  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.
