Eur. Phys. J. B 52, 61-82 (2006)
https://doi.org/10.1140/epjb/e2006-00268-8

Relaxation of a test particle in systems with long-range interactions: diffusion coefficient and dynamical friction

Laboratoire de Physique Théorique, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex, France

Corresponding author: ^a chavanis@irsamc.ups-tlse.fr

Received: 11 October 2005
Revised: 9 March 2006
Published online: 6 July 2006

Abstract

We study the relaxation of a test particle immersed in a bath of field particles interacting via weak long-range forces. To order 1/N in the N→+∞ limit, the velocity distribution of the test particle satisfies a Fokker-Planck equation whose form is related to the Landau and Lenard-Balescu equations in plasma physics. We provide explicit expressions for the diffusion coefficient and friction force in the case where the velocity distribution of the field particles is isotropic. We consider (i) various dimensions of space d=3,2 and 1; (ii) a discret spectrum of masses among the particles; (iii) different distributions of the bath including the Maxwell distribution of statistical equilibrium (thermal bath) and the step function (water bag). Specific applications are given for self-gravitating systems in three dimensions, Coulombian systems in two dimensions and for the HMF model in one dimension.