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Kinetic theory of two-dimensional point vortices with collective effects

We develop a kinetic theory of point vortices in two-dimensional hydrodynamics taking collective effects into account. We first recall the approach of Dubin & O'Neil [Phys. Rev. Lett. 60, 1286 (1988)] that leads to a Lenard-Balescu-type kinetic equation for axisymmetric flows. When collective effects are neglected, it reduces to the Landau-type kinetic equation obtained independently in our previous papers [P.H. Chavanis, Phys. Rev. E 64, 026309 (2001); Physica A 387, 1123 (2008)]. We also consider the relaxation of a test vortex in a "sea" (bath) of field vortices. Its stochastic motion is described in terms of a Fokker-Planck equation. We determine the diffusion coefficient and the drift term by explicitly calculating the first and second order moments of the radial displacement of the test vortex 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 vortex in a bath.