# A Note on Effects of Generalized and Extended Uncertainty Principles on Jüttner Gas

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## Abstract

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## 1. Introduction

## 2. The Maxwell–Boltzmann and Jüttner Distribution Functions

#### 2.1. Non-Relativistic Gas

#### 2.2. Relativistic Gas

## 3. Generalized Uncertainty Principle, Partition and Distribution Functions

#### 3.1. Maxwell–Boltzmann Statistics

#### 3.2. Jüttner Statistics

## 4. Extended Uncertainty Principle, Partition and Distribution Functions

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Maxwell–Boltzmann (MB) distribution versus energy for $\eta =0.5,1,1.5$. The ordinary MB distribution is denoted by the solid curve. Here, we have set the units so that $\hslash =c={K}_{B}=1$.

**Figure 2.**Behavior of Jüttner distribution against energy for $\eta =0.5,1,1.5$. The solid curve belongs to the ordinary Jüttner distribution and we have set the units so that $\hslash =c={K}_{B}=1$.

HUP | GUP | EUP | |
---|---|---|---|

The volume of phase space element | 1 | ${\left(1+\eta {P}^{2}\right)}^{3}$ | ${\left(1+\alpha {X}^{2}\right)}^{3}$ |

Density of States | $\sqrt{E}$ | $\frac{\sqrt{E}}{{\left(1+2m\eta E\right)}^{3}}$ | $\sqrt{E}$ |

Single Partition Function | $V{\left(\frac{2\pi m}{\beta}\right)}^{\frac{3}{2}}$ | $V{\left(\frac{2\pi m}{\beta}\right)}^{\frac{3}{2}}I\left(\frac{2m\eta}{\beta},3\right)$ | ${V}_{eff}\left(\alpha ,r\right){\left(\frac{2\pi m}{\beta}\right)}^{\frac{3}{2}}$ |

HUP | GUP | EUP | |
---|---|---|---|

The volume of phase space element | 1 | ${\left(1+\eta {P}^{2}\right)}^{3}$ | ${\left(1+\alpha {X}^{2}\right)}^{3}$ |

Density of States | $E\sqrt{{E}^{2}-{m}^{2}}$ | $\frac{E\sqrt{{E}^{2}-{m}^{2}}}{{\left(1+\eta \left({E}^{2}-{m}^{2}\right)\right)}^{3}}$ | $E\sqrt{{E}^{2}-{m}^{2}}$ |

Single Partition Function | $V{\left(\frac{2\pi m}{\beta}\right)}^{\frac{3}{2}}\Psi (\sigma )$ | $V{\left(\frac{2\pi m}{\beta}\right)}^{\frac{3}{2}}\Psi (\sigma )\left(1-\eta \frac{15}{2}\frac{1}{\beta m}\right),$ when $m\gg \frac{1}{\beta}$ | ${V}_{eff}\left(\alpha ,r\right){\left(\frac{2\pi m}{\beta}\right)}^{\frac{3}{2}}\Psi (\sigma {)}^{}$ |

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**MDPI and ACS Style**

Moradpour, H.; Aghababaei, S.; Ziaie, A.H.
A Note on Effects of Generalized and Extended Uncertainty Principles on Jüttner Gas. *Symmetry* **2021**, *13*, 213.
https://doi.org/10.3390/sym13020213

**AMA Style**

Moradpour H, Aghababaei S, Ziaie AH.
A Note on Effects of Generalized and Extended Uncertainty Principles on Jüttner Gas. *Symmetry*. 2021; 13(2):213.
https://doi.org/10.3390/sym13020213

**Chicago/Turabian Style**

Moradpour, Hooman, Sarah Aghababaei, and Amir Hadi Ziaie.
2021. "A Note on Effects of Generalized and Extended Uncertainty Principles on Jüttner Gas" *Symmetry* 13, no. 2: 213.
https://doi.org/10.3390/sym13020213