Abstract
In this note, we formulate and study a Hamilton–Jacobi approach for describing thermodynamic transformations. The thermodynamic phase space assumes the structure of a contact manifold with the points representing equilibrium states being restricted to certain submanifolds of this phase space. We demonstrate that Hamilton–Jacobi theory consistently describes thermodynamic transformations on the space of externally controllable parameters or equivalently the space of equilibrium states. It turns out that in the Hamilton–Jacobi description, the choice of the principal function is not unique but, the resultant dynamical description for a given transformation remains the same irrespective of this choice. Some examples involving thermodynamic transformations of the ideal gas are discussed where the characteristic curves on the space of equilibrium states completely describe the dynamics. The geometric Hamilton–Jacobi formulation which has emerged recently is also discussed in the context of thermodynamics.
Similar content being viewed by others
References
H Goldstein, C Poole and J Safko, Classical mechanics, 3rd Edn (Addison-Wesley, 2001)
V I Arnold, Mathematical methods of classical mechanics, Graduate texts in mathematics, 2nd Edn (Springer-Verlag, New York, 1989) Vol. 60
H Geiges, An introduction to contact topology (Cambridge University Press, 2008)
V I Arnold, Singularities of caustics and wave fronts (Springer, Netherlands, 1990)
M de Leon and C Sardon, J. Phys. A: Math. Theor. 50, 255205 (2017)
A Bravetti, H Cruz and D Tapias, Ann. Phys. 376, 17 (2017)
R Mrugala, J D Nulton, J C Schon and P Salamon, Rep. Math. Phys. 29,109 (1991)
R Mrugala, Rep. Math. Phys. 33, 149 (1993)
G Hernandez and E A Lacomba, J. Differ. Geom. Appl. 8, 205 (1998)
R Mrugala, Rep. Math. Phys. 46, 461 (2000)
R Mrugala, RIMS Kokyuroku 1142, 167 (2000)
S G Rajeev, Ann. Phys. 323, 2265 (2008)
A Bravetti, C S Lopez-monsalvo and F Nettel, Ann. Phys. 361, 377 (2015)
J M Isidro and P F de Cardoba, Entropy 20, 247 (2018)
R Hermann, Geometry, physics, and systems (M Dekker Publisher, 1973)
M J Peterson, Am. J. Phys. 47, 488 (1979)
G Ruppeiner, Phys. Rev. A 20, 1608 (1979)
G Ruppeiner, Rev. Mod. Phys. 67, 605 (1995); Erratum, Rev. Mod. Phys. 68, 313 (1996)
F Weinhold, J. Chem. Phys. 63, 2479 (1975)
R Mrugala, J D Nulton, J C Schon and P Salamon, Phys. Rev. A 41, 3156 (1990)
J Shen, R-G Cai, B Wang and R-K Su, Int. J. Mod. Phys. A 22, 11 (2007)
H Quevedo, Gen. Rel. Grav. 40, 971 (2008)
S-W Wei, Y-X Liu and R B Mann, Phys. Rev. Lett. 123, 071103 (2019)
S-W Wei, Y-X Liu and R B Mann, Phys. Rev. D 100, 124033 (2019)
A Ghosh and C Bhamidipati, Phys. Rev. D 100, 126020 (2019)
M C Baldiotti, R Fresneda and C Molina, Ann. Phys. 382, 22 (2017)
M de Leon, M Lainz and A Muniz-Brea, Mathematics 9, 1993 (2021)
O Esen, M de Leon, M Lainz, C Sardon and M Zajac, J. Phys. A: Math. Theor. 55, 403001(2022)
T Wada, A M Scarfone and H Matsuzoe, Physica A 570, 125820 (2021)
G E Crooks, Phys. Rev. Lett. 99, 100602 (2007)
D A Sivak and G E Crooks, Phys. Rev. Lett. 108, 190602 (2012)
P R Zulkowski, D A Sivak, G E Crooks and M R DeWeese, Phys. Rev. E 86, 041148 (2012)
M Grmela and H C Ottinger, Phys. Rev. E 56, 6620 (1997)
M Grmela and H C Ottinger, Phys. Rev. E 56, 6633 (1997)
M Grmela, Physica A 309, 304 (2002)
M Grmela, Entropy 16(3), 1652 (2014)
M Grmela, Entropy 23(2), 165 (2021)
P Guha, J. Math. Anal. Appl. 326, 121 (2007)
Acknowledgements
The paper is dedicated to Dr Asok K Das, on the occasion of his 50th birthday. The author is grateful to Chandrasekhar Bhamidipati for useful discussions. The financial assistance received from the Ministry of Education (MoE), Government of India, in the form of a Prime Minister’s Research Fellowship (PMRF ID: 1200454) is gratefully acknowledged. The author is also thankful to the anonymous referees for their suggestions which have lead to an improvement of the article.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Dr Asok Kumar Das on the occasion of his 50th birthday with deep respect and admiration.
Rights and permissions
About this article
Cite this article
Ghosh, A. Hamilton–Jacobi approach to thermodynamic transformations. Pramana - J Phys 97, 49 (2023). https://doi.org/10.1007/s12043-023-02523-2
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12043-023-02523-2