{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,22]],"date-time":"2026-10-22T11:38:54Z","timestamp":1792669134885,"version":"4.5.0"},"reference-count":54,"publisher":"AIP Publishing","issue":"16","license":[{"start":{"date-parts":[[2026,10,22]],"date-time":"2026-10-22T00:00:00Z","timestamp":1792627200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2026,10,28]]},"abstract":"<jats:p>The hard-disk model is a fundamental reference system for excluded-volume physics, liquid structure, and two-dimensional ordering. Yet, despite its apparent simplicity, an analytical formulation of its thermodynamics has remained difficult because the free volume available to disks\u2019 centers has a highly nontrivial many-body geometry. Here, we show that this geometry admits a closed and exact representation: for any given hard-disk configuration, the free volume can be expressed analytically through the intersection areas of up to five exclusion disks. This provides a direct geometrical route from particle coordinates to the configurational partition function and entropy. We prove that the N-disk partition function factorizes into a product of conditional free volumes and identify its two limiting asymptotic forms: the low-density fluid regime controlled by extensive cavities and the high-density fluid regime controlled by intensive private cells. Using the geometric measures computed from representative equilibrium particle configurations, the resulting theory reproduces the established hard-disk equation of state over almost the entire density range up to close packing. At intermediate densities, the analysis reveals a mixed-fluid regime in which localized caged defects provide an additional configurational entropy within the density interval preceding liquid\u2013hexatic coexistence. Finally, the five-disk intersection area is shown to be a sensitive local scalar measure of hexagonal ordering, complementing the orientational order parameter and emphasizing the central role of many-body correlations.<\/jats:p>","DOI":"10.1063\/5.0346090","type":"journal-article","created":{"date-parts":[[2026,10,22]],"date-time":"2026-10-22T10:48:11Z","timestamp":1792666091000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["Closed analytical form of many-body free volume and thermodynamics of monodisperse hard disks"],"prefix":"10.1063","volume":"165","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4804-5801","authenticated-orcid":false,"given":"Victor M.","family":"Pergamenshchik","sequence":"first","affiliation":[{"name":"Institute of Physics of the National Academy of Sciences of Ukraine 1 , Prospekt Nauky, 46, Kyiv 03039,","place":["Ukraine"]},{"name":"Center for Theoretical Physics, Polish Academy of Sciences 2 , Al. Lotnik\u00f3w 32\/46, 02-668 Warsaw,","place":["Poland"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4360-0634","authenticated-orcid":false,"given":"Taras","family":"Bryk","sequence":"additional","affiliation":[{"name":"Yukhnovskii Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine 3 , 1 Svientsitskii Str., Lviv 79011,","place":["Ukraine"]},{"name":"Institute of Applied Mathematics and Fundamental Sciences, Lviv National Polytechnic University 4 , 79013 Lviv,","place":["Ukraine"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2257-0779","authenticated-orcid":false,"given":"Andrij","family":"Trokhymchuk","sequence":"additional","affiliation":[{"name":"Yukhnovskii Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine 3 , 1 Svientsitskii Str., Lviv 79011,","place":["Ukraine"]},{"name":"Faculty of Chemistry and Chemical Technology, University of Ljubljana 5 , Ve\u010dna pot 113, 1000 Ljubljana,","place":["Slovenia"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2026,10,22]]},"reference":[{"key":"2026102206480886900_c1","doi-asserted-by":"publisher","first-page":"045003","DOI":"10.1103\/revmodphys.96.045003","article-title":"Colloidal hard spheres: Triumphs, challenges, and mysteries","volume":"96","year":"2024","journal-title":"Rev. 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