  A local excitation in a quantum many-particle system evolves
deterministically. A time-reversal procedure, involving the invertion of the
signs of every energy and interaction, should produce an excitation revival:
the Loschmidt echo (LE). If somewhat imperfect, only a fraction of the
excitation will refocus. We use such a procedure to show how non-inverted weak
disorder and interactions, when assisted by the natural reversible dynamics,
fully degrade the LE. These perturbations enhance diffusion and evenly
distribute the excitation throughout the system. Such a dynamical paradigm,
called ergodicity, breaks down when either the disorder or the interactions are
too strong. These extreme regimes give rise to the well known Anderson
localization and Mott insulating phases, where quantum diffusion becomes
restricted. Accordingly, regardless of the kinetic energy terms, the excitation
remains mainly localized and out-of-equilibrium, and the system behaves
non-ergodically. The LE constitutes a fair dynamical witness for the whole
phase diagram since it evidences a surprising topography in which ergodic and
non-ergodic phases interpenetrate each other. Furthermore, we provide an
estimation for the critical lines separating the ergodic and non-ergodic phases
around the Mott and Anderson transitions. The energy uncertainties introduced
by disorder and interaction shift these thresholds towards stronger
perturbations. Remarkably, the estimations of the critical lines are in good
agreement with the phase diagram derived from the LE dynamics.
