Gravitational dynamics of an infinite shuffled lattice of particles

Abstract

We study, using numerical simulations, the dynamical evolution of self-gravitating point particles in static euclidean space, starting from a simple class of infinite "shuffled lattice" initial conditions. These are obtained by applying independently to each particle on an infinite perfect lattice a small random displacement, and are characterized by a power spectrum (structure factor) of density fluctuations which is quadratic in the wave number k, at small k. For a specified form of the probability distribution function of the "shuffling" applied to each particle, and zero initial velocities, these initial configurations are characterized by a single relevant parameter: the variance δ 2 of the "shuffling" normalized in units of the lattice spacing ℓ. The clustering, which develops in time starting from scales around ℓ, is qualitatively very similar to that seen in cosmological simulations, which begin from lattices with applied correlated displacements and incorporate an expanding spatial background. From very soon after the formation of the first non-linear structures, a spatio-temporal scaling relation describes well the evolution of the two-point correlations. At larger times the dynamics of these correlations converges to what is termed "self-similar" evolution in cosmology, in which the time dependence in the scaling relation is specified entirely by that of the linearized fluid theory. Comparing simulations with different δ, different resolution, but identical large scale fluctuations, we are able to identify and study features of the dynamics of the system in the transient phase leading to this behavior. In this phase, the discrete nature of the system explicitly plays an essential role.

Gravitational dynamics of an infinite shuffled lattice: Early time evolution and universality of nonlinear correlations

“…As remarked in [3,4] the form of this early time correlation function coincides, to a very good approximation, with that which is also observed in the asymptotic scaling regime attained by the system at longer times. This suggests strongly (but does not prove) that the physical mechanism leading to the former correlations, which we identify here, is also that which gives rise to the latter correlations.…”

“…not predicted analytically or semi-analytically). We have noted in [3,4], however, that this form emerges, to a good approximation, in our simulations prior to the asymptotic bscaling regime, in the preceding transient phase in which strong non-linear correlations first develop. In this paper we extend and detail our analysis of this phase.…”

“…The parameter δ we refer to as the normalized shuffling parameter. As discussed in [3,4], for the case of purely gravitational interactions, the system is completely characterized, in the infinite volume limit, by the single parameter δ. The Poisson distribution corresponds to the limit in which each particle's position is completely randomized in the infinite volume, i.e., δ = ∞.…”

“…II we discuss some of the relevant properties of the initial conditions of our simulations. In the next section we give the details of the simulations considered here and summarize briefly the main relevant results of [3,4]. In Sec.…”

“…Progress in understanding would be very useful in providing analytical guidance for numerical simulations, and in particular better control on their precision in representing the relevant continuum limit. In a series of recent papers [3,4] we have studied a reduced version of the full cosmological problem, considering a very simple class of randomly perturbed lattices as initial conditions, and evolution in a static universe 1 . One of our primary results is that, despite the simplifications, the system we study has qualitative behavior very similar to that observed in the more complex cosmological simulations (correlated perturbations in initial conditions, expanding universe).…”