Statistical mechanics and thermodynamic limit of self-gravitating fermions in $D$ dimensions

We discuss the statistical mechanics of a system of self-gravitating fermions in a space of dimension $D$. We plot the caloric curves of the self-gravitating Fermi gas giving the temperature as a function of energy and investigate the nature of phase transitions as a function of the dimension of space. We consider stable states (global entropy maxima) as well as metastable states (local entropy maxima). We show that for $D⩾4$, there exists a critical temperature (for sufficiently large systems) and a critical energy below which the system cannot be found in statistical equilibrium. Therefore, for $D⩾4$, quantum mechanics cannot stabilize matter against gravitational collapse. This is similar to a result found by Ehrenfest (1917) at the atomic level for Coulomb forces. This makes the dimension $D=3$ of our Universe very particular with possible implications regarding the anthropic principle. Our study joins a long tradition of scientific and philosophical papers that examined how the dimension of space affects the laws of physics.

Figure 1

The mass-radius relation for complete white dwarf stars $\left(T=0\right)$ in different dimensions of space. It clearly shows that the dimension $D=3$ in surrounded by two critical dimensions $D=2$ and $D=4$ at which either the radius or the mass is constant.

Figure 2

The mass-energy relation for white dwarf stars $\left(T=0\right)$ in $D=3$. There exists an equilibrium state for all mass. The white dwarf star is self-confined if $M>M_{*}\left(R\right)$ and box-confined if $M<M_{*}\left(R\right)$.

Figure 3

Caloric curve in $D=3$ for different values of the degeneracy parameter (various system sizes).

Figure 4

Caloric curve in $D=3$ for small values of the degeneracy parameter (small system sizes). For $D<4$, there exists an equilibrium state for all temperatures $T$ and all accessible energies $E⩾E_{\text{ground}}$.

Figure 5

The mass-energy relation for white dwarf stars $\left(T=0\right)$ in $4<D<2(1+\sqrt{2})$ (specifically $D=4.1$). Self-confined white dwarf stars are always unstable. Box-confined white dwarf stars exist only for $M<M_c\left(R\right)$. For $M>M_c\left(R\right)$, there is no equilibrium state.

Figure 6

Caloric curve in $D=4.1$ for different values of the degeneracy parameter. For $\mu >{\mu }_c\left(D\right)$, there is no equilibrium state if the temperature and the energy are too low. The reason for the “gap” at $k_c$ is explained in Fig. 7.

Figure 7

Graphical construction determining the value of $\alpha$ for given $\mu$ and $k$ (in $D=4.1$). According to Eq. (25), the normalized box radius $\alpha$ is solution of $f\left(\alpha \right)=0$, where $f\left(\xi \right)={\xi }^{(D+2)∕(D-2)}{\psi }_k^{\prime }\left(\xi \right)-{\mu }^{2∕(D-2)}$. We see that $\alpha$ undergoes a discontinuity as $k\to k_c$. This gives rise to the “gap” in Fig. 6 for $\mu =23$. However, this gap is essentially a mathematical curiosity since the lower part of the curve (small $k$) is unstable anyway.

Figure 8

The mass-energy relation for white dwarf stars $\left(T=0\right)$ in $D=4$. Self-confined white dwarf stars are marginally stable. They have a unique mass $M_{\text{limit}}$ independent of their radius. For $M<M_{\text{limit}}$, the white dwarf star is box-confined. There is no equilibrium state with $M>M_{\text{limit}}$.

Figure 9

Caloric curves in $D=4$ for different values of the degeneracy parameter. For $\mu >{\mu }_c=22.4$, there is no equilibrium state if the temperature and the energy are too low.

Figure 10

Same as Fig. 9 for larger values of $\mu$ showing the developement of the classical spiral recovered for $\mu \to +\infty$.

Figure 11

The mass-energy relation for white dwarf stars $\left(T=0\right)$ in $D=2(1+\sqrt{2})$.

Figure 12

The mass-energy relation for white dwarf stars $\left(T=0\right)$ in $D=5.1$.

Figure 13

Caloric curve in $D=2$ for different values of the degeneracy parameter: $\mu =2,$ $10,100,$ {10}^3, ${10}^4,$ ${10}^5$, and ${10}^6$. For $\mu \to +\infty$, we recover the classical caloric curve displaying a critical temperature $T_c$. Below $T_c$, the system is expected to collapse and create a Dirac peak (“black hole”). When quantum mechanics is accounted for, the “black hole” is replaced by a “fermion ball.” This result is generally valid for $2⩽D<4$.

Figure 14

The mass-energy relation for white dwarf stars $\left(T=0\right)$ in $D<2$ (specifically $D=1$). There exists an equilibrium state for all mass. The white dwarf star is self-confined if $M<M_{*}\left(R\right)$ and box-confined if $M>M_{*}\left(R\right)$.

Figure 15

Caloric curve in $D=1$ for different values of the degeneracy parameter (various system sizes).