  Within the broad theme of understanding the dynamics of disordered quantum
many-body systems, one of the simplest questions one can ask is: given an
initial state, how does it evolve in time on the associated Fock-space graph,
in terms of the distribution of probabilities thereon? A detailed quantitative
description of the temporal evolution of out-of-equilibrium disordered quantum
states and probability transport on the Fock space, is our central aim here. We
investigate it in the context of a disordered quantum spin chain which hosts a
disorder-driven many-body localisation transition. Real-time
dynamics/probability transport is shown to exhibit a rich phenomenology, which
is markedly different between the ergodic and many-body localised phases. The
dynamics is for example found to be strongly inhomogeneous at intermediate
times in both phases, but while it gives way to homogeneity at long times in
the ergodic phase, the dynamics remain inhomogeneous and multifractal in nature
for arbitrarily long times in the localised phase. Similarly, we show that an
appropriately defined dynamical lengthscale on the Fock-space graph is directly
related to the local spin-autocorrelation, and as such sheds light on the
(anomalous) decay of the autocorrelation in the ergodic phase, and lack of it
in the localised phase.
