Explicit formula of energy-conserving, Fokker-Planck type collision term for nonaxisymmetric, single species point vortex system

TL;DR

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

Contribution

It provides the first explicit formula for a collision term in this context that conserves energy and satisfies the H theorem without assuming axisymmetry.

Findings

Collision term includes nonlocal effects

It conserves mean field energy

Collision effect vanishes at equilibrium

Abstract

This paper considers a kinetic equation for an unbounded two-dimensional single species point vortex system. No axisymmetric flow is assumed. Using the kinetic theory based on the Klimontovich formalism, we derive a collision term consisting of a diffusion 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 converge to zero all over the system.