Caloric curves of self-gravitating fermions in general relativity

The thermodynamics of a nonrelativistic classical self-gravitating gas has been studied in detail in antonov ; lbw ; katzpoincare1 ; paddyapj ; aa . The caloric curve $\eta(\Lambda)$ forms a spiral (see Fig. 1). In the microcanonical ensemble, there is no equilibrium state below a critical energy $E_{c}$ corresponding to $\Lambda_{c}=0.335$. In that case, the system undergoes a gravothermal catastrophe (core collapse) leading to a binary star surrounded by a hot halo lbe ; inagaki ; cohn . In the canonical ensemble, there is no equilibrium state below a critical temperature $T_{c}$, corresponding to $\eta_{c}=2.52$. In that case, the system undergoes an isothermal collapse leading to a Dirac peak containing all the mass post .

The thermodynamics of the nonrelativistic self-gravitating Fermi gas has been studied in detail in ijmpb . It is shown that the caloric curves $\eta(\Lambda)$ depend on a single control parameter (it should not be confused with the chemical potential):

It can be written as ijmpb :

or as

where $R_{0}$ (resp. $M_{0}$) is the radius (resp. mass) of a fermion star of mass $M$ (resp. radius $R$) at $T=0$ (see Appendix F). Introducing the normalized variables of Appendix B, this parameter becomes

Some caloric curves are represented in Fig. 2. They display a canonical critical point at $\mu_{\rm CCP}=83$ and a microcanonical critical point at $\mu_{\rm MCP}=2670$. When $\mu<\mu_{\rm CCP}=83$ there is no phase transition. When $\mu_{\rm CCP}=83<\mu<\mu_{\rm MCP}=2670$ the system displays zeroth and first order canonical phase transitions. When $\mu>\mu_{\rm MCP}=2670$ the system displays zeroth and first order canonical and microcanonical phase transitions. When $\mu\rightarrow+\infty$ we recover the caloric curve of the nonrelativistic classical self-gravitating gas (spiral) represented in Fig. 1. When $\mu<+\infty$ there is a statistical equilibrium state for any accessible value of energy and temperature. The gravitational collapse of the nonrelativistic classical self-gravitating gas (gravothermal catastrophe in the microcanonical ensemble and isothermal collapse in the canonical ensemble) is prevented by quantum mechanics (Pauli’s exclusion principle).

For a given box radius, the nonrelativistic canonical phase transition appears when

If we consider the general relativistic problem, we must require $N\ll N_{\rm OV}$, where $N_{\rm OV}=0.39853$ is the OV limit, for the validity of the nonrelativistic treatment. Therefore, we will see the nonrelativistic canonical phase transition for $N_{\rm CCP}(R)<N\ll N_{\rm OV}$ provided that

In comparison $R_{\rm OV}=3.3569$. This argument just provides an order of magnitude of the radius $R_{\rm CCP}$ above which a canonical phase transition appears for $N>N_{\rm CCP}(R)$. By solving the general relativistic equations, we find that the exact value is $R_{\rm CCP}^{\rm exact}=12.0$ (see Sec. X).

For a given box radius, the nonrelativistic microcanonical phase transition appears when

If we consider the general relativistic problem, using the same argument as before, we will see the nonrelativistic microcanonical phase transition for $N_{\rm MCP}(R)<N\ll N_{\rm OV}$ provided that

This argument just provides an order of magnitude of the radius $R_{\rm MCP}$ above which a microcanonical phase transition appear for $N>N_{\rm MCP}(R)$. By solving the general relativistic equations, we find that the exact value is $R_{\rm MCP}^{\rm exact}=92.0$ (see Sec. X).

The thermodynamics of a classical self-gravitating gas in general relativity has been studied in detail in Refs. roupas and paper2 . This corresponds to the nondegenerate limit of the general relativistic Fermi gas. It is shown that the caloric curves $\eta(\Lambda)$ depend on a single control parameter

It can be written as

or as

where $R^{*}_{S}$ can be interpreted as a sort of Schwarzschild radius defined with the rest mass $Nm$ instead of the mass $M$ (reciprocally, $N^{*}_{S}m$ is a sort of Schwarzschild rest mass of an object of radius $R$). Introducing the normalized variables of Appendix B this parameter becomes

Some caloric curves are represented in Fig. 3. When $\nu\rightarrow 0$ ($N\ll N_{S}^{*}$ or $R\gg R_{S}^{*}$), we recover the caloric curve of the nonrelativistic classical self-gravitating gas (spiral) represented in Fig. 1. When $0<\nu<\nu^{\prime}_{S}=0.128$ the caloric curve has the form of a double spiral exhibiting a collapse at low energies and low temperatures (cold spiral) and at high energies and high temperatures (hot spiral).¹⁶¹⁶16The hot spiral corresponds to an ultrarelativistic classical gas roupas which is similar to a form of radiation described by an equation of state $P=\epsilon/3$ sorkin ; aarelat1 ; aarelat2 (see paper2 for a detailed discussion). When $\nu^{\prime}_{S}=0.128<\nu<\nu_{S}=0.1415$ the two spirals are amputed (truncated) and touch each other. When $\nu_{S}=0.1415<\nu<\nu_{\rm max}=0.1764$ the two spirals disappear and the caloric curve makes a loop resembling to the symbol “$\infty$”. As $\nu$ increases, the loop shrinks more and more and, when $\nu=\nu_{\rm max}=0.1764$, it reduces to a point located at $(\Lambda_{*},\eta_{*})=(-0.9829,1.2203)$. When $\nu>\nu_{\rm max}=0.1764$, no equilibrium state is possible.

For a given box radius, the spirals touch each other when

and they form a loop when

The caloric curve reduces to a point when

If we consider the truly quantum problem, we must require $N\gg N_{\rm OV}=0.39853$ for the validity of the classical (nondegenerate) treatment. Therefore, we will see the double spiral and its evolution described previously for $N_{\rm OV}\ll N<N_{\rm max}$ provided that

We note that $N_{\rm OV}/\nu_{\rm max}=2.259$ is of the order of $R_{\rm OV}=3.3569$.

Remark: For a given box radius $R$, coming back to dimensional variables, equilibrium states exist only when $N\leq N_{\rm max}(R)=0.1764\,Rc^{2}/Gm=0.3528\,N_{S}^{*}$. Inversely, for a given number of particles $N$, equilibrium states exist only when $R\geq R_{\rm min}(N)=5.669\,GNm/c^{2}=2.834\,R^{*}_{S}$. The nonrelativistic limit corresponds to $N\ll N_{\rm max}(R)\sim Rc^{2}/2Gm\sim N_{S}^{*}$ or $R\gg R_{\rm min}(N)\sim 2GNm/c^{2}\sim R^{*}_{S}$. These results are valid in the classical limit. For small systems, quantum effects will come into play. If we argue that $N_{\rm max}=\nu_{\rm max}Rc^{2}/Gm\sim N_{\rm OV}$ when $R\sim R_{\rm OV}$, or equivalently $R_{\rm min}=GNm/\nu_{\rm max}c^{2}\sim R_{\rm OV}$ when $N\sim N_{\rm OV}$, we find that $\nu_{\rm max}\sim GN_{\rm OV}m/R_{\rm OV}c^{2}\sim 0.1187$. This may justify the order of magnitude of this constant. Alternatively, we may just remark that $\nu_{\rm max}=GN_{\rm max}m/Rc^{2}=0.1764$ is of the same order as $GN_{\rm OV}m/R_{\rm OV}c^{2}=0.1187$.

Before treating the general case, let us summarize the previous results.

Nonrelativistic $+$ classical limit. For a given box radius $R$ and particle number $N$ the system undergoes a catastrophic collapse towards a singularity at low temperatures in the canonical ensemble and at low energies in the microcanonical ensemble.

Nonrelativistic limit. For a given box radius $R$ there is no phase transition when $N<N_{\rm CCP}(R)$, the system can undergo a canonical phase transition when $N_{\rm CCP}(R)<N<N_{\rm MCP}(R)$, and the system can undergo a canonical and a microcanonical phase transition when $N>N_{\rm MCP}(R)$. For a given particle number $N$, there is no phase transition when $R<R_{\rm CCP}(N)$, the system can undergo a canonical phase transition when $R_{\rm CCP}(N)<R<R_{\rm MCP}(N)$, and the system can undergo a canonical and a microcanonical phase transition when $R>R_{\rm MCP}(N)$. Here, $R_{\rm CCP}(N)=12.9/N^{1/3}$ and $R_{\rm MCP}(N)=130/N^{1/3}$ are the reciprocal of $N_{\rm CCP}(R)=2125/R^{3}$ and $N_{\rm MCP}(R)=2.20\times 10^{6}/R^{3}$. There is an equilibrium state at all temperatures $T\geq 0$ in the canonical ensemble and at all accessible energies $E\geq E_{\rm min}$ (where $E_{\rm min}$ is the energy of the ground state) in the microcanonical ensemble.

Classical limit. For a given box radius $R$, the caloric curve has the form of a double spiral when $N<N_{\rm S}^{\prime}(R)$, the spirals touch each other when $N_{\rm S}^{\prime}(R)<N<N_{\rm S}(R)$, the caloric curve makes a loop when $N_{\rm S}(R)<N<N_{\rm max}(R)$, and there is no equilibrium state when $N>N_{\rm max}(R)$. For a given particle number $N$, the caloric curve has the form of a double spiral when $R>R_{\rm S}^{\prime}(N)$, the spirals touch each other when $R_{\rm S}(N)<R<R^{\prime}_{\rm S}(N)$, the caloric curve makes a loop when $R_{\rm min}(N)<R<R_{\rm S}(N)$, and there is no equilibrium state when $R<R_{\rm min}(N)$. Here, $R_{\rm S}^{\prime}(N)=7.81\,N$, $R_{\rm S}(N)=7.07\,N$ and $R_{\rm min}(N)=5.67\,N$ are the reciprocal of $N_{\rm S}^{\prime}(R)=0.128\,R$, $N_{\rm S}(R)=0.1415\,R$ and $N_{\rm max}(R)=0.1764\,R$. The system undergoes a catastrophic collapse towards a singularity at both low and high temperatures in the canonical ensemble and at both low and high energies in the microcanonical ensemble.

In Fig. 4 we have plotted the caloric curve for $N<N_{\rm CCP}=0.0170$. Since $N\ll N_{\rm OV}=0.39853$, this caloric curve coincides with the one obtained in the nonrelativistic limit ijmpb except at very high energies and very high temperatures (see the Remark at the end of this section).¹⁷¹⁷17As discussed in Sec. XI the nonrelativistic limit corresponds to $N\rightarrow 0$ and $R\rightarrow+\infty$ in such a way that $NR^{3}$ is fixed (in more physical terms $N\ll N_{\rm OV}$ and $R\gg R_{\rm OV}$ with $NR^{3}$ fixed).

The series of equilibria $\eta(\Lambda)$ is monotonic. According to the Poincaré theory of linear series of equilibria, all the equilibrium states are stable. The statistical ensembles (microcanonical and canonical) are equivalent. The caloric curve presents the following features:

(i) There is no phase transition and no gravitational collapse.

(ii) The specific heat is always positive. The entropy versus energy curve (not represented) is concave.

The evolution of the system is the following. At high energies and high temperatures, the system is nondegenerate (Boltzmannian). As the energy and the temperature are reduced, the system becomes more and more centrally condensed. At intermediate energies and intermediate temperatures, the Fermi gas is partially degenerate (see Appendix D). At $T=0$, the Fermi gas is completely degenerate. This cold nonrelativistic fermion ball, equivalent to a polytrope of index $n=3/2$, is similar to a nonrelativistic white dwarf. This is the state of minimum energy $E_{\rm min}$ (ground state). Since there is a stable equilibrium state at $T=0$ (i.e. $\eta\rightarrow+\infty$) with a finite energy $E_{\rm min}$, the caloric curve $\eta(\Lambda)$ presents a vertical asymptote at $\Lambda=\Lambda_{\rm max}$.¹⁸¹⁸18In the nonrelativistic limit $\Lambda_{\rm max}=0.0950N^{1/3}R$ (see Appendix F). More generally, a complete characterization of the ground state of the self-gravitating Fermi gas, in the nonrelativistic and relativistic regimes, taking into account the presence of the box is given in paper1 .

Remark: At very high energies and very high temperatures, the system is relativistic even though $N\ll N_{\rm OV}$. In that case, we recover the hot spiral studied in roupas ; paper2 . As a result, the complete caloric curve of the general relativistic Fermi gas presents a region of negative specific heat and a region of ensemble inequivalence at very high energies and very high temperatures. The system undergoes a gravitational collapse above $E_{\rm max}$ in the microcanonocal ensemble and above $T_{\rm max}$ in the canonical ensemble. We note that quantum mechanics cannot prevent such a gravitational collapse since it takes place at very high energies and very high temperatures where the system is nondegenerate. As a result, the system is expected to collapse towards a black hole. For $N\rightarrow 0$, it is shown in paper2 that $\Lambda_{\rm min}\sim-0.246/N^{2}\rightarrow-\infty$ and $\eta_{\min}\sim 18.3N^{2}\rightarrow 0$ so that the hot spiral is rejected at infinity.¹⁹¹⁹19In terms of dimensional variables this correponds to $E_{\rm max}\rightarrow 0.24631Rc^{4}/G$ and $k_{B}T_{\rm max}\sim 0.0547Rc^{4}/NG$. For small values of $N$ ($N\ll N_{\rm max}$) the hot spiral occurs at very negative values of $\Lambda$ and at very small values of $\eta$. This is why we do not see it in Fig. 4 (it is outside of the frame since $\Lambda_{\rm min}\simeq-1708$ and $\eta_{\min}\simeq 2.63\times 10^{-3}$). The hot spiral becomes visible only for larger values of $N$ ($N\lesssim N_{\rm max}$) as in Fig. 22 below. In this paper, we shall not discuss the hot spiral specifically since it has been described in detail in roupas ; paper2 .

In Fig. 5 we have plotted the caloric curve for $N_{\rm CCP}=0.0170<N<N_{1}=0.18131$. Since $N\ll N_{\rm OV}=0.39853$, the caloric curve coincides with the one obtained in the nonrelativistic limit ijmpb . The novely with respect to the previous case is that the caloric curve has a $N$-shape structure leading to canonical phase transitions and ensembles inequivalence. This $N$-shape structure appears at $N=N_{\rm CCP}=0.0170$ where the caloric curve presents a horizontal inflexion point. Let us consider the microcanonical and canonical ensembles successively (see ijmpb for a more detailed discussion).

In Fig. 7 we have plotted the caloric curve for $N_{1}=0.18131<N<N_{\rm OV}=0.39853$. The novelty with respect to the previous case is the existence of a secondary branch presenting an asymptote at $\Lambda^{\prime}_{\rm max}$. This secondary branch appears suddently at $N=N_{1}=0.18131$ (at that point $\Lambda^{\prime}_{\rm max}=-0.536\,R$ and $\Lambda_{\rm max}=0.0570\,R$). As detailed in paper1 , for $N_{1}<N<N_{\rm OV}$, there exists another equilibrium state at $T=0$ (i.e. $\eta\rightarrow+\infty$) corresponding to a completely degenerate fermion ball distinct from the ground state. This secondary equilibrium state is unstable.²⁰²⁰20Actually, for $N>N_{1}$, there can exist several unstable equilibrium states at $T=0$ (up to an infinity) that have more and more modes of instability. They are related to the spiral structure of the mass-radius relation of the general relativistic Fermi gas at $T=0$ shapiroteukolsky ; paper1 . They give rise to additional branches (with vertical asymptotes) in the caloric curve. We shall not consider these unstable solutions here, except for the less unstable one already mentioned. Its mass is larger than the mass of the stable ground state so that $\Lambda^{\prime}_{\rm max}\leq\Lambda_{\rm max}$. According to the Poincaré theory, all the configurations of the secondary branch are unstable.²¹²¹21The spiral present on the left of this secondary branch will ultimately become the cold spiral of Refs. roupas ; paper2 when $N$ will be sufficiently large (see below). Therefore, the presence of this secondary branch does not qualitatively change the description of the caloric curve made in Sec. V.2.

However, for $N>N_{1}$, relativistic effects start to become important. This has some consequences on the interpretation of the density profiles. In Fig. 8 we have plotted the different density profiles at $\eta_{t}$. We see that the energy density is low for the gaseous state (G) and for the core-halo state (CH) indicating that we are in the nonrelativistic regime. By contrast, the energy density is relatively high for the stable condensed state (C) and for the unstable condensed state (U) indicating that we are in the relativistic regime. The condensed states almost coincide with a general relativistic fermion ball at $T=0$ containing all the mass (see Appendix E.2.1). They are similar to stable and unstable neutron stars ov . At small but finite temperatures, we see in Fig. 8 that the dashed line obtained from the OV theory provides a good fit to the core of the distribution. There is a small atmosphere (containing a little mass) that becomes thinner and thinner as the temperature is reduced.

Remark: In Fig. 7, when the temperature is low enough, we find four solutions. The solutions (G) and (C) are stable (local minima of free energy) while the solutions (CH) and (U) are unstable (saddle points of free energy). Since we have an even number of extrema, this suggests that there is no global minimum of free energy (naively, this results from simple topological arguments if we plot a curve $f(x)$ with two minima and two maxima). The stable equilibrium state with the lowest value of free energy may be only metastable, not fully stable. This is consistent with the result of Zel’dovich zel446 who showed that, at $T=0$, the OV equilibrium states are only metastable. In Fig. 5, when $\eta_{*}<\eta<\eta_{c}$, we find three solutions. The solutions (G) and (C) are stable (local minima of free energy) while the solution (CH) is unstable (saddle point of free energy). Since we have an odd number of extrema, this suggests that the solution with the lowest value of free energy is a global minimum. This is the case in Newtonian gravity ijmpb . However, this is not quite clear in general relativity since the result of Zel’dovich zel446 still applies for $N<N_{1}$. Therefore, the existence of a global minimum of free energy (fully stable state) in general relativity is not trivial and would require a more careful study. Anyway, for practical purposes, metastable states are very relevant (possibly more relevant than fully stable states) so we shall determine all types of stable equilibrium states, disregarding whether they are fully stable or just metastable.

In Fig. 9 we have plotted the caloric curve for $N_{\rm OV}=0.39853<N<N_{e}=0.40002$. The novelty with respect to the previous case is that the two branches have merged. The merging occurs at $N=N_{\rm OV}$ at which the two asymptotes $\Lambda^{\prime}_{\rm max}$ and $\Lambda_{\rm max}$ coincide (at that point $\Lambda_{\rm max}=\Lambda^{\prime}_{\rm max}=0.08985\,R$). This is the highest value of $N$ at which there exist an equilibrium state at $T=0$ (ground state). When $N>N_{\rm OV}$ there is no equilibrium state at $T=0$ (no ground state) anymore ov . In that case, the caloric curve presents a turning point of temperature at $\eta^{\prime}_{c}$ and a turning point of energy at $\Lambda^{\prime}_{c}$. As a result, there is no equilibrium state at $\eta>\eta^{\prime}_{c}$ in the canonical ensemble, i.e., below a critical temperature. Similarly, there is no equilibrium state at $\Lambda>\Lambda^{\prime}_{c}$ in the microcanonical ensemble, i.e., below a critical energy. This means that when the system becomes strongly relativistic (i.e. when $N>N_{\rm OV}$) quantum mechanics is not able to prevent gravitational collapse at low temperatures and low energies. This is a generalization of the result first obtained at $T=0$ by Oppenheimer and Volkoff ov in the context of neutron stars.

In Fig. 16 we have plotted the caloric curve for $N_{e}=0.40002<N<N^{\prime}_{e}=0.40469$. The novelty with respect to the previous case is that now $\eta^{\prime}_{c}$ is smaller than $\eta_{c}$ (they become equal when $N=N_{e}=0.40002$).

The description in the microcanonical ensemble is the same as before.

In the canonical ensemble, the caloric curve presents the following features:

(i) When $\eta<\eta_{*}$ and when $\eta^{\prime}_{c}<\eta<\eta_{c}$ there are only gaseous states. When $\eta_{*}<\eta<\eta^{\prime}_{c}$ there exist gaseous and condensed states at the same temperature. A first order phase transition is expected at a transition temperature $\eta_{t}$ determined by the Maxwell construction (see Fig. 16) or by the equality of the free energy of the two phases (see Fig. 17). When $\eta_{*}<\eta<\eta_{t}$ the gaseous states have a lower free energy than the condensed states. When $\eta_{t}<\eta<\eta^{\prime}_{c}$ the condensed states have a lower free energy than the gaseous states. However, the first order phase transition does not take place in practice because of the very long lifetime of the metastable states.

(ii) There is a catastrophic collapse at $\eta_{c}$ from the gaseous phase to a black hole.

(iii) There is a catastrophic collapse at $\eta^{\prime}_{c}$ from the condensed phase to a black hole.

(iv) There is a zeroth order phase transition at $\eta_{*}$ from the condensed phase to the gaseous phase. It correspond to an explosion ultimately halted by the boundary of the box.

The evolution of the system in the canonical ensemble is the following. Let us start from high temperature states and decrease the temperature. At high temperatures, the system is in the gaseous phase. At $\eta=\eta_{t}$ the system is expected to undergo a first order phase transition from the gaseous phase to the condensed phase. However, this phase transition does not take place in practice. At $\eta=\eta_{c}$ the system undergoes a catastrophic collapse towards a black hole. A condensed phase exists for $\eta_{*}<\eta<\eta^{\prime}_{c}$ but it is not clear how it can be reached in practice.

In Fig. 18 we have plotted the caloric curve for $N^{\prime}_{e}=0.40469<N<N_{*}=0.41637$, where $N^{\prime}_{e}$ is defined such that $\eta^{\prime}_{c}=\eta_{t}$.

The description in the microcanonical ensemble is the same as before.

In the canonical ensemble, the caloric curve presents the following features:

(i) When $\eta<\eta_{*}$ and when $\eta^{\prime}_{c}<\eta<\eta_{c}$ there are only gaseous states. When $\eta_{*}<\eta<\eta^{\prime}_{c}$ there exist gaseous and condensed states at the same temperature. However, there is no first order phase transition, even in theory, because we cannot satisfy the Maxwell construction (see Fig. 18) or the equality of the free energy of the gaseous and condensed phases (see Fig. 19). When $\eta_{*}<\eta<\eta^{\prime}_{c}$ the gaseous states always have a lower free energy than the condensed states (see Fig. 19). Therefore, although there are several stable equilibrium states when $\eta_{*}<\eta<\eta^{\prime}_{c}$ there is no phase transition from one phase to the other. This is a particularity of the relativistic situation.

(ii) There is a catastrophic collapse at $\eta_{c}$ from the gaseous phase to a black hole.

(iii) There is a catastrophic collapse at $\eta^{\prime}_{c}$ from the condensed phase to a black hole.

(iv) There is a zeroth order phase transition at $\eta_{*}$ from the condensed phase to the gaseous phase. It corresponds to an explosion ultimately halted by the boundary of the box.

The evolution of the system is the same as described previously.

In Fig. 20 we have plotted the caloric curve for $N>N_{*}=0.41637$, where $N_{*}$ is defined such that $\eta^{\prime}_{c}=\eta_{*}$. From that moment, we denote the minimum energy by $\Lambda_{c}$ instead of $\Lambda^{\prime}_{c}$.

In Figs. 21 and 22 we have plotted the caloric curves for larger values of $N$. When $N\gg N_{\rm OV}=0.39853$, the system is nondegenerate and we recover the results of roupas ; paper2 for a classical general relativistic gas described by the Boltzmann distribution.²³²³23As discussed in Sec. XI the classical limit corresponds to $N\rightarrow+\infty$ and $R\rightarrow+\infty$ in such a way that $N/R$ is fixed (in more physical terms $N\gg N_{\rm OV}$ and $R\gg R_{\rm OV}$ with $N/R$ fixed). The caloric curve exhibits a double spiral. When $N<N^{\prime}_{S}=6.40$ (see Fig. 7 of paper2 ) the two spirals are separated. When $N^{\prime}_{S}=6.40<N<N_{S}=7.08$ (see Fig. 8 of paper2 ) the two spirals are amputed (truncated) and touch each other. When $N_{S}=7.08<N<N_{\rm max}=8.821$ (see Fig. 9 of paper2 ) the spirals disappear and the caloric curve makes a “loop”. When $N\rightarrow N_{\rm max}$, the caloric curve reduces to a “point” located at $(\Lambda_{*},\eta_{*})=(-0.9829,1.2203)$.

In Figs. 23 and 24 we have represented the canonical phase diagram corresponding to $R_{\rm CCP}=12.0<R<R_{\rm MCP}=92.0$ (specifically $R=50$). It shows the evolution of the critical temperatures $\eta_{\rm min}$, $\eta_{c}$, $\eta_{*}$, $\eta_{t}$, $\eta^{\prime}_{c}$ with $N$. We can clearly see the canonical critical point at $N_{\rm CCP}=0.0170$ at which the canonical phase transition appears. We also see the point $N_{\rm OV}=0.39853$ above which quantum mechanics is not able to prevent gravitational collapse above $\eta^{\prime}_{c}(N)$. Finally, we see the point $N_{\rm max}=8.821$ above which there is no equilibrium state anymore.

The nonrelativistic limit ijmpb corresponds to the dashed lines. It provides a very good approximation of $\eta_{c}$, $\eta_{*}$ and $\eta_{t}$ for $N\ll N_{\rm OV}$. As we approach $N_{\rm OV}$ general relativity must be taken into account.

The classical limit roupas ; paper2 corresponds to the dotted lines. It provides a very good approximation of $\eta_{\rm min}$ (hot spiral) for any $N$. It also provides a very good approximation of $\eta_{c}$ (cold spiral) for $N\gg N_{\rm OV}$. As we approach $N_{\rm OV}$ quantum mechanics must be taken into account.

In Figs. 25 and 26 we have represented the microcanonical phase diagram corresponding to $R_{\rm CCP}=12.0<R<R_{\rm MCP}=92.0$ (specifically $R=50$). It shows the evolution of the critical energies $\Lambda_{\rm min}$, $\Lambda_{\rm max}$, $\Lambda^{\prime}_{\rm max}$, $\Lambda_{\rm gas}$, $\Lambda_{\rm cond}$, $\Lambda^{\prime}_{c}$, $\Lambda_{c}$ with $N$. We can clearly see the canonical critical point at $N_{\rm CCP}=0.0170$ at which the region of negative specific heat (associated with the canonical phase transition) appears. We also see the point $N_{\rm OV}=0.39853$ above which quantum mechanics is not able to prevent gravitational collapse above $\Lambda^{\prime}_{c}(N)$, and the point $N_{\rm max}=8.821$ above which there is no equilibrium state anymore.

The nonrelativistic limit ijmpb corresponds to the dashed lines. It provides a very good approximation of $\Lambda_{\rm max}$, $\Lambda_{\rm gas}$ and $\Lambda_{\rm cond}$ for $N\ll N_{\rm OV}$. As we approach $N_{\rm OV}$ general relativity must be taken into account.

The classical limit roupas ; paper2 corresponds to the dotted lines. It provides a very good approximation of $\Lambda_{\rm min}$ (hot spiral) for any $N$. It also provides a very good approximation of $\Lambda_{c}$ (cold spiral) for $N\gg N_{\rm OV}$. As we approach $N_{\rm OV}$ quantum mechanics must be taken into account.