  We have conducted the first comprehensive numerical investigation of the
relative velocity distribution of dust particles in self-gravitating
protoplanetary discs with a view to assessing the viability of planetesimal
formation via direct collapse in such environments. The viability depends
crucially on the large sizes that are preferentially collected in pressure
maxima produced by transient spiral features (Stokes numbers, $St \sim 1$);
growth to these size scales requires that collision velocities remain low
enough that grain growth is not reversed by fragmentation. We show that, for a
single sized dust population, velocity driving by the disc's gravitational
perturbations is only effective for $St > 3$, while coupling to the gas
velocity dominates otherwise. We develop a criterion for understanding this
result in terms of the stopping distance being of order the disc scale height.
Nevertheless, the relative velocities induced by differential radial drift in
multi-sized dust populations are too high to allow the growth of silicate dust
particles beyond $St \sim 10^{-2}$ or $10^{-1}$ ($10\,\mathrm{cm}$ to
$\mathrm{m}$ sizes at $30\,\mathrm{au}$), such Stokes numbers being
insufficient to allow concentration of solids in spiral features. However, for
icy solids (which may survive collisions up to several
$10\,\mathrm{m\,s}^{-1}$), growth to $St \sim 1$ ($10\,\mathrm{m}$ size) may be
possible beyond $30\,\mathrm{au}$ from the star. Such objects would be
concentrated in spiral features and could potentially produce larger icy
planetesimals/comets by gravitational collapse. These planetesimals would
acquire moderate eccentricities and remain unmodified over the remaining
lifetime of the disc.
