# Density of States for the Unitary Fermi Gas and the Schwarzschild Black Hole

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## Abstract

**:**

## 1. Introduction

## 2. General Properties of the Density of States

## 3. Unitary Fermi Gas

#### 3.1. Attempt of Direct Evaluation of the Many-Body Density of States

#### 3.2. Canonical Ensemble

#### 3.3. Numerical Calculation of the Many-Body Density of States

## 4. Schwarzschild Black Hole

## 5. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Huang, K. Statistical Mechanics; Wiley: Hoboken, NJ, USA, 2008. [Google Scholar]
- Landau, L.D.; Lifshitz, E.M.; Pitaevskii, L.P. Course of Theoretical Physics, Vol. 9, Statistical Physics: Theory of the Condensed State; Butterworth-Heineman: Oxford, UK, 1980. [Google Scholar]
- Zwerger, W. (Ed.) The BCS-BEC Crossover and the Unitary Fermi Gas; Springer: Berlin/Heidelberg, Germany, 2011. [Google Scholar]
- Son, D.T.; Wingate, M. General coordinate invariance and conformal invariance in nonrelativistic physics: Unitary Fermi gas. Ann. Phys.
**2006**, 321, 197. [Google Scholar] [CrossRef][Green Version] - Sachdev, S. Statistical mechanics of strange metals and black hole. ICTS News
**2022**, 8, 1–7. [Google Scholar] - Cercignani, C. Ludwig Boltzmann: The Man Who Trusted Atoms; Oxford University Press: Oxford, UK, 1988. [Google Scholar]
- Regal, C.A.; Greiner, M.; Jin, D.S. Observation of Resonance Condensation of Fermionic Atom Pairs. Phys. Rev. Lett.
**2004**, 92, 040403. [Google Scholar] [CrossRef] [PubMed][Green Version] - Zwierlein, M.W.; Stan, C.A.; Schunck, C.H.; Raupach, S.M.F.; Kerman, A.J.; Ketterle, W. Condensation of pairs of fermionic atoms near a Feshbach resonance. Phys. Rev. Lett.
**2004**, 92, 120403. [Google Scholar] [CrossRef][Green Version] - Kinast, J.; Hemmer, S.L.; Gehm, M.E.; Turlapov, A.; Thomas, J.E. Evidence for Superfluidity in a Resonantly Interacting Fermi Gas. Phys. Rev. Lett.
**2004**, 92, 150402. [Google Scholar] [CrossRef][Green Version] - Giorgini, S.; Pitaevskii, L.P.; Stringari, S. Theory of ultracold atomic Fermi gases. Rev. Mod. Phys.
**2008**, 80, 1215. [Google Scholar] [CrossRef][Green Version] - Bulgac, A.; Drut, J.E.; Magierski, P. Spin 1/2 Fermions in the Unitary Regime: A Superfluid of a New Type. Phys. Rev. Lett.
**2006**, 96, 090404. [Google Scholar] [CrossRef][Green Version] - Salasnich, L. Low-temperature thermodynamics of the unitary Fermi gas: Superfluid fraction, first sound, and second sound. Phys. Rev. A
**2010**, 82, 063619. [Google Scholar] [CrossRef][Green Version] - Bighin, G.; Cappellaro, A.; Salasnich, L. Unitary Fermi superfluid near the critical temperature: Thermodynamics and sound modes from elementary excitations. Phys. Rev. A
**2022**, 105, 063329. [Google Scholar] [CrossRef] - Magierski, P.; Wlazlowski, G.; Bulgac, A.; Drut, J.E. Finite-Temperature Pairing Gap of a Unitary Fermi Gas by Quantum Monte Carlo Calculations. Phys. Rev. Lett.
**2009**, 103, 210403. [Google Scholar] [CrossRef][Green Version] - Carlson, J.; Reddy, S. Asymmetric Two-Component Fermion Systems in Strong Coupling. Phys. Rev. Lett.
**2005**, 95, 060401. [Google Scholar] [CrossRef] [PubMed][Green Version] - Salasnich, L.; Toigo, F. Extended Thomas-Fermi density functional for the unitary Fermi gas. Phys. Rev. A
**2010**, 78, 053626. [Google Scholar] [CrossRef][Green Version] - Bighin, G.; Salasnich, L.; Marchetti, P.A.; Toigo, F. Beliaev damping of the Goldstone mode in atomic Fermi superfluids. Phys. Rev. A
**2015**, 92, 023638. [Google Scholar] [CrossRef][Green Version] - Tempere, J.; Devreese, J.P. Superconductors: Materials, Properties and Applications; InTech: London, UK, 2012; Volume 383. [Google Scholar]
- Bulgac, A.; Drut, J.E.; Magierski, P. Quantum Monte Carlo simulations of the BCS-BEC crossover at finite temperature. Phys. Rev. A
**2008**, 78, 023625. [Google Scholar] [CrossRef][Green Version] - Horikoshi, M.; Nakajima, S.; Ueda, M.; Mukaiyama, T. Measurement of universal thermodynamic functions for a unitary Fermi gas. Science
**2010**, 327, 442–445. [Google Scholar] [CrossRef][Green Version] - Calmet, X. (Ed.) Quantum Aspects of Black Holes; Springer: Berlin/Heidelberg, Germany, 2014. [Google Scholar]
- Bekenstein, A. Black Holes and the Second Law. Lett. Nuovo Cim.
**1972**, 4, 99. [Google Scholar] [CrossRef] - Hawking, S.W. Particle creation by black holes. Comm. Math. Phys.
**1975**, 43, 199–220. [Google Scholar] - Ha, Y.K. The Gravitational Energy of a Black Hole. Gen. Rel. Grav.
**2003**, 35, 2045. [Google Scholar] [CrossRef] - Hawking, S.W. Black hole explosions? Nature
**1974**, 248, 5443. [Google Scholar] [CrossRef] - Gibbons, G.W.; Hawking, S.W. Action integrals and partition functions in quantum gravity. Phys. Rev. D
**1977**, 15, 2752. [Google Scholar] [CrossRef] - Carroll, S.M. Spacetime and Geometry: An Introduction to General Relativity; Addison-Wesley: Boston, MA, USA, 2004. [Google Scholar]
- Hilbert, D. Die Grundlagen der Physik. In Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen-Mathematisch-Physikalische Klasse; Vandenhoeck & Ruprecht: Göttingen, Germany, 1915; Volume 3, pp. 395–408. [Google Scholar]
- Schwarzschild, K. Sitzungsberichte der Koniglich Preussischen Akademie der Wissenschaften zu Berlin. Phys.-Math. Klasse
**1916**, 189. [Google Scholar]

**Figure 1.**Unitary Fermi gas: Scaled free energy $F\left(T\right)/\left(N{\u03f5}_{F}\right)$, scaled entropy $S\left(T\right)/\left(N{k}_{B}\right)$, and scaled internal energy $E\left(T\right)/\left(N{\u03f5}_{F}\right)$ deduced from our model, as a function of the scaled temperature $T/{T}_{F}$ with ${T}_{F}={\u03f5}_{F}/{k}_{B}$ the Fermi temperature.

**Figure 3.**Unitary Fermi gas: The dashed line is the scaled entropy $S\left(E\right)/\left(N{k}_{B}\right)$, as a function of the scaled internal energy $E/\left(N{\u03f5}_{F}\right)$. The solid line is the adimensional many-body density of states $W\left(E\right)$, as a function of the scaled internal energy $E/\left(N{\u03f5}_{F}\right)$.

**Figure 4.**Unitary Fermi gas. Upper panel: The dashed line is the scaled entropy ${S}_{col}\left({E}_{col}\right)/\left(N{k}_{B}\right)$ of bosonic collective elementary excitations, as a function of the scaled internal energy ${E}_{col}/\left(N{\u03f5}_{F}\right)$ of the collective elementary excitations. The solid line is the adimensional density of states ${W}_{col}\left({E}_{col}\right)$ of collective elementary excitations, as a function of the scaled internal energy ${E}_{col}/\left(N{\u03f5}_{F}\right)$ of collective elementary excitations. Lower panel: The dashed line is the scaled entropy ${S}_{sp}\left({E}_{sp}\right)/\left(N{k}_{B}\right)$ of fermionic single-particle excitations, as a function of the scaled internal energy ${E}_{sp}/\left(N{\u03f5}_{F}\right)$ of single-particle elementary excitations. The solid line is the adimensional density of states ${W}_{sp}\left({E}_{sp}\right)$ of single-particle excitations, as a function of the scaled internal energy ${E}_{sp}/\left(N{\u03f5}_{F}\right)$ of single-particle excitations.

**Figure 5.**Schwarzschild black hole: The dashed line is the scaled entropy $S\left(E\right)/{k}_{B}$, as a function of the scaled internal energy $E/{E}_{P}$, with ${E}_{P}=\sqrt{\hslash {c}^{5}/G}$ the Planck energy. The solid line is the adimensional density of states $W\left(E\right)$, as a function of the scaled internal energy $E/{E}_{P}$.

**Figure 6.**Schwarzschild black hole: Scaled free energy $F\left(T\right)/{E}_{P}$, scaled entropy $S\left(T\right){k}_{B}$, and scaled internal energy $E/{E}_{P}$ as a function of the scaled temperature $T/{T}_{P}$ with ${T}_{P}={E}_{P}/{k}_{B}$ being the Planck temperature and ${E}_{P}=\sqrt{\hslash {c}^{5}/G}$ being the Planck energy.

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**MDPI and ACS Style**

Salasnich, L.
Density of States for the Unitary Fermi Gas and the Schwarzschild Black Hole. *Symmetry* **2023**, *15*, 350.
https://doi.org/10.3390/sym15020350

**AMA Style**

Salasnich L.
Density of States for the Unitary Fermi Gas and the Schwarzschild Black Hole. *Symmetry*. 2023; 15(2):350.
https://doi.org/10.3390/sym15020350

**Chicago/Turabian Style**

Salasnich, Luca.
2023. "Density of States for the Unitary Fermi Gas and the Schwarzschild Black Hole" *Symmetry* 15, no. 2: 350.
https://doi.org/10.3390/sym15020350