New treatment of the Vlasov equation with application to the electrical conductivity of a plasma having electron-ion collisions and Coulomb correlations

The Vlasov equation for a plasma subjected to an oscillatory electric field with finite wavelength is shown to be separable into two equations. One of these considers the long-range, coherent interaction effects and the other contains the incoherent effects of electron-ion collisions. The fact that the applied field causes each electron to be shifted relative to the set of fixed ions by a position- and velocity-dependent displacement is explicitly taken into account. The collision-dependent Vlasov equation is formally modified to place this effect in the internal fields by considering the ions to be shifted instead. The collision-free Vlasov equation is solved and the current density is formed, from which the collisionless part of the electrical conductivity is found. The modified collision-dependent equation is then solved and the average force per unit charge on an electron is calculated; the equation of motion then yields the collision-dependent conductivity. The displacement of an electron relative to the fixed ions introduces a coupling between the collision-free and collision-dependent parts of the conductivity. A new integral, which is related to the complex error function, is introduced and studied