Force distribution in a randomly perturbed lattice of identical particles with $1/r^2$ pair interaction

Abstract

We study the statistics of the force felt by a particle in the class of spatially correlated distribution of identical point-like particles, interacting via a $1/r^2$ pair force (i.e. gravitational or Coulomb), and obtained by randomly perturbing an infinite perfect lattice. In the first part we specify the conditions under which the force on a particle is a well defined stochastic quantity. We then study the small displacements approximation, giving both the limitations of its validity, and, when it is valid, an expression for the force variance. In the second part of the paper we extend to this class of particle distributions the method introduced by Chandrasekhar to study the force probability density function in the homogeneous Poisson particle distribution. In this way we can derive an approximate expression for the probability distribution of the force over the full range of perturbations of the lattice, i.e., from very small (compared to the lattice spacing) to very large where the Poisson limit is recovered. We show in particular the qualitative change in the large-force tail of the force distribution between these two limits. Excellent accuracy of our analytic results is found on detailed comparison with results from numerical simulations. These results provide basic statistical information about the fluctuations of the interactions (i) of the masses in self-gravitating systems like those encountered in the context of cosmological N-body simulations, and (ii) of the charges in the ordered phase of the One Component Plasma.

arXiv comment23 pages, 10 figures

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• Gravitational Dynamics of an Infinite Shuffled Lattice: Particle Coarse-grainings, Non-linear Clustering and the Continuum Limit
• A dynamical classification of the range of pair interactions
• Gravitational dynamics of an infinite shuffled lattice: early time evolution and universality of non-linear correlations
• Gravitational collapse in an expanding background and the role of substructure II: Excess power at small scales and its effect of collapse of structures at larger scales
• Gravitational force in an infinite one-dimensional Poisson distribution
• Statistical physics for cosmic structures
• Gravitational fluctuations of the galaxy distribution
• On standardised moments of force distribution in simple liquids
• On the origin of transient features in cosmological N-Body Simulations