Secular diffusion in discrete self-gravitating tepid discs II:
  accounting for swing amplification via the matrix method
  The secular evolution of an infinitely thin tepid isolated galactic disc made
of a finite number of particles is investigated using the inhomogeneous
Balescu-Lenard equation expressed in terms of angle-action variables. The
matrix method is implemented numerically in order to model the induced
gravitational polarization. Special care is taken to account for the
amplification of potential fluctuations of mutually resonant orbits and the
unwinding of the induced swing amplified transients. Quantitative comparisons
with ${N-}$body simulations yield consistent scalings with the number of
particles and with the self-gravity of the disc: the fewer particles and the
colder the disc, the faster the secular evolution. Secular evolution is driven
by resonances, but does not depend on the initial phases of the disc. For a
Mestel disc with ${Q \sim 1.5}$, the polarization cloud around each star boosts
up its secular effect by a factor of the order of a thousand or more, promoting
accordingly the dynamical relevance of self-induced collisional secular
evolution. The position and shape of the induced resonant ridge are found to be
in very good agreement with the prediction of the Balescu-Lenard equation,
which scales with the square of the susceptibility of the disc.
  In astrophysics, the inhomogeneous Balescu-Lenard equation may 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 centre. It
could be used as a valuable check of the accuracy of ${N-}$body integrators
over secular timescales.
