  Cosmological N-body simulations have revealed many empirical relationships of
dark matter halos, yet the physical origin of these halo properties still
remains unclear. On the other hand, the attempts to establish the statistical
mechanics for self-gravitating systems have encountered many formal
difficulties, and little progress has been made for about fifty years. The aim
of this work is to strengthen the validity of the statistical-mechanical
approach we have proposed previously to explain the dark matter halo
properties. By introducing an effective pressure instead of the radial pressure
to construct the specific entropy, we use the entropy principle and proceed in
a similar way as previously to obtain an entropy stationary equation. An
equation of state for equilibrated dark halos is derived from this entropy
stationary equation, by which the dark halo density profiles with finite mass
can be obtained. We also derive the anisotropy parameter and pseudo-phase-space
density profile. All these predictions agree well with numerical simulations in
the outer regions of dark halos. Our work provides further support to the idea
that statistical mechanics for self-gravitating systems is a viable tool for
investigation.
