{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T11:01:59Z","timestamp":1774263719037,"version":"3.50.1"},"reference-count":46,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2018,12,4]],"date-time":"2018-12-04T00:00:00Z","timestamp":1543881600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/e20120925","type":"journal-article","created":{"date-parts":[[2018,12,4]],"date-time":"2018-12-04T11:56:18Z","timestamp":1543924578000},"page":"925","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":53,"title":["Geometry of Thermodynamic Processes"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2383-9234","authenticated-orcid":false,"given":"Arjan","family":"Van der Schaft","sequence":"first","affiliation":[{"name":"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"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0221-2843","authenticated-orcid":false,"given":"Bernhard","family":"Maschke","sequence":"additional","affiliation":[{"name":"Laboratoire d\u2019automatique et de g\u00e9nie des proc\u00e9d\u00e9s (LAGEP) (UMR CNRS 5007), Universit\u00e9 Claude Bernard Lyon 1, CNRS, 69622 Villeurbanne, France"}]}],"member":"1968","published-online":{"date-parts":[[2018,12,4]]},"reference":[{"key":"ref_1","unstructured":"Hermann, R. (1973). Geometry, Physics and Systems, Marcel Dekker."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1016\/0034-4877(78)90010-1","article-title":"Geometric formulation of equilibrium phenomenological thermodynamics","volume":"14","year":"1978","journal-title":"Rep. Math. 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