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The Geometry of Some Thermodynamic Systems

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Geometric Structures of Statistical Physics, Information Geometry, and Learning (SPIGL 2020)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 361))

Abstract

In this article, we continue the program started in [2] of exploring an important class of thermodynamic systems from a geometric point of view. The contents of this paper and the one already published in [2] provide a geometrical formulation, which tries to shed more light on the properties of thermodynamic systems without claiming to be a definitive theory. In order to model the time evolution of systems verifying the two laws of thermodynamics, we show that the notion of evolution vector field is adequate to appropriately describe such systems. Our formulation naturally arises from the introduction of a skew-symmetric bracket to which numerical methods based on discrete gradients fit nicely. Moreover, we study the corresponding Lagrangian and Hamiltonian formalism, discussing the fundamental principles from which the equations are derived. An important class of systems that is naturally covered by our formalism are composed thermodynamic systems, which are described by at least two thermal variables and exchange heat between its components.

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Acknowledgements

The authors acknowledge financial support from the Spanish Ministry of Science and Innovation, under grants PID2019-106715GB-C21, MTM2016-76702-P, “Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2019-000904-S) and from the Spanish National Research Council, through the “Ayuda extraordinaria a Centros de Excelencia Severo Ochoa” (20205CEX001). A. Simoes is supported by the FCT (Portugal) research fellowship SFRH/BD/129882/2017 partially funded by the European Union (ESF). The authors would also like to thank the referees for their useful comments and remarks that helped to improve the content of the paper.

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Authors and Affiliations

  1. Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Calle Nicolás Cabrera, 13-15, Campus Cantoblanco, UAM, 28049, Madrid, Spain

    Alexandre Anahory Simoes, David Martín de Diego, Manuel Lainz Valcázar & Manuel de León

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  1. Alexandre Anahory Simoes
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  2. David Martín de Diego
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  3. Manuel Lainz Valcázar
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  4. Manuel de León
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Correspondence to Manuel de León .

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  1. Thales Land & Air Systems, Technical Directorate, Thales, Limours, France

    Frédéric Barbaresco

  2. Sony Computer Science Laboratories Inc., Tokyo, Japan

    Frank Nielsen

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Simoes, A.A., de Diego, D.M., Valcázar, M.L., de León, M. (2021). The Geometry of Some Thermodynamic Systems. In: Barbaresco, F., Nielsen, F. (eds) Geometric Structures of Statistical Physics, Information Geometry, and Learning. SPIGL 2020. Springer Proceedings in Mathematics & Statistics, vol 361. Springer, Cham. https://doi.org/10.1007/978-3-030-77957-3_13

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