{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:04:35Z","timestamp":1760241875283,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2018,10,9]],"date-time":"2018-10-09T00:00:00Z","timestamp":1539043200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003141","name":"Consejo Nacional de Ciencia y Tecnolog\u00eda","doi-asserted-by":"publisher","award":["257919, 258982"],"award-info":[{"award-number":["257919, 258982"]}],"id":[{"id":"10.13039\/501100003141","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006054","name":"Universidad de Guanajuato","doi-asserted-by":"publisher","award":["CIIC 130\/2018"],"award-info":[{"award-number":["CIIC 130\/2018"]}],"id":[{"id":"10.13039\/501100006054","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We explore some important consequences of the quantum ideal Bose gas, the properties of which are described by a non-extensive entropy. We consider in particular two entropies that depend only on the probability. These entropies are defined in the framework of superstatistics, and in this context, such entropies arise when a system is exposed to non-equilibrium conditions, whose general effects can be described by a generalized Boltzmann factor and correspondingly by a generalized probability distribution defining a different statistics. We generalize the usual statistics to their quantum counterparts, and we will focus on the properties of the corresponding generalized quantum ideal Bose gas. The most important consequence of the generalized Bose gas is that the critical temperature predicted for the condensation changes in comparison with the usual quantum Bose gas. Conceptual differences arise when comparing our results with the ones previously reported regarding the q-generalized Bose\u2013Einstein condensation. As the entropies analyzed here only depend on the probability, our results cannot be adjusted by any parameter. Even though these results are close to those of non-extensive statistical mechanics for     q \u223c 1    , they differ and cannot be matched for any q.<\/jats:p>","DOI":"10.3390\/e20100773","type":"journal-article","created":{"date-parts":[[2018,10,9]],"date-time":"2018-10-09T11:10:44Z","timestamp":1539083444000},"page":"773","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["On Quantum Superstatistics and the Critical Behavior of Nonextensive Ideal Bose Gases"],"prefix":"10.3390","volume":"20","author":[{"given":"Octavio","family":"Obreg\u00f3n","sequence":"first","affiliation":[{"name":"Departamento de F\u00edsica, Divisi\u00f3n de Ciencias e Ingenier\u00edas Campus Le\u00f3n, Universidad de Guanajuato, A.P. E-143, C.P. 37150 Le\u00f3n, Guanajuato, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jos\u00e9 Luis","family":"L\u00f3pez","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Divisi\u00f3n de Ciencias e Ingenier\u00edas Campus Le\u00f3n, Universidad de Guanajuato, A.P. E-143, C.P. 37150 Le\u00f3n, Guanajuato, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marco","family":"Ortega-Cruz","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Divisi\u00f3n de Ciencias e Ingenier\u00edas Campus Le\u00f3n, Universidad de Guanajuato, A.P. E-143, C.P. 37150 Le\u00f3n, Guanajuato, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,10,9]]},"reference":[{"key":"ref_1","unstructured":"R\u00e9nyi, A. (1970). Probability Theory, North Holland."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1142\/S0217984996000547","article-title":"Generalized fractional statistics","volume":"10","author":"Kaniadakis","year":"1996","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of Boltzmann-Gibbs statistics","volume":"52","author":"Tsallis","year":"1998","journal-title":"J. Stat. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1016\/S0375-9601(96)00832-8","article-title":"A note on the q-deformation-theoretic aspect of the generalized entropies in nonextensive physics","volume":"224","author":"Abe","year":"1997","journal-title":"Phys. Lett. A"},{"key":"ref_5","first-page":"28","article-title":"New nonadditive measures of entropy for discrete probability distributions","volume":"10","author":"Sharman","year":"1975","journal-title":"J. Math. Sci."},{"key":"ref_6","first-page":"337","article-title":"A summary on entropy statistics","volume":"31","author":"Esteban","year":"1995","journal-title":"Kybernetica"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1145","DOI":"10.3390\/e12051145","article-title":"Entropy: The Markov ordering approach","volume":"12","author":"Gorban","year":"2010","journal-title":"Entropy"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1080\/00107510902823517","article-title":"Generalized information and entropy measures in physics","volume":"50","author":"Beck","year":"2009","journal-title":"Contemp. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2770","DOI":"10.1103\/PhysRevLett.84.2770","article-title":"On the interpretation of nonextensive parameter q in Tsallis statistics and Levy distributions","volume":"84","author":"Wilk","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1016\/S0378-4371(03)00019-0","article-title":"Superstatistics","volume":"322","author":"Beck","year":"2003","journal-title":"Phys. A"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"026106","DOI":"10.1103\/PhysRevE.67.026106","article-title":"Constructing a statistical mechanics for Beck-Cohen superstatistics","volume":"67","author":"Tsallis","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2067","DOI":"10.3390\/e12092067","article-title":"Superstatistics and gravitation","volume":"12","year":"2010","journal-title":"Entropy"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"062146","DOI":"10.1103\/PhysRevE.88.062146","article-title":"Generalized information entropies depending only on the probability distribution","volume":"88","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Obreg\u00f3n, O., and Ortega-Cruz, M. (2018). Generalized entropies depending only on the probability and their quantum statistics. Proceedings, 2.","DOI":"10.3390\/ecea-4-05020"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1023\/B:ASTR.0000032529.67037.21","article-title":"Foundations of Nonextensive Statistical Mechanics and Its Cosmological Applications","volume":"290","author":"Plastino","year":"2004","journal-title":"Astrophys. Space Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.physleta.2005.06.026","article-title":"On the cut-off prescriptions associated with power-law generalized thermostatistics","volume":"343","author":"Teweldeberhan","year":"2005","journal-title":"Phys. Lett. A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/j.physa.2017.06.030","article-title":"Generalized ensemble theory with non-extensive statistics","volume":"487","author":"Shen","year":"2017","journal-title":"Phys. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1530039","DOI":"10.1142\/S0217751X15300392","article-title":"Generalized information and entanglement entropy, gravitation and holography","volume":"30","year":"2015","journal-title":"Int. J. Mod. Phys. A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1016\/j.molliq.2017.10.027","article-title":"Computer simulation of effective potentials for generalized Boltzmann-Gibbs statistics","volume":"248","year":"2017","journal-title":"J. Mol. Liq."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1140\/epjp\/i2018-11883-5","article-title":"Exploring gauge\/gravity duality of a generalized von-Neumann entropy","volume":"133","author":"Bizet","year":"2018","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1133","DOI":"10.1016\/j.physleta.2018.02.024","article-title":"Thermodynamic geometry for a non-extensive ideal gas","volume":"382","year":"2018","journal-title":"Phys. Lett. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3281","DOI":"10.1142\/S0217979294001378","article-title":"\u03bd-Dimensional Ideal quantum q-gas Bose Einstein condensation and \u03bb-point transition","volume":"8","author":"Roditi","year":"1994","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/0375-9601(94)00941-H","article-title":"A statistical mechanical approach to generalized statistics of quantum and classical gases","volume":"197","author":"Buyukkilic","year":"1995","journal-title":"Phys. Lett. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/0375-9601(96)00384-2","article-title":"On the Generalized Bose\u2013Einstein condensation","volume":"218","author":"Curlief","year":"1996","journal-title":"Phys. Lett. A"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1016\/S0378-4371(98)00397-5","article-title":"Generalized quantal distribution functions within factorization approach: Some general results for bosons and fermions","volume":"26","author":"Torres","year":"1998","journal-title":"Phys. A"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1142\/S0217979200000388","article-title":"BEC in Nonextensive statistical mechanics","volume":"14","author":"Salasnich","year":"2000","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1590\/S0103-97332001000200026","article-title":"Note on BEC in nonextensive statistical mechanics","volume":"31","author":"Lenzi","year":"2001","journal-title":"Br. J. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/S0375-9601(02)00781-8","article-title":"q-Generalized Bose\u2013Einstein condensation based on Tsallis entropy","volume":"300","author":"Chen","year":"2002","journal-title":"Phys. Lett. A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/S0378-4371(02)01330-4","article-title":"Bose\u2013Einstein and Fermi-Dirac distributions in nonextensive Tsallis statistics: An exact study","volume":"317","author":"Soares","year":"2003","journal-title":"Phys. A"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/S0921-4526(02)01425-4","article-title":"Bose\u2013Einstein condensation in the framework of \u03ba-statistics","volume":"325","author":"Aliano","year":"2003","journal-title":"Phys. B Condens. Matter"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1016\/j.physleta.2006.06.074","article-title":"Generalized thermostatistics and Bose\u2013Einstein condensation","volume":"359","author":"Miller","year":"2006","journal-title":"Phys. Lett. A"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1574","DOI":"10.1016\/j.physleta.2007.10.009","article-title":"More accurate theory of Bose\u2013Einstein condensation fraction","volume":"372","author":"Biswas","year":"2008","journal-title":"Phys. Lett. A"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1007\/s10909-007-9596-2","article-title":"A nonextensive approach to Bose\u2013Einstein condensation of trapped interacting boson gas","volume":"150","author":"Lawani","year":"2008","journal-title":"J. Low. Temp. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"011126","DOI":"10.1103\/PhysRevE.80.011126","article-title":"Bose\u2013Einstein and Fermi-Dirac distributions in nonextensive quantum statistics: Exact and interpolation approaches","volume":"80","author":"Hasegawa","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"814","DOI":"10.1007\/s10701-009-9363-0","article-title":"Thermostatistics of deformed bosons and fermions","volume":"40","author":"Lavagno","year":"2010","journal-title":"Found. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1142\/S0217984910022299","article-title":"Bose\u2013Einstein condensation of a q-deformed Bose gas in a random box","volume":"24","author":"Wang","year":"2010","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1650066","DOI":"10.1142\/S021830131650066X","article-title":"Chiral phase transitions in the linear sigma model in the Tsallis nonextensive statistics","volume":"25","author":"Ishihara","year":"2016","journal-title":"Int. J. Mod. Phys. E"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"4135329","DOI":"10.1155\/2017\/4135329","article-title":"Chiral phase transition in linear sigma model with nonextensive statistical mechanics","volume":"2017","author":"Shen","year":"2017","journal-title":"Adv. High Energy Phys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"272","DOI":"10.1016\/j.physa.2018.01.020","article-title":"An extensive study of Bose\u2013Einstein condensation in liquid helium using Tsallis statistics","volume":"497","author":"Guha","year":"2018","journal-title":"Phys. A"},{"key":"ref_40","unstructured":"Pathria, R.K. (2001). Statistical Mechanics, Butterworth-Heinemann."},{"key":"ref_41","unstructured":"Goodstein, D.L. (1985). States of Matter, Dover Publications."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/10\/773\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:24:31Z","timestamp":1760196271000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/10\/773"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,10,9]]},"references-count":41,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2018,10]]}},"alternative-id":["e20100773"],"URL":"https:\/\/doi.org\/10.3390\/e20100773","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2018,10,9]]}}}