  We investigate how the dynamical fluctuations of many-body quantum systems
out of equilibrium can be mitigated when they are opened to a dephasing
environment. We consider the survival probability (spectral form factor with a
filter) evolving under different kinds of random matrices and under a spin-1/2
model with weak and strong disorder. In isolated many-body quantum systems, the
survival probability is non-self-averaging at any timescale, that is, the
relative variance of its fluctuations does not decrease with system size. By
opening the system, we find that the fluctuations are always reduced, but
self-averaging can only be ensured away from critical points. Self-averaging is
achieved for the long-time dynamics of full random matrices, power-law banded
random matrices deep in the delocalized phase, and the Rosenzweig-Porter
ensemble in all the phases except at the delocalization-localization transition
point. For the spin model, the survival probability becomes self-averaging only
in the chaotic regime provided the initial states are in the middle of the
spectrum. Overall, a strongly non-self-averaging survival probability in open
systems is an indicator of criticality.
