Normal approach to the linearized Fokker-Planck equation for the inverse-square force

Abstract

It is found that the customary approach to Fokker-Planck coefficients for the inverse-square force has three defects. First, a small scattering angle cannot guarantee a small Taylor expansion argument. Second, a cutoff on the scattering angle did not fulfill Debye cutoff theory because it cannot exclude distant (weak) collisions with small relative velocity nor include close (effective) collisions with large relative velocity. Third, a singularity attributed to zero relative velocity had been overlooked. These defects had been vaguely covered up by the artificial treatment of replacing a variable relative velocity in a Coulomb logarithm by the constant thermal velocity. Therefore, the customary approach is questionable because one cannot regard the replacement as some kind of "average" or "approximation." In this paper, the difference between small-angle scattering and small-momentum-transfer collisions of the inverse-square force has been clarified. The probability function $P(\mathrm{v}, \ensuremath{\Delta}\mathrm{v})$ for Maxwellian scatters is derived by choosing velocity transfer $\ensuremath{\Delta}\mathrm{v}$, which is the true measure of collision strength, as an independent variable. With the help of the probability function, Fokker-Planck coefficients can be obtained by the normal original Fokker-Planck approach. The previous unproved treatment of the replacement of the relative velocity is naturally avoided, and the completed linearized Fokker-Planck coefficients are generated as a uniform expression.