Statistics of the gravitational force in various dimensions of space: from Gaussian to Levy laws

Abstract

We discuss the distribution of the gravitational force created by a Poissonian distribution of field sources (stars, galaxies,...) in different dimensions of space d. In d=3, it is given by a Levy law called the Holtsmark distribution. It presents an algebraic tail for large fluctuations due to the contribution of the nearest neighbor. In d=2, it is given by a marginal Gaussian distribution intermediate between Gaussian and Levy laws. In d=1, it is exactly given by the Bernouilli distribution (for any particle number N) which becomes Gaussian for N>>1. Therefore, the dimension d=2 is critical regarding the statistics of the gravitational force. We generalize these results for inhomogeneous systems with arbitrary power-law density profile and arbitrary power-law force in a d-dimensional universe.

References in corpus (4)

• Relativistic stars with a linear equation of state: analogy with classical isothermal spheres and black holes
• White dwarf stars in D dimensions
• Hamiltonian and Brownian systems with long-range interactions: III. The BBGKY hierarchy for spatially inhomogeneous systems
• Force distribution in a randomly perturbed lattice of identical particles with $1/r^2$ pair interaction

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