Kinetic theory of point vortex systems from the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy

TL;DR

This paper derives kinetic equations for point vortex systems from the BBGKY hierarchy, connecting them to plasma physics equations and confirming their consistency with known large-N limits.

Contribution

It introduces a new derivation of kinetic equations for point vortex systems from the BBGKY hierarchy, linking vortex dynamics to plasma kinetic theories.

Findings

Collision term matches Chavanis (2001) for the Landau equation

Derived a kinetic equation analogous to the Balescu-Lenard equation for plasmas

Reduced to the Landau equation in the large N limit

Abstract

Kinetic equations are derived from the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy for point vortex systems in an infinite plane. As the level of approximation for the Landau equation, the collision term of the kinetic equation derived coincides with that by Chavanis ({\it Phys. Rev. E} {\bf 64}, 026309 (2001)). Furthermore, we derive a kinetic equation corresponding to the Balescu-Lenard equation for plasmas, using the theory of the Fredholm integral equation. For large $N$, this kinetic equation is reduced to the Landau equation above.