On the A.C. electrical conductivity of a Lorentz gas

Often one is interested in the electrical conductivity of a plasma for frequencies of the applied electric field comparable to or greater than the effective collision frequency. For the purpose of obtaining an order of magnitude estimate of the conductivity a Lorentz gas model with a Fokker–Planck collision term is used. For this simple model a formula for the a.c. electrical conductivity for two limiting cases is derived in closed form. The first term of the expression for the conductivity in the case where w/k is very large agrees with that obtained by Bernstein & Trehan (1960) for the spatially homogeneous case, while the effect of the spatial gradients is indicated in the higher order terms. In the case where w and k are both equal to zero the expression for the d.c. electrical conductivity of Spitzer (1962) is recovered.