{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T20:01:27Z","timestamp":1773259287288,"version":"3.50.1"},"reference-count":63,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2008,9,19]],"date-time":"2008-09-19T00:00:00Z","timestamp":1221782400000},"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>We revisit the equilibrium properties of a classical one-dimensional system of hardcore particles in the framework provided by the multiparticle correlation expansion of the configurational entropy. The vanishing of the cumulative contribution of more-than-two-particle correlations to the excess entropy is put in relation with the onset of a solidlike behavior at high densities.<\/jats:p>","DOI":"10.3390\/e10030248","type":"journal-article","created":{"date-parts":[[2008,9,29]],"date-time":"2008-09-29T05:43:40Z","timestamp":1222667020000},"page":"248-260","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":26,"title":["Entropy and Ordering of Hard Rods in One Dimension"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4851-5105","authenticated-orcid":false,"given":"Paolo V.","family":"Giaquinta","sequence":"first","affiliation":[{"name":"Universit\u00e0 degli Studi di Messina, Dipartimento di Fisica, Contrada Papardo, 98166 Messina, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2008,9,19]]},"reference":[{"key":"ref_1","unstructured":"Green, H.S. (1952). The molecular theory of fluids."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1365","DOI":"10.1063\/1.1744724","article-title":"Expression in terms of molecular distribution functions for the entropy density in an infinite system","volume":"29(6)","author":"Nettleton","year":"1958","journal-title":"The Journal of Chemical Physics"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3817","DOI":"10.1103\/PhysRevA.40.3817","article-title":"Direct entropy calculation from computer simulation of liquids","volume":"40(7)","author":"Baranyai","year":"1989","journal-title":"Physical Review A"},{"key":"ref_4","unstructured":"Yvon, J. (1969). Correlations and entropy in classical statistical mechanics, Pergamon Press."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"548","DOI":"10.1119\/1.1975671","article-title":"Entropy differences between ideal and nonideal systems","volume":"37(5)","author":"Leff","year":"1969","journal-title":"American Journal of Physics"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"849","DOI":"10.1103\/PhysRevA.42.849","article-title":"Three-particle contribution to the configurational entropy of simple fluids","volume":"42(2)","author":"Baranyai","year":"1990","journal-title":"Physical Review A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1515\/zna-1991-1-205","article-title":"On the entropy of the hard sphere fluid","volume":"46","author":"Baranyai","year":"1991","journal-title":"Zeitschrift f\u00fcr Naturforschung A"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/0378-4371(92)90415-M","article-title":"About entropy and correlations in a fluid of hard spheres","volume":"187","author":"Giaquinta","year":"1992","journal-title":"Physica A"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"R6966","DOI":"10.1103\/PhysRevA.45.R6966","article-title":"Entropy and the freezing of simple liquids","volume":"45(10)","author":"Giaquinta","year":"1992","journal-title":"Phys. Rev. A"},{"key":"ref_10","first-page":"B75","article-title":"Phase diagram of simple fluids: a comprehensive theo-retical approach","volume":"5(34B)","author":"Caccamo","year":"1993","journal-title":"Journal of Physics: Condensed Matter"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1200","DOI":"10.1103\/PhysRevLett.71.1200","article-title":"Prediction of the phase diagram of rigid C60 molecules","volume":"71(8)","author":"Cheng","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"8367","DOI":"10.1063\/1.466781","article-title":"Role of the interaction range in the shaping of phase diagrams in simple fluids. The hard sphere Yukawa fluid as a case study","volume":"100(11)","author":"Lomba","year":"1994","journal-title":"The Journal of Chemical Physics"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"5164","DOI":"10.1103\/PhysRevE.49.5164","article-title":"Phase transitions in simple fluids: An application of a one-phase entropic criterion to Lennard-Jones and point Yukawa fluids","volume":"49(6)","author":"Lomba","year":"1994","journal-title":"Phys. Rev. E"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/S0378-4371(97)00526-8","article-title":"Entropy versus correlations in simple fluids: The gas-liquid and freezing transitions","volume":"250","author":"Giaquinta","year":"1998","journal-title":"Physica A"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"7586","DOI":"10.1063\/1.1406528","article-title":"Scaling of local density correlations in a fluid close to freezing","volume":"115(16)","author":"Saija","year":"2001","journal-title":"The Journal of Chemical Physics"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1768","DOI":"10.1002\/cphc.200400565","article-title":"Re-entrant melting in the Gaussian-core model: The entropy imprint","volume":"6(9)","author":"Giaquinta","year":"2005","journal-title":"ChemPhysChem"},{"key":"ref_17","first-page":"050102","article-title":"Phase diagram of the Gaussian-core model","volume":"71(5)","author":"Prestipino","year":"2005","journal-title":"Physical Review E (Statistical, Nonlinear, and Soft Matter Physics)"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"244504","DOI":"10.1063\/1.2208357","article-title":"Evaluation of phenomenological one-phase criteria for the melting and freezing of softly repulsive particles","volume":"124(24)","author":"Saija","year":"2006","journal-title":"The Journal of Chemical Physics"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"136101","DOI":"10.1063\/1.2901040","article-title":"An entropy-based approach to the freezing of the generalized exponential model","volume":"128(13)","author":"Saija","year":"2008","journal-title":"The Journal of Chemical Physics"},{"key":"ref_20","first-page":"C1A0802002","article-title":"Statistical entropy and clustering in absence of attractive terms in the interparticle potential","volume":"86(2)","author":"Saija","year":"2008","journal-title":"Atti della Accademia Peloritana dei Pericolanti - Classe di Scienze Fisiche, Matematiche e Naturali"},{"key":"ref_21","first-page":"9853","article-title":"On entropy and ordering in binary hard-sphere mixtures","volume":"6(46)","author":"Saija","year":"1994","journal-title":"Journal of Physics: Condensed Matter"},{"key":"ref_22","first-page":"8137","article-title":"Statistical entropy of a binary hard-sphere mixture: the low-density limit","volume":"8(43)","author":"Saija","year":"1996","journal-title":"Journal of Physics: Condensed Matter"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2806","DOI":"10.1063\/1.1305887","article-title":"Entropy, correlations, and ordering in two dimensions","volume":"113(7)","author":"Saija","year":"2000","journal-title":"The Journal of Chemical Physics"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1023\/A:1004520432275","article-title":"Statistical entropy of a lattice-gas model: Multiparticle correlation expansion","volume":"96(1)","author":"Prestipino","year":"1999","journal-title":"Journal of Statistical Physics"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1023\/A:1018603728546","article-title":"Errata: Statistical entropy of a lattice-gas model: Multiparticle correlation expansion","volume":"98(1)","author":"Prestipino","year":"2000","journal-title":"Journal of Statistical Physics"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1007\/s100510051190","article-title":"Entropy and multi-particle correlations in two-dimensional lattice gases","volume":"11(4)","author":"Donato","year":"1999","journal-title":"The European Physical Journal B"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"021602","DOI":"10.1103\/PhysRevE.66.021602","article-title":"Analog of surface preroughening in a two-dimensional lattice Coulomb gas","volume":"66(2)","author":"Prestipino","year":"2002","journal-title":"Physical Review E"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"10368","DOI":"10.1021\/jp982202b","article-title":"Entropy and fluid-fluid separation in nonadditive hard-sphere mixtures","volume":"102(50)","author":"Saija","year":"1998","journal-title":"Journal of Physical Chemistry B"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2035","DOI":"10.1021\/jp013150k","article-title":"Entropy and fluid-fluid separation in nonadditive hard-sphere mixtures: The asymmetric case","volume":"106(8)","author":"Saija","year":"2002","journal-title":"Journal of Physical Chemistry B"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"5780","DOI":"10.1063\/1.1501126","article-title":"Monte Carlo simulation and phase behavior of nonadditive hard-core mixtures in two dimensions","volume":"117(12)","author":"Saija","year":"2002","journal-title":"The Journal of Chemical Physics"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/S0009-2614(97)01299-2","article-title":"Angular correlations and statistical entropy of hard spherocylin-ders: the isotropic-nematic transition","volume":"283(1-2)","author":"Costa","year":"1998","journal-title":"Chemical Physics Letters"},{"key":"ref_32","first-page":"252","article-title":"Erratum to: \u201cAngular correlations and statistical entropy of hard spherocylinders: the isotropic-nematic transition\"","volume":"290(2)","author":"Costa","year":"1999","journal-title":"Chemical Physics Letters"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"12297","DOI":"10.1021\/jp0259317","article-title":"Entropy and correlations in a fluid of hard sphe-rocylinders: The onset of nematic and smectic order","volume":"106(47)","author":"Costa","year":"2002","journal-title":"Journal of Physical Chemistry B"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"9514","DOI":"10.1021\/jp0345001","article-title":"Smectic ordering of parallel hard spherocylinders: An entropy-based Monte Carlo study","volume":"107(35)","author":"Costa","year":"2003","journal-title":"Journal of Physical Chemistry B"},{"key":"ref_35","first-page":"P09008","article-title":"The entropy multiparticle-correlation expansion for a mixture of spherical and elongated particles","volume":"2004(09)","author":"Prestipino","year":"2004","journal-title":"Journal of Statistical Mechanics: Theory and Experiment"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"3587","DOI":"10.1063\/1.1598431","article-title":"Statistical entropy and density maximum anomaly in liquid water","volume":"119(7)","author":"Saija","year":"2003","journal-title":"The Journal of Chemical Physics"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Lord Rayleigh (1891). On the virial of a system of hard colliding bodies. Nature, 45(1152), 80\u201382.","DOI":"10.1038\/045080a0"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1063\/1.1749355","article-title":"On the states of aggregation","volume":"2(1)","author":"Herzfeld","year":"1934","journal-title":"The Journal of Chemical Physics"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"955","DOI":"10.1103\/PhysRev.50.955","article-title":"The complete equation of state of one, two and three-dimensional gases of hard elastic spheres","volume":"50(10)","author":"Tonks","year":"1936","journal-title":"Phys. Rev."},{"key":"ref_40","unstructured":"Lieb, E.H., and Mattis, D.C. (1966). Mathematical Physics in One Dimension, Academic Press. Perspectives in Physics."},{"key":"ref_41","unstructured":"Lebowitz, J.L. (1987). Simple models of equilibrium and nonequilibrium phenomena, volume 13 of Studies in Statistical Mechanics, North-Holland Physics Publishing. chapter 1."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1098","DOI":"10.1063\/1.1699116","article-title":"Molecular distribution functions in a one-dimensional fluid","volume":"21(6)","author":"Salsburg","year":"1953","journal-title":"The Journal of Chemical Physics"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1422","DOI":"10.1063\/1.1699263","article-title":"The pair distribution function for a one-dimensional gas","volume":"21(8)","author":"Sells","year":"1953","journal-title":"The Journal of Chemical Physics"},{"key":"ref_44","first-page":"227","article-title":"On the convergence of Green\u2019s entropy expansion","volume":"165(2-3)","author":"Baranyai","year":"1992","journal-title":"Chemical Physics"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"161101","DOI":"10.1063\/1.2916697","article-title":"Residual multiparticle entropy does not generally change sign near freezing","volume":"128(16)","author":"Krekelberg","year":"2008","journal-title":"The Journal of Chemical Physics"},{"key":"ref_46","unstructured":"Hoover, W.G. (1991). Computational Statistical Mechanics, Elsevier."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"4819","DOI":"10.1088\/0305-4470\/26\/19\/015","article-title":"Harmonic properties of hard-sphere crystals: a one-dimensional study","volume":"26(19)","author":"Piasecki","year":"1993","journal-title":"Journal of Physics A: Mathematical and General"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"11390","DOI":"10.1063\/1.479080","article-title":"Exact solution for the singlet density distributions and second-order correlations of normal-mode coordinates for hard rods in one dimension","volume":"110","author":"Barnes","year":"1999","journal-title":"The Journal of Chemical Physics"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"9111","DOI":"10.1063\/1.1516592","article-title":"A comparison of some variational formulas for the free energy as applied to hard-sphere crystals","volume":"117","author":"Barnes","year":"2002","journal-title":"The Journal of Chemical Physics"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"4969","DOI":"10.1063\/1.435654","article-title":"Computer studies of dynamics in one dimension: Hard rods","volume":"68(11)","author":"Haus","year":"1978","journal-title":"The Journal of Chemical Physics"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1225","DOI":"10.1063\/1.434014","article-title":"On the stability of a fluid toward solid formation","volume":"66(3)","author":"Lovett","year":"1977","journal-title":"The Journal of Chemical Physics"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"2059","DOI":"10.1103\/PhysRevA.41.2059","article-title":"Nearest-neighbor distribution functions in many-body systems","volume":"41(4)","author":"Torquato","year":"1990","journal-title":"Physical Review A"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"7369","DOI":"10.1103\/PhysRevE.58.7369","article-title":"Density fluctuations in many-body systems","volume":"58(6)","author":"Truskett","year":"1998","journal-title":"Physical Review E"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"074508","DOI":"10.1063\/1.1848091","article-title":"The structure of hyperspherical fluids in various dimensions","volume":"122(7)","author":"Bishop","year":"2005","journal-title":"The Journal of Chemical Physics"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"2361","DOI":"10.1063\/1.1727947","article-title":"Cell theories for hard particles","volume":"45(7)","author":"Hoover","year":"1966","journal-title":"The Journal of Chemical Physics"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"685","DOI":"10.1007\/BF01009028","article-title":"One-dimensional rigorous hole theory of fluids: Internally constrained ensembles","volume":"41(3-4)","author":"Elkoshi","year":"1985","journal-title":"Journal of Statistical Physics"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"896","DOI":"10.1063\/1.1749960","article-title":"A theory of liquid structure","volume":"5(11)","author":"Hirschfelder","year":"1937","journal-title":"The Journal of Chemical Physics"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"300","DOI":"10.1063\/1.1749657","article-title":"Statistical mechanics of fluid mixtures","volume":"3(5)","author":"Kirkwood","year":"1935","journal-title":"The Journal of Chemical Physics"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1063\/1.1723737","article-title":"The radial distribution function in liquids","volume":"10(6)","author":"Kirkwood","year":"1942","journal-title":"The Journal of Chemical Physics"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1088\/0034-4885\/31\/2\/301","article-title":"Classical fluids and the superposition approximation","volume":"31(2)","author":"Cole","year":"1968","journal-title":"Reports on Progress in Physics"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"1817","DOI":"10.1063\/1.1671277","article-title":"Comment on Kirkwood instability","volume":"50(4)","author":"Kunkin","year":"1969","journal-title":"The Journal of Chemical Physics"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"897","DOI":"10.1080\/00268978200100711","article-title":"The influence of closure on the behaviour of the Yvon-Born-Green equation for a system of hard rods","volume":"45(4)","author":"Green","year":"1982","journal-title":"Molecular Physics"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"4594","DOI":"10.1063\/1.447391","article-title":"The Kirkwood superposition approximation for hard rods at high pressure","volume":"81(10)","author":"Siders","year":"1984","journal-title":"The Journal of Chemical Physics"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/10\/3\/248\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T22:20:58Z","timestamp":1760221258000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/10\/3\/248"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,9,19]]},"references-count":63,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2008,9]]}},"alternative-id":["e10030248"],"URL":"https:\/\/doi.org\/10.3390\/e10030248","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,9,19]]}}}