  Radiation-reaction in the interaction of ultra-relativistic electrons with a
strong external electromagnetic field is investigated using a kinetic approach
in the weakly quantum regime ($\chi \lesssim 1$, with $\chi$ the electron
quantum parameter). Three complementary descriptions are considered, their
domain of applicability discussed and their predictions on average properties
of an electron population compared. The first description relies on the
radiation reaction force in the Landau and Lifschitz (LL) form. The second
relies on the linear Boltzmann equation for the electron and photon
distribution functions. It is valid for any $\chi \lesssim 1$, and usually
implemented numerically using a Monte-Carlo (MC) procedure. The third
description relies on a Fokker-Planck (FP) expansion and is rigorously derived
for any ultra-relativistic, otherwise arbitrary configuration. Our study shows
that the evolution of the average energy of an electron population is described
with good accuracy in many physical situations by the leading term of the LL
equation with the so-called quantum correction, even for large values of the
$\chi$. The leading term of the LL friction force (with quantum correction) is
actually recovered naturally by taking the FP limit. The FP description is
necessary to correctly describe the evolution of the energy variance (second
order moment) of the distribution function, while the full linear Boltzmann
(MC) description allows to describe the evolution of higher order moments whose
contribution can become important when $\chi \rightarrow 1$. This analysis
allows further insight on the effect of particle straggling in the deformation
of the particle distribution function. A general criterion for the limit of
validity of each description is proposed, as well as a numerical scheme for
inclusion of the FP description in Particle-In-Cell codes.
