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

TL;DR

This paper analyzes how a test particle relaxes in systems with long-range interactions, deriving diffusion and friction expressions across different dimensions and distributions, with applications to gravitational, Coulomb, and HMF models.

Contribution

It provides explicit formulas for diffusion and friction in long-range systems, extending understanding across various dimensions and particle distributions, including specific physical models.

Findings

Derived explicit diffusion coefficients for isotropic velocity distributions.

Analyzed relaxation in systems with different spatial dimensions and mass spectra.

Applied results to gravitational, Coulomb, and HMF models.

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 $N\to +\infty$ 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 explict 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.