  We present a general master equation, correct to second order in the
system-environment coupling strength, that takes into account the initial
system-environment correlations. We assume that the system and its environment
are in a joint thermal equilibrium state, and thereafter a unitary operation is
performed to prepare the desired initial system state, with the system
Hamiltonian possibly changing thereafter as well. We show that the effect of
the initial correlations shows up in the second-order master equation as an
additional term, similar in form to the usual second-order term describing
relaxation and decoherence in quantum systems. We apply this master equation to
a generalization of the paradigmatic spin-boson model, namely a collection of
two-level systems interacting with a common environment of harmonic
oscillators, as well as a collection of two-level systems interacting with a
common spin environment. We demonstrate that, in general, the initial
system-environment correlations need to be accounted for in order to accurately
obtain the system dynamics.
