  In the stochastic mean-field (SMF) approach, an ensemble of initial values
for a selected set of one-body observables is formed by stochastic sampling
from a phase-space distribution that reproduces the initial quantum
fluctuations. Independent mean-field evolutions are performed with each set of
initial values followed by averaging over the resulting ensemble. This approach
has been recently shown to be rather versatile and accurate in describing the
correlated dynamics beyond the independent particle picture. In the original
formulation of SMF, it was proposed to use a Gaussian assumption for the
phase-space distribution. This assumption turns out to be rather effective when
the dynamics of an initially uncorrelated state is considered, which was the
case in all applications of this approach up to now. Using the
Lipkin-Meshkov-Glick (LMG) model, we show that such an assumption might not be
adequate if the quantum system under interest is initially correlated and
presents configuration mixing between several Slater determinants. In this
case, a more realistic description of the initial phase-space is necessary. We
show that the SMF approach can be advantageously combined with standard methods
to describe phase-space in quantum mechanics. As an illustration, the Husimi
distribution function is used here to obtain a realistic representation of the
phase-space of a quantum many-body system. This method greatly improves the
description of initially correlated fermionic many-body states. In the LMG
model, while the Gaussian approximation failed to describe these systems in all
interaction strength range, the novel approach gives a perfect agreement with
the exact evolution in the weak coupling regime and significantly improves the
description of correlated systems in the strong coupling regime.
