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

Abstract

We study the statistical mechanics of classical self-gravitating systems confined within a box of radius R in general relativity. It has been found that the caloric curve $T_{\infty }(E)$ has the form of a double spiral whose shape depends on the compactness parameter $\nu =GNm/Rc^2$ (Roupas in Class Quantum Grav 32:135023, 2015; Alberti and Chavanis in arXiv:1908.10316). The double spiral shrinks as $\nu$ increases and finally disappears when $\nu _{{\mathrm{max}}}=0.1764$. Therefore, general relativistic effects render the system more unstable. On the other hand, the cold spiral and the hot spiral move away from each other as $\nu$ decreases. Using a normalization $\varLambda =-ER/GN^2m^2$ and $\eta =GNm^2/R k_\mathrm{B} T_{\infty }$ appropriate to the nonrelativistic limit, and considering $\nu \rightarrow 0$, the hot spiral goes to infinity and the caloric curve tends to a limit curve (determined by the Emden equation) exhibiting a single cold spiral, as found in former works. Using another normalization ${{{\mathcal {M}}}}=GM/Rc^2$ and ${{{\mathcal {B}}}}={Rc^4}/{GNk_\mathrm{B} T_{\infty }}$ appropriate to the ultrarelativistic limit, and considering $\nu \rightarrow 0$, the cold spiral goes to infinity and the caloric curve tends to a limit curve (determined by the general relativistic Emden equation) exhibiting a single hot spiral. This result is new. We discuss the analogies and the differences between this asymptotic caloric curve and the caloric curve of the self-gravitating black-body radiation. Finally, we compare box-confined isothermal models with heavily truncated isothermal distributions in Newtonian gravity and general relativity.

Appendices

Appendix A: Series of equilibria of truncated isothermal distributions

In this Appendix, we discuss the series of equilibria of truncated isothermal distributions in Newtonian gravity and general relativity.

1.1 Nonrelativistic systems

The series of equilibria of globular clusters described by truncated isothermal distributions (Woolley [36] and King [37] models) have been determined by Lynden-Bell and Wood [40], Katz [41] and Chavanis et al. [32]. The caloric curve $\beta (E)$ of the King model is reproduced in Fig. 4. It has the form of a spiral. It is parametrized by the concentration parameter k that increases monotonically along the series of equilibria. The curves $\beta (k)$ and E(k) giving the inverse temperature and the energy as a function of the concentration parameter k display damped oscillations [32].

The thermodynamical stability of truncated isothermal distributions was analysed in Refs. [32, 40, 41] by using the Poincaré turning point criterion [73]. For $E\rightarrow 0^-$ and $\beta \rightarrow 0$, we know that the system is stable because it is equivalent to a polytrope of index $n=5/2$ that is both canonically and microcanonically stable [139]. In the canonical ensemble (fixed temperature), the series of equilibria is stable up to the first turning point of temperature (corresponding to $k_{{\mathrm{CE}}}=1.34$) and becomes unstable afterwards. This is when the specific heat becomes infinite, passing from positive to negative values. In the microcanonical ensemble (fixed energy), the series of equilibria is stable up to the first turning point of energy (corresponding to $k_{\mathrm{MCE}}=7.44$) and becomes unstable afterwards. This is when the specific heat vanishes, passing from negative to positive values. The statistical ensembles are inequivalent in the region of negative specific heat ($C<0$), between the turning point of temperature CE and the turning point of energy MCE. From general arguments, it can be shown that canonical stability implies microcanonical stability [139, 140]. Basically, this is because the microcanonical ensemble is more constrained, hence more stable, than the canonical ensemble. In the present case, this manifests itself (in conjunction with the Poincaré theory) by the fact that the turning point of temperature occurs before the turning point of energy. For isolated stellar systems, only the microcanonical ensemble makes sense physically.²¹

Let us now consider the dynamical stability of the system with respect to a collisionless evolution described by the Vlasov–Poisson equations. In Newtonian gravity, it has been shown that all isotropic stellar systems with a distribution function of the form $f=f(\epsilon )$ with $f'(\epsilon )<0$ are dynamically stable [52,53,54,55,56,57]. Therefore, the whole series of equilibria of isothermal stellar systems is dynamically stable, even the equilibrium states deep into the spiral that are thermodynamically unstable. We know from general arguments that thermodynamical stability implies dynamical stability [106, 140]. The results of [52,53,54,55,56,57] show that the converse is wrong in Newtonian gravity: before the first turning point of energy, the system is both thermodynamically stable (in the microcanonical ensemble) and dynamically stable; after the first turning point of energy, the system is thermodynamically unstable while it is still dynamically stable.

Fig. 4

Series of equilibria (caloric curve) of the classical King model. The units for the curve $\beta (E)$ are $(m v_0^{2})^{-1}$ for $\beta$ and $Mv_0^2$ for E, where $v_0$ is a typical velocity defined in [32, 41]. The series of equilibria is parametrized by the dimensionless concentration parameter $k=\beta [\varPhi (R)-\varPhi (0)]$. It has a snail-like (spiral) structure, but only the part of the curve up to CE ($k_{\mathrm{CE}}=1.34$) is stable in the canonical ensemble and only the part up to MCE ($k_{\mathrm{MCE}}=7.44$) is stable in the microcanonical ensemble. (The region between CE and MCE where the specific heat is negative corresponds to a region of ensembles inequivalence.) For $k\rightarrow 0$, the King model is equivalent to a polytrope of index $n=5/2$ [32, 41] that is both canonically and microcanonically stable [139]. (The same is true for the Wooley model which is equivalent, for $k\rightarrow 0$, to a polytrope $n=3/2$ [41].) The equilibrium states are all dynamically (Vlasov) stable

We also know from general arguments that the thermodynamical stability of a stellar system in the canonical ensemble is equivalent to the dynamical stability of the corresponding barotropic star with respect to the Euler–Poisson equations [139]. Therefore, barotropic stars with the equation of state corresponding to the truncated isothermal distribution function are dynamically stable before the first turning point of temperature and dynamically unstable after the first turning point of temperature.

The dynamical evolution of globular clusters is discussed in the introduction (see also [30, 31] and Appendix D of [32]). Because of collisions (more precisely close encounters) between stars and evaporation, the system evolves quasistatically along the series of equilibria. The natural evolution corresponds to an increase of the central density $\rho _0$ that parametrizes the series of equilibria.²² During the evolution, the energy decreases. On the other hand, the temperature decreases in the region of positive specific heat $C=dE/dT>0$ and increases in the region of negative specific heat $C=dE/dT<0$. At the first turning point of energy (minimum energy state), the system becomes unstable and undergoes the gravothermal catastrophe (core collapse). This is a thermodynamical instability taking place on a relaxation (secular) timescale. It ultimately leads to the formation of a binary star surrounded by a hot halo.

1.2 General relativistic systems

The series of equilibria of relativistic star clusters described by the heavily truncated Maxwell–Boltzmann distribution (relativistic Woolley model) was first determined by Zel’dovich and Podurets [58]. They plotted the temperature measured by an observer at infinity $T_{\infty }$ as a function of the central density $\rho _0$ and found that the curve $T_{\infty }(\rho _0)$ displays damped oscillations.²³ As a result, there exists a maximum temperature $k_\mathrm{B} T_{{\mathrm{max}}}/mc^2=0.273$ above which there is no equilibrium.²⁴ They argued (without rigorous justification) that the series of equilibria should become unstable at that point and that the system should experience a gravitational collapse that they called an “avalanche-type catastrophic contraction of the system”. Considering the same distribution function, Ipser [63] plotted the fractional binding energy $E/Nmc^2$ as a function of the central redshift $z_0$ and found that the curve $E/Nmc^2(z_0)$ displays damped oscillations. Using a rigorous instability criterion based on the equation of pulsations derived by Ipser and Thorne [61], he found that the series of equilibria becomes dynamically (Vlasov) unstable above a critical redshift $z_\mathrm{c}= 0.516$ and that this critical value happens to coincide with the turning point of fractional binding energy. At that point $[(M-Nm)/Nm]_{\mathrm{c}}=-0.0357$, $(Rc^2/2GNm)_{\mathrm{c}}=4.42$ and $(k_\mathrm{B} T_\infty /mc^2)_{\mathrm{c}}=0.23$. The turning point of binding energy found by Ipser [63] is different from the turning point of temperature found by Zel’dovich and Podurets [58] corresponding to $z_0=1.08$, $(M-Nm)/Nm=-0.0133$, $Rc^2/2GNm=3.92$, and $k_\mathrm{B}(T_{\infty })_{\mathrm{max}}/mc^2=0.27$. In particular, the gravitational instability occurs sooner than predicted by Zel’dovich and Podurets [58]. Using the values of the fractional binding energy and of the temperature measured by an observer at infinity tabulated by Ipser [63] (see his Table 1), we have plotted in Fig. 5 the caloric curve of general relativistic heavily truncated Maxwell–Boltzmann distributions giving $mc^2/k_\mathrm{B} T_{\infty }$ as a function of $E/Nmc^2$. This curve has not been plotted before. It has the form of a spiral of which we only see the beginning.

Fig. 5

Series of equilibria (caloric curve) of heavily truncated Maxwell–Boltzmann distributions in general relativity (according to the data of Ipser [63]). The series of equilibria is parametrized by the central redshift $z_0$. It has a snail-like (spiral) structure of which we only see the beginning. The equilibrium states are all canonically unstable. In the microcanonical ensemble, only the part of the curve up to $z_\mathrm{c}=0.516$ (corresponding to the turning point of binding energy) is stable. In this region, the specific heat is negative. The onset of thermodynamical instability coincides with the onset of dynamical instability

We can now investigate the dynamical and thermodynamical stability of the equilibrium states of relativistic star clusters in more detail. Let us first consider their thermodynamical stability. In a sense, we can say that Zel’dovich and Podurets [58] considered the canonical ensemble (fixed $T_{\infty }$), while Ipser [63] considered the microcanonical ensemble (fixed E). The fact that they obtained different results is related to the notion of ensembles inequivalence for systems with long-range interactions like self-gravitating systems.

Let us first consider the microcanonical ensemble which is the rigorous statistical ensemble that we should use in order to describe isolated star clusters. Let us assume that the system is stable at sufficiently high energies (i.e. close to $0^{-}$). Using the Poincaré criterion [73], we conclude that the system is microcanonically stable on the upper branch of Fig. 5 up to the first turning point of fractional binding energy and becomes unstable afterwards. We note that the system becomes microcanonically unstable when the specific heat becomes positive. This result was first stated by Horwitz and Katz [10].

Let us now turn to the canonical ensemble. If the system were stable at sufficiently low temperatures, using the Poincaré criterion [73], we would conclude that the system is canonically stable up to the first turning point of temperature and becomes canonically unstable afterwards. However, considering the topology of the series of equilibria in Fig. 5, this is not possible. Indeed, we know from general arguments that canonical stability implies microcanonical stability [139, 140]. Now, we note that the first turning point of temperature occurs after the first turning point of binding energy. Therefore, if the series of equilibria were canonically stable up to the first turning point of temperature it would also be microcanonically stable up to that point. But, we know that the series of equilibria becomes unstable after the first turning point of binding energy. Therefore, we arrive at a contradiction. Since a change of stability in the canonical ensemble can occur only at a turning point of temperature, we conclude that the whole series of equilibria is canonically unstable.²⁵ This is consistent with the fact that the beginning of the series of equilibria has a negative specific heat which is forbidden in the canonical ensemble.

Let us finally consider the dynamical stability of the system with respect to a collisionless evolution described by the Vlasov–Einstein equations. We know from general arguments that thermodynamical stability implies dynamical stability [11, 140]. From the previous results obtained in the microcanonical ensemble, we conclude that the system is both thermodynamically and dynamically stable up to the first turning point of fractional binding energy. This is a particular case (for isothermal systems) of the general binding energy criterion derived by Ipser [11]. After the first turning point of fractional binding energy, the system is thermodynamically unstable. On the other hand, Ipser [63] has shown numerically that the system is also dynamically unstable after that point. This lead Ipser [11] to the conclusion²⁶ that, in general relativity, dynamical stability and microcanonical thermodynamical stability coincide contrary to the case of Newtonian systems (see Appendix A 1).

From the previous results obtained in the canonical ensemble, we cannot conclude anything regarding the dynamical stability of collisionless isothermal star clusters since all equilibria are canonically unstable. However, we know that the canonical stability of a collisionless star cluster is equivalent to the dynamical stability of the corresponding barotropic star with respect to the Euler–Einstein equations [20, 21, 23, 26]. Therefore, our observation that all the equilibria of isothermal relativistic star clusters are canonically unstable is consistent with Ipser’s [63] finding that all the relativistic stars with the same equation of state as the isothermal star clusters are dynamically unstable.

The dynamical evolution of relativistic star clusters is reviewed in the Introduction (see also the introduction of [30]). Because of collisions between stars and evaporation, the system evolves quasistatically along the series of equilibria. The natural evolution corresponds to an increase of the central density $\rho _0$ (or central redshift $z_0$) that parametrizes the series of equilibria. During the evolution, the energy decreases. On the other hand, the temperature increases since the specific heat is negative $C=dE/dT_{\infty }<0$. At the turning point of energy (minimum energy state), the system becomes unstable and undergoes a gravitational collapse. This is a dynamical (and thermodynamical) instability of general relativistic origin taking place on a dynamical (short) timescale. It ultimately leads to the formation of a black hole surrounded by a halo of stars.

Appendix B: Analogies and differences between box models and truncated distributions

1.1 Newtonian gravity

In Newtonian gravity, the series of equilibria of truncated isothermal distributions (see Fig. 4) is qualitatively similar to the series of equilibria of box-confined isothermal systems (see Fig. 2 and the “cold spiral” of Fig. 1). The main difference stems from the fact that, for open clusters, equilibrium states necessarily have a negative energy $E<0$.²⁷ Indeed, when the energy is positive ($E>0$) the stars are unbounded and disperse away. If the system is confined within a box, equilibrium states with a positive energy are possible because the stars bounce off the wall. Apart from this difference, the series of equilibria in Figs. 2 and 4 have a similar spiralling shape. They display a turning point of temperature before the turning point of energy. For the two systems, equilibrium states are canonically stable up to the turning point of temperature and microcanonically stable up to the turning point of energy. We note that the dimensionless temperature and the dimensionless energy in the two models are related by

$$\begin{aligned} \eta =\sigma \beta m v_0^2\quad \mathrm{and}\quad \varLambda =-\frac{1}{\sigma } \frac{E}{Mv_0^2}\quad \mathrm{where}\quad \sigma =\frac{GM}{Rv_0^2}. \end{aligned}$$

(B1)

In practice, $\sigma \sim 1$ since $v_0$ corresponds to the virial velocity of the cluster. We see that the orders of magnitude of the dimensionless energies and dimensionless temperatures in Figs. 2 and 4 are consistent with each other.

1.2 General relativity

We now turn to the general relativistic case. The values of E and $T_{\infty }$ tabulated by Ipser [63] correspond to strongly relativistic stellar systems. Therefore, the caloric curve of Fig. 5 describes only the strongly relativistic part of the caloric curve (hot spiral). We need to complete this caloric curve with the nonrelativistic part from Fig. 4 (cold spiral). To that purpose, we first recall that, in the nonrelativistic regime, $\beta m v_0^2$ and ${E}/{Mv_0^2}$ are of the same order of magnitude as $\eta$ and $\varLambda$ in the box model (see Appendix B 1). On the other hand, in the relativistic regime, we have the relations

$$\begin{aligned} \eta =\nu \frac{mc^2}{k_\mathrm{B} T_{\infty }}\quad \mathrm{and} \quad \varLambda =-\frac{1}{\nu } \frac{E}{Nmc^2}\quad \mathrm{where}\quad \nu =\frac{GNm}{Rc^2}. \end{aligned}$$

(B2)

As shown in Refs. [30, 128], the parameter $\nu$ belongs to the interval [0, 0.1764]. In order to construct the complete caloric curve of truncated isothermal star clusters, we first determine the relativistic curve $\eta (\varLambda )$ from Fig. 5 by multiplying ${mc^2}/{k_\mathrm{B} T_{\infty }}$ by $\nu$ and dividing ${E}/{Nmc^2}$ by $\nu$, and then, we add the nonrelativistic caloric curve of Fig. 4. We note that the result depends on the parameter $\nu$ so we actually have a family of caloric curves. For illustration, we have taken $\nu =0.015$. This leads to the complete caloric curve reported in Fig. 6.²⁸ Our procedure is of course very approximate, but it is sufficient to show the idea. It will be important in future works to improve this procedure and determine the family of caloric curves exactly.

From the complete caloric curve of Fig. 6, it is now easy to understand the thermodynamical stability/instability of isothermal clusters in both nonrelativistic and relativistic regimes. We again advocate the Poincaré turning point criterion [73].

In the microcanonical ensemble, the nonrelativistic branch is stable up to the first fundamental turning point of energy MCE (see Appendix A 1). At that point, the series of equilibria $\beta (-E)$ rotates clockwise so that a mode of stability is lost. A new mode of stability is lost at each subsequent turning point of energy where the series of equilibria rotates clockwise. At some point, the series of equilibria unwinds and rotates anticlockwise. (The unwinding of the spiral is represented schematically by a dashed line in Fig. 6.) A mode of stability is gained at each turning point of energy where the series of equilibria rotates anticlockwise. Finally, the series of equilibria on the relativistic branch becomes stable again (after rotating anticlockwise an even number of times) and remains stable until the second fundamental turning point of energy MCE’ at which it becomes unstable again. After that point, the series of equilibria rotates only clockwise so that it remains unstable until the end. This is consistent with the results from Appendix A 2.

In the canonical ensemble, the nonrelativistic branch is stable up to the first turning point of temperature CE (see Appendix A 1). At that point, the series of equilibria $\beta (-E)$ rotates clockwise so that a mode of stability is lost. The series of equilibria thus becomes unstable. It remains unstable until the end because it can never recover the first mode of stability that it has lost. (It rotates anticlockwise an odd number of times.) This is why we found in Appendix A 2 that the relativistic branch is always unstable in the canonical ensemble.

The caloric curve of Fig. 6 provides a nice illustration of the scenario discussed in the Introduction. Because of collisions (close encounters) and evaporation, a Newtonian stellar system evolves along the upper branch of the series of equilibria. When it reaches the first fundamental turning point of energy, it becomes thermodynamically unstable (while remaining dynamically stable) and undergoes a gravothermal catastrophe. It then evolves towards very hot and very dense configurations and becomes relativistic. It may then reach a stable relativistic isothermal equilibrium distribution. Again, because of collisions and evaporation, a relativistic star cluster evolves along the lower branch of the series of equilibria. When it reaches the second fundamental turning point of energy, it becomes thermodynamically and dynamically unstable and undergoes a catastrophic collapse towards a black hole.²⁹

Fig. 6

Complete caloric curve of truncated isothermal star clusters showing the nonrelativistic cold spiral and the relativistic hot spiral. (For illustration, we have taken $\nu =0.015$.) The units of $\beta$ and E are the same as in Fig. 4. Nonrelativistic clusters undergo a gravothermal catastrophe at MCE and become relativistic. They can then undergo a catastrophic collapse at MCE’ leading to the formation of a black hole

We can now compare the caloric curve of truncated isothermal star clusters (see Fig. 6) with the caloric curve of box-confined isothermal systems (see Fig. 1). As we have seen in Appendix B 1, the nonrelativistic parts of the caloric curves are relatively similar. In particular, they present a “cold” spiral at negative energies. We now consider the relativistic parts of the caloric curves. In that case, a “hot” spiral arises because, in the ultrarelativistic limit (corresponding to very high energy densities), the equation of state of an isothermal gas is $P=\epsilon /3$ like for the self-gravitating black-body radiation. However, in the relativistic domain, the series of equilibria of truncated isothermal distributions (see Figs. 5 and 6) and the series of equilibria of box-confined isothermal systems (see Figs. 1 and 3) are very different.

(i) The origin of the difference stems from the fact that, for open clusters, stable equilibrium states necessarily have a negative energy $E<0$.³⁰ By contrast, when the system is confined within a box, stable equilibrium states with a positive energy are allowed because of the pressure exerted by the box. For box-confined systems, the “hot spiral” appears at very large positive energies. At such energies, the relativistic isothermal gas behaves as a form of radiation confined within a cavity like in the studies of Sorkin et al. [12] and Chavanis [17]. There are no such equilibrium states with very large positive energies in open star clusters. In open star clusters, the “hot spiral” appears at energies (positive and negative) around $E=0$. In addition, the stable equilibrium states necessarily have a negative energy.

(ii) For box-confined isothermal systems, the cold and hot spirals are quite distinct. The cold spiral appears at negative energies, and the hot spiral appears at large positive energies. For open star clusters, the cold and hot spirals both appear at approximately the same range of (negative) energies. When general relativity is taken into account, the cold spiral of Fig. 4 unwinds and connects the hot spiral of Fig. 5 as sketched in Fig. 6.

As a result, a relativistic isothermal cluster in a box behaves very differently from a truncated relativistic isothermal cluster. This remark may question the physical relevance of the “hot spiral” at high energies for realistic isothermal star clusters except if one can justify a form of confinement playing the role of the box. This may be the case for relativistic stars. The box could be caused by a medium exerting a tremendously large pressure on an ultrarelativistic gas, like in the supernova phenomenon where the system achieves a “core–halo” structure. The core could correspond to the ultrarelativistic isothermal gas, and the halo could play the role of the box. In that case, the core would be sustained by the external pressure of the halo. However, this scenario remains to be put on a more rigorous basis (see [29, 30, 151, 153,154,155] and [156, 157] for different scenarios of supernova formation).

Appendix C: Entropy and free energy as functionals of the density for nonrelativistic self-gravitating classical particles

We consider a nonrelativistic system of self-gravitating classical particles. In Appendix C.2 of Paper I, we have introduced entropy and free energy functionals of the distribution function $f({\mathbf{r}},\mathbf{v})$. In Sect. 2.2 of this paper and in Appendix C.1.a of Paper I, 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 Boltzmann 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 $E_{\mathrm{kin}}[f]$ and particle density n(r). The variational problem for the first variations (extremization) can be written as

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

(C1)

where $\beta$ is a global (uniform) Lagrange multiplier and $\alpha (r)$ is a local (position dependent) Lagrange multiplier. This variational problem yields

$$\begin{aligned} f({\mathbf{r}},{\mathbf{p}})=\frac{g}{h^3}\hbox {e}^{-\beta p^2/2m+\alpha (r)}. \end{aligned}$$

(C2)

As in Appendix C.1 of Paper I, we can show that this distribution is the global maximum of S[f] at fixed $E_{{\mathrm{kin}}}[f]$ and n(r). Substituting Eq. (C2) into Eqs. (I-18), (I-19) and (I-21), we get

$$\begin{aligned} n(r)= & {} \frac{g}{h^3}( 2\pi m k_\mathrm{B} T)^{3/2}\hbox {e}^{\alpha (r)}, \end{aligned}$$

(C3)

$$\begin{aligned} \epsilon _{{\mathrm{kin}}}(r)= & {} \frac{3}{2}n(r)k_\mathrm{B} T, \end{aligned}$$

(C4)

$$\begin{aligned} P(r)= & {} \frac{2}{3}\epsilon _{{\mathrm{kin}}}(r)=n(r)k_\mathrm{B} T. \end{aligned}$$

(C5)

The Lagrange multiplier $\alpha (r)$ is determined by the density n(r) according to Eq. (C3). As a result, the distribution function (C2) can be written in terms of the density as

$$\begin{aligned} f({\mathbf{r}},{\mathbf{p}})=n({r}) \left( \frac{\beta }{2\pi m}\right) ^{3/2} \hbox {e}^{-\beta {p}^2/2m}. \end{aligned}$$

(C6)

On the other hand, since T is uniform, we see from Eq. (C5) that the equation of state is isothermal: $P(r)=n(r) k_\mathrm{B} T$. The temperature T is determined by the kinetic energy $E_{\mathrm{kin}}[n(r),T]=E-W[n(r)]$ using Eq. (C4) integrated over the volume giving $E_{{\mathrm{kin}}}=(3/2)Nk_\mathrm{B} T$. In other words, the temperature is determined by the energy constraint

$$\begin{aligned} E=\frac{3}{2}Nk_\mathrm{B} T+W[n(r)]. \end{aligned}$$

(C7)

We note that T is a functional of the density n(r) but, for brevity, we shall not write this dependence explicitly. Substituting the Maxwell–Boltzmann distribution (C6) into the entropy density (1), we obtain

$$\begin{aligned} \frac{s(r)}{k_\mathrm{B}}=\frac{5}{2}n(r)-n(r)\ln n(r)-n(r)\ln \left( \frac{h^3}{g}\right) +\frac{3}{2}n(r)\ln (2\pi m k_\mathrm{B} T), \end{aligned}$$

(C8)

which is equivalent to the integrated Gibbs–Duhem relation (9). The entropy is given by

$$\begin{aligned} \frac{S}{k_\mathrm{B}}=\frac{5}{2}N-\int n(r)\ln n(r)\, 4\pi r^2\, \hbox {d}r-N\ln \left( \frac{h^3}{g}\right) +\frac{3}{2}N\ln (2\pi m k_\mathrm{B} T). \end{aligned}$$

(C9)

Finally, the statistical equilibrium state in the microcanonical ensemble is obtained by maximizing the entropy S[n(r), T] at fixed particle number N, the energy constraint being taken into account in the determination of the temperature T[n] through Eq. (C7). The variational problem for the first variations (extremization) can be written as

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

(C10)

The conservation of energy implies [see Eq. (C7)]:

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

(C11)

Using Eqs. (C1) and (C11), and proceeding as in Appendix F1 of Paper I, or performing the variations over $\delta n$ and $\delta T$ directly on the explicit expressions (C7) and (C9), we obtain

$$\begin{aligned} \alpha (r)=\alpha _0-\beta m\varPhi (r)\qquad \mathrm{and}\qquad n(r)=\frac{g}{h^3}( 2\pi m k_\mathrm{B} T)^{3/2}\hbox {e}^{\alpha _0-\beta m\varPhi (r)}. \end{aligned}$$

(C12)

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 entropy. This problem has been studied in detail in [112, 113, 122]. It has also been studied in [16] within the framework of special relativity.

1.2 Canonical ensemble

In the canonical ensemble, the statistical equilibrium state is obtained by minimizing the Boltzmann 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

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

(C13)

where $\alpha (r)$ is a local (position dependent) Lagrange multiplier. Since $\beta$ is constant in the canonical ensemble, this is equivalent to the condition (C1) yielding the distribution function (C2). 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 Appendix C 1, except that T is fixed while it was previously determined by the conservation of energy.

We can now simplify the expression of the free energy. The entropy is given by Eq. (C9) and the energy by Eq. (C7). Since $F=E-TS$, we obtain

$$\begin{aligned} F= & {} W[n(r)]-Nk_\mathrm{B} T+k_\mathrm{B} T\int n(r)\ln n(r)\, 4\pi r^2\, \hbox {d}r+Nk_\mathrm{B}T\ln \left( \frac{h^3}{g}\right) \nonumber \\&-\frac{3}{2}Nk_\mathrm{B}T\ln (2\pi m k_\mathrm{B} T). \end{aligned}$$

(C14)

Therefore, the free energy can be written as a functional of the density as

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

(C15)

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

$$\begin{aligned} U[n(r)]= & {} -Nk_\mathrm{B} T+k_\mathrm{B} T\int n(r)\ln n(r)\, 4\pi r^2\, \hbox {d}r+Nk_\mathrm{B}T\ln \left( \frac{h^3}{g}\right) \nonumber \\&-\frac{3}{2}Nk_\mathrm{B}T\ln (2\pi m k_\mathrm{B} T). \end{aligned}$$

(C16)

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

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

(C17)

Decomposing the Massieu function as $J[f]=S[f]/k_\mathrm{B}-\beta E_{\mathrm{kin}}[f]-\beta W[n]$ and proceeding as in Appendix F2 of Paper I, or performing the variations over $\delta n$ directly on the explicit expression (C14) of the free energy, we get

$$\begin{aligned} \alpha (r)=\alpha _0-\beta m\varPhi (r)\qquad \mathrm{and}\qquad n(r)=\frac{g}{h^3}( 2\pi m k_\mathrm{B} T)^{3/2}\hbox {e}^{\alpha _0-\beta m\varPhi (r)}. \end{aligned}$$

(C18)

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 variation of free energy. This problem has been studied in detail in [120, 122]. It has also been studied in [16] within the framework of special relativity.

Remark

We note that the free energy $F[\rho ]$ coincides with the energy functional associated with the Euler–Poisson equations describing a gas with an isothermal equation of state $P=\rho k_\mathrm{B} T/m$ (see Sec. V.A.1 of Paper I). As a result, the thermodynamical stability of a classical self-gravitating system in the canonical ensemble is equivalent to the dynamical stability of the corresponding isothermal gas described by the Euler–Poisson equations [120]. This is a particular case of the general result established in [139] which is valid for an arbitrary form of entropy. According to the Poincaré turning point criterion, the series of equilibria 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).