  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.
