This contribution presents an outline of a new mathematical formulation for Classical Non-Equilibrium Thermodynamics (CNET) based on a contact structure in differential geometry. First a non-equilibrium state space is introduced as the third key element besides the first and second law of thermodynamics. This state space provides the mathematical structure to generalize the Gibbs fundamental relation to non-equilibrium thermodynamics. A unique formulation for the second law of thermodynamics is postulated and it showed how the complying concept for non-equilibrium entropy is retrieved. The foundation of this formulation is a physical quantity, which is in non-equilibrium thermodynamics nowhere equal to zero. This is another perspective compared to the inequality, which is used in most other formulations in the literature. Based on this mathematical framework, it is proven that the thermodynamic potential is defined by the Gibbs free energy. The set of conjugated coordinates in the mathematical structure for the Gibbs fundamental relation will be identified for single component, closed systems. Only in the final section of this contribution will the equilibrium constraint be introduced and applied to obtain some familiar formulations for classical (equilibrium) thermodynamics.
References
[1]
Gibbs, J.W. (1928) On the Equilibrium of Heterogeneous Substances. In: Collected Works, Volume I: Thermodynamics, Chapter III, Longmans, Green, New York, 55-353. (Originally published in Transactions of the Connecticut Academy, III, pp. 108-248, 1876, and pp. 343-524, 1878).
[2]
Haupt, P. (2002) Continuum Mechanics and Theory of Materials. 2nd Edition, Springer, Berlin. https://doi.org/10.1007/978-3-662-04775-0
[3]
Moran, M.J. and Shapiro, H.N. (1998) Fundamentals of Engineering Thermodynamics. 3rd Edition, John Wiley & Sons, Chichester.
[4]
Müller, I. (2007) A History of Thermodynamics: The Doctrine of Energy and Entropy. Springer-Verlag, Berlin.
[5]
Nickel, U. (2011) Lehrbuchder Thermodynamik: Einverstandliche Einfuhrung. 2. Auflage Edition, PhysChemVerlag, Erlangen. (In German)
[6]
Nolting, W. (2012) Grundkurs Theoretische Physik 4: Spezielle Relativitatstheorie, Thermodynamik. 8. Auflage Edition, Springer-Verlag, Berlin. (In German)
[7]
Smith, J.M., Van Ness, H.C. and Abbott, M.M. (2005) Introduction to Chemical Engineering Thermodynamics. 7th International Edition, McGraw-Hill Chemical Engineering Series. McGraw-Hill, Boston.
[8]
Kjelstrup, S. and Bedeaux, D. (2008) Non-Equilibrium Thermodynamics of Heterogeneous Systems. Series on Advances in Statistical Mechanics (Vol. 16), World Scientific Publishing Co. Pte. Ltd, Singapore. https://doi.org/10.1142/6672
[9]
Knobbe, E.M. (2010) On the Integration of the Arbitrary Lagrangian-Eulerian Concept and Non-Equilibrium Thermodynamics. PhD Thesis, Delft University of Technology, Delft.
[10]
Kondepudi, D. and Prigogine, I. (1998) Modern Thermodynamics: From Heat Engine to Dissipative Structure. John Wiley & Sons, Chicester.
[11]
Attard, P. (2012) Non-Equilibrium Thermodynamics and Statistical Mechanics: Foundations and Applications. Oxford University Press, Oxford. https://doi.org/10.1093/acprof:oso/9780199662760.001.0001
[12]
Olla, P. (2015) An Introduction to Thermodynamics and Statistical Physics. UNITEXT for Physics. Springer International Publishing, Cham. https://doi.org/10.1007/978-3-319-06188-7
[13]
Scheck, F. (2008) Theoretische Physik: Statistische Theorie der Warmeleitungstheorie. Springer-Verlag, Berlin. (In German)
[14]
Zorich, V. (2011) Mathematical Analysis of Problems in the Natural Sciences. Springer-Verlag, Berlin. https://doi.org/10.1007/978-3-642-14813-2
[15]
Hermann, R. (1973) Geometry, Physics, and Systems. Pure and Applied Mathematics. Marcel Dekker, Inc., New York.
[16]
Arnol’d, V.I. (1989) Mathematical Methods of Classical Mechanics. Vol. 60, 2nd Edition, Springer-Verlag, New York. https://doi.org/10.1007/978-1-4757-2063-1
[17]
McInerney, A. (2013) First Steps in Differential Geometry: Riemannian, Contact, Symplectic. Undergraduate Texts in Mathematics. Springer Science + Business, Media.
[18]
Edelen, D.G.B. (1985) Applied Exterior Calculus. John Wiley & Sons, Inc., New York.
[19]
Kushner, A., Lychagin, V. and Rubtsov, V. (2007) Contact Geometry and Nonlinear Differential Equations. Volume 101 of Encyclopedia of Mathematics and It’s Applications, Cambridge University Press, Cambridge.
[20]
Lang, S. (2002) Introduction to Differentiable Manifolds. Universitext. 2nd Edition, Springer, New York.
[21]
Mendoza (1988) Reflections on the Motive Power of Fire. Dover Publications, Inc., Mineola.
[22]
Clausius, R.J.E. (1854) Uebereineveranderte Form des zweitenHauptsatzes der mechanischen Warmetheorie. Annalen der Physik und Chemie, 169, 481-506. (In German)
[23]
Clausius, R.J.E. (1865) Ueberverschiedenefür die Anwendungenbequeme Formen der Hauptgleichungen der mechanischen Warmetheorie. Poggendorff’s Annalen der Physik, 125, 353-400. (In German)
[24]
Bamberg, P.G. and Sternberg, S. (1990) A Course in Mathematics for Students of Physics. Volume 2, Cambridge University Press, Cambridge.
[25]
Frankel, T. (2004) The Geometry of Physics: An Introduction. 2nd Edition, Cambridge University Press, Cambridge.
[26]
Szekeres, P. (2004) A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511607066
[27]
Eu, B.C. (1995) Form of Uncompensated Heat Giving Rise to a Pfaffan Differential Form in Thermodynamic Space. Physical Review E, 51, 768-771.
[28]
Eu, B.C. (1999) Generalized Thermodynamics of Global Irreversible Processes in a Finite Macroscopic System. Journal of Physical Chemistry B, 103, 8583-8594.
[29]
Eu, B.C. (2004) Generalized Hydrodynamics in the Transient Regime and Irreversible Thermodynamics. Philosophical Transactions of the Royal Society A, 362, 1553-1565. https://doi.org/10.1098/rsta.2004.1404
[30]
Born, M. (1921) Kritische Betrachtungen zur traditionellen Darstellung der Thermodynamik. Physikzeitschrift, 22, 218-224, 249-254, 282-286. (In German)
[31]
Carathéodory, C. (1909) Untersuchungenüber die Grundlagen der Thermodynamik. Mathematische Annalen, 67, 355-386. (In German) https://doi.org/10.1007/BF01450409
[32]
Caratheodory, C. (1925) Uber die Bestimmung der Energie und der absoluten Temperature mitHilfe von reversible Prozessen. Sitzungsberichte der Praußischen Akademie der Wissenschaftenzum, Berlin, 39-47. (In German)
[33]
Pogliani, L. and Berberan-Santos, M.N. (2000) Constantin Carathéodory and the Axiomatic Thermodynamics. Journal of Mathematical Chemistry, 28, 313-324.
[34]
Truesdell, C. (1984) Rational Thermodynamics. 2nd Edition, Springer, New York. https://doi.org/10.1007/978-1-4612-5206-1
[35]
Uffink, J. (2001) Bluff Your Way in the Second Law of Thermodynamics. Studies in History and Philosophy of Modern Physics, 32B, 305-394. https://doi.org/10.1016/S1355-2198(01)00016-8
[36]
Uffink, J. (2003) Entropy, Chapter 7: Irreversibility and the Second Law of Thermodynamics. Princeton Series in Applied Mathematics, Princeton University Press, Princeton, 121-146.
[37]
Salamon, P., Andresen, B., Nulton, J.D. and Konopka, A.K. (2006) Systems Biology: Principles, Methods and Concepts, Chapter 9: The Mathematical Structure of Thermodynamics. CRC Press, Boca Raton, 207-221.
[38]
Lebon, G., Jou, D. and Casas-V’azquez, J. (2008) Understanding Non-Equilibrium Thermodynamics: Foundations, Applications, Frontiers. Springer-Verlag, Berlin. https://doi.org/10.1007/978-3-540-74252-4
[39]
Reichl, L.E. (2009) A Modern Course in Statistical Physics. 3rd Revised and Updated Edition, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.
[40]
Shell, M.S. (2015) Thermodynamics and Statistical Mechanics: An Integrated Approach. Cambridge University Press, Santa Barbara. https://doi.org/10.1017/CBO9781139028875
[41]
Haslach Jr., H.W. (1997) Geometric Structure of the Non-Equilibrium Thermodynamics of Homogeneous Systems. Reports on Mathematical Physics, 39, 147-162.
[42]
Mrugala, R. (1993) Continuous Contact Transformations in Thermodynamics. Reports on Mathematical Physics, 33, 149-154. https://doi.org/10.1016/0034-4877(93)90050-O
[43]
Mrugala, R., Nulton, J.D., Schon, J.C. and Salamon, P. (1991) Contact Structure in Thermodynamic Theory. Reports on Mathematical Physics, 29, 109-121.
[44]
Mrugala, R. (1978) Geometrical Formulation of Equilibrium Phenomenological Thermodynamics. Reports on Mathematical Physics, 14, 419-427. https://doi.org/10.1016/0034-4877(78)90010-1
[45]
Mrugala, R. (1985) Submanifold in the Thermodynamic Phase Space. Reports on Mathematical Physics, 21, 197-203. https://doi.org/10.1016/0034-4877(85)90059-X
[46]
Mrugala, R., Nulton, J.D., Schön, J.C. and Salamon, P. (1990) Statistical Approach to the Geometric Structure of Thermodynamics. Physical Review A, 41, 3156-3160. https://doi.org/10.1103/PhysRevA.41.3156
[47]
Tu, L.W. (2008) An Introduction to Manifolds. Universitext. Springer Science + Business Media, LLC, New York.
[48]
Kluge, G. and Neugebauer, G. (1994) Grundlagen der Thermodynamik. Spektrum Akademischer Verlag, Heidelberg. (In German).
[49]
Abraham, R., Marsden, J.E. and Ratiu, T. (1988) Manifolds, Tensor Analysis, and Applications. 2nd Edition, Vol. 75, Springer-Verlag, New York.
[50]
Flanders, H. (1963) Differential Forms with Applications to the Physical Sciences. Vol. 11, Academic Press, New York.
[51]
Lang, C.B. and Pucker, N. (2005) Mathematische Methoden in der Physik. 2nd Edition, Spektrum Akademischer Verlag, München. (In German) https://doi.org/10.1007/978-3-8274-3125-7
