The boltzmann equations in electron-phonon systems

The present paper derives conditions on which the Boltzmann equations for electron-phonon systems (two coupled equations) can be derived without the use of the objectionable random phase approximation. The treatment uses von Neumann's density matrix based on the Hamiltonian introduced some time ago by one of us in connection with superconductivity. Formally exact expressions for the rate of change of coarse-grained one-particle distribution functions are derived. They become identical with the collision terms in the Boltzmann equations if (i) the influence of an external field on them is negligible, (ii a ) particle correlation is absent at an initial time, and (ii b ) the relevant series in terms of the interaction parameter converges and is at all times dominated by a certain term. (ii b ) excludes the possibility of extensive correlations as they occur in superconductivity.