# 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

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**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