The grazing collisions limit from the linearized Boltzmann equation to the Landau equation for short-range potentials

The Landau equation and the Boltzmann equation are connected through the limit of grazing collisions. This has been proved rigorously for certain families of Boltzmann operators concentrating on grazing collisions. In this contribution, we study the collision kernels associated to the two-particle scattering via a finite range potential $ \Phi(x) $ in three dimensions. We then consider the limit of weak interaction given by $ \Phi_ \epsilon(x) = \epsilon \Phi(x) $. Here $ \epsilon\rightarrow 0 $ is the grazing parameter, and the rate of collisions is rescaled to obtain a non-trivial limit. The grazing collisions limit is of particular interest for potentials with a singularity of order $ s\geq 0 $ at the origin, i.e. $ \phi(x) \sim |x|^{-s} $ as $ |x|\rightarrow 0 $. For $ s\in [0,1] $, we prove the convergence to the Landau equation with diffusion coefficient given by the Born approximation, as predicted in the works of Landau and Balescu. On the other hand, for potentials with $ s>1 $ we obtain the non-cutoff Boltzmann equation in the limit. The Coulomb singularity $ s = 1 $ appears as a threshold value with a logarithmic correction to the diffusive timescale, the so-called Coulomb logarithm.