Explicit formula of energy-conserving Fokker-Planck type collision term for single species point vortex systems with weak mean flow

Abstract

This paper derives a kinetic equation for a two-dimensional single species point vortex system. We consider a situation (different from the ones considered previously) of weak mean flow where the time scale of the macroscopic motion is longer than the decorrelation time so that the trajectory of the point vortices can be approximated by a straight line on the decorrelation time scale. This may be the case when the number $N$ of point vortices is not too large. Using a kinetic theory based on the Klimontovich formalism, we derive a collision term consisting of a diffusion term and a drift term, whose structure is similar to the Fokker-Planck equation. The collision term exhibits several important properties: (a) it includes a nonlocal effect; (b) it conserves the mean field energy; (c) it satisfies the H theorem; (d) its effect vanishes in each local equilibrium region with the same temperature. When the system reaches a global equilibrium state, the collision term completely converges to zero all over the system.

12 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:1303.2793

References in corpus (4)

• Kinetic theory of point vortices in two dimensions: analytical results and numerical simulations
• Kinetic theory of spatially inhomogeneous stellar systems without collective effects
• Kinetic theory of two dimensional point vortices from a BBGKY-like hierarchy
• Slow relaxation in the two dimensional electron plasma under the strong magnetic field

Cited by in corpus (3)

• Statistical Mechanics of Systems with Negative Temperature
• Functional integral derivation of the kinetic equation of two-dimensional point vortices
• Mechanism of self-organization in point vortex system