Subsystem dynamics in mixed quantum-classical systems

Starting with the mixed quantum-classical Liouville equation, projection operator methods are used to derive an equation of motion for a quantum subsystem dissipatively interacting with a classical bath. The resulting generalized master equation is reduced to the Lindblad equation after making a Markovian approximation in the weak coupling limit. The bath subsystem dynamics is studied from a similar perspective. For situations where the classical subsystem consists of few degrees of freedom, or one is interested in the nature of the modifications of its dynamics as a result of coupling to a large, rapidly relaxing quantum bath, a classical analog of the Lindblad equation is derived and discussed.