# American Institute of Mathematical Sciences

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 & 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.  Google Scholar [2] W. M. Boothby and H. C. Wang, On contact manifolds, Ann. of Math., 68 (1958), 721-734. 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, New York, 1960. Google Scholar [4] Y. Hatakeyama, Some notes on differentiable manifolds with almost contact structures, Tohoku Math. J., 15 (1963), 176-181. doi: 10.2748/tmj/1178243844.  Google Scholar [5] R. Hermann, "Geometry, Physics, and Systems," Pure and Applied Mathematics, Vol. 18, Marcel Dekker, INc., New York, 1973.  Google Scholar [6] S. Kobayashi, Principal fibre bundles with the 1-dimensional toroidal group, Toku Math. J., 8 (1956), 29-45.  Google Scholar [7] A. Morimoto, On normal almost contact structures with a regularity, Toku Math. J., 16 (1964), 90-104. doi: 10.2748/tmj/1178243735.  Google Scholar [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.  Google Scholar [9] R. Mrugala, Submanifolds in the thermodynamic phase space, Math. Phys., 21 (1985), 197-203.  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), 3156-3160. doi: 10.1103/PhysRevA.41.3156.  Google Scholar [11] J. Nulton and P. Salamon, Geometry of the ideal gas, Phys. Rev. A, 31 (1985), 2520-2524. doi: 10.1103/PhysRevA.31.2520.  Google Scholar [12] G. Ruppeiner, Thermodynamics: A riemannian geometric model, Phys. Rev. A, 20 (1979), 1608-1613. 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), 2515-2520. doi: 10.1063/1.525629.  Google Scholar [14] H. Shima, "The Geometry of Hessian Structures," World Scienfific, Singapore, 2007.  Google Scholar [15] F. Weinhold, Metric geometry of equilibrium thermodynamics, J. Chem. Phys., 63 (1975), 2479-2483. 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), 2484-2487. 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), 2488-2495. 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), 2496-2501. doi: 10.1063/1.431637.  Google Scholar

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##### References:
 [1] D. E. Blair, "Riemannian Geometry of Contact and Symplectic Manifolds," $2^{nd}$ edition, Birkhser, Boston, MA, 2010.  Google Scholar [2] W. M. Boothby and H. C. Wang, On contact manifolds, Ann. of Math., 68 (1958), 721-734. 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, New York, 1960. Google Scholar [4] Y. Hatakeyama, Some notes on differentiable manifolds with almost contact structures, Tohoku Math. J., 15 (1963), 176-181. doi: 10.2748/tmj/1178243844.  Google Scholar [5] R. Hermann, "Geometry, Physics, and Systems," Pure and Applied Mathematics, Vol. 18, Marcel Dekker, INc., New York, 1973.  Google Scholar [6] S. Kobayashi, Principal fibre bundles with the 1-dimensional toroidal group, Toku Math. J., 8 (1956), 29-45.  Google Scholar [7] A. Morimoto, On normal almost contact structures with a regularity, Toku Math. J., 16 (1964), 90-104. doi: 10.2748/tmj/1178243735.  Google Scholar [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.  Google Scholar [9] R. Mrugala, Submanifolds in the thermodynamic phase space, Math. Phys., 21 (1985), 197-203.  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), 3156-3160. doi: 10.1103/PhysRevA.41.3156.  Google Scholar [11] J. Nulton and P. Salamon, Geometry of the ideal gas, Phys. Rev. A, 31 (1985), 2520-2524. doi: 10.1103/PhysRevA.31.2520.  Google Scholar [12] G. Ruppeiner, Thermodynamics: A riemannian geometric model, Phys. Rev. A, 20 (1979), 1608-1613. 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), 2515-2520. doi: 10.1063/1.525629.  Google Scholar [14] H. Shima, "The Geometry of Hessian Structures," World Scienfific, Singapore, 2007.  Google Scholar [15] F. Weinhold, Metric geometry of equilibrium thermodynamics, J. Chem. Phys., 63 (1975), 2479-2483. 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), 2484-2487. 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), 2488-2495. 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), 2496-2501. doi: 10.1063/1.431637.  Google Scholar
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