  The Markovian evolution of an open quantum system is characterized by a
positive entropy production, while the global entropy gets redistributed
between the system and the environment degrees of freedom. Starting from these
premises, we analyze the entropy variation of an open quantum system in terms
of two distinct relations: the Clausius inequality, that provides an intrinsic
bound for the entropy variation in terms of the heat absorbed by the system,
and an extrinsic inequality, which instead relates the former to the
corresponding entropy increment of the environment. By modeling the
thermalization process with a Markovian collisional model, we compare and
discuss the two bounds, showing that the latter is asymptotically saturated in
the limit of large interaction time. In this regime not only the reduced
density matrix of the system reaches an equilibrium configuration, but it also
factorizes from the environment degrees of freedom. This last result is proven
analytically when the system-bath coupling is sufficiently strong and through
numerical analysis in the weak-coupling regime.
