Gravitational dynamics of an infinite shuffled lattice: Particle coarse-graining, nonlinear clustering, and the continuum limit

One-dimensional gravity in infinite point distributions

The dynamics of infinite asymptotically uniform distributions of purely self-gravitating particles in one spatial dimension provides a simple and interesting toy model for the analogous three dimensional problem treated in cosmology. In this paper we focus on a limitation of such models as they have been treated so far in the literature: the force, as it has been specified, is well defined in infinite point distributions only if there is a centre of symmetry (i.e., the definition requires explicitly the breaking of statistical translational invariance). The problem arises because naive background subtraction (due to expansion, or by "Jeans swindle" for the static case), applied as in three dimensions, leaves an unregulated contribution to the force due to surface mass fluctuations. Following a discussion by Kiessling of the Jeans swindle in three dimensions, we show that the problem may be resolved by defining the force in infinite point distributions as the limit of an exponentially screened pair interaction. We show explicitly that this prescription gives a well defined (finite) force acting on particles in a class of perturbed infinite lattices, which are the point processes relevant to cosmological N -body simulations. For identical particles the dynamics of the simplest toy model (without expansion) is equivalent to that of an infinite set of points with inverted harmonic oscillator potentials which bounce elastically when they collide. We discuss and compare with previous results in the literature and present new results for the specific case of this simplest (static) model starting from "shuffled lattice" initial conditions. These show qualitative properties of the evolution (notably its "self-similarity") like those in the analogous simulations in three dimensions, which in turn resemble those in the expanding universe.

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

In two recent papers, a detailed study has been presented of the out-of-equilibrium dynamics of an infinite system of self-gravitating points initially located on a randomly perturbed lattice. In this paper, we extend the treatment of the early time phase during which strong nonlinear correlations first develop, prior to the onset of "self-similar" scaling in the two-point correlation function. We establish more directly, using appropriate modifications of the numerical integration, that the development of these correlations can be well described by an approximation of the evolution in two phases: a first perturbative phase in which particle displacements are small compared to the lattice spacing, and a subsequent phase in which particles interact only with their nearest neighbors. For the range of initial amplitudes considered, we show that the first phase can be well approximated as a transformation of the perturbed lattice configuration into a Poisson distribution at the relevant scales. This appears to explain the universality of the spatial dependence of the asymptotic nonlinear clustering observed from both shuffled lattice and Poisson initial conditions.