{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T15:42:50Z","timestamp":1787240570176,"version":"build-2736575974"},"reference-count":26,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2026,7,12]],"date-time":"2026-07-12T00:00:00Z","timestamp":1783814400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002850","name":"Fondo Nacional de Desarrollo Cient\u00edfico y Tecnol\u00f3gico","doi-asserted-by":"publisher","award":["1251928"],"award-info":[{"award-number":["1251928"]}],"id":[{"id":"10.13039\/501100002850","id-type":"DOI","asserted-by":"publisher"}]},{"award":["1251928"],"award-info":[{"award-number":["1251928"]}],"id":[{"id":"https:\/\/ror.org\/016nafs32","id-type":"ROR","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The onset of quantum behavior in gases is traditionally established through a criterion that is external to classical statistical mechanics. One introduces the thermal de Broglie wavelength and compares it with the mean intermolecular separation, concluding that quantum effects become relevant when n\u03bbT3\u223c1. This condition originates in quantum statistical mechanics and is absent from the classical ideal-gas or van der Waals partition functions. In this work, we show that a grand-canonical treatment of the van der Waals fluid naturally generates an interaction-corrected crossover temperature T3(a,b,m,n) determined by the particle mass, density, and van der Waals interaction parameters. While the thermal de Broglie wavelength provides the standard quantum crossover scale, the interaction-induced correction leading to T3 is obtained without invoking the explicit form of the Bose\u2013Einstein or Fermi\u2013Dirac distributions. Instead, T3 follows from a self-consistent condition within the grand-canonical van der Waals description. We demonstrate that, below this temperature, the statistical assumptions underlying the classical theory become self-inconsistent, indicating the breakdown of the classical description and the onset of the quantum-degenerate regime. The resulting temperature scale therefore provides an interaction-corrected boundary of validity of the classical van der Waals description. These findings provide a new perspective on how intermolecular interactions modify the crossover to the quantum-degenerate regime and clarify the limits of applicability of the classical van der Waals theory.<\/jats:p>","DOI":"10.3390\/e28070789","type":"journal-article","created":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T09:51:09Z","timestamp":1783936269000},"page":"789","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["An Endogenous Quantum\u2013Classical Crossover Temperature in the van der Waals Fluid: Quantumness as an Emergent Behavior"],"prefix":"10.3390","volume":"28","author":[{"given":"Flavia","family":"Pennini","sequence":"first","affiliation":[{"name":"Departamento de F\u00edsica, Universidad Cat\u00f3lica del Norte, Av. Angamos 0610, Antofagasta 1270709, Chile"},{"name":"Departamento de F\u00edsica, Facultad de Ingenier\u00eda, Universidad Nacional de Mar del Plata (UNMDP), CONICET, Av. J.B. Justo 4302, Mar del Plata 7600, Argentina"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5934-2783","authenticated-orcid":false,"given":"Angelo","family":"Plastino","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica La Plata\u2014CCT-CONICET, Universidad Nacional de La Plata, C.C. 727, La Plata 1900, Argentina"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,7,12]]},"reference":[{"key":"ref_1","unstructured":"Pathria, R.K., and Beale, P.D. (2021). Statistical Mechanics, Academic Press. [4th ed.]."},{"key":"ref_2","unstructured":"Huang, K. (1987). Statistical Mechanics, Wiley. [2nd ed.]."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1051\/anphys\/192510030022","article-title":"Recherches sur la th\u00e9orie des quanta","volume":"10","year":"1925","journal-title":"Ann. Phys."},{"key":"ref_4","unstructured":"Landau, L.D., and Lifshitz, E.M. (1980). Statistical Physics, Pergamon Press. [3rd ed.]."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1103\/PhysRev.40.749","article-title":"On the Quantum Correction For Thermodynamic Equilibrium","volume":"40","author":"Wigner","year":"1932","journal-title":"Phys. Rev."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1103\/PhysRev.44.31","article-title":"Quantum Statistics of Almost Classical Assemblies","volume":"44","author":"Kirkwood","year":"1933","journal-title":"Phys. Rev."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"729","DOI":"10.1016\/S0031-8914(36)80346-2","article-title":"The Quantum Theory of the Non-Ideal Gas. I. Deviations from the Classical Theory","volume":"3","author":"Uhlenbeck","year":"1936","journal-title":"Physica"},{"key":"ref_8","unstructured":"van der Waals, J.D. (1873). On the Continuity of the Gaseous and Liquid States. [Ph.D. Thesis, Leiden University]."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Rowlinson, J.S. (2002). Cohesion: A Scientific History of Intermolecular Forces, Cambridge University Press.","DOI":"10.1017\/CBO9780511535420"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1119\/1.12877","article-title":"Parametric solution of the van der Waals liquid\u2013vapor coexistence curve","volume":"50","author":"Lekner","year":"1982","journal-title":"Am. J. Phys."},{"key":"ref_11","unstructured":"Stanley, H.E. (1971). Introduction to Phase Transitions and Critical Phenomena, Oxford University Press."},{"key":"ref_12","unstructured":"Hill, T.L. (1987). Statistical Mechanics, Dover Publications."},{"key":"ref_13","unstructured":"Reif, F. (2009). Fundamentals of Statistical and Thermal Physics, Waveland Press. [1st ed.]."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1063\/1.881293","article-title":"Decoherence and the Transition from Quantum to Classical","volume":"44","author":"Zurek","year":"1991","journal-title":"Phys. Today"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1103\/RevModPhys.75.715","article-title":"Decoherence, Einselection, and the Quantum Origins of the Classical","volume":"75","author":"Zurek","year":"2003","journal-title":"Rev. Mod. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1267","DOI":"10.1103\/RevModPhys.76.1267","article-title":"Decoherence, the Measurement Problem, and Interpretations of Quantum Mechanics","volume":"76","author":"Schlosshauer","year":"2005","journal-title":"Rev. Mod. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1372","DOI":"10.1007\/s10701-019-00309-4","article-title":"The Correspondence Principle and the Understanding of Decoherence","volume":"49","author":"Fortin","year":"2019","journal-title":"Found. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Pennini, F., and Plastino, A. (2022). The Classical\u2013Quantum Passage: A van derWaals Description. Entropy, 24.","DOI":"10.3390\/e24020182"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"073124","DOI":"10.1063\/5.0274274","article-title":"Statistical complexity as indicator of the classical\u2013quantum crossover","volume":"35","author":"Escobar","year":"2025","journal-title":"Chaos"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Pethick, C.J., and Smith, H. (2008). Bose\u2013Einstein Condensation in Dilute Gases, Cambridge University Press. [2nd ed.].","DOI":"10.1017\/CBO9780511802850"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1215","DOI":"10.1103\/RevModPhys.80.1215","article-title":"Theory of ultracold atomic Fermi gases","volume":"80","author":"Giorgini","year":"2008","journal-title":"Rev. Mod. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"915","DOI":"10.1016\/S0031-8914(37)80189-5","article-title":"The quantum theory of the non-ideal gas. II. Behaviour at low temperatures","volume":"4","author":"Beth","year":"1937","journal-title":"Physica"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1016\/j.physa.2018.07.037","article-title":"Statistical complexity and classical-quantum frontier","volume":"511","author":"Branada","year":"2018","journal-title":"Phys. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"108680","DOI":"10.1016\/j.rinp.2026.108680","article-title":"Statistical complexity as a universal indicator of thermodynamic crossovers in ideal and van der Waals gases","volume":"86","author":"Pennini","year":"2026","journal-title":"Results Phys."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Pennini, F., and Plastino, A. (2026). How interactions affect many-body states distinguishability: A geometric viewpoint. Preprints, 216688.","DOI":"10.20944\/preprints202606.0289.v1"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1028","DOI":"10.1021\/acs.jced.9b00443","article-title":"Molecular Calculation of the Critical Parameters of Classical Helium","volume":"65","author":"Messerly","year":"2020","journal-title":"J. Chem. Eng. 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