Local virial relation for self-gravitating system

We demonstrate that the quasi-equilibrium state in a self-gravitating $N$-body system after cold collapse is uniquely characterized by the local virial relation using numerical simulations. Conversely, assuming the constant local virial ratio and Jeans equation for a spherically steady-state system, we investigate the full solution space of the problem under the constant anisotropy parameter and obtain some relevant solutions. Specifically, the local virial relation always provides a solution which has a power-law density profile in both the asymptotic regions $r\to 0$ and $\infty$. This type of solution is commonly observed in many numerical simulations. Only the anisotropic velocity dispersion controls this asymptotic behavior of density profile.