Statistical mechanics of self-gravitating systems in general relativity: II. The classical Boltzmann gas

Abstract

We study the statistical mechanics of classical self-gravitating systems confined within a box of radius R in general relativity. It has been found that the caloric curve T∞(E) has the form of a double spiral whose shape depends on the compactness parameter ν = GN m/Rc 2 . The double spiral shrinks as ν increases and finally disappears when νmax = 0.1764. Therefore, general relativistic effects render the system more unstable. On the other hand, the cold spiral and the hot spiral move away from each other as ν decreases. Using a normalization Λ = −ER/GN 2 m 2 and η = GN m 2 /RkBT∞ appropriate to the nonrelativistic limit, and considering ν → 0, the hot spiral goes to infinity and the caloric curve tends towards a limit curve (determined by the Emden equation) exhibiting a single cold spiral, as found in former works. Using another normalization M = GM/Rc 2 and B = Rc 4 /GN kBT∞ appropriate to the ultrarelativistic limit, and considering ν → 0, the cold spiral goes to infinity and the caloric curve tends towards a limit curve (determined by the general relativistic Emden equation) exhibiting a single hot spiral. This result is new. We discuss the analogies and the differences between this asymptotic caloric curve and the caloric curve of the selfgravitating black-body radiation. Finally, we compare box-confined isothermal models with heavily truncated isothermal distributions in Newtonian gravity and general relativity.PACS numbers: 04.40. Dg, 05.70.Fh, 95.30.Sf, 95.35.+d I. INTRODUCTIONIn a preceding paper [1] (Paper I), relying on previous works on the subject [2-25], we have developed a general formalism to study the statistical mechanics of self-gravitating systems in general relativity. This formalism is valid for an arbitrary form of entropy. The statistical equilibrium state is obtained by maximizing the entropy S at fixed mass-energy M c 2 and particle number N . The extremization problem yields the Tolman-Oppenheimer-Volkoff (TOV) equations [2,26] expressing the condition of hydrostatic equilibrium together with the Tolman-Klein relations [2,27]. The precise form of entropy determines the distribution function and the equation of state of the system. For illustration, in Paper I, we considered a system of self-gravitating fermions. In that case, the statistical equilibrium state is obtained by maximizing the Fermi-Dirac entropy at fixed mass-energy and particle number. This yields the relativistic Fermi-Dirac distribution function. In the present paper, we consider the statistical mechanics of classical self-gravitating particles. 1 In that case, the statistical equilibrium state is obtained by maximizing the Boltzmann entropy at fixed mass-energy and particle number. This yields the relativistic Maxwell-Boltzmann (or Maxwell-Juttner) distribution function. We start by a brief history of the subject giving an exhaustive list of references. 2 The statistical mechanics of nonrelativistic classical self-gravitating systems was developed in connection with the dynamical evolution of globular clusters (see...

Thermal equilibrium states and instability of self-gravitating particles in an asymptotically AdS spacetime

“…This structure in Fig. 3a is similar to a curve in a relativistic particle system confined by an artificial wall [15,16]. Having this similarity, following Refs.…”

“…Futhermore, the strong instability also sets in at the turning point because the gravothermal energy is bounded by the value at the point. In the relativistic system [15,16], as is explicitly shown in Sec. 2 for the asymptotically AdS case, the series of equilibria draws a two-dimensional surface.…”

“…This expectation was shown to be correct in Refs. [11][12][13][14][15][16] for massive particle systems (see also [17,18] for radiation). They showed a curve for the series of equilibria has a double spiral structure.…”

On thermodynamics of compact objects

“…( 26), with the help of Tolman's law, given in Eq. (13). It is in general larger than the geometric volume V geo , given in Eq.…”

“…It was shown that the maximum entropy principle of statistical mechanics could be used to derive the basic equations describing a static and spherically symmetric self-gravitating gas in GR [10,11] (see Ref. [12,13] for recent reviews). Explicit examples include the self-gravitating black-body radiation [14,15] and fermion gas [16], corresponding to the equilibrium solutions for photon stars and neutron stars.…”