  The stochastic mean-field (SMF) approach allows to treat correlations beyond
mean-field using a set of independent mean-field trajectories with appropriate
choice of fluctuating initial conditions. We show here, that this approach is
equivalent to a simplified version of the Bogolyubov-Born-Green-Kirkwood-Yvon
(BBGKY) hierarchy between one-, two-, ..., N-body degrees of freedom. In this
simplified version, one-body degrees of freedom are coupled to fluctuations to
all orders while retaining only specific terms of the general BBGKY hierarchy.
The use of the simplified BBGKY is illustrated with the Lipkin-Meshkov-Glick
(LMG) model. We show that a truncated version of this hierarchy can be useful,
as an alternative to the SMF, especially in the weak coupling regime to get
physical insight in the effect beyond mean-field. In particular, it leads to
approximate analytical expressions for the quantum fluctuations both in the
weak and strong coupling regime. In the strong coupling regime, it can only be
used for short time evolution. In that case, it gives information on the
evolution time-scale close to a saddle point associated to a quantum
phase-transition. For long time evolution and strong coupling, we observed that
the simplified BBGKY hierarchy cannot be truncated and only the full SMF with
initial sampling leads to reasonable results.
