  The conditions under which quantum-classical Liouville dynamics may be
reduced to a master equation are investigated. Systems that can be partitioned
into a quantum-classical subsystem interacting with a classical bath are
considered. Starting with an exact non-Markovian equation for the diagonal
elements of the density matrix, an evolution equation for the subsystem density
matrix is derived. One contribution to this equation contains the bath average
of a memory kernel that accounts for all coherences in the system. It is shown
to be a rapidly decaying function, motivating a Markovian approximation on this
term in the evolution equation. The resulting subsystem density matrix equation
is still non-Markovian due to the fact that bath degrees of freedom have been
projected out of the dynamics. Provided the computation of non-equilibrium
average values or correlation functions is considered, the non-Markovian
character of this equation can be removed by lifting the equation into the full
phase space of the system. This leads to a trajectory description of the
dynamics where each fictitious trajectory accounts for decoherence due to the
bath degrees of freedom. The results are illustrated by computations of the
rate constant of a model nonadiabatic chemical reaction.
