sufficient. If this term grows above the value of the other terms in the Vlasov equation, then the approximation introduced in this paper is not valid any more. Reducing the value of $\hbar$ or increasing the number $N$ of wavefunctions used in the initial discretization will decrease the contribution of the correction terms but this will lead to a higher computational cost. The correction terms as well as the terms entering the Vlasov equation are expensive to compute but need not be computed at each time step.

Once the final time has been reached, the distribution function can be recovered by computing the integral (\[eq:initial\_distribution\]) or if one is only interested in the density field only, then equation (\[eq:2\]) is sufficient and straightforward to compute.

Formal derivation {#sec:framework}
=================

In the previous section, we described the problem we were interested and the usual schemes used in the literature. We also presented a brief description of the route we intend to follow in order to tackle the issues outlined. In this section, we describe the whole formulation in detail, derive its main equations and discuss its limits. For completeness, we start with a review of a formulation of quantum mechanics and 