  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 [P.H. Chavanis, Physica A 377, 469
(2007)]. 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 the relaxation time is shorter for inhomogeneous
(or high-dimensional) systems than for homogeneous (or low-dimensional) systems
because there are potentially more resonances.
