A statistical-mechanical explanation of dark matter halo properties

Kang, Dong-Biao; He, Pinjia

doi:10.1051/0004-6361/201015057

Cited by 15 publications

(14 citation statements)

References 26 publications

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“…Based on the dynamical similarities between self‐gravitating systems and fluids, we further confirm the validity of fluid‐like entropy for self‐gravitating systems. Also, we improve the variational calculation of our previous work (Kang & He 2011), and propose an index‐varying polytropic structure, which has finite mass and can be obtained under the assumption of incomplete relaxation. If we treat the equation of dynamical equilibrium as a differential constraint to the entropy, we derive a similar equation to White & Narayan (1987).…”

Section: Discussionmentioning

confidence: 99%

“…However, in this paper we will argue about the validity of their calculation. Our previous works (He & Kang 2010; Kang & He 2011, hereafter KH11) employed this entropy form to study the equilibrium configurations of dark matter haloes. Moreover, KH11 considered the general anisotropic situation and found an equation of state whose density profile has finite mass and extent.…”

Section: Introductionmentioning

confidence: 99%

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Fluid-like entropy and equilibrium statistical mechanics of self-gravitating systems

Kang

2011

Monthly Notices of the Royal Astronomical Society

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The statistical mechanics of self‐gravitating systems has not been well understood, and still remains an open question so far. In a previous study by Kang & He, we showed that the fluid approximation may give a clue to investigate this problem further. In fact, there are indeed many dynamical similarities between self‐gravitating and fluid systems. Based on a fluid‐like entropy, that work explained successfully the outer density profiles of dark matter haloes, but there remain some drawbacks with the calculation concerning the extremizing process of the entropy. In the current paper, with the improved extremizing calculation – including an additional differential constraint of dynamical equilibrium and without any other assumptions, we confirm that statistical‐mechanical methods can give a density profile with finite mass and finite energy. Moreover, this density profile is also consistent with the observational surface brightness of the elliptical galaxy NGC 3379. In our methods, the density profile is derived from the equation of state, which is obtained from the entropy principle but does not correspond to the maximum entropy of the system. Finally, we suggest an alternative entropy form, a hybrid of Boltzmann–Gibbs and Tsallis entropy, whose global maximum may give rise to this equation of state.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fluid-like entropy and equilibrium statistical mechanics of self-gravitating systems

Kang

2011

Monthly Notices of the Royal Astronomical Society

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“…Hence, it is necessary to return to the fundamentals of statistical mechanics and develop special techniques to handle the long‐range nature of gravity (Padmanabhan 1990; Chavanis 2006). In our earlier two works (He & Kang 2010; Kang & He 2011), we performed preliminary investigations into the statistical mechanics of collisionless self‐gravitating systems, and found that both the concept of entropy and the entropy principle are still valid for self‐gravitating systems, which implies that an entropy‐based statistical mechanics of self‐gravitating systems is feasible.…”

Section: Introductionmentioning

confidence: 99%

Second-order solutions of the equilibrium statistical mechanics for self-gravitating systems

He¹

2012

Monthly Notices of the Royal Astronomical Society

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In a previous study, we formulated a framework for the entropy‐based equilibrium statistical mechanics for self‐gravitating systems. This theory is based on the Boltzmann–Gibbs entropy and includes the generalized virial equations as additional constraints. With the truncated distribution function to the lowest order, we derived a set of second‐order equations for the equilibrium states of the system. In this work, the numerical solutions of these equations are investigated. It is found that there are three types of solutions for these equations. Both the isothermal and divergent solutions are thermally unstable and have unconfined density profiles with infinite mass, energy and spatial extent. The convergent solutions, however, seem to be reasonable. Although the results cannot match the simulation data well, because of the truncations of the distribution function and its moment equations, these lowest‐order convergent solutions show that the density profiles of the system are confined, the velocity dispersions are variable functions of the radius, and the velocity distributions are also anisotropic in different directions. The convergent solutions also indicate that the statistical equilibrium of self‐gravitating systems is by no means the thermodynamic equilibrium. These solutions are just the lowest‐order approximation, but they have already manifested the qualitative success of our theory. We expect that higher‐order solutions of our statistical‐mechanical theory will give much better agreement with the simulation results concerning dark matter haloes.

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“…In a follow‐up work (Kang & He 2011), by introducing an effective pressure instead of a radial pressure to construct the specific entropy, we used the entropy principle. We proceeded in a similar way as in He & Kang (2010) and we obtained a new entropy stationary equation.…”

Section: Introductionmentioning

confidence: 99%

“…Entropy is extensive. We first constructed the specific entropy, and then obtained the total entropy in the additive and hence extensive way, as in of He & Kang (2010) or of Kang & He (2011). The fact that such an extensively defined entropy works very well indicates that a non‐extensive entropy, such as the entropy of Tsallis (1988), is not necessary. The equilibrium states also consist of mechanical equilibrium.…”

Section: Introductionmentioning

confidence: 99%

Equilibrium statistical mechanics for self-gravitating systems: local ergodicity and extended Boltzmann-Gibbs/White-Narayan statistics

He¹

2011

Monthly Notices of the Royal Astronomical Society

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The long‐standing puzzle surrounding the statistical mechanics of self‐gravitating systems has not yet been solved successfully. We formulate a systematic theoretical framework of entropy‐based statistical mechanics for spherically symmetric collisionless self‐gravitating systems. We use an approach that is very different from that of the conventional statistical mechanics of short‐range interaction systems. We demonstrate that the equilibrium states of self‐gravitating systems consist of both mechanical and statistical equilibria, with the former characterized by a series of velocity‐moment equations and the latter by statistical equilibrium equations, which should be derived from the entropy principle. The velocity‐moment equations of all orders are derived from the steady‐state collisionless Boltzmann equation. We point out that the ergodicity is invalid for the whole self‐gravitating system, but it can be re‐established locally. Based on the local ergodicity, using Fermi–Dirac‐like statistics, with the non‐degenerate condition and the spatial independence of the local microstates, we rederive the Boltzmann–Gibbs entropy. This is consistent with the validity of the collisionless Boltzmann equation, and should be the correct entropy form for collisionless self‐gravitating systems. Apart from the usual constraints of mass and energy conservation, we demonstrate that the series of moment or virialization equations must be included as additional constraints on the entropy functional when performing the variational calculus; this is an extension to the original prescription by White & Narayan. Any possible velocity distribution can be produced by the statistical‐mechanical approach that we have developed with the extended Boltzmann–Gibbs/White–Narayan statistics. Finally, we discuss the questions of negative specific heat and ensemble inequivalence for self‐gravitating systems.

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A statistical-mechanical explanation of dark matter halo properties

Cited by 15 publications

References 26 publications

Fluid-like entropy and equilibrium statistical mechanics of self-gravitating systems

Fluid-like entropy and equilibrium statistical mechanics of self-gravitating systems

Second-order solutions of the equilibrium statistical mechanics for self-gravitating systems

Equilibrium statistical mechanics for self-gravitating systems: local ergodicity and extended Boltzmann-Gibbs/White-Narayan statistics

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