# Restricted Phase Space Thermodynamics of Einstein-Power-Yang–Mills AdS Black Hole

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## Abstract

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## 1. Introduction

## 2. EPYM AdS Black Hole and Hawking Temperature

## 3. Restricted Phase Space Formulism of EPYM AdS Black Hole

## 4. Critical Curves of EPYM AdS Black Hole

## 5. First-Order Phase Transition in Restricted Phase Space

## 6. Phase Transition from Ehrenfest’s Equations

## 7. Discussions and Conclusions

- The first law of thermodynamics for the EPYM AdS black hole in the restricted phase space conforms to the standard description of ordinary thermodynamic systems: the mass parameter is to be understood as the internal energy, and the Euler relation of this system in the restricted phase space is restored as in an ordinary thermodynamic system.
- In these two different phase spaces, the property of phase transition including the first-order and second-order phase transitions for the EPYM AdS black hole does not change. That means that the thermodynamic property of AdS black holes is independent of the adoption of corresponding phase spaces.
- From the PD ratio perspective, this charged non-linear black hole is indeed in an equilibrium state at $T={T}_{c}$ as well as ordinary thermodynamic systems. This also indicates that black holes can be indeed regarded as thermodynamic systems.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The critical curves in ${T}_{c}-{\overline{P}}_{c}$ and ${T}_{c}-{\overline{C}}_{c}$ diagrams with the non-linear charge parameter $\gamma $. The YM charge is set to $\overline{q}=1$.

**Figure 2.**The phase diagrams of ${V}_{eff}-\overline{P}$ and $\mu -\overline{C}$ with different values of temperature.

**Figure 3.**The first-order phase diagrams of ${V}_{eff}-\overline{P}$ with different values of temperature. The parameters are set to $\overline{q}=1,\phantom{\rule{3.33333pt}{0ex}}T=0.033<{T}_{c}$, and the non-linear YM charge parameter varies from 0.8 to 1 from the black line to the red one.

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**MDPI and ACS Style**

Du, Y.-Z.; Li, H.-F.; Zhang, Y.; Zhou, X.-N.; Zhao, J.-X.
Restricted Phase Space Thermodynamics of Einstein-Power-Yang–Mills AdS Black Hole. *Entropy* **2023**, *25*, 687.
https://doi.org/10.3390/e25040687

**AMA Style**

Du Y-Z, Li H-F, Zhang Y, Zhou X-N, Zhao J-X.
Restricted Phase Space Thermodynamics of Einstein-Power-Yang–Mills AdS Black Hole. *Entropy*. 2023; 25(4):687.
https://doi.org/10.3390/e25040687

**Chicago/Turabian Style**

Du, Yun-Zhi, Huai-Fan Li, Yang Zhang, Xiang-Nan Zhou, and Jun-Xin Zhao.
2023. "Restricted Phase Space Thermodynamics of Einstein-Power-Yang–Mills AdS Black Hole" *Entropy* 25, no. 4: 687.
https://doi.org/10.3390/e25040687