{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:38:57Z","timestamp":1760150337087,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2023,11,8]],"date-time":"2023-11-08T00:00:00Z","timestamp":1699401600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Brazilian agencies Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de S\u00e3o Paulo (FAPESP)","award":["464898\/2014-5"],"award-info":[{"award-number":["464898\/2014-5"]}]},{"name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado do Rio de Janeiro (FAPERJ)","award":["464898\/2014-5"],"award-info":[{"award-number":["464898\/2014-5"]}]},{"name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico (CNPq)","award":["464898\/2014-5"],"award-info":[{"award-number":["464898\/2014-5"]}]},{"name":"Coordena\u00e7\u00e3o de Aperfei\u00e7oamento de Pessoal de N\u00edvel Superior (CAPES)","award":["464898\/2014-5"],"award-info":[{"award-number":["464898\/2014-5"]}]},{"name":"National Natural Science Foundation of China (NSFC)","award":["464898\/2014-5"],"award-info":[{"award-number":["464898\/2014-5"]}]},{"name":"Institutos Nacionais de Ci\u00eancias e Tecnologia\u2014F\u00edsica Nuclear e Aplica\u00e7\u00f5es (INCT\/FNA)","award":["464898\/2014-5"],"award-info":[{"award-number":["464898\/2014-5"]}]},{"name":"Center for Scientific Computing (NCC\/GridUNESP) of S\u00e3o Paulo State University (UNESP)","award":["464898\/2014-5"],"award-info":[{"award-number":["464898\/2014-5"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Although partition temperature derived using the Darwin\u2013Fowler method is exact for simple scenarios, the derivation for complex systems might reside in specific approximations whose viability is not ensured if the thermodynamic limit is not attained. This work elaborates on a related problem relevant to relativistic high-energy collisions. On the one hand, it is simple enough that closed-form expressions can be obtained precisely for the one-particle distribution function. On the other hand, the resulting expression is not an exponential form, and therefore, it is not straightforward that the notion of partition function could be implied. Specifically, we derive the one-particle distribution function for massless particles where the phase-space integration is performed exactly for the underlying canonical ensemble consisting of a given number of particles. We discuss the viability of the partition temperature in this case. Possible implications of the obtained results regarding the observed Tsallis distribution in transverse momentum spectra in high-energy collisions are also addressed.<\/jats:p>","DOI":"10.3390\/sym15112035","type":"journal-article","created":{"date-parts":[[2023,11,9]],"date-time":"2023-11-09T03:42:08Z","timestamp":1699501328000},"page":"2035","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Partition Temperature of Massless Particles in High-Energy Collisions"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3450-1984","authenticated-orcid":false,"given":"Wei-Liang","family":"Qian","sequence":"first","affiliation":[{"name":"Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, Yangzhou 225009, China"},{"name":"Escola de Engenharia de Lorena, Universidade de S\u00e3o Paulo, Lorena 12602-810, Brazil"},{"name":"Faculdade de Engenharia de Guaratinguet\u00e1, Universidade Estadual Paulista, Guaratinguet\u00e1 12516-410, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kai","family":"Lin","sequence":"additional","affiliation":[{"name":"Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, Yangzhou 225009, China"},{"name":"Hubei Subsurface Multi-Scale Imaging Key Laboratory, School of Geophysics and Geomatics, China University of Geosciences, Wuhan 430074, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rui-Hong","family":"Yue","sequence":"additional","affiliation":[{"name":"Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, Yangzhou 225009, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yogiro","family":"Hama","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica, Universidade de S\u00e3o Paulo, C.P. 66318, S\u00e3o Paulo 05315-970, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Takeshi","family":"Kodama","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica, Universidade Federal do Rio de Janeiro, C.P. 68528, Rio de Janeiro 21945-970, Brazil"},{"name":"Instituto de F\u00edsica, Universidade Federal Fluminense, Niter\u00f3i 24210-346, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2291","DOI":"10.1103\/PhysRevD.9.2291","article-title":"Vacuum Stability and Vacuum Excitation in a Spin 0 Field Theory","volume":"9","author":"Lee","year":"1974","journal-title":"Phys. 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