  We analyze two approaches to elimination of a fast variable (velocity) in
stochastic systems: moment and cumulant formalisms. With these approaches, we
obtain the corresponding Smoluchovski-type equations, which contain only the
coordinate/phase variable. The adiabatic elimination of velocity in terms of
cumulants and moments requires the first three elements. However, for the case
of small inertia, the corrected Smoluchowski equation in terms of moments
requires five elements, while in terms of cumulants the same first three
elements are sufficient. Compared to the method based on the expansion of the
velocity distribution in Hermite functions, the considered approaches have
comparable efficiency, but do not require individual mathematical preparation
for the case of active Brownian particles, where one has to construct a new
basis of eigenfunctions instead of the Hermite ones.
