Gravitational Dynamics of an Infinite Shuffled Lattice of Particles

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 ${\delta }^2$ of the shuffling'' normalized in units of the lattice spacing $\ell$. The clustering, which develops in time starting from scales around $\ell$, 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 $\delta$, 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.