Epilogue

Dmitriev, Sviatoslav

doi:10.4324/9781315112244-11

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“…We can now investigate the dynamical and thermodynamical stability of the equilibrium states of relativistic star 20 They mentioned that the damped oscillations of T (ρ 0 ) are similar to the damped oscillations of M (ρ 0 ) for neutron stars discovered by Dmitriev and Kholin [135]. This is because, in the ultrarelativistic limit where the density (or the redshift) is large, the equation of state of a classical isothermal gas takes the form P = /3 = Kn 4/3 , where is the energy density and n the particle number, like the ultrarelativistic equation of state of a Fermi gas at T = 0 (or like the black-body radiation).…”

Section: General Relativistic Systemsmentioning

confidence: 68%

“…According to Eqs. (135) and (139), the caloric curve η(Λ) presents a clear scaling. Indeed, when ν → 0, the caloric curve defined in terms of the variables η/ν 2 and Λν 2 − ν (instead of η and Λ) tends towards a limit curve given in parametric form by…”

Section: H the Asymptotic Caloric Curve B(m)mentioning

confidence: 91%

“…From these functions, using Eqs. (135) and (139), we can obtain the caloric curve (series of equilibria) η(Λ) of the classical ultrarelativistic gas for a given value of ν. It has the form of a spiral parametrized by a or, equivalently, by the energy density contrast R = (0)/ (R) = e ψ(a) .…”

Section: F Entropy and Free Energymentioning

confidence: 99%

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

Chavanis

2020

Eur. Phys. J. Plus

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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...

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Section: General Relativistic Systemsmentioning

confidence: 68%

Section: H the Asymptotic Caloric Curve B(m)mentioning

confidence: 91%