{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T00:44:23Z","timestamp":1776732263387,"version":"3.51.2"},"reference-count":23,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2014,3,19]],"date-time":"2014-03-19T00:00:00Z","timestamp":1395187200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The principal methods for the definition of thermodynamic entropy are discussed with special reference to those developed by Carath\u00e9odory, the Keenan School, Lieb and Yngvason, and the present authors. An improvement of the latter method is then presented. Seven basic axioms are employed: three Postulates, which are considered as having a quite general validity, and four Assumptions, which identify the domains of validity of the definitions of energy (Assumption 1) and entropy (Assumptions 2, 3, 4). The domain of validity of the present definition of entropy is not restricted to stable equilibrium states. For collections of simple systems, it coincides with that of the proof of existence and uniqueness of an entropy function which characterizes the relation of adiabatic accessibility proposed by Lieb and Yngvason. However, our treatment does not require the formation of scaled copies so that it applies not only to collections of simple systems, but also to systems contained in electric or magnetic fields and to small and few-particle systems.<\/jats:p>","DOI":"10.3390\/e16031547","type":"journal-article","created":{"date-parts":[[2014,3,19]],"date-time":"2014-03-19T13:33:16Z","timestamp":1395235996000},"page":"1547-1570","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":31,"title":["Recent Progress in the Definition of Thermodynamic Entropy"],"prefix":"10.3390","volume":"16","author":[{"given":"Enzo","family":"Zanchini","sequence":"first","affiliation":[{"name":"Department of Industrial Engineering, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy"}]},{"given":"Gian","family":"Beretta","sequence":"additional","affiliation":[{"name":"Department of Mechanical and Industrial Engineering, University of Brescia, Via Branze 38, 25123 Brescia, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2014,3,19]]},"reference":[{"key":"ref_1","unstructured":"Callen, H.B. (1960). Thermodynamics, Wiley."},{"key":"ref_2","unstructured":"Pippard, A.B. (1957). The Elements of Classical Thermodynamics, Cambridge University Press."},{"key":"ref_3","unstructured":"Zemansky, M.W. (1968). Heat and Thermodynamics, Mc Graw-Hill."},{"key":"ref_4","unstructured":"Planck, M. (1927). Treatise on Thermodynamics, Longmans, Green, and Co."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1007\/BF01450409","article-title":"Untersuchungen \u00fcber die Grundlagen der Thermodynamik","volume":"67","year":"1909","journal-title":"Math. Ann"},{"key":"ref_6","unstructured":"Fermi, E. (1937). Thermodynamics, Prentice-Hall."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"781","DOI":"10.1119\/1.1936001","article-title":"Simplification of Carath\u00e9odory\u2019s treatment of thermodynamics","volume":"28","author":"Turner","year":"1960","journal-title":"Am. J. Phys"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1002\/pssb.19610010203","article-title":"On suggested simplifications of Carath\u00e9odory\u2019s thermodynamics","volume":"1","author":"Landsberg","year":"1961","journal-title":"Phys. Status Solidi B"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"747","DOI":"10.1119\/1.1969067","article-title":"A simplified simplification of Carath\u00e9odory\u2019s thermodynamics","volume":"31","author":"Saers","year":"1963","journal-title":"Am. J. Phys"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Giles, R. (1964). Mathematical Foundations of Thermodynamics, Pergamon.","DOI":"10.1016\/B978-0-08-010071-5.50019-1"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0370-1573(98)00082-9","article-title":"The physics and mathematics of the second law of thermodynamics","volume":"310","author":"Lieb","year":"1999","journal-title":"Phys. Rep"},{"key":"ref_12","unstructured":"Keenan, J.H. (1941). Thermodynamics, Wiley."},{"key":"ref_13","unstructured":"Hatsopoulos, G.N., and Keenan, J.H. (1965). Principles of General Thermodynamics, Wiley."},{"key":"ref_14","unstructured":"Gyftopoulos, E.P., and Beretta, G.P. (2005). Thermodynamics: Foundations and Applications, Dover."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1007\/BF00765339","article-title":"On the definition of extensive property energy by the first postulate of thermodynamics","volume":"16","author":"Zanchini","year":"1986","journal-title":"Found. Phys"},{"key":"ref_16","first-page":"147","article-title":"The entropy of classical thermodynamics","volume":"Chapter 8","author":"Greven","year":"2003","journal-title":"Entropy"},{"key":"ref_17","first-page":"9408","article-title":"The entropy concept for nonequilibrium states","volume":"469","author":"Lieb","year":"2013","journal-title":"Proc. Royal Society A"},{"key":"ref_18","first-page":"67","article-title":"Removing Heat and Conceptual Loops from the Definition of Entropy","volume":"12","author":"Zanchini","year":"2010","journal-title":"Int. J. Thermodynamics"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Tadashi, M. (2011). Thermodynamics, InTech.","DOI":"10.5772\/558"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1103\/RevModPhys.62.745","article-title":"Boundary conditions for open quantum systems driven far from equilibrium","volume":"62","author":"Frensley","year":"1990","journal-title":"Rev. Mod. Phys"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1103\/RevModPhys.83.771","article-title":"Quantum fluctuation relations: Foundations and applications","volume":"83","author":"Campisi","year":"2011","journal-title":"Rev. Mod. Phys"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2012.11.001","article-title":"Understanding quantum measurement from the solution of dynamical models","volume":"525","author":"Allahverdyan","year":"2013","journal-title":"Phys. Reps"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2059","DOI":"10.1038\/ncomms3059","article-title":"Fundamental limitations for quantum and nanoscale thermodynamics","volume":"4","author":"Horodecki","year":"2013","journal-title":"Nature Comm"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/3\/1547\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:09:23Z","timestamp":1760216963000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/3\/1547"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,3,19]]},"references-count":23,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2014,3]]}},"alternative-id":["e16031547"],"URL":"https:\/\/doi.org\/10.3390\/e16031547","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,3,19]]}}}