Investigation Of Thermodynamic Geometry And Christoffel Coefficients of Modified Chaplygin Gas Using Weinhold Metric

Emine Canan Günay Demirel

doi:10.17798/bitlisfen.1927264

Investigation Of Thermodynamic Geometry And Christoffel Coefficients of Modified Chaplygin Gas Using Weinhold Metric

Abstract

In this article, the dynamic components and temperature values of modified Chaplygin gas, which has the equation of state P=Aρ-B/ρ^α were obtained as a function of entropy and volume for the universe filled with modified Chaplygin gas, first at α=0 and then at α=1. All symbols used in the derivation are defined explicitly, and the entropy-dependent integration functions are fixed through the thermodynamic relation T=(∂U/∂S)V. The original contribution of this work is the combined derivation of the Weinhold curvature and the corresponding Christoffel coefficients for modified Chaplygin gas within the same thermodynamic phase-space framework. The signs of the thermodynamic curvature (R=0, R>0, R<0) were interpreted only as thermodynamic-geometric indicators of the dominant interaction regime and were compared with the related literature. A numerical illustration is also included to show how the curvature sign changes with model parameters. Subsequently, all components of the Weinhold metric and the thermodynamic curvature for modified Chaplygin gas were obtained for both α=0 and α=1. Finally, the Christoffel coefficients were obtained on the Weinhold thermodynamic phase space for the same conditions.

Keywords

Ethical Statement

The study is complied with research and publication ethics.

References

E. Hubble, “A relation between distance and radial velocity among extra-galactic nebulae,” *Proceedings of the National Academy of Sciences of the United States of America*, vol. 15, p. 168, 1929.
P. M. Garnavich, R. P. Kirshner, P. Challis, *et al*., “Constraints on cosmological models from Hubble Space Telescope observations of high-z supernovae,” *The Astrophysical Journal*, vol. 493, p. L53, 1998.
S. Perlmutter, G. Aldering, M. Della Valle, *et al*., “Discovery of a supernova explosion at half the age of the universe,” *Nature*, vol. 391, p. 51, 1998.
S. Perlmutter, G. Aldering, G. Goldhaber, *et al*., “Measurements of Ω and Λ from 42 high-redshift supernovae,” *The Astrophysical Journal*, vol. 517, p. 565, 1999.
A. G. Riess, A. V. Filippenko, P. Challis, *et al*., “Observational evidence from supernovae for an accelerating universe and a cosmological constant,” *The Astronomical Journal*, vol. 116, p. 1009, 1998.
A. G. Riess, L. G. Strolger, J. Tonry, *et al*., “Type Ia supernova discoveries at z > 1 from the Hubble Space Telescope: Evidence for past deceleration and constraints on dark energy evolution,” *The Astrophysical Journal*, vol. 607, p. 665, 2004.
R. Amanullah, C. Lidman, D. Rubin, *et al*., “Spectra and Hubble Space Telescope light curves of six Type Ia supernovae at 0.511 < z < 1.12 and the Union2 compilation,” *The Astrophysical Journal*, vol. 716, p. 712, 2010.
N. A. Bahcall, J. P. Ostriker, S. Perlmutter, and P. J. Steinhardt, “The cosmic triangle: Revealing the state of the universe,” *Science*, vol. 284, p. 1481, 1999.

M. Tegmark, M. A. Strauss, M. R. Blanton, *et al*., “Cosmological parameters from SDSS and WMAP,” *Physical Review D*, vol. 69, Art. no. 103501, 2004.
A. D. Miller, R. Caldwell, M. J. Devlin, *et al*., “A measurement of the angular power spectrum of the cosmic microwave background from l = 100 to 400,” *The Astrophysical Journal*, vol. 524, p. L1, 1999.
C. L. Bennett, M. Halpern, G. Hinshaw, *et al*., “First-year Wilkinson Microwave Anisotropy Probe observations: Preliminary maps and basic results,” *The Astrophysical Journal Supplement Series*, vol. 148, p. 1, 2003.
S. L. Bridle, O. Lahav, J. P. Ostriker, and P. J. Steinhardt, “Precision cosmology? Not just yet...,” *Science*, vol. 299, p. 1532, 2003.
D. N. Spergel, L. Verde, H. V. Peiris, *et al*., “First-year Wilkinson Microwave Anisotropy Probe observations: Determination of cosmological parameters,” *The Astrophysical Journal Supplement Series*, vol. 148, p. 175, 2003.
L. M. Krauss and M. S. Turner, “The cosmological constant is back,” *General Relativity and Gravitation*, vol. 27, p. 1137, 1995.
E. J. Copeland, M. Sami, and S. Tsujikawa, “Dynamics of dark energy,” *International Journal of Modern Physics D*, vol. 15, p. 1753, 2006.
J. Yoo and Y. Watanabe, “Theoretical models of dark energy,” *International Journal of Modern Physics D*, vol. 21, Art. no. 1230002, 2012.
E. O. Kâhya and B. Pourhassan, “The universe dominated by the extended Chaplygin gas,” *Modern Physics Letters A*, vol. 30, Art. no. 1550070, 2015.
S. M. Carroll, “The cosmological constant,” *Living Reviews in Relativity*, vol. 4, p. 1, 2001.
P. J. E. Peebles and B. Ratra, “The cosmological constant and dark energy,” *Reviews of Modern Physics*, vol. 75, p. 559, 2003.
B. Ratra and P. J. E. Peebles, “Cosmological consequences of a rolling homogeneous scalar field,” *Physical Review D*, vol. 37, p. 3406, 1988.
A. Sen, “Rolling tachyon,” *Journal of High Energy Physics*, vol. 2002, Art. no. 065, 2002.
F. Piazza and S. Tsujikawa, “Dilatonic ghost condensate as dark energy,” *Journal of Cosmology and Astroparticle Physics*, vol. 2004, Art. no. 004, 2004.
Z. K. Guo, Y. S. Piao, X. Zhang, and Y. Z. Zhang, “Cosmological evolution of a quintom model of dark energy,” *Physics Letters B*, vol. 608, p. 177, 2005.
V. Sahni and Y. Shtanov, “Braneworld models of dark energy,” *Journal of Cosmology and Astroparticle Physics*, vol. 2003, Art. no. 014, 2003.
A. Kamenshchik, U. Moschella, and V. Pasquier, “An alternative to quintessence,” *Physics Letters B*, vol. 511, p. 265, 2001.
C. Armendariz-Picon and V. Mukhanov, “Essentials of k-essence,” *Physical Review D*, vol. 63, Art. no. 103510, 2001.
T. Chiba, “Tracking k-essence,” *Physical Review D*, vol. 66, Art. no. 063514, 2002.
R. J. Scherrer, “Purely kinetic k-essence as unified dark matter,” *Physical Review Letters*, vol. 93, Art. no. 011301, 2004.
S. Bhattacharya and U. Debnath, “Thermodynamics of modified Chaplygin gas and tachyonic field,” *International Journal of Theoretical Physics*, vol. 51, p. 565, 2012.
D. Panigrahi, “Thermodynamical behavior of the variable Chaplygin gas,” *International Journal of Modern Physics D*, vol. 24, Art. no. 1550030, 2015.
F. C. Santos, M. L. Bedran, and V. Soares, “On the thermodynamic stability of the modified Chaplygin gas,” *Physics Letters B*, vol. 646, p. 215, 2007.
H. B. Benaoum, “Modified Chaplygin gas cosmology,” *Advances in High Energy Physics*, vol. 2012, Art. no. 357802, 2012.
C. L. Degen, F. Reinhard, and P. Cappellaro, “Quantum sensing,” *Reviews of Modern Physics*, vol. 89, Art. no. 035002, 2017.
S. Pirandola, B. R. Bardhan, T. Gehring, C. Weedbrook, and S. Lloyd, “Advances in photonic quantum sensing,” *Nature Photonics*, vol. 12, p. 724, 2018.
Y. Yao, X. Xiao, L. Ge, X. G. Wang, and C. P. Sun, “Quantum Fisher information in noninertial frames,” *Physical Review A*, vol. 89, Art. no. 042336, 2014.
E. Türkgenç and İ. Yilmaz, “Some information from thermodynamic curvature of Chaplygin gas,” *Modern Physics Letters A*, vol. 34, Art. no. 1950180, 2019.
F. Weinhold, “Metric geometry of equilibrium thermodynamics,” *The Journal of Chemical Physics*, vol. 63, p. 2479, 1975.
P. Salamon, J. D. Nulton, and R. S. Berry, “Length in statistical thermodynamics,” *The Journal of Chemical Physics*, vol. 82, p. 2433, 1985.
M. Santoro, “Weinhold length in an isentropic ideal and quasi-ideal gas,” *Chemical Physics*, vol. 310, p. 269, 2005.
J. A. Frieman, M. S. Turner, and D. Huterer, “Dark energy and the accelerating universe,” *Annual Review of Astronomy and Astrophysics*, vol. 46, p. 385, 2008.
J. C. Fabris, H. E. S. Velten, C. Ogouyandjou, and J. Tossa, “Ruling out the modified Chaplygin gas cosmologies,” *Physics Letters B*, vol. 694, p. 289, 2011.
M. Jamil and U. Debnath, “Interacting modified Chaplygin gas in loop quantum cosmology,” *Astrophysics and Space Science*, vol. 333, p. 3, 2011.
B. Chakraborty, T. Mukhopadhyay, D. Mondal, and U. Debnath, “Thermodynamics of modified Chaplygin-Jacobi gas and modified Chaplygin-Abel gas: Stability analysis and observational constraints,” *Nuclear Physics B*, vol. 1010, Art. no. 116770, 2025.
N. Bilić, “Thermodynamics of dark energy,” *Fortschritte der Physik*, vol. 56, p. 363, 2008.
T. Bandyopadhyay and S. Chakraborty, “The laws of thermodynamics and thermodynamic stability of the modified Chaplygin gas,” *Modern Physics Letters A*, vol. 25, p. 2779, 2010.
R. Mrugała, “On equivalence of two metrics in classical thermodynamics,” *Physica A: Statistical Mechanics and its Applications*, vol. 125, p. 631, 1984.
M. Santoro and S. Preston, “Curvature of the Weinhold metric for thermodynamical systems with two degrees of freedom,” *arXiv preprint math-ph/0505010*, 2005.
S. W. Wei, Y. X. Liu, and R. B. Mann, “Ruppeiner geometry, phase transitions, and the microstructure of charged AdS black holes,” *Physical Review D*, vol. 100, Art. no. 124033, 2019.
G. Ruppeiner, “Thermodynamics: A Riemannian geometric model,” *Physical Review A*, vol. 20, p. 1608, 1979.
H. Janyszek and R. Mrugała, “Riemannian geometry and stability of ideal quantum gases,” *Journal of Physics A: Mathematical and General*, vol. 23, p. 467, 1990.
H. Oshima, T. Obata, and H. Hara, “Riemann scalar curvature of ideal quantum gases obeying Gentile’s statistics,” *Journal of Physics A: Mathematical and General*, vol. 32, p. 6373, 1999.
G. Ruppeiner, “Thermodynamic curvature measures interactions,” *American Journal of Physics*, vol. 78, p. 1170, 2010.

Details

Primary Language

English

Subjects

General Physics

Journal Section

Research Article

Authors

Emine Canan Günay Demirel ^*
0000-0003-2968-9780
Türkiye

Publication Date

September 30, 2026

Submission Date

April 10, 2026

Acceptance Date

June 26, 2026

Published in Issue

Year 2026 Volume: 15 Number: 3

DOI

https://doi.org/10.17798/bitlisfen.1927264

IZ

https://izlik.org/JA26EJ33RB

Cite

RIS / Bibtex

APA

Günay Demirel, E. C. (2026). Investigation Of Thermodynamic Geometry And Christoffel Coefficients of Modified Chaplygin Gas Using Weinhold Metric. Bitlis Eren Üniversitesi Fen Bilimleri Dergisi, 15(3), 1523-1541. https://doi.org/10.17798/bitlisfen.1927264

AMA

1.Günay Demirel EC. Investigation Of Thermodynamic Geometry And Christoffel Coefficients of Modified Chaplygin Gas Using Weinhold Metric. Bitlis Eren Üniversitesi Fen Bilimleri Dergisi. 2026;15(3):1523-1541. doi:10.17798/bitlisfen.1927264

Chicago

Günay Demirel, Emine Canan. 2026. “Investigation Of Thermodynamic Geometry And Christoffel Coefficients of Modified Chaplygin Gas Using Weinhold Metric”. Bitlis Eren Üniversitesi Fen Bilimleri Dergisi 15 (3): 1523-41. https://doi.org/10.17798/bitlisfen.1927264.

EndNote

Günay Demirel EC (September 1, 2026) Investigation Of Thermodynamic Geometry And Christoffel Coefficients of Modified Chaplygin Gas Using Weinhold Metric. Bitlis Eren Üniversitesi Fen Bilimleri Dergisi 15 3 1523–1541.

IEEE

[1]E. C. Günay Demirel, “Investigation Of Thermodynamic Geometry And Christoffel Coefficients of Modified Chaplygin Gas Using Weinhold Metric”, Bitlis Eren Üniversitesi Fen Bilimleri Dergisi, vol. 15, no. 3, pp. 1523–1541, Sept. 2026, doi: 10.17798/bitlisfen.1927264.

ISNAD

Günay Demirel, Emine Canan. “Investigation Of Thermodynamic Geometry And Christoffel Coefficients of Modified Chaplygin Gas Using Weinhold Metric”. Bitlis Eren Üniversitesi Fen Bilimleri Dergisi 15/3 (September 1, 2026): 1523-1541. https://doi.org/10.17798/bitlisfen.1927264.

JAMA

1.Günay Demirel EC. Investigation Of Thermodynamic Geometry And Christoffel Coefficients of Modified Chaplygin Gas Using Weinhold Metric. Bitlis Eren Üniversitesi Fen Bilimleri Dergisi. 2026;15:1523–1541.

MLA

Günay Demirel, Emine Canan. “Investigation Of Thermodynamic Geometry And Christoffel Coefficients of Modified Chaplygin Gas Using Weinhold Metric”. Bitlis Eren Üniversitesi Fen Bilimleri Dergisi, vol. 15, no. 3, Sept. 2026, pp. 1523-41, doi:10.17798/bitlisfen.1927264.

Vancouver

1.Emine Canan Günay Demirel. Investigation Of Thermodynamic Geometry And Christoffel Coefficients of Modified Chaplygin Gas Using Weinhold Metric. Bitlis Eren Üniversitesi Fen Bilimleri Dergisi. 2026 Sep. 1;15(3):1523-41. doi:10.17798/bitlisfen.1927264