Kinetic Equation to Higher Orders in the Plasma Parameter

Abstract

A kinetic equation for a homogeneous electron gas is derived to all orders in the plasma parameter λ (the reciprocal number of electrons per Debye sphere). It has the form ∂φ1∂t= ∑ n=1∞ (e2)n(R̄(n)+β̄n−R̄n(n)),where R̄(n) is a Fokker-Planck type collision integral which rigorously describes distant collisions between (n+1) electrons and diverges logarithmically at small impact parameter; β̄n is a Boltzmann-like collision integral for close collisions between (n+1) electrons and diverges at large impact parameter; and R̄n(n) accounts for intermediate collisions between (n+1) electrons and diverges at both large and small impact parameter. Whether or not the various divergent parts of (R̄(n) + β̄n − R̄n(n)) exactly cancel each other out has not yet been proven for all n. The infinite sum, however, is a direct consequence of Liouville's equation and is exact at all t. The first term (R̄(1) + β̄1 − R̄1(1)) has been proven convergent elsewhere. The second-order term (R̄(2) + β̄2 − R̄2(2)), which corresponds to ternary correlations (ternary collisions), is examined in detail and found to have its order of magnitude given by O(λ2 log λ) + O(λ2).