{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,3]],"date-time":"2026-02-03T17:36:53Z","timestamp":1770140213286,"version":"3.49.0"},"reference-count":31,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2020,9,1]],"date-time":"2020-09-01T00:00:00Z","timestamp":1598918400000},"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 Hamiltonian-based approach to the nonextensive thermodynamics of small systems, where small is a relative term comparing the size of the system to the size of the effective interaction region around it. We show that the effective Hamiltonian approach gives easy accessibility to the thermodynamic properties of systems strongly coupled to their surroundings. The theory does not rely on the classical concept of dividing surface to characterize the system\u2019s interaction with the environment. Instead, it defines an effective interaction region over which a system exchanges extensive quantities with its surroundings, easily producing laws recently shown to be valid at the nanoscale.<\/jats:p>","DOI":"10.3390\/e22090975","type":"journal-article","created":{"date-parts":[[2020,9,1]],"date-time":"2020-09-01T08:53:43Z","timestamp":1598950423000},"page":"975","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Strong Coupling and Nonextensive Thermodynamics"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4057-3570","authenticated-orcid":false,"given":"Rodrigo","family":"de Miguel","sequence":"first","affiliation":[{"name":"Department of Teacher Education, Norwegian University of Science and Technology, 7491 Trondheim, Norway"}]},{"given":"J. Miguel","family":"Rub\u00ed","sequence":"additional","affiliation":[{"name":"Department of Condensed Matter Physics, University of Barcelona, 08007 Barcelona, Spain"},{"name":"PoreLab\u2014Center of Excellence, Norwegian University of Science and Technology, 7491 Trondheim, Norway"}]}],"member":"1968","published-online":{"date-parts":[[2020,9,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Campa, A., Dauxois, T., Fanelli, D., and Ruffo, S. (2014). Physics of Long-Range Interacting Systems, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780199581931.001.0001"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"6210","DOI":"10.1038\/ncomms7210","article-title":"Visualization and thermodynamic encoding of single-molecule partition function projections","volume":"6","author":"Palma","year":"2015","journal-title":"Nat. Commun."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"10345","DOI":"10.1021\/jp0722657","article-title":"Thermodynamics of Atomic Clusters Using Variational Quantum Hydrodynamics","volume":"111","author":"Derrickson","year":"2007","journal-title":"J. Phys. Chem. A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"S215","DOI":"10.1088\/0953-8984\/18\/14\/S06","article-title":"Unzipping DNA by force: Thermodynamics and finite size behaviour","volume":"18","author":"Kapri","year":"2006","journal-title":"J. Phys. Condens. Matter"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1021\/ar0001719","article-title":"The physics of molecular motors","volume":"34","author":"Bustamante","year":"2001","journal-title":"Acc. Chem. Res."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1007\/s11705-019-1800-9","article-title":"Preparation of adsorptive nanoporous membrane using powder activated carbon: Isotherm and thermodynamic studies","volume":"14","author":"Peyravi","year":"2020","journal-title":"Front. Chem. Sci. Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"60","DOI":"10.3389\/fphy.2019.00060","article-title":"Pressures Inside a Nano-Porous Medium. The Case of a Single Phase Fluid","volume":"7","author":"Galteland","year":"2019","journal-title":"Front. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"21502","DOI":"10.1021\/jp052904i","article-title":"The Mesoscopic Dynamics of Thermodynamic Systems","volume":"109","author":"Reguera","year":"2005","journal-title":"J. Phys. Chem. B"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1191","DOI":"10.1103\/PhysRevLett.86.1191","article-title":"Negative Heat Capacity for a Cluster of 147 Sodium Atoms","volume":"86","author":"Schmidt","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1140\/epje\/e2006-00012-9","article-title":"The \u201cmacromolecular tourist\u201d: Universal temperature dependence of thermal diffusion in aqueous colloidal suspensions","volume":"19","author":"Iacopini","year":"2006","journal-title":"Eur. Phys. J. E"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"230601","DOI":"10.1103\/PhysRevLett.114.230601","article-title":"Thermodynamics of Nonadditive Systems","volume":"114","author":"Latella","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"022143","DOI":"10.1103\/PhysRevE.94.022143","article-title":"Open system trajectories specify fluctuating work but not heat","volume":"94","author":"Talkner","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"020601","DOI":"10.1103\/PhysRevLett.116.020601","article-title":"First and Second Law of Thermodynamics at Strong Coupling","volume":"116","author":"Seifert","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_14","first-page":"011008","article-title":"Stochastic and Macroscopic Thermodynamics of Strongly Coupled Systems","volume":"7","author":"Jarzynski","year":"2017","journal-title":"Phys. Rev. X"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"200602","DOI":"10.1103\/PhysRevLett.123.200602","article-title":"Negative thermophoretic force in the strong coupling regime","volume":"123","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1088\/0370-1301\/70\/2\/301","article-title":"Temperature Dependent Energy Levels in Statistical Mechanics","volume":"70","author":"Elcock","year":"1957","journal-title":"Proc. Phys. Soc. Lond. Sect. B"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"10006","DOI":"10.1209\/0295-5075\/85\/10006","article-title":"Shannon meets Carnot: Generalized second thermodynamic law","volume":"85","author":"Shental","year":"2009","journal-title":"Europhys. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"9180","DOI":"10.1021\/acs.jpcb.6b05591","article-title":"Finite Systems in a Heat Bath: Spectrum Perturbations and Thermodynamics","volume":"120","year":"2016","journal-title":"J. Phys. Chem. B"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"7011","DOI":"10.1039\/C5CP07572F","article-title":"Efficiencies of thermodynamics when temperature-dependent energy levels exist","volume":"18","author":"Yamano","year":"2016","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"10855","DOI":"10.1038\/s41598-019-47288-0","article-title":"Heat capacities of thermally manipulated mechanical oscillator at strong coupling","volume":"9","author":"Ryabov","year":"2019","journal-title":"Sci. Rep."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3182","DOI":"10.1063\/1.1732447","article-title":"Thermodynamics of Small Systems","volume":"36","author":"Hill","year":"1962","journal-title":"J. Chem. Phys."},{"key":"ref_22","unstructured":"Hill, T.L. (2013). Thermodynamics of Small Systems, Parts I & II, Dover."},{"key":"ref_23","unstructured":"Gibbs, J.W., Bumstead, H.A., and Van Name, R.G. (1906). Scientific Papers of J. Willard Gibbs: Thermodynamics, Longmans, Green and Co."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.25103\/jestr.011.01","article-title":"What is a surface excess?","volume":"1","year":"2008","journal-title":"J. Eng. Sci. Tech. Rev."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"23936","DOI":"10.1038\/srep23936","article-title":"Effectiveness of the Young-Laplace equation at nanoscale","volume":"6","author":"Liu","year":"2016","journal-title":"Sci. Rep."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1021\/acs.jpclett.7b03003","article-title":"Capillary Condensation in 8 nm Deep Channels","volume":"9","author":"Zhong","year":"2018","journal-title":"J. Phys. Chem. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"096101","DOI":"10.1103\/PhysRevLett.111.096101","article-title":"Young\u2019s Equation at the Nanoscale","volume":"111","author":"Seveno","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1063\/1.1747247","article-title":"The Effect of Droplet Size on Surface Tension","volume":"17","author":"Tolman","year":"1949","journal-title":"J. Chem. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"8795","DOI":"10.1021\/acs.jpcc.0c00816","article-title":"Interfacial Free Energy and Tolman Length of Curved Liquid\u2013Solid Interfaces from Equilibrium Studies","volume":"124","author":"Bianco","year":"2020","journal-title":"J. Phys. Chem. C"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1080\/07388551.2019.1616668","article-title":"Prospects and application of nanobiotechnology in food preservation: Molecular perspectives","volume":"39","author":"Bhuyan","year":"2019","journal-title":"Crit. Rev. Biotechnol."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Chandra, P., and Prakash, R. (2020). Nanobiomaterials in Drug Delivery: Designing Strategies and Critical Concepts for Their Potential Clinical Applications. Nanobiomaterial Engineering, Springer.","DOI":"10.1007\/978-981-32-9840-8"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/9\/975\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:05:26Z","timestamp":1760177126000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/9\/975"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9,1]]},"references-count":31,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2020,9]]}},"alternative-id":["e22090975"],"URL":"https:\/\/doi.org\/10.3390\/e22090975","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,9,1]]}}}