Abstract
In this paper, we make a review of our recent development of Dirac structures and the associated variational formulation for nonequilibrium thermodynamics (see, [15, 16]). We specifically focus on the case of simple and open systems, in which the thermodynamic state is represented by one single entropy and the transfer of matter and heat with the exterior is included. We clarify the geometric structure by introducing an induced Dirac structure on the covariant Pontryagin bundle and then develop the associated dynamical system (the port-Dirac systems) in the context of time-dependent nonholonomic systems with nonlinear constraints of thermodynamic type. We also present the variational structure associated with the Dirac formulation in the context of the generalized Lagrange-d’Alembert-Pontryagin principle.
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Acknowledgements
H.Y. is partially supported by JSPS Grant-in-Aid for Scientific Research (17H01097), JST CREST Grant Number JPMJCR1914, the MEXT Top Global University Project, Waseda University (SR 2020C-194) and the Organization for University Research Initiatives (Evolution and application of energy conversion theory in collaboration with modern mathematics). F.G.B. is partially supported by the ANR project GEOMFLUID, ANR-14-CE23-0002-01.
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Yoshimura, H., Gay-Balmaz, F. (2021). Dirac Structures and Variational Formulation of Thermodynamics for Open 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_12
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