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

TL;DR

This paper derives an explicit energy-conserving Fokker-Planck type collision term for a single species point vortex system under weak mean flow, capturing nonlocal effects and satisfying key physical properties.

Contribution

It provides a novel kinetic equation with an explicit collision term that conserves energy and satisfies the H theorem for point vortex systems with weak mean flow.

Findings

Collision term includes nonlocal effects

Energy conservation is maintained by the collision term

Collision effect vanishes at local equilibrium

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.