{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,15]],"date-time":"2026-10-15T23:41:50Z","timestamp":1792107710555,"version":"4.4.0"},"reference-count":25,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2018,3,4]],"date-time":"2018-03-04T00:00:00Z","timestamp":1520121600000},"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>We propose a variational formulation for the nonequilibrium thermodynamics of discrete open systems, i.e., discrete systems which can exchange mass and heat with the exterior. Our approach is based on a general variational formulation for systems with time-dependent nonlinear nonholonomic constraints and time-dependent Lagrangian. For discrete open systems, the time-dependent nonlinear constraint is associated with the rate of internal entropy production of the system. We show that this constraint on the solution curve systematically yields a constraint on the variations to be used in the action functional. The proposed variational formulation is intrinsic and provides the same structure for a wide class of discrete open systems. We illustrate our theory by presenting examples of open systems experiencing mechanical interactions, as well as internal diffusion, internal heat transfer, and their cross-effects. Our approach yields a systematic way to derive the complete evolution equations for the open systems, including the expression of the internal entropy production of the system, independently on its complexity. It might be especially useful for the study of the nonequilibrium thermodynamics of biophysical systems.<\/jats:p>","DOI":"10.3390\/e20030163","type":"journal-article","created":{"date-parts":[[2018,3,6]],"date-time":"2018-03-06T07:37:25Z","timestamp":1520321845000},"page":"163","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":28,"title":["A Variational Formulation of Nonequilibrium Thermodynamics for Discrete Open Systems with Mass and Heat Transfer"],"prefix":"10.3390","volume":"20","author":[{"given":"Fran\u00e7ois","family":"Gay-Balmaz","sequence":"first","affiliation":[{"name":"Centre National de la Recherche Scientifique (CNRS), Le Laboratoire de M\u00e9t\u00e9orologie Dynamique (LMD), Ecole Normale Sup\u00e9rieure, 75005 Paris, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hiroaki","family":"Yoshimura","sequence":"additional","affiliation":[{"name":"School of Science and Engineering, Waseda University, Tokyo 169-8555, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2018,3,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1088\/0143-0807\/20\/4\/303","article-title":"Thermodynamics of systems with internal adiabatic constraints: Time evolution of the adiabatic piston","volume":"20","author":"Gruber","year":"1999","journal-title":"Eur. 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