  A central problem in the theory of the dynamics of open quantum systems is
the derivation of a rigorous and computationally tractable master equation for
the reduced system density matrix. Most generally, the evolution of an open
quantum system is described by a completely positive linear map. We show how to
derive a completely positive Markovian master equation (the Lindblad equation)
from such a map by a coarse graining procedure. We provide a novel and explicit
recipe for calculating the coefficients of the master equation, using
perturbation theory in the weak-coupling limit. The only parameter external to
our theory is the coarse-graining time-scale. We illustrate the method by
explicitly deriving the master equation for the spin-boson model. The results
are evaluated for the exactly solvable case of pure dephasing, and an excellent
agreement is found within the timescale where the Markovian approximation is
expected to be valid. The method can be extended in principle to include
non-Markovian effects.
