Abstract
We develop a general formalism to determine the statistical equilibrium states of self-gravitating systems in general relativity and complete previous works on the subject. Our results are valid for an arbitrary form of entropy but, for illustration, we explicitly consider the FermiâDirac entropy for fermions. The maximization of entropy at fixed mass energy and particle number determines the distribution function of the system and its equation of state. It also implies the TolmanâOppenheimerâVolkoff equations of hydrostatic equilibrium and the TolmanâKlein relations. Our paper provides all the necessary equations that are needed to construct the caloric curves of self-gravitating fermions in general relativity obtained in recent works (Roupas and Chavanis in Class Quant Grav 36:065001, 2019; Chavanis and Alberti in Phys Lett B 801:135155, 2020). We consider the nonrelativistic limit \(c\rightarrow +\infty \) and recover the equations obtained within the framework of Newtonian gravity. We also discuss the inequivalence of statistical ensembles as well as the relation between the dynamical and thermodynamical stability of self-gravitating systems in Newtonian gravity and general relativity.
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Notes
It is not well known that, at the same period, Zwicky [35, 36] also attempted to determine the maximum mass of neutron stars that he interpreted as Schwarzschildâs mass. Using heuristic arguments, he obtained an expression of the form \(M_{\mathrm{max}}^{\mathrm{Zwicky}}=k\, m_n(e^2/Gm_\mathrm{p}m_\mathrm{e})^{3/2}=91\, k\, M_{\odot }\), where k is a dimensionless number assumed to be of order unity. He mentioned the need to take into account the equation of state of the neutrons but did not quote Oppenheimer and Volkoff [34] (actually, Oppenheimer and Zwicky refused to acknowledge each otherâs papers, see [37]). The maximum mass obtained by Oppenheimer and Volkoff [34] is \(M_{\mathrm{max}}^{\mathrm{OV}}=0.384\, (\hbar c/G)^{3/2}m_n^{-2}=0.710\, M_{\odot }\). Matching the two expressions of the maximum mass obtained by Zwicky and Oppenheimer and Volkoff, we get \(k=7.80\times 10^{-3}\) and \(\alpha \equiv e^2/\hbar c=13.4 m_\mathrm{p}m_\mathrm{e}/m_n^2\simeq 13.4 {m_\mathrm{e}}/{m_\mathrm{p}}\). Interestingly, this equation provides a relation between the fine-structure constant \(\alpha \simeq 1/137\) and the ratio \(m_\mathrm{e}/m_\mathrm{p}\) between the electron mass and the proton mass (we have used \(m_n\simeq m_\mathrm{p}\)). This type of relationships has been proposed in the past by several authors [38] using heuristic arguments or pure numerology (see [39] for a recent discussion).
These equations are equivalent to the FermiâDiracâPoisson equations. They constitute the so-called finite-temperature ThomasâFermi (TF) model where the quantum potential is neglected. At \(T=0\), the FermiâDirac distribution reduces to a step function and we recover the TF model. We recall that the original TF model [167,168,169,170] describes the distribution of electrons in an atom which results from the balance between the quantum pressure (Pauliâs exclusion principle), the electrostatic repulsion of the electrons and the electrostatic attraction of the nucleus. For self-gravitating systems, the TF model describes the distribution of fermions in a star or in a dark matter halo, which results from the balance between the quantum pressure (Pauliâs exclusion principle) and the gravitational attraction of the fermions. The rigorous mathematical justification of the thermodynamic limit for self-gravitating fermions in a box leading to the finite-temperature TF model is given in Refs. [171,172,173,174,175,176,177,178,179,180,181].
A functional of the form \(S=-\int C({\overline{f}})\, \mathrm{d}{\mathbf{r}}\mathrm{d}{\mathbf{v}}\), where \(C({\overline{f}})\) is any convex function of the coarse-grained distribution function \({\overline{f}}({\mathbf{r}},{\mathbf{v}},t)\), was introduced by Antonov [220] for collisionless stellar systems, and called âquasi-entropyâ (see Ref. [222]). The same functional was reintroduced independently by Tremaine et al. [221] in relation to the theory of violent relaxation [160] and called âH-function.â A functional of the form \(S=-\int C({f})\, \mathrm{d}{\mathbf{r}}\mathrm{d}{\mathbf{v}}\), without the bar on f, was introduced by Ipser [187, 196] in his study on the dynamical stability of collisionless stellar systems with respect to the VlasovâPoisson equations. An effective thermodynamical formalism involving generalized entropic functionals of the form \(S=-\int C({f})\, \mathrm{d}{\mathbf{r}}\mathrm{d}{\mathbf{v}}\) was developed by Chavanis [223,224,225,226,227,228,229,230,231,232,233] who gave various (dynamical and thermodynamical) interpretations of these functionals.
There is debate [242,243,244] on the name that should be given to these equations: OV or TOV? As far as we can judge, Eq. (108) was first written by Tolman [241], while Eq. (110) was first written by Oppenheimer and Volkoff [34]. Therefore, Eq. (108) should be called the Tolman equation and Eq. (110) should be called the OV equation. If we consider that Eq. (110) is a rather direct consequence of Eq. (108), then it may be called the TOV equation as well [243]. However, Ref. [244] stresses some fundamental differences between Eqs. (108) and (110). To complement this discussion, we note that Chandrasekhar [51, 245] rederived Eq. (110) without referring to Oppenheimer and Volkoff [34], maybe considering that this equation results almost immediately from the Einstein field equations (107)â(109) with the metric (103). In his review on the first 30 years of general relativity [246], he writes: âEquations (60) and (61) are often referred to as the OppenheimerâVolkoff equations though they are contained in Schwarzschildâs paper in very much these forms.â
The equation of state of a completely degenerate (\(T=0\)) Fermi gas at arbitrary densities (i.e., for any degree of relativistic motion) was first derived by Frenkel [14] in a not-well-known paper. It was rederived independently by Stoner [16, 21] and Chandrasekhar [26]. The equation of state of a gas of fermions at arbitrary temperature was first derived by Juttner [139] extending his earlier work on the relativistic theory of an ideal classical gas [247, 248]. These different results are exposed in the classical monograph of Chandrasekhar [140] on stellar structure.
The case of classical particles described by the Boltzmann entropy is specifically considered in our companion paper [3] (Paper II).
We use this âtwo-stepâ procedure because (i) it can be easily extended to general relativity and (ii) it is useful for studying the sign of the second variations of entropy determining the thermodynamical stability of the system (see Appendix F).
According to the Jeans theorem [282], a spherical stellar system in collisionless equilibrium has a distribution function of the form \(f=f(\epsilon ,L)\) where \(\epsilon \) is the energy and L is the angular momentum. We note that an extremum of entropy S at fixed energy E and particle number N leads to a distribution function that depends only on \(\epsilon \), not on L. Therefore, an extremum of entropy at fixed energy and particle number is necessarily isotropic.
This result is very general (being valid for an arbitrary entropic functional and for any long-range potential of interaction) and stems from the fact that the entropy (which is a particular Casimir), the energy and the particle number are conserved by the VlasovâPoisson equations (see [231] and Sect. 5.2).
This is at variance with the VlasovâPoisson and EulerâPoisson equations which are reversible and do not satisfy an H-theorem.
Since \(\rho (R)=0\) implies \(\mu (R)=0\), we find that \(\mu _0=m\Phi (R)=-GMm/R\). Therefore, \(\mu _0<0\).
See Appendix C of [289] for a more general discussion.
This inequality was previously derived by Schwarzschild [78] in the case of equilibrium configurations with uniform energy density.
Equation (111) is valid at equilibrium (for a steady state). Therefore, expressions (128) and (129) of S and N are justified only at equilibrium. However, it is shown in [51] that Eq. (111) remains valid for small perturbations about equilibrium up to second order in the motion. This justifies using Eqs. (128) and (129) when we make perturbations about the equilibrium state as in the variational problem considered below.
The binding energy is usually defined by \(E_\mathrm{b}=(Nm-M)c^2\) so that \(E=-E_\mathrm{b}\). It is the difference between the rest mass energy \(Nmc^2\) (the energy that the matter of the star would have if dispersed to infinity) and the total mass energy \(Mc^2\). In order to simplify the discussion, we shall define the binding energy as \(E_\mathrm{b}=(M-Nm)c^2\) so that \(E=E_\mathrm{b}\).
We note that Bilic and Viollier [262] work in the canonical ensemble while we work in the microcanonical ensemble. However, as we have already indicated (see Sect. 2.8), the statistical ensembles are equivalent at the level of the first-order variations (extremization problem) so they determine the same equilibrium states. For systems with long-range interactions, like self-gravitating systems, ensembles inequivalence may occur at the level of the second-order variations of the thermodynamical potential, i.e., regarding the stability of the equilibrium states.
A similar equation \(\mathrm{d}P/{\mathrm{d}T}=(\epsilon +P)/T\) is used in cosmology in order to relate the temperature T of a cosmic fluid described by an equation of state \(P=P(\epsilon )\) to its energy density \(\epsilon \) (see, e.g., [299]). In that context, it is derived from thermodynamical arguments [290] by assuming that \(\mu =0\) like in the case of the black-body radiation. By contrast, in the present calculation, we have simply used the fact that \(\mu /T\) is constant, not that \(\mu \) is necessarily equal to zero.
Furthermore, an entropy maximum at fixed energy and particle number is dynamically stable.
We have seen in Sect. 3.5 that the inverse Tolman temperature \(\beta _{\infty }\) is the conjugate variable to the energy. Therefore, this is the quantity to keep constant in the canonical ensemble.
This equation can be directly obtained by taking the derivative of Eq. (197).
According to the relativistic Jeans theorem [258, 300,301,302,303], a spherical stellar system in collisionless equilibrium has a distribution function of the form \(f=f(E e^{\nu (r)/2},L)\) where \(E e^{\nu (r)/2}\) is the energy at infinity and L is the angular momentum. We note that an extremum of entropy S at fixed mass energy \(Mc^2\) and particle number N leads to a distribution function that depends only on \(E e^{\nu (r)/2}\), not on L. Therefore, an extremum of entropy at fixed mass energy and particle number is necessarily isotropic. We also note that the relativistic Jeans theorem implies the Tolman relation (165) when f is the Fermi-Dirac or MaxwellâBoltzmann distribution. Thus, the Tolman relation is contained in the relativistic Jeans theorem.
We note that \(\alpha \sim mc^2/k_\mathrm{B} T\rightarrow +\infty \) in the nonrelativistic limit \(c\rightarrow +\infty \) (i.e., \(k_\mathrm{B} T\ll mc^2\)). Therefore, \(\alpha \) is positive while \(\alpha ^{\mathrm{NR}}_0\) may be of any sign.
We note that \(\mu (r)\sim mc^2\rightarrow +\infty \) in the nonrelativistic limit \(c\rightarrow +\infty \). Therefore, \(\mu (r)\) is positive while \(\mu _{\mathrm{NR}}(r)\) may be of any sign.
We can use this method to show graphically (without calculation) that gaseous polytropes with index \(n<3\) are stable, while gaseous polytropes with index \(3<n\le 5\) are unstable (polytropes with index \(n>5\) have infinite mass) [228]. Note that in certain situations (e.g., for self-gravitating fermions confined within a box), a mode of stability can be regained after a turning point of mass (see [117] for detail).
The optimization problems (251) and (252) are equivalent to \(\min \lbrace E | N\, \mathrm{fixed} \rbrace \) and \(\min \lbrace E | M\, \mathrm{fixed} \rbrace \), where \(E=(M-Nm)c^2\) is the binding energy. In the nonrelativistic limit, they reduce to the optimization problem (247). Indeed, when \(c\rightarrow +\infty \), repeating the steps of Sect. 4.6 with u in place of \(\epsilon _{\mathrm{kin}}\), we get \(E\rightarrow U+W={{{\mathcal {W}}}}\).
The PoincarĂŠ turning point criterion [281] is equivalent to the massâradius theorem of Wheeler [66] introduced in the physics of compact objects like white dwarfs and neutron stars. Note that in certain situations (e.g., for neutron stars described by the HarrisonâWakanoâWheeler equation of state [40]), a mode of stability can be regained after a turning point of mass (see [66] for detail).
Here, the analogy with thermodynamics is effective (hence the quotation marks).
There is an exception. In the case of an infinite and homogeneous medium, collisionless stellar systems (Vlasov) and self-gravitating fluids (Euler) behave in the same way with respect to the Jeans instability in the sense that they lead to the same criterion for instability [328]. Note, however, that the evolution of the perturbations is different in the two systems.
Following the work of Tsallis [234], there has been a large activity in statistical mechanics about the use of âgeneralized entropies.â We are not claiming that all entropies are equally relevant or equally significant. The Boltzmann entropy for an ideal gas of classical particles and the FermiâDirac entropy for an ideal gas of quantum particles remain the fundamental ones because they can be obtained from first principles by a combinatorial analysis (see, e.g., [334]). However, it has been noticed by many authors (see, e.g., [230, 333] and references therein) that the formalism of statistical mechanics and kinetic theory can be extended to an almost arbitrary class of entropies. This fact is now well established. The present paper shows that this property remains true in general relativity. Indeed, the variational principles discussed in this paper can be developed for an arbitrary entropy of the form of Eq. (C1), not just for the Boltzmann and FermiâDirac entropies. It is convenient to treat the case of a general entropy because (i) this allows us to unify the treatments of classical (Boltzmann) and quantum (FermiâDirac) statistics; (ii) this allows us to draw a connection between dynamical and thermodynamical stabilities (see Sect. 5); and (iii) generalized forms of entropies may have applications in certain situations, e.g., for nonideal (complex) fluids with short-range interactions and correlations (see [230, 232, 233]), and for tidally truncated self-gravitating systems (the generalized entropies associated with the classical and fermionic King models are derived in [153, 215]).
This relation determines the chemical potential \(\mu \) up to an additive constant that may depend on the temperature T. The complete expression of the chemical potential can be obtained from Eq. (F4).
A more direct derivation of this result is given in âNewtonian self-gravitating gases at statistical equilibriumâ section in Appendix I.
It can also be obtained by substituting Eq. (G1) into the BoseâEinstein entropy \(s=-k_\mathrm{B}\frac{1}{h^3}\int \lbrace \frac{f}{f_{*}}\ln \frac{f}{f_{*}} - (1+\frac{f}{f_{*}} )\ln (1+\frac{f}{f_{*}})\rbrace \, \mathrm{d}{\mathbf{p}}\) where \(f_*=1/h^3\).
It is important to point out, however, that, for the self-gravitating black-body radiation, dynamical stability refers to the EulerâEinstein equations while, for the collisionless star clusters considered by Ipser [258], it refers to the VlasovâEinstein equations. This is a difference of fundamental importance.
Comparing the right term of Eq. (G16) with the right term of Eq. (253), we note that \(\mu _{\infty }^{\mathrm{eff}}\equiv \lambda k_\mathrm{B} T_{\infty }\) corresponds to an âeffectiveâ chemical potential. Recall, however, that the true chemical potential of the photons is equal to zero (\(\mu =0\)).
The calculations of Tolman [241] based on the maximum entropy principle are comparatively much more complicated.
In the isentropic case (\(s/n=\lambda \)), we have \(S=\lambda N\). Therefore, a minimum of \({{{\mathcal {W}}}}\) at fixed N is also a minimum of \({{{\mathcal {W}}}}\) at fixed S or a maximum of S at fixed \({{{\mathcal {W}}}}\).
In the isentropic case (\(s/n=\lambda \)), we have \(S=\lambda N\). Therefore, a minimum of E at fixed N is also a minimum of E at fixed S or a maximum of S at fixed E (or \({{{\mathcal {E}}}}\)).
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Appendices
Appendix A: General relations between the pressure and the energy density
In this appendix, we provide general relations between the pressure and the energy density in the nonrelativistic and ultrarelativistic limits. They are valid for an arbitrary distribution function.
1.1 Nonrelativistic limit
In the nonrelativistic limit, the kinetic energy of a particle is
In that case, the kinetic energy density and the pressure are given by
We have the general relations
1.2 Ultrarelativistic limit
In the ultrarelativistic limit, the energy of a particle is
In that case, the energy density and the pressure are given by
We have the general relations
Appendix B: Virial theorem for Newtonian systems
In this appendix, we establish the general expression of the equilibrium scalar virial theorem for Newtonian systems. For the sake of generality, we allow the particles to be relativistic in the sense of special relativity.
It can be shown that the virial of the gravitational force is equal to the gravitational energy (see, e.g., Appendix G of [341]):
This equation is generally valid for steady and unsteady configurations. It does not depend whether the system is spherically symmetric or not. If we now consider a spherically symmetric system (still allowed to be unsteady), using the Newton law (3), we find from Eq. (B1) that
This formula is useful to calculate the gravitational potential energy of a spherically symmetric distribution of matter. It can be directly obtained by approaching from infinity a succession of spherical shells of mass \({\mathrm{d}M}(r)=\rho (r) 4\pi r^{2}\mathrm{d}r\) with potential energy \(-GM(r){\mathrm{d}M}(r)/r\) in the field of the mass M(r) already present, and integrating over r (see, e.g., Ref. [334]).
We now consider a self-gravitating system at equilibrium. Substituting the condition of hydrostatic equilibrium
into Eq. (B1), and integrating by parts, we get
If the system is not submitted to an external pressure, the second term on the right-hand side vanishes. On the other hand, if the external pressure is uniform on the boundary of the system, i.e., \(P({\mathbf{r}})=P_{b}=\mathrm{cst}\), we have
More generally, this relation can be taken as a definition of \(P_{b}\). Combining the foregoing relations, we obtain the general form of the equilibrium scalar virial theorem
For nonrelativistic particles, using Eq. (A3), the equilibrium scalar virial theorem becomes
Using \(E=E_{\mathrm{kin}}+W\), we get
When \(P_\mathrm{b}=0\), we obtain \(E=-E_{\mathrm{kin}}<0\). For ultrarelativistic particles, using Eq. (A6), the equilibrium scalar virial theorem becomes
When \(P_\mathrm{b}=0\), we obtain \(E=0\).
Appendix C: Derivation of the statistical equilibrium state for a general form of entropy
In this appendix, we derive the statistical equilibrium state of a self-gravitating system for a general form of entropy
where C(f) is any convex function (i.e., \(C''(f)>0\)). This is what we call a generalized entropy [223, 230, 232]. These functionals appeared in relation to the notion of âgeneralized thermodynamicsâ pioneered by Tsallis [234] who introduced a particular form of non-Boltzmannian entropy (of a power-law type) called the Tsallis entropy. These functionals (which are particular Casimir integrals) are also useful to obtain sufficient conditions of nonlinear dynamical stability [187, 196, 228, 231, 258] with respect to the Vlasov equation describing a collisionless evolution of the system (see Sect. 5.2). Below, we show that the maximum entropy principle can be applied to an arbitrary form of entropy. We consider both Newtonian and general relativistic systems. We first present a two-step derivation as in the main text, then a one-step derivation.
1.1 Two-step derivation
To maximize the entropy S at fixed energy E and particle number N, we proceed in two steps as in Sects. 2 and 3. We first maximize the entropy density s(r) at fixed energy density \(\epsilon (r)\) and particle number density n(r) with respect to variations of \(f({\mathbf{r}},{\mathbf{p}})\) repeating the calculations of Sects. 2.4 and 3.4. The variational principle (132) for the extremization problem yields
Since C is convex, this relation can be inverted. It determines a distribution function of the form
where
Since
this distribution function is the global maximum of the entropy density at fixed energy density and particle number density. This corresponds to the condition of local thermodynamical equilibrium. Using the integrated GibbsâDuhem relation (E16) which is valid for a general form of entropy (see Appendix E), we can express the entropy S as a functional of n(r) and \(\epsilon (r)\). We now maximize the entropy S at fixed energy E and particle number N with respect to variations of n(r) and \(\epsilon (r)\).
1.1.1 Newtonian gravity
We first consider the Newtonian gravity regime but, for the sake of generality, we allow the particles to be relativistic in the sense of special relativity. In Newtonian gravity, Eq. (C3) is replaced by
where \(E_{\mathrm{kin}}(p)\) is given by Eq. (123). Repeating the calculations of Sect. 2.5, we obtain Eqs. (47) and (58) expressing the uniformity of the temperature and of the total chemical potential (Gibbs law). We also obtain the condition of hydrostatic equilibrium (61). As a result, the distribution function at statistical equilibrium is given by
This is a function of the form
where \(\epsilon ({\mathbf{r}},{\mathbf{p}})=E_{\mathrm{kin}}(p)/m+\Phi (r)\) is the energy of a particle by unit of mass (in the nonrelativistic regime where \(E_{\mathrm{kin}}=p^2/2m\) we have \(\epsilon ({\mathbf{r}},{\mathbf{v}})=v^2/2+\Phi (r)\) where we have introduced the velocity \({\mathbf{v}}={\mathbf{p}}/m\) instead of the impulse \({\mathbf{p}}\)). We note that an extremum of entropy at fixed energy and particle number is necessarily isotropic. Repeating the arguments of Sect. 2.5, we can show that the gas corresponding to the distribution function (C7) is described by a barotropic equation of state \(P(r)=P[\rho (r),T]\), where the function \(P(\rho ,T)\) is determined by the function C(f) characterizing the entropy.
1.1.2 General relativity
We now consider the general relativity case. Repeating the steps of Sect. 3.5, we obtain Eq. (151). We also obtain the TOV equations (162) and (163) expressing the condition of hydrostatic equilibrium and the TolmanâKlein relations (165) and (166). As a result, the distribution function at statistical equilibrium is given by
Using Eq. (165), it can be written as
This is a function of the form
where E(p) is the energy of a particle. We note that an extremum of entropy at fixed mass energy and particle number is necessarily isotropic. Repeating the arguments of Sect. 3.7.1, we can show that the gas corresponding to the distribution function (C9) is described by a barotropic equation of state \(P(r)=P[\epsilon (r),\alpha ]\), where the function \(P(\epsilon ,\alpha )\) is determined by the function C(f) characterizing the entropy.
1.2 One-step derivation
We now present a one-step derivation of the preceding results. We first consider the Newtonian gravity case. The generalized entropy is
The particle number and the mass are given by
The energy is given by
where \(E_{\mathrm{kin}}\) is the kinetic energy and W is the potential (gravitational) energy.
In the microcanonical ensemble, the statistical equilibrium state is obtained by maximizing the entropy S at fixed energy E and particle number N with respect to variations of \(f({\mathbf{r}},{\mathbf{p}})\). We write the variational problem for the first variations (extremization) as
where \(\beta \) and \(\alpha _0\) are global Lagrange multipliers. Taking the variations with respect to \(f({\mathbf{r}},{\mathbf{p}})\), we obtain
leading to
where F(x) is defined by Eq. (C4). This returns the result from Eq. (C7). The temperature T and the chemical potential \(\mu _0\) are related to \(\beta \) and \(\alpha _0\) by
We can then rewrite Eq. (C17) as
For the FermiâDirac entropy
we obtain the mean field FermiâDirac distribution
or, equivalently,
For the Boltzmann entropy
we obtain the mean field MaxwellâBoltzmann distribution
or, equivalently,
This one-step derivation of the statistical equilibrium state, valid for a generalized entropy of form (C1) and directly leading to Eq. (C7), was given in Newtonian gravity by Ipser [187, 196], Tremaine et al. [221] and Chavanis [223]. It was extended in general relativity by Ipser [258], directly leading to Eq. (C10).
Remark
If we multiply Eq. (C16) by f and integrate over phase space, we get
If we compare this expression with the relation
obtained from Eqs. (39), (48) and (58), we obtain the identity
The right-hand side can be expressed in different manners by using the virial theorem (B6) or relations (A3) and (A6) valid for nonrelativistic and ultrarelativistic particles, respectively.
Appendix D: Condition of hydrostatic equilibrium for a general form of entropy
In this appendix, we show by a direct calculation that the condition of statistical equilibrium, obtained by extremizing the entropy at fixed energy and particle number, implies the condition of hydrostatic equilibrium. We consider a general form of entropy given by Eq. (C1).
1.1 Newtonian gravity
We first consider the Newtonian gravity case but, for the sake of generality, we allow the particles to be relativistic in the sense of special relativity. The extremization of the entropy S at fixed particle number N and energy E leads to a distribution function of the form (see Appendix C)
where F is defined by Eq. (C4) and where \(\beta \) and \(\alpha _0\) are constant. According to Eqs. (126) and (D1), the pressure is given by
Taking its gradient with respect to \({\mathbf{r}}\), we get
This can also be written as
Integrating by parts, we can rewrite the foregoing equation as
Since the density is given by
we finally obtain the condition of hydrostatic equilibrium
1.2 General relativity
We now consider the general relativity case. The extremization of the entropy S at fixed mass energy \({{{\mathcal {E}}}}\) and particle number N leads to a distribution function of the form (see Appendix C)
where F is defined by Eq. (C4) and where \(\alpha \) is constant. According to Eqs. (126) and (D8), the pressure is given by
Taking its derivative with respect to r, we obtain
This can also be written as
Integrating by parts, we can rewrite the foregoing equation as
Since the pressure is given by Eq. (D9) and the energy density by
we finally obtain the equation
Combined with Eq. (161), it leads to the OV equation (162). From Eqs. (107), (109) and (162), we then obtain the Tolman equation (163) which expresses the condition of hydrostatic equilibrium.
Appendix E: GibbsâDuhem relation
In this appendix, we derive the GibbsâDuhem relation. We first recall the usual derivation of the global GibbsâDuhem relation for extensive systems. Then, we provide a direct derivation of the local GibbsâDuhem relation (the one used in our paper) for an arbitrary form of entropy without using the extensivity assumption. This shows that our thermodynamical formalism is valid for an arbitrary form of entropy.
1.1 Standard derivation
The first law of thermodynamics can be written as
An extensive variable (energy, entropy, ...) is proportional to the absolute size of the system. In other words, if one doubles all extensive variables, all other extensive quantities also become twice as large. For example,
where \(\alpha \) is the enlargement factor. One calls functions which have this property homogeneous functions of first order. All extensive variables are homogeneous functions of first order of the other extensive variables. On the other hand, the intensive variables (temperature, pressure ...) are homogenous functions of zeroth order of the extensive variables, i.e., they do not change if we divide or duplicate the system. For example,
According to the Euler theorem, we have
Differentiating this expression and using the first law of thermodynamics (E1), we get the GibbsâDuhem relation
We note that the energy does not appear in this expression. The Euler relation (E4) for thermodynamic variables is also called the integrated GibbsâDuhem relation.
Defining \(s=S/V\), \(n=N/V\) and \(\epsilon =E/V\), the local GibbsâDuhem and integrated GibbsâDuhem relations can be written as
and
Introducing \(S=sV\), \(N=nV\) and \(E=\epsilon V\) in the first law of thermodynamics (E1), developing the expression and using the integrated GibbsâDuhem relation (E7), we obtain
We note that the pressure does not explicitly appear in this expression. This is the local form of the first law of thermodynamics [see Eq. (136)].
Remark
In âDirect derivation of the integrated GibbsâDuhem relation for a general form of entropyâ section in Appendix E, we establish Eq. (E7) directly from the local condition of thermodynamical equilibrium for an arbitrary form of entropy. Therefore, the local GibbsâDuhem relations (E6) and (E7) are always valid. By contrast, the global GibbsâDuhem relations (E4) and (E5) are valid only for extensive systems with short-range interactions. In particular, the global GibbsâDuhem relation is different for extensive systems and for nonextensive systems with long-range interactions. In Sect. 2.5.5, we generalize the global GibbsâDuhem relations (E4) and (E5) to the case of self-gravitating systems and recover the results of Latella et al. [276,277,278].
1.2 Direct derivation of the integrated GibbsâDuhem relation for a general form of entropy
The local condition of thermodynamical equilibrium, obtained by maximizing the local entropy at fixed energy density and particle number density, is given by Eq. (C3) with Eq. (C4). Substituting Eq. (C3) into Eqs. (119), (120) and (126), we find that the particle number density, the energy density and the pressure are given by
On the other hand, the entropy density [see Eq. (C1)] is given by
Integrating this equation by parts and using \(C'[F(x)]=-x\), we get
Integrating by parts on more time, we obtain
Comparing Eq. (E14) with Eqs. (E9)â(E11), we find that
Using Eq. (134), we finally obtain the integrated GibbsâDuhem relation
This calculation emphasizes the fact that the integrated GibbsâDuhem relation is valid for an arbitrary form of entropyFootnote 39 and for an arbitrary level of relativity.
Remark
The calculations presented in this appendix are equivalent to those performed in Appendix B of [228], in Appendix C of [289] and in Appendix D of [230] although the connection with the integrated GibbsâDuhem relation was not realized at that time.
Appendix F: Entropy and free energy as functionals of the density for Newtonian self-gravitating systems
We consider a Newtonian self-gravitating system but, for the sake of generality, we allow the particles to be relativistic in the sense of special relativity. We also consider a general form of entropy. In âOne-step derivationâ section in Appendix C, we have introduced entropy and free energy functionals of the distribution function \(f({\mathbf{r}},{\mathbf{v}})\). In Sect. 2.5 and in âNewtonian gravityâ section in Appendix C, we have introduced entropy and free energy functionals of the local density n(r) and local kinetic energy \(\epsilon _{\mathrm{kin}}(r)\). In this appendix, we introduce entropy and free energy functionals of the local density n(r).
1.1 Microcanonical ensemble
In the microcanonical ensemble, the statistical equilibrium state is obtained by maximizing the entropy S[f] at fixed energy E and particle number N. To solve this maximization problem, we proceed in two steps. We first maximize S[f] at fixed E, Nand particle density n(r). Since n(r) determines the particle number N[n] and the gravitational energy W[n], this is equivalent to maximizing S[f] at fixed kinetic energy \(E_{\mathrm{kin}}\) and particle density n(r). The variational problem for the first variations (extremization) can be written as
where \(\beta \) is a global (uniform) Lagrange multiplier and \(\alpha (r)\) is a local (position-dependent) Lagrange multiplier. This variational problem, which is equivalent to
returns the results of âTwo-step derivationâ section in Appendix C, except that \(\beta (r)\) is replaced by \(\beta \). Therefore, it yields
In this manner, we immediately find that T is uniform at statistical equilibrium. This results from the conservation of energy. As in âTwo-step derivationâ section in Appendix C, we can show that the distribution (F3) is the global maximum of S[f] at fixed \(E_{\mathrm{kin}}\) and n(r). Substituting Eq. (F3) into Eqs. (18), (19) and (21), we get
The Lagrange multiplier \(\alpha (r)\) is determined by the density n(r) according to Eq. (F4). On the other hand, the temperature T is determined by the kinetic energy \(E_{\mathrm{kin}}[n(r),T]=E-W[n(r)]\) using Eq. (F5) integrated over the volume. In other words, the temperature is determined by the energy constraint
We note that T is a functional of the density n(r) but, for brevity, we shall not write this dependence explicitly.
Repeating the steps of âDirect derivation of the integrated GibbsâDuhem relation for a general form of entropyâ section in Appendix E, we can derive the integrated GibbsâDuhem relation (E16), except that T(r) is replaced by T. Therefore, we get
Since T is uniform, Eq. (41) reduces to
On the other hand, eliminating formally \(\alpha (r)\) between Eqs. (F4) and (F6), we see that the equation of state is barotropic: \(P(r)=P[n(r),T]\) (we have explicitly written the temperature T because it is uniform but not constant when we consider variations of n(r) as explained above). Therefore, according to Eq. (F9), we have \(\mu (r)=\mu [n(r),T]\) with
where the derivative is with respect to n.Footnote 40
We can now simplify the expression of the entropy. Using the integrated GibbsâDuhem relation (F8), we have
The entropy can be written as a functional of the density as
or, using Eq. (F7), as
where U[n(r), T] is the internal energy given by
Combining Eqs. (F10) and (F14), we get
Therefore, the pressure P(n, T) is related to the density of internal energy V(n, T) by
Inversely, the density of internal energy is determined by the equation of state P[n(r), T] according to the relation
We note the identities
The internal energy can be written explicitly as
Finally, the statistical equilibrium state in the microcanonical ensemble is obtained by maximizing the entropy S[n] at fixed particle number N, the energy constraint being taken into account in the determination of the temperature T[n] through relation (F7). The variational problem for the first variations (extremization) can be written as
The conservation of energy implies [see Eq. (F7)]:
Using Eqs. (F1) and (F21), we get
As a result, the variational problem (F20) yields
We then recover all the results of Sect. 2. The interest of this formulation it that it allows us to solve more easily the stability problem related to the sign of the second variations of entropy [342]. This problem has been studied in detail in [204, 211, 216] for the Boltzmann entropy and in [225, 237, 238] for the Tsallis entropy. It has also been studied in [263] for the Boltzmann entropy within the framework of special relativity.
1.2 Canonical ensemble
In the canonical ensemble, the statistical equilibrium state is obtained by minimizing the free energy \(F[f]=E[f]-TS[f]\) at fixed particle number N, or equivalently, by maximizing the Massieu function \(J[f]=S[f]/k_\mathrm{B}-\beta E[f]\) at fixed particle number N. To solve this maximization problem, we proceed in two steps. We first maximize \(J[f]=S[f]/k_\mathrm{B}-\beta E[f]\) at fixed Nand particle density n(r). Since n(r) determines the particle number N[n] and the gravitational energy W[n], this is equivalent to maximizing \(S[f]/k_\mathrm{B}-\beta E_{\mathrm{kin}}[f]\) at fixed particle density n(r). The variational problem for the first variations (extremization) can be written as
where \(\alpha (r)\) is a local (position dependent) Lagrange multiplier. Since \(\beta \) is constant in the canonical ensemble, this is equivalent to conditions (F1) and (F2) yielding the distribution function (F3). This distribution is the global maximum of \(S[f]/k_\mathrm{B}-\beta E_{\mathrm{kin}}[f]\) at fixed n(r). We then obtain the same results as in âMicrocanonical ensembleâ section in Appendix F, except that T is fixed while it was previously determined by the conservation of energy (F7).
We can now simplify the expression of the free energy. The entropy is given by Eq. (F12) and the energy by Eq. (F7). Since \(F=E-TS\), we obtain
where U[n] is the internal energy given by Eq. (F14).
Finally, the statistical equilibrium state in the canonical ensemble is obtained by minimizing the free energy F[n] at fixed particle number N. The variational problem for the first variations (extremization) can be written as
Decomposing the Massieu function as \(J[f]=S[f]/k_\mathrm{B}-\beta E_{\mathrm{kin}}[f]-\beta W[n]\) and using Eq. (F24), we get
As a result, the variational problem (F26) yields
We then recover all the results of Sect. 2. The interest of this formulation is that it allows us to solve more easily the stability problem related to the sign of the second variations of free energy [342]. This problem has been studied in detail in [209, 211] for the Boltzmann free energy and in [225, 236, 238] for the Tsallis free energy. It has also been studied in [263] for the Boltzmann free energy within the framework of special relativity.
Remark
Using Eq. (F19), we see that the free energy (F25) can be written asFootnote 41
We have not explicitly written the temperature T since it is a constant in the canonical ensemble. Up to the kinetic term, Eq. (F29) coincides with the energy functional (246) associated with the EulerâPoisson equations describing a gas with a barotropic equation of state \(P=P(\rho )\) (see Sect. 5.1.1). As a result, the thermodynamical stability of a self-gravitating system in the canonical ensemble is equivalent to the dynamical stability of the corresponding barotropic gas described by the EulerâPoisson equations. This returns the general result established in [228]. It is valid for an arbitrary form of entropy. According to the PoincarĂŠ turning point criterion, the series of equilibria generically becomes both thermodynamically unstable in the canonical ensemble and dynamically unstable with respect to the EulerâPoisson equations at the first turning point of temperature (or, equivalently, at the first turning point of mass).
1.3 Scaling of the equation of state in the nonrelativistic and ultrarelativistic limits
We have seen that the equation of state implied by the distribution function (F3) is of the form \(P(r)=P[n(r),T]\). A simple scaling of this equation of state can be obtained in the nonrelativistic and ultrarelativistic limits.
In the nonrelativistic limit, using \(E_{\mathrm{kin}}=p^2/2m\), Eqs. (F4)â(F6) reduce to
Making the change of variables \(\mathbf{x}=(\beta /m)^{1/2}{\mathbf{p}}\), we obtain the scaling
Therefore, the internal energy (F19) takes the form
For the Boltzmann entropy in phase space \(S_{B}[f]\), leading to the isothermal equation of state \(P(n,T)=n k_\mathrm{B} T\), the free energy (F25) with U given by Eq. (F19) is of the form \(F[n]=W[n]-T S_\mathrm{B}[n]\) where \(S_\mathrm{B}[n]\) is the Boltzmann entropy in configuration space (see [211] for details). For the Tsallis entropy in phase space \(S_{q}[f]\), leading to the polytropic equation of state \(P(n,T)=K(T) n^{\gamma }\), the free energy (F25) with U given by Eq. (F19) is of the form \(F[n]=W[n]-K(T) S_{\gamma }[n]\) where \(S_{\gamma }[n]\) is the Tsallis entropy in configuration space (see [225] for details). In general, we do not have \(F[n]=W[n]-T S[n]\) (except for the Boltzmann entropy) nor \(F[n]=W[n]-\Theta (T) S[n]\) (except for the Tsallis entropy).
In the ultrarelativistic limit, using \(E_{\mathrm{kin}}=pc\), Eqs. (F4)â(F6) reduce to
Making the change of variables \(\mathbf{x}=\beta {\mathbf{p}} c\), we obtain the scaling
Therefore, the internal energy (F19) takes the form
1.4 General relativity
Let us briefly consider the general relativity case. In the microcanonical ensemble, the statistical equilibrium state is obtained by maximizing the entropy S[f] at fixed mass energy \(Mc^2\) and particle number N. To solve this maximization problem, we proceed in two steps. We first maximize S[f] at fixed \(Mc^2\), Nand energy density \(\epsilon (r)\). Since \(\epsilon (r)\) determines \(Mc^2\), this is equivalent to maximizing S[f] at fixed N and \(\epsilon (r)\). The variational problem for the first variations (extremization) can be written as
where \({{\tilde{\beta }}}(r)\) is a local (position dependent) Lagrange multiplier and \(\alpha \) is a global (uniform) Lagrange multiplier. Noting that M(r)âwhich appears in the expressions of S and Nâis fixed since it is determined by \(\epsilon (r)\), this variational problem yields
with \(\beta (r)\equiv {{{\tilde{\beta }}}}(r)[1-2GM(r)/rc^2]^{1/2}\), leading to Eq. (C9). In this manner, we immediately find that \(\alpha \) is uniform at statistical equilibrium. This results from the conservation of N. Substituting Eq. (F41) into the expressions of S, M and N may help solving the stability problem related to the sign of the second variations of entropy [342].
Appendix G: Black-body radiation in general relativity
In this appendix, we consider a gas of photons (black-body radiation) that is so intense that general relativity must be taken into account. This leads to the concept of âphoton starsâ or self-gravitating black-body radiation. This problem has been studied in detail in [259, 264]. Below, we recall the basic equations determining the statistical equilibrium state of a gas of photons in general relativity and compare these results with those obtained in Sect. 3 for material particles such as self-gravitating fermions.
1.1 Thermodynamics of the black-body radiation
The distribution function of a gas of photons is
This corresponds to the BoseâEinstein statistics in the ultrarelativistic limit (\(E=pc\)) and with a vanishing chemical potential (\(\mu =0\)). These simplifications arise because the photons have no rest mass. Using Eq. (120), we find that the energy density is related to the temperature by the StefanâBoltzmann law
The factor in front of \(T^4\) is the StefanâBoltzmann constant. Using Eq. (126), we find that the pressure is given by
It is related to the energy density by the linear equation of state
This linear relationship, with a coefficient 1/3, is valid for an arbitrary ultrarelativistic gas (see âUltrarelativistic limitâ section in Appendix A). Using Eq. (119), we find that the particle density is given by
where \(\zeta (3)=1.202056 \ldots \) is the ApĂŠry constant (value of the Riemann zeta function \(\zeta (x)\) in \(x=3\)). The pressure is related to the particle density through the polytropic equation of state
Finally, using the integrated GibbsâDuhem relation (141) with \(\mu =0\), we find that the entropy density is given byFootnote 42
We see that the entropy density is proportional to the particle density:
More details about these relations and their consequences can be found in Ref. [264].
1.2 Mechanical derivation of the Tolman relation
Substituting the relation \(\epsilon =3P\) from Eq. (G4) into Tolmanâs equation of hydrostatic equilibrium (108), we get
On the other hand, according to Eq. (G3), we have
These two equations directly imply the Tolman relation
This derivation is valid only for the black-body radiation. It was given by Tolman [241] as a particular example of his relation before considering the general case of an arbitrary perfect fluid.
Remark
This derivation presupposes the condition of hydrostatic equilibrium (108). In the following section, we show that this equation can be obtained from the condition of thermodynamical equilibrium (extremization of the entropy S at fixed mass energy \(Mc^2\)).
1.3 Equivalence between dynamical and thermodynamical stabilities for the self-gravitating black-body radiation
According to Eq. (G8), the entropy of the black-body radiation is proportional to the particle number:
The condition of thermodynamical stability, corresponding to the maximization of the entropy at fixed mass energy:
turns out to be equivalent to the maximization of the particle number at fixed mass energy:
which is itself equivalent to the minimization of the mass energy at fixed particle number:
corresponding to the condition of dynamical stability for a barotropic fluid in general relativity (see Sect. 5.1.2). Therefore, in the case of the self-gravitating black-body radiation, it is straightforward to show the equivalence between dynamical and thermodynamical stability. This is a particular case where Ipserâs conjecture [258] (see Sect. 5.2.2) can be easily demonstrated.Footnote 43
The maximization problem (G13) determining the thermodynamical stability of the self-gravitating black-body radiation in general relativity was first studied by Tolman [241], and later by Cocke [250] (in a more general context), Sorkin et al. [259] and Chavanis [264]. The variational principle for the first variations (extremization) can be written as
where \(1/T_{\infty }\) is a Lagrange multiplier.Footnote 44 It leads to the TOV equations (equivalent to the condition of hydrostatic equilibrium) and to the Tolman relation (the Lagrange multiplier \(T_{\infty }\) corresponds to the Tolman temperature). Then, considering the second variations of the entropy S, it can be shown that the self-gravitating black-body radiation is linearly stable with respect to the EulerâEinstein equations if, and only if, it is a local maximum of S at fixed M. This is also equivalent to its spectral stability. Indeed, the complex pulsations \(\omega \) of the normal modes of the linearized EulerâEinstein equations [51, 322] satisfy \(\omega ^2>0\) for all modes if, and only if, \(\delta ^2S<0\) for all perturbations that conserve M. Using the PoincarĂŠ criterion [281], we can show [264] that the series of equilibria is thermodynamically and dynamically stable before the turning points of mass energy M, particle number N, binding energy E or entropy S (they all coincide) and that it becomes thermodynamically and dynamically unstable afterward. Furthermore, the curve \(S({{{\mathcal {E}}}})\) displays cusps at its extremal points (since \(\delta S=0\Leftrightarrow \delta {{{\mathcal {E}}}}=0\)). We refer to [250, 259, 264] for the derivation of these results.
Remark
In the case of material particles, the statistical equilibrium state is obtained by maximizing the entropy at fixed mass energy and particle number. In the case of the self-gravitating black-body radiation, the statistical equilibrium state is obtained by maximizing the entropy, which is proportional to the particle number, at fixed mass energy. How can we understand this difference? First, we have to realize that, in the case of the black-body radiation, the particle number is not fixed. What is fixed instead is the ratio between the chemical potential and the temperature. Therefore, the correct manner to treat the thermodynamics of the self-gravitating black-body radiation is to work in the grand microcanonical ensemble [199, 212] where \(\alpha =\mu /k_\mathrm{B} T\) and \({{{\mathcal {E}}}}=Mc^2\) are fixed. In that ensemble, the thermodynamic potential is \({{{\mathcal {K}}}}=S+\alpha k_\mathrm{B} N\). The statistical equilibrium state is then obtained by maximizing \(\mathcal{K}\) at fixed mass energy:
The extremization problem (first variations) yields
Now, for (massless) photons, the chemical potential vanishes: \(\mu =0\). This implies \(\alpha =0\) and \({{{\mathcal {K}}}}=S\). In that case, the maximization problem (G17) reduces to (G13).
Appendix H: The TolmanâKlein relations
In this appendix, we review the main results given in the seminal papers of Tolman [241] and Klein [249].
1.1 Tolmanâs (1930) paper
In a paper published in 1930, Tolman [241] investigated âthe weight of heat and thermal equilibrium in general relativity.â His main finding is that, even at thermodynamic equilibrium, the temperature is inhomogeneous in the presence of gravitation. He discovered a definite relation connecting the distribution of temperature T(r) throughout the system to the gravitational potential (or metric coefficient) \(\nu (r)\). Tolmanâs relation [see Eq. (165)] between equilibrium temperature and gravitational potential was something essentially new in thermodynamics since, until his work, uniform temperature throughout any system which has come to thermal equilibrium had hitherto been taken as an inescapable part of thermodynamic theory.
Tolman first considered the case of a weak gravitational field described by Newtonian gravitation. By maximizing the entropy for an isolated system, he obtained an approximate relation between the temperature distribution and the Newtonian gravitational potential [see Eq. (222)]. This can be viewed as a post-Newtonian relation since the temperature gradient is inversely proportional to the square of the velocity of light.
He then considered the case of the black-body radiation. By performing a purely mechanical treatment of temperature distribution based on the Einstein equations (using Eq. (108) representing the general relativistic extension of the condition of hydrostatic equilibrium), he obtained an exact general relativistic relation between the proper temperature and the metric coefficient \(\nu \) [see Eq. (G11)].
He then recovered this result by maximizing the entropy of the self-gravitating black-body radiation by using the formalism of relativistic thermodynamics that he had developed a few years earlier. Therefore, in the simple case of the black-body radiation where a mechanical treatment can be given, the thermodynamical and mechanical treatments of temperature distribution under the action of gravity lead to the same result.
Finally, he generalized his thermodynamical approach (maximum entropy principle) to the case of any perfect fluid and obtained the Tolman relation (165) in a general setting.
He noted at the end of his paper that the maximum entropy principle implies the general relativistic condition of hydrostatic equilibrium [see Eq. (108)] contained in the Einstein equations. He wrote: âIt may seem strange that this purely mechanical equation holding within the interior of the system should be derivable from the application of thermodynamics to the system as a whole. The result, however, is the relativity analogue to the equation for change in pressure with height obtained by Gibbs (âScientific Papers,â Longmans, Green 1906, equation 230, p. 145) in his thermodynamic treatment of the conditions of equilibrium under the influence of gravity. Indeed the whole treatment of this article may be regarded as the relativistic extension of this part of Gibbsâ work.â
1.2 Kleinâs (1949) paper
In a paper entitled âOn the thermodynamical equilibrium of fluids in gravitational fieldsâ published in 1949, Klein [249] managed to derive the Tolman relation (165), together with a similar relation between the chemical potential \(\mu (r)\) and the metric coefficient \(\nu (r)\) [see Eq. (166)] with almost no calculation,Footnote 45 by using essentially the GibbsâDuhem relation and the first principle of thermodynamics. We give below a summary of Kleinâs calculations.
Klein started from the first principle of thermodynamics
Since E is a homogeneous function of the first degree in the three variables V, S and N, the Euler theorem implies that
which is the GibbsâDuhem relation (see Appendix E). From Eqs. (H1) and (H2), we get
Written under a local form, with the variables \(s=S/V\), \(n=N/V\) and \(\epsilon =E/V\), Eqs. (H1)â(H3) return Eqs. (136) and (141)â(143). In turn, Eq. (142) can be written as
Combined with the condition of hydrostatic equilibrium from Eq. (108), we get
At that point, Klein considered several independent substances present in the same gravitational field and argued that an equation of the type (H5) holds for each of them separately with the same values of \(\nu \) and T. As one such substance, we always have the radiation for which \(\mu =0\). Thus, we get
which is Tolmanâs relation. Then, for all other substances
which is Kleinâs relation. As emphasized by Klein [249], this relation constitutes the relativistic generalization of the well-known Gibbs [343] condition for the equilibrium in a gravitational field.
Appendix I: Thermodynamic identities
In this appendix, we regroup useful thermodynamic identities valid for Newtonian and general relativistic barotropic gases.
1.1 Newtonian isentropic or cold barotropic gases
We consider a Newtonian self-gravitating gas. The first principle of thermodynamics writes
where u is the density of internal energy. If we introduce the enthalpy per particle
we can rewrite Eq. (I1) as
We now assume that \(T\mathrm{d}(s/n)=0\). This corresponds to cold (\(T=0\)) or isentropic (\(s/n=\lambda =\mathrm{cst}\)) gases. In that case, Eq. (I1) reduces to
We also have
The foregoing equations can be rewritten equivalently as
For a cold or isentropic gas, using the GibbsâDuhem relation (E7), we see that the chemical potential coincides with the enthalpy
For a barotropic equation of state \(P=P(n)\), Eq. (I4) can be integrated into
The internal energy is
We note the identities
and
We also have
and
The energy of a Newtonian isentropic or cold barotropic self-gravitating gas is \({{{\mathcal {W}}}}=U+W\) where U is the internal energy and W is the gravitational energy. A stable equilibrium state of the EulerâPoisson equations is a minimum of energy \(\mathcal{W}\) at fixed particle number N (see Sect. 5.1.1).Footnote 46 If the pressure can be written as \(P(n)=T\, \Pi (n)\), we get \({{{\mathcal {W}}}}=W-TS_{\mathrm{eff}}\) where \(S_{\mathrm{eff}}=-\int n\int ^n \frac{\Pi (n')}{{n'}^2}\, \mathrm{d}n'\, \mathrm{d}{\mathbf{r}}\) is a generalized entropy of the density n [230, 232].
Remark
For an ideal gas at \(T=0\), the thermodynamic identities of Sect. 2 reduce to \(\mathrm{d}\epsilon _{\mathrm{kin}}=\mu \, \mathrm{d}n\), \(\epsilon _{\mathrm{kin}}+P-\mu n=0\) and \(\mathrm{d}P=n \, {\mathrm{d}\mu }\). They coincide with Eq. (I5) with \(u=\epsilon _{\mathrm{kin}}\) and \(h=\mu \).
1.2 General relativistic isentropic or cold barotropic gases
We consider a general relativistic self-gravitating gas. The first principle of thermodynamics writes
where \(\epsilon =\rho c^2+u\) is the mass energy density and \(\rho =n m\) is the rest mass density. The relations of âNewtonian isentropic or cold barotropic gasesâ section in Appendix I remain valid with u or with \(\epsilon \). When \(T\mathrm{d}(s/n)=0\), Eq. (I14) reduces to
For a barotropic equation of state of the form \(P=P(n)\), we obtain
Since \(\epsilon \) is a function of n, the pressure is a function \(P=P(\epsilon )\) of the energy density. Therefore, Eq. (I15) can be integrated into
The binding energy of a general relativistic isentropic or cold barotropic gas is \(E=(M-Nm)c^2\) where M is the mass and N is the particle number. A stable equilibrium state of the EulerâEinstein equations is a minimum of energy E at fixed particle number N (see Sect. 5.1.2).Footnote 47 In the Newtonian limit, \(E\rightarrow U+W={{{\mathcal {W}}}}\) (see Sect. 4.6 with u in place of \(\epsilon _{\mathrm{kin}}\)) and we recover the results of âNewtonian isentropic or cold barotropic gasesâ section in Appendix I.
Remark
For a polytropic equation of state of the form \(P=Kn^{\gamma }\), we getFootnote 48
For a linear equation of state of the form \(P=q\epsilon \) with \(q=\gamma -1\), we obtain
where K is a constant of integration. When \(\mu =0\), the integrated GibbsâDuhem relation (E7) reduces to
Two cases may be considered. (i) When \(T=0\), we obtain \(P=-\epsilon \). This is the equation of state of dark energy. Therefore, the equation of state of dark energy corresponds to a relativistic gas at \(T=0\) with \(\mu =0\). (ii) When \(s=\lambda n\), we obtain
The case \(\mu =0\) applies in particular to the black-body radiation for which \(q=1/3\). In that case, we recover the relations of Appendix G but the constant K is not determined by the present method.
1.3 Newtonian self-gravitating gases at statistical equilibrium
We consider a Newtonian self-gravitating gas at statistical equilibrium (see Sect. 2). The first principle of thermodynamics writes
Since the temperature T is uniform at statistical equilibrium, we can rewrite the foregoing equation as
On the other hand, we have seen in Sect. 2 that the gas has a barotropic equation of state \(P=P(n)\). Therefore, Eq. (I24) can be integrated into
Introducing the internal energy defined by Eq. (I8), we obtain the important relation
Integrating this relation over the whole configuration, we find that the entropy is given by
which returns Eq. (F12). On the other hand, the total energy is given by \(E=E_{\mathrm{kin}}+W\). In the microcanonical ensemble, a stable equilibrium state is a maximum of entropy S at fixed energy E and particle number N. In the canonical ensemble, a stable equilibrium state is a minimum of free energy F at fixed particle number N. Using Eq. (I27), we find that the free energy is given by
which returns Eq. (F25). We see that
Therefore, the criterion of thermodynamical stability in the canonical ensemble (minimum of F at fixed N) coincides with the criterion of dynamical stability with respect to the EulerâPoisson equations (minimum of \({{{\mathcal {W}}}}\) at fixed N). The calculations of this appendix provide a direct proof of the nonlinear Antonov first law obtained in [228].
Remark
At \(T=0\), we see from Eq. (I26) that \(\epsilon _{\mathrm{kin}}(n)=u(n)\). This implies that \(E_{\mathrm{kin}}=U\) so that \(E=E_{\mathrm{kin}}+W=U+W={{{\mathcal {W}}}}\). This is a particular case of the general relation (I29). At \(T=0\), the equilibrium state is obtained either by minimizing \(E=E_{\mathrm{kin}}+W\) at fixed particle number N or by minimizing \({{{\mathcal {W}}}}=U+W\) at fixed particle number N (see Sect. 2.10.1).
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Chavanis, PH. Statistical mechanics of self-gravitating systems in general relativity: I. The quantum Fermi gas. Eur. Phys. J. Plus 135, 290 (2020). https://doi.org/10.1140/epjp/s13360-020-00268-0
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DOI: https://doi.org/10.1140/epjp/s13360-020-00268-0