  When it comes to applying the adiabatic theorem in practice, the key question
to be answered is how slow "slowly enough" is. This question can be an
intricate one, especially for many-body systems, where the limits of slow
driving and large system size may not commute. Recently we have shown how the
quantum adiabaticity in many-body systems is related to the generalized
orthogonality catastrophe [Phys. Rev. Lett. 119, 200401 (2017)]. We have proven
a rigorous inequality relating these two phenomena and applied it to establish
conditions for the quantized transport in the topological Thouless pump. In the
present contribution we (i) review these developments and (ii) apply the
inequality to establish the conditions for adiabaticity in a one-dimensional
system consisting of a quantum fluid and an impurity particle pulled through
the fluid by an external force. The latter analysis is vital for the correct
quantitative description of the phenomenon of quasi Bloch oscillations in a
one-dimensional translation invariant impurity-fluid system.
