Kinetic theory with angle-action variables

TL;DR

This paper develops a kinetic theory for inhomogeneous systems with long-range interactions using angle-action variables, deriving equations that describe system relaxation and test particle dynamics.

Contribution

It introduces a novel kinetic equation in angle-action variables for inhomogeneous systems, extending the understanding of long-range interaction dynamics.

Findings

Derived a closed kinetic equation conserving mass and energy.

Established a Fokker-Planck equation for test particle relaxation.

Highlighted analogies with vortex dynamics in hydrodynamics.

Abstract

We present a kinetic theory for inhomogeneous systems with weak long-range interactions. Starting from the Klimontovich equation and using a quasilinear theory, we obtain a closed kinetic equation describing the relaxation of the distribution function of the system as a whole due to resonances between different orbits. This equation is written in angle-action variables. It conserves mass and energy and increases the Boltzmann entropy (H-theorem). Using a thermal bath approximation, we derive a Fokker-Planck equation that describes the relaxation of a test particle towards the Boltzmann distribution under the combined effect of diffusion and friction terms. We mention some analogies with the kinetic theory of point vortices in two-dimensional hydrodynamics. We also stress the limitations of our approach and the connection with recent works.