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Kinetic theory of long-range interacting systems with angle-action variables and collective effects

We develop a kinetic theory of systems with long-range interactions taking collective effects and spatial inhomogeneity into account. Starting from the Klimontovich equation and using a quasilinear approximation, we derive a Lenard-Balescu-type kinetic equation written in angle-action variables. We confirm the result obtained by Heyvaerts [Mon. Not. R. Astron. Soc. {\bf 407}, 355 (2010)] who started from the Liouville equation and used the BBGKY hierarchy truncated at the level of the two-body distribution function. When collective effects are neglected, we recover the Landau-type kinetic equation obtained in our previous papers [P.H. Chavanis, Physica A {\bf 377}, 469 (2007); J. Stat. Mech., P05019 (2010)]. We also consider the relaxation of a test particle in a bath of field particles. Its stochastic motion is described by a Fokker-Planck equation written in angle-action variables. We determine the diffusion tensor and the friction force by explicitly calculating the first and second order moments of the increment of action of the test particle from its equations of motion, taking collective effects into account. This generalizes the expressions obtained in our previous works. We discuss the scaling with $N$ of the relaxation time for the system as a whole and for a test particle in a bath.