Dust dynamics in RAMSES -- II. Equilibrium drift velocity distributions of charged dust grains

We investigate the gas-grain relative drift velocity distributions of charged astrophysical dust grains in MHD turbulence. We do this using a range of MHD-PIC simulations spanning different plasma-$\beta$, sonic/Alfv\'en Mach number, and with grains of varying size and charge-to-mass ratio. We find that the root-mean-square drift velocity is a strong function of the grain size, following a power law with a 1/2 slope. The r.m.s. value has only a very weak dependence on the charge-to-mass ratio. On the other hand, the shape of the distribution is a strong function of the grain charge-to-mass ratio, and in compressible turbulence, also the grain size. We then compare these results to simple analytic models based upon time-domain quasi-linear theory and solutions to the Fokker-Planck equation. These models explain qualitatively the r.m.s. drift velocity's lack of charge-to-mass ratio dependence, as well as why the shape of the distribution changes as the charge-to-mass ratio increases. Finally we scale our results to astrophysical conditions. As an example, at a length scale of one parsec in the cold neutral medium, 0.1 $\mu$m grains should be drifting at roughly 40% of the turbulent velocity dispersion. These findings may serve as a basis for a model for grain velocities in the context of grain-grain collisions, non-thermal sputtering, and accretion of metals. These findings also have implications for the transport of grains through the galaxy, suggesting that grains may have non-negligible random motions at length-scales that many modern galaxy simulations approach.