Functional integral derivation of the kinetic equation of two-dimensional point vortices

TL;DR

This paper derives the kinetic equation for two-dimensional point vortices using a functional integral approach, offering a new perspective that complements traditional derivations in fluid dynamics.

Contribution

It introduces a novel functional integral formalism to derive the kinetic equation, providing deeper insight into the vortex system's evolution beyond classical methods.

Findings

Derivation of the kinetic equation via functional integrals

Reveals new insights into vortex dynamics

Provides a complementary perspective to traditional derivations

Abstract

We present a brief derivation of the kinetic equation describing the secular evolution of point vortices in two-dimensional hydrodynamics, by relying on a functional integral formalism. We start from Liouville's equation which describes the exact dynamics of a two-dimensional system of point vortices. At the order ${1/N}$, the evolution of the system is characterised by the first two equations of the BBGKY hierarchy involving the system's 1-body distribution function and its 1-body correlation function. Thanks to the introduction of auxiliary fields, these two evolution constraints may be rewritten as a functional integral. When functionally integrated over the 2-body correlation function, this rewriting leads to a new constraint coupling the 1-body distribution function and the two auxiliary fields. Once inverted, this constraint provides, through a new route, the closed non-linear kinetic equation satisfied by the 1-body distribution function. Such a method sheds new lights on the origin of these kinetic equations complementing the traditional derivation methods.