  Large scale simulations and analytical theory have been combined to obtain
the non-equilibrium velocity distribution, $f(v)$, of randomly accelerated
particles in suspension. The simulations are based on an event-driven
algorithm, generalised to include friction. They reveal strongly anomalous but
largely universal distributions which are independent of volume fraction and
collision processes, which suggests a one-particle model should capture all the
essential features. We have formulated this one-particle model and solved it
analytically in the limit of strong damping, where we find that $f(v)$ decays
as $1/v$ for multiple decades, eventually crossing over to a Gaussian decay for
the largest velocities. Many particle simulations and numerical solution of the
one-particle model agree for all values of the damping.
