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

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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

Abstract
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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 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.

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