American Institute of Mathematical Sciences

Advanced
March  2013, 33(3): 1113-1116. doi: 10.3934/dcds.2013.33.1113

Are the geometries of the first and second laws of thermodynamics compatible?

Gerardo Hernández 1, and  Ernesto A. Lacomba 2,

1. 

Sección de Metodología y Teoría de la Ciencia, Cinvestav, Av. IPN 2508, C.P. 07360, México, D.F., Mexico
2. 

Departamento de Matemáticas, Universidad Autónoma Metropolitana–Iztapalapa, Av. San Rafael Atlixco 186, C.P. 09340 México, D.F., Mexico

Received  April 2011 Revised  November 2011 Published  October 2012

First and second laws of thermodynamics are naturally associated, respectively, to contact and Hessian geometries. In this paper we seek for a unique geometric setting that might account for both thermodynamic laws. Using Riemannian metrics that are compatible with the contact structure, we prove that the Hessian manifold of thermodynamic states cannot isometrically be embedded as Legendre submanifold of a contact manifold. Well known fibrations suggest the nature of the obstruction for such embedding.
Keywords: Boothby fibration, classical thermodynamics., Riemannian contact geometry, Hessian geometry.
Mathematics Subject Classification: Primary: 53D10, 53Z05; Secondary: 80A0.
Citation: Gerardo Hernández, Ernesto A. Lacomba. Are the geometries of the first and second laws of thermodynamics compatible?. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1113-1116. doi: 10.3934/dcds.2013.33.1113
References:
[1]

D. E. Blair, "Riemannian Geometry of Contact and Symplectic Manifolds,", $2^{nd}$ edition, (2010).   Google Scholar

[2]

W. M. Boothby and H. C. Wang, On contact manifolds,, Ann. of Math., 68 (1958), 721.  doi: 10.2307/1970165.  Google Scholar

[3]

H. B. Callen, "Thermodynamics: An Introduction to the Physical Theories of Equilibrium Thermostatics and Irreversible Thermodynamics,", John Wiley and Sons, (1960).   Google Scholar

[4]

Y. Hatakeyama, Some notes on differentiable manifolds with almost contact structures,, Tohoku Math. J., 15 (1963), 176.  doi: 10.2748/tmj/1178243844.  Google Scholar

[5]

R. Hermann, "Geometry, Physics, and Systems,", Pure and Applied Mathematics, (1973).   Google Scholar

[6]

S. Kobayashi, Principal fibre bundles with the 1-dimensional toroidal group,, Toku Math. J., 8 (1956), 29.   Google Scholar

[7]

A. Morimoto, On normal almost contact structures with a regularity,, Toku Math. J., 16 (1964), 90.  doi: 10.2748/tmj/1178243735.  Google Scholar

[8]

R. Mrugala, Geometrical formulation of equilibrium phenomenological thermodynamics,, Reports on Mathematical Physics, 14 (1978), 419.  doi: 10.1016/0034-4877(78)90010-1.  Google Scholar

[9]

R. Mrugala, Submanifolds in the thermodynamic phase space,, Math. Phys., 21 (1985), 197.   Google Scholar

[10]

R. Mrugala, J. D. Nulton, J. C. Schön and P. Salamon, Statistical approach to the geometric structure of thermodynamics,, Phys. Rev. A, 41 (1990), 3156.  doi: 10.1103/PhysRevA.41.3156.  Google Scholar

[11]

J. Nulton and P. Salamon, Geometry of the ideal gas,, Phys. Rev. A, 31 (1985), 2520.  doi: 10.1103/PhysRevA.31.2520.  Google Scholar

[12]

G. Ruppeiner, Thermodynamics: A riemannian geometric model,, Phys. Rev. A, 20 (1979), 1608.  doi: 10.1103/PhysRevA.20.1608.  Google Scholar

[13]

P. Salamon, E. Ihrig and R. S. Berry, A group of coordinate transformations which preserve the metric of Weinhold,, J. Math. Phys., 24 (1983), 2515.  doi: 10.1063/1.525629.  Google Scholar

[14]

H. Shima, "The Geometry of Hessian Structures,", World Scienfific, (2007).   Google Scholar

[15]

F. Weinhold, Metric geometry of equilibrium thermodynamics,, J. Chem. Phys., 63 (1975), 2479.  doi: 10.1063/1.431635.  Google Scholar

[16]

F. Weinhold, Metric geometry of equilibrium thermodynamics. II. Scaling, homogeneity, and generalized Gibbs-Duhem relations,, J. Chem. Phys., 63 (1975), 2484.  doi: 10.1063/1.431635.  Google Scholar

[17]

F. Weinhold, Metric geometry of equilibrium thermodynamics. III. Elementary formal structure of a vector-algebraic representation of equilibrium thermodynamics,, J. Chem. Phys., 63 (1975), 2488.  doi: 10.1063/1.431636.  Google Scholar

[18]

F. Weinhold, Metric geometry of equilibrium thermodynamics. IV. Vector-algebraic evaluation of thermodynamic derivatives,, J. Chem. Phys., 63 (1975), 2496.  doi: 10.1063/1.431637.  Google Scholar

show all references

References:
[1]

D. E. Blair, "Riemannian Geometry of Contact and Symplectic Manifolds,", $2^{nd}$ edition, (2010).   Google Scholar

[2]

W. M. Boothby and H. C. Wang, On contact manifolds,, Ann. of Math., 68 (1958), 721.  doi: 10.2307/1970165.  Google Scholar

[3]

H. B. Callen, "Thermodynamics: An Introduction to the Physical Theories of Equilibrium Thermostatics and Irreversible Thermodynamics,", John Wiley and Sons, (1960).   Google Scholar

[4]

Y. Hatakeyama, Some notes on differentiable manifolds with almost contact structures,, Tohoku Math. J., 15 (1963), 176.  doi: 10.2748/tmj/1178243844.  Google Scholar

[5]

R. Hermann, "Geometry, Physics, and Systems,", Pure and Applied Mathematics, (1973).   Google Scholar

[6]

S. Kobayashi, Principal fibre bundles with the 1-dimensional toroidal group,, Toku Math. J., 8 (1956), 29.   Google Scholar

[7]

A. Morimoto, On normal almost contact structures with a regularity,, Toku Math. J., 16 (1964), 90.  doi: 10.2748/tmj/1178243735.  Google Scholar

[8]

R. Mrugala, Geometrical formulation of equilibrium phenomenological thermodynamics,, Reports on Mathematical Physics, 14 (1978), 419.  doi: 10.1016/0034-4877(78)90010-1.  Google Scholar

[9]

R. Mrugala, Submanifolds in the thermodynamic phase space,, Math. Phys., 21 (1985), 197.   Google Scholar

[10]

R. Mrugala, J. D. Nulton, J. C. Schön and P. Salamon, Statistical approach to the geometric structure of thermodynamics,, Phys. Rev. A, 41 (1990), 3156.  doi: 10.1103/PhysRevA.41.3156.  Google Scholar

[11]

J. Nulton and P. Salamon, Geometry of the ideal gas,, Phys. Rev. A, 31 (1985), 2520.  doi: 10.1103/PhysRevA.31.2520.  Google Scholar

[12]

G. Ruppeiner, Thermodynamics: A riemannian geometric model,, Phys. Rev. A, 20 (1979), 1608.  doi: 10.1103/PhysRevA.20.1608.  Google Scholar

[13]

P. Salamon, E. Ihrig and R. S. Berry, A group of coordinate transformations which preserve the metric of Weinhold,, J. Math. Phys., 24 (1983), 2515.  doi: 10.1063/1.525629.  Google Scholar

[14]

H. Shima, "The Geometry of Hessian Structures,", World Scienfific, (2007).   Google Scholar

[15]

F. Weinhold, Metric geometry of equilibrium thermodynamics,, J. Chem. Phys., 63 (1975), 2479.  doi: 10.1063/1.431635.  Google Scholar

[16]

F. Weinhold, Metric geometry of equilibrium thermodynamics. II. Scaling, homogeneity, and generalized Gibbs-Duhem relations,, J. Chem. Phys., 63 (1975), 2484.  doi: 10.1063/1.431635.  Google Scholar

[17]

F. Weinhold, Metric geometry of equilibrium thermodynamics. III. Elementary formal structure of a vector-algebraic representation of equilibrium thermodynamics,, J. Chem. Phys., 63 (1975), 2488.  doi: 10.1063/1.431636.  Google Scholar

[18]

F. Weinhold, Metric geometry of equilibrium thermodynamics. IV. Vector-algebraic evaluation of thermodynamic derivatives,, J. Chem. Phys., 63 (1975), 2496.  doi: 10.1063/1.431637.  Google Scholar

[1]

Yunlong Huang, P. S. Krishnaprasad. Sub-Riemannian geometry and finite time thermodynamics Part 1: The stochastic oscillator. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1243-1268. doi: 10.3934/dcdss.2020072
[2]

Abbas Bahri. Recent results in contact form geometry. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 21-30. doi: 10.3934/dcds.2004.10.21
[3]

Entisar A.-L. Ali, G. Charlot. Local contact sub-Finslerian geometry for maximum norms in dimension 3. Mathematical Control & Related Fields, 2021, 11 (2) : 373-401. doi: 10.3934/mcrf.2020041
[4]

Erlend Grong, Alexander Vasil’ev. Sub-Riemannian and sub-Lorentzian geometry on $SU(1,1)$ and on its universal cover. Journal of Geometric Mechanics, 2011, 3 (2) : 225-260. doi: 10.3934/jgm.2011.3.225
[5]

Daniel Genin, Serge Tabachnikov. On configuration spaces of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards. Journal of Modern Dynamics, 2007, 1 (2) : 155-173. doi: 10.3934/jmd.2007.1.155
[6]

Stefan Sommer, Anne Marie Svane. Modelling anisotropic covariance using stochastic development and sub-Riemannian frame bundle geometry. Journal of Geometric Mechanics, 2017, 9 (3) : 391-410. doi: 10.3934/jgm.2017015
[7]

Alex L Castro, Wyatt Howard, Corey Shanbrom. Bridges between subriemannian geometry and algebraic geometry: Now and then. Conference Publications, 2015, 2015 (special) : 239-247. doi: 10.3934/proc.2015.0239
[8]

Joachim Escher, Boris Kolev, Marcus Wunsch. The geometry of a vorticity model equation. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1407-1419. doi: 10.3934/cpaa.2012.11.1407
[9]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003
[10]

Janina Kotus, Mariusz Urbański. The dynamics and geometry of the Fatou functions. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 291-338. doi: 10.3934/dcds.2005.13.291
[11]

Katarzyna Grabowska, Paweƚ Urbański. Geometry of Routh reduction. Journal of Geometric Mechanics, 2019, 11 (1) : 23-44. doi: 10.3934/jgm.2019002
[12]

Jean-Marc Couveignes, Reynald Lercier. The geometry of some parameterizations and encodings. Advances in Mathematics of Communications, 2014, 8 (4) : 437-458. doi: 10.3934/amc.2014.8.437
[13]

Len G. Margolin, Roy S. Baty. Conservation laws in discrete geometry. Journal of Geometric Mechanics, 2019, 11 (2) : 187-203. doi: 10.3934/jgm.2019010
[14]

Yong Lin, Gábor Lippner, Dan Mangoubi, Shing-Tung Yau. Nodal geometry of graphs on surfaces. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1291-1298. doi: 10.3934/dcds.2010.28.1291
[15]

Klas Modin. Geometry of matrix decompositions seen through optimal transport and information geometry. Journal of Geometric Mechanics, 2017, 9 (3) : 335-390. doi: 10.3934/jgm.2017014
[16]

Matteo Novaga, Enrico Valdinoci. The geometry of mesoscopic phase transition interfaces. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 777-798. doi: 10.3934/dcds.2007.19.777
[17]

Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040
[18]

Giuseppe Gaeta. On the geometry of twisted prolongations, and dynamical systems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1209-1227. doi: 10.3934/dcdss.2020070
[19]

François Lalonde, Yasha Savelyev. On the injectivity radius in Hofer's geometry. Electronic Research Announcements, 2014, 21: 177-185. doi: 10.3934/era.2014.21.177
[20]

Oğul Esen, Partha Guha. On the geometry of the Schmidt-Legendre transformation. Journal of Geometric Mechanics, 2018, 10 (3) : 251-291. doi: 10.3934/jgm.2018010

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences