  Dust plays a key role during star, disk and planet formation. Yet, its
dynamics during the protostellar collapse remains a poorly investigated field.
Recent studies seem to indicate that dust may decouple efficiently from the gas
during these early stages. We aim to understand how much and in which regions
dust grains concentrate during the early phases of the protostellar collapse,
and see how it depends on the properties of the initial cloud and of the solid
particles. We use the multiple species dust dynamics solver of the grid-based
code RAMSES to perform various simulations of dusty collapses. We perform
hydrodynamical and MHD simulations where we vary the maximum grain size, the
thermal-to-gravitational energy ratio and the magnetic properties of the cloud.
We simulate the simultaneous evolution of ten neutral dust grains species with
grain sizes varying from a few nm to a few hundredth of microns. We obtain a
significant decoupling between the gas and the dust for grains of typical sizes
a few 10 microns. This decoupling strongly depends on the
thermal-to-gravitational energy ratio, the grain sizes or the inclusion of a
magnetic field. With a semi-analytic model calibrated on our results, we show
that the dust ratio mostly varies exponentially with the initial Stokes number
at a rate that depends on the local cloud properties. We find that larger
grains tend to settle and drift efficiently in the first-core and in the newly
formed disk. This can produce dust-to-gas ratios of several times the initial
value. Dust concentrates in high density regions and is depleted in low density
regions. The size at which grains decouple from the gas depends on the initial
properties of the clouds. Since dust can not necessarily be used as a proxy for
gas during the collapse, we emphasize on the necessity of including the
treatment of its dynamics in collapse simulations.
