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The secular evolution of an infinitely thin tepid isolated galactic disc madeof a finite number of particles is described using the inhomogeneousBalescu-Lenard equation. Assuming that only tightly wound transient spirals arepresent in the disc, a WKB approximation provides a simple and tractablequadrature for the corresponding drift and diffusion coefficients. It providesinsight into the physical processes at work during the secular diffusion of aself-gravitating discrete disc and makes quantitative predictions on theinitial variations of the distribution function in action space.  When applied to the secular evolution of an isolated stationaryself-gravitating Mestel disc, this formalism predicts initially the importanceof the corotation resonance in the inner regions of the disc leading to aregime involving radial migration and heating. It predicts in particular theformation of a "ridge like" feature in action space, in agreement withsimulations, but over-estimates the timescale involved in its appearance. Swingamplification is likely to resolve this discrepancy.  In astrophysics, the inhomogeneous Balescu-Lenard equation and its WKB limitmay also describe the secular diffusion of giant molecular clouds in galacticdiscs, the secular migration and segregation of planetesimals inproto-planetary discs, or even the long-term evolution of population of starswithin the Galactic center.

Secular diffusion in discrete self-gravitating tepid discs I : analytic solution in the tightly wound limit