is clear that there cannot be a unique answer to this question: the deviations from the unitary case may depend strongly on the form of $\mathbb{D}[\rho]$ in relation to the type of evolution imposed by $H(t)$. This variety of possible answers is reflected in the wide spectrum of cases already considered in the literature. Within the plethora of possible scenarios, it is however important to establish some general trends that may serve as guidelines in going deeper in this formidable problem.

This type of analysis was first performed [@Patane_2008] in the context of the Kibble-Zurek (KZ) mechanism for defect formation [@Kibble; @Zurek]. The KZ mechanism is related to the fact that, when crossing a gapless critical point of $H(t)$, no matter how slow the variation of the Hamiltonian in time is, the adiabatic theorem is violated and a finite density of defects will be produced. More than thirty years ago, Kibble put forward a scaling argument aimed at predicting the size and the number of such defects [@Kibble], while the mechanism yielding the correct scaling was found later by Zurek [@Zurek], roughly dividing the dynamics in either adiabatic or impulsive, according to the distance from the critical point. The KZ 