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Entropy 2018, 20(12), 925;

Geometry of Thermodynamic Processes

Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, Jan C. Willems Center for Systems and Control, University of Groningen, P.O. Box 407, 9700 AK Groningen, The Netherlands
Laboratoire d’automatique et de génie des procédés (LAGEP) (UMR CNRS 5007), Université Claude Bernard Lyon 1, CNRS, 69622 Villeurbanne, France
Author to whom correspondence should be addressed.
Received: 9 November 2018 / Revised: 29 November 2018 / Accepted: 30 November 2018 / Published: 4 December 2018
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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. View Full-Text
Keywords: thermodynamics; symplectization; metrics; non-equilibrium processes; interconnection thermodynamics; symplectization; metrics; non-equilibrium processes; interconnection
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Van der Schaft, A.; Maschke, B. Geometry of Thermodynamic Processes. Entropy 2018, 20, 925.

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