Kinetic theory of collisionless relaxation for systems with long-range interactions

TL;DR

This paper develops a kinetic theory framework for collisionless relaxation in systems with long-range interactions, extending previous models to multi-level cases and connecting kinetic equations with maximum entropy principles.

Contribution

It introduces a method to close the hierarchy of kinetic equations for multi-level systems, generalizing existing models to include a broader class of entropies and applications.

Findings

Derived a generalized kinetic equation for multi-level systems.

Connected kinetic equations with maximum entropy production principles.

Discussed applications to dark matter halos and turbulence.

Abstract

We develop the kinetic theory of collisionless relaxation for systems with long-range interactions in relation to the statistical theory of Lynden-Bell. We treat the multi-level case. We make the connection between the kinetic equation obtained from the quasilinear theory of the Vlasov equation and the relaxation equation obtained from a maximum entropy production principle. We propose a method to close the infinite hierarchy of kinetic equations for the phase level moments and obtain a kinetic equation for the coarse-grained distribution function in the form of a generalized Landau, Lenard-Balescu or Kramers equation associated with a generalized form of entropy [P.H. Chavanis, Physica A {\bf 332}, 89 (2004)]. This allows us to go beyond the two-level case associated with a Fermi-Dirac-type entropy. We discuss the numerous analogies with two-dimensional turbulence. We also mention possible applications of the present formalism to fermionic and bosonic dark matter halos.