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?

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.
Citation: Gerardo Hernández, Ernesto A. Lacomba. Are the geometries of the first and second laws of thermodynamics compatible?. Discrete and Continuous Dynamical Systems, 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, Birkhser, Boston, MA, 2010.

[2]

W. M. Boothby and H. C. Wang, On contact manifolds, Ann. of Math., 68 (1958), 721-734. doi: 10.2307/1970165.

[3]

H. B. Callen, "Thermodynamics: An Introduction to the Physical Theories of Equilibrium Thermostatics and Irreversible Thermodynamics," John Wiley and Sons, New York, 1960.

[4]

Y. Hatakeyama, Some notes on differentiable manifolds with almost contact structures, Tohoku Math. J., 15 (1963), 176-181. doi: 10.2748/tmj/1178243844.

[5]

R. Hermann, "Geometry, Physics, and Systems," Pure and Applied Mathematics, Vol. 18, Marcel Dekker, INc., New York, 1973.

[6]

S. Kobayashi, Principal fibre bundles with the 1-dimensional toroidal group, Toku Math. J., 8 (1956), 29-45.

[7]

A. Morimoto, On normal almost contact structures with a regularity, Toku Math. J., 16 (1964), 90-104. doi: 10.2748/tmj/1178243735.

[8]

R. Mrugala, Geometrical formulation of equilibrium phenomenological thermodynamics, Reports on Mathematical Physics, 14 (1978), 419-427. doi: 10.1016/0034-4877(78)90010-1.

[9]

R. Mrugala, Submanifolds in the thermodynamic phase space, Math. Phys., 21 (1985), 197-203.

[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), 3156-3160. doi: 10.1103/PhysRevA.41.3156.

[11]

J. Nulton and P. Salamon, Geometry of the ideal gas, Phys. Rev. A, 31 (1985), 2520-2524. doi: 10.1103/PhysRevA.31.2520.

[12]

G. Ruppeiner, Thermodynamics: A riemannian geometric model, Phys. Rev. A, 20 (1979), 1608-1613. doi: 10.1103/PhysRevA.20.1608.

[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), 2515-2520. doi: 10.1063/1.525629.

[14]

H. Shima, "The Geometry of Hessian Structures," World Scienfific, Singapore, 2007.

[15]

F. Weinhold, Metric geometry of equilibrium thermodynamics, J. Chem. Phys., 63 (1975), 2479-2483. doi: 10.1063/1.431635.

[16]

F. Weinhold, Metric geometry of equilibrium thermodynamics. II. Scaling, homogeneity, and generalized Gibbs-Duhem relations, J. Chem. Phys., 63 (1975), 2484-2487. doi: 10.1063/1.431635.

[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), 2488-2495. doi: 10.1063/1.431636.

[18]

F. Weinhold, Metric geometry of equilibrium thermodynamics. IV. Vector-algebraic evaluation of thermodynamic derivatives, J. Chem. Phys., 63 (1975), 2496-2501. doi: 10.1063/1.431637.

show all references

References:
[1]

D. E. Blair, "Riemannian Geometry of Contact and Symplectic Manifolds," $2^{nd}$ edition, Birkhser, Boston, MA, 2010.

[2]

W. M. Boothby and H. C. Wang, On contact manifolds, Ann. of Math., 68 (1958), 721-734. doi: 10.2307/1970165.

[3]

H. B. Callen, "Thermodynamics: An Introduction to the Physical Theories of Equilibrium Thermostatics and Irreversible Thermodynamics," John Wiley and Sons, New York, 1960.

[4]

Y. Hatakeyama, Some notes on differentiable manifolds with almost contact structures, Tohoku Math. J., 15 (1963), 176-181. doi: 10.2748/tmj/1178243844.

[5]

R. Hermann, "Geometry, Physics, and Systems," Pure and Applied Mathematics, Vol. 18, Marcel Dekker, INc., New York, 1973.

[6]

S. Kobayashi, Principal fibre bundles with the 1-dimensional toroidal group, Toku Math. J., 8 (1956), 29-45.

[7]

A. Morimoto, On normal almost contact structures with a regularity, Toku Math. J., 16 (1964), 90-104. doi: 10.2748/tmj/1178243735.

[8]

R. Mrugala, Geometrical formulation of equilibrium phenomenological thermodynamics, Reports on Mathematical Physics, 14 (1978), 419-427. doi: 10.1016/0034-4877(78)90010-1.

[9]

R. Mrugala, Submanifolds in the thermodynamic phase space, Math. Phys., 21 (1985), 197-203.

[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), 3156-3160. doi: 10.1103/PhysRevA.41.3156.

[11]

J. Nulton and P. Salamon, Geometry of the ideal gas, Phys. Rev. A, 31 (1985), 2520-2524. doi: 10.1103/PhysRevA.31.2520.

[12]

G. Ruppeiner, Thermodynamics: A riemannian geometric model, Phys. Rev. A, 20 (1979), 1608-1613. doi: 10.1103/PhysRevA.20.1608.

[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), 2515-2520. doi: 10.1063/1.525629.

[14]

H. Shima, "The Geometry of Hessian Structures," World Scienfific, Singapore, 2007.

[15]

F. Weinhold, Metric geometry of equilibrium thermodynamics, J. Chem. Phys., 63 (1975), 2479-2483. doi: 10.1063/1.431635.

[16]

F. Weinhold, Metric geometry of equilibrium thermodynamics. II. Scaling, homogeneity, and generalized Gibbs-Duhem relations, J. Chem. Phys., 63 (1975), 2484-2487. doi: 10.1063/1.431635.

[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), 2488-2495. doi: 10.1063/1.431636.

[18]

F. Weinhold, Metric geometry of equilibrium thermodynamics. IV. Vector-algebraic evaluation of thermodynamic derivatives, J. Chem. Phys., 63 (1975), 2496-2501. doi: 10.1063/1.431637.

[1]

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

[2]

Abbas Bahri. Recent results in contact form geometry. Discrete and Continuous Dynamical Systems, 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 and 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 and 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 and Continuous Dynamical Systems, 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]

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

[13]

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

[14]

Yong Lin, Gábor Lippner, Dan Mangoubi, Shing-Tung Yau. Nodal geometry of graphs on surfaces. Discrete and Continuous Dynamical Systems, 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]

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

[17]

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

[18]

Giuseppe Gaeta. On the geometry of twisted prolongations, and dynamical systems. Discrete and 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

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (116)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]