Quantum-Classical Dynamics in a Classical Bath

A derivation of the equations of motion for a mixed quantum-classical system in a dissipative environment is presented. The classical environment of the quantum subsystem is taken to be composed of two parts: the first part, termed the classical subsystem, comprises those degrees of freedom that are coupled directly to the quantum subsystem, whereas the second part is a large bath that is coupled only to the classical subsystem. Projection operator methods are used to eliminate the bath degrees of freedom yielding a dissipative equation of motion for the coupled quantum and classical subsystems.