The statistical mechanics of isolated collisionless self‐gravitating systems is a long‐held puzzle, which has not been successfully resolved for nearly 50 years. In this work, we employ a phenomenological entropy form of ideal gas, first proposed by White & Narayan, to revisit this issue. By calculating the first‐order variation of the entropy, subject to the usual mass‐ and energy‐conservation constraints, we obtain an entropy stationary equation. Incorporated with the Jeans equation, and by specifying some functional form for the anisotropy parameter β, we numerically solve the two equations, and demonstrate that the velocity anisotropy parameter plays an important role in attaining a density profile that is finite in mass, energy and spatial extent. If incorporated again with some empirical density profile from simulations, our theoretical predictions of the anisotropy parameter, and the radial pseudo‐phase‐space density ρ/σ3r in the outer non‐gravitationally degenerate region of the dark matter halo, agree rather well with the simulation data, and the predictions are also acceptable in the middle weak‐degenerate region of the dark halo. The disagreements occur just in the inner strong‐degenerate region because of the neglect of gravitational degeneracy. As far as we know, our results may be the first theoretical predictions based on the entropy principle that can partially match the empirical data. The second‐order variational calculus reveals the seemingly paradoxical but actually complementary consequence that the equilibrium state of self‐gravitating systems is the global minimum entropy state for the whole system under long‐range violent relaxation, but simultaneously the local maximum entropy state for every and any small part of the system under short‐range two‐body relaxation and Landau damping. This minimum–maximum entropy duality means that the standard second law of thermodynamics needs to be re‐expressed or generalized for self‐gravitating systems. We believe that our findings, especially the complementary second law of thermodynamics, may provide crucial clues to the development of the statistical physics of self‐gravitating systems as well as other long‐range interaction systems.

Context. 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. Aims. 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. Methods. 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.Results. 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. Conclusions. Our work provides further support to the idea that statistical mechanics for self-gravitating systems is a viable tool for investigation.

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