Linearized Boltzmann collision integral with the correct cutoff

Abstract

In the calculation of the linearized Boltzmann collision operator for an inverse-square force law interaction (Coulomb interaction) F(r)=κ/r2, we found the widely used scattering angle cutoff θ≥θmin is a wrong practise since the divergence still exists after the cutoff has been made. When the correct velocity change cutoff |v′−v|≥δmin is employed, the scattering angle can be integrated. A unified linearized Boltzmann collision operator for both inverse-square force law and rigid-sphere interactions is obtained. Like many other unified quantities such as transition moments, Fokker-Planck expansion coefficients and energy exchange rates obtained recently [Y. B. Chang and L. A. Viehland, AIP Adv. 1, 032128 (2011)], the difference between the two kinds of interactions is characterized by a parameter, γ, which is 1 for rigid-sphere interactions and −3 for inverse-square force law interactions. When the cutoff is removed by setting δmin=0, Hilbert's well known kernel for rigid-sphere interactions is recovered for γ = 1.