Kinetic theory of twodimensional point vortices with collective effects
Abstract
We develop a kinetic theory of point vortices in twodimensional hydrodynamics taking collective effects into account. We first recall the quasilinear theory of Dubin and O'Neil (1988 Phys. Rev. Lett. 60 1286) based on the Klimontovich equation and leading to a LenardBalescutype kinetic equation for axisymmetric flows. When collective effects are neglected, it reduces to the Landautype kinetic equation obtained independently in our previous papers (Chavanis 2001 Phys. Rev. E 64 026309; 2008 Physica A 387 1123) for more general flows. 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 FokkerPlanck equation. We determine the diffusion coefficient and the drift term by explicitly calculating the firstand secondorder 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 papers. We discuss the scaling with N of the relaxation time for the system as a whole and for a test vortex in a bath.
 Publication:

Journal of Statistical Mechanics: Theory and Experiment
 Pub Date:
 February 2012
 DOI:
 10.1088/17425468/2012/02/P02019
 arXiv:
 arXiv:1107.1447
 Bibcode:
 2012JSMTE..02..019C
 Keywords:

 Condensed Matter  Statistical Mechanics
 EPrint:
 J. Stat. Mech. (2012) 02019