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

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