Density matrix formalism for coupled dynamical systems

A density matrix first principles formalism is extended for use in coupled dynamical systems within the framework of the Zwanzig projection operator technique. Coupled linear integro-differential equations for the reduced density operators of two (or more) dynamical subsystems interacting with one (or more) dissipative subsystem(s) and weak driving fields are obtained. These coupled equations, which are highly problem independent and rendered in a form simple for applications, are applied to the well-known s-d exchange model in metals where the coupled Bloch-like equations obtained by others using many-body techniques are recovered. The problem of the instantaneous destination vector is discussed within the framework of our formalism using superoperator resolvent techniques.