https://doi.org/10.1140/epjb/e2009-00232-2
Statistics of the gravitational force in various dimensions of space: from Gaussian to Lévy laws
Laboratoire de Physique Théorique (CNRS UMR 5152),
Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France
Corresponding author: a chavanis@irsamc.ups-tlse.fr
Received:
7
October
2008
Revised:
18
May
2009
Published online:
8
July
2009
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
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2009
