  In the weak-coupling limit approach to open quantum systems, the presence of
the bath is eliminated and accounted for by a master equation that introduces
dissipative contributions to the system reduced dynamics. Within this
framework, there are no bath entropy contributions to the entropy balance. We
show that, as a consequence, the entropy production fails to be positive for a
class of physically legitimate (i.e., completely positive and trace preserving)
non-Markovian dynamical maps. Moreover, in the absence of the semigroup
property, if the reduced dynamics has a thermal asymptotic state, this need not
be stationary. In this case, even the integrated entropy production becomes
negative. These observations imply that, when the conditions leading to reduced
dynamics of semigroup type are relaxed, a consistent formulation of the second
law of thermodynamics requires that the environment contribution to the entropy
balance be explicitly taken into account.
