Weak-Coupling Rate Equations and Initial Conditions

Irreversible rate equations for the diagonal and off-diagonal elements of the density matrix are derived in the weak-coupling, long-time approximation for two types of crystal perturbations---electrons scattering from random impurities, and spin-spin interactions. No initial random phase assumption on the density matrix is necessary in the derivation using the random perturbation. For the spin-spin interactions, a "partial" initial random phase assumption is introduced and used in a natural way. It is shown, alternatively, that if one takes into account the very extensive cancellation of terms for the vast majority of unperturbed energy states, no initial restrictions need be placed on the majority of the density matrix elements. In all three cases the equations obtained are the same, being Pauli equations for the diagonal density matrix elements, but simpler relaxation-oscillation equations for the off-diagonal elements.