The electrical conductivity for inhomogeneous electric fields by the Liouville operator method

A derivation is given of the electrical conductivity for a metal in a space- and time-dependent electric field. Thereby it is assumed that the electrons are scattered elastically by randomly distributed impurities. The derivation starts from the Kubo-Nakajima formula and is based on a perturbation expansion with Liouville operators, where use is made of van Hove's diagonal singularity property of the scattering potential. The formulae obtained are compact and the method is simpler and more transparent than the perturbation formalism developed by van Hove. It is shown that the lowest order approximation corresponds to the Boltzmann equation for electrons in inhomogeneous electric fields.