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

Abstract

(abbreviated) The statistical mechanics of self-gravitating systems is a long-held puzzle. 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 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 beta, we numerically solve the two equations, and demonstrate that the velocity anisotropy parameter plays an important role to attain 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 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 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.

11 pages, 6 eps figures, to be published in MNRAS, final version

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Cited by in corpus (15)

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• Testing DARKexp against energy and density distributions of Millennium-II halos
• Equilibrium statistical mechanics for self-gravitating systems: local ergodicity and extended Boltzmann-Gibbs/White-Narayan statistics
• Saddle-point entropy states of equilibrated self-gravitating systems
• Ergodicity in a two-dimensional self gravitating many body system
• Power law pseudo phase-space density profiles of dark matter halos: fluke of physics?
• A Universal Power-law Profile of Pseudo-Phase-Space Density-like Quantities in Elliptical Galaxies
• Structural and dynamical modeling of WINGS clusters. III. The pseudo phase-space density profile
• Cored DARKexp systems with finite size: numerical results
• Second-order solutions of the equilibrium statistical mechanics for self-gravitating systems
• Statistical computation of Boltzmann entropy and estimation of the optimal probability density function from statistical sample
• A derivation of (half) the dark matter distribution function
• Fluid-like entropy and equilibrium statistical mechanics of self-gravitating systems
• Cored density profiles in the DARKexp model
• An approach to the quasi-equilibrium state of a self-gravitating system