  We establish a theoretical framework for exploring the quantum dynamics of
finite ultracold bosonic ensembles based on the
Born-Bogoliubov-Green-Kirkwood-Yvon (BBGKY) hierarchy of equations of motion
for few-particle reduced density matrices (RDMs). The theory applies to zero as
well as low temperatures and is formulated in a highly efficient way by
utilizing dynamically optimized single-particle basis states and representing
the RDMs in terms of permanents with respect to those. An energy, RDM
compatibility and symmetry conserving closure approximation is developed on the
basis of a recursively formulated cluster expansion for these finite systems.
In order to enforce necessary representability conditions, two novel,
minimal-invasive and energy-conserving correction algorithms are proposed,
involving the dynamical purification of the solution of the truncated BBGKY
hierarchy and the correction of the equations of motion themselves,
respectively. For gaining conceptual insights, the impact of two-particle
correlations on the dynamical quantum depletion is studied analytically. We
apply this theoretical framework to both a tunneling and an interaction-quench
scenario. Due to our efficient formulation of the theory, we can reach
truncation orders as large as twelve and thereby systematically study the
impact of the truncation order on the results. While the short-time dynamics is
found to be excellently described with controllable accuracy, significant
deviations occur on a longer time-scale in sufficiently far off-equilibrium
situations. Theses deviations are accompanied by exponential-like instabilities
leading to unphysical results. The phenomenology of these instabilities is
investigated in detail and we show that the minimal-invasive correction
algorithm of the equation of motion can indeed stabilize the BBGKY hierarchy
truncated at the second order.
