# On a Self-Consistency Thermodynamical Criterion for Equations of the State of Gases in Relativistic Frames

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## Abstract

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## 1. Introduction

## 2. Relativistic Transformations and Equations of State

## 3. van der Waals Gas

## 4. Virial Coefficients

## 5. Ideal Fermi and Bose Gases and the Blackbody

#### 5.1. Ideal Fermi and Bose Gases

#### 5.2. The Blackbody

#### 5.3. Summary

## 6. Equation of State for Nuclear Matter

## 7. The Rubber Band

## 8. Concluding Remarks

## Acknowledgments

## References

- Tykodi, R.J.; Hummel, E.P. On the equation of state for gases. Am. J. Phys.
**1973**, 41, 340–343. [Google Scholar] - Angulo-Brown, F.; Olivar-Romero, F. About the thermodynamical self-consistency of some equations of state 2012. MSc.Thesis, ESFM IPN, Mexico City, Mexico, 2012. [Google Scholar]
- Carathéodory, C. Untersuchungen ber die Grundlagen der Thermodynamik. Math. Ann.
**2012**, 67, 355–386. [Google Scholar] [CrossRef] - Frankel, T. The Geometry of Physics: An Introduction; Cambridge University Press: Cambridge, UK, 2003; Chapter 6; p. 181. [Google Scholar]
- Ares de Parga, G.; López-Carrera, B. Relativistic statistical mechanics vs. relativistic thermodynamics. Entropy
**2011**, 13, 1664–1693. [Google Scholar] [CrossRef] - Planck, M. Zur dynamik bewegter systeme. Ann. Phys.
**1908**, 331, 1–34. [Google Scholar] [CrossRef] - Einstein, A. Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen. Jahrbuch der Radioaktivität und Elektronik
**1907**, 4, 411–462. [Google Scholar] - Landsberg, P.T.; Matsas, G.E. Laying the ghost of the relativistic temperature transformation. Phys. Lett. A
**1996**, 223, 401–403. [Google Scholar] [CrossRef] - Ott, H. Lorentz-transformtion der warme und der temperatur. Z. Phys.
**1963**, 175, 70–104. [Google Scholar] [CrossRef] - Landsberg, P.T. Does a moving body appear cool? Nature
**1966**, 212, 571–572. [Google Scholar] [CrossRef] - Rohrlich, F. True and apparent transformations, classical electrons, and relativistic thermodynamics. Il Nuovo Cimento B
**1966**, XLV, 6200–6207. [Google Scholar] [CrossRef] - Van Kampen, N.G. Relativistic thermodynamics of moving systems. Phys. Rev.
**1968**, 173, 295–301. [Google Scholar] [CrossRef] - Israel, W. Nonstationary irreversible thermodynamics: A causal relativistic theory. Ann. Phys.
**1976**, 100, 310–331. [Google Scholar] [CrossRef] - Israel, W. Thermodynamics of relativistic systems. Physica A
**1981**, 106, 204–214. [Google Scholar] [CrossRef] - Balescu, R. Relativistic statistical thermodynamics. Physica
**1968**, 40, 309–338. [Google Scholar] [CrossRef] - Staruszkiewicz, A. Relativistic transformation laws for thermodynamical variables with applications to classical electron theory. Il Nuovo Cimento A
**1966**, XLV, 6632–6636. [Google Scholar] [CrossRef] - Nakamura, T.K. Three views of a secret in relativistic thermodynamics. Prog. Theor. Phys.
**2012**, 128, 463–475. [Google Scholar] [CrossRef] - Möller, C. The theory of Relativity; International Series of Monographs on Physics; Clarendon Press: Oxford, UK, 1972; Chapter 7. [Google Scholar]
- Requardt, M. Thermodynamics meets special relativity—Or what is real in physics? arXiv
**2008**, 08012639, 2639:1–2639:27. [Google Scholar] - Przanowski, M.; Tosiek, J. Notes on thermodynamics in special relativity. Phys. Scripta
**2011**, 84. [Google Scholar] [CrossRef] - Jüttner, F. Das Maxwellsche Gesetz der Geschwindigkeitsverteilung in der Relativtheorie. Ann. Phys.
**1911**, 339, 856–882. [Google Scholar] [CrossRef] - Cubero, D.; Casado-Pascual, J.; Dunkel, J.; Talkner, P.; Hänggi, P. Thermal equilibrium and statistical thermometers in special relativity. Phys. Rev. Lett.
**2007**, 99, 170601:1–170601:4. [Google Scholar] - Ares de Parga, G.; López-Carrera, B. Redefined relativistic thermodynamics based on the Nakamura formalism. Physica A
**2009**, 388, 4345–4356. [Google Scholar] [CrossRef] - Nakamura, T.K. Relativistic energy-momentum of a body with a finite volume. Space Sci. Rev.
**2006**, 122, 271–278. [Google Scholar] [CrossRef] - Dunkel, J.; Hänggi, P.; Hilbert, S. Non-local observables and lightcone-averaging in relativistic thermodynamics. Nat. Phys.
**2009**, 5, 741–747. [Google Scholar] [CrossRef] - Greiner, W.; Neise, L.; Stcker, H. Thermodynamics and Statistical Mechanics; Springer-Verlag: New York, NY, USA, 1995; Chapter 8. [Google Scholar]
- Ares de Parga, G.; López-Carrera, B.; Angulo-Brown, F. A proposal for relativistic transformations in thermodynamics. J. Phys. A Math. Gen.
**2005**, 38, 2821–2834. [Google Scholar] [CrossRef] - Callen, H.B. Thermodynamics and an Introduction to Thermostatistics; John Wiley & Sons: Hoboken, NJ, USA, 1985; Chapters 1 and 2; pp. 28, 37. [Google Scholar]
- Currie, D.G.; Jordan, T.F.; Sudarshan, E.C.G. Relativistic invariance and hamiltonian theories of interacting particles. Rev. Mod. Phys.
**1963**, 35, 350–375. [Google Scholar] [CrossRef] - Pryce, M.H.L. The mass-centre in the restricted theory of relativity and its connexion with the quantum theory of elementary particles. Proc. R. Soc. Lond. A
**1948**, 195, 62–81. [Google Scholar] [CrossRef] - Thomas, L.H. The relativistic dynamics of a system of particles interacting at a distance. Phys. Rev.
**1952**, 85, 868–872. [Google Scholar] [CrossRef] - Bakamjian, B.; Thomas, L.H. Relativistic particle dynamics. II. Phys. Rev.
**1953**, 92, 1300–1310. [Google Scholar] [CrossRef] - Gill, T.; Zachary, W. Two mathematically equivalent versions of maxwells equations. Found. Phys.
**2011**, 41, 99–128. [Google Scholar] [CrossRef] - Reference [26], a comparison between last expression in p 403, Equation (16.3) of p 404, the van der Waals equation and the virial expansion, Equation (16.61) of p 414, shows that the van der Waals equation is a virial expansion to second order.
- Dobado, A.; Peláez, J.R. Chiral symmetry and the pion gas virial expansion. Phys. Rev. D
**1998**, 59, 034004:1–034004:9. [Google Scholar] [CrossRef] - Eletsky, V.L.; Kapusta, J.I.; Venugopalan, R. Screening mass from chiral perturbation theory, virial expansion, and the lattice. Phys. Rev. D
**1993**, 48, 4398–4407. [Google Scholar] [CrossRef] - Huang, K. Statistical Mechanics; John Wiley & Sons: Hoboken, NJ, USA, 1963; Chapters 9 and 12; pp. 200–201, 266–267. [Google Scholar]
- López-Carrera, B.; Antonio, R.M.; Ares de Parga, G. The 2.7 K black-body radiation background reference frame. Chin. Phys. B
**2010**, 19, 040203:1–040203:5. [Google Scholar] - Landsberg, P.T.; Matsas, G.E.A. The impossibility of a universal relativistic temperature transformation. Phys. A: Stat. Mech. Appl.
**2004**, 340, 92–94. [Google Scholar] [CrossRef] - Shen, H.; Toki, H.; Oyamatsu, K.; Sumiyoshi, K. Relativistic equation of state of nuclear matter for supernova and neutron star. Nucl. Phys. A
**1998**, 637, 435–450. [Google Scholar] [CrossRef] - Wigner, E.; Seitz, F. On the constitution of metallic sodium. Phys. Rev.
**1933**, 43, 804–810. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

De Parga, G.A.; Vargas, A.Á.; López-Carrera, B.
On a Self-Consistency Thermodynamical Criterion for Equations of the State of Gases in Relativistic Frames. *Entropy* **2013**, *15*, 1271-1288.
https://doi.org/10.3390/e15041271

**AMA Style**

De Parga GA, Vargas AÁ, López-Carrera B.
On a Self-Consistency Thermodynamical Criterion for Equations of the State of Gases in Relativistic Frames. *Entropy*. 2013; 15(4):1271-1288.
https://doi.org/10.3390/e15041271

**Chicago/Turabian Style**

De Parga, Gonzalo Ares, Adriana Ávalos Vargas, and Benjamín López-Carrera.
2013. "On a Self-Consistency Thermodynamical Criterion for Equations of the State of Gases in Relativistic Frames" *Entropy* 15, no. 4: 1271-1288.
https://doi.org/10.3390/e15041271