this fluid-like behaviour at shorter wavelengths arising from the discrete nature of the system. This analysis should be a useful step towards a precise quantitative understanding, which is currently lacking, of the role of discreteness in cosmological NBS (see e.g. [4,5,6]). One simple conclusion, for example, is that a body centred cubic (bcc) lattice may be a better choice of discretisation, as its spectrum has only growing modes with exponents bounded above by that of fluid linear theory.The equation of motion of particles moving under their mutual self-gravity is [1]Here dots denote derivatives with respect to time t, x i is the comoving position of the ith particle, of mass m i , related to the physical coordinate by r i = a(t)x i , where a(t) is the scale factor of the background cosmology with Hubble constant H(t) =ȧ a . We treat a system of N point particles, of equal mass m, initially placed on a Bravais lattice, with periodic boundary conditions. Perturbations from the Coulom...