Let us consider a supermassive black hole surrounded by a spherical cluster of stars. Without the other stars, each individual star would simply move on a Keplerian orbit (we neglect the effects of general relativity). However, due to the mutual interaction, stars diffuse between different Keplerian orbits. When stars approach too close to the central black hole they get accreted. In order to maintain a steady state, there must be a constant stream of stars moving inward. This condition determines the density profile of stars around the black hole.

The time scale over which the orbital elements change considerably is

$ \frac{1}{\tau} \approx n \Sigma v $

where $ n $ is the number density of stars, $ v \propto r^{-1/2} $ and $ \Sigma \propto r^2 $ is the effective time between collisions between two stars. The inward current of stars is given by $ I \approx \frac{n r^3 v}{\tau} \propto n^2 r^{9/2} \Rightarrow n \propto r^{-9/4} $