Hopping conductivity for localized electronic states Liouville space formalism

We calculate the conductivity for a system of localized electrons, with conduction via phonon assisted transitions, using a Liouville space formalism. The result is in agreement with lowest order rate equation calculations, leading to an equivalent random resistance network. Our work is in contradiction with a similar work of Capek. We illustrate the differences between the two calculations and show that ours is the correct procedure to follow. We also comment on the criticism that the predictions of the random resistance network model do not agree with experimental data.