  The well-known Rutherford differential cross section, denoted by $
d\Omega/d\sigma$, corresponds to a two body interaction with Coulomb potential.
It leads to the logarithmically divergence of the momentum transfer (or the
transport cross section) which is described by $$\int_{{\mathbb S}^2}
(1-\cos\theta) \frac{d\Omega}{d\sigma} d\sigma\sim \int_0^{\pi}
\theta^{-1}d\theta. $$ Here $\theta$ is the deviation angle in the scattering
event. Due to screening effect, physically one can assume that $\theta_{\min}$
is the order of magnitude of the smallest angles for which the scattering can
still be regarded as Coulomb scattering. Under ad hoc cutoff $\theta \geq
\theta_{\min}$ on the deviation angle, L. D. Landau derived a new equation in
\cite{landau1936transport} for the weakly interacting gas which is now referred
to as the Fokker-Planck-Landau or Landau equation. In the present work, we
establish a unified framework to justify Landau's formal derivation in
\cite{landau1936transport} and the so-called Landau approximation problem
proposed in \cite{alexandre2004landau} in the close-to-equilibrium regime.
Precisely, (i). we prove global well-posedness of the Boltzmann equation with
cutoff Rutherford cross section which is perhaps the most singular kernel both
in relative velocity and deviation angle. (ii). we prove a global-in-time error
estimate between solutions to Boltzmann and Landau equations with logarithm
accuracy, which is consistent with the famous Coulomb logarithm. Key
ingredients into the proofs of these results include a complete coercivity
estimate of the linearized Boltzmann collision operator, a uniform spectral gap
estimate and a novel linear-quasilinear method.
