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

Abstract

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.

A Derivation of (Half) the Dark Matter Distribution Function

“…The origin of the universalities may lie in some fundamental property of dark matter, be it some statistical mechanics (Lynden-Bell 1967;Hjorth & Williams 2010) or optimization of some generalized entropy (Plastino & Plastino 1993;Hansen et al 2005;He & Kang 2010;He 2012). It may also be associated with dynamical effects, like radial orbit instability (Henriksen 2009;Bellovary et al 2008), or phase mixing or violent relaxation (Lynden-Bell 1967; Kandrup et al 2003).…”

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

“…First, we integrate the steady‐state CBE (i.e. ∂ f /∂ t = 0 of with Γ[ f ] = 0) to obtain its various orders of velocity‐moment equations (see equation A11 of He 2012): with the even integers m , k ≥ 0. These moment equations are then translated into their equivalent virialization forms (see of He 2012) as These generalized virial equations are just the proper forms for the additional macroscopic constraints besides mass and energy conservation, which is an extension to the original prescription by White & Narayan (1987).…”

“…is the differential form of the mass function, . See section 4.3 of He (2012) for details about these equations.…”

“…Based on these findings, He (2012) formulated a systematic theoretical framework of the statistical mechanics for spherically symmetric collisionless self‐gravitating systems. It was demonstrated that the ergodicity is invalid globally for the whole self‐gravitating system but can be re‐established locally, and the Boltzmann–Gibbs entropy was re‐derived, which should be the correct entropy form for collisionless gravitating systems.…”

“…In this work, the aim is to study the lowest‐order solutions of the statistical‐mechanical theory developed in He (2012). The paper is organized as follows.…”