Geometry of Thermodynamic Processes

Authors

Arjan Van der Schaft, Bernhard Maschke

Abstract

Since the 1970s, contact geometry has been recognized as an appropriate framework for the geometric formulation of thermodynamic systems, and in particular their state properties. More recently it has been shown how the symplectization of contact manifolds provides a new vantage point; enabling, among other things, to switch easily between the energy and entropy representations of a thermodynamic system. In the present paper, this is continued towards the global geometric definition of a degenerate Riemannian metric on the homogeneous Lagrangian submanifold describing the state properties, which is overarching the locally-defined metrics of Weinhold and Ruppeiner. Next, a geometric formulation is given of non-equilibrium thermodynamic processes, in terms of Hamiltonian dynamics defined by Hamiltonian functions that are homogeneous of degree one in the co-extensive variables and zero on the homogeneous Lagrangian submanifold. The correspondence between objects in contact geometry and their homogeneous counterparts in symplectic geometry, is extended to the definition of port-thermodynamic systems and the formulation of interconnection ports. The resulting geometric framework is illustrated on a number of simple examples, already indicating its potential for analysis and control.

Citation

• Journal: Entropy
• Year: 2018
• Volume: 20
• Issue: 12
• Pages: 925
• Publisher: MDPI AG
• DOI: 10.3390/e20120925

BibTeX

@article{Van_der_Schaft_2018,
  title={{Geometry of Thermodynamic Processes}},
  volume={20},
  ISSN={1099-4300},
  DOI={10.3390/e20120925},
  number={12},
  journal={Entropy},
  publisher={MDPI AG},
  author={Van der Schaft, Arjan and Maschke, Bernhard},
  year={2018},
  pages={925}
}

Download the bib file

References

• MrugaŁa, R. Geometrical formulation of equilibrium phenomenological thermodynamics. Reports on Mathematical Physics vol. 14 419–427 (1978) – 10.1016/0034-4877(78)90010-1
• On equivalence of two metrics in classical thermodynamics. Physica (1984)
• Mrugała, R. Submanifolds in the thermodynamic phase space. Reports on Mathematical Physics vol. 21 197–203 (1985) – 10.1016/0034-4877(85)90059-x
• On contact and metric structures on thermodynamic spaces. RIMS Kokyuroku (2000)
• On a special family of thermodynamic processes and their invariants. Rep. Math. Phys. (2000)
• Mrugala, R., Nulton, J. D., Schön, J. C. & Salamon, P. Statistical approach to the geometric structure of thermodynamics. Physical Review A vol. 41 3156–3160 (1990) – 10.1103/physreva.41.3156
• Mrugala, R., Nulton, J. D., Christian Schön, J. & Salamon, P. Contact structure in thermodynamic theory. Reports on Mathematical Physics vol. 29 109–121 (1991) – 10.1016/0034-4877(91)90017-h
• Bravetti, A., Lopez-Monsalvo, C. S. & Nettel, F. Contact symmetries and Hamiltonian thermodynamics. Annals of Physics vol. 361 377–400 (2015) – 10.1016/j.aop.2015.07.010
• Bravetti, A., Lopez-Monsalvo, C. & Nettel, F. Conformal Gauge Transformations in Thermodynamics. Entropy vol. 17 6150–6168 (2015) – 10.3390/e17096150
• Eberard, D., Maschke, B. M. & van der Schaft, A. J. An extension of Hamiltonian systems to the thermodynamic phase space: Towards a geometry of nonreversible processes. Reports on Mathematical Physics vol. 60 175–198 (2007) – 10.1016/s0034-4877(07)00024-9
• Favache, A., Dos Santos Martins, V. S., Dochain, D. & Maschke, B. Some Properties of Conservative Port Contact Systems. IEEE Transactions on Automatic Control vol. 54 2341–2351 (2009) – 10.1109/tac.2009.2028973
• Favache, A., Dochain, D. & Maschke, B. An entropy-based formulation of irreversible processes based on contact structures. Chemical Engineering Science vol. 65 5204–5216 (2010) – 10.1016/j.ces.2010.06.019
• Gay-Balmaz, F. & Yoshimura, H. A Lagrangian variational formulation for nonequilibrium thermodynamics. Part I: Discrete systems. Journal of Geometry and Physics vol. 111 169–193 (2017) – 10.1016/j.geomphys.2016.08.018
• Gromov, D. Two Approaches to the Description of the Evolution of Thermodynamic Systems. IFAC-PapersOnLine vol. 49 34–39 (2016) – 10.1016/j.ifacol.2016.10.749
• Merker, J. & Krüger, M. On a variational principle in thermodynamics. Continuum Mechanics and Thermodynamics vol. 25 779–793 (2012) – 10.1007/s00161-012-0277-2
• Ramirez, H., Maschke, B. & Sbarbaro, D. Feedback equivalence of input–output contact systems. Systems & Control Letters vol. 62 475–481 (2013) – 10.1016/j.sysconle.2013.02.008
• Ramirez, H., Maschke, B. & Sbarbaro, D. Irreversible port-Hamiltonian systems: A general formulation of irreversible processes with application to the CSTR. Chemical Engineering Science vol. 89 223–234 (2013) – 10.1016/j.ces.2012.12.002
• Ramirez, H., Maschke, B. & Sbarbaro, D. Partial Stabilization of Input-Output Contact Systems on a Legendre Submanifold. IEEE Transactions on Automatic Control vol. 62 1431–1437 (2017) – 10.1109/tac.2016.2572403
• Gromov, D. & Castaños, F. The geometric structure of interconnected thermo-mechanical systems. IFAC-PapersOnLine vol. 50 582–587 (2017) – 10.1016/j.ifacol.2017.08.083
• Balian, R. & Valentin, P. Hamiltonian structure of thermodynamics with gauge. The European Physical Journal B vol. 21 269–282 (2001) – 10.1007/s100510170202
• Arnold, V. I. Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics (Springer New York, 1989). doi:10.1007/978-1-4757-2063-1 – 10.1007/978-1-4757-2063-1
• Libermann, P. & Marle, C.-M. Symplectic Geometry and Analytical Mechanics. (Springer Netherlands, 1987). doi:10.1007/978-94-009-3807-6 – 10.1007/978-94-009-3807-6
• Maschke, B. & van der Schaft, A. Homogeneous Hamiltonian Control Systems Part II: Application to thermodynamic systems. IFAC-PapersOnLine vol. 51 7–12 (2018) – 10.1016/j.ifacol.2018.06.002
• Maschke, Homogeneous Hamiltonian control systems, Part I: Geometric formulation. IFAC-Papers OnLine (2018)
• Weinhold, F. Metric geometry of equilibrium thermodynamics. The Journal of Chemical Physics vol. 63 2479–2483 (1975) – 10.1063/1.431689
• Ruppeiner, G. Thermodynamics: A Riemannian geometric model. Physical Review A vol. 20 1608–1613 (1979) – 10.1103/physreva.20.1608
• Maschke, B. M. & van der Schaft, A. J. Port-Controlled Hamiltonian Systems: Modelling Origins and Systemtheoretic Properties. IFAC Proceedings Volumes vol. 25 359–365 (1992) – 10.1016/s1474-6670(17)52308-3
• Maschke, The Hamiltonian formulation of energy conserving physical systems with external ports. Archiv für Elektronik und Übertragungstechnik (1995)
• van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control vol. 1 173–378 (2014) – 10.1561/2600000002
• Tisza, L. The thermodynamics of phase equilibrium. Annals of Physics vol. 13 1–92 (1961) – 10.1016/0003-4916(61)90027-6
• Amari, S. Information Geometry and Its Applications. Applied Mathematical Sciences (Springer Japan, 2016). doi:10.1007/978-4-431-55978-8 – 10.1007/978-4-431-55978-8
• Bullo, F. & Lewis, A. D. Geometric Control of Mechanical Systems. Texts in Applied Mathematics (Springer New York, 2005). doi:10.1007/978-1-4899-7276-7 – 10.1007/978-1-4899-7276-7
• Variational and Hamiltonian Control Systems. Lecture Notes in Control and Information Sciences (Springer Berlin Heidelberg, 1987). doi:10.1007/bfb0042858 – 10.1007/bfb0042858
• Grmela, M. Contact Geometry of Mesoscopic Thermodynamics and Dynamics. Entropy vol. 16 1652–1686 (2014) – 10.3390/e16031652
• Morrison, P. J. A paradigm for joined Hamiltonian and dissipative systems. Physica D: Nonlinear Phenomena vol. 18 410–419 (1986) – 10.1016/0167-2789(86)90209-5
• Hamiltonian dynamics with external forces and observations. Math. Syst. Theory (1982)
• van der Schaft, A. J. System theory and mechanics. Lecture Notes in Control and Information Sciences 426–452 (1989) doi:10.1007/bfb0008472 – 10.1007/bfb0008472
• van der Schaft, A. L2-Gain and Passivity Techniques in Nonlinear Control. Communications and Control Engineering (Springer International Publishing, 2017). doi:10.1007/978-3-319-49992-5 – 10.1007/978-3-319-49992-5
• Willems, J. C. Dissipative dynamical systems part I: General theory. Archive for Rational Mechanics and Analysis vol. 45 321–351 (1972) – 10.1007/bf00276493
• Nijmeijer, H. & van der Schaft, A. Nonlinear Dynamical Control Systems. (Springer New York, 1990). doi:10.1007/978-1-4757-2101-0 – 10.1007/978-1-4757-2101-0