As the title says, if I have a system of particles interacting only due to gravity, over what timescale do we expect them to fall into a virial equilibrium?
By virial equilibrium I mean a system that satisfies the virial theorem: $$ 2K + W =0 $$
where $K$ and $W$ are the kinetic and potential energy of the system, respectively.
According to this lecture from the University of Edinburgh, numerical simulations of N-body systems suggest a half-mass relaxation time: $$ t_\text{rh} = 0.138\frac{N^{1/2}r_\text{h}^{3/2}}{m^{1/2}G^{1/2}\ln(\gamma N)} $$
where $r_\text{h}$ is the radius that initially contains half the mass of the system, $G$ is the gravitational constant, $m$ is the average particle mass, $N$ is the number of particles, and $\gamma$ is a constant, approximately 0.11 for a system containing equal masses.
Since the half-mass radius is a good approximation to the virial radius, this is a good approximation (at least a lower bound) to the virial equilibrium timescale.
There is also a version that depends on velocity dispersion $v$ and density $\rho$: $$ t_\text{r} = 0.065\frac{v^3}{\rho m G^2\ln(\gamma N)} $$
