Kinetic theory of spatially homogeneous systems with long-range interactions: III. Application to power-law potentials, plasmas, stellar systems, and to the HMF model

TL;DR

This paper applies kinetic theory to systems with long-range power-law interactions, deriving explicit expressions for diffusion, friction, and relaxation times, and clarifying the regimes where different kinetic equations are valid.

Contribution

It provides a unified analytical framework for understanding the kinetic behavior of systems with power-law interactions across various dimensions and models.

Findings

Derived explicit formulas for diffusion and friction coefficients.

Identified regimes where Boltzmann, Landau, or Lenard-Balescu equations apply.

Clarified regularization mechanisms for divergences in kinetic equations.

Abstract

We apply the general results of the kinetic theory of systems with long-range interactions to particular systems of physical interest. We consider repulsive and attractive power-law potentials of interaction r^{-\gamma} with \gamma<d in a space of dimension d. For \gamma>\gamma_c= (d-1)/2, strong collisions must be taken into account and the evolution of the system is governed by the Boltzmann equation or by a modified Landau equation; for \gamma<\gamma_c, strong collisions are negligible and the evolution of the system is governed by the Lenard-Balescu equation. In the marginal case \gamma=\gamma_c, we can use the Landau equation (with appropriately justified cut-offs) as a relevant approximation of the Boltzmann and Lenard-Balescu equations. The divergence at small scales that appears in the original Landau equation is regularized by the effect of strong collisions. In the case of repulsive interactions with a neutralizing background (e.g. plasmas), the divergence at large scales that appears in the original Landau equation is regularized by collective effects accounting for Debye shielding. In the case of attractive interactions (e.g. gravity), it is regularized by the spatial inhomogeneity of the system and its finite extent. We provide explicit analytical expressions of the diffusion and friction coefficients, and of the relaxation time, depending on the value of the exponent \gamma and on the dimension of space d. We treat in a unified framework the case of Coulombian plasmas and stellar systems in various dimensions of space, and the case of the attractive and repulsive HMF models.