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id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Cambiar a la tabla de contenidos</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Estadística de Bose-Einstein</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Ir a un artículo en otro idioma. Disponible en 49 idiomas" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-49" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">49 idiomas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A5%D8%AD%D8%B5%D8%A7%D8%A1_%D8%A8%D9%88%D8%B2_%D9%88%D8%A3%D9%8A%D9%86%D8%B4%D8%AA%D8%A7%D9%8A%D9%86" title="إحصاء بوز وأينشتاين (árabe)" lang="ar" hreflang="ar" data-title="إحصاء بوز وأينشتاين" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%AC%E0%A6%B8%E0%A7%81-%E0%A6%86%E0%A6%87%E0%A6%A8%E0%A6%B7%E0%A7%8D%E0%A6%9F%E0%A6%BE%E0%A6%87%E0%A6%A8_%E0%A6%AA%E0%A7%B0%E0%A6%BF%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="বসু-আইনষ্টাইন পৰিসংখ্যা (asamés)" lang="as" hreflang="as" data-title="বসু-আইনষ্টাইন পৰিসংখ্যা" data-language-autonym="অসমীয়া" data-language-local-name="asamés" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D0%BA%D0%B0_%D0%BD%D0%B0_%D0%91%D0%BE%D0%B7%D0%B5-%D0%90%D0%B9%D0%BD%D1%89%D0%B0%D0%B9%D0%BD" title="Статистика на Бозе-Айнщайн (búlgaro)" lang="bg" hreflang="bg" data-title="Статистика на Бозе-Айнщайн" data-language-autonym="Български" data-language-local-name="búlgaro" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%B8%E0%A7%81-%E0%A6%86%E0%A6%87%E0%A6%A8%E0%A6%B8%E0%A7%8D%E0%A6%9F%E0%A6%BE%E0%A6%87%E0%A6%A8_%E0%A6%AA%E0%A6%B0%E0%A6%BF%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%A8" title="বসু-আইনস্টাইন পরিসংখ্যান (bengalí)" lang="bn" hreflang="bn" data-title="বসু-আইনস্টাইন পরিসংখ্যান" data-language-autonym="বাংলা" data-language-local-name="bengalí" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Bose%E2%80%93Einsteinova_statistika" title="Bose–Einsteinova statistika (bosnio)" lang="bs" hreflang="bs" data-title="Bose–Einsteinova statistika" data-language-autonym="Bosanski" data-language-local-name="bosnio" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Estad%C3%ADstica_de_Bose-Einstein" title="Estadística de Bose-Einstein (catalán)" lang="ca" hreflang="ca" data-title="Estadística de Bose-Einstein" data-language-autonym="Català" data-language-local-name="catalán" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Boseho%E2%80%93Einsteinovo_rozd%C4%9Blen%C3%AD" title="Boseho–Einsteinovo rozdělení (checo)" lang="cs" hreflang="cs" data-title="Boseho–Einsteinovo rozdělení" data-language-autonym="Čeština" data-language-local-name="checo" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Bose-Einstein-Statistik" title="Bose-Einstein-Statistik (alemán)" lang="de" hreflang="de" data-title="Bose-Einstein-Statistik" data-language-autonym="Deutsch" data-language-local-name="alemán" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statistics" title="Bose–Einstein statistics (inglés)" lang="en" hreflang="en" data-title="Bose–Einstein statistics" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Statistiko_de_Bose-Einstein" title="Statistiko de Bose-Einstein (esperanto)" lang="eo" hreflang="eo" data-title="Statistiko de Bose-Einstein" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Bose-Einsteini_statistika" title="Bose-Einsteini statistika (estonio)" lang="et" hreflang="et" data-title="Bose-Einsteini statistika" data-language-autonym="Eesti" data-language-local-name="estonio" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Bose-Einstein_estatistika" title="Bose-Einstein estatistika (euskera)" lang="eu" hreflang="eu" data-title="Bose-Einstein estatistika" data-language-autonym="Euskara" data-language-local-name="euskera" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A2%D9%85%D8%A7%D8%B1_%D8%A8%D9%88%D8%B2-%D8%A7%DB%8C%D9%86%D8%B4%D8%AA%DB%8C%D9%86" title="آمار بوز-اینشتین (persa)" lang="fa" hreflang="fa" data-title="آمار بوز-اینشتین" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Bosen%E2%80%93Einsteinin_statistiikka" title="Bosen–Einsteinin statistiikka (finés)" lang="fi" hreflang="fi" data-title="Bosen–Einsteinin statistiikka" data-language-autonym="Suomi" data-language-local-name="finés" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Statistique_de_Bose-Einstein" title="Statistique de Bose-Einstein (francés)" lang="fr" hreflang="fr" data-title="Statistique de Bose-Einstein" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Staitistic_Bose-Einstein" title="Staitistic Bose-Einstein (irlandés)" lang="ga" hreflang="ga" data-title="Staitistic Bose-Einstein" data-language-autonym="Gaeilge" data-language-local-name="irlandés" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Estat%C3%ADstica_de_Bose-Einstein" title="Estatística de Bose-Einstein (gallego)" lang="gl" hreflang="gl" data-title="Estatística de Bose-Einstein" data-language-autonym="Galego" data-language-local-name="gallego" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%AC%E0%AB%8B%E0%AA%9D-%E0%AA%86%E0%AA%88%E0%AA%A8%E0%AB%8D%E0%AA%B8%E0%AB%8D%E0%AA%9F%E0%AA%BE%E0%AA%88%E0%AA%A8_%E0%AA%86%E0%AA%82%E0%AA%95%E0%AA%A1%E0%AA%BE%E0%AA%B6%E0%AA%BE%E0%AA%B8%E0%AB%8D%E0%AA%A4%E0%AB%8D%E0%AA%B0" title="બોઝ-આઈન્સ્ટાઈન આંકડાશાસ્ત્ર (guyaratí)" lang="gu" hreflang="gu" data-title="બોઝ-આઈન્સ્ટાઈન આંકડાશાસ્ત્ર" data-language-autonym="ગુજરાતી" data-language-local-name="guyaratí" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%AA%D7%A4%D7%9C%D7%92%D7%95%D7%AA_%D7%91%D7%95%D7%96-%D7%90%D7%99%D7%99%D7%A0%D7%A9%D7%98%D7%99%D7%99%D7%9F" title="התפלגות בוז-איינשטיין (hebreo)" lang="he" hreflang="he" data-title="התפלגות בוז-איינשטיין" data-language-autonym="עברית" data-language-local-name="hebreo" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AC%E0%A5%8B%E0%A4%B8-%E0%A4%86%E0%A4%87%E0%A4%A8%E0%A5%8D%E0%A4%B8%E0%A5%8D%E0%A4%9F%E0%A4%BE%E0%A4%87%E0%A4%A8_%E0%A4%B8%E0%A4%BE%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BF%E0%A4%95%E0%A5%80" title="बोस-आइन्स्टाइन सांख्यिकी (hindi)" lang="hi" hreflang="hi" data-title="बोस-आइन्स्टाइन सांख्यिकी" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Bose-Einsteinova_statistika" title="Bose-Einsteinova statistika (croata)" lang="hr" hreflang="hr" data-title="Bose-Einsteinova statistika" data-language-autonym="Hrvatski" data-language-local-name="croata" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Bose%E2%80%93Einstein-statisztika" title="Bose–Einstein-statisztika (húngaro)" lang="hu" hreflang="hu" data-title="Bose–Einstein-statisztika" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B2%D5%B8%D5%A6%D5%A5-%D4%B1%D5%B5%D5%B6%D5%B7%D5%BF%D5%A1%D5%B5%D5%B6%D5%AB_%D5%BE%D5%AB%D5%B3%D5%A1%D5%AF%D5%A1%D5%A3%D6%80%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Բոզե-Այնշտայնի վիճակագրություն (armenio)" lang="hy" hreflang="hy" data-title="Բոզե-Այնշտայնի վիճակագրություն" data-language-autonym="Հայերեն" data-language-local-name="armenio" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Statistik_Bose%E2%80%93Einstein" title="Statistik Bose–Einstein (indonesio)" lang="id" hreflang="id" data-title="Statistik Bose–Einstein" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesio" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Statistica_di_Bose-Einstein" title="Statistica di Bose-Einstein (italiano)" lang="it" hreflang="it" data-title="Statistica di Bose-Einstein" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%9C%E3%83%BC%E3%82%B9%E5%88%86%E5%B8%83%E9%96%A2%E6%95%B0" title="ボース分布関数 (japonés)" lang="ja" hreflang="ja" data-title="ボース分布関数" data-language-autonym="日本語" data-language-local-name="japonés" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%91%E1%83%9D%E1%83%96%E1%83%94-%E1%83%90%E1%83%98%E1%83%9C%E1%83%A8%E1%83%A2%E1%83%90%E1%83%98%E1%83%9C%E1%83%98%E1%83%A1_%E1%83%A1%E1%83%A2%E1%83%90%E1%83%A2%E1%83%98%E1%83%A1%E1%83%A2%E1%83%98%E1%83%99%E1%83%90" title="ბოზე-აინშტაინის სტატისტიკა (georgiano)" lang="ka" hreflang="ka" data-title="ბოზე-აინშტაინის სტატისტიკა" data-language-autonym="ქართული" data-language-local-name="georgiano" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%91%D0%BE%D0%B7%D0%B5%E2%80%93%D0%AD%D0%B9%D0%BD%D1%88%D1%82%D0%B5%D0%B9%D0%BD_%D1%81%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D0%BA%D0%B0%D1%81%D1%8B" title="Бозе–Эйнштейн статистикасы (kazajo)" lang="kk" hreflang="kk" data-title="Бозе–Эйнштейн статистикасы" data-language-autonym="Қазақша" data-language-local-name="kazajo" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AC%E0%B3%8B%E0%B2%B8%E0%B3%8D-%E0%B2%90%E0%B2%A8%E0%B3%8D%E2%80%8D%E0%B2%B8%E0%B3%8D%E0%B2%9F%E0%B3%88%E0%B2%A8%E0%B3%8D_%E0%B2%B8%E0%B2%82%E0%B2%96%E0%B3%8D%E0%B2%AF%E0%B2%BE%E0%B2%95%E0%B2%B2%E0%B2%A8%E0%B2%B5%E0%B2%BF%E0%B2%9C%E0%B3%8D%E0%B2%9E%E0%B2%BE%E0%B2%A8" title="ಬೋಸ್-ಐನ್‍ಸ್ಟೈನ್ ಸಂಖ್ಯಾಕಲನವಿಜ್ಞಾನ (canarés)" lang="kn" hreflang="kn" data-title="ಬೋಸ್-ಐನ್‍ಸ್ಟೈನ್ ಸಂಖ್ಯಾಕಲನವಿಜ್ಞಾನ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="canarés" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B3%B4%EC%8A%A4-%EC%95%84%EC%9D%B8%EC%8A%88%ED%83%80%EC%9D%B8_%ED%86%B5%EA%B3%84" title="보스-아인슈타인 통계 (coreano)" lang="ko" hreflang="ko" data-title="보스-아인슈타인 통계" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Bose-Einsteinstatistiek" title="Bose-Einsteinstatistiek (neerlandés)" lang="nl" hreflang="nl" data-title="Bose-Einsteinstatistiek" data-language-autonym="Nederlands" data-language-local-name="neerlandés" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Bose%E2%80%93Einstein-statistikk" title="Bose–Einstein-statistikk (noruego nynorsk)" lang="nn" hreflang="nn" data-title="Bose–Einstein-statistikk" data-language-autonym="Norsk nynorsk" data-language-local-name="noruego nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Bose%E2%80%93Einstein-statistikk" title="Bose–Einstein-statistikk (noruego bokmal)" lang="nb" hreflang="nb" data-title="Bose–Einstein-statistikk" data-language-autonym="Norsk bokmål" data-language-local-name="noruego bokmal" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%AC%E0%AD%8B%E0%AC%B7-%E0%AC%86%E0%AC%87%E0%AC%A8%E0%AD%8D%E2%80%8C%E0%AC%B7%E0%AD%8D%E0%AC%9F%E0%AC%BE%E0%AC%87%E0%AC%A8%E0%AD%8D_%E0%AC%AA%E0%AC%B0%E0%AC%BF%E0%AC%B8%E0%AC%82%E0%AC%96%E0%AD%8D%E0%AD%9F%E0%AC%BE%E0%AC%A8" title="ବୋଷ-ଆଇନ୍‌ଷ୍ଟାଇନ୍ ପରିସଂଖ୍ୟାନ (oriya)" lang="or" hreflang="or" data-title="ବୋଷ-ଆଇନ୍‌ଷ୍ଟାଇନ୍ ପରିସଂଖ୍ୟାନ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="oriya" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Statystyka_Bosego-Einsteina" title="Statystyka Bosego-Einsteina (polaco)" lang="pl" hreflang="pl" data-title="Statystyka Bosego-Einsteina" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Estat%C3%ADstica_de_Bose-Einstein" title="Estatística de Bose-Einstein (portugués)" lang="pt" hreflang="pt" data-title="Estatística de Bose-Einstein" data-language-autonym="Português" data-language-local-name="portugués" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Statistica_Bose-Einstein" title="Statistica Bose-Einstein (rumano)" lang="ro" hreflang="ro" data-title="Statistica Bose-Einstein" data-language-autonym="Română" data-language-local-name="rumano" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D0%BA%D0%B0_%D0%91%D0%BE%D0%B7%D0%B5_%E2%80%94_%D0%AD%D0%B9%D0%BD%D1%88%D1%82%D0%B5%D0%B9%D0%BD%D0%B0" title="Статистика Бозе — Эйнштейна (ruso)" lang="ru" hreflang="ru" data-title="Статистика Бозе — Эйнштейна" data-language-autonym="Русский" data-language-local-name="ruso" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Bose-Einsteinova_statistika" title="Bose-Einsteinova statistika (serbocroata)" lang="sh" hreflang="sh" data-title="Bose-Einsteinova statistika" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbocroata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Bose-Einstein_statistics" title="Bose-Einstein statistics (Simple English)" lang="en-simple" hreflang="en-simple" data-title="Bose-Einstein statistics" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Boseho-Einsteinovo_rozdelenie" title="Boseho-Einsteinovo rozdelenie (eslovaco)" lang="sk" hreflang="sk" data-title="Boseho-Einsteinovo rozdelenie" data-language-autonym="Slovenčina" data-language-local-name="eslovaco" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Bose-Einsteinova_statistika" title="Bose-Einsteinova statistika (esloveno)" lang="sl" hreflang="sl" data-title="Bose-Einsteinova statistika" data-language-autonym="Slovenščina" data-language-local-name="esloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Bose%E2%80%93Einstein-statistik" title="Bose–Einstein-statistik (sueco)" lang="sv" hreflang="sv" data-title="Bose–Einstein-statistik" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AF%8B%E0%AE%B8%E0%AF%8D%E2%80%93%E0%AE%90%E0%AE%A9%E0%AF%8D%E0%AE%B8%E0%AF%8D%E0%AE%9F%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%AA%E0%AF%81%E0%AE%B3%E0%AF%8D%E0%AE%B3%E0%AE%BF%E0%AE%AF%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D" title="போஸ்–ஐன்ஸ்டின் புள்ளியியல் (tamil)" lang="ta" hreflang="ta" data-title="போஸ்–ஐன்ஸ்டின் புள்ளியியல்" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%91%D0%BE%D0%B7%D0%B5_%E2%80%94_%D0%AD%D0%B9%D0%BD%D1%88%D1%82%D0%B5%D0%B9%D0%BD_%D0%B1%D2%AF%D0%BB%D0%B5%D0%BD%D0%B5%D1%88%D0%B5" title="Бозе — Эйнштейн бүленеше (tártaro)" lang="tt" hreflang="tt" data-title="Бозе — Эйнштейн бүленеше" data-language-autonym="Татарча / tatarça" data-language-local-name="tártaro" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D0%BA%D0%B0_%D0%91%D0%BE%D0%B7%D0%B5_%E2%80%94_%D0%95%D0%B9%D0%BD%D1%88%D1%82%D0%B5%D0%B9%D0%BD%D0%B0" title="Статистика Бозе — Ейнштейна (ucraniano)" lang="uk" hreflang="uk" data-title="Статистика Бозе — Ейнштейна" data-language-autonym="Українська" data-language-local-name="ucraniano" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Boze_Eynshteyn_statistikasi" title="Boze Eynshteyn statistikasi (uzbeko)" lang="uz" hreflang="uz" data-title="Boze Eynshteyn statistikasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbeko" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li 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class="vector-pinnable-header-label">Apariencia</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mover a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ocultar</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">De Wikipedia, la enciclopedia libre</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="es" dir="ltr"><p>La <b>estadística de Bose-Einstein</b> es un tipo de <a href="/wiki/Mec%C3%A1nica_estad%C3%ADstica" title="Mecánica estadística">mecánica estadística</a> aplicable a la determinación de las propiedades <a href="/wiki/Estad%C3%ADstica" title="Estadística">estadísticas</a> de conjuntos grandes de partículas indistinguibles capaces de coexistir en el mismo <a href="/wiki/Estado_cu%C3%A1ntico" title="Estado cuántico">estado cuántico</a> (<a href="/wiki/Bos%C3%B3n" title="Bosón">bosones</a>) en <a href="/wiki/Equilibrio_t%C3%A9rmico" title="Equilibrio térmico">equilibrio térmico</a>.<sup id="cite_ref-1" class="reference separada"><a href="#cite_note-1"><span class="corchete-llamada">[</span>1<span class="corchete-llamada">]</span></a></sup><sup id="cite_ref-2" class="reference separada"><a href="#cite_note-2"><span class="corchete-llamada">[</span>2<span class="corchete-llamada">]</span></a></sup> A bajas temperaturas, los <a href="/wiki/Bos%C3%B3n" title="Bosón">bosones</a> tienden a tener un comportamiento cuántico similar que puede llegar a ser idéntico a temperaturas cercanas al cero absoluto en un estado de la materia conocido como <a href="/wiki/Condensado_de_Bose-Einstein" title="Condensado de Bose-Einstein">condensado de Bose-Einstein</a> y producido por primera vez en laboratorio en el año 1995. El condensador Bose-Einstein funciona a temperaturas cercanas al cero absoluto, -273,15 °C (0 kelvin). La estadística de Bose-Einstein fue introducida para estudiar las propiedades estadísticas de los <a href="/wiki/Fot%C3%B3n" title="Fotón">fotones</a> en 1920 por el físico indio <a href="/wiki/Satyendra_Nath_Bose" title="Satyendra Nath Bose">Satyendra Nath Bose</a> y generalizada para <a href="/wiki/%C3%81tomo" title="Átomo">átomos</a> y otros bosones por <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> en 1924. Este tipo de estadística está íntimamente relacionada con la <a href="/wiki/Estad%C3%ADstica_de_Maxwell-Boltzmann" title="Estadística de Maxwell-Boltzmann">estadística de Maxwell-Boltzmann</a> (derivada inicialmente para <a href="/wiki/Gas" title="Gas">gases</a>) y a las <a href="/wiki/Estad%C3%ADstica_de_Fermi-Dirac" title="Estadística de Fermi-Dirac">estadísticas de Fermi-Dirac</a> (aplicables a partículas denominadas <a href="/wiki/Fermi%C3%B3n" title="Fermión">fermiones</a> sobre las que rige el <a href="/wiki/Principio_de_exclusi%C3%B3n_de_Pauli" title="Principio de exclusión de Pauli">principio de exclusión de Pauli</a> que impide que dos fermiones compartan el mismo estado cuántico). </p><p>La estadística de Bose-Einstein se reduce a la estadística de Maxwell-Boltzmann para energías suficientemente elevadas. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formulación_matemática"><span id="Formulaci.C3.B3n_matem.C3.A1tica"></span>Formulación matemática</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Estad%C3%ADstica_de_Bose-Einstein&action=edit&section=1" title="Editar sección: Formulación matemática"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>El <a href="/wiki/N%C3%BAmero_de_part%C3%ADculas" title="Número de partículas">número de partículas</a> en un estado de energía <i>i</i> es: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{i}\left(\varepsilon _{i}{\text{, }}T\right)={\frac {g_{i}}{e^{{\left(\varepsilon _{i}-\mu \right)}/{k_{B}T}\;}-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>, </mtext> </mrow> <mi>T</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> <mspace width="thickmathspace" /> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{i}\left(\varepsilon _{i}{\text{, }}T\right)={\frac {g_{i}}{e^{{\left(\varepsilon _{i}-\mu \right)}/{k_{B}T}\;}-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a347baa329227d9c5f6bcdf204926296eeae7522" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.989ex; height:5.343ex;" alt="{\displaystyle n_{i}\left(\varepsilon _{i}{\text{, }}T\right)={\frac {g_{i}}{e^{{\left(\varepsilon _{i}-\mu \right)}/{k_{B}T}\;}-1}}}"></span> </p> </blockquote> <p>donde: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{i}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{i}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d636fa67ff31e695d9568062ea814cf29c47fc35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.581ex; height:2.009ex;" alt="{\displaystyle n_{i}\,}"></span> es el número de partículas en un estado <i>i</i>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{i}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{i}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/129b851fa0c30bdec8b5f99c642de9e28df617a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.296ex; height:2.009ex;" alt="{\displaystyle g_{i}\,}"></span> es la degeneración cuántica del estado <i>i</i> o número de funciones de onda diferentes que poseen dicha energía,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{i}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{i}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cef0a884f63dda389fe95a535edcaa8d19fd0731" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.27ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{i}\,}"></span> es la energía del estado <i>i</i>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d20addf0d9f04e185714134b97726c4bf17d340" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.789ex; height:2.176ex;" alt="{\displaystyle \mu \,}"></span> es el potencial químico,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{B}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{B}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cd643a7f8b267149fdbb5e47dd02914548b672f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.078ex; height:2.509ex;" alt="{\displaystyle k_{B}\,}"></span> es la constante de Boltzmann,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/476a8389064c06ab89963a2467aef525838da0cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.023ex; height:2.176ex;" alt="{\displaystyle T\,}"></span> es la temperatura.</li></ul> <p>La estadística de Bose-Einstein se reduce a la estadística de Maxwell-Boltzmann para energías: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\epsilon _{i}-\mu )>>k_{B}T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>>></mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\epsilon _{i}-\mu )>>k_{B}T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b01cfc28562769338f77d9100eab661bb467088" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.029ex; height:2.843ex;" alt="{\displaystyle (\epsilon _{i}-\mu )>>k_{B}T}"></span> </p> </blockquote> <div class="mw-heading mw-heading2"><h2 id="Derivación"><span id="Derivaci.C3.B3n"></span>Derivación</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Estad%C3%ADstica_de_Bose-Einstein&action=edit&section=2" title="Editar sección: Derivación"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dado que los sistemas bosónicos son sistemas de partículas indistinguibles, los estados cuya única diferencia es la permutación de estados de dos partículas son idénticos. De este modo, un estado del sistema estará unívocamente definido por el número de partículas que se encuentren en un determinado estado energético. Se denotará por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon _{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ff36e034ac43c6b67b68ed1b87ca7a5b4e42f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.918ex; height:2.009ex;" alt="{\displaystyle \epsilon _{r}}"></span> el estado energético r-ésimo, por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eb80b8d27c081a9f436cd327f481425e58c6b9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.368ex; height:2.009ex;" alt="{\displaystyle n_{r}}"></span> el número de partículas en el estado r-ésimo y R cada una de las posibles combinaciones de números de ocupación. La función de partición resulta: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Z}}=\sum _{l}e^{-\beta (E_{l}-\mu n_{l})}=\sum _{R}e^{-\beta \sum _{r}(\epsilon _{r}n_{r}-\mu n_{r})}=\sum _{R}\prod _{r}e^{-\beta (\epsilon _{r}n_{r}-\mu n_{r})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">Z</mi> </mrow> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </munder> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mo stretchy="false">(</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Z}}=\sum _{l}e^{-\beta (E_{l}-\mu n_{l})}=\sum _{R}e^{-\beta \sum _{r}(\epsilon _{r}n_{r}-\mu n_{r})}=\sum _{R}\prod _{r}e^{-\beta (\epsilon _{r}n_{r}-\mu n_{r})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c38f33fb8a815f75a5e5738536acb05c93bbf23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:63.006ex; height:5.509ex;" alt="{\displaystyle {\mathcal {Z}}=\sum _{l}e^{-\beta (E_{l}-\mu n_{l})}=\sum _{R}e^{-\beta \sum _{r}(\epsilon _{r}n_{r}-\mu n_{r})}=\sum _{R}\prod _{r}e^{-\beta (\epsilon _{r}n_{r}-\mu n_{r})}}"></span> </p> </blockquote> <p>La anterior expresión contiene todas las combinaciones posibles de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eb80b8d27c081a9f436cd327f481425e58c6b9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.368ex; height:2.009ex;" alt="{\displaystyle n_{r}}"></span> entre 0 e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }"></span> (puesto que en un sistema bosónico el número de partículas por estado cuántico no está limitado) de forma que puede ser reescrita de la siguiente forma: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Z}}=\prod _{r}\sum _{n_{r}=0}^{\infty }e^{-\beta (\epsilon _{r}n_{r}-\mu n_{r})}=\prod _{r}{\frac {1}{1-e^{-\beta (\epsilon _{r}-\mu )}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">Z</mi> </mrow> </mrow> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </munder> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mo stretchy="false">(</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mo stretchy="false">(</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Z}}=\prod _{r}\sum _{n_{r}=0}^{\infty }e^{-\beta (\epsilon _{r}n_{r}-\mu n_{r})}=\prod _{r}{\frac {1}{1-e^{-\beta (\epsilon _{r}-\mu )}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0438a69915ea63bec46a8bddc0f959079511cdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:44.315ex; height:7.176ex;" alt="{\displaystyle {\mathcal {Z}}=\prod _{r}\sum _{n_{r}=0}^{\infty }e^{-\beta (\epsilon _{r}n_{r}-\mu n_{r})}=\prod _{r}{\frac {1}{1-e^{-\beta (\epsilon _{r}-\mu )}}}}"></span> </p> </blockquote> <p>Aplicando que: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi =k_{B}Tln{\mathcal {Z}}\quad y\quad {\frac {\partial \Phi }{\partial \mu }}=-N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> <mi>l</mi> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">Z</mi> </mrow> </mrow> <mspace width="1em" /> <mi>y</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>μ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi =k_{B}Tln{\mathcal {Z}}\quad y\quad {\frac {\partial \Phi }{\partial \mu }}=-N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5eb99b2d988ab55c88657538a84e770000f1e7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.577ex; height:6.009ex;" alt="{\displaystyle \Phi =k_{B}Tln{\mathcal {Z}}\quad y\quad {\frac {\partial \Phi }{\partial \mu }}=-N}"></span> </p> </blockquote> <p>Se tiene que: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi =k_{B}Tln{\mathcal {Z}}=-k_{B}T\sum _{r}ln(1-e^{-\beta (\epsilon _{r}-\mu )})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> <mi>l</mi> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">Z</mi> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </munder> <mi>l</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mo stretchy="false">(</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi =k_{B}Tln{\mathcal {Z}}=-k_{B}T\sum _{r}ln(1-e^{-\beta (\epsilon _{r}-\mu )})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/593a98fbf048c2237cdeb4901a1bd7dc736f76fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.756ex; height:5.509ex;" alt="{\displaystyle \Phi =k_{B}Tln{\mathcal {Z}}=-k_{B}T\sum _{r}ln(1-e^{-\beta (\epsilon _{r}-\mu )})}"></span> </p> </blockquote> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \Rightarrow \quad {\frac {\partial \Phi }{\partial \mu }}=-N=-\sum _{r}n_{r}=-\sum _{r}{\frac {e^{-\beta (\epsilon _{r}-\mu )}}{1-e^{-\beta (\epsilon _{r}-\mu )}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mo stretchy="false">⇒</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>μ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>N</mi> <mo>=</mo> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </munder> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mo stretchy="false">(</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mo stretchy="false">(</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \Rightarrow \quad {\frac {\partial \Phi }{\partial \mu }}=-N=-\sum _{r}n_{r}=-\sum _{r}{\frac {e^{-\beta (\epsilon _{r}-\mu )}}{1-e^{-\beta (\epsilon _{r}-\mu )}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92e7ecbf0e8463a521b1e4181fec211c14579af5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:52.215ex; height:6.843ex;" alt="{\displaystyle \quad \Rightarrow \quad {\frac {\partial \Phi }{\partial \mu }}=-N=-\sum _{r}n_{r}=-\sum _{r}{\frac {e^{-\beta (\epsilon _{r}-\mu )}}{1-e^{-\beta (\epsilon _{r}-\mu )}}}}"></span> </p> </blockquote> <p>De modo que: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{r}={\frac {1}{e^{\beta (\epsilon _{r}-\mu )}-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mo stretchy="false">(</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{r}={\frac {1}{e^{\beta (\epsilon _{r}-\mu )}-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5da1cf21966eb60f340ef7d355e17f566a2a2e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.546ex; height:5.676ex;" alt="{\displaystyle n_{r}={\frac {1}{e^{\beta (\epsilon _{r}-\mu )}-1}}}"></span> </p> </blockquote> <p>Debido a que pueden existir diferentes estados cuánticos con una misma energía el número de partículas con una determinada energía vendrá dado por: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{\epsilon }={\frac {g_{\epsilon }}{e^{\beta (\epsilon -\mu )}-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{\epsilon }={\frac {g_{\epsilon }}{e^{\beta (\epsilon -\mu )}-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8046b50080541b1ab131269d71acfe20a0cc9410" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.706ex; height:5.343ex;" alt="{\displaystyle n_{\epsilon }={\frac {g_{\epsilon }}{e^{\beta (\epsilon -\mu )}-1}}}"></span> </p> </blockquote> <p>siendo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\epsilon }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{\epsilon }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cacc748e06bb52e983f28f312b9fecde4d4fcf94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.009ex; height:2.009ex;" alt="{\displaystyle g_{\epsilon }}"></span> la degeneración de tal energía. </p><p>En la anterior expresión se observa que el potencial químico ha de ser menor que todas las energías, de lo contrario el número medio de partículas en un estado podría ser negativo. Este hecho también se pudo haber observado al sumar la serie geométrica, ye que la anterior condición es la condición para su convergencia. </p> <div class="mw-heading mw-heading2"><h2 id="Aplicaciones">Aplicaciones</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Estad%C3%ADstica_de_Bose-Einstein&action=edit&section=3" title="Editar sección: Aplicaciones"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>La distribución de energía de la radiación del <a href="/wiki/Cuerpo_negro" title="Cuerpo negro">cuerpo negro</a> se deduce de la aplicación de la estadística de Bose-Einstein a los <a href="/wiki/Fot%C3%B3n" title="Fotón">fotones</a> que componen la radiación electromagnética.</li> <li>La <a href="/wiki/Capacidad_calor%C3%ADfica#Sólidos_cristalinos" title="Capacidad calorífica">capacidad calorífica</a> de los sólidos tanto a altas como a bajas temperaturas puede ser deducida a partir de la estadística de Bose-Einstein aplicada a los <a href="/wiki/Fon%C3%B3n" title="Fonón">fonones</a>, <a href="/wiki/Cuasipart%C3%ADcula" title="Cuasipartícula">cuasipartículas</a> que dan cuenta de las excitaciones de la red cristalina. En particular la <a href="/wiki/Ley_de_Dulong-Petit" title="Ley de Dulong-Petit">ley de Dulong-Petit</a> puede ser deducida de la estadística de Bose-Einstein.</li> <li>La estadística de Bose-Einstein predice el fenómeno de la condensación de Bose-Einstein, también conocido como el quinto estado de la materia.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Véase_también"><span id="V.C3.A9ase_tambi.C3.A9n"></span>Véase también</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Estad%C3%ADstica_de_Bose-Einstein&action=edit&section=4" title="Editar sección: Véase también"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Estad%C3%ADstica_de_Fermi-Dirac" title="Estadística de Fermi-Dirac">Estadística de Fermi-Dirac</a></li> <li><a href="/wiki/Estad%C3%ADstica_de_Maxwell-Boltzmann" title="Estadística de Maxwell-Boltzmann">Estadística de Maxwell-Boltzmann</a></li> <li><a href="/wiki/Condensado_de_Bose-Einstein" title="Condensado de Bose-Einstein">Condensado de Bose-Einstein</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Referencias">Referencias</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Estad%C3%ADstica_de_Bose-Einstein&action=edit&section=5" title="Editar sección: Referencias"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="listaref" style="-moz-column-count:2; -webkit-column-count:2; column-count:2; list-style-type: decimal;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><span id="CITAREFDirac1981-01-01" class="citation libro">Dirac, Paul Adrien Maurice (1 de enero de 1981). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=XehUpGiM6FIC"><i>The Principles of Quantum Mechanics</i></a> <span style="color:var(--color-subtle, #555 );">(en inglés)</span>. Clarendon Press. p. 149. <small><a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Especial:FuentesDeLibros/9780198520115" title="Especial:FuentesDeLibros/9780198520115">9780198520115</a></small>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEstad%C3%ADstica+de+Bose-Einstein&rft.au=Dirac%2C+Paul+Adrien+Maurice&rft.aufirst=Paul+Adrien+Maurice&rft.aulast=Dirac&rft.btitle=The+Principles+of+Quantum+Mechanics&rft.date=1981-01-01&rft.genre=book&rft.isbn=9780198520115&rft.pages=149&rft.pub=Clarendon+Press&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXehUpGiM6FIC&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><span id="CITAREFPauli1980-01-01" class="citation libro">Pauli, Wolfgang (1 de enero de 1980). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=A84NAQAAIAAJ"><i>General principles of quantum mechanics</i></a> <span style="color:var(--color-subtle, #555 );">(en inglés)</span>. Springer-Verlag. <small><a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Especial:FuentesDeLibros/9783540098423" title="Especial:FuentesDeLibros/9783540098423">9783540098423</a></small>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEstad%C3%ADstica+de+Bose-Einstein&rft.au=Pauli%2C+Wolfgang&rft.aufirst=Wolfgang&rft.aulast=Pauli&rft.btitle=General+principles+of+quantum+mechanics&rft.date=1980-01-01&rft.genre=book&rft.isbn=9783540098423&rft.pub=Springer-Verlag&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DA84NAQAAIAAJ&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Bibliografía"><span id="Bibliograf.C3.ADa"></span>Bibliografía</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Estad%C3%ADstica_de_Bose-Einstein&action=edit&section=6" title="Editar sección: Bibliografía"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span id="CITAREFLandau,_L.D.;_Lifshitz,_E._M.1980" class="citation libro">Landau, L.D.; Lifshitz, E. M. (1980). <i>Statistical Physics</i>. Pergamon Press Ltd. <small>0-08-023039-3</small>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEstad%C3%ADstica+de+Bose-Einstein&rft.au=Landau%2C+L.D.%3B+Lifshitz%2C+E.+M.&rft.aulast=Landau%2C+L.D.%3B+Lifshitz%2C+E.+M.&rft.btitle=Statistical+Physics&rft.date=1980&rft.genre=book&rft.pub=Pergamon+Press+Ltd&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFPathria_R._K.2001" class="citation libro">Pathria R. K. (2001). <i>Statistical Mechanics</i>. Butterworth Heinemann. <small>0 7506 2469 8</small>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AEstad%C3%ADstica+de+Bose-Einstein&rft.au=Pathria+R.+K.&rft.aulast=Pathria+R.+K.&rft.btitle=Statistical+Mechanics&rft.date=2001&rft.genre=book&rft.pub=Butterworth+Heinemann&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li></ul> <style data-mw-deduplicate="TemplateStyles:r161257576">.mw-parser-output .mw-authority-control{margin-top:1.5em}.mw-parser-output .mw-authority-control .navbox table{margin:0}.mw-parser-output .mw-authority-control .navbox hr:last-child{display:none}.mw-parser-output .mw-authority-control .navbox+.mw-mf-linked-projects{display:none}.mw-parser-output .mw-authority-control .mw-mf-linked-projects{display:flex;padding:0.5em;border:1px solid var(--border-color-base,#a2a9b1);background-color:var(--background-color-neutral,#eaecf0);color:var(--color-base,#202122)}.mw-parser-output .mw-authority-control .mw-mf-linked-projects ul li{margin-bottom:0}.mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#a2a9b1);background-color:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .mw-authority-control .navbox-list{border-color:#f8f9fa}.mw-parser-output .mw-authority-control .navbox th{background-color:#eeeeff}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .mw-mf-linked-projects{border:1px solid var(--border-color-base,#72777d);background-color:var(--background-color-neutral,#27292d);color:var(--color-base,#eaecf0)}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral-subtle,#202122)!important}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox-list{border-color:#202122!important}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox th{background-color:#27292d!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .mw-mf-linked-projects{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral,#27292d)!important;color:var(--color-base,#eaecf0)!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral-subtle,#202122)!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox-list{border-color:#202122!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox th{background-color:#27292d!important}}</style><div class="mw-authority-control"><div role="navigation" class="navbox" aria-label="Navbox" style="width: inherit;padding:3px"><table class="hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width: 12%; 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