Failure of the Fokker--Planck approximation to the Boltzmann integral for (1/r) potentials

It is shown that the widely used Landau–RMJ collision term provides an inadequate description of collisional relaxation caused by Coulomb or gravitational interactions because of its inability to correctly describe large angle scattering events. This is established by extending the Taylor series expansion of the Boltzmann integral to fourth order thereby obtaining an approximation of the form JB[ fa, fb]ΔΩχ =ln(xmax/xmin)L2[ fa, fb] +(x2max−x2min)L4[ fa,  fb] +O(x4max), where xmax=sin(χmax/2), xmin=sin(χmin/2), and χ is the scattering angle in the center of mass frame. For χmax =180°, the first term in the series is the familiar Landau–RMJ collision term. The higher‐order collision term L4 is derived and its properties are discussed. In particular it is shown that for like‐particle and light–heavy particle collision, L4 can exceed L2 ln Λ over large regions of velocity space whenever the particle distribution functions fa, fb differ significantly from common equilibrium distributions. The conclusion drawn...