After rescaling such quantity as $\Delta(\tau) \to \Delta(\tau) / \kappa$, we observe a fairly good data collapse with $\tau$, as plotted in Fig. \[fig:Rscl\_FreeF-Pump\]. In addition, our data obey a linear scaling as a function of the annealing time except for deviations induced by bigger values of $\kappa$ (rather than by longer annealing times $\tau$) in the regime where the excess energy is nearly saturated to its maximal value (see also Fig. \[fig:ExcT\_FreeF-Pump\]). We have checked that the behaviour of $\varepsilon(\tau) - \varepsilon(+\infty)$ towards saturation decays with a power law as $\sim \tau^{-1}$, which is in accordance with Ref. [@CamposVenuti].

![Final excess-energy difference $\Delta$ as a function of $\tau$, once rescaled by $\kappa$. The various data sets stand for different values of $\kappa$, and correspond to those of Fig. \[fig:ExcT\_FreeF-Pump\], where the same colour code has been used. A straight line indicating the linear scaling in the annealing time $\tau$ is shown in black.[]{data-label="fig:Rscl_FreeF-Pump"}](RsclK_FreeF-Pump){width="\columnwidth"}

The observations made above point toward a substantial independence of the role played by the dissipation, with respect to the KZ mechanism. The incoherent coupling to the external bath acts uniformly and irrespective of the adiabaticity condition ruled by the ground-state energy gap.

Interplay between pumping 