  The secular evolution of an infinitely thin tepid isolated galactic disc made
of a finite number of particles is described using the inhomogeneous
Balescu-Lenard equation. Assuming that only tightly wound transient spirals are
present in the disc, a WKB approximation provides a simple and tractable
quadrature for the corresponding drift and diffusion coefficients. It provides
insight into the physical processes at work during the secular diffusion of a
self-gravitating discrete disc and makes quantitative predictions on the
initial variations of the distribution function in action space.
  When applied to the secular evolution of an isolated stationary
self-gravitating Mestel disc, this formalism predicts initially the importance
of the corotation resonance in the inner regions of the disc leading to a
regime involving radial migration and heating. It predicts in particular the
formation of a "ridge like" feature in action space, in agreement with
simulations, but over-estimates the timescale involved in its appearance. Swing
amplification is likely to resolve this discrepancy.
  In astrophysics, the inhomogeneous Balescu-Lenard equation and its WKB limit
may also describe the secular diffusion of giant molecular clouds in galactic
discs, the secular migration and segregation of planetesimals in
proto-planetary discs, or even the long-term evolution of population of stars
within the Galactic center.
