We investigate the merging process in N-body self-gravitating system from the viewpoints of the local virial relation which is the relation between the local kinetic energy and the local potential .
We compare both the density profile and the phase space density profile in cosmological simulations with the critical solutions of collisionless static state satisfying the local virial ( LV ) relation .
We got the results that the critical solution can explain the characteristic density profile with the appropriate value of anisotropy parameter \beta \sim 0.5 .
It can also explain the power law of the phase space density profile in the outer part of a bound state .
However , it fails in explaining the central low temperature part which is connected to the scale invariant phase space density .
It can be well fitted to the critical solution with the higher value of \beta \sim 0.75 .
These results indicate that the LV relation is not compatible with the scale invariant phase space density in cosmological simulation .