{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T19:53:28Z","timestamp":1762113208956,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2010,3,16]],"date-time":"2010-03-16T00:00:00Z","timestamp":1268697600000},"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 differential Shannon entropy of information theory can change under a change of variables (coordinates), but the thermodynamic entropy of a physical system must be invariant under such a change. This difference is puzzling, because the Shannon and Gibbs entropies have the same functional form. We show that a canonical change of variables can, indeed, alter the spatial component of the thermodynamic entropy just as it alters the differential Shannon entropy. However, there is also a momentum part of the entropy, which turns out to undergo an equal and opposite change when the coordinates are transformed, so that the total thermodynamic entropy remains invariant. We furthermore show how one may correctly write the change in total entropy for an isothermal physical process in any set of spatial coordinates.<\/jats:p>","DOI":"10.3390\/e12030578","type":"journal-article","created":{"date-parts":[[2010,3,16]],"date-time":"2010-03-16T14:30:07Z","timestamp":1268749807000},"page":"578-590","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":37,"title":["Thermodynamic and Differential Entropy under a Change of Variables"],"prefix":"10.3390","volume":"12","author":[{"given":"Vladimir","family":"Hnizdo","sequence":"first","affiliation":[{"name":"Health Effects Laboratory Division, National Institute for Occupational Safety and Health, Morgantown, West Virginia 26505, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael K.","family":"Gilson","sequence":"additional","affiliation":[{"name":"Center for Advanced Research in Biotechnology, University of Maryland Biotechnology Institute, Rockville, Maryland 20850, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2010,3,16]]},"reference":[{"key":"ref_1","unstructured":"Campbell, J. (1982). Grammatical Man; Information, Entropy, Language & Life, Simon and Schuster."},{"key":"ref_2","unstructured":"Cover, T.M., and Thomas, J.A. (1991). Elements of Information Theory, Wiley."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1007\/BF02477270","article-title":"The entropy of a continuous distribution","volume":"27","author":"White","year":"1965","journal-title":"Bull. Math. Biophys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1021\/ma60052a001","article-title":"On the use of classical statistical mechanics in the treatment of polymer chain conformation","volume":"9","author":"Scheraga","year":"1976","journal-title":"Macromolecules"},{"key":"ref_5","unstructured":"Landau, L.D., and Lifshitz, E.M. (1989). Statistical Physics, Part 1, Butterworth-Heinemann."},{"key":"ref_6","unstructured":"Landau, L.D., and Lifshitz, E.M. (1981). Quantum Mechanics, Butterworth-Heinemann."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1011","DOI":"10.1080\/00268978400100661","article-title":"Entropy estimation from simulations of non-diffusive systems","volume":"51","author":"Edholm","year":"1984","journal-title":"Mol. Phys."},{"key":"ref_8","unstructured":"Landau, L.D., and Lifshitz, E.M. (1976). Mechanics, Butterworth-Heinemann."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1047","DOI":"10.1016\/S0006-3495(97)78756-3","article-title":"The statistical-thermodynamic basis for computation of binding affinites: A critical review","volume":"72","author":"Gilson","year":"1997","journal-title":"Biophys. J."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"024107","DOI":"10.1063\/1.2746329","article-title":"Extraction of configurational entropy from molecular simulations via an expansion approximation","volume":"127","author":"Killian","year":"2007","journal-title":"J. Chem. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1016\/j.jmb.2009.04.003","article-title":"Configurational entropy in protein-protein binding: Computational study of Tsg101 ubiquitin E2 variant domain with an HIV-derived PTAP nonapeptide","volume":"389","author":"Killian","year":"2009","journal-title":"J. Mol. Biol."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1021\/ma50003a019","article-title":"Method for estimating the configurational entropy of macromolecules","volume":"14","author":"Karplus","year":"1981","journal-title":"Macromolecules"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1510","DOI":"10.1063\/1.1670272","article-title":"General internal motion of molecules, classical and quantum-mechanical Hamiltonian","volume":"49","author":"Meyer","year":"1969","journal-title":"J. Chem. Phys."},{"key":"ref_14","unstructured":"Turrell, G. (2002). Mathematics for Chemistry and Physics, Academic Press."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/12\/3\/578\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T22:01:46Z","timestamp":1760220106000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/12\/3\/578"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,3,16]]},"references-count":14,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2010,3]]}},"alternative-id":["e12030578"],"URL":"https:\/\/doi.org\/10.3390\/e12030578","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2010,3,16]]}}}