Unified Theory of Relaxations in Plasmas, I. Basic Theorem

Abstract

Two approximate theories–the impact theory and wave theory– of relaxation phenomena in hot plasmas are united into an exact theory, in which no cut-off procedure of the diverging integrals is needed, and which gives Coulomb logarithms with exact numerical factors in the arguments. When a relaxation rate is given by a diverging integral \({\int}B(b)db\) with respect to the impact parameter b in the impact theory and by a diverging integral \({\int}K(k)dk\) with respect to the wave number k in the wave theory, then the present theory gives the rate in the form \begin{aligned} \int_{0}^{\infty}B(b)\exp\left(-\frac{1}{2}b^{2}/{b_{0}}^{2}\right)db+\int_{0}^{\infty}K(k)\exp\left(-\frac{1}{2}k^{2}/{b_{0}}^{2}\right)dk. \end{aligned} Here b 0 is any length much longer than the close impact radius but much shorter than the Debye radius; and the final results are independent of b 0 . Simple examples are treated.