  The possibility to apply phase-space methods to many-body interacting systems
might provide accurate descriptions of correlations with a reduced numerical
cost. For instance, the so--called stochastic mean-field phase-space approach,
where the complex dynamics of interacting fermions is replaced by a statistical
average of mean-field like trajectories is able to grasp some correlations
beyond the mean-field. We explore the possibility to use alternative equations
of motion in the phase-space approach. Guided by the BBGKY hierarchy, equations
of motion that already incorporate part of the correlations beyond mean-field
are employed along each trajectory. The method is called Hybrid Phase-Space
(HPS) because it mixes phase-space techniques and the time-dependent reduced
density matrix approach. The novel approach is applied to the one-dimensional
Fermi-Hubbard model. We show that the predictive power is improved compared to
the original stochastic mean-field method. In particular, in the weak-coupling
regime, the results of the HPS theory can hardly be distinguished from the
exact solution even for long time.
