Brownian Motion in Gravitationally-Interacting Systems

We derive a model that describes the motion of a Brownian particle in a system which is dominated by gravitational forces. An example of such a system is a massive black hole immersed in a cluster of stars. We compute the dispersion in the position and velocity of such a black hole, and examine whether it achieves a state of equipartition of kinetic energy with the stars. This problem has been considered before only for stellar systems with an isothermal Maxwellian distribution of velocities; here we study other examples and confirm our calculations with N-body simulations. In certain cases, depending on the stellar distribution function, the black hole can acquire a steady state kinetic energy which is very far from equipartition relative to the stars.