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

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 (Chavanis, 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.(c) 2022 Elsevier B.V. All rights reserved.

Automated summary

This study develops the kinetic theory of collisionless relaxation for systems with long-range interactions, specifically in relation to Lynden-Bell's statistical theory. The authors discuss the multi-level case and establish the connection between the kinetic equation derived from the quasilinear theory of the Vlasov equation and the relaxation equation obtained from a maximum entropy production principle. They propose a method to close the infinite hierarchy of kinetic equations and obtain a generalized Landau, Lenard-Balescu, or Kramers equation for the coarse-grained distribution function. The authors also explore the analogies with two-dimensional turbulence and potential applications to fermionic and bosonic dark matter halos.