Functional integral approach to the kinetic theory of inhomogeneous systems

TL;DR

This paper derives the inhomogeneous Landau kinetic equation for long-range interacting systems using a functional integral approach, offering an alternative to traditional methods and paving the way for more precise kinetic models.

Contribution

It introduces a novel functional integral formalism to derive the kinetic equation, providing an alternative to existing methods like the BBGKY hierarchy or Klimontovich equation.

Findings

Derivation of the inhomogeneous Landau equation using functional integrals.

Provides a new constraint linking the 1-body distribution and auxiliary fields.

Framework potentially extendable to include collective effects and higher-order correlations.

Abstract

We present a derivation of the kinetic equation describing the secular evolution of spatially inhomogeneous systems with long-range interactions, the so-called inhomogeneous Landau equation, by relying on a functional integral formalism. We start from the BBGKY hierarchy derived from the Liouville equation. At the order ${1/N}$, where $N$ is the number of particles, the evolution of the system is characterised by its 1-body distribution function and its 2-body correlation function. Introducing associated auxiliary fields, the evolution of these quantities may be rewritten as a traditional functional integral. By functionally integrating over the 2-body autocorrelation, one obtains a new constraint connecting the 1-body DF and the auxiliary fields. When inverted, this constraint allows us to obtain the closed non-linear kinetic equation satisfied by the 1-body distribution function. This derivation provides an alternative to previous methods, either based on the direct resolution of the truncated BBGKY hierarchy or on the Klimontovich equation. It may turn out to be fruitful to derive more accurate kinetic equations, e.g., accounting for collective effects, or higher order correlation terms.