  A quantum system interacting with a dilute gas experiences irreversible
dynamics. The corresponding master equation can be derived within two different
approaches: The fully quantum description in the low-density limit and the
semiclassical collision model, where the motion of gas particles is classical
whereas their internal degrees of freedom are quantum. The two approaches have
been extensively studied in the literature, but their predictions have not been
compared. This is mainly due to the fact that the low-density limit is
extensively studied for mathematical physics purposes, whereas the collision
models have been essentially developed for quantum information tasks such as a
tractable description of the open quantum dynamics. Here we develop and for the
first time compare both approaches for a spin system interacting with a gas of
spin particles. Using some approximations, we explicitly find the corresponding
master equations including the Lamb shifts and the dissipators. The low density
limit in the Born approximation for fast particles is shown to be equivalent to
the semiclassical collision model in the stroboscopic approximation. We reveal
that both approaches give exactly the same master equation if the gas
temperature is high enough. This allows to interchangeably use complicated
calculations in the low density limit and rather simple calculations in the
collision model.
