1-d gravity in infinite point distributions

Abstract

The dynamics of infinite, asymptotically uniform, distributions of self-gravitating particles in one spatial dimension provides a simple toy model for the analogous three dimensional problem. We focus here on a limitation of such models as 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, 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 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 is equivalent to that of an infinite set of points with inverted harmonic oscillator potentials which bounce elastically when they collide. We discuss 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 (notably its "self-similarity") of the evolution very similar to those in the analogous simulations in three dimensions, which in turn resemble those in the expanding universe.

20 pages, 8 figures, small changes (section II shortened, added discussion in section IV), matches final version to appear in PRE

References in corpus (8)

• Linear perturbative theory of the discrete cosmological N-body problem
• Towards quantitative control on discreteness error in the non-linear regime of cosmological N body simulations
• Relaxation of a 1-D gravitational system
• Cosmological N-Body simulation: Techniques, Scope and Status
• Gravitational Dynamics of an Infinite Shuffled Lattice of Particles
• Gravitational Dynamics of an Infinite Shuffled Lattice: Particle Coarse-grainings, Non-linear Clustering and the Continuum Limit
• Force distribution in a randomly perturbed lattice of identical particles with $1/r^2$ pair interaction
• Gravitational dynamics of an infinite shuffled lattice: early time evolution and universality of non-linear correlations

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• Exponents of non-linear clustering in scale-free one dimensional cosmological simulations
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• A dynamical classification of the range of pair interactions
• Thermodynamics of a one-dimensional self-gravitating gas with periodic boundary conditions
• Gravitational force in an infinite one-dimensional Poisson distribution
• One-Dimensional Relativistic Self-Gravitating Systems
• Self-similarity and stable clustering in a family of scale-free cosmologies
• Projection of the gravitational dynamics on a subspace of probability distributions: curl-free Gaussian ansatz
• Expansion into the vacuum of stochastic gases with long-range interactions
• Lagrangian-space Gaussian ansatz for the matter redshift-space power spectrum and correlation function