Master equation for the density matrix and Boltzmann equation of localized electrons in crystals

A Master equation for the density matrix and a Boltzmann equation for the occupation number of electrons in a crystal are derived. The wave function of an electron is localized within a sub crystal by a superposition of the corresponding Wannier functions. The electrons are coupled to an external electrical field, to phonons, and to a particle reservoir, describing phenomena as transport, damping, and electron sources on the surface of contact.