Journal Article

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

Ping He

in Monthly Notices of the Royal Astronomical Society

Published on behalf of The Royal Astronomical Society

Volume 419, issue 2, pages 1667-1681
Published in print January 2012 | ISSN: 0035-8711
Published online December 2011 | e-ISSN: 1365-2966 | DOI:
Equilibrium statistical mechanics for self-gravitating systems: local ergodicity and extended Boltzmann–Gibbs/White–Narayan statistics

More Like This

Show all results sharing this subject:

  • Astronomy and Astrophysics


Show Summary Details


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.

Keywords: methods: analytical; galaxies: kinematics and dynamics; cosmology: theory; dark matter; large-scale structure of Universe

Journal Article.  9429 words.  Illustrated.

Subjects: Astronomy and Astrophysics

Full text: subscription required

How to subscribe Recommend to my Librarian

Users without a subscription are not able to see the full content. Please, subscribe or login to access all content.