In open quantum systems decoherence occurs through interaction of a quantum
subsystem with its environment. The computation of expectation values requires
a knowledge of the quantum dynamics of operators and sampling from initial
states of the density matrix describing the subsystem and bath. We consider
situations where the quantum evolution can be approximated by quantum-classical
Liouville dynamics and examine the circumstances under which the evolution can
be reduced to surface-hopping dynamics, where the evolution consists of
trajectory segments evolving exclusively on single adiabatic surfaces, with
probabilistic hops between these surfaces. The justification for the reduction
depends on the validity of a Markovian approximation on a bath averaged memory
kernel that accounts for quantum coherence in the system. We show that such a
reduction is often possible when initial sampling is from either the quantum or
classical bath initial distributions. If the average is taken only over the
quantum dispersion that broadens the classical distribution, then such a
reduction is not always possible.Comment: 11, pages, 8 figure