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The European Physical Journal B

, Volume 70, Issue 3, pp 413–433 | Cite as

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

  • P. H. ChavanisEmail author
Statistical and Nonlinear Physics

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, when the particle number N →+∞, it is given by a Lévy law called the Holtsmark distribution. It presents an algebraic tail for large fluctuations due to the contribution of the nearest neighbor. In d = 2, for large but finite values of N, it is given by a marginal Gaussian distribution intermediate between Gaussian and Lévy laws. It presents a Gaussian core and an algebraic tail. 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.

PACS

05.20.-y Classical statistical mechanics 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion 05.10.Gg Stochastic analysis methods 05.90.+m Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems 

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Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Laboratoire de Physique Théorique (CNRS UMR 5152), Université Paul SabatierToulouseFrance

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