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.
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