VIRIAL THEOREM FOR ROTATING SELF-GRAVITATING BROWNIAN PARTICLES AND TWO-DIMENSIONAL POINT VORTICES

Abstract

We derive the virial theorem for an overdamped system of rotating self-gravitating Brownian particles. We show that, in the two-dimensional case, it takes a closed form that can be used to obtain general results about the dynamics without being required to solve the Smoluchowski–Poisson system explicitly. In particular, we obtain the exact analytical expression of the mean square displacement 〈r2〉(t) of the interacting Brownian particles. We exhibit a critical temperature below which the system collapses, and above which it evaporates, and we determine how this temperature is affected by a solid rotation. We also develop an analogy between self-gravitating systems and two-dimensional point vortices. We derive a virial-like relation for point vortices at statistical equilibrium relating the angular velocity to the angular momentum and the temperature.