Two-step BEC coming from a temperature dependent energy gap

Two-step BEC coming from a temperature dependent energy gap | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Two-step BEC coming from a temperature dependent energy gap Juan José Valencia Acevedo, Miguel Angel Solís Atala This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3852351/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract We report the effects on the thermodynamic properties of a 3D Bose gas caused by a temperature dependent energy gap ∆ ( T ) at the lower edge of the energy spectrum of the particles constituting the Bose gas which behaves like an ideal Bose gas when the gap is removed. Explicit formulae are given for the critical temperature, the condensate fraction, the internal energy and the isochoric specific heat, which are calculated for three different gaps that abruptly go to zero at temperature T B , as well as for the damped counterparts whose drop to zero we have smoothed. In particular, for the undamped BCS (Bardeen, Cooper and Schrieffer) gap it is observed that the Bose-Einstein condensation (BEC) critical temperature T c is equal to that of the ideal Bose gas T 0 , for all T B ≤ T 0 ; surprisingly, the condensate fraction presents two different filling rates of the ground state at T c = T 0 and at T B < T 0 while the specific heat shows a finite jump at T c as well as a divergence at T B . Three-dimensional infinite Bose gas results are recovered when the temperature independent gap is either a constant or equal to zero. PACS numbers: 03.75.Fi; 05.30.Jp; 67.40.Kh Bose-Einstein condensation phase-transition thermodynamic properties Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3852351","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":266935698,"identity":"c1fe5932-35fe-481a-94e1-a20e75119ca3","order_by":0,"name":"Juan José Valencia Acevedo","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABD0lEQVRIiWNgGAWjYBACxgYILQMiDjBUQIV5iNDCA9FyhggtMABRw9hGhBbmGcnPHjC22fHwSzc/PPBz3mF5/mkHGB+8bWOINjiAw2Ez0swNGNuSeSTnHDM42LvtsOGM2wnMhnPbGHJnNuDSkmAmwXCGmcfgRg7DAd5thxk3SCewSfMCtfTjcBjjjPRvQC31PPZALQf/zjlsD9TC/hukpQ2nlhygLRWHeQwkchgO8zYcTgTZwozXlp43ZRIJFcd5JG6kGRyWOZaePON2YrPknHMSOP1i2J6+TeKDQbUc/4zkxx/f1Fjb9s9OPvjhTZlN7gYcIWYIMioBzWaQmAQOZzEwyOOUGQWjYBSMglEAAwCaSljuxucmSwAAAABJRU5ErkJggg==","orcid":"","institution":"Universidad Autónoma de la Ciudad de México","correspondingAuthor":true,"prefix":"","firstName":"Juan","middleName":"José Valencia","lastName":"Acevedo","suffix":""},{"id":266935699,"identity":"b9aa28ed-8bcc-480c-b915-fce8ca438e72","order_by":1,"name":"Miguel Angel Solís Atala","email":"","orcid":"","institution":"National Autonomous University of Mexico","correspondingAuthor":false,"prefix":"","firstName":"Miguel","middleName":"Angel Solís","lastName":"Atala","suffix":""}],"badges":[],"createdAt":"2024-01-11 04:59:16","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3852351/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3852351/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":68600939,"identity":"e9575004-04b3-4abc-9144-b7448fb5c68a","added_by":"auto","created_at":"2024-11-09 06:16:56","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1575639,"visible":true,"origin":"","legend":"","description":"","filename":"msBGasWithGapVsT.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3852351/v1_covered_711947ee-6ab1-462f-a7c4-c24a90897fbf.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eTwo-step BEC coming from a temperature dependent energy gap\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Bose-Einstein condensation, phase-transition, thermodynamic properties","lastPublishedDoi":"10.21203/rs.3.rs-3852351/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3852351/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe report the effects on the thermodynamic properties of a 3D Bose gas caused by a temperature dependent energy gap \u003cem\u003e∆\u003c/em\u003e (\u003cem\u003eT\u003c/em\u003e) at the lower edge of the energy spectrum of the particles constituting the Bose gas which behaves like an ideal Bose gas when the gap is removed. Explicit formulae are given for the critical temperature, the condensate fraction, the internal energy and the isochoric specific heat, which are calculated for three different gaps that abruptly go to zero at temperature \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003eB\u003c/em\u003e\u003c/sub\u003e, as well as for the damped counterparts whose drop to zero we have smoothed. In particular, for the undamped BCS (Bardeen, Cooper and Schrieffer) gap it is observed that the Bose-Einstein condensation (BEC) critical temperature \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003ec\u003c/em\u003e\u003c/sub\u003e is equal to that of the ideal Bose gas \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e, for all \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003eB\u003c/em\u003e\u003c/sub\u003e \u0026nbsp;≤ \u0026nbsp;\u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e; surprisingly, the condensate fraction presents two different filling rates of the ground state at \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003ec\u003c/em\u003e\u003c/sub\u003e = \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e and at \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003eB\u003c/em\u003e\u003c/sub\u003e \u0026lt; \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e while the specific heat shows a finite jump at \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003ec\u003c/em\u003e\u003c/sub\u003e as well as a divergence at \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003eB\u003c/em\u003e\u003c/sub\u003e. Three-dimensional infinite Bose gas results are recovered when the temperature independent gap is either a constant or equal to zero.\u003c/p\u003e\n\u003cp\u003ePACS numbers: 03.75.Fi; 05.30.Jp; 67.40.Kh\u003c/p\u003e","manuscriptTitle":"Two-step BEC coming from a temperature dependent energy gap","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-01-16 18:51:06","doi":"10.21203/rs.3.rs-3852351/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"decf41db-037e-4d05-b24b-40aa21df065c","owner":[],"postedDate":"January 16th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-11-09T06:08:36+00:00","versionOfRecord":[],"versionCreatedAt":"2024-01-16 18:51:06","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3852351","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3852351","identity":"rs-3852351","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}