Kinetic theory of spatially homogeneous systems with long-range interactions: II. Basic equations

TL;DR

This paper derives and discusses the fundamental kinetic equations governing the evolution of spatially homogeneous systems with long-range interactions across different dimensions, highlighting how relaxation times scale with particle number and system inhomogeneity.

Contribution

It provides a general formulation of kinetic equations for long-range interacting systems in arbitrary dimensions, extending previous work and analyzing relaxation time scaling.

Findings

Relaxation time scales as N in higher dimensions for homogeneous systems.

In one dimension, relaxation time scales as N^2 for homogeneous systems.

Test particle relaxation time always scales linearly with N.

Abstract

We provide a short historic of the early development of kinetic theory in plasma physics and synthesize the basic kinetic equations describing the evolution of systems with long-range interactions derived in Paper I. We describe the evolution of the system as a whole and the relaxation of a test particle in a bath at equilibrium or out-of-equilibrium. We write these equations for an arbitrary long-range potential of interaction in a space of arbitrary dimension d. We discuss the scaling of the relaxation time with the number of particles for non-singular potentials. For always spatially homogeneous systems, the relaxation time of the system as a whole scales like N in d>1 and like N^2 (presumably) in d=1. For always spatially inhomogeneous systems, the relaxation time of the system as a whole scales like N in any dimension of space. For one dimensional systems undergoing a dynamical phase transition from a homogeneous to an inhomogeneous phase, we expect a relaxation time of the form N^{\delta} with 1<\delta<2 intermediate between the two previous cases. The relaxation time of a test particle in a bath always scales like N. We also briefly discuss the kinetic theory of systems with long-range interactions submitted to an external stochastic potential. This paper gathers basic equations that are applied to specific systems in Paper III.