Statistical mechanics of self-gravitating systems in general relativity: I. The quantum Fermi gas

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 Fermi–Dirac 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 Tolman–Oppenheimer–Volkoff equations of hydrostatic equilibrium and the Tolman–Klein 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.

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

$$\begin{aligned} E_{\mathrm{kin}}(p)=\frac{p^2}{2m}. \end{aligned}$$

(A1)

In that case, the kinetic energy density and the pressure are given by

$$\begin{aligned} \epsilon _{\mathrm{kin}}=\int f\frac{p^2}{2m} \, \mathrm{d}{\mathbf{p}}\qquad \mathrm{and}\qquad P=\frac{1}{3}\int f \frac{p^2}{m}\, \mathrm{d}{\mathbf{p}}. \end{aligned}$$

(A2)

We have the general relations

$$\begin{aligned} P=\frac{2}{3}\epsilon _{\mathrm{kin}},\qquad E_{\mathrm{kin}}=\frac{3}{2}\int P\, \mathrm{d}{\mathbf{r}}. \end{aligned}$$

(A3)

1.2 Ultrarelativistic limit

In the ultrarelativistic limit, the energy of a particle is

$$\begin{aligned} E(p)=E_{\mathrm{kin}}(p)=pc. \end{aligned}$$

(A4)

In that case, the energy density and the pressure are given by

$$\begin{aligned} \epsilon =\epsilon _{\mathrm{kin}}=\int f pc \, \mathrm{d}{\mathbf{p}}\qquad \mathrm{and}\qquad P=\frac{1}{3}\int f p c\, \mathrm{d}{\mathbf{p}}. \end{aligned}$$

(A5)

We have the general relations

$$\begin{aligned} P=\frac{1}{3}\epsilon =\frac{1}{3}\epsilon _{\mathrm{kin}},\qquad \mathcal{E}=E_{\mathrm{kin}}=3\int P\, \mathrm{d}{\mathbf{r}}. \end{aligned}$$

(A6)

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]):

$$\begin{aligned} -\int \rho {\mathbf{r}}\cdot \nabla \Phi \, \mathrm{d}{\mathbf{r}}=W. \end{aligned}$$

(B1)

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

$$\begin{aligned} W=-\int \rho \frac{GM(r)}{r}\, {\mathrm{d}}V. \end{aligned}$$

(B2)

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

$$\begin{aligned} \nabla P+\rho \nabla \Phi =\mathbf{0} \end{aligned}$$

(B3)

into Eq. (B1), and integrating by parts, we get

$$\begin{aligned} W=-3\int P\, \mathrm{d}{\mathbf{r}}+\oint P{\mathbf{r}}\cdot d\mathbf{S}. \end{aligned}$$

(B4)

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

$$\begin{aligned} \oint P{\mathbf{r}}\cdot d\mathbf{S}=P_{b}\oint {\mathbf{r}}\cdot d\mathbf{S}=P_{b}\int \nabla \cdot {\mathbf{r}} \, \mathrm{d}{\mathbf{r}}=3 P_{b}V. \end{aligned}$$

(B5)

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

$$\begin{aligned} 3\int P\, \mathrm{d}{\mathbf{r}}+W=3 P_{b}V. \end{aligned}$$

(B6)

For nonrelativistic particles, using Eq. (A3), the equilibrium scalar virial theorem becomes

$$\begin{aligned} 2 E_{\mathrm{kin}}+ W=3 P_{b}V. \end{aligned}$$

(B7)

Using $E=E_{\mathrm{kin}}+W$, we get

$$\begin{aligned} E=-E_{\mathrm{kin}}+3 P_{b}V. \end{aligned}$$

(B8)

When $P_\mathrm{b}=0$, we obtain $E=-E_{\mathrm{kin}}<0$. For ultrarelativistic particles, using Eq. (A6), the equilibrium scalar virial theorem becomes

$$\begin{aligned} E=E_{\mathrm{kin}}+ W=3 P_{b}V. \end{aligned}$$

(B9)

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

$$\begin{aligned} s=-k_\mathrm{B}\int C(f)\, \mathrm{d}{\mathbf{p}}, \end{aligned}$$

(C1)

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 “generalized thermodynamics” 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

$$\begin{aligned} C'(f)=-\beta (r)E(p)+\alpha (r). \end{aligned}$$

(C2)

Since C is convex, this relation can be inverted. It determines a distribution function of the form

$$\begin{aligned} f({\mathbf{r}},{\mathbf{p}})=F\left[\beta (r)E(p)-\alpha (r)\right], \end{aligned}$$

(C3)

where

$$\begin{aligned} F(x)=(C')^{-1}(-x). \end{aligned}$$

(C4)

Since

$$\begin{aligned} \delta ^2 s=-k_\mathrm{B}\int C''(f)\frac{(\delta f)^2}{2}\, \mathrm{d}{\mathbf{p}}<0, \end{aligned}$$

(C5)

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 Gibbs–Duhem 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

$$\begin{aligned} f({\mathbf{r}},{\mathbf{p}})=F\left[\beta (r)E_{\mathrm{kin}}(p)-\alpha (r)\right], \end{aligned}$$

(C6)

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

$$\begin{aligned} f({\mathbf{r}},{\mathbf{p}})=F\left[\beta (E_{\mathrm{kin}}(p)+m\Phi (r))-\alpha _0\right]. \end{aligned}$$

(C7)

This is a function of the form

$$\begin{aligned} f({\mathbf{r}},{\mathbf{v}})=f[\epsilon ({\mathbf{r}},{\mathbf{v}})] \quad \mathrm{with} \quad f'(\epsilon )<0, \end{aligned}$$

(C8)

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 Tolman–Klein relations (165) and (166). As a result, the distribution function at statistical equilibrium is given by

$$\begin{aligned} f({\mathbf{r}},{\mathbf{p}})=F\left[\frac{E(p)}{k_\mathrm{B} T(r)}-\alpha \right]. \end{aligned}$$

(C9)

Using Eq.  (165), it can be written as

$$\begin{aligned} f({\mathbf{r}},{\mathbf{p}})=F\left[\beta _{\infty } e^{\nu (r)/2} E(p)-\alpha \right]. \end{aligned}$$

(C10)

This is a function of the form

$$\begin{aligned} f({\mathbf{r}},{\mathbf{p}})=f\left[e^{\nu (r)/2} E(p)\right]\quad \mathrm{with} \quad f'\left[e^{\nu (r)/2} E(p)\right]<0, \end{aligned}$$

(C11)

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

$$\begin{aligned} S=-k_\mathrm{B}\int C(f)\, \mathrm{d}{\mathbf{r}}\mathrm{d}{\mathbf{p}}. \end{aligned}$$

(C12)

The particle number and the mass are given by

$$\begin{aligned} M=Nm=m\int f\, \mathrm{d}{\mathbf{r}} \mathrm{d}{\mathbf{p}}=\int m n\, \mathrm{d}{\mathbf{r}}=\int \rho \, \mathrm{d}{\mathbf{r}}. \end{aligned}$$

(C13)

The energy is given by

$$\begin{aligned} E=E_{\mathrm{kin}}+W=\int f E_{\mathrm{kin}}(p) \,\mathrm{d}{\mathbf{r}} \mathrm{d}{\mathbf{p}}+\frac{1}{2}\int \rho \Phi \, \mathrm{d}{\mathbf{r}}, \end{aligned}$$

(C14)

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

$$\begin{aligned} \frac{\delta S}{k_\mathrm{B}}-\beta \delta E+\alpha _0\delta N=0, \end{aligned}$$

(C15)

where $\beta$ and $\alpha _0$ are global Lagrange multipliers. Taking the variations with respect to $f({\mathbf{r}},{\mathbf{p}})$, we obtain

$$\begin{aligned} C'(f)=-\beta (E_{\mathrm{kin}}(p)+m\Phi ({\mathbf{r}}))+\alpha _0, \end{aligned}$$

(C16)

leading to

$$\begin{aligned} f({\mathbf{r}},{\mathbf{p}})=F\left[\beta (E_{\mathrm{kin}}(p)+m\Phi ({\mathbf{r}}))-\alpha _0\right], \end{aligned}$$

(C17)

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

$$\begin{aligned} \beta =\frac{1}{k_\mathrm{B} T},\qquad \alpha _0=\frac{\mu _0}{k_\mathrm{B} T}. \end{aligned}$$

(C18)

We can then rewrite Eq. (C17) as

$$\begin{aligned} f({\mathbf{r}},{\mathbf{p}})=F\left[\frac{1}{k_\mathrm{B} T}\left( E_{\mathrm{kin}}(p)+m\Phi ({\mathbf{r}})-\mu _0\right) \right]. \end{aligned}$$

(C19)

For the Fermi–Dirac entropy

$$\begin{aligned} S=-k_\mathrm{B}\frac{g}{h^3}\int \Biggl \lbrace \frac{f}{f_{\mathrm{max}}}\ln \frac{f}{f_{\mathrm{max}}} +\left( 1-\frac{f}{f_{\mathrm{max}}}\right) \ln \left( 1-\frac{f}{f_{\mathrm{max}}}\right) \Biggr \rbrace \, \mathrm{d}{\mathbf{r}} \mathrm{d}{\mathbf{p}}, \end{aligned}$$

(C20)

we obtain the mean field Fermi–Dirac distribution

$$\begin{aligned} f({\mathbf{r}},{\mathbf{p}})=\frac{g}{h^3}\frac{1}{1+e^{-\alpha _0}e^{\beta (E_{\mathrm{kin}}(p)+m\Phi ({\mathbf{r}}))}} \end{aligned}$$

(C21)

or, equivalently,

$$\begin{aligned} f({\mathbf{r}},{\mathbf{p}})=\frac{g}{h^3}\frac{1}{1+e^{(E_{\mathrm{kin}}(p)+m\Phi ({\mathbf{r}})-\mu _0)/k_\mathrm{B} T}}. \end{aligned}$$

(C22)

For the Boltzmann entropy

$$\begin{aligned} S= & {} -k_\mathrm{B}\int f\left[\ln \left( \frac{f}{f_{\mathrm{max}}}\right) -1\right]\, \mathrm{d}{\mathbf{r}} \mathrm{d}{\mathbf{p}}, \end{aligned}$$

(C23)

we obtain the mean field Maxwell–Boltzmann distribution

$$\begin{aligned} f({\mathbf{r}},{\mathbf{p}})=\frac{g}{h^3}e^{\alpha _0}e^{-\beta (E_{\mathrm{kin}}(p)+m\Phi ({\mathbf{r}}))} \end{aligned}$$

(C24)

or, equivalently,

$$\begin{aligned} f({\mathbf{r}},{\mathbf{p}})=\frac{g}{h^3}e^{-(E_{\mathrm{kin}}(p)+m\Phi ({\mathbf{r}})-\mu _0)/k_\mathrm{B} T}. \end{aligned}$$

(C25)

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

$$\begin{aligned} \int f C'(f)\, \mathrm{d}{\mathbf{r}}\mathrm{d}{\mathbf{p}}=\frac{\mu _0 N-2W-E_{\mathrm{kin}}}{k_\mathrm{B} T}. \end{aligned}$$

(C26)

If we compare this expression with the relation

$$\begin{aligned} S=\frac{E_{\mathrm{kin}}+\int P\, \mathrm{d}{\mathbf{r}}-\mu _0 N+2W}{T} \end{aligned}$$

(C27)

obtained from Eqs. (39), (48) and (58), we obtain the identity

$$\begin{aligned} -k_\mathrm{B}\int f C'(f)\, \mathrm{d}{\mathbf{r}}\mathrm{d}{\mathbf{p}}=S-\frac{1}{T}\int P\, \mathrm{d}{\mathbf{r}}. \end{aligned}$$

(C28)

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)

$$\begin{aligned} f({\mathbf{r}},{\mathbf{p}})=F\left[\beta \left( E_{\mathrm{kin}}(p)+m\Phi ({\mathbf{r}})\right) -\alpha _0\right], \end{aligned}$$

(D1)

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

$$\begin{aligned} P=\frac{1}{3}\int f p \frac{\mathrm{d}E_{\mathrm{kin}}}{\mathrm{d}p}\, \mathrm{d}{\mathbf{p}}=\frac{1}{3}\int F\left[\beta \left( E_{\mathrm{kin}}(p)+m\Phi ({\mathbf{r}})\right) -\alpha _0\right]p \frac{\mathrm{d}E_{\mathrm{kin}}}{\mathrm{d}p}\, \mathrm{d}{\mathbf{p}}. \end{aligned}$$

(D2)

Taking its gradient with respect to ${\mathbf{r}}$, we get

$$\begin{aligned} \nabla P=\frac{1}{3}\beta m \nabla \Phi \int F'\left[\beta \left( E_{\mathrm{kin}}(p)+m\Phi ({\mathbf{r}})\right) -\alpha _0\right]p \frac{\mathrm{d}E_{\mathrm{kin}}}{\mathrm{d}p} \, \mathrm{d}{\mathbf{p}}. \end{aligned}$$

(D3)

This can also be written as

$$\begin{aligned} \nabla P=\frac{1}{3} m \nabla \Phi \int {\mathbf{p}}\cdot \frac{\partial F}{\partial {\mathbf{p}}}\left[\beta \left( E_{\mathrm{kin}}(p)+m\Phi ({\mathbf{r}})\right) -\alpha _0\right]\, \mathrm{d}{\mathbf{p}}. \end{aligned}$$

(D4)

Integrating by parts, we can rewrite the foregoing equation as

$$\begin{aligned} \nabla P=- m \nabla \Phi \int F\left[\beta \left( E_{\mathrm{kin}}(p)+m\Phi ({\mathbf{r}})\right) -\alpha _0\right]\, \mathrm{d}{\mathbf{p}}. \end{aligned}$$

(D5)

Since the density is given by

$$\begin{aligned} \rho = m\int f\, \mathrm{d}{\mathbf{p}}=m \int F\left[\beta \left( E_{\mathrm{kin}}(p)+m\Phi ({\mathbf{r}})\right) -\alpha _0\right]\, \mathrm{d}{\mathbf{p}}, \end{aligned}$$

(D6)

we finally obtain the condition of hydrostatic equilibrium

$$\begin{aligned} \nabla P=-\rho \nabla \Phi . \end{aligned}$$

(D7)

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)

$$\begin{aligned} f({\mathbf{r}},{\mathbf{p}})=F\left[\frac{E(p)}{k_\mathrm{B} T(r)}-\alpha \right], \end{aligned}$$

(D8)

where F is defined by Eq. (C4) and where $\alpha$ is constant. According to Eqs. (126) and (D8), the pressure is given by

$$\begin{aligned} P=\frac{1}{3}\int f p \frac{\mathrm{d}E}{\mathrm{d}p}\, \mathrm{d}{\mathbf{p}}=\frac{1}{3}\int F\left[\frac{E(p)}{k_\mathrm{B} T(r)}-\alpha \right]p \frac{\mathrm{d}E}{\mathrm{d}p}\, \mathrm{d}{\mathbf{p}}. \end{aligned}$$

(D9)

Taking its derivative with respect to r, we obtain

$$\begin{aligned} \frac{\mathrm{d}P}{\mathrm{d}r}=-\frac{1}{3} \frac{1}{k_\mathrm{B} T(r)^2}\frac{\mathrm{d}T}{\mathrm{d}r} \int F'\left[\frac{E(p)}{k_\mathrm{B} T(r)}-\alpha \right]p \frac{\mathrm{d}E}{\mathrm{d}p}E(p)\, \mathrm{d}{\mathbf{p}}. \end{aligned}$$

(D10)

This can also be written as

$$\begin{aligned} \frac{\mathrm{d}P}{\mathrm{d}r}=-\frac{1}{3} \frac{1}{T(r)}\frac{\mathrm{d}T}{\mathrm{d}r} \int _{0}^{+\infty } \frac{\partial F}{\partial p}\left[\frac{E(p)}{k_\mathrm{B} T(r)}-\alpha \right]p E(p) 4\pi p^2\, \mathrm{d}p. \end{aligned}$$

(D11)

Integrating by parts, we can rewrite the foregoing equation as

$$\begin{aligned} \frac{\mathrm{d}P}{\mathrm{d}r}=\frac{1}{3} \frac{1}{T(r)}\frac{\mathrm{d}T}{\mathrm{d}r} \int _{0}^{+\infty } F\left[\frac{E(p)}{k_\mathrm{B} T(r)}-\alpha \right]\left( 3 E(p) p^2+p^3\frac{\mathrm{d}E}{\mathrm{d}p}\right) 4\pi \, \mathrm{d}p. \end{aligned}$$

(D12)

Since the pressure is given by Eq. (D9) and the energy density by

$$\begin{aligned} \epsilon =\int f E(p) \, \mathrm{d}{\mathbf{p}}=\int F\left[\frac{E(p)}{k_\mathrm{B} T(r)}-\alpha \right]E(p)\, \mathrm{d}{\mathbf{p}}, \end{aligned}$$

(D13)

we finally obtain the equation

$$\begin{aligned} \frac{\mathrm{d}P}{\mathrm{d}r}=\frac{\epsilon (r)+P(r)}{T(r)} \frac{\mathrm{d}T}{\mathrm{d}r}. \end{aligned}$$

(D14)

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: Gibbs–Duhem relation

In this appendix, we derive the Gibbs–Duhem relation. We first recall the usual derivation of the global Gibbs–Duhem relation for extensive systems. Then, we provide a direct derivation of the local Gibbs–Duhem 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

$$\begin{aligned} \mathrm{d}E=-P\mathrm{d}V+T\mathrm{d}S+\mu \mathrm{d}N. \end{aligned}$$

(E1)

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,

$$\begin{aligned} E(\alpha S,\alpha V,\alpha N)=\alpha E(S,V,N), \end{aligned}$$

(E2)

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,

$$\begin{aligned} T(\alpha S,\alpha V,\alpha N)=T(S,V,N). \end{aligned}$$

(E3)

According to the Euler theorem, we have

$$\begin{aligned} E=-PV+TS+\mu N. \end{aligned}$$

(E4)

Differentiating this expression and using the first law of thermodynamics (E1), we get the Gibbs–Duhem relation

$$\begin{aligned} S{\mathrm{d}T}-V\mathrm{d}P+N{\mathrm{d}\mu }=0. \end{aligned}$$

(E5)

We note that the energy does not appear in this expression. The Euler relation (E4) for thermodynamic variables is also called the integrated Gibbs–Duhem relation.

Defining $s=S/V$, $n=N/V$ and $\epsilon =E/V$, the local Gibbs–Duhem and integrated Gibbs–Duhem relations can be written as

$$\begin{aligned} s{\mathrm{d}T}-\mathrm{d}P+n{\mathrm{d}\mu }=0 \end{aligned}$$

(E6)

and

$$\begin{aligned} \epsilon =-P+Ts+\mu n. \end{aligned}$$

(E7)

Introducing $S=sV$, $N=nV$ and $E=\epsilon V$ in the first law of thermodynamics (E1), developing the expression and using the integrated Gibbs–Duhem relation (E7), we obtain

$$\begin{aligned} \mathrm{d}\epsilon =Tds+\mu \mathrm{d}n. \end{aligned}$$

(E8)

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 “Direct derivation of the integrated Gibbs–Duhem relation for a general form of entropy” 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 Gibbs–Duhem relations (E6) and (E7) are always valid. By contrast, the global Gibbs–Duhem relations (E4) and (E5) are valid only for extensive systems with short-range interactions. In particular, the global Gibbs–Duhem relation is different for extensive systems and for nonextensive systems with long-range interactions. In Sect. 2.5.5, we generalize the global Gibbs–Duhem 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 Gibbs–Duhem 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

$$\begin{aligned} n(r)= & {} \int _0^{+\infty } F\left[\beta (r)E(p)-\alpha (r)\right]4\pi p^2\, \mathrm{d}p, \end{aligned}$$

(E9)

$$\begin{aligned} \epsilon (r)= & {} \int _0^{+\infty } F\left[\beta (r)E(p)-\alpha (r)\right]E(p) 4\pi p^2\, \mathrm{d}p, \end{aligned}$$

(E10)

$$\begin{aligned} P(r)= & {} \frac{1}{3}\int _0^{+\infty } F\left[\beta (r)E(p)-\alpha (r)\right]p E'(p) 4\pi p^2\, \mathrm{d}p. \end{aligned}$$

(E11)

On the other hand, the entropy density [see Eq. (C1)] is given by

$$\begin{aligned} s(r)=-k_\mathrm{B} \int _0^{+\infty } C\left\{ F\left[\beta (r)E(p)-\alpha (r)\right]\right\} 4\pi p^2\, \mathrm{d}p. \end{aligned}$$

(E12)

Integrating this equation by parts and using $C'[F(x)]=-x$, we get

$$\begin{aligned} s(r)=-k_\mathrm{B} \int _0^{+\infty } \left[\beta (r)E(p)-\alpha (r)\right]F'\left[\beta (r)E(p)-\alpha (r)\right]\beta (r) E'(p) \frac{4\pi }{3} p^3\, \mathrm{d}p.\nonumber \\ \end{aligned}$$

(E13)

Integrating by parts on more time, we obtain

$$\begin{aligned} s(r)= & {} k_\mathrm{B} \int _0^{+\infty } \beta (r) E'(p) F\left[\beta (r)E(p)-\alpha (r)\right]\frac{4\pi }{3} p^3\, \mathrm{d}p\nonumber \\&+k_\mathrm{B} \int _0^{+\infty } \left[\beta (r)E(p)-\alpha (r)\right]F\left[\beta (r)E(p)-\alpha (r)\right]4\pi p^2\, \mathrm{d}p. \end{aligned}$$

(E14)

Comparing Eq. (E14) with Eqs. (E9)–(E11), we find that

$$\begin{aligned} \frac{s(r)}{k_\mathrm{B}}=\beta (r)P(r)+\beta (r)\epsilon (r)-\alpha (r)n(r). \end{aligned}$$

(E15)

Using Eq. (134), we finally obtain the integrated Gibbs–Duhem relation

$$\begin{aligned} s(r)=\frac{ \epsilon (r)+P(r)-\mu (r)n(r)}{T(r)}. \end{aligned}$$

(E16)

This calculation emphasizes the fact that the integrated Gibbs–Duhem relation is valid for an arbitrary form of entropy³⁹ 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 Gibbs–Duhem 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 “One-step derivation” 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 “Newtonian gravity” 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

$$\begin{aligned} \frac{\delta S}{k_\mathrm{B}}-\beta \delta E_{\mathrm{kin}}+\int \alpha (r)\delta n \, \mathrm{d}{\mathbf{r}}=0, \end{aligned}$$

(F1)

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

$$\begin{aligned} \frac{\delta s}{k_\mathrm{B}}-\beta \delta \epsilon _{\mathrm{kin}}+\alpha (r)\delta n=0, \end{aligned}$$

(F2)

returns the results of “Two-step derivation” section in Appendix C, except that $\beta (r)$ is replaced by $\beta$. Therefore, it yields

$$\begin{aligned} f({\mathbf{r}},{\mathbf{p}})=F\left[\beta E_{\mathrm{kin}}(p)-\alpha (r)\right]. \end{aligned}$$

(F3)

In this manner, we immediately find that T is uniform at statistical equilibrium. This results from the conservation of energy. As in “Two-step derivation” 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

$$\begin{aligned} n(r)= & {} \int F\left[\beta E_{\mathrm{kin}}(p)-\alpha (r)\right]\, \mathrm{d}{\mathbf{p}}, \end{aligned}$$

(F4)

$$\begin{aligned} \epsilon _{\mathrm{kin}}(r)= & {} \int F\left[\beta E_{\mathrm{kin}}(p)-\alpha (r)\right]E_{\mathrm{kin}}(p)\, \mathrm{d}{\mathbf{p}}, \end{aligned}$$

(F5)

$$\begin{aligned} P(r)= & {} \frac{1}{3}\int F\left[\beta E_{\mathrm{kin}}(p)-\alpha (r)\right]p E_{\mathrm{kin}}'(p)\, \mathrm{d}{\mathbf{p}}. \end{aligned}$$

(F6)

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

$$\begin{aligned} E=E_{\mathrm{kin}}[n(r),T]+W[n(r)]. \end{aligned}$$

(F7)

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 “Direct derivation of the integrated Gibbs–Duhem relation for a general form of entropy” section in Appendix E, we can derive the integrated Gibbs–Duhem relation (E16), except that T(r) is replaced by T. Therefore, we get

$$\begin{aligned} s(r)=\frac{\epsilon _{\mathrm{kin}}(r)+P(r)-\mu (r)n(r)}{T}\quad \mathrm{with}\quad \mu (r)=\alpha (r) k_\mathrm{B} T. \end{aligned}$$

(F8)

Since T is uniform, Eq. (41) reduces to

$$\begin{aligned} {\mathrm{d}\mu }=\frac{\mathrm{d}P}{n}. \end{aligned}$$

(F9)

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

$$\begin{aligned} \mu '(n,T)=\frac{P'(n,T)}{n}, \quad \mathrm{i.e.,} \quad \mu (n,T)=\int ^{n}\frac{P'(n',T)}{n'}\, \mathrm{d}n', \end{aligned}$$

(F10)

where the derivative is with respect to n.⁴⁰

We can now simplify the expression of the entropy. Using the integrated Gibbs–Duhem relation (F8), we have

$$\begin{aligned} S=\frac{1}{T}\left( E_{\mathrm{kin}}+\int P(r)\, \mathrm{d}{\mathbf{r}}-\int \mu (r)n(r)\, \mathrm{d}{\mathbf{r}}\right) . \end{aligned}$$

(F11)

The entropy can be written as a functional of the density as

$$\begin{aligned} S[n(r),T]=\frac{1}{T}\left( E_{\mathrm{kin}}[n(r),T]-U[n(r),T]\right) , \end{aligned}$$

(F12)

or, using Eq. (F7), as

$$\begin{aligned} S[n(r),T]=\frac{1}{T}\left( E-W[n(r)]-U[n(r),T]\right) , \end{aligned}$$

(F13)

where U[n(r), T] is the internal energy given by

$$\begin{aligned} U[n(r),T]=\int V(n(r),T)\, \mathrm{d}{\mathbf{r}}\quad \mathrm{with}\quad V(n,T)=n \mu (n,T)-P(n,T). \end{aligned}$$

(F14)

Combining Eqs. (F10) and (F14), we get

$$\begin{aligned} V'(n,T)=\mu (n,T). \end{aligned}$$

(F15)

Therefore, the pressure P(n, T) is related to the density of internal energy V(n, T) by

$$\begin{aligned} P(n,T)=n\mu (n,T)-V(n,T)=n V'(n,T)-V(n,T)=n^2\left[\frac{V(n,T)}{n}\right]'. \end{aligned}$$

(F16)

Inversely, the density of internal energy is determined by the equation of state P[n(r), T] according to the relation

$$\begin{aligned} V(n,T)=n\int ^n \frac{P(n',T)}{{n'}^2}\, \mathrm{d}n'. \end{aligned}$$

(F17)

We note the identities

$$\begin{aligned} V'(n,T)=\int ^{n} \frac{P'(n',T)}{n'}\, \mathrm{d}n'\qquad \mathrm{and}\qquad V''(n,T)=\frac{P'(n,T)}{n}. \end{aligned}$$

(F18)

The internal energy can be written explicitly as

$$\begin{aligned} U[n(r),T]=\int n\int ^n \frac{P(n',T)}{{n'}^2}\, \mathrm{d}n' \mathrm{d}{\mathbf{r}}. \end{aligned}$$

(F19)

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

$$\begin{aligned} \frac{\delta S}{k_\mathrm{B}}+\alpha _0\delta N=0. \end{aligned}$$

(F20)

The conservation of energy implies [see Eq. (F7)]:

$$\begin{aligned} 0=\delta E_{\mathrm{kin}}+\int m\Phi \delta n\, \mathrm{d}{\mathbf{r}}. \end{aligned}$$

(F21)

Using Eqs. (F1) and (F21), we get

$$\begin{aligned} \frac{\delta S}{k_\mathrm{B}}=-\beta \int m\Phi \delta n\, \mathrm{d}{\mathbf{r}}-\int \alpha (r)\delta n \, \mathrm{d}{\mathbf{r}}. \end{aligned}$$

(F22)

As a result, the variational problem (F20) yields

$$\begin{aligned} \alpha (r)=\alpha _0-\beta m\Phi (r). \end{aligned}$$

(F23)

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

$$\begin{aligned} \delta \left( \frac{S}{k_\mathrm{B}}-\beta E_{\mathrm{kin}}\right) +\int \alpha (r)\delta n \, \mathrm{d}{\mathbf{r}}=0, \end{aligned}$$

(F24)

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 “Microcanonical ensemble” 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

$$\begin{aligned} F[n(r),T]=U[n(r),T]+W[n(r)], \end{aligned}$$

(F25)

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

$$\begin{aligned} \delta J+\alpha _0\delta N=0. \end{aligned}$$

(F26)

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

$$\begin{aligned} \delta J=-\int \alpha (r)\delta n \, \mathrm{d}{\mathbf{r}}-\beta \int m\Phi \delta n\, \mathrm{d}{\mathbf{r}}. \end{aligned}$$

(F27)

As a result, the variational problem (F26) yields

$$\begin{aligned} \alpha (r)=\alpha _0-\beta m\Phi (r). \end{aligned}$$

(F28)

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 as⁴¹

$$\begin{aligned} F[\rho ]=\int \rho \int ^{\rho } \frac{P(\rho ')}{{\rho '}^2}\, \mathrm{d}\rho '\, \mathrm{d}{\mathbf{r}}+\frac{1}{2}\int \rho \Phi \, \mathrm{d}{\mathbf{r}}. \end{aligned}$$

(F29)

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 Euler–Poisson 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 Euler–Poisson equations. This returns the general result established in [228]. It is valid for an arbitrary form of entropy. According to the Poincaré turning point criterion, the series of equilibria generically becomes both thermodynamically unstable in the canonical ensemble and dynamically unstable with respect to the Euler–Poisson 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)–(F6) reduce to

$$\begin{aligned} n(r)= & {} \int F\left[\frac{\beta p^2}{2m}-\alpha (r)\right]\, \mathrm{d}{\mathbf{p}}, \end{aligned}$$

(F30)

$$\begin{aligned} \epsilon _{\mathrm{kin}}(r)= & {} \int F\left[\frac{\beta p^2}{2m}-\alpha (r)\right]\frac{p^2}{2m}\, \mathrm{d}{\mathbf{p}}, \end{aligned}$$

(F31)

$$\begin{aligned} P(r)= & {} \frac{1}{3}\int F\left[\frac{\beta p^2}{2m}-\alpha (r)\right]\frac{p^2}{m}\, \mathrm{d}{\mathbf{p}}. \end{aligned}$$

(F32)

Making the change of variables $\mathbf{x}=(\beta /m)^{1/2}{\mathbf{p}}$, we obtain the scaling

$$\begin{aligned} P(n,T)=T^{5/2}{\Pi }_{\mathrm{NR}}\left( \frac{n}{T^{3/2}}\right) . \end{aligned}$$

(F33)

Therefore, the internal energy (F19) takes the form

$$\begin{aligned} U[n(r),T]=T\int n\int ^{n/T^{3/2}} \frac{\Pi _{\mathrm{NR}}(x)}{{x}^2}\, \mathrm{d}x \mathrm{d}{\mathbf{r}}. \end{aligned}$$

(F34)

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)–(F6) reduce to

$$\begin{aligned} n(r)= & {} \int F\left[\beta p c -\alpha (r)\right]\, \mathrm{d}{\mathbf{p}}, \end{aligned}$$

(F35)

$$\begin{aligned} \epsilon _{\mathrm{kin}}(r)= & {} \int F\left[\beta p c -\alpha (r)\right]pc \, \mathrm{d}{\mathbf{p}}, \end{aligned}$$

(F36)

$$\begin{aligned} P(r)= & {} \frac{1}{3}\int F\left[\beta p c-\alpha (r)\right]p c \, \mathrm{d}{\mathbf{p}}. \end{aligned}$$

(F37)

Making the change of variables $\mathbf{x}=\beta {\mathbf{p}} c$, we obtain the scaling

$$\begin{aligned} P(n,T)=T^{4}{\Pi }_{\mathrm{UR}}\left( \frac{n}{T^{3}}\right) . \end{aligned}$$

(F38)

Therefore, the internal energy (F19) takes the form

$$\begin{aligned} U[n(r),T]=T\int n\int ^{n/T^{3}} \frac{\Pi _{\mathrm{UR}}(x)}{{x}^2}\, \mathrm{d}x \mathrm{d}{\mathbf{r}}. \end{aligned}$$

(F39)

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

$$\begin{aligned} \frac{\delta S}{k_\mathrm{B}}-\int {{\tilde{\beta }}}(r)\delta \epsilon \, \mathrm{d}{\mathbf{r}}+\alpha \delta N=0, \end{aligned}$$

(F40)

where ${{\tilde{\beta }}}(r)$ is a local (position dependent) Lagrange multiplier and $\alpha$ is a global (uniform) Lagrange multiplier. Noting that M(r)—which appears in the expressions of S and N—is fixed since it is determined by $\epsilon (r)$, this variational problem yields

$$\begin{aligned} C'(f)=-\beta (r)E(p)+\alpha \end{aligned}$$

(F41)

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 “photon stars” 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

$$\begin{aligned} f({\mathbf{p}})=\frac{1}{h^3}\frac{1}{e^{\beta pc}-1}. \end{aligned}$$

(G1)

This corresponds to the Bose–Einstein 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 Stefan–Boltzmann law

$$\begin{aligned} \epsilon =\frac{24\pi }{h^3c^3}(k_\mathrm{B} T)^4\frac{\pi ^4}{90}. \end{aligned}$$

(G2)

The factor in front of $T^4$ is the Stefan–Boltzmann constant. Using Eq. (126), we find that the pressure is given by

$$\begin{aligned} P=\frac{8\pi }{h^3c^3}(k_\mathrm{B} T)^4\frac{\pi ^4}{90}. \end{aligned}$$

(G3)

It is related to the energy density by the linear equation of state

$$\begin{aligned} P=\frac{1}{3}\epsilon . \end{aligned}$$

(G4)

This linear relationship, with a coefficient 1/3, is valid for an arbitrary ultrarelativistic gas (see “Ultrarelativistic limit” section in Appendix A). Using Eq. (119), we find that the particle density is given by

$$\begin{aligned} n=\frac{8\pi }{h^3c^3}(k_\mathrm{B} T)^3\zeta (3), \end{aligned}$$

(G5)

where $\zeta (3)=1.202056 \ldots$ is the Apéry 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

$$\begin{aligned} P=Kn^{4/3}\quad \mathrm{with}\quad K=\frac{\pi ^4}{90}\frac{hc}{[8\pi \zeta (3)^4]^{1/3}}. \end{aligned}$$

(G6)

Finally, using the integrated Gibbs–Duhem relation (141) with $\mu =0$, we find that the entropy density is given by⁴²

$$\begin{aligned} s=k_\mathrm{B} \frac{32\pi ^5}{90 h^3c^3}(k_\mathrm{B} T)^3. \end{aligned}$$

(G7)

We see that the entropy density is proportional to the particle density:

$$\begin{aligned} s=\lambda n k_\mathrm{B} \quad \mathrm{with}\quad \lambda =\frac{4\pi ^4}{90\zeta (3)}. \end{aligned}$$

(G8)

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 Tolman’s equation of hydrostatic equilibrium (108), we get

$$\begin{aligned} \frac{\mathrm{d}\ln P}{\mathrm{d}r}=-2\frac{\mathrm{d}\nu }{\mathrm{d}r}. \end{aligned}$$

(G9)

On the other hand, according to Eq. (G3), we have

$$\begin{aligned} \frac{\mathrm{d}\ln P}{\mathrm{d}r}=4\frac{\mathrm{d}\ln T}{\mathrm{d}r}. \end{aligned}$$

(G10)

These two equations directly imply the Tolman relation

$$\begin{aligned} \frac{\mathrm{d}\ln T}{\mathrm{d}r}=-\frac{1}{2}\frac{\mathrm{d}\nu }{\mathrm{d}r} \qquad \Rightarrow \qquad T(r) e^{\nu (r)/2}=\mathrm{cst}. \end{aligned}$$

(G11)

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:

$$\begin{aligned} S=\lambda N k_\mathrm{B}\quad \mathrm{with}\quad \lambda =\frac{4\pi ^4}{90\zeta (3)}. \end{aligned}$$

(G12)

The condition of thermodynamical stability, corresponding to the maximization of the entropy at fixed mass energy:

$$\begin{aligned} \max \, \lbrace S\, |\, {{{\mathcal {E}}}}=Mc^2\,\, \mathrm{fixed}\rbrace , \end{aligned}$$

(G13)

turns out to be equivalent to the maximization of the particle number at fixed mass energy:

$$\begin{aligned} \max \, \lbrace N\, |\, M\,\, \mathrm{fixed}\rbrace , \end{aligned}$$

(G14)

which is itself equivalent to the minimization of the mass energy at fixed particle number:

$$\begin{aligned} \min \, \lbrace M\, |\, N\,\, \mathrm{fixed}\rbrace , \end{aligned}$$

(G15)

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 Ipser’s conjecture [258] (see Sect. 5.2.2) can be easily demonstrated.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

$$\begin{aligned} \delta S-\frac{1}{T_{\infty }}\delta {{{\mathcal {E}}}}=0 \qquad \Rightarrow \qquad \delta N-\frac{1}{\lambda }\beta _{\infty }c^2\delta M=0, \end{aligned}$$

(G16)

where $1/T_{\infty }$ is a Lagrange multiplier.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 Euler–Einstein 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 Euler–Einstein 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 Poincaré 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:

$$\begin{aligned} \max \, \lbrace {{{\mathcal {K}}}}\, |\, {{{\mathcal {E}}}}=Mc^2\,\, \mathrm{fixed}\rbrace . \end{aligned}$$

(G17)

The extremization problem (first variations) yields

$$\begin{aligned} \delta {{{\mathcal {K}}}}-\frac{1}{T_{\infty }}\delta {{{\mathcal {E}}}}=0. \end{aligned}$$

(G18)

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 Tolman–Klein relations

In this appendix, we review the main results given in the seminal papers of Tolman [241] and Klein [249].

1.1 Tolman’s (1930) paper

In a paper published in 1930, Tolman [241] investigated “the weight of heat and thermal equilibrium in general relativity.” 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) $\nu (r)$. Tolman’s 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: “It 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 (‘Scientific Papers,’ 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 Gibbs’ work.”

1.2 Klein’s (1949) paper

In a paper entitled “On the thermodynamical equilibrium of fluids in gravitational fields” 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,⁴⁵ by using essentially the Gibbs–Duhem relation and the first principle of thermodynamics. We give below a summary of Klein’s calculations.

Klein started from the first principle of thermodynamics

$$\begin{aligned} \mathrm{d}E=-P\mathrm{d}V+T\mathrm{d}S+\mu \mathrm{d}N. \end{aligned}$$

(H1)

Since E is a homogeneous function of the first degree in the three variables V, S and N, the Euler theorem implies that

$$\begin{aligned} E=-PV+TS+\mu N, \end{aligned}$$

(H2)

which is the Gibbs–Duhem relation (see Appendix E). From Eqs. (H1) and (H2), we get

$$\begin{aligned} \mathrm{d}\left( \frac{P}{T}\right) =\frac{N}{V}\mathrm{d}\left( \frac{\mu }{T}\right) -\frac{E}{V}\mathrm{d}\left( \frac{1}{T}\right) . \end{aligned}$$

(H3)

Written under a local form, with the variables $s=S/V$, $n=N/V$ and $\epsilon =E/V$, Eqs. (H1)–(H3) return Eqs. (136) and (141)–(143). In turn, Eq. (142) can be written as

$$\begin{aligned} \frac{\mathrm{d}P}{\mathrm{d}r}=\frac{\epsilon +P}{T}\frac{\mathrm{d}T}{\mathrm{d}r}+nT\frac{\mathrm{d}}{\mathrm{d}r}\left( \frac{\mu }{T}\right) . \end{aligned}$$

(H4)

Combined with the condition of hydrostatic equilibrium from Eq. (108), we get

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{d}\nu }{\mathrm{d}r}+\frac{1}{T}\frac{\mathrm{d}T}{\mathrm{d}r}=-\frac{nT}{\epsilon +P}\frac{ d }{ \mathrm{d}r } \left( \frac{\mu }{T}\right) . \end{aligned}$$

(H5)

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

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{d}\nu }{\mathrm{d}r}+\frac{1}{T}\frac{\mathrm{d}T}{\mathrm{d}r}=0\qquad \Rightarrow \qquad T(r) e^{\nu (r)/2}=\mathrm{cst}, \end{aligned}$$

(H6)

which is Tolman’s relation. Then, for all other substances

$$\begin{aligned} \frac{\mu (r)}{k_\mathrm{B} T(r)}=\mathrm{cst} \qquad \Rightarrow \qquad \mu (r) e^{\nu (r)/2}=\mathrm{cst}, \end{aligned}$$

(H7)

which is Klein’s 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

$$\begin{aligned} \mathrm{d}\left( \frac{u}{n}\right) =-P\mathrm{d}\left( \frac{1}{n}\right) +T\mathrm{d}\left( \frac{s}{n}\right) , \end{aligned}$$

(I1)

where u is the density of internal energy. If we introduce the enthalpy per particle

$$\begin{aligned} h=\frac{P+u}{n}, \end{aligned}$$

(I2)

we can rewrite Eq. (I1) as

$$\begin{aligned} \mathrm{d}h=\frac{\mathrm{d}P}{n}+T \mathrm{d}\left( \frac{s}{n}\right) . \end{aligned}$$

(I3)

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

$$\begin{aligned} \mathrm{d}\left( \frac{u}{n}\right) =-P\mathrm{d}\left( \frac{1}{n}\right) =\frac{P}{n^2}\mathrm{d}n. \end{aligned}$$

(I4)

We also have

$$\begin{aligned} \mathrm{d}u=\frac{P+u}{n} \, \mathrm{d}n=h\, \mathrm{d}n\qquad \mathrm{and}\qquad \mathrm{d}h=\frac{\mathrm{d}P}{n}. \end{aligned}$$

(I5)

The foregoing equations can be rewritten equivalently as

$$\begin{aligned} P=-\frac{\mathrm{d}(u/n)}{\mathrm{d}(1/n)}=n\frac{du}{\mathrm{d}n}-u,\qquad h=\frac{du}{\mathrm{d}n}=\frac{P+u}{n}. \end{aligned}$$

(I6)

For a cold or isentropic gas, using the Gibbs–Duhem relation (E7), we see that the chemical potential coincides with the enthalpy

$$\begin{aligned} \mu =h. \end{aligned}$$

(I7)

For a barotropic equation of state $P=P(n)$, Eq. (I4) can be integrated into

$$\begin{aligned} u(n)=n\int ^n \frac{P(n')}{{n'}^2}\, \mathrm{d}n'. \end{aligned}$$

(I8)

The internal energy is

$$\begin{aligned} U=\int u(n)\, \mathrm{d}{\mathbf{r}}=\int n\int ^n \frac{P(n')}{{n'}^2}\, \mathrm{d}n'\, \mathrm{d}{\mathbf{r}}. \end{aligned}$$

(I9)

We note the identities

$$\begin{aligned} u'(n)=h(n)=\frac{P(n)+u(n)}{n}, \qquad u''(n)=h'(n)=\frac{P'(n)}{n},\qquad \left( \frac{u}{n}\right) '=\frac{P(n)}{n^2}, \end{aligned}$$

(I10)

and

$$\begin{aligned} P(n)=n h(n)-u(n)=n u'(n)-u(n)=n^2 \left( \frac{u}{n}\right) '. \end{aligned}$$

(I11)

We also have

$$\begin{aligned} h(n)=\int ^n \frac{P'(n')}{n'}\, \mathrm{d}n',\qquad u(n)=\int ^n h(n')\, \mathrm{d}n', \end{aligned}$$

(I12)

and

$$\begin{aligned} u(n)=nh(n)-P(n)=n\int ^n \frac{P'(n')}{n'}\, \mathrm{d}n'-P(n). \end{aligned}$$

(I13)

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 Euler–Poisson equations is a minimum of energy $\mathcal{W}$ at fixed particle number N (see Sect. 5.1.1).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

$$\begin{aligned} \mathrm{d}\left( \frac{\epsilon }{n}\right) =-P\mathrm{d}\left( \frac{1}{n}\right) +T\mathrm{d}\left( \frac{s}{n}\right) , \end{aligned}$$

(I14)

where $\epsilon =\rho c^2+u$ is the mass energy density and $\rho =n m$ is the rest mass density. The relations of “Newtonian isentropic or cold barotropic gases” section in Appendix I remain valid with u or with $\epsilon$. When $T\mathrm{d}(s/n)=0$, Eq. (I14) reduces to

$$\begin{aligned} \mathrm{d}\epsilon =\frac{P+\epsilon }{n} \, \mathrm{d}n. \end{aligned}$$

(I15)

For a barotropic equation of state of the form $P=P(n)$, we obtain

$$\begin{aligned} \epsilon (n)=nmc^2+n\int ^n \frac{P(n')}{{n'}^2}\, \mathrm{d}n'=\rho c^2+u(n). \end{aligned}$$

(I16)

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

$$\begin{aligned} n(\epsilon )=e^{\int ^{\epsilon } \frac{\mathrm{d}\epsilon '}{P(\epsilon ')+\epsilon '}}. \end{aligned}$$

(I17)

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 Euler–Einstein equations is a minimum of energy E at fixed particle number N (see Sect. 5.1.2).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 “Newtonian isentropic or cold barotropic gases” section in Appendix I.

Remark

For a polytropic equation of state of the form $P=Kn^{\gamma }$, we get⁴⁸

$$\begin{aligned} \epsilon= & {} nmc^2+\frac{P}{\gamma -1}\qquad (\gamma \ne 1), \end{aligned}$$

(I18)

$$\begin{aligned} \epsilon= & {} nmc^2+Kn\ln n\qquad (\gamma =1). \end{aligned}$$

(I19)

For a linear equation of state of the form $P=q\epsilon$ with $q=\gamma -1$, we obtain

$$\begin{aligned} P=Kn^{\gamma }\qquad \mathrm{and}\qquad \epsilon =\frac{K}{q} n^{\gamma }, \end{aligned}$$

(I20)

where K is a constant of integration. When $\mu =0$, the integrated Gibbs–Duhem relation (E7) reduces to

$$\begin{aligned} s=\frac{\epsilon +P}{T}. \end{aligned}$$

(I21)

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

$$\begin{aligned} T=\frac{q+1}{q} \frac{K}{\lambda } n^{\gamma -1}. \end{aligned}$$

(I22)

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

$$\begin{aligned} \mathrm{d}\left( \frac{\epsilon _{\mathrm{kin}}}{n}\right) =-P\mathrm{d}\left( \frac{1}{n}\right) +T\mathrm{d}\left( \frac{s}{n}\right) . \end{aligned}$$

(I23)

Since the temperature T is uniform at statistical equilibrium, we can rewrite the foregoing equation as

$$\begin{aligned} \mathrm{d}\left( \frac{\epsilon _{\mathrm{kin}}-Ts}{n}\right) =-P\mathrm{d}\left( \frac{1}{n}\right) =\frac{P}{n^2}\mathrm{d}n. \end{aligned}$$

(I24)

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

$$\begin{aligned} \epsilon _{\mathrm{kin}}(n)-Ts(n)=n\int ^n \frac{P(n')}{{n'}^2}\, \mathrm{d}n'. \end{aligned}$$

(I25)

Introducing the internal energy defined by Eq. (I8), we obtain the important relation

$$\begin{aligned} \epsilon _{\mathrm{kin}}(n)-Ts(n)=u(n). \end{aligned}$$

(I26)

Integrating this relation over the whole configuration, we find that the entropy is given by

$$\begin{aligned} S=\frac{E_{\mathrm{kin}}-U}{T}, \end{aligned}$$

(I27)

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

$$\begin{aligned} F=E-TS=E_{\mathrm{kin}}+W-TS=U+W, \end{aligned}$$

(I28)

which returns Eq. (F25). We see that

$$\begin{aligned} F={{{\mathcal {W}}}}. \end{aligned}$$

(I29)

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 Euler–Poisson 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).