  This paper deals with the convergence of the Doi-Navier-Stokes model of
liquid crystals to the Ericksen-Leslie model in the limit of the Deborah number
tending to zero. While the literature has investigated this problem by means of
the Hilbert expansion method, we develop the moment method, i.e. a method that
exploits conservation relations obeyed by the collision operator. These are
non-classical conservation relations which are associated with a new concept,
that of Generalized Collision Invariant (GCI). In this paper, we develop the
GCI concept and relate it to geometrical and analytical structures of the
collision operator. Then, the derivation of the limit model using the GCI is
performed in an arbitrary number of spatial dimensions and with non-constant
and non-uniform polymer density. This non-uniformity generates new terms in the
Ericksen-Leslie model.
