Brownian Motion in Gravitationally Interacting Systems

We derive a model that describes the dynamics of a Brownian particle, such as a massive black hole, in a stellar system dominated by gravitational forces, 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. We show that in certain cases the black hole’s steady state kinetic energy can be very far from equipartition.