  The definition of a consistent evolution equation for statistical hybrid
quantum-classical systems is still an open problem. In this paper we analyze
the case of Ehrenfest dynamics on systems defined by a probability density and
identify the relations of the non-linearity of the dynamics with the
obstructions to define a consistent dynamics for the first quantum moment of
the distribution. This first quantum moment represents the physical states as a
family of classically-parametrized density matrices $\hat \rho(\xi)$, for $\xi$
a classical point; and it is the most common representation of hybrid systems
in the literature. Due to this obstruction, we consider higher order quantum
moments, and argue that only a finite number of them are physically measurable.
Because of this, we propose an effective solution for the hybrid dynamics
problem based on approximating the distribution by those moments and
representing the states by them.
