Kinetic equations for systems with long-range interactions: a unified description

Abstract

We complete the existing literature on the kinetic theory of systems with long-rangeinteractions. Starting from the BBGKY hierarchy, or using projection operator technics ora quasilinear theory, a general kinetic equation can be derived when collectiveeffects are neglected. This equation (which is not well known) applies to possiblyspatially inhomogeneous systems, which is specific to systems with long-rangeinteractions. Interestingly, the structure of this kinetic equation bears a clearphysical meaning in terms of generalized Kubo relations. Furthermore, this equationtakes a very similar form for stellar systems and two-dimensional point vortices,providing therefore a unified description of the kinetic theory of these systems. Ifwe assume that the system is spatially homogeneous (or axisymmetric for pointvortices), this equation can be simplified and reduces to the Landau equation (or itscounterpart for point vortices). Our formalism thus offers a simple derivation ofLandau-type equations. We also use this general formalism to derive a kinetic equation,written in angle-action variables, describing spatially inhomogeneous systemswith long-range interactions. This new derivation solves the shortcomings of ourprevious derivation (Chavanis 2007 Physica A 377 469). Finally, we consider a testparticle approach and derive general expressions for the diffusion and friction(or drift) coefficients of a test particle evolving in a bath of field particles. Wemake contact with the expressions previously obtained in the literature. As anapplication of the kinetic theory, we argue that, for one-dimensional systems andtwo-dimensional point vortices, the relaxation time is shorter for inhomogeneous (ornon-axisymmetric) distributions than for homogeneous (or axisymmetric) distributionsbecause there are potentially more resonances. We compare this prediction with existingnumerical results. For the HMF model, we argue that the relaxation time scales likeN for inhomogeneousdistributions and like eN for permanently homogeneous distributions. Phase-space structures can reduce the relaxationtime by creating some inhomogeneities and resonances. Similar results are expected for 2Dpoint vortices. For systems with higher dimension, the relaxation time scales likeN. The relaxation time of a test particle in a bath also scales likeN in any dimension.