The Fokker-Planck Asymptotics of the Boltzmann Collision Operator in the Coulomb Case

The Fokker-Planck collision operator is usually considered as an approximation of the Boltzmann collision operator when the collisions become grazing. A mathematical framework to this approach has recently been given in Ref. 2, by assuming that the scattering cross-section is smooth and depends upon a small parameter ε which tends to zero. However, the connection between ε and the physical quantities is unclear. In the present paper, our main concern is the Boltzmann operator for Coulomb collisions and its Fokker-Planck approximation. In the case of Coulomb collisions, the scattering cross-section has a non-integrable singularity when the relative velocity of the colliding particles tends to zero and a careful analysis is required. Furthermore, by a scaling of the collision operator, the small parameter which is involved in the Fokker-Planck asymptotics is clearly identified to the plasma parameter, and an expansion which is consistent with the physical observations is derived.