Fokker-Planck-Rosenbluth-Type Equations for Self-gravitating Systems in 1PN Approximation

TL;DR

This paper develops two formulations of Fokker-Planck-Rosenbluth-type equations incorporating first-order relativistic corrections for self-gravitating systems, advancing understanding of relativistic effects in stellar dynamics.

Contribution

It introduces two novel derivations of 1PN Fokker-Planck equations, including a covariant approach and a hierarchy-based method, highlighting the role of ternary correlations.

Findings

Inclusion of ternary correlations affects diffusion coefficients.

Both formulations extend previous non-relativistic models.

Relativistic effects are significant in dense stellar systems.

Abstract

We present two formulations of Fokker-Planck-Rosenbluth-type (FPR) equations for many-particle self-gravitating systems, with first order relativistic corrections in the post-Newtonian approach (1PN). The first starts from a covariant Fokker-Planck equation for a simple gas, introduced recently by G. Chacon-Acosta and G. Kremer (Phys. Rev. E 76, 021201, 2007). The second derivation is based on the establishment of an 1PN-BBGKY hierarchy, developed systematically from the 1PN microscopic law of force and using the Klimontovich-Dupree (KD) method. We close the hierarchy by the introduction of a two-point correlation function that describes adequately the relaxation process. This picture reveals an aspect that is not considered in the first formulation: the contribution of ternary correlation patterns to the diffusion coefficients, as a consequence of the nature of 1PN interaction. Both formulations can be considered as a generalization of the equation derived by Rezania and Sobouti (Astron. Astrophys. 54, 1110-1114, 2000), to stellar systems where the relativistic effects of gravitation play a significant role.