Entropy principle and complementary second law of thermodynamics for self-gravitating systems

Abstract

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.

A statistical-mechanical explanation of dark matter halo properties

“…As expected, the resulting Q(r) profile closely follows a power law, r −1.875 , near the center of the dark halo. However, the predictions do show a manifest curve-up deviation from this power law at the outskirts of dark halos, which has been observed by the simulation result of Ludlow et al (2010), and has already been reproduced by our previous study (He & Kang 2010).…”

“…As mentioned in the introduction, in He & Kang (2010) we employed the entropy principle and derived an entropy stationary equation by introducing the specific entropy,…”

“…The left-hand side of this equation is formally the same as that of He & Kang (2010), except that p r is replaced by the effective pressure P. Equation (7) can be readily transformed into dρ dP = 3 5…”

“…In a recent work (He & Kang 2010), we employed a phenomenological entropy form of ideal gas, first proposed by White & Narayan (1987), to revisit this question. First, subject to the usual mass-and energy-conservation constraints, we calculated the first-order variation of the entropy, and obtained an entropy stationary equation.…”

“…With the analogy of the isotropic Jeans equation, we introduce the effective pressure instead of the radial pressure to define the entropy. In a similar procedure, we derive a new entropy stationary equation, which can be easily integrated to attain the equation of state of the equilibrated dark halos, which He & Kang (2010) failed to obtain with the radial pressure. It is this state equation A&A 526, A147 (2011) that we employ to derive the halo density profile.…”