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

We complete the existing literature on the kinetic theory of systems with long-range interactions. Starting from the BBGKY hierarchy, or using projection operator technics or a quasilinear theory, a general kinetic equation can be derived when collective effects are neglected. This equation (which is not well known) applies to possibly spatially inhomogeneous systems, which is specific to systems with long-range interactions. Interestingly, the structure of this kinetic equation bears a clear physical meaning in terms of generalized Kubo relations. Furthermore, this equation takes a very similar form for stellar systems and two-dimensional point vortices, providing therefore a unified description of the kinetic theory of these systems. If we assume that the system is spatially homogeneous (or axisymmetric for point vortices), this equation can be simplified and reduces to the Landau equation (or its counterpart for point vortices). Our formalism thus offers a simple derivation of Landau-type equations. We also use this general formalism to derive a kinetic equation, written in angle-action variables, describing spatially inhomogeneous systems with long-range interactions. This new derivation solves the shortcomings of our previous derivation (Chavanis 2007 Physica A 377 469). Finally, we consider a test particle approach and derive general expressions for the diffusion and friction (or drift) coefficients of a test particle evolving in a bath of field particles. We make contact with the expressions previously obtained in the literature. As an application of the kinetic theory, we argue that, for one-dimensional systems and two-dimensional point vortices, the relaxation time is shorter for inhomogeneous (or non-axisymmetric) distributions than for homogeneous (or axisymmetric) distributions because there are potentially more resonances. We compare this prediction with existing numerical results. For the HMF model, we argue that the relaxation time scales like N for inhomogeneous distributions and like e(N) for permanently homogeneous distributions. Phase-space structures can reduce the relaxation time by creating some inhomogeneities and resonances. Similar results are expected for 2D point vortices. For systems with higher dimension, the relaxation time scales like N. The relaxation time of a test particle in a bath also scales like N in any dimension.