Resonant thickening of self-gravitating discs: imposed or self-induced
  orbital diffusion in the tightly wound limit
  The secular thickening of a self-gravitating stellar galactic disc is
investigated using the dressed collisionless Fokker-Planck equation and the
inhomogeneous multicomponent Balescu-Lenard equation. The thick WKB limits for
the diffusion fluxes are found using the epicyclic approximation, while
assuming that only radially tightly wound transient spirals are sustained by
the disc. This yields simple quadratures for the drift and diffusion
coefficients, providing a clear understanding of the positions of maximum
vertical orbital diffusion within the disc, induced by fluctuations either
external or due to the finite number of particles. These thick limits also
offer a consistent derivation of a thick disc Toomre parameter, which is shown
to be exponentially boosted by the ratio of the vertical to radial scale
heights. Dressed potential fluctuations within the disc statistically induce a
vertical bending of a subset of resonant orbits, triggering the corresponding
increase in vertical velocity dispersion. When applied to a tepid stable
tapered disc perturbed by shot noise, these two frameworks reproduce
qualitatively the formation of ridges of resonant orbits towards larger
vertical actions, as found in direct numerical simulations, but overestimates
the time-scale involved in their appearance. Swing amplification is likely
needed to resolve this discrepancy, as demonstrated in the case of razor-thin
discs. Other sources of thickening are also investigated, such as fading
sequences of slowing bars, or the joint evolution of a population of giant
molecular clouds within the disc.
