  We analyze the dynamics of a spin 1/2 subsystem coupled to a spin chain. We
simulate numerically the full quantum many-body system for various sets of
parameters and initial states of the chain, and characterize the divisibility
of the subsystem dynamics, i.e. whether it is Markovian and can be described by
a (time dependent) master equation. We identify regimes in which the subsystem
admits such Markovian description, despite the many-body setting, and provide
insight about why the same is not possible in other regimes. Interestingly,
coupling the subsystem at the edge, instead of the center, of the chain gives
rise to qualitatively distinct behavior.
