Functional integral approach to the kinetic theory of inhomogeneous
  systems
  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.
