  In this work, we analyze the Born, Bogoliubov, Green, Kirkwood and Yvon
(BBGKY) hierarchy of equations for describing the full time-evolution of a
many-body fermionic system in terms of its reduced density matrices (at all
orders). We provide an exhaustive study of the challenges and open problems
linked to the truncation of such hierarchy of equations to make them
practically applicable. We restrict our analysis to the coupled evolution of
the one- and two-body reduced density matrices, where higher order correlation
effects are embodied into the approximation used to close the equations. We
prove that within this approach, the number of electrons and total energy are
conserved, regardless of the employed approximation. Further, we demonstrate
that although most of the truncation schemes available in the literature give
acceptable ground state energy, when applied to describe driven electron
dynamics exhibit undesirable and unphysical behavior, e.g., violation and even
divergence in local electronic density, both in weakly- and strongly-correlated
regimes. We illustrate and analyze these problems within the few-site Hubbard
model. The model can be solved exactly and provides a unique reference for our
detailed study of electron dynamics for different values of interaction,
different initial conditions, and large set of approximations considered here.
Moreover, we study the role of compatibility between two hierarchical
equations, and positive- semidefiniteness of reduced density matrices in
instability of electron dynamics. We show that even if the used approximation
holds the compatibility, electron dynamics can still diverge when positive-
definitiveness is violated. We propose some partial solutions of such problem
and point the main paths for future work in order to make this approach
applicable for the description of the correlated electron dynamics in complex
systems.
